The English Works of Thomas Hobbes of Malmesbury, vol. 7 of 11

SIX LESSONS TO THE PROFESSORS OF THE MATHEMATICS, — LESSON I. — LESSON II. — LESSON III. — LESSON IV. — LESSON V. — LESSON VI.

My most noble Lord,

Not knowing on my own part any cause of the favour your Lordship has been pleased to express towards me, unless it be the principles, method, and manners you have observed and approved in my writings; and seeing these have all been very much reprehended by men, to whom the name of public professors hath procured reputation in the university of Oxford, I thought it would be a forfeiture of your Lordship’s good opinion, not to justify myself in public also against them, which, whether I have sufficiently performed or not in the six following Lessons addressed to the same professors, I humbly pray your Lordship to consider. The volume itself is too small to be offered to you as a present, but to be brought before you as a controversy it is perhaps the better for being short. Of arts, some are demonstrable, others indemonstrable; and demonstrable are those the construction of the subject whereof is in the power 187of the artist himself, who, in his demonstration, does no more but deduce the consequences of his own operation. The reason whereof is this, that the science of every subject is derived from a precognition of the causes, generation, and construction of the same; and consequently where the causes are known, there is place for demonstration, but not where the causes are to seek for. Geometry therefore is demonstrable, for the lines and figures from which we reason are drawn and described by ourselves; and civil philosophy is demonstrable, because we make the commonwealth ourselves. But because of natural bodies we know not the construction, but seek it from the effects, there lies no demonstration of what the causes be we seek for, but only of what they may be.

And where there is place for demonstration, if the first principles, that is to say, the definitions contain not the generation of the subject, there can be nothing demonstrated as it ought to be. And this in the three first definitions of Euclid sufficiently appeareth. For seeing he maketh not, nor could make any use of them in his demonstrations, they ought not to be numbered among the principles of geometry. And Sextus Empiricus maketh use of them (misunderstood, yet so understood as the said professors understand them) to the overthrow of that so much renowned evidence of geometry. In that part therefore of my book where I treat of geometry, I thought it necessary in my definitions to express those motions by 188which lines, superficies, solids, and figures, were drawn and described, little expecting that any professor of geometry should find fault therewith, but on the contrary supposing I might thereby not only avoid the cavils of the sceptics, but also demonstrate divers propositions which on other principles are indemonstrable. And truly, if you shall find those my principles of motion made good, you shall find also that I have added something to that which was formerly extant in geometry.

For first, from the seventh chapter of my book De Corpore, to the thirteenth, I have rectified and explained the principles of the science; id est, I have done that business for which Dr. Wallis receives the wages. In the seventh, I have exhibited and demonstrated the proportion of the parabola and parabolasters to the parallelograms of the same height and base; which, though some of the propositions were extant without that demonstration, were never before demonstrated, nor are by any other than this method demonstrable.

In the eighteenth, as it is now in English, I have demonstrated, for anything I yet perceive, equation between the crooked line of a parabola or any parabolaster and a straight line.

In the twenty-third I have exhibited the centre of gravity of any sector of a sphere.

Lastly, the twenty-fourth, which is of the nature of refraction and reflection, is almost all new.

But your Lordship will ask me what I have done in the twentieth, about the quadrature of 189the circle. Truly, my Lord, not much more than before. I have let stand there that which I did before condemn, not that I think it exact, but partly because the division of angles may be more exactly performed by it than by any organical way whatsoever; and I have attempted the same by another method, which seemeth to me very natural, but of calculation difficult and slippery. I call them only aggressions, retaining nevertheless the formal manner of assertion used in demonstration. For I dare not use such a doubtful word as videtur, because the professors are presently ready to oppose me with a videtur quod non. Nor am I willing to leave those aggressions out, but rather to try if it may be made pass for lawful, (in spite of them that seek honour, not from their own performances, but from other men’s failings), amongst many difficult undertakings carried through at once to leave one and the greatest for a time behind; and partly because the method is such as may hereafter give farther light to the finding out of the exact truth.

But the principles of the professors that reprehend these of mine, are some of them so void of sense, that a man at the first hearing, whether geometrician or not geometrician, must abhor them. As for example:

1. That two equal proportions are not double to one of the same proportions.

2. That a proportion is double, triple, &c. of a number, but not of a proportion.

3. That the same body, without adding to it, or 190taking from it, is sometimes greater, and sometimes less.

4. That a quantity may grow less and less eternally, so as at last to be equal to another quantity; or, which is all one, that there is a last in eternity.

5. That the nature of an angle consisteth in that which lies between the lines that comprehend the angle in the very point of their concourse, that is to say, an angle is the superficies which lies between the two points which touch, or, as they understand a point, the superficies that lies between the two nothings which touch.

6. That the quotient is the proportion of the division to the dividend.

Upon these and some such other principles is grounded all that Dr. Wallis has said, not only in his Elenchus of my geometry, but also in his treatises of the Angle of Contact, and in his Arithmetica Infinitorum; which two last I have here in two or three leaves wholly and clearly confuted. And I verily believe that since the beginning of the world, there has not been, nor ever shall be, so much absurdity written in geometry, as is to be found in those books of his; with which there is so much presumption joined, that an ἀποκατάϛασις of the like conjunction cannot be expected in less than a Platonic year. The cause whereof I imagine to be this, that he mistook the study of symbols for the study of geometry, and thought symbolical writing to be a new kind of method, and other 191men’s demonstrations set down in symbols new demonstrations. The way of analysis by squares, cubes, &c., is very ancient, and useful for the finding out whatsoever is contained in the nature and generation of rectangled planes, which also may be found without it, and was at the highest in Vieta; but I never saw anything added thereby to the science of geometry, as being a way wherein men go round from the equality of rectangled planes to the equality of proportion, and thence again to the equality of rectangled planes, wherein the symbols serve only to make men go faster about, as greater wind to a windmill.

It is in sciences as in plants; growth and branching is but the generation of the root continued; nor is the invention of theorems anything else but the knowledge of the construction of the subject prosecuted. The unsoundness of the branches are no prejudice to the roots, nor the faults of theorems to the principles. And active principles will correct false theorems if the reasoning be good; but no logic in the world is good enough to draw evidence out of false or unactive principles. But I detain your Lordship too long. For all this will be much more manifest in the following discourses, wherein I have not only explained and rectified many of the most important principles of geometry, but also by the examples of those errors which have been committed by my reprehenders, made manifest the evil consequence of the principles they now proceed on. So that it is not only 192my own defence that I here bring before you, but also a positive doctrine concerning the true grounds, or rather atoms of geometry, which I dare only say are very singular, but whether they be very good or not, I submit to your Lordship’s judgment. And seeing you have been pleased to bestow so much time, with great success, in the reading of what has been written by other men in all kinds of learning, I humbly pray your Lordship to bestow also a little time upon the reading of these few and short lessons; and if your Lordship find them agreeable to your reason and judgment, let me, notwithstanding the clamour of my adversaries, be continued in your good opinion, and still retain the honour of being,

London, June 10, 1656.

I suppose, most egregious professors, you know already that by geometry, though the word import no more but the measuring of land, is understood no less the measuring of all other quantity than that of bodies. And though the definition of geometry serve not for proof, nor enter into any geometrical demonstration, yet for understanding of the principles of the science, and for a rule to judge by, who is a geometrician, and who is not, I hold it necessary to begin therewith.

Geometry is the science of determining the quantity of anything, not measured, by comparing it with some other quantity or quantities measured. Which science therefore whosoever shall go about to teach, must first be able to tell his disciple what measuring or dimension is; by what each several kind of quantity is measured; what quantity is, and what are the several kinds thereof. Therefore as they, who handle any one part of geometry, determine by definition the signification of 195every word which they make the subject or predicate of any theorem they undertake to demonstrate; so must he which intendeth to write a whole body of geometry, define and determine the meaning of whatsoever word belongeth to the whole science. The design of Euclid was to demonstrate the properties of the five regular bodies mentioned by Plato; in which demonstrations there was no need to allege for argument the definition of quantity, which it may be was the cause he hath not anywhere defined it, but done what he undertook without it. And though having perpetually occasion to speak of measure, he hath not defined measure; yet instead thereof he hath, in the beginning of his first elements, assumed an axiom which serveth his turn sufficiently as to the measure of lines, which is the eighth axiom; that those things which lie upon one another all the way (called by him ἐφαρμόζονταἐφαρμόζοντα) are equal. Which axiom is nothing else but a description of the art of measuring length and superficies. For this ἐφάρμοσις can have no place in solid bodies, unless two bodies could at the same time be in one place. But amongst the principles of geometry universal, the definitions are necessary, both of quantity and dimensions.

Quantity is that which is signified by what we answer to him that asketh, how much any thing is? and thereby determine the magnitude thereof. For magnitude being a word indefinite, if a man ask of a thing, quantum est? that is, how much it is, we do not satisfy him by saying it is magnitude or quantity, but by saying it is tantum, so much. And they that first called it in Greek, 196πηλικότης, and in Latin quantity, might more properly have called it in Latin tantity, and in Greek τηλικότης; and we, if we allowed ourselves the eloquence of the Greeks and Latins, should call it the so-muchness.

There is therefore to everything concerning which a man may ask without absurdity, how much it is, a certain quantity belonging, determining the magnitude to be so much. Also wheresoever there is more and less, there is one kind of quantity or other. And first there is the quantity of bodies, and that of three kinds: length, which is by one way of measuring; superficies, made of the complication of two lengths, or the measure taken two ways; and solid, which is the complication of three lengths, or of the measure taken three ways, for breadth or thickness are but other lengths. And the science of geometry, so far forth as it contemplateth bodies only, is no more but by measuring the length of one or more lines, and by the position of others known in one and the same figure, to determine by ratiocination, how much is the superficies; and by measuring length, breadth, and thickness, to determine the quantity of the whole body. Of this kind of magnitudes and quantities the subject is body.

And because for the computing of the magnitudes of bodies, it is not necessary that the bodies themselves should be present, the ideas and memory of them supplying their presence, we reckon upon those imaginary bodies, which are the quantities themselves, and say the length is so great, the breadth so great, &c. which in truth is no better than to say the length is so long, or the breadth so 197broad, &c. But in the mind of an intelligent man it breedeth no mistake.

Besides the quantity of bodies, there is a quantity of time. For seeing men, without absurdity, do ask how much it is; by answering tantum, so much, they make manifest there is a quantity that belongeth unto time, namely, a length. And because length cannot be an accident of time, which is itself an accident, it is the accident of a body; and consequently the length of the time, is the length of the body; by which length or line, we determine how much the time is, supposing some body to be moved over it.

Also we not improperly ask with how much swiftness a body is moved; and consequently there is a quantity of motion or swiftness, and the measure of that quantity is also a line. But then again, we must suppose another motion, which determineth the time of the former. Also of force, there is a question of how much, which is to be answered by so much; that is, by quantity. If the force consist in swiftness, the determination is the same with that of swiftness, namely, by a line; if in swiftness and quantity of body jointly, then by a line and a solid; or if in quantity of body only, as weight, by a solid only.

So also is number quantity; but in no other sense than as a line is quantity divided into equal parts.

Of an angle, which is of two lines whatsoever they be, meeting in one point, the digression or openness in other points, it may be asked how great is that digression? This question is answered also by quantity. An angle therefore hath quantity, 198though it be not the subject of quantity; for the body only can be the subject, in which body those straddling lines are marked.

And because two lines may be made to divaricate by two causes; one, when having one end common and immoveable, they depart one from another at the other ends circularly, and this is called simply an angle; and the quantity thereof is the quantity of the arch, which the two lines intercept.

The other cause is the bending of a straight line into a circular or other crooked line, till it touch the place of the same line, whilst it was straight, in one only point. And this is called an angle of contingence. And because the more it is bent, the more it digresseth from the tangent, it may be asked how much more? And therefore the answer must be made by quantity; and consequently an angle of contingence hath its quantity as well as that which is called simply an angle. And in case the digression of two such crooked lines from the tangent be uniform, as in circles, the quantity of their digression may be determined. For, if a straight line be drawn from the point of contact, the digression of the lesser circle will be to the digression of the greater circle, as the part of the line drawn from the point of contact, and intercepted by the circumference of the greater circle is to the part of the same line intercepted by the circumference of the lesser circle, or, which is all one, as the greater radius is to the lesser radius. You may guess by this what will become of that part of your last book, where you handle the question of the quantity of an angle of contingence.

199Also there lieth a question of how much comparatively one magnitude is to another magnitude, as how much water is in a tun in respect of the ocean, how much in respect of a pint; little in the first respect, much in the latter. Therefore the answer must be made by some respective quantity. This respective quantity is called ratio and proportion, and is determined by the quantity of their differences; and if their differences be compared also with the quantities themselves that differ, it is called simply proportion, or proportion geometrical. But if the differences be not so compared, then it is called proportion arithmetical. And where the difference is none, there is no quantity of the proportion, which in this case is but a bare comparison.

Also concerning heat, light, and divers other qualities, which have degrees, there lieth a question of how much, to be answered by a so much, and consequently they have their quantities, though the same with the quantity of swiftness: because the intensions and remissions of such qualities are but the intensions and remissions of the swiftness of that motion by which the agent produceth such a quality. And as quantity may be considered in all the operations of nature, so also doth geometry run quite through the whole body of natural philosophy.

To the principles of geometry the definition appertaineth also of a measure, which is this, one quantity is the measure of another quantity, when it, or the multiple of it, is coincident in all points with the other quantity. In which definition, instead of that ἐφαρμογὴ of Euclid, I put coincidence. 200For the superposition of quantities, by which they render the word ἐφαρμογὴ, cannot be understood of bodies, but only of lines and superficies. Nevertheless many bodies may be coincident successively with one and the same place, and that place will be their measure, as we see practised continually in the measuring of liquid bodies, which art of measuring may properly be called ἐφάρμοσις, but not superposition.

Also the definitions of greater, less, and equal, are necessary principles of geometry. For it cannot be imagined than any geometrician should demonstrate to us the equality and inequality of magnitudes, except he tell us first what those words do signify. And it is a wonder to me, that Euclid hath not anywhere defined what are equals, or at least, what are equal bodies, but serveth his turn throughout with that forementioned ἐφάρμοσις, which hath no place in solids, nor in time, nor in swiftness, nor in circular, or other crooked lines; and therefore no wonder to me, why geometry hath not proceeded to the calculation neither of crooked lines, nor sufficiently of motion, nor of many other things, that have proportion to one another.

Equal bodies, superficies, and lines, are those of which every one is capable of being coincident with the place of every one of the rest: and equal times, wherein with one and the same motion equal lines are described. And equally swift are those motions by which we run over equal spaces in any time determined by any other motion. And universally all quantities are equal, that are measured by the same number of the same measures.

201It is necessary also to the science of geometry, to define what quantities are of one and the same kind, which they call homogeneous, the want of which definitions hath produced those wranglings (which your book De Angulo Contactus will not make to cease) about the angle of contingence.

Homogeneous quantities are those which may be compared by (ἐφάρμοσις) application of their measures to one another; so that solids and superficies are heterogeneous quantities, because there is no coincidence or application of those two dimensions.

No more is there of line and superficies, nor of line and solid, which are therefore heterogeneous. But lines and lines, superficies and superficies, solids and solids, are homogeneous.

Homogeneous also are line, and the quantity of time; because the quantity of time is measured by the application of a line to a line; for though time be no line, yet the quantity of time is a line, and the length of two times is compared by the length of two lines.

Weight and solid have their quantity homogeneous, because they measure one another by application, to the beam of a balance. Line and angle simply so called, have their quantity homogeneous, because their measure is an arch or arches of a circle applicable in every point to one another.

The quantity of an angle simply so called, and the quantity of an angle of contingence are heterogeneous. For the measures by which two angles simply so called are compared, are in two coincident arches of the same circle; but the measure by which an angle of contingence is measured, is a straight line intercepted between the point of 202contact and the circumference of the circle; and therefore one of them is not applicable to the other; and consequently of these two sorts of angles the quantities are heterogeneous. The quantities of two angles of contingence are homogeneous; for they may be measured by the ἐφάρμοσις of two lines, whereof one extreme is common, namely, the point of contact, the other extremes are in the arches of the two circles.

Besides this knowledge of what is quantity and measure, and their several sorts, it behoveth a geometrician to know why, and of what, they are called principles. For not every proposition that is evident is therefore a principle. A principle is the beginning of something. And because definitions are the beginnings or first propositions of demonstration, they are therefore called principles, principles, I say, of demonstration. But there be also necessary to the teaching of geometry other principles, which are not the beginnings of demonstration, but of construction, commonly called petitions; as that it may be granted that a man can draw a straight line, and produce it; and with any radius, on any centre describe a circle, and the like. For that a man may be able to describe a square, he must first be able to draw a straight line; and before he can describe an equilateral triangle, he must be able first to describe a circle. And these petitions are therefore properly called principles, not of demonstration, but of operation. As for the commonly received third sort of principles, called common notions, they are principles, only by permission of him that is the disciple; who being ingenuous, and coming not to cavil 203but to learn, is content to receive them, though demonstrable, without their demonstrations. And though definitions be the only principles of demonstration, yet it is not true that every definition is a principle. For a man may so precisely determinedetermine the signification of a word as not to be mistaken, yet may his definition be such as shall never serve for proof of any theorem, nor ever enter into any demonstration, such as are some of the definitions of Euclid, and consequently can be no beginnings of demonstration, that is to say, no principles.

All that hitherto hath been said, is so plain and easy to be understood, that you cannot, most egregious professors, without discovering your ignorance to all men of reason, though no geometricians, deny it. And the same (saving that the words are all to be found in dictionaries) new; also to him that means to learn, not only the practice, but also the science of geometry necessary, and, though it grieve you, mine. And now I come to the definitions of Euclid.

The first is of a point: Σημεῖον, &c. “Signum est, cujus est pars nulla,” that is to say, a mark is that of which there is no part. Which definition, not only to a candid, but also to a rigid construer, is sound and useful. But to one that neither will interpret candidly, nor can interpret accurately, is neither useful nor true. Theologers say the soul hath no part, and that an angel hath no part, yet do not think that soul or angel is a point. A mark or as some put instead of it ϛίγμη, which is a mark with a hot iron, is visible; if visible, then it hath quantity, and consequently may be divided 204into parts innumerable. That which is indivisible is no quantity; and if a point be not quantity, seeing it is neither substance nor quality, it is nothing. And if Euclid had meant it so in his definition, as you pretend he did, he might have defined it more briefly, but ridiculously, thus, a point is nothing. Sir Henry Savile was better pleased with the candid interpretation of Proclus, that would have it understood respectively to the matter of geometry. But what meaneth this respectively to the matter of geometry? It meaneth this, that no argument in any geometrical demonstration should be taken from the division, quantity, or any part of a point; which is as much as to say, a point is that whose quantity is not drawn into the demonstration of any geometrical conclusion; or, which is all one, whose quantity is not considered.

An accurate interpreter might make good the definition thus, a point is that which is undivided; and this is properly the same with cujus non est pars: for there is a great difference between undivided and indivisible, that is, between cujus non est pars, and cujus non potest esse pars. Division is an act of the understanding; the understanding is therefore that which maketh parts, and there is no part where there is no consideration but of one. And consequently Euclid’s definition of a point is accurately true, and the same with mine, which is, that a point is that body whose quantity is not considered. And considered is that, as I have defined it chap. I. at the end of the third article, which is not put to account in demonstration.

Euclid therefore seemeth not to be of your 205opinion, that say a point is nothing. But why then doth he never use this definition in the demonstration of any proposition? Whether he useth it expressly or no, I remember not; but the sixteenth proposition of the third book without the force of this definition is undemonstrated.

The second definition is of a line: γραμμὴ δὲ μῆκος ἂπλατες. “Linea est longitudo latitudinis expers; a line is length which hath no breadth;” and if candidly interpreted, sound enough, though rigorously not so. For to what purpose is it to say length not broad, when there is no such thing as a broad length. One path may be broader than another path, but not one mile than another mile; and it is not the path but the mile which is the way’s length. If therefore a man have any ingenuity he will understand it thus, that a line is a body whose length is considered without its breadth, else we must say absurdly a broad length; or untruly, that there be bodies which have length and yet no breadth; and this is the very sense which Apollonius, saith Proclus, makes of this definition; “when we measure,” says he, “the length of a way, we take not in the breadth or depth, but consider only one dimension.” See this of Proclus cited by Sir Henry Savile, where you shall find the very word consider.

The fourth definition is of a straight line, thus Ἐυθεῖα γραμμή ἐϛιν, &c. “Recta linea est quæ ex æquo sua ipsius puncta inter jacet.” A straight line is that which lieth equally (or perhaps evenly) between its own points. This definition is inexcusable. Between what points of its own can a straight line lie but between its extremes? And how lies it 206evenly between them, unless it swerve no more from some other line which hath the same extremes, one way than another? And then why are not between the same points both the lines straight? How bitterly, and with what insipid jests would you have reviled Euclid for this, if living now he had written a Leviathan . And yet there is somewhat in this definition to help a man, not only to conceive the nature of a straight line (for who doth not conceive it?) but also to express it. For he meant perhaps to call a straight line that which is all the way from one extreme to another, equally distant from any two or more such lines as being like and equal have the same extremes. So the axis of the earth is all the way equally distant from the circumference of any two or more meridians. But then before he had defined a straight line, he should have defined what lines are like, and what are equal. But it had been best of all, first to have defined crooked lines, by the possibility of a deduction or setting further asunder of their extremes; and then straight lines, by the impossibility of the same.

The seventh definition, which is that of a plain superficies, hath the same faults.

The eighth is of a plane angle, Ἑπὶπεδος γωνία ἑϛὶν ἡ ἐν ἐπιπέδω, &c. “Angulus planus est duarum linearum in plano se mutuo tangentium, et non in directum jacentium, alterius ad alteram inclinatio.” A plane angle is the inclination one towards another of two lines that touch one another in the same plane, and lie not in the same straight line. Besides the faults here observed by Sir Henry Savile, as the clause of not lying in the same 207straight line, and the obscurity or equivocation of the word inclination, there is yet another, which is, that by this definition two right angles together taken, are no angle; which is a fault which you somewhere (asking leave to use the word angle, καταχριϛικώς acknowledge, but avoid not. For in geometry, where you confess there is required all possible accurateness, every καταχρῆσις is a fault. Besides you see by this definition, that Euclid is not of your, but of Clavius’s opinion. For it is manifest that the two lines which contain an angle of contact incline one towards another, and come together in a point, and lie not in the same straight line, and consequently make an angle.

The thirteenth definition is exact, but makes against your doctrine, that a point is nothing. Examine it. Ὅρος ἐϛὴν ὅ τινός ἐϛῖ πέρας. “Terminus est quod alicujus extremum est.” A term or bound is that which is the extreme of anything. We had before, the extremes of a line are points. But what is in a line the extreme, but the first or last part, though you may make that part as small as you will? A point is therefore a part, and nothing is no extreme.

The fourteenth, Σχῆμα ἐϛὶ τὸ ὑπὸ τινος ἤ τινῶν ὅρων περιεχόμενον. “Figura est (subaudi quantitas) quæ ab aliquo, vel aliquibus terminis undique continetur sive clauditur.” A figure is quantity contained within some bound or bounds. Or shortly thus, a figure is quantity every way determined, is in my opinion as exact a definition of a figure as can possibly be given, though it must not be so in yours. For this determination is the same thing with circumscription; and whatsoever is anywhere 208(ubicunque) definitivè is there also circumscriptivè; and by this means the distinction is lost, by which theologers, when they deny God to be in any place, save themselves from being accused of saying he is nowhere; for that which is nowhere is nothing. This definition of Euclid cannot therefore possibly be embraced by you that carry double, namely, mathematics and theology. For if you reject it, you will be cast out of all mathematic schools; and if you maintain it, from the society of all school-divines, and lose the thanks of the favour you have shown (you the astronomer) to Bishop Bramhall.

The fifteenth is of a circle. Κοὐκλος ἐστὶ σχῆμα ἐπίπεδον, &c. A circle is a plain figure comprehended by one line which is called the circumference, to which circumference all the straight lines drawn from one of the points within the figure are equal to one another. This is true. But if a man had never seen the generation of a circle by the motion of a compass or other equivalent means, it would have been hard to persuade him that there was any such figure possible. It had been therefore not amiss first to have let him see that such a figure might be described. Therefore so much of geometry is no part of philosophy, which seeketh the proper passions of all things in the generation of the things themselves.

After the fifteenth till the last or thirty-fifth definition, all are most accurate, but the last which is this, parallel straight lines are those which being in the same plane, though infinitely produced both ways, shall never meet. Which is less accurate. For how shall a man know that there be straight 209lines whichwhich shall never meet, though both ways infinitely produced? Or how is the definition of parallels, that is, of lines perpetually equidistant, good, wherein the nature of equidistance is not signified? Or if it were signified, why should it not comprehend as well the parallelism of circular and other crooked lines, as of straight, and as well of superficies, as of lines? By parallels is meant equidistant both lines and superficies, and the word is therefore not well defined without defining first equality of distance. And because the distance between two lines or superficies, is the shortest line that can join them, there either ought to be in the definition the shortest distance, which is that of the perpendicular and without inclination, or the distance in equal inclination, that is, in equal angles. Therefore if parallels be defined to be those lines or superficies, where the lines drawn from one to another in equal angles be equal, the definition, as to like lines, or like superficies, will be universal and convertible. And if we add to this definition, that the equal angles be drawn not opposite ways, it will be absolute, and convertible in all lines and superficies; and the definition will be this: parallels are those lines and superficies between which every line drawn, in any angle, is equal to any other line drawn in the same angle the same way. For by this definition the distance between them will perpetually be equal, and consequently they will never come nearer together, how much, or which way soever they be produced. And the converse of it will be also true, if two lines, or two superficies be parallel, and a straight line be drawn from one to the other, any other 210straight line, drawn from one to the other in the same angle, and the same way, will be equal to it. This is manifestly true, and, most egregious professors, new, at least to you.

And thus much for the definitions placed before the first of Euclid’s Elements.

Before the third of his Elements is this definition: “In circulo æqualiter distare a centro rectæ lineæ dicuntur, cum perpendiculares quæ a centro in ipsas ducuntur sunt æquales.” In a circle two straight lines are said to be equally distant from the centre, upon which the perpendiculars drawn from the centre are equal. This is true; but it is rather an axiom than a definition, as being demonstrable that the perpendicular is the measure of the distance between a point and a straight or a crooked line.

Before the fifth Element the first definition is of a part: Pars est magnitudo magnitudinis, minor majoris, cum minor metitur majorem. A part is one magnitude of another, the less of the greater, when the less measureth the greater. From which definition it followeth, that more than a half is not a part of the whole. But because Euclid meaneth here an aliquot part, as a half, a third, or a fourth, &c., it may pass for the definition of a measure under the name of part, as thus: a measure is a part of the whole, when multiplied it may be equal to the whole, though properly a measure is external to the thing measured, and not the aliquot part itself, but equal to an aliquot part.

But the third definition is intolerable; it is the definition of λόγος, in Latin ratio, in English, proportion, in this manner, λόγος ἐςὶ δύο μεγεθῶν ὁμογενῶν ῆ 211κατὰ πηλικότητα προς ἄλληλα ποιὰ σχέσις. “Ratio est duarum magnitudinum ejusdem generis mutua quædam secundum quantitatem habitudo.” Proportion is a certain mutual habitude in quantity, of two magnitudes of the same kind, one to another. First, we have here ignotum per ignotius; for every man understandeth better what is meant by proportion than by habitude. But it was the phrase of the Greeks when they named like proportions, to say, the first to the second, οὕτωςοὕτως ἔχει, id est, ita se habet, and in English, is as, the third to the fourth. As for example, in the proportions of two to four, and three to six, to say two to four, οὕτωςοὕτως ἔχει, id est, ita se habet, id est, is as, three to six. From which phrase Euclid made this his definition of proportion by ποιὰ σχέσις, which the Latins translate quædam habitudo. Quædam in a definition is a most certain note of not understanding the word defined; and in Greek, ποιὰ σχέσις is much worse; for to render rightly the Greek definition, we are to say in English, that proportion is a what-shall-I-call-it-isness, or soness of two magnitudes, &c.; than which nothing can be more unworthy of Euclid. It is as bad as anything was ever said in geometry by Orontius, or by Dr. Wallis. That proportion is quantity compared, that is to say, little or great in respect of some other quantity, as I have above defined it, is I think intelligible.

The fourth is, Ἀναλογία δέ ὲστιν ῆ των λόγων ὁμοιότης. “Proportio vero est rationum similitudo.” Here we have no one word by which to render Ἀναλογία; for our word proportion is already bestowed upon the rendering of λόγος. Nevertheless the Greek may be translated into English thus, iterated proportions. 212But iterated proportion is the same with eadem ratio. To what purpose then serveth the sixth definition, which is of eadem ratio? For Ἀναλογία and eadem ratio and similitudo rationum, are the same thing, as appeareth by Euclid himself, where he defines those quantities, that are in the same proportion by ἀνάλογον. Therefore the sixth definition is but a lemma, and assumed without demonstration.

The fourteenth, “Compositio rationis est sumptio antecedentis cum consequente, ceu unius, ad ipsum consequentem,” To compound proportion, is to take both antecedent and consequent together as one magnitude, and compare it to the consequent, is good; though he might have compared it as well with the antecedent; for both ways it had been a composition of proportion. We are to note here, that the composition defined in this place by Euclid is not adding together of proportions, but of two quantities that have proportion. And therefore it is not the same composition which he defineth in the fourth place before the sixth element, for there he defineth the addition of one proportion to another proportion in this manner: λόγος ἐκ λόγων συγκεῖσθαι λέγεται, &c. A proportion is said to be compounded of proportions, when their quantities multiplied into one another make a proportion; as when we would compound or add together the proportions of three to two, and of four to five, we must multiply three and four, which maketh twelve, and two and five, which maketh ten. And then the proportion of twelve to ten is the sum of the proportions of three to two, and of four to five, which is true, but not a definition; for it may 213and ought to be demonstrated. For to define what is addition of two proportions (which are always in four quantities, though sometimes one of them be twice named) we are to say, that they are then added together when we make the second to another in the same proportion, which the third hath to the fourth.

And thus much of the definitions; of which some, very few, you see are faulty; the rest either accurate, or good enough if well interpreted. For the rest of the elements all are accurate, notwithstanding that you allow not for good any definition in geometry that hath in it the word motion, of which there be divers before the eleventh Element. But I must here put you in mind, that geometry being a science, and all science proceeding from a precognition of causes, the definition of a sphere, and also of a circle, by the generation of it, that is to say, by motion, is better than by the equality of distance from a point within.

The second sort of principles are those of construction, usually called postulata, or petitions. As for those notiones communes, called axioms, they are from the definitions of their terms demonstrable, though they be so evident as they need not demonstration. These petitions are by Euclid called Ἀιτήματα, such as are granted by favour, that is, simply petitions, whereas by axiom is understood that which is claimed as due. So that between Ἀξίωμα and Ἀίτημα there is this other difference, that this latter is simply a petition, the former a petition of right.

Of petitions simply, the first is, that from any point to any point may be drawn a straight line. 214The second, that a finite straight line may be produced. The third, that upon any centre at any distance may be described a circle. All which are both evident and necessary to be granted.

And by all these a man may easily perceive that Euclid in the definitions of a point, a line, and a superficies, did not intend that a point should be nothing, or a line be without latitude, or a superficies without thickness; for if he did, his petitions are not only unreasonable to be granted, but also impossible to be performed. For lines are not drawn but by motion, and motion is of body only. And therefore his meaning was, that the quantity of a point, the breadth of a line, and the thickness of a superficies were not to be considered, that is to say, not to be reckoned in the demonstration of any theorems concerning the quantity of bodies, either in length, superficies, or solid.

There be but two causes from which can spring an error in the demonstration of any conclusion in any science whatsoever; and those are ignorance or want of understanding, and negligence. For as in the adding together of many and great numbers, he cannot fail that knoweth the rules of addition, and is also all the way so careful, as not 215to mistake one number or one place for another; so in any other science, he that is perfect in the rules of logic, and is so watchful over his pen, as not to put one word for another, can never fail of making a true, though not perhaps the shortest and easiest demonstration.

The rules of demonstration are but of two kinds: one, that the principles be true and evident definitions; the other, that the inferences be necessary. And of true and evident definitions, the best are those which declare the cause or generation of that subject, whereof the proper passions are to be demonstrated. For science is that knowledge which is derived from the comprehension of the cause. But when the cause appeareth not, then may, or rather must we define some known property of the subject, and from thence derive some possible way, or ways, of the generation. And the more ways of generation are explicated, the more easy will be the derivation of the properties; whereof some are more immediate to one, some to another generation. He therefore that proceedeth from untrue, or not understood definitions, is ignorant of that he goes about; which is an ill-favoured fault, be the matter he undertaketh easy or difficult, because he was not forced to undergo a greater charge than he could carry through. But he that from right definitions maketh a false conclusion, erreth through human frailty, as being less awake, more troubled with other thoughts, or more in haste when he was in writing. Such faults, unless they be very frequent, are not attended with shame, as being common to all men, or are at least less ugly than the former, except 216then, when he that committeth them reprehendeth the same in other men. For that is in every man intolerable, which he cannot tolerate in another. But to the end that the faults of both kinds may by every man be well understood, it will not be amiss to examine them by some such demonstrations as are publicly extant. And for this purpose I will take such as are in mine and in your books, and begin with your Elenchus of the geometry contained in my book De Corpore; to which I will also join your book lately set forth concerning the Angle of Contact, Conic Sections , and your Arithmetica Infinitorum; and then examine the rest of my philosophy, and yours that oppugn it. For I will take leave to consider you both everywhere as one author, because you publicly declare your approbation of one another’s doctrine.

My first definition is of a line, of length, and of a point. “The way,” say I, “of a body moved, in which magnitude (though it always have some magnitude) is not considered, is called a line; and the space gone over by that motion, length, or one and a simple dimension.” To this definition you say, first, “what mathematician did ever thus define a line or length?” Whether you call in others for help or testimony, it is not done like a geometrician; for they use not to prove their conclusions by witnesses, but rely upon the strength of their own reason; and when your witnesses appear, they will not take your part. Secondly, you grant that what I say is true, but not a definition. But to tell you truly what it is which we call a line, is to define a line. Why then is not this a definition? “Because,” say you in the first place, “it 217is not a reciprocal proposition.” But by your favour it is reciprocal. For not only the way of a body whose quantity is not considered is a line, but also every line is, or may be conceived to be, the way of a body so moved. And if you object that there is a difference between is and may be conceived to be, Euclid, whom you call to your aid, will be against you in the fourteenth definition before his eleventh Element; where he defines a sphere just as convertibly as I define a line; except you think the globes of the sun and stars cannot be globes, unless they were made by the circumduction of a semicircle; and again in the eighteenth definition, which is of a cone, unless you admit no figure for a cone, which is not generated by the revolution of a triangle; and again, in the twentieth definition, which is of a cylinder, except it be generated by the circumvolution of a parallelogram. Euclid saw that what proper passion soever should be derived from these his definitions, would be true of any other cylinder, sphere, or cone, though it were otherwise generated; and the description of the generation of any one being by the imagination applicable to all, which is equivalent to convertible, he did not believe that any rational man could be misled by learning logic to be offended with it. Therefore this exception proceedeth from want of understanding, that is, from ignorance of the nature, and use of a definition.

Again, you object and ask: “What need is there of motion, or of body moved, to make a man understand what is a line? Are not lines in a body at rest, as well as in a body moved? And is not the distance of two resting points length, as well 218as the measure of the passage? Is not length one and a simple dimension, and one and a simple dimension line? Why then is not line and length all one?” See how impertinent these questions are. Euclid defines a sphere to be a solid figure described by the revolution of a semicircle about the unmoved diameter. Why do you not ask, what need there is to the understanding of what a sphere is, to bring in the motion of a semicircle? Is not a sphere to be understood without such motion? Is not the figure so made a sphere without this motion? And where he defines the axis of a sphere to be that unmoved diameter, may not you ask, whether there be no axis of a sphere, when the whole sphere, diameter and all, is in motion? But it is not to my purpose to defend my definition by the example of that of Euclid. Therefore first, I say, to me, howsoever it may be to others, it was fit to define a line by motion. For the generation of a line is the motion that describes it. And having defined philosophy in the beginning, to be the knowledge of the properties from the generation, it was fit to define it by its generation. And to your question, is not distance length? I answer, that though sometimes distance be equivalent to length, yet certainly the distance between the two ends of a thread wound up into a clue is not the length of the thread; for the length of the thread is equal to all the windings whereof the clue is made. But if you will needs have distance and length to be all one, tell me of what the distance between any two points is the length. Is it not the length of the way? And how is that called way, which is not defined by 219some motion? And have not several ways between the same places, as by land and by water, several lengths? But they have but one distance, because the distance is the shortest way. Therefore between the length of the path, and the distance of places, there is a real difference in this case, and in all cases a difference of the consideration. Your objection, that line is longitude, proceeds from want of understanding English. Do men ever ask what is the line of a thread, or the line of a table, or of any other body? Do they not always ask what is the length of it? And why, but because they use their own judgments, not yet corrupted by the subtlety of mistaken professors. Euclid defines a line to be length without breadth. If those terms be all one, why said he not that a line is a line without breadth? But what definition of a line give you? None. Be contented then with such as you receive, and with this of mine, which you shall presently see is not amiss.

Your next objections are to my definition of a point. Which definition adhereth to the former in these words, “and the body itself is called a point.” Here again you call for help: “Quis unquum mortalium, etc. What mortal man, what sober man, did ever so define a point?” It is well, and I take it to be an honour to be the first that do so. But what objection do you bring against it. This: “That a point added to a point, if it have magnitude, makes it greater.” I say it doth so, but then presently it loseth the name of a point, which name was given to signify that it was not the meaning of him that used it in demonstration to add, subtract, multiply, divide, or any way 220compute it. Then you come in with, “perhaps you will say though it have magnitude, that magnitude is not considered.” You need not say perhaps. You know I affirm it; and therefore your argument might have been left out, but that it gave you an occasion of a digression into scurvy language.

And whereas you ask why I defined not a point thus: “Punctum est corpus quod non consideratur esse corpus, et magnum quod non consideratur esse magnum.” I will tell you why. First, because it is not Latin. Secondly, because when I had defined it by corpus, there was no need to define it again by magnum. I understand very well this language, “punctum est corpus, quod non consideratur ut corpus.” A point is a body not considered as body. But punctum est corpus, quod non consideratur esse corpus, vel esse magnum, is not Latin; nor the version of it, a point is a body which is not considered to be a body, English. My definition was, that a point is that body whose magnitude is not considered, not reckoned, not put to account in demonstration. And I exemplified the same by the body of the earth describing the ecliptic line; because the magnitude is not there reckoned nor chargeth the ecliptic line with any breadth. But I perceive you understand not what the word consideration signifieth, but take it for comparison or relation; and say I ought to define a point simply, and not by relation to a great body; as if to reckon and to compare were the same thing. “Omnia mihi,” saith Cicero, “provisa et considerata sunt.” I have provided and reckoned everything. There is a great difference between reckoning and relation.

221Again, you ask, why corpus motum, a body moved? I will tell you; because the motion was necessary for the generation of a line. And though after the generation of the line the point should rest, yet it is not necessary from this definition that it should be no more a point; nor when Euclid defines a sphere by the circumduction of a semicircle upon an axis that resteth, doth it follow from thence when the sphere, axis, centre and all, as that of the earth, is moved from place to place, that it is no more an axis.

Lastly, you object “that motion is accidentary to a point, and consequently not essential, nor to be put into the definition.” And is not the circumduction of a semicircle accidentary to a sphere? Or do you think the sphere of the sun was generated by the revolution of a semicircle? And yet it was thought no fault in Euclid to put the motion into the definition of a sphere.

The conceit you have concerning definitions, that they must explicate the essence of the thing defined, and must consist of a genus and a difference, is not so universally true as you are made believe, or else there be very many insufficient definitions that pass for good with you in Euclid. You are much deceived if you think these woful notions of yours, and the language that doth everywhere accompany them, show handsomely together. Or that such grounds as these be able to sustain so many, and so haughty reproaches as you advance upon them, so as they fall not, as you shall see immediately, upon your own head. I say a point hath quantity, but not to be reckoned in demonstrating the properties of lines, solids, or 222superficies; you say it hath no quantity at all, but is plainly nothing.

The first of the petitions of Euclid is, “that a line may be drawn from point to point at any distance.” The second, “that a straight line may be produced.” The third, “that on any centre a circle may be described at any distance.” And the eighth axiom (which Sir H. Savile observes to be the foundation of all geometry) is this, “Quæ sibi mutuo congruunt, etc. Those things that are applied to one another in all points are equal.” All or any of these principles being taken away, there is not in Euclid one proposition demonstrated or demonstrable. If a point have no quantity, a line can have no latitude; and because a line is not drawn but by motion, by motion of a body, and body imprinteth latitude all the way, it is impossible to draw or produce a straight line, or to describe a circular line without latitude. Also if a line have no latitude, one straight line cannot be applied to another. To them therefore that deny a point to have quantity, that is, a line to have latitude, the forenamed principles are not possible, and consequently no proposition in geometry is demonstrated or demonstrable. You therefore that deny a point to have quantity, and a line to have breadth, have nothing at all of the science of geometry. The practice you may have, but so hath any man that hath learned the bare propositions by heart; but they are not fit to be professors either of geometry or of any other science that dependeth on it. Some man perhaps may say that this controversy is not much worth, and that we both mean the same thing. But that man, though 223in other things prudent enough, knoweth little of science and demonstration. For definitions are not only used to give us the notions of those things whose appellations are defined, for many times they that have no science have the ideas of things more perfect than such as are raised by definitions. As who is there that understandeth not better what a straight line is, or what proportion is, and what many other things are, without definition, than some that set down the definitions of them. But their use is, when they are truly and clearly made, to draw arguments from them for the conclusions to be proved. And therefore you that in your following censures of my geometry, take your argument so often from this, that a point is nothing, and so often revile me for the contrary, are not to be allowed such an excuse. He that is here mistaken, is not to be called negligent in his expression, but ignorant of the science.

In the next place, you take exceptions to my definition of equal bodies, which is this: “Corpora æqualia sunt quæ eundem locum possidere possunt. Equal bodies are those which may have the same place.” To which you object impertinently, that I may as well define a man to be, he that may be prince of Transylvania, wittily, as you count wit. Formerly in every definition, you exacted an explication of the essence. You are therefore of opinion that the possibility of being prince of Transylvania is no less essential to a man, than the possibility of the being of two bodies successively in the same place, is essential to bodies equal.

You take no notice of the twenty-third article 224of this same chapter, where I define what it is we call essence, namely, that accident for which we give the thing its name. As the essence of a man is his capacity of reasoning; the essence of a white body, whiteness, &c., because we give the name of man to such bodies as are capable of reasoning, for that their capacity; and the name of white to such bodies as have that colour, for that colour. Let us now examine why it is that men say bodies are one to another equal; and thereby we shall be able to determine whether the possibility of having the same place be essential or not to bodies equal, and consequently whether this definition be so like to the defining of a man by the possibility of being prince of Transylvania as you say it is. There is no man, besides such egregious geometricians as yourselves, that inquireth the equality of two bodies, but by measure. And for liquid bodies, or the aggregates of innumerable small bodies, men (men, I say) measure them by putting them one after another into the same vessel, that is to say, into the same place, as Aristotle defines place, or into the space determined by the vessel, as I define place. And the bodies that so fill the vessel, they acknowledge and receive for equal. But though, when hard bodies cannot be so measured, without the incommodity or trouble of altering their figure, they then enquire, if the bodies are both of the same kind, their equality by weight, knowing, without your teaching, that equal bodies of the same nature weigh proportionably to their magnitudes; yet they do it not for fear of missing of the equality, but to avoid inconvenience or trouble. But you (you, I say), that 225have no definition of equals, neither received from others, nor framed by yourselves, out of your shallow meditation and deep conceit of your own wits, contend against the common light of nature. So much is unheedy learning a hinderance to the knowledge of the truth, and changeth into elves those that were beginning to be men.

Again, when men inquire the equality of two bodies in length, they measure them by a common measure; in which measure they consider neither breadth nor thickness, but how the length of it agreeth, first with the length of one of the bodies, then with the length of the other. And both the bodies whose lengths are measured, are successively in the same place under their common measure. Place therefore in lines also, is the proper index and discoverer of equality and inequality. And as in length, so it is in breadth and thickness, which are but lengths otherwise taken in the same solid body. But now when we come from this equality and inequality of lengths known by measure, to determine the proportions of superficies and of solids, by ratiocination, then it is that we enter into geometry; for the making of definitions, in whatsoever science they are to be used, is that which we call philosophia prima. It is not the work of a geometrician, as a geometrician, to define what is equality, or proportion, or any other word he useth, though it be the work of the same man, as a man. His geometrical part is, to draw from them as many true and useful theorems as he can.

You object secondly, that a pyramis may be equal to a cube whilst it is a pyramis. True. And 226so also whilst it is a pyramis it hath a possibility by flexion and transposition of parts to become a cube, and to be put into the place where another cube equal to it was before. This is to argue like a child that hath not yet the perfect understanding of any language.

In the third and fourth objection, you teach me to define equal bodies (if I will needs define them by place) by the equality of place, and to say, that bodies are equal that have equal places. Teach others, if you can, to measure their grain, not by the same, but equal bushels.

In the fifth objection, you except against the the word can, in that I say that bodies are equal which can fill the same place. For the greater body can, you say, fill the place of the less, though not reciprocally the less of the greater. It is true, that though the place of the less can never be the place of the greater, yet it may be filled by a part of the greater. But it is not then the greater body that filleth the place of the less, but a part of it, that is to say, a less body. Howsoever, to take away from simple men this straw they stumble at, I have now put the definition of equal bodies into these words: equal bodies are those whereof every one can fill the place of every other. And if my definition displease you, propound your own, either of equal bodies, or of equals simply. But you have none. Take therefore this of mine.

The sixth is a very admirable exception. “What,” say you, “if the same body can sometimes take up a greater, sometimes a lesser place, as by rarefaction and condensation?” I understand very well that bodies may be sometimes thin and sometimes 227thick, as they chance to stand closer together or further from one another. So in the mathematic schools, when you read your learned lectures, you have a thick or thronging audience of disciples, which in a great church would be but a very thin company. I understand how thick and thin may be attributed to bodies in the plural, as to a company; but I understand not how any one of them is thicker in the school than in the church; or how any one of them taketh up a greater room in the school, when he can get in, than in the street. For I conceive the dimensions of the body, and of the place, whether the place be filled with gold or with air, to be coincident and the same; and consequently both the quantity of the air, and the quantity of the gold, to be severally equal to the quantity of the place; and therefore also, by the first axiom of Euclid, equal to one another; insomuch as if the same air should be by condensation contained in a part of the place it had, the dimensions of it would be the same with the dimensions of part of the place, that is, should be less than they were, and by consequence the quantity less. And then either the same body must be less also, or we must make a difference between greater bodies and bodies of greater quantity; which no man doth that hath not lost his wits by trusting them with absurd teachers. When you receive salary, if the steward give you for every shilling a piece of sixpence, and then say, every shilling is condensed into the room of sixpence, I believe you would like this doctrine of yours much the worse. You see how by your ignorance you confound the affairs of mankind, as far forth as they give credit 228to your opinions, though it be but little. For nature abhors even empty words, such as are (in the meaning you assign them), rarefying and condensing. And you would be as well understood if you should say (coining words by your own power), that the same body might take up sometimes a greater, sometimes a lesser place, by wallifaction and wardensation, as by rarefactionrarefaction and condensation. You see how admirable this your objection is.

In the seventh objection you bewray another kind of ignorance, which is the ignorance of what are the proper works of the several parts of philosophy. “Though it were out of doubt,” say you, “that the same body cannot have several magnitudes, yet seeing it is matter of natural philosophy, nor hath anything to do with the present business, to what purpose is it to mention it in a mathematical definition?” It seems by this, that all this while you think it is a piece of the geometry of Euclid, no less to make the definitions he useth, than to infer from them the theorems he demonstrateth. Which is not true. For he that telleth you in what sense you are to take the appellations of those things which he nameth in his discourse, teacheth you but his language, that afterwards he may teach you his art. But teaching of language is not mathematic, nor logic, nor physic, nor any other science; and therefore to call a definition, as you do, mathematical, or physical, is a mark of ignorance, in a professor inexcusable. All doctrine begins at the understanding of words, and proceeds by reasoning till it conclude in science. He that will learn geometry must understand 229the terms before he begin, which that he may do, the master demonstrateth nothing, but useth his natural prudence only, as all men do when they endeavour to make their meaning clearly known. For words understood are but the seed, and no part of the harvest of philosophy. And this seed was it, which Aristotle went about to sow in his twelve books of metaphysics, and in his eight books concerning the hearing of natural philosophy. And in these books he defineth time, place, substance or essence, quantity, relation, &c., that from thence might be taken the definitions of the most general words for principles in the several parts of science. So that all definitions proceed from common understanding; of which, if any man rightly write, he may properly call his writing philosophia prima, that is, the seeds, or the grounds of philosophy. And this is the method I have used, defining place, magnitude, and the other the most general appellations in that part which I entitle philosophia prima. But you now, not understanding this, talk of mathematical definitions. You will say perhaps that others do the same as well as you. It may be so. But the appeaching of others does not make your ignorance the less.

In the eighth place you object not, but ask me why I define equal bodies apart? I will tell you. Because all other things which are said to be equal, are said to be so from the equality of bodies; as two lines are said to be equal, when they be coincident with the length of one and the same body; and equal times, which are measured by equal lengths of body, by the same motion. And the reason is, because there is no subject of quantity, 230or of equality, or of any other accident but body; all which I thought certainly was evident enough to any uncorrupted judgment; and therefore that I needed first to define equality in the subject thereof, which is body, and then to declare in what sense it was attributed to time, motion, and other things that are not body.

The ninth objection is an egregious cavil. Having set down the definition of equal bodies, I considered that some men might not allow the attribute of equality to any things but those which are the subjects of quantity, because there is no equality, but in respect of quantity. And to speak rigidly, magnum et magnitudo are not the same thing; for that which is great, is properly a body, whereof greatness is an accident. In what sense therefore, might you object, can an accident have quantity? For their sakes therefore that have not judgment enough to perceive in what sense men say the length is so long, or the superficies so broad, &c. I added these words: “Eadem ratione (qua scilicet corpora dicuntur æqualia) magnitudo magnitudini æqualis dicitur,” that is, in the same manner, as bodies are said to be equal, their magnitudes also are said to be equal. Which is no more than to say, when bodies are equal, their magnitudes also are called equal. When bodies are equal in length, their lengths are also called equal. And when bodies are equal in superficies, their superficies are also called equal. All which is common speech, as well amongst mathematicians, as amongst common people; and, though improper, cannot be altered, nor needeth to be altered to intelligent men. Nevertheless I did 231think fit to put in that clause, that men might know what it is we call equality, as well in magnitudes as in magnis, that is, in bodies. Which you so interpret, as if it bore this sense, that when bodies are equal their superficies also must be equal, contrary to your own knowledge, only to take hold of a new occasion of reviling. How unhandsome and unmanly this is, I leave to be judged by any reader that hath had the fortune to see the world, and converse with honest men.

Against the fourteenth article, where I prove that the same body hath always the same magnitude, you object nothing but this, that though it be granted, that the same body hath the same magnitude, while it resteth, yet I bring nothing to prove that when it changeth place, it may not also change its magnitude by being enlarged or contracted. There is no doubt but to a body, whether at rest or in motion, more body may be added, or part of it taken away. But then it is not the same body, unless the whole and the part be all one. If the schools had not set your wit awry, you could never have been so stupid as not to see the weakness of such objections. That which you add in the end of your objections to this eighth chapter, that I allow not Euclid this axiom gratis, that the whole is greater than a part, you know to be untrue.

At my eleventh chapter, you enter into dispute with me about the nature of proportion. Upon the truth of your doctrine therein, and partly upon the truth of your opinions concerning the definitions of a point, and of a line, dependeth the question whether you have any geometry or none; 232and the truth of all the demonstrations you have in your other books, namely of the Angle of Contact , and Arithmetica Infinitorum. Here I say you enter, how you will get out, your reputation saved, we shall see hereafter.

When a man asketh what proportion one quantity hath to another, he asketh how great or how little the one is comparatively to, or in respect of the other. When a geometrician prefixeth before his demonstrations a definition, he doth it not as a part of his geometry, but of natural evidence, not to be demonstrated by argument, but to be understood in understanding the language wherein it is set down; though the matter may nevertheless, if besides geometry he have wit, be of some help to his disciple to make him understand it the sooner. But when there is no significant definition prefixed, as in this case, where Euclid’s definition of proportion, that it is a whatshicalt habitude of two quantities, &c., is insignificant, and you allege no other, every one that will learn geometry, must gather the definition from observing how the word to be defined is most constantly used in common speech. But in common speech if a man shall ask how much, for example, is six in respect of four, and one man answer that it is greater by two, and another that it is greater by half of four, or by a third of six, he that asked the question will be satisfied by one of them, though perhaps by one of them now, and by the other another time, as being the only man that knoweth why he himself did ask the question. But if a man should answer, as you would do, that the proportion of six to two is of those numbers a certain 233quotient, he would receive but little satisfaction. Between the said answers to this question, how much is six in respect of four? there is this difference. He that answereth that it is more by two, compareth not two with four, nor with six, for two is the name of a quantity absolute. But he that answereth it is more by half of four, or by a third of six, compareth the difference with one of the differing quantities. For halfs and thirds, &c. are names of quantity compared.

From hence there ariseth two species or kinds of (ratio) proportion, into which the general word proportion may be divided. The one whereof, namely, that wherein the difference is not compared with either of the differing quantities, is called ratio arithmetica, arithmetical proportion; the other ratio geometrica, geometrical proportion; and, because this latter is only taken notice of by the name of proportion, simply proportion. Having considered this, I defined proportion, chapter II. article 3, in this manner: “Ratio est relatio antecedentis ad consequens secundum magnitudinem:” Proportion is the relation of the antecedent to the consequent in magnitude; having immediately before defined relatives, antecedent, and consequent, in the same article, and by way of explication added, that such relation was nothing else but that one of the quantities was equal to the other, or exceeded it by some quantity, or was by some quantity exceeded by it. And for exemplification of the same, I added further, that the proportion of three to two was, that three exceeded two by a unity; but said not that the unity, or the difference whatsoever it were, was their proportion, 234for unity, and to exceed another by unity, is not the same thing. This is clear enough to others; let us therefore see why it is not so to you. You say I make proportion to consist in that which remaineth after the lesser quantity is subtracted out of the greater; and that you make it to consist in the quotient, when one number is divided by the other. Wherein you are mistaken; first, in that you say, I make the proportion to consist in the remainder. For I make it to consist in the act of exceeding, or of being exceeded, or of being equal; whereas the remainder is always an absolute quantity, and never a proportion. To be more or less than another number by two, is not the number two; likewise to be equal to two, where the difference is nothing, is not that nothing? Again, you mistake in saying the proportion consisteth in the quotient. For divide twenty by five, the quotient is four. Is it not absurd to say that the proportion of five to twenty, or of twenty to five, is four? You may say the proportion of five to twenty, is the proportion of one to four. And so say I. And you may therefore also say, that the proportion of one to four is a measure of the proportion of five to twenty, as being equal. And so say I. But that is only in geometrical proportion, and not in proportion universally. For though the species obtain the denomination of the genus, yet it is not the genus. But as the quotient giveth us a measure of the proportion of the dividend to the divisor in geometrical proportion, so also the remainder after subtraction is the measure of proportion arithmetical.

You object in the next place, “that if the proportion 235of one quantity to another be nothing but the excess or defect, then, wheresoever the excess or defect is the same, there the proportion is the same.” This you say follows in your logic, and from thence, that the proportion of three to two, and five to four is the same. But is not three to two, and five to four, where the excess is the same number, the same proportion arithmetical? And is not arithmetical proportion, proportion? You take here (ratio) proportion, which is the genus, for that species of it which is called geometrical, because usually this species has the name of proportion simply. Also that the proportion of three to two, is the same with that of nine to six; is it not because the excesses are one and three, the same portions of three and nine, that is to say the same excesses comparatively? I wonder you ask me not what is the genus of arithmetical and geometrical proportions, and what the difference? The genus is (ratio) proportion, or comparison in magnitude, and the difference is that one comparison is made by the absolute quantity, the other by the comparative quantity, of the excess or defect, if there be any. Can anything be clearer than this? You after come in with ignosce habitudini to no purpose. I am not so inhuman as not to pardon dulness or madness: they are not voluntary faults. But when men adventure voluntarily to talk of that they understand not censoriously and scornfully, I may tell them of it.

This difference between the excesses or defects, as they are simply or comparatively reckoned, being thus explained, all the rest of that you say in your objections to this eleventh chapter (saving 236that art. 5 for ratio binarii ad quinarium est superari ternario, as it is in other places, I have put too hastily ratio binarii ad quinarium est ternarius), will be understood by every reader to be frivolous, and to proceed from the ignorance of what proportion is.

At the twelfth chapter you only note that I say, that the proportion of inequality is quantity, but the proportion of equality not quantity, and refer that which you have to say against it to the chapter following; to which place I shall also come in the following lesson.

You begin your reprehension of my thirteenth chapter with a question; whereas I divide proportion into arithmetical and geometrical. You ask me what proportion it is I so divide. Euclid divides an angle into right, obtuse, and acute. I may ask you as pertinently, what angle it is he so divides? Or, when you divide animal into homo and brutum, what animal that is, which you so divide? You see by this, how absurd your question is. But you say the definition of proportion which I make at Chap. II. art. 3., namely, that proportion is the comparison of two magnitudes, one to another, agreeth not, neither with arithmetical, 237nor with geometrical proportion. I believe you thought so then, but having read what I have said in the end of the last lesson, if you think so still, your fault will be too great to be pardoned easily. But why did you think so before? Is it not because there was no definition in Euclid of proportion universal, and because he maketh no mention of proportion arithmetical, and because you had not in your minds a sufficient notion thereof yourselves to supply that defect? And is not this the cause also, why you put in this parenthesis (if arithmetical proportion ought to be called proportion)? Which is a confession that you know not whether there be such a thing as arithmetical proportion or not, notwithstanding that on all occasions you speak of arithmetical proportionals. Yes, this was it that made you think that proportion universally, and proportion geometrical, is the same, and yet to say you cannot tell whether they be the same or not. It is no wonder, therefore, if in such confusion of the understanding, you apprehend not that the proportions of two to five, and nine to twelve, are the same; so you are blinded by seeing that they are not the same proportions geometrical. Nor doth it help you that I say the difference is the proportion; for by difference you might, if you would, have understood the act of differing.

At the second article you note for a fault in method, that after I had used the words antecedent and consequent of a proportion in some of the precedent chapters, I define them afterwards. I do not believe you say this against your knowledge, but that the eagerness of your malice made you 238oversee; therefore go back again to the third article of chapter II. where, having defined correlatives, I add these words, of which the first is called the antecedent, the second the consequent. This is but an oversight, though such as in me you would not have excused.

At the thirteenth article you find fault with, that I say that the proportion of inequality, whether it be of excess or of defect, is quantity, but the proportion of equality is not quantity. Whether that which you say, or that which I say, be the truth, is a question worthy of a very strict examination. The first time I heard it argued, was in Mersennus’ chamber at Paris, at such time as the first volume of his Cogitata Physico-Mathematica was almost printed; in which, because he had not said all he would say of proportion, he was forced to put the rest into a general preface, which, as was his custom, he did read to his friends before he sent it to the press. In that general preface, under the title De Rationibus atque Proportionibus, at the numbers twelve, thirteen, fourteen, he maintaineth against Clavius, that the composition of proportion is (as of all other things) a composition of the parts to make a total, and that the proportion of equality answereth in quantity to non-ens, or nothing; the proportion of excess, to ens, or quantity; and the proportion of defect, to less than nothing; because equality (he says) is a term of middle signification between excess and defect. And there also he refuteth the arguments which Clavius, at the end of the ninth Element of Euclid, bringeth to the contrary. And though this were approved by divers good geometricians then 239present, and never gainsaid by any since, yet do not I say it upon the credit of them, but upon sufficient grounds. For it hath been demonstrated by Eutocius, that if there be three magnitudes, the proportion of the first to the third is compounded of the proportions of the first to the second, and of the second to the third; which also I demonstrate in this article. And if there were never so many magnitudes ranked, it might be likewise demonstrated, that the proportion of the first to the last is compounded of the proportions of the first to the second, and of the second to the third, and of the third to the fourth, and so on to the last. If, therefore, we put in order any three numbers, whereof the two last be equal, as four, seven, seven, the proportion of four the first to seven the last, will be compounded of the proportions of four the first to seven the second, and of seven the second to seven the third. Wherefore the proportion of seven to seven (which is of equality) addeth nothing to the proportion of four the first, to seven the second; and consequently the proportion of seven to seven hath no quantity; but that the proportion of inequality hath quantity, I prove it from this, that one inequality may be greater than another.

But for the clearing of this doctrine (which Mersennus calls intricate) of the composition of proportions, I observed, first, that any two quantities, being exposed to sense, their proportion was also exposed; which is not intricate. Again, I observed that if besides the two exposed quantities, there were exposed a third, so as the first were the least, and the third the greatest, or the first the 240greatest, and the third the least, that not only the proportions of the first to the second, but also (because the differences and the quantities proceed the same way) the proportion of the first to the last is exposed by composition, or addition of the differences; nor is there any intricacy in this. But when the first is less than the second, and the second greater than the third, or the first greater than the second, and the second less than the third, so that to make the first and second equal, if we use addition, we must, to make the second and third equal, use subtraction; then comes in the intricacy, which cannot be extricated, but by such as know the truth of this doctrine which I now delivered out of Mersennus, namely, that the proportions of excess, equality, and defect, are as quantity, not-quantity, nothing want quantity; or as symbolists mark them 0+1 . 0 . 0-1. And upon this ground I thought depended the universal truth of this proposition, that in any rank of magnitudes of the same kind, the proportion of the first to the last, was compounded of all the proportions (in order) of the intermediate quantities; the want of the proof thereof, Sir Henry Savile calls (nævus) a mole in the body of geometry. This proposition is demonstrated at the thirteenth article of this chapter.

But before we come thither, I must examine the arguments you bring to confute this proposition, that the proportion of inequality is quantity, of equality, not quantity.

And first, you object that equality and inequality are in the same predicament: a pretty argument to flesh a young scholar in the logic school, 241that but now begins to learn the predicaments. But what do you mean by æquale and inequale? Do you mean corpus æquale, and corpus inequale? They are both in the predicament of substance, neither of them in that of quantity. Or do you mean æqualitas and inæqualitas? They are both in the predicament of relation, neither of them in that of quantity; and yet both corpus and inæqualitas, though neither of them be quantity, may be quanta, that is, both of them have quantity. And when men say body is quantity, or inequality is quantity, they are no otherwise understood, than if they had said corpus est tantum, and inæqualitas tanta, not tantitas; that is, bodies and inequalities are so much, not somuchness; and all intelligent men are contented with that expression, and yourselves use it. And the quantity of inequality is in the predicament of quantity, because the measure of it is in that line by which one quantity exceeds the other. But when neither exceedeth the other, then there is no line of excess, or defect by which the equality can be measured, or said to be so much, or be called quantity. Philosophy teacheth us how to range our words; but Aristotle’s ranging them in his predicaments doth not teach philosophy; and therefore no argument taken from thence, can become a doctor and a professor of geometry.

To prove that the proportion of inequality was quantity, but the proportion of equality not quantity, my argument was this: that because one inequality may be greater or less than another, but one equality cannot be greater nor less than another: therefore inequality hath quantity, or is 242tanta, and equality not. Here you come in again with your predicaments, and object, that to be susceptible of magis and minus, belongs not to quantity, but to quality; but without any proof, as if you took it for an axiom. But whether true or false, you understand not in what sense it is true or false. It is true that one inequality is inequality, as well as another; as one heat is heat as well as another, but not as great. Tam, but not tantus. But so it is also in the predicament of quantity; one line is as well a line as another, but not so great. All degrees, intentions, and remissions of quality, are greater or less quantity of force, and measured by lines, superficies, or solid quantity, which are properly in the predicament of quantity. You see how wise a thing it is to argue from the predicaments of Aristotle, which you understand not; and yet you pretend to be less addicted to the authority of Aristotle now than heretofore.

In the next place you say, I may as well conclude from the not susception of greater and less, that a right angle is not quantity, but an oblique one is. Very learnedly. As if to be greater or less, could be attributed to a quantity once determined. Number (that is, number indefinitively taken) is susceptible of greater and less, because one number may be greater than another; and this is a good argument to prove that number is quantity. And do you think the argument the worse for this, that one six cannot be greater than another six? After all these childish arguments which you have hitherto urged, can you persuade any man, or yourselves, that you are logicians?

243To the fifth and sixth article you object, first, that if I had before sufficiently defined (ratio) proportion, I needed not again define what is (eadem ratio) the same proportion; and ask me whether when I have defined man, I use to define anew what is the same man? You think when you have the definition of homo, you have also the definition of idem homo, when it is harder to conceive what idem signifies, than what homo. Besides, idem hath not the same signification always, and with whatsoever it be joined; it doth not signify the same with homo, that it doth with ratio. For with homo it signifies the same individual man, but with ratio it signifies a like, or an equal proportion: and both (ratio) proportion and (idem) the same, being defined, there will still be need of another definition for (eadem ratio) the same proportion; and this is enough to defend both myself and Euclid, against this objection: for Euclid also, after he had defined (ratio) proportion, and that sufficiently, as he believed, yet he defines the same proportion again apart. I know you did not mean in this place to object anything against Euclid, but you saw not what you were doing. There is within you some special cause of intenebration, which you should do well to look to.

In the next place you say, when I had defined arithmetical proportions to be the same when the difference is the same; it was to be expected I should define geometrical proportions to be then the same, when the antecedents are of their consequents totuple or tantuple, that is, equimultiple (for tantuplum signifies nothing). In plain words, you expected, that as I defined one by subtraction, 244I should define the other by the quotient in division. But why should you expect a definition of the same proportion by the quotient? Neither reason nor the authority of Euclid could move you to expect it. Or why should you say it was to be expected? But it seems you have the vanity to place the measure of truth in your own learning. In lines incommensurable there may be the same proportion, when, nevertheless, there is no quotient; for setting their symbols one above another doth not make a quotient: for quotient there is none, but in aliquot parts. It is therefore impossible to define proportion universally, by comparing quotients. This incommensurability of magnitudes was it that confounded Euclid in the framing of his definition of proportion at the fifth Element. For when he came to numbers, he defined the same proportion irreprehensibly thus: numbers are then proportional, when the first of the second and the third of the fourth are equimultiple, or the same part, or the same parts; and yet there is in this definition no mention at all of a quotient. For though it be true, that if in dividing two numbers you make the same quotient, the dividends and the divisors are proportional, yet that is not the definition of the same proportion, but a theorem demonstrable from it. But this definition Euclid could not accommodate to proportion in general, because of incommensurability.

To supply this want, I thought it necessary to seek out some way, whereby the proportion of two lines, commensurable or incommensurable, might be continued perpetually the same. And this I 245found might be done by the proportion of two lines described by some uniform motion, as by an efficient cause both of the said lines, and also of their proportions; which motions continuing, the proportions must needs be all the way the same. And therefore I defined those magnitudes to have the same geometrical proportion, when some cause producing in equal times equal effects, did determine both the proportions. This, you say, needs an Œdipus to make it understood. You are, I see, no Œdipus; but I do not see any difficulty, neither in the definition nor in the demonstration. That which you call perplexity in the explication, is your prejudice, arising from the symbols in your fancy. For men that pretend no less to natural philosophy than to geometry, to find fault with bringing motion and time into a definition, when there is no effect in nature which is not produced in time by motion, is a shame. But you swim upon other men’s bladders in the superficies of geometry, without being able to endure diving, which is no fault of mine; and therefore I shall, without your leave, be bold to say, I am the first that hath made the grounds of geometry firm and coherent. Whether I have added anything to the edifice or not, I leave to be judged by the readers. You see, you that profess with the pricking of bladders the letting out of their vapour, how much you are deceived. You make them swell more than ever.

For the corollaries that follow this sixth article, you say they contain nothing new. Which is not true. For the ninth is new, and the demonstrations of all the rest are new, being grounded upon a new definition of proportion; and the corollaries 246themselves, for want of a good definition of proportion, were never before exactly demonstrated. For the truth of the sixth definition of the fifth Element of Euclid cannot be known but by this definition of mine; because it requires a trial in all numbers possible, that is to say, an infinite time of trial, whether the quimultiples of the first and third, and of the second and fourth, in all multiplications, do together exceed, together come short, and are together equal; which trial is impossible.

In objecting against the thirteenth and sixteenth article, I observe that you bewray together, both the greatest ignorance and the greatest malice; and it is well, for they are suitable to one another, and fit for one and the same man. In the thirteenth article my proposition is this: If there be three magnitudes that have proportion one to another, the proportions of the first to the second, and of the second to the third, taken together (as one proportion), are equal to the proportion of the first to the third. This demonstrated, there is taken away one of those moles which Sir Henry Savile complaineth of in the body of geometry. Let us see now what you say, both against the enunciation and against the demonstration.

Against the enunciation you object, that other men would say (not the proportions of the first to the second, and of the second to the third, taken together, &c. but) the proportion which is compounded of the proportion of the first to the second, and of the second to the third, &c. Is not the compounding of any two things whatsoever the finding of the sum of them both, or the taking of them together as one total? This is that absurdity 247of which Mersennus, in the general preface to his Cogitata Physico-Mathematica, hath convinced Clavius, who, at the end of Euclid’s ninth Element, denieth the composition of proportion to be a composition of parts to make a total; which, therefore, he denied, because he did not observe, that the addition of a proportion of defect to a proportion of excess, was a subtraction of magnitude; and because he understood not that to say, composition is not the making a whole of parts, was contradiction; which all but too learned men would as soon as they heard abhor. Therefore, in saying that other men would not speak in that manner, you say in effect they would speak absurdly. You do well to mark what other geometricians say; but you would do better if you could by your own meditation upon the things themselves, examine the truth of what they say. But you have no mind, you say, to contend about the phrase. Let us see, therefore, what it is you contend about.

The proportion, you say, which is compounded of double and triple proportion, is not, as I would have it, quintuple, but sextuple, as in these numbers, six, three, one; where the proportion of six to three is double, the proportion of three to one triple, and the proportion of six to one sextuple, not quintuple. Tell me, egregious professors, how is six to three double proportion? Is six to three the double of a number, or the double of some proportion? All men know the number six is double to the number three, and the number three triple to an unity. But is the question here of compounding numbers, or of compounding proportions? 248Euclid, at the last proposition of his ninth Element, says indeed, that these numbers, one, two, four, eight, are ἐν διπλασίονι ἀναλογία, in double proportion; yet there is no man that understands it otherwise, than if he had said in proportion of the single quantity to the double quantity; and after the same rate, if he had said three, nine, twenty-seven, &c. had been in triple proportion, all men would have understood it, of the proportion of any quantity to its triple. Your instance, therefore, of six, three, one, is here impertinent, there being in them no doubling, no tripling, no sextupling of proportions, but of numbers. You may observe also, that Euclid never distinguished between double and duplicate, as you do. One word διπλάσιον serves him every where for either. Though, I confess, some curious grammarians take διπλάσιον for duplicate in number, and διπλοῦν for double in quantity; which will not serve your turn. Your geometry is not your own, but you case yourselves with Euclid’s; in which, as I have showed you, there be some few great holes; and you by misunderstanding him, as in this place, have made them greater. Though the beasts that think your railing roaring, have for a time admired you; yet now that through these holes of your case I have showed them your ears, they will be less affrighted. But to exemplify the composition of proportions, take these numbers, thirty-two, eight, one, and then you shall see that the proportion of thirty-two to one is the sum of the proportions of thirty-two to eight, and of eight to one. For the proportion of thirty-two to eight is double the proportion of thirty-two to sixteen; and the 249proportion of eight to one, is triple the proportion of thirty-two to sixteen; and the proportion of thirty-two to one is quintuple of thirty-two to sixteen; but double and triple added together maketh quintuple. What can be here denied?

My demonstration consisteth of three cases: the first is when both the proportions are of defect, which is then when the first quantity is the least; as in these three quantities, A B, A C, A D. The first case I demonstrated thus: A B C D
a Let it be supposed that the point A were moved uniformly through the whole line A D. The proportions, therefore, of A B to A C, and of A C to A D, are determined by the difference of the times in which they are described. And the proportion also of A B to A D, is that which is determined by the difference of the times in which they are described; but the difference of the times in which A B and A C are described, together with the difference of the times wherein A C and A D are described, is the same with the difference of the times wherein are described A B and A D. The same cause, therefore, which determines both the proportions of A B to A C, and of A C to A D, determines also the proportion of A B to A D. Wherefore, by the definition of the same proportion, article six, the proportion of A B to A C, together with the proportion of A C to A D, is the same with the proportion of A B to A D.

Consider now your argumentation against it. “Let there be taken,” say you, “between A and B the point a; and then in your own words, I argue thus: The difference of the times wherein are described A B and A C, together with the difference 250of the times wherein are described A C and A D, is the same with the difference of the times in which are described a B and a C (namely, B D, or B C + C D); wherefore, the same cause which determines the two proportions of A B to A C, and of A C to A D, determines also the proportion of a B to a D.” Let me ask you here whether you suppose the motion from a to B, or from a to D, to have the same swiftness with the motion from A to B, or from A to D? If you do not, then you deny the supposition. If you do, then B C, which is the difference of the times A B and A C, cannot be the difference of the times in which are described a B and a C, except A B and a B are equal. Let any man judge now whether there be any paralogism in Orontius that can equal this. And whether all that follows in the rest of this, and the next two whole pages, be not all a kind of raving upon the ignorance of what is the meaning of proportion, which you also make more ill-favoured by writing it; not in language, but in gambols; I mean in the symbols, which have made you call those demonstrations short, which put into words so many as a true demonstration requires, would be longer than any of those of Clavius upon the twelfth Element of Euclid.

To the sixteenth article you bring no argument, but fall into a loud oncethmus (the special figure wherewith you grace your oratory), offended with my unexpected crossing of the doctrine you teach, that proportion consisteth in a quotient. For that being denied you, your a
b - c
d + e
f - g
h + i
k comes to nothing, that is, to just as much as it is worth. But are not you very simple men, to say that all 251mathematicians speak so, when it is not speaking? When did you see any man but yourselves publish his demonstrations by signs not generally received, except it were not with intention to demonstrate, but to teach the use of signs? Had Pappus no analytics? or wanted he the wit to shorten his reckoning by signs? Or has he not proceeded analytically in a hundred problems (especially in his seventh book), and never used symbols? Symbols are poor unhandsome, though necessary, scaffolds of demonstration; and ought no more to appear in public, than the most deformed necessary business which you do in your chambers. “But why,” say you, “is this limitation to the proportion of the greater to the less?” I will tell you; because iterating of the proportion of the less to the greater, is a making of the proportion less, and the defect greater. And it is absurd to say that the taking of the same quantity twice should make it less. And thence it is, that in quantities which begin with the less, as one, two, four, the proportion of one to two is greater than that of one to four, as is demonstrated by Euclid, Elem. 5, prop. 8; and by consequent the proportion of one to four, is a proportion of greater littleness than that of one to two. And who is there, that when he knoweth that the respective greatness of four to one, is double to that of the respective greatness of four to two, or of two to one, will not presently acknowledge that the respective greatness of one to two, or two to four, is double to the respective greatness of one to four? But this was too deep for such men as take their opinions, not from weighing, but from reading.

252Lastly you object against the corollary of art. 28; which you make absurd enough by rehearsing it thus: Si quantitas aliqua divisa supponatur in partes aliquot æquales numero infinitas, &c. Do you think that of partes aliquot, or of partes aliquotæ, it can be said without absurdity, that they are numero infinitæ? And then you say I seem to mean, that if of the quantity A B, there be supposed a part C B, infinitely little; and that between A C and A B be taken two means, one arithmetical, A E, the other geometrical, A D, the difference between A D and A E, will be infinitely little. My meaning is, and is sufficiently expressed, that the said means taken everywhere (not in one place only) will be the same throughout: and you that say there needed not so much pains to prove it, and think you do it shorter, prove it not at all. For why may not I pretend against your demonstration, that B E, the arithmetical difference, is greater than B D, the geometrical difference. You bring nothing to prove it; and if you suppose it, you suppose the thing you are to prove. Hitherto you have proceeded in such manner with your Elenchus, as that so many objections as you have made, so many false propositions you have advanced. Which is a peculiar excellence of yours, that for so great a stipend as you receive, you will give place to no man living for the number and grossness of errors you teach your scholars.

At the fourteenth chapter your first exception is to the second article; where I define a plane in this manner: A plane superficies is that which is described by a straight line so moved, as that every point thereof describe a several straight line. In 253which you require, first, that instead of describe, I should have said can describe. Why do you not require of Euclid, in the definition of a cone, instead of continetur, is contained, he say contineri potest, can be contained ? If I tell you how one plane is generated, cannot you apply the same generation to any other plane? But you object, that the plane of a circle may be generated by the motion of the radius, whose every point describeth, not a straight, but a crooked line, wherein you are deceived; for you cannot draw a circle (though you can draw the perimeter of a circle) but in a plane already generated. For the motion of a straight line, whose one point resting, describeth with the other points several perimeters of circles, may as well describe a conic superficies, as a plane. The question, therefore, is, how you will, in your definition, take in the plane which must be generated before you begin to describe your circle, and before you know what point to make your centre. This objection, therefore, is to no purpose; and besides, that it reflecteth upon the perfect definitions of Euclid before the eleventh Element, it cannot make good his definition (which is nothing worth) of a plane superficies, before his first Element.

In the next place, you reprehend briefly this corollary, that two planes cannot enclose a solid. I should, indeed, have added, with the base on whose extremes they insist: but this is not a fault to be ashamed of; for any man, by his own understanding, might have mended my expression without departing from my meaning. But from your doctrine, that a superficies has no thickness, it is 254impossible to include a solid, with any number of planes whatsoever, unless you say that solid is included which nothing at all includes.

At the third article, where I say of crooked lines, some are everywhere crooked, and some have parts not crooked. You ask me what crooked line has parts not crooked; and I answer, it is that line which with a straight line makes a rectilineal triangle. But this, you say, cannot stand with what I said before, namely, that a straight and crooked line cannot be coincident; which is true, nor is there any contradiction; for that part of a crooked line which is straight, may with a straight line be coincident.

To the fourth article, where I define the centre of a circle to be that point of the radius, which in the description of the circle is unmoved; you object as a contradiction, that I had before defined a point to be the body which is moved in the description of a line: foolishly, as I have already shown at your objection to Chap. VIII. art. 12.

But at the sixth article, where I say, that crooked and incongruous lines touch one another but in one point, you make a cavil from this, that a circle may touch a parabola in two points. Tell me truly, did you read and understand these words that followed? “A crooked line cannot be congruent with a straight line; because if it could, one and the same line should be both straight and crooked.” If you did, you could not but understand the sense of my words to be this: when two crooked lines which are incongruous, or a crooked and a straight line touch one another, the contact is not in a line, but only in one point; and then 255your instance of a circle and a parabola was a wilful cavil, not befitting a doctor. If you either read them not, or understood them not, it is your own fault. In the rest that followeth upon this article, with your diagram, there is nothing against me, nor anything of use, novelty, subtlety, or learning.

At the seventh article, where I define both an angle, simply so called, and an angle of contingence, by their several generations; namely, that the former is generated when two straight lines are coincident, and one of them is moved, and distracted from the other by circular motion upon one common point resting, &c.; you ask me “to which of these kinds of angle I refer the angle made by a straight line when it cuts a crooked line?” I answer easily and truly, To that kind of angle which is called simply an angle. This you understand not. “For how”, will you say, “can that angle which is generated by the divergence of two straight lines, be other than rectilineal? or how can that angle which is not comprehended by two straight lines, be other than curvilineal?” I see what it is that troubles you; namely, the same which made you say before, that if the body which describes a line be a point, then there is nothing which is not moved that can be called a point. So you say here, “If an angle be generated by the motion of a straight line, then no angle so generated can be curvilineal;” which is as well argued, as if a man should say, the house was built by the carriage and motion of stone and timber, therefore, when the carriage and that motion is ended, it is no more a house. Rectilineal and curvilineal hath 256nothing to do with the nature of an angle simply so called, though it be essential to an angle of contact. The measure of an angle, simply so called, is a circumference of a circle; and the measure is always the same kind of quantity with the thing measured. The rectitude or curvity of the lines, which drawn from the centre, intercept the arch, is accidentary to the angle, which is the same, whether it be drawn by the motion circular of a straight line or of a crooked. The diameter and the circumference of a circle make a right angle, and the same which is made by the diameter and the tangent. And because the point of contact is not, as you think, nothing, but a line unreckoned, and common both to the tangent and the circumference; the same angle computed in the tangent is rectilineal, but computed in the circumference, not rectilineal, but mixed: or, if two circles cut one another, curvilineal. For every chord maketh the same angle with the circumference which it maketh with the line that toucheth the circumference at the end of the chord. And, therefore, when I divide an angle, simply so called, into rectilineal and curvilineal, I respect no more the generation of it, than when I divide it into right and oblique. I then respect the generation, when I divide an angle into an angle simply so called, and an angle of contact. This that I have now said, if the reader remember when he reads your objections to this, and to the ninth article, he will need no more to make him see that you are utterly ignorant of the nature of an angle; and that if ignorance be madness, not I, but you, are mad: and when an angle is comprehended between a straight 257and a crooked line (if I may compute the same angle as comprehended between the same straight line and the point of contact), that it is consonant to my definition of a point by a magnitude not considered. But when you, in your treatise, De Angulo Contactus (chap. III. p. 6, l. 8) have these words: “Though the whole concurrent lines incline to one another, yet they form no angle anywhere but in the very point of concourse:” you, that deny a point to be anything, tell me how two nothings can form an angle; or if the angle be not formed, neither before the concurrent lines meet, nor in the point of concourse, how can you apprehend that any angle can possibly be framed? But I wonder not at this absurdity; because this whole treatise of yours is but one absurdity, continued from the beginning to the end, as shall then appear when I come to answer your objections to that which I have briefly and fully said of that subject in my 14th chapter.

At the twelfth article, I confess your exception to my universal definition of parallels to be just, though insolently set down. For it is no fault of ignorance (though it also infect the demonstration next it), but of too much security. The definition is this: Parallels are those lines or superficies, upon which two straight lines falling, and wheresoever they fall, making equal angles with them both, are equal; which is not, as it stands, universally true. But inserting these words the same way, and making it stand thus: parallel lines or superficies, are those upon which two straight lines falling the same way, and wheresoever they fall, making equal angles, are equal, it is both true 258and universal; and the following consectary, with very little change, as you may see in the translation, perspicuously demonstrated. The same fault occurreth once or twice more; and you triumph unreasonably, as if you had given therein a very great proof of your geometry.

The same was observed also upon this place by one of the prime geometricians of Paris, and noted in a letter to his friend in these words (Chap. XIV. art. 12): “The definition of parallels wanteth somewhat to be supplied.” And of the consectary he says, “It concludeth not, because it is grounded on the definition of parallels.” Truly and severely enough, though without any such words as savour of arrogance, or of malice, or of the clown.

At the thirteenth article you recite the demonstration by which I prove the perimeters of two circles to be proportional to their semidiameters; and with esto, fortasse, recte, omnino, noddying to the several parts thereof, you come at length to my last inference: Therefore, by Chap. XIII. art. 6, the perimeters and semidiameters of circles are proportional; which you deny; and therefore deny, because you say it followeth by the same ratiocination, that circles also and spheres are proportional to their semidiameters. “For the same distance, you say, of the perimeter from the centre which determines the magnitude of the semidiameter, determines also the magnitude both of the circle and of the sphere.” You acknowledge that perimeters and semidiameters have the cause of their determination such as in equal times make equal spaces. Suppose now a sphere generated by the semidiameters, whilst the semicircle 259is turned about. There is but one radius of an infinite number of radii, which describes a great circle; all the rest describe lesser circles parallel to it, in one and the same time of revolution. Would you have men believe, that describing greater and lesser circles, is according to the supposition (temporibus æqualibus æqualia facere) to make equal spaces in equal times? Or, when by the turning about of the semidiameter is described the plane of a circle, does it, think you, in equal times make the planes of the interior circles equal to the planes of the exterior? Or is the radius that describes the inner circles equal to the radius that describes the exterior? It does not, therefore, follow from anything I have said in this demonstration, that either spheres or planes of circles, are proportional to their radii; and consequently, all that you have said, triumphing in your own incapacity, is said imprudently by yourselves to your own disgrace. They that have applauded you, have reason by this time to doubt of all the rest that follows, and if they can, to dissemble the opinion they had before of your geometry. But they shall see before I have done, that not only your whole Elenchus , but also your other books of the Angle of Contact , &c. are mere ignorance and gibberish.

To the fourteenth article you object, that (in the sixth figure) I assume gratis, that F G, D E, B C, are proportional to A F, A D, A B; and you refer it to be judged by the reader: and to the reader I refer it also. The not exact drawing of the figure (which is now amended) is it that deceived you. For A F, F D, D B, are equal by construction. Also, 260A G, G E, E C, are equal by construction. And F G, D K, B H, K E, H I, I C, are equal by parallelism. And because A F, F G, are as the velocities wherewith they are described; also 2 A F (that is A D) and 2 F G (that is D E) are as the same velocities. And finally, 3 A F (that is A B) and 3 F G (that is B C) are as the same velocities. It is not therefore assumed gratis, that F G, D E, B C are proportional to A F, A D, A B, but grounded upon the sixth article of the thirteenth chapter; and consequently your objection is nothing worth. You might better have excepted to the placing of D E, first at adventure, and then making A D two-thirds of A B; for that was a fault, though not great enough to trouble a candid reader; yet great enough to be a ground, to a malicious reader, of a cavil.

That which you object to the third corollary of art. 15, was certainly a dream. There is no assuming of an angle C D E, for an angle H D E, or B D E, neither in the demonstration, nor in any of the corollaries. It may be you dreamt of somewhat in the twentieth article of chapter XVI. But because that article, though once printed, was afterwards left out, as not serving to the use I had designed it for, I cannot guess what it is: for I have no copy of that article, neither printed nor written; but am very sure, though it were not useful, it was true.

Article the sixteenth. Here we come to the controversy concerning the angle of contact, which, you say, you have handled, in a special treatise published; and that you have clearly demonstrated, in your public lectures, that Peletarius 261was in the right. But that I agree not sufficiently, neither with Peletarius nor with Clavius. I confess I agree not in all points with Peletarius, nor in all points with Clavius. It does not thence follow that I agree not with the truth. I am not, as you, of any faction, neither in geometry nor in politics. If I think that you, or Peletarius, or Clavius, or Euclid, have erred, or been too obscure, I see no cause for which I ought to dissemble it. And in this same question I am of opinion that Peletarius did not well in denying the angle of contingence to be an angle. And that Clavius did not well to say, the angle of a semicircle was less than a right-lined right angle. And that Euclid did not well to leave it so obscure what he meant by inclination in the definition of a plane angle, seeing elsewhere he attributeth inclination only to acute angles; and scarce any man ever acknowledged inclination in a straight line, to any other line to which it was perpendicular. But you, in this question of what is inclination, though you pretend not to depart from Euclid, are, nevertheless, more obscure than he; and also are contrary to him. For Euclid by inclination meaneth the inclination of one line to another; and you understand it of the inclination of one line from another; which is not inclination, but declination. For you make two straight lines, when they lie one on another, to lie ἁκλινῶς, that is, without any inclination (because it serves your turn); not observing that it followeth thence, that inclination is a digression of one line from another. This is in your first argument in the behalf of Peletarius (p. 10, l. 22), and destroys his opinion. For, according to Euclid, 262the greatest angle is the greatest inclination; and an angle equal to two right angles by this ἀκλισία, should not be the greatest inclination, as it is, but the least that can be. But if by the inclination of two lines, we understand that proceeding of them to a common point, which is caused by their generation, which, I believe, was Euclid’s meaning; then will the angle of contact be no less an angle than a rectilineal angle, but only (as Clavius truly says it is) heterogeneous to it; and the doctrine of Clavius more conformable to Euclid than that of Peletarius. Besides, if it be granted you, that there is no inclination of the circumference to the tangent, yet it does not follow that their concourse doth not form some kind of angle; for Euclid defineth there but one of the kinds of a plane angle. And then you may as much in vain seek for the proportion of such angle to the angle of contact, as seek for the focus or parameter of the parabola of Dives and Lazarus. Your first argument therefore is nothing worth, except you make good that which in your second argument you affirm, namely, that all plane angles, not excepting the angle of contact, are (homogeneous) of the same kind. You prove it well enough of other curvilineal angles; but when you should prove the same of an angle of contact, you have nothing to say but (p. 17, l. 15), “Unde autem illa quam somniet heterogenia oriatur, neque potest ille ullatenus ostendere, neque ego vel somniare:” “Whence should arise that diversity of kind which he dreams of, neither can he at all show, nor I dream;” as if you knew what he could do if he were to answer you; or all were false which you 263cannot dream of. So that besides your customary vanity, here is nothing hitherto proved, neither for the opinion of Peletarius, nor against that of Clavius. I have, I think, sufficiently explicated, in the first lesson, that the angle of contact is quantity, namely, that it is the quantity of that crookedness or flexion, by which a straight line is bent into an arch of a circle equal to it; and that because the crookedness of one arch may be greater than the crookedness of another arch of another circle equal to it; therefore the question quanta est curvitas, how much is the crookedness, is pertinent, and to be answered by quantity. And I have also shown you in the same lesson, that the quantity of one angle of contact is compared with that of another angle of contact by a line drawn from the point of contact, and intercepted by their circumferences; and that it cannot be compared by any measure with a rectilineal angle.

But let us see how you answer to that which Clavius has objected already. “They are heterogeneous,” says he, “because the angle of contact, how oft soever multiplied, can never exceed a rectilineal angle.” To answer which, you allege it is no angle at all; and that therefore, it is no angle at all, because the lines have no inclination one to another. How can lines that have no inclination one to another, ever come together? But you answer, at least they have no inclination in the point of contact. And why have two straight lines inclination before they come to touch, more than a straight line and an arch of a circle? And in the point of contact itself, how can it be that there is less inclination of the two points of a 264straight line and an arch of a circle, than of the points of two straight lines? But the straight lines, you say, will cut; which is nothing to the question; and yet this also is not so evident, but that it may receive an objection. Suppose two circles, A G B and C F B, to touch in B, and have a common tangent through B. Is not the line C F B G A a crooked line? and is it not cut by the common tangent D B E? What is the quantity of the two angles F B E and G B D, seeing you say neither D B G nor E B F is an angle? It is not, therefore, the cutting of a crooked line, and the touching of it, that distinguisheth an angle simply, from an angle of contact. That which makes them differ, and in kind, is, that the one is the quantity of a revolution, and the other, the quantity of flexion.

In the seventh chapter of the same treatise, you think you prove the angle of contact, if it be an angle, and a rectilineal angle to be (homogeneous) of the same kind; when you prove nothing but that you understand not what you say. Those quantities which can be added together, or subtracted one from another, are of the same kind; but an angle of contact may be subtracted from a right angle, and the remainder will be the angle of a semicircle, &c. So you say, but prove it not, unless you think a man must grant you that the superficies contained between the tangent and the arch, which is it you subtract, is the angle of contact; and that the plane of the semicircle is the angle of the semicircle, which is absurd; though, as absurd as it is, you say it directly in your Elenchus , 265p. 35, l. 14, in these words: “When Euclid defines a plane angle to be the inclination of two lines, he meaneth not their aggregate, but that which lies between them.” It is true, he meaneth not the aggregate of the two lines; but that he means that which lies between them, which is nothing else but an indeterminate superficies, is false, or Euclid was as foolish a geometrician as either of you two.

Again, you would prove the angle of contact, if it be an angle, to be of the same kind with a rectilineal angle, out of Euclid (III. 16); where he says, it is less than any acute angle. And it follows well, that if it be an angle, and less than any rectilineal angle, it is also of the same kind with it. But, to my understanding, Euclid meant no more, but that it was neither greater nor equal; which is as truly said of heterogeneous, as of homogeneous quantities. If he meant otherwise, he confirms the opinion of Clavius against you, or makes the quantity of an angle to be a superficies, and indefinite. But I wonder how you dare venture to determine whether two quantities be homogeneous or not, without some definition of homogeneous (which is a hard word), that men may understand what it meaneth.

In your eighth chapter you have nothing but Sir H. Savile’s authority, who had not then resolved what to hold; but esteeming the angle of contact, first, as others falsely did, by the superficies that lies between the tangent and the arch, makes the angle of contact and a rectilineal angle homogeneous; and afterwards, because no multiplication of the angle of contact 266can make it equal to the least rectilineal angle, with great ingenuity returneth to his former uncertainty.

In your ninth and tenth chapters you prove with much ado, that the angles of like segments are equal; as if that might not have been taken gratis by Peletarius, without demonstration. And yet your argument, contained in the ninth chapter, is not a demonstration, but a conjectural discourse upon the word similitude. And in the eleventh chapter, wherein you answer to an objection, which might be made to your argument in the precedent page, taken from the parallelism of two concentric circles, though objection be of no moment, yet you have in the same treatise of yours that which is much more foolish, which is this, (p. 38, l. 12): “Non enim magnitudo anguli,” &c. “The“The magnitude of an angle is not to be estimated by that straddling of the legs, which it hath without the point of concourse, but by that straddling which it hath in the point of the concourse itself.” I pray you tell me what straddling there is of two coincident points, especially such points as you say are nothing? When did you ever see two nothings straddle?

The arguments in your twelfth and thirteenth chapters are grounded all on this untruth, that an angle is that which is contained between the lines that make it; that is to say, is a plane superficies, which is manifestly false; because the measure of an angle is an arch of a circle, that is to say, a line; which is no measure of a superficies. Besides this gross ignorance, your way of demonstration, by putting N for a great number of sides of an equilateral 267polygon, is not to be admitted; for, though you understand something by it, you demonstrate nothing to anybody but those who understand your symbolic tongue, which is a very narrow language. If you had demonstrated it in Irish or Welsh, though I had not read it, yet I should not have blamed you, because you had written to a considerable number of mankind, which now you do not.

In your last chapters you defend Vitellio without need; for there is no doubt but that whatsoever crooked line be touched by a straight line, the angle of contingence will neither add anything to, nor take anything from, a rectilineal right angle; but that it is because the angle of contact is no angle, or no quantity, is not true. For it is therefore an angle, because an angle of contact; and therefore quantity, because one angle of contact may be greater than another; and therefore heterogeneal, because the measure of an angle of contact cannot (congruere) be applied to the measure of a rectilineal angle, as they think it may, who affirm with you that the nature of an angle consisteth in that which is contained between the lines that comprehend it, viz., in a plane superficies. And thus you see in how few lines, and without brachygraphy, your treatise of the angle of contingence is discovered for the greatest part to be false, and for the rest, nothing but a detection of some errors of Clavius grounded on the same false principles with your own. To return now from your treatise of the angle of contact back again to your Elenchus .

The fault you find at art. 18, is, that I understand 268not that Euclid makes a plane angle to be that which is contained between the two lines that form it. It is true, that I do not understand that Euclid was so absurd, as to think the nature of an angle to consist in superficies; but I understand that you have not had the wit to understand Euclid.

The nineteenth article of mine in this fourteenth chapter, is this: “All respect or variety of position of two lines, seemeth to be comprehended in four kinds. For they are either parallel, or (being if need be produced) make an angle; or, (if drawn out far enough) touch; or, lastly, they are asymptotes”; in which you are first offended with the word It seems. But I allow you, that never err, to be more peremptory than I am. For to me it seemed (I say again seemed) that such a phrase, in case I should leave out something in the enumeration of the several kinds of position, would save me from being censured for untruth; and yet your instance of two straight lines in divers planes, does not make my enumeration insufficient. For those lines, though not parallels, nor cutting both the planes, yet being moved parallelly from one plane to another, will fall into one or other of the kinds of position by me enumerated; and consequently, are as much that position, as two straight lines in the same plane, not parallel, make the same angle, though not produced till they meet, which they would make if they were so produced: for you have nowhere proved, nor can prove, that two such lines do not make an angle. It is not the actual concurrence of the lines, but the arch of a circle, drawn upon that point for centre, in which 269they would meet if they were produced, and intercepted between them, that constitutes the angle.

Also your objection concerning asymptotes in general is absurd. You would have me add, that their distance shall at last be less than any distance that can be assigned; and so make the definition of the genus the same with that of the species. But because you are not professors of logic, it is not necessary for me to follow your counsel. In like manner, if we understand one line to be moved towards another always parallelly to itself, which is, though not actually, yet potentially the same position, all the rest of your instances will come to nothing.

At the two-and-twentieth article you object to me the use of the word figure, before I had defined it: wherein also you do absurdly; for I have nowhere before made such use of the word figure, as to argue anything from it; and therefore your objection is just as wise as if you had found fault with putting the word figure in the titles of the chapters placed before the book. If you had known the nature of demonstration, you had not objected this.

You add further, that by my definition of figure, a solid sphere, and a sphere made hollow within, is the same figure; but you say not why, nor can you derive any such thing from my definition. That which deceived your shallowness, is, that you take those points that are in the concave superficies of a hollowed sphere, not to be contiguous to anything without it, because that whole concave superficies is within the whole sphere. Lastly, for the fault you find with the definition of like 270figures in like positions, I confess there wants the same word which was wanting in the definition of parallels; namely, ad easdem partes (the same way) which should have been added in the end of the definition of like figures, &c., and may easily be supplied by any student of geometry, that is not otherwise a fool.

At the fifteenth chapter, art. 1, number 6, you object as a contradiction, that I make motion to be the measure of time; and yet, in other places, do usually measure motion and the affections thereof by time. If your thoughts were your own, and not taken rashly out of books, you could not but, (with all men else that see time measured by clocks, dials, hour-glasses, and the like), have conceived sufficiently, that there cannot be of time any other measure besides motion; and that the most universal measure of motion, is a line described by some other motion; which line being once exposed to sense, and the motion whereby it was described sufficiently explicated, will serve to measure all other motions and their time: for time and motion (time being but the mental image or remembrance of the motion) have but one and the same dimension, which is a line. But you, that would have me measure swiftness and slowness by longer and shorter motion, what do you mean by longer and shorter motion? Is longer and shorter in the motion, or in the duration of the motion, which is time? Or is the motion, or the duration of the motion, that which is exposed, or designed by a line? Geometricians say often, let the line A B be the time; but never say, let the line A B be the motion. There is no unlearned man that understandeth 271not what is time, and motion, and measure; and yet you, that undertake to teach it (most egregious professors) understand it not.

At the second article you bring another argument (which it seems in its proper place you had forgotten), to prove that a point is not quantity not considered, but absolutely nothing; which is this, That if a point be not nothing, then the whole is greater than its two halves. How does that follow? Is it impossible when a line is divided into two halves, that the middle point should be divided into two halves also, being quantity?

At the seventh article, I have sufficiently demonstrated, that all motion is infinitely propagated, as far as space is filled with body. You allege no fault in the demonstration, but object from sense, that the skipping of a flea is not propagated to the Indies. If I ask you how you know it, you may wonder perhaps, but answer you cannot. Are you philosophers, or geometricians, or logicians, more than are the simplest of rural people? or are you not rather less, by as much as he that standeth still in ignorance, is nearer to knowledge, than he that runneth from it by erroneous learning?

And, lastly, what an absurd objection is it which you make to the eighth article, where I say that when two bodies of equal magnitude fall upon a third body, that which falls with greater velocity, imprints the greater motion? You object, that not so much the magnitude is to be considered as the weight; as if the weight made no difference in the velocity, when notwithstanding weight is nothing else but motion downward. Tell me, when a weighty body thrown upwards worketh on the 272body it meeteth with, do you not then think it worketh the more for the greatness, and the less for the weight.

Of twenty articles which you say (of nineteen which I say) make the sixteenth chapter, you except but three, and confidently affirm the rest are false. On the contrary, except three or four faults, such as any geometrician may see proceed not from ignorance of the subject, or from want of the art of demonstration, (and such as any man might have mended of himself) but from security; I affirm that they are all true, and truly demonstrated; and that all your objections proceed from mere ignorance of the mathematics.

The first fault you find is this, that I express not (art. 1) what impetus it is, which I would have to be multiplied into the time.

The last article of my thirteenth chapter was this, “If there be a number of quantities propounded, howsoever equal or unequal to one another; and there be another quantity which so often taken as there be quantities propounded, is equal to their whole sum; that quantity I call the mean arithmetical of them all.” Which definition I did there insert to serve me in the explication of those 273propositions of which the sixteenth chapter consisted, but did not use it here as I intended. My first proposition therefore as it standeth yet in the Latin, being this, “the velocity of any body moved during any time, is so much as is the product of the impetus in one point of time, multiplied into the whole time;” to a man that hath not skill enough to supply what is wanting, is not intelligible. Therefore I have caused it in the English to go thus: “the velocity of any body in whatsoever time moved, hath its quantity determined by the sum of all the several (impetus) quicknesses, which it hath in the several points of the time of the body’s motion. And added, that all the impetus together taken through the whole time is the same thing with the mean impetus (which mean is defined (Chapter XIII. art. 29) multiplied into the whole time.” To this first article, as it is uncorrected in the Latin, you object, that meaning by impetus some middle impetus, and assigning none, I determine nothing. And it is true. But if you had been geometricians sufficient to be professors, you would have shewed your skill much better, by making it appear that this middle impetus could be none but that, which being taken so often, as there be points in the line of time, would be equal to the sum of all the several impetus taken in the points of time respectively; which you could not do.

To the corollary, you ask first how impetus can be ordinately applied to a line; absurdly. For does not Archimedes sometimes say, and with him many other excellent geometricians, let such a line be the time? And do they not mean, that that 274line, or the motion over it, is the measure of the time? And may not also a line serve to measure the swiftness of a motion? You thought, you say, only lines ought to be said to be ordinately applied to lines. Which I easily believe; for I see you understand not that a line, though it be not the time itself, may be the quantity of a time. You thought also, all you have said in your Elenchus , in your doctrine of the angle of contact, in your Arithmetica Infinitorum, and in your Conics , is true; and yet it is almost all proved false, and the rest nothing worth.

Secondly, you object, that I design a parallelogram by one only side. It was indeed a great oversight, and argueth somewhat against the man, but nothing against his art. For he is not worthy to be thought a geometrician that cannot supply such a fault as that, and correct his book himself. Though you could not do it, yet another from beyond sea took notice of the same fault in this manner, “He maketh a parallelogram of but one side; it should be thus: vel denique per parallelogrammum cujus unum latus est medium proportionale inter impetum maximum (sive ultimo acquisitum) et impetus ejusdem maximi semissem; alterum vero latus, medium proportionale, inter totum tempus, et ejusdem totius temporis semissem.” Which I therefore repeat, that you may learn good manners; and know, that they who reprehend, ought also, when they can, to add to their reprehension the correction.

At the second article, you are pleased to advise me, instead of in omni motu uniformi, to put in in omnibus motibus uniformibus. You have a strange 275opinion of your own judgment, to think you know to what end another man useth any word, better than himself. My intention was only to consider motions uniform, and motions from rest uniformly, or regularly accelerated, that I might thereby compute the lengths of crooked lines, such as are described by any of those motions. And therefore it was enough to prove this theorem to be true in all uniform or uniformly accelerated motion, not motions; though it be true also in the plural. It seems you think a man must write all he knows, whether it conduce, or not, to his intended purpose. But that you may know that I was not (as you think), ignorant how far it might be extended, you may read it demonstrated at the same article in the English universally. Against the demonstration itself you run into another article, namely, the thirteenth, which is this problem: “the length being given, which is passed over in a given time by uniform motion, to find the length which shall be passed over by motion uniformly accelerated in the same time, so as that the impetus last acquired be equal to the time.” Which you recite imperfectly, thereby to make it seem that such a length is not determined. Whether you did this out of ignorance, or on purpose, thinking it a piece of wit, as your pretended mystery which goes immediately before, I cannot tell, for in neither place can any wit be espied by any but yourselves. To imagine motions with their times and ways, is a new business, and requires a steady brain, and a man that can constantly read in his own thoughts, without being diverted by the noise of words. The want of this ability, made you stumble and fall unhandsomely 276in the very first place (that is in Chap. XIII. art. 13), where you venture to reckon both motion and time at once; and hath made you in this chapter to stumble in the like manner at every step you go. As, for example, when I say, as the product of the time, and impetus, to the product of the time and impetus, so the space to the space when the motion is uniform; you come in with, nay, rather as the time to the time; as if the parallelograms A I, and A H, were not also as the times A B, and A F. Thus it is, when men venture upon ways they never had been in before, without a guide.

In the corollary, you are offended with the permutation of the proportion of times and lines, because you think, (you that have scarce one right thought of the principles of geometry), that line and time are heterogeneous quantities. I know time and line are of divers natures; and more, that neither of them is quantity. Yet they may be both of them quanta, that is, they may have quantity; but that their quantities are heterogeneous is false. For they are compared and measured both of them by straight lines. And to this there is nothing contrary in the place cited by you out of Clavius; or if there were, it were not to be valued. And to your question, what is the proportion of an hour to an ell? I answer, it is the same proportion that two hours have to two ells. You see your question is not so subtle as you thought it. By and bye you confess that in times and lines there is quid homogeneum (this quid is an infallible sign of not fully understanding what you say); which is false if you take it of the lines 277themselves; though if you take it of their quantities, it is true without a quid. Lastly, you tell m”e how I might have expressed myself so as it might have been true. But because your expressions please me not, I have not followed your advice.

To the third article, which is this: “In motu uniformiter a quiete accelerato,” etc. “In motion uniformly accelerated from rest, that is, when the impetus increaseth in proportion to the times, the length run over in one time is to the length run over in another time, as the product of the impetus multiplied by the time, to the product of the impetus multiplied by the time;” you object, “that the lengths run over are in that proportion which the impetus hath to the impetus; not that which the impetus hath to the time, because impetus to time has no proportion, as being heterogeneous.” First, when you say the impetus, do you mean some one impetus designed by some one of the unequal straight lines parallel to the base B I? That is manifestly false. You mean the aggregate of all those unequal parallels. But that is the same thing with the time multiplied into the mean impetus. And so you say the same that I do. Again, I ask, where it is that I say or dream that the lengths run over are in the proportion of the impetus to the times? Is it you or I that dream? And for your heterogeneity of the quantities of time and of swiftness, I have already in divers places showed you your error. Again, why do you make B I represent the lengths run over, which I make to be represented by D E, a line taken at pleasure; and you also a few lines before make the same 278B I to design the greatest acquired impetus? These are things which show that you are puzzled and entangled with the unusual speculation of time and motion, and yet are thrust on with pride and spite to adventure upon the examination of this chapter.

Secondly, you grant the demonstration to be good, supposing I mean it, as I seem to speak, of one and the same motion. But why do I not mean it of one and the same motion, when I say not in motions, but in motion uniform? Because, say you, in that which follows, I draw it also to different motions. You should have given at least one instance of it; but there is no such matter. And yet the proposition is in that case also true; though then it must not be demonstrated by the similitude of triangles, as in the case present. And therefore the objections you make from different impetus acquired in the same time, and from other cases which you mention, are nothing worth.

At the fourth article, you allow the demonstration all the way (except the faults of the third, which I have already proved to be none) till I come to say, “that because the proportion of F K to B I is double to the proportion of A F to A B, therefore the proportion of A B to A F is double to the proportion of B I to F K.” This you deny, and wonder at as strange, (for it is indeed strange to you), and in many places you exclaim against it as extreme ignorance in geometry. In this place you only say, “no such matter; for though one proportion be double to another, yet it does not follow that the converse is the double of the converse.” So that this is the issue to which the 279question is reduced, whether you have any or no geometry. I say, if there be three quantities in continual proportion, and the first be the least, the proportion of the first to the second is double to the proportion of the first to the third; and you deny it. The reason of our dissent consisteth in this, that you think the doubling of a proportion to be the doubling of the quantity of the proportion, as well in proportions of defect, as in proportions of excess; and I think that the doubling of a proportion of defect, is the doubling of the defect of the quantity of the same. As for example in these three numbers, 1, 2, 4, which are in continual proportion, I say the quantity of the proportion of one to two, is double the quantity of the proportion of one to four. And the quantity of the proportion of one to four, is half the quantity of the proportion of one to two. And yet deny not but that the quantity of the defect in the proportion of one to two is doubled in the proportion of one to four. But because the doubling of defect makes greater defect, it maketh the quantity of the proportion less. And as for the part which I hold in this question, first, there is thus much demonstrated by Euclid, El. v. prop. 8; that the proportion of one to two, is greater than the proportion of one to four, though how much it is greater be not there demonstrated. Secondly, I have demonstrated (Chap, XIII. art. 16); that it is twice as great, that is to say, (to a man that speaks English), double. The introducing of duplicate, triplicate, &c. instead of double, triple, &c. (though now they be words well understood by such as understand what proportion is), proceeded at first from such as durst not for 280fear of absurdity, call the half of any thing double to the whole, though it be manifest that the half of any defect is a double quantity to the whole defect; for want added to want maketh greater want, that is, a less positive quantity. This difference between double and duplicate, lighting upon weak understandings, has put men out of the way of true reasoning in very many questions of geometry. Euclid never used but one word both for double and duplicate. It is the same fault when men call half a quantity subduplicate, and a third part subtriplicate of the whole, with intention (as in this case) to make them pass for words of signification different from the half and the third part. Besides, from my definition of proportion (which is clear, and easy to be understood by all men, but such as have read the geometry of others unluckily) I can demonstrate the same evidently and briefly thus. My definition is this, proportion is the quantity of one magnitude taken comparatively to another. Let there be therefore three quantities, 1, 2, 4, in continual proportion. Seeing therefore the quantity of four in respect of one, is twice as great as the quantity of the same four in respect of 2, it followeth manifestly that the quantity of 1 in respect of 4, is twice as little as the quantity of the same 1 in respect of 2; and consequently the quantity of 1 in respect of 2, is twice as great as the quantity of the same 1 in respect of 4; which is the thing I maintain in this question. Would not a man that employs his time at bowls, choose rather to have the advantage given him of three in nine, thanthan of one in nine? And why, but that three is a greater quantity in respect of nine, than 281is one? Which is as much as to say, three to nine hath a greater proportion than one to nine; as is demonstrated by Euclid, El. v. prop. 8. Is it not therefore (you that profess mathematics, and theology, and practise the depression of the truth in both) well owled of you, to teach the contrary? But where you say “that the point K (in the second figure of the table belonging to this sixteenth chapter) is not in the parabolical line whose diameter is A B, and base B I, but in the parabolical line of the complement of my semiparabola (as I may learn from the twenty-third proposition of your Arithmetica Infinitorum) whose diameter is A C, and base I C.” What line is that? Is it the same line with that of my semiparabola, or not the same? If the same, why find you fault? If not the same, you ought to have made a semiparabola on the diameter A C, and base I C, and following my construction made it appear that K is not in the line wherein I say it is; which you have not done, nor could do.

Then again, running on in the same blindness of passion, you pretend I make the proportion of B I to F K double to that of A B to A F, and then confute it; when you knew I made the proportion of F K to B I, double to that of F N, to B I, that is, of A F to A B; and this was it you should have confuted. That which followeth is but a triumphing in your own ignorance, wherein you also say, “that all that I afterwards build upon this doctrine is false.” You see whether it be like to prove so or not. As for your Arithmetica Infinitorum, I shall then read to you a piece of a lesson on it when I come to your objections against the next 282Chapter. In the mean time let me tell you, it is not likely you should be great geometricians, that know not what is quantity, nor measure, nor straight, nor angle, nor homogeneous, nor heterogeneous, nor proportion, as I have already made appear in this and the former lessons.

To the first corollary of this fourth article your exception I confess is just, and (which I wonder at) without any incivility. But this argues not ignorance, but security. For who is there that ever read any thing in the Conics, that knows not that the parts of a parabola cut off by lines parallel to the base, are in triplicate proportion to their bases? But having hitherto designed the time by the diameter, and the impetus by the base; and in the next chapter (where I was to calculate the proportion of the parabola, to the parallelogram) intending to design the time by the base, I mistook and put the diameter again for the time; which any man but you might as easily have corrected as reprehended.

To the second corollary, which is this, that the lengths run over in equal times by motion so accelerated, as that the impetus increase in double proportion to their times, are as the differences of the cubic numbers beginning at unity, that is, as seven, nineteen, thirty-seven, &c. you say it is false. But why? “Because” say you “portions of the parabola of equal altitude, taken from the beginning, are not as those numbers seven, nineteen, thirty-seven, &c.” Does this, think you, contradict any thing in this proposition of mine? Yes, because, you think, the lengths gone over in equal times, are the same with the parts of the 283diameter cut off from the vertex, and proportional to the numbers one, two, three, &c. Whereas the lengths run over, are as the aggregates of their velocities, that is, as the parts of the parabola itself, that is, as the cubes of their bases, that is, as the numbers one, eight, twenty-seven, sixty-four, &c., and consequently the lengths run over in equal times, are as the differences of those cubic numbers, one, eight, twenty-seven, sixty-four, whose differences are seven, nineteen, thirty-seven, &c. The cause of your mistake was, that you cannot yet, nor perhaps ever will, contemplate time and motion (which requireth a steady brain) without confusion.

The third corollary you also say is false, “whether it be meant of motion uniformly accelerated (as the words are) or (as perhaps, you say, I meant it) of such motion as is accelerated in double proportion to the time.” You need not say perhaps I meant it. The words of the proposition are enough to make the meaning of the corollary understood. But so also you say it is false. Methinks you should have offered some little proof to make it seem so. You think your authority will carry it. But on the contrary I believe rather that they that shall see how your other objections hitherto have sped, will the rather think it true, because you think it false. The demonstration as it is, is evident enough; and therefore I saw no cause to change a word of it.

To the fifth article you object nothing, but that it dependeth on this proposition (Chap. XIII. art. 16): “That when three quantities are in continual proportion, and the first is the least, as in 284these numbers, four, six, nine, the proportion of the first to the second, is double to the proportion of the same first to the last;” which is there demonstrated, and in the former lessons so amply explicated, as no man can make any further doubt of the truth of it. And you will, I doubt not, assent unto it. But in what estate of mind will you be then? A man of a tender forehead after so much insolence, and so much contumelious language grounded upon arrogance and ignorance, would hardly endure to outlive it. In this vanity of yours, you ask me whether I be angry, or blush, or can endure to hear you. I have some reason to be angry; for what man can be so patient as not to be moved with so many injuries? And I have some reason to blush, considering the opinion men will have beyond sea, (when they shall see this in Latin) of the geometry taught in Oxford. But to read the worst you can say against me, I can endure, as easily at least, as to read any thing you have written in your treatises of the Angle of Contact, of the Conic Sections, or your Arithmetica Infinitorum.

The sixth, seventh, eighth articles, you say are sound. True. But never the more to be thought so for your approbation, but the less; because you are not fit, neither to reprehend, nor praise; and because all that you have hitherto condemned as false, hath been proved true. Then you show me how you could demonstrate the sixth and seventh articles a shorter way. But though there be your symbols, yet no man is obliged to take them for demonstration. And though they be granted to be dumb demonstrations, yet when they are taught 285to speak as they ought to do, they will be longer demonstrations than these of mine.

To the ninth article, which is this, “If a body be moved by two movents at once, concurring in what angle soever, of which, one is moved uniformly, the other, with motion uniformly accelerated from rest, till it acquire an impetus equal to that of the uniform motion, the line in which the body is carried, shall be the crooked line of a semiparabola,” you lift up your voice again, and ask, what latitude? what diameter? what inclination of the diameter to the ordinate lines? If your founder should see this, or the like objections of yours, he would think his money ill bestowed. When I say, in what angle soever, you ask, in what angle? When I say two movents, one uniform, the other uniformly accelerated, make the body describe a semiparabolical line; you ask, which is the diameter? as not knowing that the accelerated motion describes the diameter, and the other a parallel to the base. And when I say the two movents meet in a point, from which point both the motions begin, and one of them from rest, you ask me what is the altitude? As if that point where the motion begins from rest were not the vertex; or that the vertex and base being given, you had not wit enough to see that the altitude of the parabola is determined? When Galileo’s proposition, which is the same with this of mine, supposed no more but a body moved by these two motions, to prove the line described to be the crooked line of a semiparabola, I never thought of asking him what altitude, nor what diameter, nor what angle, nor what base, had his parabola. And when 286Archimedes said, let the line A B be the time, I should never have said to him, do you think time to be a line, as you ask me whether I think impetus can be the base of a parabola. And why, but because I am not so egregious a mathematician, as you are. In this giddiness of yours, caused by looking upon this intricate business of motion, and of time, and the concourse of motion uniform, and uniformly accelerated, you rave upon the numbers 1, 4, 9, 16, &c. without reference to any thing that I had said; insomuch as any one that had seen how much you have been deceived in them before, in your scurvy book of Arithmetica Infinitorum, would presently conclude, that this objection was nothing else but a fit of the same madness which possessed you there.

My tenth article is like my ninth; and your objections to it are the same which are to the former. Therefore you must take for answer just the same which I have given to your objection there.

To the eleventh, you say first, you have done it better at the sixty-fourth article of your Arithmetica Infinitorum. But what you have done there, shall be examined when I come to the defence of my next chapter. And whereas I direct the reader for the finding of the proportions of the complements of those figures to the figures themselves, to the table of art. 3, Chap, XVII., you say that if the increase of the spaces, were to the increase of the times, as one to two, then the complement should be to the parallelogram as one to three, and say you find not 1
3 in the table. Did you not see that the table is only of those figures which are 287described by the concourse of a motion uniform with a motion accelerated? You had no reason therefore to look for 1
3 in that table; for your case is of motion uniform concurring with motion retarded, because you make not the proportions of the spaces to the proportions of the times as two to one, but the contrary; so that your objection ariseth from want of observing what you read. But I “may learn” you say, “these, and greater matters than these, in your twenty-third and sixty-fourth propositions of your Arithmetica Infinitorum.” This, which you say here is a great absurdity; but if you mean I shall find greater there, I will not say against you. This 1
3 you looked for, belongs to the complements of the figures calculated in that table; which because you are not able to find out of yourselves, I will direct you to them. Your case is of 1
3 for the complement of a parabola. Take the denominator of the fraction which belongs to the parabola, namely three, and for numerator take the numerator of the fraction which belongs to the triangle, namely one, and you have the fraction sought. And in like manner for the complement of any other figure. As, for example, of the second parabolaster, whose fraction hath for denominator five, take the numerator of the fraction of the same triangle which is one, and you have 1
3 for the fraction sought for; and so of the rest, taking always one for the numerator.

The twelfth article, which you say is miserably false, I have left standing unaltered. For not comprehending the sense of the proposition, you make a figure of your own, and fight against your 288own fancied motions, different from mine. Other geometricians that understand the construction better, find no fault. And if you had in your own fifth figure drawn a line through N parallel to A E, and upon that line supposed your accelerated motion, you would quickly have seen that in the time A E, the body moved from rest in A, would have fallen short of the diagonal A D; and that all your extravagant pursuing of your own mistake had been absurd.

My thirteenth article you say is ridiculous. But why? “The impetus last acquired cannot” you say, “be equal to a time.” But the quantity of the impetus may be equal to the quantity of a time, seeing they are both measured by line. And when they are measured by the same described line, each of their quantities is equal to that same line, and consequently to one another. But when I meet with this kind of objection again, since I have so often already shown it to be frivolous, and no less to be objected against all the ancients that ever demonstrated any thing by motion, than against me, I purpose to neglect it.

Secondly, you object “that motion uniformly accelerated does no more determine swiftness, than motion uniform.” True; you needed not have used sixteen lines to set down that. But suppose I add, as I do, so as the last acquired impetus be equal to the time. But that, you say, is not sense; which is the objection I am to neglect. But, you say again, supposing it sense, this limitation helps me nothing. Why? Because, you say, a parabola may be described upon a base given, and yet have any altitude, or any diameter 289one will. Who doubts it? But how follows it from thence, that when a parabolical line is described by two motions, one uniform, the other uniformly accelerated from rest, that the determining of the base does not also determine the whole parabola? But fifthly, you say, that this equality of the impetus to the time does not determine the base. Why not? Because, you say, it is an error proceeding from this, that I understand not what is ratio subduplicata. I looked for this. I have shown and inculcated sufficiently before, but the error is on your side; and therefore must tell you, that this objection, and also a great part of the rest of your errors in geometry, proceedeth from this, that you know not what proportion is. But see how wisely you argue about this duplication of proportion. For thus you say verbatim. “Stay a little. What proportion has duplicate proportion to single proportion? Is it always the same? I think not for example, duplicate proportion 4
1 = (2 in 2)/(1 in 1) is double to the single 2
1. Duplicate proportion 9
1 = (3 in 3)/(1 in 1) is triple to its single 3
1.” Let any man, even of them that are most ready in your symbols, say in your behalf (if he be not ashamed) that the proportion of nine to one is triple to the proportion of three to one, as you do.

In the fourteenth, fifteenth, and sixteenth articles, you bid me repeat your objections to the thirteenth. I have done it; and find that what you have objected to the thirteenth, may as well be objected to these; and consequently, that my answer there will also serve me here. Therefore, if you can endure it, read the same answer over again.

290But you have not yet done, you say, with these articles. Therefore (after you had for a while spoken perplextly, conjecturing, not without just cause, that I could not understand you) you say that to the end I may the better perceive your meaning, I should take the example following. “Let a movent (in the first figure of this chapter) be moved uniformly in the time A B, with the continual impetus A C, or B I, whose whole velocity shall therefore be the parallelogram A C I B. And another movent be uniformly accelerated, so as in the time A B it acquire the same impetus B I. Now as the whole velocity, is to the whole velocity, so is the length run over, to the length run over.” All this I acknowledge to be according to my sense, saving that your putting your word movens instead of my word mobile hath corrupted this article. For in the first article, I meddle not with motion by concourse, wherein only I have to do with two movents to make one motion; but in this I do, wherein my word is not movens but mobile; by which it is easy to perceive you understand not this proposition. Then you proceed: “But the length run over by that accelerated motion is greater than the length run over by that uniform motion.” Where do I say that? You answer, “in the ninth and thirteenth article, in making A B (in the fifth figure) greater than A C; and A H (in the eighth figure) greater than A B; and consequently, the triangle A B I, greater than the parallelogram A C I B.” That consequently is without consequence; for it importeth nothing at all in this demonstration, whether A B, or A C in the fifth figure be the greater. 291Besides I speak there of the concourse of two movents, that describe the parabolical line A G D; where the increasing impetus (because it increaseth as the times) will be designed by the ordinate lines in the parabola A G D B. And if both the motions in A B and A C were uniform, the aggregate of the impetus would be designed by the triangle A B D, which is less than the parallelogram A C D B. But you thought that the motion made by A C uniformly, is the same with the motion made uniformly in the same time by the motions in A B and A C concurring; so likewise, in the eighth figure, there is nothing hinders A H from being greater than A B, unless I had said that A B had been described in the time A C with the whole impetus A C maintained entire; of which there is nothing in the proposition, nor would at all have been pertinent to it. Therefore all this new undertaking of the thirteenth, fourteenth, fifteenth, and sixteenth articles, is to as little purpose as your former objections. But I perceive that these new and hard speculations, though they turn the edge of your wit, turn not the edge of your malice.

At the seventeenth article, you show again the same confusion. Return to the eighth figure: “if in a time given a body run over two lengths, one with uniform, the other with accelerated motion”; as for example, if in the same time A C, a body, run over the line A B with uniform motion, and the line A H with motion accelerated; “and again in a part of that time it run over a part of the length A H, with uniform motion, and another part of the same with motion accelerated;” as for 292example, in the time A M it run over with uniform motion the line A I, and with motion accelerated the line A B. I say the excess of the whole A H above the part A B, is to the excess of the whole A B above the part A I, as the whole A H to the whole A B. But first you will say, that these words as the whole A H to the whole A B, are left out in the proposition. But you acknowledge that it was my meaning; and you see it is expressed before I come to the demonstration. And therefore it was absurdly done to reprehend it. Let us therefore pass to the demonstration. Draw I K parallel to A C, and make up the parallelogram A I K M. And supposing first the acceleration to be uniform, divide I K in the midst at N; and between I N, and I K, take a mean proportional I L. And the straight line A L, drawn and produced, shall cut the line B D in F, and the line C G in G (which lines C G, and B D, as also H G and B F, are determined, though you could not carry it so long in memory, by the demonstration of the thirteenth article). For seeing A B is described by motion uniformly accelerated, and A I by motion uniform in the same time A M; and I L is a mean proportional between I N (the half of I K) and I K; therefore by the demonstration of the thirteenth article, A I is a mean proportional between A B and the half of A B, namely A O. Again, because A B is described by uniform motion, and A H by motion uniformly accelerated, both of them in the same time A C, B F is a mean proportional between B D and half B D, namely B E; therefore by the demonstration of the same thirteenth article, the straight line A L F produced 293will fall on G; and the line A H will be to the line A B, as the line A B to the line A I. And consequently as A H to A B, so H B to B I; which was to be demonstrated. And by the like demonstration the same may be proved, where the acceleration is in any other proportion that can be assigned in numbers, saving that whereas this demonstration dependeth on the construction of the thirteenth article, if the motion had been accelerated in double proportion to the times, it would have depended on the fourteenth, where the lines are determined. Which determinations being not repeated, but declared before, in the thirteenth article, to which this diagram belongeth, you take no notice of, but go back to a figure belonging to another article, where there was no use of these determinations. But because I see that the words of the proposition, are as of four motions, and not of two motions made by twice two movents, I must pardon them that have not rightly understood my meaning; and I have now made the proposition according to the demonstration. Which being done, all that you have said in very near two leaves of your Elenchus comes to nothing; and the fault you find comes to no more than a too much trusting to the skill and diligence of the reader. And whereas after you had sufficiently troubled yourself upon this occasion, you add, “that if Sir H. Savile had read my Geometry, he had never given that censure of Joseph Scaliger, in his lecture upon Euclid, that he was the worst geometrician of all mortal men, not exceptioning so much as Orontius, but that praise should have been kept for me.” You see by this time, at least 294others do, how little I ought to value that opinion; and that though I be the least of geometricians, yet my geometry is to yours as 1 to 0. I recite these words of yours, to let the world see your indiscretion in mentioning so needlessly that passage of your founder. It is well known that Joseph Scaliger deserved as well of the state of learning, as any man before or since him; and that though he failed in his ratiocination concerning the quadrature of the circle, yet there appears in that very failing so much knowledge of geometry, that Sir H. Savile could not but see that there were mortal men very many that had less; and consequently he knew that that censure of his in a rigid sense (without the license of an hyperbole) was unjust. But who is there that will approve of such hyperboles to the dishonour of any but of unworthy persons, or think Joseph Scaliger unworthy of honour from learned men? Besides, it was not Sir H. Savile that confuted that false quadrature, but Clavius. What honour was it then for him to triumph in the victory of another? When a beast is slain by a lion, is it not easy for any of the fowls of the air to settle upon, and peck him? Lastly, though it were a great error in Scaliger, yet it was not so great a fault as the least sin; and I believe that a public contumely done to any worthy person after his death, is not the least of sins. Judge therefore whether you have not done indiscreetly, in reviving the only fault, perhaps that any man living can lay to your founder’s charge; and yet this error of Scaliger’s was no greater than one of your own of the like nature, in making the true spiral of Archimedes equal to 295half the circumference of the circle of the first revolution; and then thinking to cover your fault by calling it afterwards an aggregate of arches of circles (which is no spiral at all of any kind) you do not repair but double the absurdity. What would Sir Henry Savile have said to this?

The eighteenth article is this, “in any parallelogram, if the two sides that contain the angle be moved to their opposite sides, the one uniformly, the other uniformly accelerated; the side that is moved uniformly, by its concourse through all its longitude, hath the same effect which it would have if the other motion were also uniform, and the line described were a mean proportional between the whole length, and the half of the same.”

To the proposition you object first, “that it is all one whether the other motion be uniform or not, because the effect of each of their motions, is but to carry the body to the opposite side.” But do you think that whatsoever be the motions, the body shall be carried by their concourse always to the same point of the opposite side? If not, then the effect is not all one when a motion is made by the concourse of two motions uniform and accelerated, and when it is made by the concourse of two uniform or of two accelerated motions.

Secondly, you say that these words, and the line described were a mean proportional between the whole length, and the half of the same, have no sense, or that you are deceived. True. For you are deceived; or rather you have not understanding enough distinctly to conceive variety of motions though distinctly expressed. For when a line is gone over with motion uniformly accelerated, 296you cannot understand how a mean proportional can be taken between it and its half; or if you can, you cannot conceive that that mean can be gone over with uniform motion in the same time that the whole line was run over by motion uniformly accelerated. Yet these are things conceivable, and your want of understanding must be made my fault.

My demonstration is this, in the parallelogram A B C D, (Fig. 11). Let the side A B be conceived to be moved uniformly till it lie in C D; and let the time of that motion be A C, or B D. And in the same time let it be conceived that A C is moved with uniform acceleration, till it lie in B D. To which you object, that then the acceleration last acquired must be far greater than that wherewith A B is moved uniformly: else it shall never come to the place you would have it in the same time. What proof bring you for this? None here. Where then? Nowhere that I remember. On the contrary I have proved (Art. 9 of the chapter) that the line described by the concourse of those two motions, namely, uniform from A B to C D, and uniformly accelerated from A C to B D, is the crooked line of the semiparabola A H D. And though I had not, yet it is well known that the same is demonstrated by Galileo. And seeing it is manifest that in what proportion the motion is accelerated in the line A B, in the same proportion the impetus beginning from rest in A is increased in the same times (which impetus is designed all the way by the ordinate lines of the semiparabola), the greatest impetus acquired must needs be the base of the semiparabola, namely B D, 297equal to A C, which designs the whole time. I cannot therefore imagine what should make you say without proof, that the greatest acquired impetus is greater than that which is designed by the base B D. Next you say, “you see not to what end I divide A B in the middle at E.” No wonder; for you have seen nothing all the way. Others would see it is necessary for the demonstration; as also that the point F is not to be taken arbitrarily; and likewise that the thirteenth article, which you admit not for proof, is sufficiently demonstrated, and your objections to it answered. By the way you advise me, where I say percursam eodem motu uniformi, cum impetu ubique, &c. to blot out cum; because the impetus is not a companion in the way, but the cause. Pardon me in that I cannot take your learned counsel; for the word motu uniformi is the ablative of the cause, and impetu the ablative of the manner. But to come again to your objections, you say, I make “a greater space run over in the same time by the slower motion than by the swifter.” How does that appear? because there is no doubt, but the swiftness is greater where the greatest impetus is always maintained, than where it is attained to in the same time from rest. True, but that is, when they are considered asunder without concourse, but not then when by the concourse they debilitate one another, and describe a third line different from both the lines, which they would describe singly. In this place I compare their effects as contributing to the description of the parabolical line A H D. What the effects of their several motions are, when they are considered asunder, is sufficiently shown before in 298the first article. You should first have gotten into your minds the perfect and distinct ideas of all the motions mentioned in this chapter, and then have ventured upon the censure of them, but not before. And then you would have seen that the body moved from A, describeth not the line A C, nor the line A B, but a third, namely the semiparabolical line A H D.

Again, where I say, Wherefore, if the whole A B be uniformly moved to C D, in the same time wherein A C is moved uniformly to F G; you ask me “whether with the same impetus or not?” How is it possible that in the same time two unequal lengths should be passed over the same impetus? “But why,” say you, “do you not tell us with what impetus A C comes to F G?” What need is there of that, when all men know that in uniform motion and the same time, impetus is to impetus, as length to length? Which to have expressed had not been pertinent to the demonstration. That which follows in the demonstration, rursus suppono quod latus A C, &c. to these words, ut ostensum est, Art. 12, you confute with saying you have proved that article to be false. But you may see now, if you please, at the same place that I have proved your objection to be frivolous.

After this you run on without any argument against the rest of the demonstration, showing nothing all the way, but that the variety and concourse of motions, the speculations whereof you have not been used to, have made you giddy.

To the nineteenth article you apply the same objection which you made to the eighteenth. 299Which having been answered, it appears that from the very beginning of your Elenchus to this place all your objections (except such as are made to three or four mistakes of small importance in setting down my mind), are mere paralogisms, and such are less pardonable than any paralogism in Orontius, both because the subject as less difficult is more easily mastered, and because the same faults are most shamefully committed by a reprehender than by any other man.

I had once added to these nineteen articles a twentieth, which was this: “If from a point in the circumference there be drawn a cord, and a tangent equal to it, the angle which they make shall be double to the aggregate of all the angles made by the cords of all the equal arches into which the arch given can possibly be divided.” Which proposition is true, and I did when I writ it think I might have use of it. But be it, or the demonstration of it true or false, seeing it was not published by me, it is somewhat barbarous to charge me with the faults thereof. No doctor of humanity but would have thought it a poor and wretched malice, publicly to examine and censure papers of geometry never published, by what means soever they came into his hands. I must confess that in these words, in such kind of progression arithmetical (that is, which begins with 0) the sum of all the numbers taken together, is equal to half the number that is made by multiplying the greatest into the least, there is a great error; for by this account these numbers, 0, 1, 2, 3, 4, taken together, should be equal to nothing. I should have said they are equal to that number which is made 300by multiplying half the greatest into the number of the terms. There was therefore, if those words were mine (for truly I have no copy of them, nor have had since the book was printed, and I have no great reason, as any man may see, to trust your faith) a great error in the writing, but not an erroneous opinion in the writer. The demonstration so corrected is true. And the angles that have the proportions of the numbers 1, 2, 3, 4, are in the table of your Elenchus , fig. 12, the angles G A D, H D E, I E F, K F B. And if the divisions were infinite, so that the first were not to be reckoned but as a cypher, the angle C A B would be double to them altogether. This mistake of mine, and the finding that I had made no use of it in the whole book, was the cause why I thought fit to leave it quite out. But your professorships, could not forbear to take occasion thereby, to commend your zeal against Leviathan to your doctorships of divinity, by censuring it.

At the seventeenthseventeenth chapter, your first exception is to the definition of proportional proportions, which is this: “Four proportions are then proportional, when the first is to the second, as the third to the 301fourth.” The reader will hardly believe that your exception is in earnest. You say, I mean not by proportionality the “quantity of the proportions.” Yes I do. Therefore I say again, that four proportions are then proportional, when the quantity of the first proportion, is to the quantity of the second proportion, as the quantity of the third proportion, to the quantity of the fourth proportion. Is not my meaning now plainly enough expressed? Or is it not the same definition with the former. But what do I mean, you will say, by the quantity of a proportion? I mean the determined greatness of it, that is, for example, in these numbers, the quantity of the proportion of two to three, is the same with the quantity of the proportion of four to six, or six to nine; and again, the quantity of the proportion of six to four, is the same with the quantity of the proportion of nine to six, or of three to two. But now what do you mean by the quantity of a proportion? You mean that two and three, are the quantities of the proportion of two to three (for so Euclid calls them) and that six and four are the quantities of the proportion of six to four, which is the same with the proportion of three to two. And by this rule, one and the same proportion shall have an infinite number of quantities; and consequently the quantity of a proportion can never be determined. I call one proportion double to another, when one is equal to twice the other; as the proportion of four to one, is double to the proportion of two to one. You call that proportion double where one number, line, or quantity absolute, is double to the other; so that with you the proportion 302of two to one is a double proportion. It is easy to understand how the number two is double to one, but to what, I pray you, is double the proportion of two to one, or of one to two? Is not every double proportion double to some proportion? See whether this geometry of yours can be taken by any man of sound mind for sense. “But it is known,” you say, “that in proportions, double is one thing, and duplicate another;” so that it seems to you, that in talking of proportion men are allowed to speak senselessly. “It is known,” you say. To whom? It is indeed in use at this day to call double duplicate, and triple triplicate. And it is well enough; for they are words that signify the same thing, but that they differ (in what subject soever) I never heard till now. I am sure that Euclid, whom you have undertaken to expound, maketh no such difference. And even there where he putteth these numbers, one, two, four, eight, &c. for numbers in double proportion (which is the last proposition of the ninth element) he meaneth not that one to two, or two to one, is a double proportion, but that every number in that progression is double to the number next before it; and yet he does not call it analogia dupla, but duplicate. This distinction in proportions between double and duplicate, proceeded long after from want of knowledge that the proportion of one to two is double to the proportion of one to four; and this from ignorance of the different nature of proportions of excess, and proportions of defect. And you that have nothing but by tradition saw not the absurdities that did hang thereon.

In the second article I make E K, (fig. 1) the 303third part of L K, which you say is false; and consequently the proposition undemonstrated. And thus you prove it false: “Let A C be to G C, or G K to G L, as eight to one (for seeing the point G is taken arbitrarily, we may place it where we will, &c.)” and upon this placing of G arbitrarily, you prove well enough that E K is not a third part of L K. But you did not then observe, that I make the altitude A G, less than any quantity given, and by consequence E K to differ from a third part by a less difference than any quantity that can be given. Therefore as yet the demonstration proceedeth well enough. But perceiving your oversight, you thought fit (though before, you thought this confutation sufficient) to endeavour to confute it another way; but with much more evidence of ignorance. For when I come to say, the proportion therefore between A C and G C is triple, in arithmetical proportion, to the proportion between G K and G E, &c. you say, “the proportion of A C to G C is the proportion of identity, as also that of G K to G E.”G E.” But why? Does my construction make it so? Do not I make G C less than A C, though with less difference than any quantity that can be assigned? And then where I say, therefore E K is the third part of L K, you come in, by parenthesis, with (or a fourth, or a fifth, &c.). Upon what ground? Because you think it will pass for current, without proof, that a point is nothing. Which if it do, geometry also shall pass for nothing, as having no ground nor beginning but in nothing. But I have already in a former lesson sufficiently showed you the consequence of that opinion. To which I may 304add, that it destroys the method of indivisibles, invented by Bonaventura; and upon which, not well understood, you have grounded all your scurvy book of Arithmetica Infinitorum; where your indivisibles have nothing to do, but as they are supposed to have quantity, that is to say, to be divisibles. You allow, it seems, your own nothings to be somethings, and yet will not allow my somethings to be considered as nothing. The rest of your objections having no other ground than this, “that a point is nothing,” my whole demonstration standeth firm; and so do the demonstrations of all such geometricians, ancient and modern, as have inferred any thing in the manner following, viz. If it be not greater nor less, then it is equal. But it is neither greater nor less. Therefore, &c. If it be greater, say by how much. By so much. It is not greater by so much. Therefore it is not greater. If it be less, say how much, &c. Which being good demonstrations are together with mine overthrown by the nothingness of your point, or rather of your understanding; upon which you nevertheless have the vanity of advising me what to do, if I demonstrate the same again; meaning I should come to your false, impossible, and absurd method of Arithmetica Infinitorum, worthy to be gilded, I do not mean with gold.

And for your question, why I set the base of my figure upwards, you may be sure it was not because I was afraid to say, that the proportions of the ordinate lines beginning at the vertex were triplicate, or otherwise multiplicate of the proportions of the intercepted parts of the diameter. For I never doubted to call double duplicate, nor 305triple triplicate, &c., or if I had, I should have avoided it afterwards at the tenth article of the same chapter. But because when I went about to compare the proportions of the ordinate lines with those of their contiguous diameters, the first thing I considered in them was in what manner the base grew less and less till it vanished into a point. And though the base had been placed below, it had not therefore required any change in the demonstration. But I was the more apt to place the base uppermost, because the motion began at the base, and ended at the vertex. To proceed which way I pleased was in my own choice; and it is of grace that I give you any account of it at all.

To the third article, together with its table, you say, “it falls in the ruin of the second; and that the same is to be understood of the sixth, seventh, eighth, and ninth.” For confutation whereof I need to say no more, but that they all stand good by the confutation of your objections to the second.

To the fourth article you say, “the description of those curvilineal figures is easy.” True, to some men; and now that I have showed you the way, it is easy enough for you also. For the way you propound is wholly transcribed out of the figure of the second article, which article you had before rejected. For seeing the lines H F, G E, A B, &c. are equal to the lines C Q, C O, C D; and the lines Q F, O E, B D, equal to the lines C H, C G, C A; the proportion of D B to O E, will be triple (that is, triplicate) to the proportion of C O to G E; and the proportion of D B to Q F, triple to the proportion of C D to C Q; and consequently, 306because the complement B D C F E B is made by the decrease of A C in triple proportion to that of the decrease of C D, it will be (by the second article) a third part of the figure A B E F C A. So that it comes all to one pass, whether we take triple proportion in decreasing to make the complement, or triple proportion in increasing to make the figure; for the proportion of H F to B A, is triple to the proportion of C H to C A. Wherefore you have done no more but what you have seen first done, saving that from your construction you prove not the figure to be triple to the complement; perhaps because you have proved the contrary in your Arithmetica Infinitorum. But your way differs from mine, in that you call the proportion subtriplicate, which I call triplicate; as if the divers naming of the same thing made it differ from itself. You might as well have said briefly, the proposition is true, but ill proved, because I call the proportion of one or two triple, or triplicate of that of one to eight; which you say is false, and hath infected the fourth, fifth, ninth, tenth, eleventh, thirteenth, fourteenth, fifteenth, sixteenth, seventeenth, and nineteenth articles of the sixteenth chapter. But I say, and you know now, that it is true; and that all those articles are demonstrated.

Lastly you add, “Tu vero, in presente articulo, &c. id est, you bid find as many mean proportionals as one will, between two given lines; as if that could not be done by the geometry of planes, &c.” You might have left out Tu vero to seek an Ego quidem. But tell me, do you think that you can find two mean proportionals (which is less than 307as many as one will) by the geometry of planes? We shall see anon how you go about it. I never said it was impossible, and if you look upon the places cited by you more attentively, you will find yourself mistaken. But I say, the way to do it has not been yet found out, and therefore it may prove a solid problem for anything you know.

The fifth article you reject, because it citeth the corollary of the twenty-eighth article of the thirteenth chapter, where there is never a word to that purpose. But there is in the twenty-sixth article; which was my own fault, though you knew not but it might have been the printer’s.

To the tenth you object for almost three leaves together, against these words of mine, because, in the sixth figure, B C is to B F in triplicate proportion of C D to F E, therefore inverting, F E is to C D in triplicate proportion of B F to B C. This you objected then. But now that I have taught you so much geometry, as to know that of three quantities, beginning at the least, if the third be to the first in triplicate proportion of the second to the first, also by conversion the first to the second shall be in triplicate proportion of the first to the third; if it were to do again, you would not object it.

My eleventh article you would allow for demonstrated, if my second had been demonstrated, upon which it dependeth. Therefore seeing your objections to that article are sufficiently answered, this article also is to be allowed.

The twelfth also is allowed upon the same reason. What falsities you shall find in such following propositions as depend upon the same second 308article, we shall then see when I come to the places where you object against them.

To the thirteenth article you object, “that the same demonstration may be as well applied to a portion of any conoeides, parabolical, hyperbolical, elliptical, or any other, as to the portion of a sphere.” By the truth of this let any man judge of your and my geometry. Your comparison of the sphere and conoeides, so far holds good, as to prove that the superficies of the conoeides is greater than the superficies of the cone described by the subtense of the parabolical, hyperbolical, or elliptical line. But when I come to say, that the cause of the excess of the superficies of the portion of the sphere above the superficies of the cone, consists in the angle D A B, and the cause of the excess of the circle made upon the tangent A D, above the superficies of the same cone, consists in the magnitude of the same angle D A B, how will you apply this to your conoeides? For suppose that the crooked line A B (in the seventh figure) were not an arch of a circle, do you think that the angles which it maketh with the subtense A B, at the points A and B, must needs be equal? Or if they be not, does the excess of the superficies of the circle upon A D above the superficies of the cone, or the excess of the superficies of the portion of the conoeides above the superficies of the same cone, consist in the angle D A B, or rather in the magnitude of the two unequal angles D A B, and A B A? You should have drawn some other crooked line, and made tangents to it through A and B, and you would presently have seen your error. See how you can answer this; for if this demonstration of 309mine stand firm, I may be bold to say, though the same be well demonstrated by Archimedes, that this way of mine is more natural, as proceeding immediately from the natural efficient causes of the effect contained in the conclusion; and besides, more brief and more easy to be followed by the fancy of the reader.

To the fourteenth article you say that I “commit a circle in that I require in the fourth article the finding of two mean proportionals, and come not till now to show how it is to be done.” Nor now neither. But in the mean time you commit two mistakes in saying so. The place cited by you in the fourth article is, in the Latin, p. 215, line 26, in the English, p. 255, line 24. Let any reader judge whether that be a requiring it, or a supposing it to be done; this is your first mistake. The second is, that in this place the proportion itself, which is, “If these deficient figures could be described in a parallelogram exquisitely, there might be found thereby between any two lines given, as many mean proportionals as one would,” is a theorem, upon supposition of these crooked lines exquisitely drawn; but you take it for a problem.

And proceeding in that error, you undertake the invention of two mean proportionals, using therein my first figure, which is of the same construction with the eighth that belongeth to this fourteenth article. Your construction is, “Let there be taken in the diameter C A, (fig. 1) the two given lines, or two others proportional to them, as C H, C G, and their ordinate lines H F, G E (which by construction are in subtriplicate proportion of the intercepted diameters). These 310lines will show the proportions which those four proportionals are to have.” But how will you find the length of H F or G E, the ordinate lines? Will you not do it by so drawing the crooked line C F E, as it may pass through both the points F and E? You may make it pass through one of them, but to make it pass through the other, you must find two mean proportionals between G K and G L, or between H I and H P; which you cannot do, unless the crooked line be exactly drawn; which it cannot be by the geometry of planes. Go shew this demonstration of yours to Orontius, and see what he will say to it.

I am now come to an end of your objections to the seventeenth chapter, where you have an epiphonema not to be passed over in silence. But because you pretend to the demonstration of some of these propositions by another method in your Arithmetica Infinitorum, I shall first try whether you be able to defend those demonstrations as well as I have done these of mine by the method of motion.

The first proposition of your Arithmetica Infinitorum is this lemma: “In a series, or row of quantities, arithmetically proportional, beginning at a point or cypher, as 0, 1, 2, 3, 4, &c. to find the proportion of the aggregate of them all, to the aggregate of so many times the greatest, as there are terms.” This is to be done by multiplying the greatest into half the number of the terms.

The demonstration is easy. But how do you demonstrate the same? “The most simple way,” say you, “of finding this and some other problems, is to do the thing itself a little way, and to 311observe and compare the appearing proportions, and then by induction to conclude it universally.” Egregious logicians and geometricians, that think an induction, without a numeration of all the particulars sufficient, to infer a conclusion universal, and fit to be received for a geometrical demonstration! But why do you limit it to the natural consecution of the numbers, 0, 1, 2, 3, 4, &c? Is it not also true in these numbers, 0, 2, 4, 6, &c. or in these, 0, 7, 14, 21, &c? Or in any numbers where the difference of nothing and the first number is equal to the difference between the first and second, and between the second and third, &c.? Again, are not these quantities, 1, 3, 5, 7, &c. in continual proportion arithmetical? And if you put before them a cypher thus, 0, 1, 3, 5, 7, do you think that the sum of them is equal to the half of five times seven? Therefore though your lemma be true, and by me (Chap. XIII. art. 5) demonstrated; yet you did not know why it is true; which also appears most evidently in the first proposition of your Conic Sections , where first you have this, “that a parallelogram whose altitude is infinitely little, that is to say, none, is scarce anything else but a line.” Is this the language of geometry? How do you determine this word scarce? The least altitude, is somewhat or nothing. If somewhat, then the first character of your arithmetical progression must not be a cypher; and consequently the first eighteen propositions of this your Arithmetica Infinitorum are all nought. If nothing, then your whole figure is without altitude, and consequently your understanding nought. Again, in the same proposition, you say thus: 312“We will sometimes call those parallelograms rather by the name of lines than of parallelograms, at least when there is no consideration of a determinate altitude; but where there is a consideration of a determinate altitude (which will happen sometimes) there that little altitude shall be so far considered, as that being infinitely multiplied it may be equal to the altitude of the whole figure.” See here in what a confusion you are when you resist the truth. When you consider no determinate altitude, that is no quantity of altitude, then you say your parallelogram shall be called a line. But when the altitude is determined, that is, when it is quantity, then you will call it a parallelogram. Is not this the very same doctrine which you so much wonder at and reprehend in me, in your objections to my eighth chapter, and your word considered used as I used it? It is very ugly in one that so bitterly reprehendeth a doctrine in another, to be driven upon the same himself by the force of truth when he thinks not on it. Again, seeing you admit in any case those infinitely little altitudes to be quantity, what need you this limitation of yours, “so far forth as that by multiplication they may be made equal to the altitude of the whole figure?” May not the half, the third, the fourth, or the fifth part, &c. be made equal to the whole by multiplication? Why could you not have said plainly, so far forth as that every one of those infinitely little altitudes be not only something but an aliquot part of the whole? So you will have an infinitely little altitude, that is to say, a point to be both nothing and something and an aliquot part. And all this proceeds from 313not understanding the ground of your profession. Well, the lemma is true. Let us see the theorems you draw from it. The first is (p. 3) “that a triangle to a parallelogram of equal base and altitude is as one to two.” The conclusion is true, but how know you that? “Because,” say you, “the triangle consists as it were [as it were, is no phrase of a geometrician] of an infinite number of straight parallel lines.” Does it so? Then by your own doctrine, which is, that “lines have no breadth,” the altitude of your triangle consisteth of an infinite number of no altitudes, that is of an infinite number of nothings, and consequently the area of your triangle has no quantity. If you say that by the parallels you mean infinitely little parallelograms, you are never the better; for if infinitely little, either they are nothing, or if somewhat, yet seeing that no two sides of a triangle are parallel, those parallels cannot be parallelograms. I see they may be counted for parallelograms by not considering the quantity of their altitudes in the demonstration. But you are barred of that plea, by your spiteful arguing against it in your Elenchus . Therefore this third proposition, and with it the fourth, is undemonstrated.

Your fifth proposition is, “the spiral line is equal to half the circle of the first revolution.” But what spiral line? We shall understand that by your construction, which is this: “The straight line M A [in your figure which I have placed at the end of the fifth lesson] turned round (the point M remaining unmoved) is supposed to describe with its point A the circle A O A, whilst some point, in the same M A, whilst it goes about, 314is supposed to be moved uniformly from M to A, describing the spiral line.” This therefore, is the spiral line of Archimedes; and your proposition affirms it to be equal to the half of the circle A O A; which you perceived not long after to be false. But thinking it had been true, you go about to prove it, “by inscribing in the circle an infinite multitude of equal angles, and consequently an infinite number of sectors, whose arches will therefore be in arithmetical proportion;” which is true. “And the aggregate of those arches equal to half the circumference A O A;” which is true also. And thence you conclude “that the spiral line is equal to half the circumference of the circle A O A;” which is false. For the aggregate of that infinite number of infinitely little arches, is not the spiral line made by your construction, seeing by your construction the line you make is manifestly the spiral of Archimedes; whereas no number, though infinite, of arches of circles, how little soever, is any kind of spiral at all; and though you call it a spiral, that is but a patch to cover your fault, and deceiveth no man but yourself. Besides, you saw not how absurd it was, for you that hold a point to be absolutely nothing, to make an infinite number of equal angles (the radius increasing as the number of angles increaseth) and then to say, “that the arches of the sectors whose angles they are, are as 0, 1, 2, 3, 4, &c.” For you make the first angle 0, and all the rest equal to it; and so make 0, 0, 0, 0, 0, &c. to be the same progression with 0, 1, 2, 3, 4, &c. The influence of this absurdity reacheth to the end of the eighteenth proposition. So many are therefore false, 315or nothing worth. And you needed not to wonder that the doctrine contained in them was omitted by Archimedes, who never was so senseless as to think a spiral line was compounded of arches of circles.

Your nineteenth proposition is this other lemma: “In a series, or a row, of quantities, beginning from a point, or cypher, and proceeding according to the order of the square numbers, as 0, 1, 4, 9, 16, &c. to find what proportion the whole series hath to so many times the greatest.” And you conclude “the proportions to be that of 1 to 3.” Which is false, as you shall presently see. First, let the series of squares with the prefixed cypher, and under every one of them the greatest 4 be 0 . 1 . 4
4 . 4 . 4. And you have for the sum of the squares 5, and for thrice the greatest 12, the third part whereof is 4. But 5 is greater than 4, by 1, that is, by one twelfth of 12; which quantity is somewhat, let it be called A. Again, let the row of squares be lengthened one term further, and the greatepm divst set under every one of them as 0 . 1 . 4 . 9
9 . 9 . 9 . 9. The sum of the squares is 14, and the sum of four times the greatest is 36, whereof the third part is 12. But 14 is greater than 12 by two unities, that is, by two twelfths of 12, that is, by 2 A. The difference therefore between the sum of the squares, and the sum of so many times the greatest square, is greater, when the cypher is followed by three squares, than when by but two. Again, let the row have five terms, as in these numbers 0 .   1 .   4 .   9 . 16
16 . 16 . 16 . 16 . 16 with the greatest five times described, and the sum of the squares will be 30, the sum of all the greatest will be 80. The third part whereof is 262
3. 316But 30 is greater than 262
3 by 31
3, that is, by three twelfths of twelve, and 1
3 of a twelfth, that is, by 31
3 A. Likewise in the series continued to six places with the greatest six times subscribed, as   0 .   1 .   4 .   9 . 16 . 25
25 . 25 . 25 . 25 . 25 . 25 the sum of the squares is 55, and the sum of the greatest six times taken is 150, the third part whereof is 50. But 55 is greater than 50 by 5, that is, by five-twelfths of 12, that is by 5 A. And so continually as the row groweth longer, the excess also of the aggregate of the squares above the third part of the aggregate of so many times the greatest square, growing greater. And consequently if the number of the squares were infinite, their sum would be so far from being equal to the third part of the aggregate of the greatest as often taken, as that it would be greater than it by a quantity greater than any that can be given or named.

That which deceived you was partly this, that you think, as you do in your Elenchus , that these fractions 1
12 1
18 1
24 1
30 1
36 &c. are proportions, as if 1
12 were the proportion of one to twelve, and consequently 2
12 double the proportion of one to twelve; which is as unintelligible as school-divinity; and I assure you, far from the meaning of Mr. Ougthred in the sixth chapter of his Clavis Mathematica, where he says that 43
7 is the proportion of 31 to 7; for his meaning is, that the proportion of 43
7 to one, is the proportion of 31 to 7; whereas if he meant as you do, then 86
7 should be double the proportion of 31 to 7. Partly also because you think (as in the end of the twentieth proposition) that if the proportion of the numerators of 317these fractions 1
12 1
18 1
24 1
30 1
36 to their denominators decrease eternally, they shall so vanish at last as to leave the proportion of the sum of all the squares to the sum of the greatest so often taken, (that is, an infinite number of times), as one to three, or the sum of the greatest to the sum of the increasing squares, as three to one; for which there is no more reason than for four to one, or five to one, or any other such proportion. For if the proportions come eternally nearer and nearer to the subtriple, they must needs also come nearer and nearer to subquadruple; and you may as well conclude thence that the upper quantities shall be to the lower quantities as one to four, or as one to five, &c. as conclude they are as one to three. You can see without admonition, what effect this false ground of yours will produce in the whole structure of your Arithmetica Infinitorum; and how it makes all that you have said unto the end of your thirty-eighth proposition, undemonstrated, and much of it false.

The thirty-ninth is this other lemma: “In a series of quantities beginning with a point or cypher, and proceeding according to the series of the cubic numbers, as O. 1. 8. 27. 64, &c. to find the proportion of the sum of the cubes to the sum of the greatest cube, so many times taken as there be terms.” And you conclude that “they have a proportion of 1 to 4;” which is false.

Let the first series be of three terms subscribed with the greatest (0. 1. 8.)/(8. 8. 8.); the sum of the cubes is nine; the sum of all the greatest is 24; a quarter whereof is 6. But 9 is greater than 6 by three unities. An unity is something. Let it be therefore 318A. Therefore the row of cubes is greater than a quarter of three times eight, by three A. Again, let the series have four terms, as (0. 1. 8. 27)/(27. 27. 27. 27); the sum of the cubes is 36; a quarter of the sum of all the greatest is twenty-seven. But thirty-six is greater than twenty-seven by nine, that is, by 9 A. The excess therefore of the sum of the cubes above the fourth part of the sum of all the greatest, is increased by the increase of the number of terms. Again, let the terms be five, as (0. 1. 8. 27. 64)/(64. 64. 64. 64. 64), the sum of the cubes is one hundred; the sum of all the greatest three hundred and twenty; a quarter whereof is eighty. But one hundred is greater than eighty by twenty, that is, by 20 A. So you see that this lemma also is false. And yet there is grounded upon it all that which you have of comparing parabolas and paraboloeides with the parallelograms wherein they are accommodated. And therefore though it be true, that the parabola is 2
3 and the cubical paraboloeides 3
4 of their parallelograms respectively, yet it is more than you were certain of when you referred me, for the learning of geometry, to this book of yours. Besides, any man may perceive that without these two lemmas (which are mingled with all your compounded series with their excesses) there is nothing demonstrated to the end of your book: which to prosecute particularly, were but a vain expense of time. Truly, were it not that I must defend my reputation, I should not have showed the world how little there is of sound doctrine in any of your books. For when I think how dejected you will be for the future, and how the grief of so much time irrecoverably lost, together 319with the conscience of taking so great a stipend, for mis-teaching the young men of the University, and the consideration of how much your friends will be ashamed of you, will accompany you for the rest of your life, I have more compassion for you than you have deserved. Your treatise of the Angle of Contact , I have before confuted in a very few leaves. And for that of your Conic Sections , it is so covered over with the scab of symbols, that I had not the patience to examine whether it be well or ill demonstrated.

Yet I observed thus much, that you find a tangent to a point given in the section by a diameter given; and in the next chapter after, you teach the finding of a diameter, which is not artificially done.

I observe also, that you call the parameter an imaginary line, as if the place thereof were less determined than the diameter itself; and then you take a mean proportional between the intercepted diameter, and its contiguous ordinate line, to find it. And it is true, you find it: but the parameter has a determined quantity, to be found without taking a mean proportional. For the diameter and half the section being given, draw a tangent through the vertex, and dividing the angle in the midst which is made by the diameter and tangent, the line that so divideth the angle, will cut the crooked line. From the intersection draw a line (if it be a parabola) parallel to the diameter, and that line shall cut off in the tangent from the vertex the parameter sought. But if the section be an ellipsis, or an hyperbole, you may use the same method, saving that the line drawn from the intersection 320must not be parallel, but must pass through the end of the transverse diameter, and then also it shall cut off a part of the tangent, which measured from the vertex is the parameter. So that there is no more reason to call the parameter an imaginary line than the diameter.

Lastly, I observe that in all this your new method of conics, you show not how to find the burning points, which writers call the foci and umbilici of the section, which are of all other things belonging to the conics most useful in philosophy. Why therefore were they not as worthy of your pains as the rest, for the rest also have already been demonstrated by others? You know the focus of the parabola is in the axis distant from the vertex a quarter of the parameter. Know also that the focus of an hyperbole, is in the axis, distant from the vertex, as much as the hypotenusal of a rectangled triangle, whose one side is half the transverse axis, the other side half the mean proportional between the whole transverse axis and the parameter, is greater than half the transverse axis.

The cause why you have performed nothing in any of your books (saving that in your Elenchus you have spied a few negligences of mine, which I need not be ashamed of) is this, that you understood not what is quantity, line, superficies, angle, and proportion; without which you cannot have the science of any one proposition in geometry. From this one and first definition of Euclid, “a point is that whereof there is no part,” understood by Sextus Empiricus, as you understand it, that is to say misunderstood, Sextus Empiricus had utterly destroyed most of the rest, and demonstrated, 321that in geometry there is no science, and by that means you have betrayed the most evident of the sciences to the sceptics. But as I understand it for that whereof no part is reckoned, his arguments have no force at all, and geometry is redeemed. If a line have no latitude, how shall a cylinder rolling on a plane, which it toucheth not but in a line, describe a superficies? How can you affirm that any of those things can be without quantity, whereof the one may be greater or less than the other? But in the common contact of divers circles the external circle maketh with the common tangent a less angle of contact than the internal. Why then is it not quantity? An angle is made by the concourse of two lines from several regions, concurring, by their generation, in one and the same point. How then can you say the angle of contact is no angle? One measure cannot be applicable at once to the angle of contact, and angle of conversion. How then can you infer, if they be both angles, that they must be homogeneous? Proportion is the relation of two quantities. How then can a quotient or fraction, which is quantity absolute, be a proportion? But to come at last to your Epiphonema , wherein, though I have perfectly demonstrated all those propositions concerning the proportion of parabolasters to their parallelograms, and you have demonstrated none of them (as you cannot now but plainly see), but committed most gross paralogisms, how could you be so transported with pride, as insolently to compare the setting of them forth as mine, to the act of him that steals a horse, and comes to the gallows for it. You have read, I 322think, of the gallows set up by Haman. Remember therefore also who was hanged upon it.

After your dejection I shall comfort you a little, a very little, with this, that whereas this eighteenth chapter containeth two problems, one, “the finding of a straight line equal to the crooked line of a semi-parabola;” the other, “the finding of straight lines equal to the crooked lines of the parabolasters, in the table of the third article of the seventeenth chapter;” you have truly demonstrated that they are both false; and another hath also demonstrated the same another way. Nevertheless, the fault was not in my method, but in a mistake of one line for another and such as was not hard to correct; and is now so corrected in the English as you shall not be able (if you can sufficiently imagine motions) to reprehend. The fault was this, that in the triangles which have the same base and altitude with the parabola and parabolaster, I take for designation of the mean uniform impetus, a mean proportional, in the first figure, between the whole diameter and its half, and, in the second figure, a mean proportional between the whole diameter and its third part; which was manifestly false, and contrary to what I had shown in the sixteenth chapter. Whereas I ought to have taken the half of the base, as now I have done, and thereby exhibited the straight lines equal to those crooked lines, as I undertook to do. Which error therefore proceeded not from want of skill, but from want of care; and what I promised (as bold as you say the promise was), I have now performed.

The rest of your exceptions to this chapter, are to these words in the end: “There be some that 323say, that though there be equality between a straight and crooked line, yet now, they say, after the fall of Adam, it cannot be found without the especial help of divine grace.” And you say you think there be none that say so. I am not bound to tell you who they are. Nevertheless, that other men may see the spirit of an ambitious part of the clergy, I will tell you where I read it. It is in the Prolegomena of Lalovera, a Jesuit, to his Quadrature of the Circle, p. 13 and 14, in these words: “Quamvis circuli tetragonismus sit φύσει possibilis, an tamen etiam πρός ἡμᾶς, hoc est, post Adæ lapsum homo ejus scientiam absque speciali divinæ gratiæ auxilio, possit comparare, jure merito inquirunt theologi, pronunciantque; hanc veritatem tanta esse caligine involutam ut illam videre nemo possit, nisi ignorantiæ ex primi parentis prævaricatione propagatas tenebras indebitus divinæ lucis radius dissipet; quod verissimum esse sentio.” Wherein I observed that he, supposing he had found that quadrature, would have us believe it was not by the ordinary and natural help of God (whereby one man reasoneth, judgeth and remembereth better than another), but by a special (which must be a supernatural) help of God, that he hath given to him of the order of Jesus above others that have attempted the same in vain. Insinuating thereby, as handsomely as he could, a special love of God towards the Jesuits. But you taking no notice of the word special, would have men think I held, that human sciences might be acquired without any help of God. And thereupon proceed in a great deal of ill language to the end of your objections to this chapter. But 324I shall take notice of your manners for altogether in my next lesson.

At the nineteenth chapter you see not, you say, the method. Like enough. In this chapter I consider not the cause of reflection, which consisteth in the resistance of bodies natural; but I consider the consequences, arising from the supposition of the equality of the angle of reflection, to that of incidence; leaving the causes both of reflection, and of refraction, to be handled together in the twenty-fourth chapter. Which method, think what you will, I still think best.

Secondly, you say I define not, here, but many chapters after, what an angle of incidence, and what an angle of reflection is. Had you not been more hasty than diligent readers, you had found that those definitions of the angle of incidence, and of reflection, were here set down in the first article, and not deferred to the twenty-fourth. Let not therefore your own oversight be any more brought in for an objection.

Thirdly, you say there is no great difficulty in the business of this chapter. It may be so, now it is down; but before it was done, I doubt not but you that are a professor would have done the same, as well as you have done that of the Angle of Contact , or the business of your Arithmetica Infinitorum . But what a novice in geometry would have done I cannot tell.

To the third, fourth, and fifth article, you object a want of determination; and show it by instance, as to the third article. But what those determinations should be, you determine not, because you could not. The words in the third 325article, are first these, if there fall two straight lines parallel, &c. which is too general. It should be, if there fall the same way two straight lines parallel, &c. Next these, their reflected lines produced inwards shall make an angle, &c. This also is too general. I should have said, their reflected lines produced inwards, if they meet within, shall make an angle, &c. Which done, both this article and the fourth and fifth are fully demonstrated. And without it, an intelligent reader had been satisfied, supplying the want himself by the construction.

To the eighth, you object only the too great length and labour of it, because you can do it a shorter way. Perhaps so now, as being easy to shorten many of the demonstrations both of Euclid, and other the best geometricians that are or have been. And this is all you had to say to my nineteenth chapter. Before I proceed, I must put you in mind that these words of yours, “adducis malleum, ut occidas muscam,” are not good Latin, malleum affers, malleum adhibes, malleo uteris, are good. When you speak of bringing bodies animate, ducere and adducere are good, for there to bring, is to guide or lead. And of bodies inanimate, adducere is good for attrahere, which is to draw to. But when you bring a hammer, will you say adduco malleum, I lead a hammer? A man may lead another man, and a ninny may be said to lead another ninny, but not a hammer. Nevertheless, I should not have thought fit to reprehend this fault upon this occasion in an Englishman, nor to take notice of it, but that I find you in some places nibbling, but causelessly, at my Latin.

326Concerning the twentieth chapter, before I answer to the objections against the propositions themselves, I must answer to the exception you first take to these words of mine, “Quæ de dimensione circuli et angulorum pronuntiata sunt tanquam exactè inventa, accipiat lector tanquam dicta problematicè.”problematicè.” To which you say thus: “We are wont in geometry to call some propositions theorems, others problems, &c. of which a theorem is that wherein some assertion is propounded to be proved; a problem that wherein something is commanded to be done.” Do you mean to be done, and not proved? By your favour, a problem in all ancient writers signifies no more but a proposition uttered, to the end to have it, by them to whom it is uttered, examined whether it be true or not true, faisable or not faisable; and differs not amongst geometricians from a theorem but in the manner of propounding. For this proposition, to make an equilateral triangle, so propounded they call a problem. But if propounded thus: If upon the ends of a straight line given be described two circles, whose radius is the same straight line, and there be drawn from the intersection of the circles to their two centres, two straight lines, there will be made an equilateral triangle, then they call it a theorem; and yet the proposition is the same. Therefore these words, accipiat lector tanquam dicta problematicè signify plainly this, that I would have the reader, take for propounded to him to examine, whether from my construction the quadrature of the circle can be truly inferred or not; and this is not to bid him, as you interpret it, to square the circle. And if 327you believe that problematicè signifies probably, you have been very negligent in observing the sense of the ancient Greek philosophers in the word problem. Therefore your solemus in geometria, &c. is nothing to the purpose; nor had it been though you had spoken more properly, and said solent, leaving out yourselves.

Six Lessons.
Vol. VII. Eng. p.310_II. 325

My first article hath this title, “from a false supposition, a false quadrature of the circle.” Seeing therefore you were resolved to show where I erred, you should have proved either that the supposition was true, and the conclusion falsely inferred, or contrarily, that though the supposition be false, yet the conclusion is true; for else you object nothing to my geometry, but only to my judgment, in thinking fit to publish it; which nevertheless you cannot justly do, seeing it was likely to give occasion to ingenious men (the practice of it being so accurate to sense) to inquire wherein the fallacy did consist. And for the problem as it was first printed, but never published, and consequently ought to have passed for a private paper stolen out of my study, your public objecting against it (in the opinion of all men that have conversed so much with honest company as to know what belongs to civil conversation), was sufficiently barbarous in divines. And seeing you knew I had rejected that proposition, it was but a poor ambition to take wing as you thought to do, like beetles from my egestions. But let that be as it will, you will think strange now I should resume, and make good, at least against your objection, that very same proposition. So much of the figure as is needful you will find noted with the 328same letters, and placed at the end of this fifth lesson. Wherein let B I, be an arch not greater than the radius of the circle, and divided into four equal parts, in L, N, O. Draw S N, the sine of the arch B N, and produce it to T, so as S T be double to S N, that is, equal to the chord B I. Draw likewise a L, the sine of the arch B L, and produce it to c, so as a c be quadruple to a L, that is, equal to the two chords B N, N I. Upon the centre N with the radius N I, draw the arch I d, cutting B U the tangent in d. Then will B N produced cut the arch I d, in the midst at o. In the line B S produced take S b, equal to B S; then draw and produce b N, and it will fall on the point d. And B d, S T, will be equal; and d T joined and produced will fall upon o, the midst of the arch I d. Join I T, and produce it to the tangent B U in U. I say, that the straight line I T U shall pass through c. For seeing B S, S b, are equal, and the angle at S a right angle, the straight lines B N, and b N, are also equal, and the triangles B N b, d N o like and equal; and the lines d T, T o equal. Draw o i parallel to d U, cutting I U in i; and the triangles d T U, o T i will also be like and equal. Produce S T to the arch d o I in e, and produce it further to f, so that the line e f be equal to T e; and then S f will be equal to a c. Therefore f c joined will be parallel to B S. In c f produced take f g equal to c f; and draw g m parallel to d U, cutting I U in m, and d o in n; and let the intersection of the two lines a c and d o be in r; which being done, the triangles m n T, r c T will be like and equal. Therefore m n and r c are equal; and consequently 329the straight line I m T U shall pass through c. Dividing therefore a c in the midst at t, and S N in the midst at l, and joining t N, L l, the lines L l, t N, and c T produced, will all meet in one and the same point of B S produced; suppose at q. Therefore the point q being given by the two known points T and I, the lines drawn from q through equal parts of the sine of the arch B I, (for example through the points P, Q, R, of the sine M I), shall cut off equal arches, as B L, L N, N O, O I. And this is enough to make good that problem, as to your objection.

The straight line therefore B U, for any thing you have said, is proved equal to the arch B I, and the division of any angle given into any proportion given, the quadrature of any sector, and the construction of any equilateral polygon is also given. And though in this also I should have erred, yet it cannot be denied but that I have used a more natural, a more geometrical, and a more perspicuous method in the search of this so difficult a problem, than you have done in your Arithmetica Infinitorum. For though it be true that the aggregate of all the mean proportionals between the radius, together with an infinitely little part of the same, and the radius wanting an infinitely little part of the same; and again, between the radius, together with two infinitely little parts, and the radius wanting two infinitely little parts, and so on eternally, will be equal to the quadrant (a thing which every mean geometrician knew before); yet it was absurd to think those means could be calculated in numbers by interpoling of a symbol; especially when you make that symbol to stand for 330a number neither true nor surd; as if there were a number that could neither be uttered in words, nor not be uttered in words. For what else is surd, but that which cannot be spoken?

To the fifth article, though your discourse be long, you object but two things. One is, that “Whereas the spiral of Archimedes is made of two motions, one straight, the other circular, both uniform, I taking the motion compounded of them both for one of those that are compounded, conclude falsely, that the generation of the spiral is like to the generation of the parabola.” What heed you use to take in your reprehensions, appears most manifestly in this objection. For I say in that demonstration of mine, that the velocity of the point A in describing the spiral increaseth continually in proportion to the times. For seeing it goes on uniformly in the semidiameter, it is impossible it should not pass into greater and greater circles, proportionally to the times, and consequently it must have a swifter and swifter motion circular, to be compounded with the uniform motion in every point of the radius as it turneth about. This objection therefore is nothing but an effect of a will, without cause, to contradict.

The other objection is, that “Granting all to be true hitherto, yet because it depends upon the finding of a straight line equal to a parabolical line in the eighteenth chapter, where I was deceived, I am also deceived here.” True. But because in the eighteenth chapter of this English edition I have found a straight line equal to the spiral line of Archimedes. I must here put you in mind that by these words in your objections to the 331fifth article at your number two, Quatenus verum est, etc., we have demonstrated prop. 10, 11, 13, Arithmetica Infinitorum; you make it appear that you thought your spiral (made of arches or circles) was the true spiral of Archimedes; which is fully as absurd as the quadrature of Joseph Scaliger, whose geometry you so much despise.

To the sixth article, which is a digression concerning the analytics of geometricians, you deny that the efficient cause of the construction ought to be contained in the demonstration. As if any problem could be known to be truly done, otherwise than by knowing first how, that is to say, by what efficient cause, and in what manner, it is to be done. Whatsoever is done without that knowledge, cannot be demonstrated to be done; as you see in your computation of the parabola, and paraboloeides, in your Arithmetica Infinitorum.

And whereas I said that the ends of all straight lines drawn from a straight line, and passing through one and the same point, if their parts be proportional, shall be in a straight line; is true and accurate; as also, if they begin in the circumference of a circle, they shall also be in the circumference of another circle. And so is this: if the proportion be duplicate, they shall be in a parabola. All this I say is true and accurately spoken. But this was no place for the demonstration of it. Others have done it. And I perceive by that you put in by parenthesis (“Intelligis credo inter duas peripherias concentricas”) that you understand not what I mean.

Hitherto reach your objections to my geometry: for the rest of your book, it containeth nothing 332but a collection of lies, wherewith you do what you can, to extenuate as vulgar, and disgrace as false, that which followeth, and to which you have made no special objection.

I shall therefore only add in this place concerning your Analytica per Potestates, that it is no art. For the rule, both in Mr. Ougthred, and in Des Cartes, is this: “When a problem or question is propounded, suppose the thing required done, and then using a fit ratiocination, put A or some other vowel for the magnitude sought.” How is a man the better for this rule without another rule, how to know when the ratiocination is fit? There may therefore be in this kind of analysis more or less natural prudence, according as the analyst is more or less wise, or as one man in choosing of the unknown quantity with which he will begin, or in choosing the way of the consequences which he will draw from the hypothesis, may have better luck than another. But this is nothing to art. A man may sometimes spend a whole day in deriving of consequences in vain, and perhaps another time solve the same problem in a few minutes.

I shall also add, that symbols, though they shorten the writing, yet they do not make the reader understand it sooner than if it were written in words. For the conception of the lines and figures (without which a man learneth nothing) must proceed from words either spoken or thought upon. So that there is a double labour of the mind, one to reduce your symbols to words, which are also symbols, another to attend to the ideas which they signify. Besides, if you but consider how none of 333the ancients ever used any of them in their published demonstrations of geometry, nor in their books of arithmetic, more than for the roots and potestates themselves; and how bad success you have had yourself in the unskilful using of them, you will not, I think, for the future be so much in love with them as to demonstrate by them that first part you promise of your Opera Mathematica. In which, if you make not amends for that which you have already published, you will much disgrace those mathematicians you address your epistles to, or otherwise have commended; as also the Universities, as to this kind of learning, in the sight of learned men beyond sea. And thus having examined your pannier of Mathematics, and finding in it no knowledge, neither of quantity, nor of measure, nor of proportion, nor of time, nor of motion, nor of any thing, but only of certain characters, as if a hen had been scraping there; I take out my hand again, to put it into your other pannier of theology, and good manners. In the mean time I will trust the objections made by you the astronomer (wherein there is neither close reasoning, nor good style, nor sharpness of wit, to impose upon any man) to the discretion of all sorts of readers.

Having in the precedent lessons maintained the truth of my geometry, and sufficiently made appear that your objections against it are but so many errors of your own, proceeding from misunderstanding of the propositions you have read in Euclid, and other masters of geometry; I leave it to your consideration to whom belong, according to your own sentence, the unhandsome attributes you so often give me upon supposition, that you yourselves are in the right, and I mistaken; and come now to purge myself of those greater accusations which concern my manners. It cannot be expected that there should be much science of any kind in a man that wanteth judgment; nor judgment in a man that knoweth not the manners due to a public disputation in writing; wherein the scope of either party ought to be no other than the examination and manifestation of the truth. For whatsoever is added of contumely, either directly or scommatically, is want of charity and uncivil, unless it be done by way of reddition from him that is first provoked to it. I say unless it be by way of reddition; for so was the judgment given by the emperor Vespasian in a quarrel between a senator and a knight of Rome which had given him ill language. For when the knight had proved that the first ill language proceeded from the senator, 335the emperor acquitted him in these words: “Maledici senatoribus non oportere; remaledicere, fas et civile esse.” Nevertheless, now-a-days, uncivil words are commonly and bitterly used by all that write in matter of controversy, especially in divinity, excepting now and then such writers as have been more than ordinarily well bred, and have observed how heinous and hazardous a thing such contumely is amongst some sorts of men, whether that which is said in disgrace be true or false. For evil words by all men of understanding are taken for a defiance, and a challenge to open war. But that you should have observed so much, who are yet in your mother’s belly, was not a thing to be much expected.

The faults in manners you lay to my charge are these: 1. Self-conceit. 2. That I will be very angry with all men that do not presently submit to my dictates. 3. That I had my doctrine concerning Vision, out of papers which I had in my hands of Mr. Warner’s. 4. That I have injured the universities. 5. That I am an enemy to religion. These are great faults; but such as I cannot yet confess. And therefore I must, as well as I can, seek out the grounds upon which you build your accusation. Which grounds (seeing you are not acquainted with my conversation) must be either in my published writings, or reported to you by honest men, and without suspicion of interest in reporting it. As for my self-conceit and ostentation, you shall find no such matter in my writings. That which you allege from thence is first, that in the epistle dedicatory I say of my book De Corpore, “though it be little, yet it is 336full; and if good may go for great, great enough.” When a man presenting a gift great or small to his betters, adorneth it the best he can to make it the more acceptable; he that thinks this to be ostentation and self-conceit, is little versed in the common actions of human life. And in the same epistle, where I say of civil philosophy: “It is no ancienter than my book De Cive;” these words are added: “I say it provoked, and that my detractors may see they lose their labour.” But that which is truly said, and upon provocation, is not boasting, but defence. A short sum of that book of mine, now publicly in French, done by a gentleman I never saw, carrieth the title of Ethics Demonstrated . The book itself translated into French, hath not only a great testimony from the translator Sorberius, but also from Gassendus, and Mersennus, who being both of the Roman religion had no cause to praise it, or the divines of England have no cause to find fault with it. Besides, you know that the doctrine therein contained is generally received by all but those of the clergy, who think their interest concerned in being made subordinate to the civil power; whose testimonies therefore are invalid. Why therefore, if I commend it also against them that dispraise it publicly, do you call it boasting? “You have heard,” you say, “that I had promised the quadrature of the circle, &c.” You heard then that which was not true. I have been asked sometimes, by such as saw the figure before me, what I was doing, and I was not afraid to say I was seeking for the solution of that problem; but not that I had done it. And afterwards being asked of the success, I have 337said, I thought it done. This is not boasting; and yet it was enough, when told again, to make a fool believe it was boasting. But you, the astronomer, in the epistle before your philosophical essay, say “You had a great expectation of my philosophical and mathematical works, before they were published.” It may be so. Is that my fault? Can a man raise a great expectation of himself by boasting? If he could, neither of you would be long before you raised it of yourselves; saving that what you have already published, has made it now too late. For I verily believe there was never seen worse reasoning than in that philosophical essay; which any judicious reader would believe proceeded from a prevaricator, rather than from a man that believed himself; nor worse principles, than those in your books of Geometry. The expectation of that which should be written by me, was raised partly by the Cogitata Physica-Mathematica of Mersennus, wherein I am often named with honour; and partly by others with whom I then conversed in Paris, without any ostentation. That no man has a great expectation of any thing that shall proceed from either of you two, I am content to let it be your praise.

Another argument of my self-conceit, you take from my contempt of the writers of metaphysics and school-divinity. If that be a sign of self-conceit, I must confess I am guilty; and if your geometry had then been published, I had contemned that as much. But yet I cannot see the consequence (unless you lend me your better logic) from despising insignificant and absurd language, to self-conceit.

338And again, in your Vindiciæ Academiarum, you put for boasting, that in my Leviathan , page 331, I would have that book by entire sovereignty imposed upon the Universities; and in my Review , p. 713, that I say of my Leviathan , “I think it may be profitably printed, and more profitably taught in the University.” The cause of my writing that book, was the consideration of what the ministers before, and in the beginning of, the civil war, by their preaching and writing did contribute thereunto. Which I saw not only to tend to the abatement of the then civil power, but also to the gaining of as much thereof as they could (as did afterwards more plainly appear) unto themselves. I saw also that those ministers, and many other gentlemen who were of their opinion, brought their doctrines against the civil power from their studies in the Universities. Seeing therefore that so much as could be contributed to the peace of our country, and the settlement of sovereign power without any army, must proceed from teaching; I had reason to wish, that civil doctrine were truly taught in the Universities. And if I had not thought that mine was such, I had never written it. And having written it, if I had not recommended it to such as had the power to cause it to be taught, I had written it to no purpose. To me therefore that never did write anything in philosophy to show my wit, but, as I thought at least, to benefit some part or other of mankind, it was very necessary to commend my doctrine to such men as should have the power and right to regulate the Universities. I say my doctrine; I say not my Leviathan . For wiser men may so digest the same 339doctrine as to fit it better for a public teaching. But as it is, I believe it hath framed the minds of a thousand gentlemen to a conscientious obedience to present government, which otherwise would have wavered in that point. This therefore was no vaunting, but a necessary part of the business I took in hand. You ought also to have considered, that this was said in the close of that part of my book which concerneth policy merely civil. Which part, if you, the astronomer, that now think the doctrine unworthy to be taught, were pleased once to honour with praises printed before it, you are not very constant nor ingenuous. But whether you did so or not, I am not certain, though it was told me for certain. If it were not you, it was somebody else whose judgment has as much weight at least as yours.

And for anything you have to say from your own knowledge, I remember not that I ever saw either of your faces. Yet you, the professor of geometry, go about obliquely to make me believe that Vindex hath discoursed with me, once at least, though I remember it not. I suppose it therefore true; but this I am sure is false, that either he or any man living did ever hear me brag of my science, or praise myself, but when my defence required it. Perhaps some of our philosophers that were at Paris at the same time, and acquainted with the same learned men that I was acquainted with, might take for bragging the maintaining of my opinions, and the not yielding to the reasons alledged against them. If that be ostentation, they tell you the truth. But you that are so wise should have considered, that even such men as 340profess philosophy are carried away with the passions of emulation and envy (the sole ground of this your accusation) as well as other men, and instanced in yourselves. And this is sufficient to shake off your aspersions of ostentation and self-conceit. For if I added, that my acquaintance know that I am naturally of modest rather than of boasting speech, you will not believe it; because you distinguish not between that which is said upon provocation, and that which is said without provocation, from vain glory.

The next accusation is: “That I will be very angry with all men that do not presently submit to my dictates; and that for advancing the reputation of my own skill, I care not what unworthy reflections I cast on others.” This is in the epistle placed before the Vindiciæ Academiarum, subscribed by N S, as the plain song for H D in the rest of the book to descant upon. I know well enough the authors’ names; and am sorry that N S has lent his name to be abused to so ill a purpose. But how does this appear? What argument, what witness is there of it? You offer none; nor am I conscious of any. I begin to suspect since you, the professor of geometry, have in your objections to the twentieth chapter these words concerning “Vindex, ocularis ille testis de quo hic agitur, erat, ni fallor, ille ipse,”--that Vindex himself, in other company, has bestowed a visit on me. Seeing you will have me believe it, let it be so; and, as it is likely, not long after my return into England. At which time (for the reputation, it seems, I had gotten by my boasting) divers persons that professed to love philosophy 341and mathematics, came to see me; and some of them to let me see them, and hear and applaud what they applauded in themselves. I see now it hath happened to me with Vindex, as it happened to Dr. Harvey with Moranus. Moranus, a jesuit, came out of Flanders hither, especially, as he says, to see what learned men in divinity, ethics, physics, and geometry, were here yet alive, to the end that by discoursing with them in these sciences, he might correct either his own, or their errors. Amongst others he was brought, he says, to that most civil and renowned old man Dr. Harvey. That is very well. And in good earnest if he had made good use of the time which was very patiently afforded him, he might have learned of him (or of no man living) very much knowledge concerning the circulation of the blood, the generation of living creatures, and many other difficult points of natural philosophy. And if he had had anything in him but common and childish learning, he could have showed it nowhere more to his advantage, than before him that was so great a judge of such matters. But what did he? That precious time (which was but little, because he was to depart again presently for Flanders) he bestowed wholly in venting his own childish opinions, not suffering the Doctor scarce to speak; losing thereby the benefit he came for, and discovering that he came not to hear what others could say, but to show to others how learned he was himself already. Why else did he take so little time, and so misspendmisspend it? Or why returned he not again? But when he had talked away his time, and found (though patiently and civilly heard) he was not much admired, he 342took occasion, writing against me, to be revenged of Dr. Harvey, by slighting his learning publicly; and tells me that his learning was only experiments; which he says I say have no more certainty than civil histories. Which is false. My words are: “Ante hos nihil certi in physica erat præter experimenta cuique sua, et historias naturales, si tamen et hæ dicendæ certæ sint, quæ civilibus historiis certiores non sunt.” Where I except expressly from uncertainty the experiments that every man maketh to himself. But you see the near cut, by which vain glory joined with ignorance passeth quickly over to envy and contumely.

Thus it seems by your own confession I was used by Vindex. He comes with some of my acquaintance in a visit. What he said I know not, but if he discoursed then, as in his philosophical essay he writeth, I will be bold to say of myself, I was so far from morosity, or, to use his phrase, from being tetrical, as I may very well have a good opinion of my own patience. And if there passed between us the discourse you mention in your Elenchus, page 116, it was an incivility in him so great, that without great civility I could not have abstained from bidding him be gone. That which passed between us you say was this: “I complained that whereas I made sense, nothing but a perception of motion in the organ, nevertheless, the philosophy schools through all Europe, led by the text of Aristotle, teach another doctrine, namely, that sensation is performed by species.” This is a little mistaken. For I do glory, not complain, that whereas all the Universities of Europe hold sensation to proceed from species, I 343hold it to be a perception of motion in the organ. The answer of Vindex, you say, was: “That the other hypothesis, whereby sense was explicated by the principles of motion, was commonly admitted here before my book came out, as having been sufficiently delivered by Des Cartes, Gassendus, and Sir Kenelm Digby, before I had published anything in this kind.” This then, it seems, was it that made me angry. Truly I remember not an angry word that ever I uttered in all my life to any man that came to see me, though some of them have troubled me with very impertinent discourse; and with those that argued with me, how impertinently soever, I always thought it more civility to be somewhat earnest in the defence of my opinion, than by obstinate and affected silence to let them see I contemned them, or hearkened not to what they said. If I were earnest in making good, that the manner of sensation by such motion as I had explicated in my Leviathan , is in none of the authors by him named, it was not anger, but a care of not offending him, with any sign of the contempt which his discourse deserved. But it was incivility in him to make use of a visit, which all men take for a profession of friendship, to tell me that that which I had already published for my own, was found before by Des Cartes, Gassendus, and Sir Kenelm Digby. But let any man read Des Cartes; he shall find, that he attributeth no motion at all to the object of sense, but an inclination to action, which inclination no man can imagine what it meaneth. And for Gassendus, and Sir Kenelm Digby, it is manifest by their writings, that their opinions are not different from 344that of Epicurus, which is very different from mine. Or if these two, or any of those I conversed with at Paris, had prevented me in publishing my own doctrine, yet since it was there known and declared for mine by Mersennus in the preface to his Ballistica (of which the three first leaves are employed wholly in the setting forth of my opinion concerning sense, and the rest of the faculties of the soul) they ought not therefore to be said to have found it out before me. And consequently this answer which you say was given me by Vindex was nothing else but untruth and envy; and, because it was done by way of visit, incivility. But you have not alleged, nor can allege, any words of mine, from which can be drawn that I am so angry as you say I am with those that submit not to my dictates. Though the discipline of the University be never so good; yet certainly this behaviour of yours and his are no good arguments to make it thought so. But you the professor of geometry, that out of my words spoken against Vindex in my twentieth chapter, argue my angry humour, do just as well, as when (in your Arithmetica Infinitorum) from the continual increase of the excess of the row of squares above the third part of the aggregate of the greatest, you conclude they shall at last be equal to it. For though you knew that Vindex had given me first the worst words that possibly can be given, yet you would have that return of mine to be a demonstration of an angry humour; not then knowing what I told you even now in the beginning of this lesson, of the sentence given by Vespasian. But to this point I shall speak again hereafter.

345Your third accusation is: “That I had my doctrine of vision, which I pretended to be my own, out of papers which I had a long time in my hands of Mr. Warner’s.” I never had sight of Mr. Warner’s papers in all my life, but that of Vision by Refraction (which by his approbation I carried with me to Park, and caused it to be printed under his own name, at the end of Mersennus his Cogitata Physico-Mathematica, which you may have there seen, and another treatise of the proportions of alloy in gold and silver coin; which is nothing to the present purpose)purpose). In all my conversation with him, I never heard him speak of anything he had written, or was writing, De penicillo optico. And it was from me that he first heard it mentioned that light and colour were but fancy. Which he embraced presently as a truth, and told me it would remove a rub he was then come to in the discovery of the place of the image. If after my going hence he made any use of it (though he had it from me, and not I from him), it was well done. But wheresoever you find my principles, make use of them, if you can, to demonstrate all the symptoms of vision; and I will do (or rather have done and mean to publish) the same; and let it be judged by that, whether those principles be of mine, or other men’s invention. I give you time enough, and this advantage besides, that much of my optics hath been privately read by others. For I never refused to lend my papers to my friends, as knowing it to be a thing of no prejudice to the advancement of philosophy, though it be, as I have found it since, some prejudice to the advancement of my own reputation in those 346sciences; which reputation I have always postposed to the common benefit of the studious.

You say further (you the geometrician) that I had the proposition of the spiral line equal to a parabolical line from Mr. Robervall: true. And if I had remembered it, I would have taken also his demonstration; though if I had published his, I would have suppressed mine. I was comparing in my thoughts those two lines, spiral and parabolical, by the motions wherewith they were described; and considering those motions as uniform, and the lines from the centre to the circumference, not to be little parallelograms, but little sectors, I saw that to compound the true motion of that point which described the spiral, I must have one line equal to half the perimeter, the other equal to half the diameter. But of all this I had not one word written. But being with Mersennus and Mr. Robervall in the cloister of the convent, I drew a figure on the wall, and Mr. Robervall perceiving the deduction I made, told me that since the motions which make the parabolical line, are one uniform, the other accelerated, the motions that make the spiral must be so also; which I presently acknowledged; and he the next day, from this very method, brought to Mersennus the demonstration of their equality. And this is the story mentioned by Mersennus, prop. 25, corol. 2, of his Hydraulica; which I know not who hath most magnanimously interpreted to you in my disgrace.

The fourth accusation is: “That I have injured the Universities.” Wherein? First, “In that I would have the doctrine of my Leviathan by entire 347sovereignty be imposed on them.” You often upbraid me with thinking well of my own doctrine; and grant by consequence, that I thought this doctrine good; I desired not therefore that anything should be imposed upon them, but what (at least in my opinion) was good both for the Commonwealth and them. Nay more, I would have the state make use of them to uphold the civil power, as the Pope did to uphold the ecclesiastical. Is it not absurdly done to call this an injury? But to question, you will say, whether the civil doctrine there taught be such as it ought to be, or not, is a disgrace to the Universities. If that be certain, it is certain also that those sermons and books, which have been preached and published, both against the former and the present government, directly or obliquely, were not made by such ministers and others as had their breeding in the Universities; though all men know the contrary. But the doctrine which I would have to be taught there, what is it? It is this: “That all men that live in a Commonwealth, and receive protection of their lives and fortunes from the supreme governor thereof, are reciprocally bound, as far as they are able, and shall be required, to protect that governor.” Is it, think you, an unreasonable thing to impose the teaching of such doctrine upon the Universities? Or will you say they taught it before, when you know that so many men which came from the Universities to preach to the people, and so many others that were not ministers, did stir the people up to resist the then supreme civil power? And was it not truly therefore said, that the Universities receiving their discipline from the 348authority of the pope, were the shops and operatories of the clergy? Though the competition of the papal and civil power be taken now away, yet the competition between the ecclesiastical and the civil power hath manifestly enough appeared very lately. But neither is this an upbraiding of an University (which is a corporation or body artificial), but of particular men, that desire to uphold the authority of a Church, as of a distinct thing from the Commonwealth. How would you have exclaimed, if, instead of recommending my Leviathan to be taught in the Universities, I had recommended the erecting of a new and lay-university, wherein lay-men should have the reading of physics, mathematics, moral philosophy, and politics, as the clergy have now the sole teaching of divinity? Yet the thing would be profitable, and tend much to the polishing of man’s nature, without much public charge. There will need but one house, and the endowment of a few professions. And to make some learn the better, it would do very well that none should come thither sent by their parents, as to a trade to get their living by, but that it should be a place for such ingenuous men, as being free to dispose of their own time, love truth for itself. In the mean time divinity may go on in Oxford and Cambridge to furnish the pulpit with men to cry down the civil power, if they continue to do as they did. If I had, I say, made such a motion in my Leviathan , though it would have offended the divines, yet it had been no injury. But it is an injury, you will say, to deny in general the utility of the ancient schools, and to deny that we have received from them our 349geometry. True, if I had not spoken distinctly of the schools of philosophy, and said expressly, that the geometricians passed not then under the name of philosophers; and that in the school of Plato (the best of the ancient philosophers) none were received that were not already in some measure geometricians. Euclid taught geometry; but I never heard of a sect of philosophers called Euclidians, or Alexandrians, or ranged with any of the other sects, as Peripatetics, Stoics, Academics, Epicureans, Pyrrhonians, &c. But what is this to the Universities of Christendom? Or why are we beholden for geometry to our universities, more than to Gresham College, or to private men in London, Paris, and other places, which never taught or learned it in a public school? For even those men that living in our Universities have most advanced the mathematics, attained their knowledge by other means than that of public lectures, where few auditors, and those of unequal proficiency, cannot make benefit by one and the same lesson. And the true use of public professors, especially in the mathematics, being to resolve the doubts, and problems, as far as they can, of such as come unto them with desire to be informed.

That the Universities now are not regulated by the Pope, but by the civil power, is true, and well. But where say I the contrary? And thus much for the first injury.

Another, you say, is this, that in my Leviathan , p. 670, I say: “The principal schools were ordained for the three professions of Roman religion, Roman law, and the art of medicine.” Thirdly, that I say: “Philosophy had no otherwise place 350there than as a hand-maid to Roman religion.” Fourthly: “Since the authority of Aristotle was received there, that study is not properly philosophy, but Aristotelity.” Fifthly: “That for geometry, till of late times it had no place there at all.” As for the second, it is too evident to be denied; the fellowships having been all ordained for those professions; and (saving the change of religion) being so yet. Nor hath this any reflection upon the Universities, either as they now are, or as they then were, seeing it was not in their own power to endow themselves, or to receive other laws and discipline than their founder and the state was pleased to ordain. For the third, it is also evident. For all men know that none but the Roman religion had any stipend or preferment in any university, where that religion was established? No, nor for a great while, in their commonwealths; but were everywhere persecuted as heretics. But you will say, the words of my Leviathan are not, philosophy “had no place,” but “hath no place.” Are you not ashamed to lay to my charge a mistake of the word hath for had? which was either a mistake of the printer, or if it were so in the copy, it could be no other than the mistake of a letter in the writing, unless you think you can make men believe that after fifty years being acquainted with what was publicly professed and practised in Oxford and Cambridge, I knew not what religion they were of. This taking of advantage from the mistake of a word, or of a letter, I find also in the Elenchus, where for prætendit se scire, there is prætendit scire, which you the geometrician sufficiently mumble, mistaking it I 351think for an anglicism, not for a fault of the impression.

To the fourth, you pretend, that men are not now so tied to Aristotle as not to enjoy a liberty of philosophising, though it were otherwise when I was conversant in Magdalen Hall. Was it so then? Then am I absolved, unless you can show some public act of the university made since that time to alter it. For it is not enough to name some few particular ingenuous men that usurp that liberty in their private discourses, or, with connivance, in their public disputations. And your doctrine, that even here you avow, of abstracted essences, immaterial substances, and of nunc-stans; and your improper language in using the word (not as mine, for I have it nowhere) successive eternity; as also your doctrine of condensation, and your arguing from natural reason the incomprehensible mysteries of religion, and your malicious writing, are very shrewd signs that you yourselves are none of those which you say do freely philosophise; but that both your philosophy and your language are under the servitude, not of the Roman religion, but of the ambition of some other doctors, that seek, as the Roman clergy did, to draw all human learning to the upholding of their power ecclesiastical. Hitherto therefore there is no injury done to the universities. For the fifth, you grant it, namely, “that till of late there was no allowance for the teaching of geometry.” But lest you should be thought to grant me anything, you say, you the astronomer, “geometry hath now so much place in the universities, that when Mr. Hobbes shall have published his philosophical and 352geometrical pieces, you assure yourself you shall be able to find a greater number in the university who will understand as much, or more, of them than he desired they should,” &c. But though this be true of the now, yet it maketh nothing against my then. I know well enough that Sir Henry Savile’s lectures were founded and endowed since. Did I deny then that there were in Oxford many good geometricians? But I deny now, that either of you is of the number. For my philosophical and geometrical pieces are published, and you have understood only so much in them, as all men will easily see by your objections to them, and by your own published geometry, that neither of you understand anything either in philosophy or in geometry. And yet you would have those books of yours to stand for an argument, and to be an index of the philosophy and geometry to be found in the universities. Which is a greater injury and disgrace to them, than any words of mine, though interpreted by yourselves.

Your last and greatest accusation, or rather railing (for an accusation should contain, whether true or false, some particular fact, or certain words, out of which it might seem at least to be inferred), is, that I am an enemy to religion. Your words are: “It is said that Mr. Hobbes is no otherwise an enemy to the Roman religion, saving only as it hath the name of religion.” This is said by Vindex. You, the geometrician, in your epistle dedicatory, say thus: “With what pride and imperiousness he tramples on all things both human and divine, uttering fearful and horrible words of God, (peace), of sin, of the holy Scripture, of all incorporeal 353substances in general, of the immortal soul of man, and of the rest of the weighty points of religion (down), it is not so much to be doubted as lamented.” And at the end of your objections to the eighteenth chapter, “Perhaps you take the whole history of the fall of Adam for a fable, which is no wonder, when you say the rules of honouring and worshipping of God are to be taken from the laws.” Down, I say; you bark now at the supreme legislative power. Therefore it is not I, but the laws which must rate you off. But do not many other men, as well as you, read my Leviathan , and my other books? And yet they all find not such enmity in them against religion. Take heed of calling them all atheists that have read and approved my Leviathan . Do you think I can be an atheist and not know it? Or knowing it, durst have offered my atheism to the press? Or do you think him an atheist, or a contemner of the Holy Scripture, that sayeth nothing of the Deity but what he proveth by the Scripture? You that take so heinously that I would have the rules of God’s worship in a Christian commonwealth taken from the laws, tell me, from whom you would have them taken? From yourselves? Why so, more than from me? From the bishops? Right, if the supreme power of the commonwealth will have it so; if not, why from them rather than from me? From a consistory of presbyters by themselves, or joined with lay-elders, whom they may sway as they please? Good, if the supreme governor of the commonwealth will have it so; if not, why from them, rather than from me, or from any man else? They are wiser and learneder 354than I. It may be so; but it has not yet appeared. Howsoever, let that be granted. Is there any man so very a fool as to subject himself to the rules of other men in those things which so nearly concern himself, for the title they assume of being wise and learned, unless they also have the sword which must protect them. But it seems you understand the sword as comprehended. If so, do you not then receive the rules of God’s worship from the civil power? Yes, doubtless; and you would expect, if your consistory had that sword, that no man should dare to exercise or teach any rules concerning God’s worship which were not by you allowed. See therefore how much you have been transported by your malice towards me, to injure the civil power by which you live.live. If you were not despised, you would in some places and times, where and when the laws are more severely executed, be shipped away for this your madness to America, I would say, to Anticyra. What luck have I, when this, of the laws being the rules of God’s public worship, was by me said and applied to the vindication of the Church of England from the power of the Roman clergy, it should be followed with such a storm from the ministers, presbyterian and episcopal, of the Church of England? Again, for those other points, namely, that I approve not of incorporeal bodies, nor of other immortality of the soul, than that which the Scripture calleth eternal life, I do but as the Scripture leads me. To the texts whereof by me alleged, you should either have answered, or else forborne to revile me for the conclusions I derived from them. Lastly, what an absurd question is it to ask me 355whether it be in the power of the magistrate, whether the world be eternal or not? It were fit you knew it is in the power of the supreme magistrate to make a law for the punishment of them that shall pronounce publicly of that question anything contrary to that which the law hath once pronounced. The truth is, you are content that the papal power be cut off, and declaimed against as much as any man will; but the ecclesiastical power, which of late was aimed at by the clergy here, being a part thereof, every violence done to the papal power is sensible to them yet; like that which I have heard say of a man, whose leg being cut off for the prevention of a gangrene that began in his toe, would nevertheless complain of a pain in his toe, when his leg was cut off.

Thus much in my defence; which I believe if you had foreseen, this accusation of yours had been left out. I come now to examine (though it be done in part already) what manners those are which I find everywhere in your writings.

And first, how came it into your minds that a man can be an atheist, I mean an atheist in his conscience? I know that David confesseth of himself, upon sight of the prosperity of the wicked, that his feet had almost slipped, that is, that he had slipped into a short doubtfulness of the Divine Providence. And if anything else can cause a man to slip in the same kind, it is the seeing such as you (who though you write nothing but what is dictated to each of you by a doctor of divinity) do break the greatest of God’s commandments, which is charity, in every line before his face. And though such forgettings of God be somewhat 356more than short doubtings, and sudden transportations incident to human passion, yet I do not for that cause think you atheists and enemies of religion, but only ignorant and imprudent Christians. But how, I say, could you think me an atheist, unless it were because finding your doubts of the Deity more frequent than other men do, you are thereby the apter to fall upon that kind of reproach? Wherein you are like women of poor and evil education when they scold; amongst whom the readiest disgraceful word is whore: why not thief, or any other ill name, but because, when they remember themselves, they think that reproach the likeliest to be true?

Secondly, tell me what crime it was which the Latins called by the name of scelus? You think not, unless you be Stoics, that all crimes are equal. Scelus was never used but for a crime of greatest mischief, as the taking away of life and honour; and besides, basely acted, as by some clandestine way, or by such a way as might be covered with a lie. But when you insinuate in a writing published that I am an atheist, you make yourselves authors to the multitude, and do all you can to stir them up to attempt upon my life; and if it succeed, then to sneak out of it by leaving the fault on them that are but actors. This is to endeavour great mischief basely, and therefore scelus. Again, to deprive a man of the honour he hath merited, is no little wickedness; and this you endeavour to do by publishing falsely that I challenge as my own the inventions of other men. This is therefore scelus publicly to tell all the world that I will be angry with all men that do not presently submit to my dictates; to deprive me of the friendship 357of all the world; great damage, and a lie, and yours. For to publish any untruth of another man to his disgrace, on hearsay from his enemy, is the same fault as if he published it on his own credit. If I should say I have heard that Dr. Wallis was esteemed at Oxford for a simple fellow, and much inferior to his fellow-professor Dr. Ward (as indeed I have heard, but do not believe it), though this be no great disgrace to Dr. Wallis, yet he would think I did him injury. Therefore public accusation upon hearsay is scelus. And whosoever does any of these things does sceleratè. But you the professors of the mathematics at Oxford, by the advice of two doctors of divinity have dealt thus with me. Therefore you have done, I say not foolishly, though no wickedness be without folly, but sceleratè, ὅπερ ἔδει δεῖξαι.

Thirdly, it is ill manners, in reprehending truth, to send a man in a boasting way to your own errors; as you the professor of geometry have often sent me to your two tractates of the Angle of Contact and Arithmetica Infinitorum.

Fourthly, it is ill manners, to diminish the just reputation of worthy men after they be dead, as you the professor of geometry have done in the case of Joseph Scaliger.

Fifthly, when I had in my Leviathan suffered the clergy of the Church of England to escape, you did imprudently in bringing any of them in again. An Ulysses upon so light an occasion would not have ventured to return again into the cave of Polyphemus.

Lastly, how ill does such levity and scurrility, which both of you have shown so often in your writings, become the gravity and sanctity requisite 358to the calling of the ministry? They are too many to be repeated. Do but consider, you the geometrician, how unhandsome it is to play upon my name, when both yours and mine are plebeian names; though from Willis by Wallis, you go from yours in Wallisius. The jest of using at every word mi Hobbi, is lost to them beyond sea. But this is not so ill as some of the rest. I will write out one of them, as it is in the fourth page of your Elenchus: “Whence it appears that your Empusa was of the number of those fairies which you call in English hob-goblins. The word is made of ἕν and πους; and thence comes the children’s play called the play of Empusa, Anglicè (hitherto in Latin all but hob-goblins, then follows in English) fox, fox, come out of your hole (then in Latin again), in which the boy that is called the fox, holds up one foot, and jumps with the other, which in English is to hop.” When a stranger shall read this, and hoping to find therein some witty conceit, shall with much ado have gotten it interpreted and explained to him, what will he think of our doctors of divinity at Oxford, that will take so much pains as to go out of the language they set forth in, for so ridiculous a purpose? You will say it is a pretty paranomasia. How you call it there I know not, but it is commonly called here a clinch; and such a one as is too insipid for a boy of twelve years old, and very unfit for the sanctity of a minister, and gravity of a doctor of divinity. But I pray you tell me where it was you read the word empusa for the boy’s play you speak of, or for any other play amongst the Greeks? In this (as you have done throughout all your other writings) you presume too much upon your first 359cogitations. There be a hundred other scoffing passages, and ill-favoured attributes given me in both your writings, which the reader will observe without my pointing to them, as easily as you would have him; and which perhaps some young students, finding them full of gall, will mistake for salt. Therefore to disabuse those young men, and to the end they may not admire such kind of wit, I have here and there been a little sharper with you than else I would have been. If you think I did not spare you, but that I had not wit enough to give you as scornful names as you give me, are you content I should try? Yes (you the geometrician will say) give me what names you please, so you call me not Arithmetica Infinitorum. I will not. Nor Angle of Contact ; nor Arch Spiral ; nor Quotient . I will not. But I here dismiss you both together. So go your ways, you Uncivil Ecclesiastics, Inhuman Divines, Dedoctors of morality, Unasinous Colleagues, Egregious pair of Issachars, most wretched Vindices and Indices Academiarum; and remember Vespasian’s law, that it is uncivil to give ill language first, but civil and lawful to return it. But much more remember the law of God, to obey your sovereigns in all things; and not only not to derogate from them, but also to pray for them, and as far as you can to maintain their authority, and therein your own protection. And, do you hear? take heed of speaking your mind so clearly in answering my Leviathan, as I have done in writing it. You should do best not to meddle with it at all, because it is undertaken, and in part published already, and will be better performed, from term to term, by one Christopher Pike.

ΣΤΙΓΜΑΙ — AN EXTRACT OF A LETTER CONCERNING THE GRAMMATICAL PART OF THE CONTROVERSY BETWEEN MR. HOBBES AND DR. WALLIS.

My most noble Lord,

I did not intend to trouble your Lordship twice with this contention between me and Dr. Wallis. But your Lordship sees how I am constrained to it; which, whatsoever reply the Doctor makes, I shall be constrained to no more. That which I have now said of his Geometry, Manners, Divinity, and Grammar, altogether is not much, though enough. As for that which I here have written concerning his Geometry, which you will look for first, is so clear, that not only your Lordship, and such as have proceeded far in that science, but also any man else that doth but know how to add and subtract proportions, (which is taught at the twenty-third proposition of the sixth of Euclid), may see the Doctor is in the wrong. That which I say of his ill language and politics is yet shorter. The rest, which concerneth grammar, is almost all another man’s, but so full of learning of that kind, as no man that taketh delight in knowing the proprieties 363of the Greek and Latin tongues, will think his time ill bestowed in the reading it. I give the Doctor no more ill words, but am returned from his manners to my own. Your Lordship may perhaps say, my compliment in my title-page is somewhat coarse; and it is true. But, my Lord, it is since the writing of the title-page, that I am returned from the Doctor’s manners to my own; which are such as I hope you will not be ashamed to own me, my Lord, for one of

Sir,

When unprovoked you addressed unto me, in your Elenchus, your harsh compliment with great security, wantonly to show your wit, I confess you made me angry, and willing to put you into a better way of considering your own forces, and to move you a little as you had moved me, which I perceive my lessons to you have in some measure done; but here you shall see how easily I can bear your reproaches, now they proceed from anger, and how calmly I can argue with you about your geometry and other parts of learning.

I shall in the first part confer with you about your Arithmetica Infinitorum, and afterwards compare our manner of elocution; then your politics; and last of all your grammar and critics. Your spiral line is condemned by him whose authority you use to prove me a plagiary, (that is, a man that stealeth other men’s inventions, and arrogates them to himself), whether it be Roberval or not that writ that paper, I am not certain. But I think I shall be shortly; but whosoever it be, his authority will serve no less to 365show that your doctrine of the spiral line, from the fifth to the eighteenth proposition of your Arithmetica Infinitorum, is all false; and that the principal fault therein (if all faults be not principal in geometry, when they proceed from ignorance of the science) is the same that I objected to you in my Lessons. And for the author of that paper, when I am certain who it is, it will be then time enough to vindicate myself concerning that name of plagiary. And whereas he challenges the invention of your method delivered in your Arithmetica Infinitorum, to have been his before it was yours, I shall, I think, by and by say that which shall make him ashamed to own it; and those that writ those encomiastic epistles to you ashamed of the honour they meant to you. I pass therefore to the nineteenth proposition, which in Latin is this: your geometry!

“Si proponatur series quantitatum in duplicata ratione arithmetice proportionalium (sive juxta seriem numerorum quadraticorum) continue crescentium, a puncto vel 0 inchoatarum, (puta ut 0. 1. 4. 9. 16. etc.), propositum sit, inquirere quam habeat illa rationem ad seriem totidem maximæ æqualium.

“Fiat investigatio per modum inductionis ut (in prop. 1)

Eritque,

0 + 1 = 1
1 + 1 = 2 = 1
3 + 1
6

0 + 1 + 4 = 5
4 + 4 + 4 = 12 = 1
3 + 1
12

0 + 1 + 4 + 9 = 14
9 + 9 + 9 + 9 = 36 = 1
3 + 1
18 et sic deinceps.

“Ratio proveniens est ubique major quam 366subtripla seu 1
3; excessus autem perpetuo decrescit prout numerus terminorum augetur (puta 1
6 1
12 1
18 1
24 etc.) aucto nimirum fractionis denominatore sive consequente rationis in singulis locis numero senario (ut patet) ut sit rationis provenientis excessus supra subtriplam, ea quam habet unitas ad sextuplum numeri terminorum post 0; adeoque.”

That is, if there be propounded a row of quantities in duplicate proportion of the quantities arithmetically proportional (or proceeding in the order of the square numbers) continually increasing; and beginning at a point or 0; let it be propounded to find what proportion the row hath; to as many quantities equal to the greatest;

Let it be sought by induction (as in the first proposition).

The proportion arising is everywhere greater than subtriple, or 1
3, and the excess perpetually decreaseth as the number of terms is augmented, as here, 1
6 1
12 1
18 1
24 1
30, &c. denominator of the fraction being in every place augmented by the number six, as is manifest; so that the excess of the rising proportion above subtriple is the same which unity hath to six times the number of terms after 0; and so.

Sir, in these your characters I understand by the cross + that the quantities on each side of it are to be added together and make one aggregate; and I understand by the two parallel lines = that the quantities between which they are placed are one to another equal; this is your meaning, or you should have told us what you meant else; I understand also, that in the first row 0 + 1 is equal to 1, 367and 1 + 1 equal to 2; and that in the second row 0 + 1 + 4 is equal to 5; and 4 + 4 + 4 equal to 12; but (which you are too apt to grant) I understand your symbols no further; but must confer with yourself about the rest.

And first I ask you (because fractions are commonly written in that manner) whether in the uppermost row (which is 0 + 1 = 1
1 + 1 = 2 = 1
3 + 1
6)0
1 be a fraction, 1
1 be a fraction, 1
2 be a fraction, that is to say, a part of an unit, and if you will, for the cypher’s sake, whether 0
1, be an infinitely little part of 1; and whether 1
1 or 1 divided by 1 signify an unity? if that be your meaning, then the fraction 0
1 added to the fraction 1
1 is equal to the fraction 1
2: But the fraction 0
1 is equal to O; therefore the fraction 0
1 + 1
1 is equal to the fraction 1
1; and 1
1 equal to 1
2 which you will confess to be an absurd conclusion, and cannot own that meaning.

I ask you therefore again, if by 0
1 you mean the proportion of 0 to 1; and consequently by 1
1 the proportion of 1 to 1, and by 1
2 the proportion of 1 to 2: if so, then it will follow, that if the proportions of 0 to 1 and of 1 to 1 be compounded by addition, the proportion arising will be the proportion of 1 to 2. But the proportion of 0 to 1 is infinitely little, that is, none. Therefore the proposition arising by composition will be that of 1 to 1, and equal (because of the symbol =) to the proportion of 1 to 2, and so 1 = 2. This also is so absurd that I dare say that you will not own it.

There may be another meaning yet: perhaps you mean that the uppermost quantity 0 + 1 is equal 368to the uppermost quantity 1; and the lowermost quantity 1 + 1 equal to the lowermost quantity 2: which is true. But how then in this equation 1
2 = 1
3 + 1
6? Is the uppermost quantity 1 equal to the uppermost quantity 1 + 1; or the lowermost quantity 2 equal to the lowermost quantity 3 + 6? Therefore neither can this be your meaning. Unless you make your symbols more significant, you must not blame me for want of understanding them.

Let us now try what better success we shall have where the places are three, as here:

0 + 1 + 4 = 5
4 + 4 + 4 = 12 = 5
12 = 1
3 + 1
12:

If your symbols be fractions, the compound of them by addition is 5
4, for 01
4 and 4
4 make 5
4; and consequently (because of the symbol = ) 5
4 equal to 5
12, which is not to be allowed, and therefore that was not your meaning. If you meant that the proportions of 0 to 4 and of 1 to 4 and of 4 to 4 compounded, is equal to the proportion of 5 to 12, you will fall again into no less an inconvenience. For the proportion arising out of that composition will be the proportion of 1 to 4. For the proportion of 0 to 4 is infinitely little. Then to compound the other two, set them in this order 1. 4. 4. and you have a proportion compounded of 1 to 4 and of 4 to 4, namely, the proportion of the first to the last, which is of 1 to 4, which must be equal, by this your meaning, to the proportion of 5 to 12, and consequently as 5 to 12, so is 1 to 4, which you must not own. Lastly, if you mean that the uppermost quantities to the uppermost, and the lowermost to the lowermost in the first 369equation are equal, it is granted, but then again in the second equation it is false. It concerns your fame in the mathematics to look about how to justify these equations which are the premises to your conclusion following, namely, that the proportion arising is every where greater than sub-triple, or a third; and that the excess (that is, the excess above subtriple) perpetually decreaseth as the number of terms is augmented, as here 1
6 1
12 1
18 1
24 1
30, &c. which I will show you plainly is false.

But first I wonder why you were so angry with me for saying you made proportion to consist in the quotient, as to tell me it was abominably false, and to justify it, cite your own words penes quotientem; do not you say here, the proportion is everywhere greater than subtriple, or 1
3? And is not 1
3 the quotient of 1 divided by 3? You cannot say in this place that penes is understood; for if it were expressed you would not be able to proceed.

But I return to your conclusion, that the excess of the proportion of the increasing quantities above the third part of so many times the greatest, decreaseth, as 1
6 1
12 1
18 1
24 1
30, &c. For by this account in this row 0 + 1
1 + 1 = 1
2 where the quantity above exceeds the third part of the quantities below by 1
3, you make 1
3 equal to 1
6, which you do not mean. It may be said your meaning is, that the proportion of 1 to the subtriple of 2 which is 2
3, exceedeth what? I cannot imagine what, nor proceed further where the terms be but two. Let us therefore take the second row, that is, 0 + 1 + 4
4 + 4 + 4 = 5
12. The 370sum above is 5, the sum below is 12, the third part whereof is 4; if you mean, that the proportion of 5 to 4 exceeds the proportion of 4 to 12 (which is subtriple) by 1
12, you are out again. For 5 exceeds 4 by unity, which is 12
12. I do not think you will own such an equation as 12
12 = 1
12 Therefore I believe you mean (and your next proposition assures me of it), that the proportion of 5 to 4 exceeds subtriple proportion by the proportion of 1 to 12; if you do so, you are yet deceived.

For if the proportion of 5 to 4 exceeds subtriple proportion by the proportion of 1 to 12, then subtriple proportion, that is, of 4 to 12 added to the proportion of 1 to 12 must make the proportion of 5 to 4. But if you look on these quantities, 4, 12, 144, you will see, and must not dissemble, that the proportion of 4 to 12 is subtriple, and the proportion of 12 to 144 is the same with that of 1 to 12. Therefore by your assertion it must be as 5 to 4 so 4 to 144, which you must not own.

And yet this is manifestly your meaning, as appeareth in these words: “Ut sit rationis provenientis excessus supra subtriplam ea quam habet unitas ad sextuplum numeri terminorum post 0, adeoque,” which cannot be rendered in English, nor need to be. For you express yourself in the twentieth proposition very clearly; I noted it only that you may be more merciful hereafter to the stumblings of a hasty pen. For excessus ea quam does not well, nor is to be well excused by subauditur ratio. Your twentieth proposition is this:

“Si proponatur series quantitatum in duplicata ratione arithmetice proportionalium (sive juxta 371seriem numerorum quadraticorum) continue crescentium, a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem totidem maximæ æqualium subtriplam superabit; eritque excessus ea ratio quam habet unitas ad sextuplum numeri terminorum post 0, sive quam habet radix quadratica termini primi post 0 ad sextuplum radicis quadraticæ termini maximi.”

That is, if there be propounded a row of quantities in duplicate proportion of arithmetically-proportionals (or according to the row of square numbers) continually increasing, and beginning with a point or O. The proportion of that row to a row of so many equals to the greatest, shall be greater than subtriple proportion, and the excess shall be that proportion which unity hath to the sextuple of the number of terms after 0, or the same which the square root of the first number after 0, hath to the sextuple of the square root of the greatest.

For proof whereof you have no more here than patet ex præcedentibus; and no more before but adeoque. You do not well to pass over such curious propositions so slightly; none of the ancients did so, nor, that I remember, any man before yourself. The proposition is false, as you shall presently see.

Take, for example, any one of your rows: as 0 + 1 + 4
4 + 4 + 4. By this proportion of yours 1 + 4, which makes 5, is to 12 in more than subtriple proportion; by the proportion of 1 to the sextuple of 2 which is 12. Put in order these three quantities 5, 4, 12, and you must see the proportion of 5 to 12 is greater than the proportion of 4 to 12, that is, subtriple proportion, by the proportion of 5 to 4. 372But by your account the proportion of 5 to 4 is greater than that of 4 to 12 by the proportion of 1 to 12. Therefore, as 5 to 4 so is 1 to 12, which is a very strange paradox.

After this you bring in this consectary: “Cum autem crescente numero terminorum excessus ille supra rationem subtriplam continue minuatur, ut tandem quovis assignabili minor evadat (ut patet) si in infinitum producatur, prorsus evaniturus est. Adeoque.”

That is, seeing as the number of terms increaseth, that excess above subtriple proportion continually decreaseth, so as at length it becomes less than any assignable (as is manifest) if it be produced infinitely, it shall utterly vanish, and so. And so what?

Sir, this consequence of yours is false. For two quantities being given, and the excess of the greater above the less, that excess may continually be decreased, and yet never quite vanish. Suppose any two unequal quantities differing by more than an unit, as 3 and 6, the excess 3, let 3 be diminished, first by an unit, and the excess will be 2, and the quantities will be 3 and 5; 5 is greater than 4, the excess 1. Again, let 1 be diminished and made 1
2, the excess 4 and the quantities 3 and 41
2, 41
2 is yet greater than 4. Again diminish the excess to 1
4, the quantities will be 3 and 41
4, yet still 41
4 is greater than 4. In the same manner you may proceed to 1
8 1
16 1
32, &c. infinitely; and yet you shall never come within an unit (though your unit stand for 100 miles) of the lesser quantity propounded 3, if that 3 stands for 300 miles. The excesses above subtriple proportion do not decrease 373in the manner you say it does, but in the manner which I now shall show you.

In the first row (0 + 1)/(1 + 1) a third of the quantities below is 2
3, set in order these three quantities 1 2
9 2
3. The first is 1, equal to the sum above, the last is 2
3, equal to the subtriple of the sum below. The middlemost is 2
9 subtriple to the last quantity 2
3. The excess of the proportion of 1 to 2
3 above the subtriple proportion of 2
9 to 2
3 is the proportion of 1 to 2
9 that is of 9 to 2, that is, of 18 to 4.

Secondly, in the second row, which is 0 + 1 + 4
4 + 4 + 4, a third of the sum below is 4, the sum above is 5. Set in order these quantities, 1, 5, 4, 12. There the proportion of 15 to 12 is the proportion of 5 to 4. The proportion of 4 to 12 is subtriple; the excess is the proportion of 15 to 4, which is less than the proportion of 18 to 4, as it ought to be; but not less by the proportion of 1
6 to 1
12 as you would have it.

Thirdly, in the third row, which is 0 + 1 + 4 + 9
9 + 9 + 9 + 9. A third of the sum below is 12, the sum above is 14. Set in order these quantities, 42, 4, 12. There the proportion of 42 to 12 is the same with that of 14 to 4. And the proportion of 4 to 12 subtriple, less than the former excess of 15 to 4. And so it goes on decreasing all the way in this manner, 18 to 4, 15 to 4, 14 to 4, &c. which differs very much from your 1 to 6, 1 to 12, 1 to 18, &c. and the cause of your mistake is this: you call the twelfth part of twelve 1
12, and the eighteenth part of thirty-six you call 1
18, and so of the rest. But what need of all those equations in symbols, to show that the proportion decreases; is there 374any man can doubt, but that the proportion of 1 to 2 is greater than that of 5 to 12, or that of 5 to 12 greater than that of 14 to 36, and so on continually forwards; or could you have fallen into this error, unless you had taken, as you have done in very many places of your Elenchus, the fractions 1
6 and 1
12, &c. which are the quotients of 1 divided by 6 and 12, for the very proportions of 1 to 6 and 1 to 12. But notwithstanding the excess of the proportions of the increasing quantities, to subtriple proportion decrease, still, as the number of terms increaseth, and that what proportions soever I shall assign, the decrement will in time (in time, I say, without proceeding in infinitum) produce a less, yet it does not follow that the row of increasing quantities shall ever be equal to the third part of the row of so many equals to the last or greatest. For it is not, I hope, a paradox to you, that in two rows of quantities the proportion of the excesses may decrease, and yet the excesses themselves increase, and do perpetually.

For in the second and third rows, which are 0 + 1 + 4 =  5
4 + 4 + 4 = 12 and 0 + 1 + 4 + 9 = 14
9 + 9 + 9 + 9 = 36 5 exceeds the third part of 12 by a quarter of the square of 4, and 14 exceeds the third part of 36 by 2 quarters of the square of 4, and proceeding on, the sum of the increasing quantities where the terms are 5 (which sum is 30) exceedeth the third part of those below, (those below are 80, and their third part 262
3) by 3 quarters and 1
2 a quarter of the square of 4, and when the terms are 6, the quantities above will exceed the third part of them below by 5 quarters of the square of 4. Would you have men believe, 375that the further they go, the excess of the increasing quantities above the third part of those below shall be so much the less? And yet the proportions of those above, to the thirds of those below, shall decrease eternally; and therefore your twenty-first proposition is false, namely this:

“Si proponatur series infinita quantitatum in duplicata ratione arithmetice proportionalium (sive juxta seriem numerorum quadraticorum), continue crescentium a puncto sive 0 inchoatarum; erit illa ad seriem totidem maximæ æqualium, ut 1 ad 3.”

That is, if an infinite row of quantities be propounded in duplicate proportion of arithmetically-proportionals (or according to the row of quadratic numbers), continually increasing and beginning from a point or 0; that row shall be to the row of as many equals to the greatest, as 1 to 3. This is false, ut patet ex præcedentibus; and, consequently, all that you say in proof of the proportion of your parabola to a parallelogram, or of the spiral (the true spiral) to a circle is in vain.

But your spiral puts me in mind of what you have under-written to the diagram of your proposition 5. The spiral, in both figures, was to be continued whole to the middle, but, by the carelessness of the graver, it is in one figure manca, in the other intercisa.

Truly, Sir, you will hardly make your reader believe that a graver could commit those faults without the help of your own copy, nor that it had been in your copy, if you had known how to describe a spiral line then as now. This I had not said, though truth, but that you are pleased to say, 376though not truth, that I attributed to the printer some faults of mine.

I come now to the thirty-ninth proposition, which is this:

“Si proponatur series quantitatum in triplicata ratione arithmetice proportionalium (sive juxta seriem numerorum cubicorum), continue crescentium a puncto sive 0 inchoatarum (puta ut 0, 1, 8, 27, etc.), propositum sit inquirere quam habeat series illa rationem ad seriem totidem maximæ æqualium:

“Fiat investigatio per modum inductionis (ut in prop. 1, et prop. 19):

Eritque

0 + 1 = 1
1 + 1 = 2 = 2
4 = 1
4 + 1
4

0 + 1 + 8 =  9
8 + 8 + 8 = 24 = 1
4 + 1
8

0 +  1 +  8 + 27 =  36
27 + 27 + 27 + 27 = 108 = 4
12 = 1
4 + 1
12

“Ratio proveniens est ubique major quam subquadrupla, sive 1
4. Excessus autem perpetuo decrescit, pro ut numerus terminorum augetur, puta 1
4 1
8 1
12 1
16 etc. Aucto nimirum fractionis denominatore sive consequente rationis in singulis locis numero quaternatio, ut patet, ut sit rationis provenientis excessus supra subquadruplam ea quam habet unitas ad quadruplum numeri terminorum post 0 adeoque.”

That is, if a row of quantities be propounded in triplicate proportion of arithmetically proportionals (or according to the row of cubic numbers), continually increasing, and beginning from a point or 0, as 0, 1, 8, 27, 64, &c., let it be propounded to 377inquire, what proportion that row hath to a row of as many equals to the greatest.

Be it sought by way of induction, as in proposition 1 and 19.

The proposition arising is everywhere greater than subquadruple, or 1
4, and the excess perpetually decreaseth as the number of terms increaseth, as 1
4 1
8 1
12 1
16 1
20 &c. The denominator of the fraction, or consequent of the proportion, being in every place augmented by the number 4, as is manifest, so that the excess of the arising proportion above subquadruple is the same with that which an unit hath to the quadruple of the number of the terms after 0, and so. Here are just the same faults which are in proposition 19.

For, if 0
1 be a fraction, and 1
1 be a fraction, and 1
2 be another fraction, then this equation 0 + 1 = 1
1 + 1 = 2 is false. For this fraction 0
1 is equal to 0; and, therefore, we have 1
1 = 1
2, that is, the whole equal to half. But perhaps you do not mean them fractions, but proportions; and, consequently, that the proportion of 0 to 1, and of 1 to 1, compounded by addition (I say by addition, not that I, but that you think there is a composition of proportions by multiplication, which I shall show you anon is false), must be equal to the proportion of 1 to 2, which cannot be. For the proportion of 0 to 1 is infinitely little, that is, none at all; and, consequently, the proportion of 1 to 1 is equal to the proportion of 1 to 2, which is again absurd. There is no doubt but the whole number of 0 + 1 is equal to 1, and the whole number of 1 + 1 equal to 2. But, reckoning them as you do, not for 378whole numbers, but for fractions or proportions, the equations are false.

Again, your second equation, 2
4 = 1
4 + 1
4, though meant of fractions, that is, of quotients, it be true, and serve nothing to your purpose, yet, if it be meant of proportions, it is false. For the proportion of 1 to 4, and of 1 to 4 being compounded, are equal to the proportion of 1 to 16, and so you make the proportion of 2 to 4 equal to the proportion of 1 to 16, where, as it is but subquaduplicate, as you call it, or the quarter of it, as I call it. And, in the same manner, you may demonstrate to yourself the same fault in all the other rows of how many terms soever they consist. Therefore, you may give for lost this thirty-ninth proposition, as well as all the other thirty-eight that went before. As for the conclusion of it, which is, that the excess of the arising proportion, &c. They are the words of your fortieth proposition, where you express yourself better, and make your error more easy to be detected.

The proposition is this:

“Si proponatur series quantitatum in triplicata ratione arithmetice proportionalium (sive juxta seriem numerorum cubicorum) continue crescentium a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem totidem maximæ æqualium subquadruplam superabit; eritque excessus ea ratio quam habet unitas ad quadruplum numeri terminorum post 0; sive quam habet radix cubica termini primi post 0 ad quadruplum radicis cubicæ termini maximi. Patet ex præcedente.

“Quum autem crescente numero terminorum excessus ille supra rationem subquadruplam ita 379continuo minuatur, ut tandem quolibet assignabili minor evadat, ut patet, si in infinitum procedatur, prorsus evaniturus est, adeoque.

“Patet ex propositione præcedente.”præcedente.”

That is, if a row of quantities be propounded in triplicate proportion of arithmetically proportionals (or according to the row of cubic numbers), continually increasing, and beginning at a point or 0; the proportion which that row hath to a row of as many equals to the greatest, is greater than subquadruple proportion; and the excess is that proportion which one unit hath to the quadruple of the number of terms after 0; or, which the cubic root of the first term after 0 hath to the quadruple of the root of the greatest term.

It is manifest by the precedent propositions.

And, seeing the number of terms increasing, that excess above quadruple proportion doth so continually decrease, as that, at length, it becomes less than any proportion that can be assigned, as is manifest, if the proceeding be infinite, it shall quite vanish. And soAnd so

This conclusion was annexed to the end of your thirty-ninth proposition, as there proved. What cause you had to make a new proposition of it, without other proof than patet ex præcedente, I cannot imagine. But, howsoever, the proposition is false.

For example, set forth any of your rows, as this of fewer terms:

0 +   1 +  8 + 27 =   36
(27 + 27 + 27 + 27 = 108

The row above is 36, the fourth part of the row below is 27. The quadruple of the number of terms after 0 is 12. Then, by your account, 380the proportion of 36 to 108 is greater than subquadruple proportion by the proportion of 1 to 12. For trial whereof, set in order these three quantities, 36, 27, 108. The proportion of 36 (the uppermost row) to 108 (the lowermost row) is compounded by addition of the proportions 36 to 27, and 27 to 108. And the proportion of 36 to 108, exceedeth the proportion of 27 to 108, by the proportion of 36 to 27. But the proportion of 27 to 108 is subquadruple proportion. Therefore, the proportion of 36 to 108 exceedeth subquadruple proportion, by the proportion of 36 to 27. And, by your account, by the proportion of 1 to 12; and, consequently, as 36 to 27, so is 1 to 12. Did you think such demonstrations as these should always pass?

Then, for your inference from the decrease of the proportions of the excess, to the vanishing of the excess itself, I have already showed it to be false; and by consequence that your next proposition, namely, the fortieth, is also false.

The proposition is this:

“Si proponatur series infinita quantitatum in triplicata ratione arithmetice proportionalium (sive juxta seriem numerorum cubicorum), continue crescentium a puncto sive 0 inchoatarum, erit illa ad seriem totidem maximæ æqualium, ut 1 ad 4, patet ex præcedente.”

That is, if there be propounded an infinite row of quantities in triplicate proportion of arithmetically proportionals (or according to the row of cubic numbers), continually increasing, and beginning at a point or 0; it shall be to the row of as many equals to the greatest as 1 to 4. Manifest out of the precedent proposition.

381Even as manifest as that 36, 27, 1, 12, are proportionals. Seeing, therefore, your doctrine of the spiral lines and the spaces is given by yourself for lost, and a vain attempt, your first forty-one propositions are undemonstrated, and the grounds of your demonstrations all false. The cause whereof is partly your taking quotient for proportion, and a point for 0, as you do in the first, sixteenth, and fortieth propositions, and in other places where you say, beginning at a point or 0, though now you deny you ever said either. There be very many places in your Elenchus, where you say both; and have no excuse for it, but that, in one of the places, you say the proportion is penes quotientem, which is to the same or no sense.

Your forty-second proposition is grounded on the fortieth; and therefore, though true, and demonstrated by others, is not demonstrated by you.

Your forty-third is this:

“Pari methodo invenietur ratio seriei infinitæ quantitatum arithmetice proportionalium in ratione quadruplicata, quintuplicata, sextuplicata, etc., arithmetice proportionalium a puncto seu 0 inchoatarum, ad seriem totidem maximæ æqualium. Nempe in quadruplicata erit, ut 1 ad 5; in quintuplicata, ut 1 ad 6; in sextuplicata, ut 1 ad 7. Et sic deinceps.”

That is, by the same method will be found, the proportion of an infinite row of arithmetically proportionals, in proportion quadruplicate, quintuplicate, sextuplicate, &c., of arithmetically proportionals, beginning at a point or 0, to the row of as many equals to the greatest; namely, in quadruplicate, 382it shall be as 1 to 5; in quintuplicate, as 1 to 6; in sextuplicate, as 1 to 7; and so forth.

But by the same method that I have demonstrated, that the propositions 19, 20, 21, 39, 40, and 41, are false: any man else, that will examine the forty-third may find it false also. And, because all the rest of the propositions of your Arithmetica Infinitorum depend on these, they may safely conclude, that there is nothing demonstrated in all that book, though it consist of 194 propositions. The proportions of your parabolocides to their parallelograms are true, but the demonstrations false, and infer the contrary. Nor were they ever demonstrated (at least the demonstrations are not extant) but by me; nor can they be demonstrated, but upon the same grounds, concerning the nature of proportion, which I have clearly laid, and you not understood. For, if you had, you could never have fallen into so gross an error as is this your book of Arithmetica Infinitorum, or that of the angle of contact. You may see by this, that your symbolic method is not only not at all inventive of new theorems, but also dangerous in expressing the old. If the best masters of symbolics think for all this you are in the right, let them declare it. I know how far the analysis by the powers of the lines extendeth, as well as the best of your half-learnt epistlers, that approve so easily of such analogisms as those, 5, 4, 1, 12, and 36, 27, 1, 12, &c.

It is well for you that they who have the disposing of the professors’ places take not upon them to be judges of geometry. For, if they did, seeing you confess you have read these doctrines in your 383school, you had been in danger of being put out of your place.

When the author of the paper wherein I am called Plagiary, and wherein the honour is taken from you of being the first inventor of these fine theorems, shall read this that I have here written, he will look to get no credit by it; especially if it be Roberval, which methinks it should not be. For he understands what proportion is, better than to make 5 to 4 the same with 1 to 12. Or to make, again, the proportion of 36 to 27 the same with that of 1 to 12; and innumerable disproportionalites that may be inferred from the grounds you go on. But if it be Roberval indeed, that snatches this invention from you, when he shall see this burning coal hanging at it, he will let it fall again, for fear of spoiling his reputation.

But what shall I answer to the authority of the three great mathematicians that sent you those encomiastic letters. For the first, whom you say I use to praise, I shall take better heed hereafter of praising any man for his learning whilst he is young, further than that he is in a good way. But it seems he was in too ready a way of thinking very well of himself, as you do of yourself. For the muddiness of my brain I must confess it; but, Sir, ought not you to confess the same of yours? No, men of your tenets use not to do so. He wonders, say you, you thought it worth the while to foul your fingers about such a piece. It is well; every man abounds in his own sense. If you and I were to be compared by the compliments that are given us in private letters, both you and your complimentors would be out of countenance; which compliments, 384besides that which has been printed and published in the commendations of my writings, if it were put together, would make a greater volume than either of your libels. And truly, Sir, I had never answered your Elenchus as proceeding from Dr. Wallis, if I had not considered you also as the minister to execute the malice of that sort of people that are offended with my Leviathan.

As for the judgment of that public Professor that makes himself a witness of the goodness of your geometry, a man may easily see by the letter itself that he is a dunce. And for the English person of quality whom I know not, I can say no more yet than I can say of all three, that he is so ill a geometrician, as not to detect those gross paralogisms as infer that 5 to 4 and 1 to 12 are the same proportion. He came into the cry of those whom your title had deceived.

And now I shall let you see that the composition of proportion by multiplication, as it is in the fifth definition of the sixth element, is but another way of adding proportions one to another. Let the proportions be of 2 to 3, and of 4 to 5. Multiply 2 into 4 and 3 into 5, the proportion arising is of 8 to 15. Put in order these three quantities, 8, 12, 15. The proportion therefore of 8 to 15, compounded of the proportions of 8 to 12, (that is, of 2 to 3) and of 12 to 15, that is, of 4 to 5 by addition. Again, let the proportion be of 2 to 3, and of 4 to 5, multiply 2 into 5 and 3 into 4, the proportions arising is of 10 to 12. Put in order these three numbers, 10, 8, 12. The proportion 10 to 12 is compounded of the proportions of 10 to 8, that is of 5 to 4, and of 8 to 12, that is, of 2 to 3 by addition. I wonder you know not this.

385I find not any more clamour against me for saying the proportion of 1 to 2 is double to that of 1 to 4.

Your book, you speak of, concerning proportion against Meibomius is like to be very useful when neither of you both do understand what proportion is.

You take exceptions, as that I say, that Euclid has but one word for double and duplicate; which nevertheless was said very truly, and that word is sometimes διπλάσιος and sometimes διπλάσιων. And you think you have come off handsomely with asking me whether διπλάσιος and διπλασίωνδιπλασίων be one word.

Nor are you now of the mind you were, that a point is not quantity unconsidered, but that in an infinite series it may be safely neglected. What is neglected but unconsidered.

Nor do you any more stand to it, that the quotient is the proportion. And yet were these the main grounds of your Elenchus.

But you will say, perhaps, I do answer to the defence you have now made in this your School Discipline: ’tis true. But ’tis not because you answer never a word to my former objections against these propositions 19, 89; but because you do so shift and wriggle, and throw out ink, that I cannot perceive which way you go, nor need I, especially in your vindication of your Arithmetica Infinitorum. Only I must take notice that in the end of it, you have these words, “Well, Arithmetica Infinitorum is come off clear” You see the contrary. For sprawling is no defence.

It is enough to me that I have clearly demonstrated both before sufficiently, and now again 386abundantly, that your book of Arithmetica Infinitorum is all nought from the beginning to the end, and that thereby I have effected that your authority shall never hereafter be taken for a prejudice. And, therefore, they that have a desire to know the truth in the questions between us, will henceforth, if they be wise, examine my geometry, by attentive reading me in my own writings, and then examine, whether this writing of yours confute or enervate mine.

There is in my fifth lesson a proposition, with a diagram to it, to make good, I dare say, at least against you, my twentieth chapter concerning the dimension of a circle. If that demonstration be not shown to be false, your objections to that chapter, though by me rejected, come to nothing. I wonder why you pass it over in silence. But you are not, you say, bound to answer it. True, nor yet to defend what you have written against me.

Before I give over the examination of your geometry, I must tell you that your words, (p. 101 of your School Discipline), against the first corollary are untrue.

Your words are these: “you affirm that the proportion of the parabola A B I to the parabola A F K is triplicate to the proportion of the time A B to A F, as it is in the English.” This is not so. Let the reader turn to the place and judge. And going on you say, “or of the impetus B I to F K as it is in the Latin.” Nay, as it is in the English, and the other in the Latin. It is but your mistake; but a mistake is not easily excused in a false accusation.

Your exception to my saying, “that the differences 387of two quantities is their proportion,” (when they differ, as the no difference, when they be equal), might have been put in amongst other marks of your not sufficiently understanding the Latin tongue. Differre and differentia differ no more than vivere and vita, which is nothing at all, but as the other words require that go with them, which other words you do not much use to consider. But differre and the quantity by which they differ, are quite of another kind. Differre (τὸ διαφέρειν, τὸ ὑπερέχειν) differing, exceeding, is not quantity, but relation. But the quantity by which they differ is always a certain and determined quantity, yet the word differentia serves for both, and is to be understood by the coherence with that which went before. But I had said before, and expressly to prevent cavil, that relation is nothing but a comparison, and that proportion is nothing but relation of quantities, and so defined them, and therefore I did there use the word differentia for differing, and not for the quantity which was left by subtraction. For a quantity is not a differing. This I thought the intelligent reader would of himself understand without putting me, instead of differentia, to use (as some do, and I shall never do) the mongrel word τὸ differre. And whereas in one only place for differre ternario I have writ ternarius, if you had understood what was clearly expressed before, you might have been sure it was not my meaning, and therefore the excepting against it was either want of understanding, or want of candour, choose which you will.

You do not yet clear your doctrine of condensation and rarefaction. But I believe you will by 388degrees become satisfied that they who say the same numerical body may be sometimes greater, sometimes less, speak absurdly, and that condensation and rarefaction here, and definitive and circumscriptive, and some other of your distinctions elsewhere are but snares, such as school divines have invented

to entangle shallow wits.

And that that distinction which you bring here, “that it is of the same quantity while it is in the same place, but it may be of a different quantity when it goes out of its place,” (as if the place added to, or took any quantity from the body placed), is nothing but mere words. It is true that the body which swells changeth place, but it is not by becoming itself a greater body, but by admixtion of air or other body, as when water riseth up in boiling, it taketh in some parts of air. But seeing the first place of the body is to the body equal, and the second place equal to the same body, the places must also be equal to one another, and consequently the dimensions of the body remain equal in both places.

Sir, when I said that such doctrine was taught in the Universities, I did not speak against the Universities, but against such as you. I have done with your geometry, which is one στιγμὴ.

As for your eloquence, let the reader judge whether yours or mine be the more muddy, though I in plain scolding should have outdone you, yet 389I have this excuse which you have not, that I did but answer your challenge at that weapon which you thought fit to choose. The catalogue of the hard language which you put in at pages 3 and 4 of your School Discipline, I acknowledge to be mine, and would have been content you had put in all. The titles you say I give you of fools, beasts, and asses, I do not give you, but drive back upon you, which is no more than not to own them; for the rest of the catalogue, I like it so well as you could not have pleased me better than by setting those passages together to make them more conspicuous; that is all the defence I will make to your accusations of that kind.

And now I would have you to consider whether you will make the like defence against the faults that I shall find in the language of your School Discipline.

I observe, first, the facetiousness of your title-page, “Due correction for Mr. Hobbes, or School Discipline, for not saying his Lessons right.” What a quibble is this upon the word lesson; besides, you know it has taken wind; for you vented it amongst your acquaintance at Oxford then when my Lessons were but upon the press. Do you think if you had pretermitted that piece of wit, the opinion of your judgment would have been ere the less? But you were not content with this, but must make this metaphor from the rod to take up a considerable part of your book, in which there is scarce anything that yourself can think wittily said besides it. Consider also these words of yours: “It is to be hoped that in time you may come to learn the language, for you be come to great A already.” And presently after, 390“were I great A, before I would be willing to be so used, I should wish myself little a a hundred times.” Sir, you are a doctor of divinity and a professor of geometry, but do not deceive yourself, this does not pass for wit in these parts, no, nor generally at Oxford; I have acquaintance there that will blush at the reading it.

Again, in another place you have these words: “Then you catechize us, ‘what is your name? Are you geometricians? Who gave you that name,’” &c. Besides in other places such abundance of the like insipid conceits, as would make men think, if they were no otherwise acquainted with the University but by reading your books, that the dearth there of salt were very great. If you have any passage more like to salt than these are (excepting now and anon) you may do well to show it to your acquaintance, lest they despise you; for, since the detection of your geometry, you have nothing left you else to defend you from contempt. But I pass over this kind of eloquence, and come to somewhat yet more rural.

Page 27, line 1, you say I have given Euclid his lurry. And again, page 129, line 11, “and now he is left to learn his lurry.” I understand not the word lurry. I never read it before, nor heard it, as I remember, but once, and that was when a clown threatening another clown said he would give him such a lurry come poop, &c. Such words as these do not become a learned mouth, much less are fit to be registered in the public writings of a doctor of divinity. In another place you have these words, “just the same to a cow’s thumb,” a pretty adage.

391Page 2, “But prithee tell me.” And again, page 95, “prithee tell me, why dost thou ask me such a question,” and the like in many other places.

You cannot but know how easy it is and was for me to have spoken to you in the same language. Why did I not? Because I thought that amongst men that were civilly bred it would have redounded to my shame, as you have cause to fear that this will redound to yours. But what moved you to speak in that manner? Were you angry? If I thought that the cause, I could pardon it the sooner, but it must be very great anger that can put a man, that professeth to teach good manners, so much out of his wits as to fall into such a language as this of yours. It was perhaps an imagination that you were talking to your inferior, which I will not grant you, nor will the heralds, I believe, trouble themselves to decide the question. But, howsoever, I do not find that civil men use to speak so to their inferiors. If you grant my learning but to be equal to yours, (which you may certainly do without very much disparaging of yourself abroad in the world), you may think it less insolence in me to speak so to you in respect of my age, than for you to speak so to me in respect of your young doctorship. You will find that for all your doctorship, your elders, if otherwise of as good repute as you, will be respected before you. But I am not sure that this language of yours proceeded from that cause; I am rather inclined to think you have not been enough in good company, and that there is still somewhat left in your manners for which the honest youths of Hedington and Hincsey may compare with you for good language, as great a doctor as you are.

392For my verses of the Peak, though they be as ill in my opinion as I believe they are in yours, and made long since, yet they are not so obscene as that they ought to be blamed by Dr. Wallis. I pray you, sir, whereas you have these words in your School Discipline, page 96, “unless you will say that one and the same motion may be now and anon too.” What was the reason you put these words, now and anon too, in a different character, that makes them to be more taken notice of? Do you think that the story of the minister that uttered his affection (if it be not a slander) not unlawfully but unseasonably, is not known to others as well as to you? What needed you then, when there was nothing that I had said could give the occasion, to use those words; there is nothing in my verses that do olere hircum so much as this of yours. I know what good you can receive by ruminating on such ideas, or cherishing of such thoughts. But I go on to other words of mine by you reproached, “you may as well seek the focus of the parabola of Dives and Lazarus,” which you say is mocking the Scripture; to which I answer only, that I intended not to mock the Scripture, but you, and that which was not meant for mocking was none. And thus you have a second στιγμὴ.

I come now to the comparison of our Grammar and Critiques. You object first against the signification I give of στιγμὴ, and say thus: “What should come into your cap (that, if you mark it, in a man that wears a square cap to one that wears a hat, is very witty) to make you think that στιγμὴ signifies 393a mark or brand with a hot iron? I perceive where the business lies, it was στίγμα run in your mind when you talked of στιγμὴ; and because the words are somewhat alike you jumble them both together.” Sir, I told you once before, you presume too much upon your first cogitations. Aristophanes, in Ranis, Act. V. Scen. 5,

The old commentator upon the word στίξας saith thus, ϛίξας ἀντὶ τοῦ ϛιγματίσας, ἠν γάρ ξένος. That is, στίξας for ϛιγματίσας, for he (Adimantus) was not a citizen. I hope the commentator does not here mock Aristophanes for jumbling ϛίξας and ϛιγματίσας together, for want of understanding Greek. No, ϛίξας and στιγματίσας signify the same, save that for branding I seldom read ϛιγματίσας but ϛίξας. For ϛίγμα does no more signify a brand with a hot iron, than ϛιγμὴ a point made also with a hot iron. They have both one common theme ϛίζω, which does not signify pungo, nor interpungo, nor inuro, for all your Lexicon, but notam imprimere, or pungendo notare, without any restriction to burning or punching. It is therefore no less proper to say that ϛιγμὴ is a mark with a hot iron, than to say the same of στίγμα. The difference is only this, that when they marked a slave, or a rascal, as you are not ignorant is usually done here at the assizes in the hand or shoulder with a hot iron, they called that ϛίγμα, not for the burning, but for the mark. And as it would have been called ϛίγμα that was imprinted on a slave, though made by staining or incision, so it is ϛιγμὴ, though done with a hot iron. And therefore there was no jumbling of those two words together, as for want of reading Greek authors, and 394by trusting too much to your dictionaries, which you say are proofs good enough for such a business, you were made to imagine. The use I have made thereof was to show that a point, both by the word ΣημεῖονΣημεῖον in Euclid, and by the word στιγμὴ in some others, was not nothing, but a visible mark, the ignorance whereof hath thrown you into so many paralogisms in geometry.

But do you think you can defend your Adducis Malleum as well as I have now defended my ϛιγμὴ? You have brought, I confess, above a hundred places of authors, where there is the word duco, or some of its compounds, but none of them will justify Adducis Malleum, and, excepting two of those places, you yourself seem to condemn them all, comparing yours with none of the rest but with these two only, both out of Plautus, by you not well understood. The first is in Casina, Act. V. Scen. 2, “Ubi intro hanc novam nuptam deduxi, via recta, clavem abduxi;” which you, presently presuming of your first thoughts, a peculiar fault to men of your principles, assure yourself is right. But if you look on the place as Scaliger reads it, cited by the commentator, you will find it should be obduxi, and that clavis is there used for the bolt of the lock. Besides, he bolted it within. Whither then could he carry away the key? The place is to be rendered thus, when I had brought in this new bride I presently locked the door, and is this as bad every whit as Adducis Malleum? The second place is in Amphytruo, Act. I. Scen. 1, “Eam (cirneam), ut a matre fuerat natum, plenam vini eduxi meri,” which you interpret I brought out a flagon of wine, unlearnedly. They are 395the words of Mercury transformed into Sosia. And to try whether Mercury were Sosia or not, Sosia asked him where he was and what he did during the battle; to which Mercury answered, who knew where Sosia then was and what he did, I was in the cellar, where I filled a cirnea, and brought it up full of wine, pure as it came from its mother. By the mother of the wine meaning the vine, and alluding to the education of children, for ebibi said eduxi, and with an emphasis in meri, because cirnea (from Κφνάω, misceo) was a vessel wherein they put water to temper to their wine. Intimating that though the vessel was cirnea, yet the wine was merum. This is the true sense of the place; but you will have eduxi to be, I brought out, though he came not out himself. You see, sir, that neither this is so bad as Adducis Malleum.

But suppose out of some one place in some one blind author you had paralleled your Adducis Malleum, do you think it must therefore presently be held for good Latin? Why more than learn his lurry must be therefore thought good English a thousand years hence, because it will be read in Dr. Wallis’s long-lived works. But how do you construe this passage (1 Tim. ii. 15) of the Greek Testament: Σωθήσεται δὲ διὰ τῆς τεκνογονίας, ἐὰν μείνωσιν ἐν πίστει? You construe it thus: she shall be saved notwithstanding child-bearing, if (the woman) remain in the faith. Is child-bearing any obstacle to the salvation of women? You might as well have translated the first verse of the fifth of Romans in this manner, Being then justified by faith, we have peace with God notwithstanding our Lord Jesus Christ. I let pass your not finding in τεκνογονίας, as good a 396grammarian as you are, a nominative case to μείνωσιν. If you had remembered the place, 1 Pet. iii. 20, εσώθησαν δὶ ὑδατος, that is, they were saved in the waters, you would have thought your construction justified then very well; but you had been deceived, for διὰ does not there signify causam, ablationem impedimenti, but transitum; not cause or removing an impediment, but passage. Being come thus far, I found a friend that hath eased me of this dispute; for he showed me a letter written to himself from a learned man, that hath out of very good authors collected enough to decide all the grammatical questions between you and me, both Greek and Latin. He would not let me know his name, nor anything of him but only this, that he had better ornaments than to be willing to go clad abroad in the habit of a grammarian. But he gave me leave to make use of so much of the letter as I thought fit in this dispute, which I have done, and have added it to the end of this writing. But before I come to that, you must not take it ill, though I have done with your School Discipline, if I examine a little some other of your printed writings as you have examined mine; for neither you in geometry, nor such as you in church politics, cannot expect to publish any unwholesome doctrine without some antidotes from me, as long as I can hold a pen. But why did you answer nothing to my sixth Lesson? Because, you say, it concerned your colleague only. No, sir, it concerned you also, and chiefly, for I have not heard that your colleague holdeth those dangerous principles which I take notice of in you, in my sixth Lesson, page 350, upon the occasion of these 397words, not his but yours: “Perhaps you take the whole history of the fall of Adam for a fable, which is no wonder, seeing you say the rules of honouring and worshipping of God are to be taken from the laws.” In answer to which I said thus: “You that take so heinously, that I would have the rule of God’s worship in a Christian commonwealth to be taken from the laws, tell me from whom you would have them taken? From yourself? Why so, more than from me? From the bishops? Right, if the supreme power of the commonwealth will have it so; if not, why from them rather than from me? From a consistory of presbyters themselves, or joined with lay elders, whom they may sway as they please? Good, if the supreme governor of the commonwealth will have it so. If not, why from them rather than from me, or from any man else? They are wiser and learneder than I; it may be so, but it has not yet appeared. Howsoever, let that be granted. Is there any man so very a fool as to subject himself to the rules of other men in those things which do so nearly concern himself, for the title they assume of being wise and learned, unless they also have the sword which must protect them? But it seems you understand the sword as comprehended. If so, do not you then receive the rules of God’s worship from the civil power? Yes, doubtless; and you would expect, if your consistory had that sword, that no man should dare to exercise or teach any rules concerning God’s worship which were not by you allowed.”

This will be thought strong arguing, if you do not answer it. But the truth is, you could say 398nothing against it without too plainly discovering your disaffection to the government. And yet you have discovered it pretty well in your second Thesis, maintained in the Act at Oxford, 1654, and since by yourself published. This Thesis I shall speak briefly to.

You define ministers of the Gospel to be those to whom the preaching of the Gospel by their office is enjoined by Christ. Pray you, first, what do you mean by saying preaching ex officio is enjoined by Christ? Are they preachers ex officio, and afterwards enjoined to preach? Ex officio adds nothing to the definition; but a man may easily see your purpose to disjoin yourself from the state by inserting it.

Secondly, I desire to know in what manner you will be able out of this definition to prove yourself a minister? Did Christ himself immediately enjoin you to preach, or give you orders? No. Who then, some bishop, or minister, or ministers? Yes; by what authority? Are you sure they had authority immediately from Christ? No. How then are you sure but that they might have none? At least, some of them through whom your authority is derived might have none. And therefore if you run back for your authority towards the Apostles’ times but a matter of sixscore years, you will find your authority derived from the Pope, which words have a sound very unlike to the voice of the laws of England. And yet the Pope will not own you. There is no man doubts but that 399you hold that your office comes to you by successive imposition of hands from the time of the Apostles; which opinion in those gentle terms passeth well enough; but to say you derive your authority from thence, not through the authority of the sovereign power civil, is too rude to be endured in a state that would live in peace. In a word, you can never prove you are a minister, but by the supreme authority of the commonwealth. Why then do you not put some such clause into your definition? As thus, ministers of the Gospel are those to whom the preaching of the Gospel is enjoined by the sovereign power in the name of Christ. What harm is there in this definition, saving only it crosses the ambition of many men that hold your principles? Then you define the power of a minister thus: “The power of a minister is that which belongeth to a minister of the Gospel in virtue of the office he holds, inasmuch as he holds a public station, and is distinguished from private Christians. Such as is the power of preaching the Gospel, administering the sacrament, the use of ecclesiastical censures, and ordaining of ministers,” &c.

Again, how will you prove out of this definition that you, or any man else, hath the power of a minister, if it be not given him by him that is the sovereign of the commonwealth? For seeing, as I have now proved, it is from him that you must derive your ministry, you can have no other power than that which is limited in your orders, nor that neither longer than he thinks fit. For if he give it you for the instruction of his subjects in their duty, he may take it from you again whensoever he shall see you instruct them with undutiful 400and seditious principles. And if the sovereign power give me command, though without the ceremony of imposition of hands, to teach the doctrine of my Leviathan in the pulpit, why am not I, if my doctrine and life be as good as yours, a minister as well as you, and as public a person as you are? For public person, primarily, is none but the civil sovereign, and so secondarily, all that are employed in the execution of any part of the public charge. For all are his ministers, and therefore also Christ’s ministers because he is so; and other ministers are but his vicars, and ought not to do or say anything to his people contrary to the intention of the sovereign in giving them their commission.

Again, if you have in your commission a power to excommunicate, how can you think that your sovereign who gave you that commission, intended it for a commission to excommunicate himself? that is, as long as he stand excommunicate, to deprive him of his kingdom. If all subjects were of your mind, as I hope they will never be, they will have a very unquiet life. And yet this has, as I have often heard, been practised in Scotland, when ministers holding your principles had power enough, though no right, to do it.

And for administration of the sacraments, if by the supreme power of the commonwealth it were committed to such of the laity as know how it ought to be done as well as you, they would ipso facto be ministers as good as you. Likewise the right of ordination of ministers depends not now on the imposition of hands of a minister or presbytery, but on the authority of the Christian sovereign, Christ’s immediate vicar and supreme 401governor of all persons and judge of all causes, both spiritual and temporal, in his own dominions, which I believe you will not deny.

This being evident, what acts are those of yours which you call authoritative, and receive not from the authority of the civil power? A constable does the acts of a constable authoritatively in that sense. Therefore you can no otherwise claim your power than a constable claimeth his, who does not exercise his office in the constabulary of another. But you forget that the Scribes and the Pharisees sit no more in Moses’ chair.

You would have every minister to be a minister of the universal Church, and that it be lawful for you to preach your doctrine at Rome; if you would be pleased to try, you would find the contrary. You bring no argument for it that looks like reason. Examples prove nothing, where persons, times, and other circumstances differ; as they differ very much now when kings are Christians, from what they were then when kings persecuted Christians. It is easy to perceive what you aim at.

You would fain have market-day lectures set up by authority, (not by the authority of the civil power, but by the authority of example of the Apostles in the emission of preachers to the infidels), not knowing that any Christian may lawfully preach to the infidels; that is to say, proclaim unto them that Jesus is the Messiah, without need of being otherways made a minister, as the deacons did in the Apostles’ time; nor that many teachers, unless they can agree better, do anything else but prepare men for faction, nay, rather you know it well enough, but it conduces to your end 402upon the market-days to dispose at once both town and country, under a false pretence of obedience to God, to a neglecting of the commandments of the civil sovereign, and make the subject to be wholly ruled by yourselves, wherein you have already found yourselves deceived. You know how to trouble and sometimes undo a slack government, and had need to be warily looked to, but are not fit to hold the reins. And how should you, being men of so little judgment as not to see the necessity of unity in the governor, and of absolute obedience in the governed, as is manifest out of the place of your Elenchus above recited. The doctrine of the duty of private men in a commonwealth is much more difficult, not only than the knowledge of your symbols, but also than the knowledge of geometry itself. How then do you think, when you err so grossly in a few equations, and in the use of most common words, you should be fit to govern so great nations as England, Ireland, and Scotland, or so much as to teach them? For it is not reading but judgment that enables one man to teach another.

I have one thing more to add, and that is the disaffection I am charged withal to the universities. Concerning the Universities of Oxford and Cambridge, I ever held them for the greatest and noblest means of advancing learning of all kinds, where they should be therein employed, as being furnished with large endowments and other helps of study, and frequented with abundance of young gentlemen of good families and good breeding from their childhood. On the other side, in case the same means and the same wits should be employed in the advancing of the doctrines that tend 403to the weakening of the public, and strengthening of the power of any private ambitious party, they would also be very effectual for that; and consequently that if any doctrine tending to the diminishing of the civil power were taught there, not that the Universities were to blame, but only those men that in the universities, either in lectures, sermons, printed books, or theses, did teach such doctrine to their hearers or readers. Now you know very well that in the time of the Roman religion, the power of the Pope in England was upheld principally by such teachers in the universities. You know also how much the divines that held the same principles in Church government with you, have contributed to our late troubles. Can I therefore be justly taxed with disaffection to the universities for wishing this to be reformed? And it hath pleased God of late to reform it in a great measure, and indeed as I thought totally, when out comes this your Thesis boldly maintained to show the contrary. Nor can I yet call this your doctrine the doctrine of the university; but surely it will not be unreasonable to think so, if by public act of the university it be not disavowed, which done, and that as often as there shall be need, there can be no longer doubt but that the universities of England are not only the noblest of all Christian universities, but also absolutely, and of the greatest benefit to this commonwealth that can be imagined, except that benefit of the head itself that uniteth and ruleth all. I have not here particularized at length all the ill consequences that may be deduced from this Thesis of yours, because I may, when further provoked, have somewhat to say that is new. So much for the third ϛιγμὴ.

Mr. Hobbes hath these words: “Longitudinem percursam motu uniformi, cum impetu ubique ipsi B D æquali.” Dr. Wallis saith cum were better out, unless you would have impetus to be only a companion, not a cause. Mr. Hobbes answered it was the ablative case of the manner. The truth is the ablative case of the manner and cause both, may be used with the conjunction cum, as may be justified. Cicero in Lib. II. De Nat. Deorum: “Moliri aliquid cum labore operoso ac molesto;” and in his oration for Cæcina: “De se autem hoc prædicat, Antiocho Ebulii servo imperasse ut in Cæcinam advenientem cum ferro invaderet.” Let us see then what Dr. Wallis objects against Tully, where a casualty is imported, though we may use with in English, yet not cum in Latin; to kill with a sword, importing this to have an instrumental or causal influence, and not only that it hangs by the man’s side whilst some other weapon is made use of, is not in Latin occidere cum gladio, but gladio occidere. This shows that the Doctor hath not forgot his grammar, for the subsequent examples as well as this rule are borrowed thence. But yet he might have known that great personages have never confined themselves to this pedantry, but have chosen to walk in a greater latitude. Most of the elegancies and idioms of every language are exceptions to his grammar. But since Mr. Hobbes saith it is the ablative case of 405the manner, there is no doubt it may be expressed with cum. The Doctor in the meantime knew no more than what Lilly had taught him; Alvarez would have taught him more; and Vossius in his book, De Constructione, cap. XLVII. expressly teacheth, “Ablativos causæ, instrumenti, vel modi, non a verbo regi sed a præpositione omissa, a vel ab, de, e vel ex, præ, aut cum, ac præpositiones eas quandoque exprimi nisi quod cum ablativis instrumenti haud temere invenias;” and afterwards he saith, “non timere imitandum.” If this be so, then did Mr. Hobbes speak grammatically, and with Tully, but not usually. And might not one retort upon the Doctor, that Vossius is as great a critic as he?

His next reflection is upon prætendit scire, this he saith is an Anglicism. If this be all his accusation, upon this score we shall lose many expressions that are used by the best authors, which I take to be good Latinisms, though they be also Anglicisms, the latter being but an imitation of the former. The Doctor therefore was too fierce to condemn upon so general an account, that which was not to have been censured for being an Anglicism, unless also it had been no Latinism. Mr. Hobbes replies, that the printer had omitted se. He saith, this mends the matter a little. It is very likely, for then it is just such another Anglicism as that of Quintilian: “Cum loricatus in foro ambularet, prætendebat se id metu facere.” The Doctor certainly was very negligent, or else he could not have missed this in Robert Stephen. Or haply he was resolved to condemn Quintilian for this and that other Anglicism, “Ignorantia prætendi non potest,” as all those that have used 406prætendo, which are many and as good authors as Dr. Wallis, that makes his own encomiasts (not an Englishman amongst them) to write Anglicisms.

Then he blames “Tractatus hujus partis tertiæ, in qua motus et magnitudo per se et abstracte consideravimus, terminum hic statuo.” Here I must confess the exception is colourable, yet I can parallel it with the like objection made by Erasmus against Tully, out of whom Erasmus quotes this passage: “Diutius commorans Athenis, quoniam venti negabant solvendi facultatem, erat animus ad te scribere;” and excuses it thus, that Tully might have had at first in his thoughts volebam or statuebam, which he afterwards relinquished for erat animus, and did not remember what he had antecedently written, which did not vary from his succeeding thoughts, but words. And this excuse may pass with any who know that Mr. Hobbes values not the study of words, but as it serves to express his thoughts, which were the same whether he wrote in qua motus et magnitudo per se at abstracte considerati sunt or consideravimus. And if the Doctor will make this so capital, he must prove it voluntary, and show that it is greater than what is legible in the puny letter of his encomiast, whom he would have to be beyond exception.

Now follows his ridiculous apology for adducis malleum, ut occidas muscam. The cause why he did use that proverb, of his own phrasing, was this. Mr. Hobbes had taken a great deal of pains to demonstrate what Dr. Wallis thought he could have proved in short; upon this occasion he objects, adducis malleum ut occidas muscam, which I shall suppose he intended to English thus, you bring 407a beetle to kill a fly. Mr. Hobbes retorted, that adduco was not used in that sense. The Doctor vindicates himself thus: duco, deduco, reduco, perduco, produco, &c. signify the same thing, ergo, adduco may be used in that sense; which is a most ridiculous kind of arguing, where we are but to take up our language from others, and not to coin new phrases. It is not the grammar that shall secure the Doctor, nor weak analogies, where elegance comes in contest. To justify his expression he must have showed it usu tritum, or alleged the authority of some author of great note for it. I have not the leisure to examine his impertinent citations about those other compounds, nor yet of that simple verb duco; nay, to justify his saying he hath not brought one parallel example. He talks indeed very high, that duco, with its compounds, is a word of a large signification, and amongst the rest to bring, fetch, carry, &c. is so exceeding frequent in all authors, Plautus, Terence, Tully, Cæsar, Tacitus, Pliny, Seneca, Virgil, Horace, Ovid, Claudian, &c. that he must needs be either maliciously blind, or a very stranger to the Latin tongue, that doth not know it, or can have the face to deny it. I read, what will be my doom for not allowing his Latin; yet I must profess I dare secure the Doctor for having read all authors, notwithstanding his assertion, and I hope he will do the like for me. And for those which he hath read, had he brought no better proofs than these, he had, I am sure, been whipped soundly in Westminster School, for his impudence as well as ignorance, by the learned master thereof at present. But I dare further affirm, the Doctor hath not 408read in this point any, but only consulted with Robert Stephen’s Thesaurus Linguæ Latinæ, whence he hath borrowed his allegations in adduco; and for the other, I had not so much idle time as to compare them. And, lest the fact might be discovered, he hath sophisticated those authors whence Stephen cites the expressions, and imposed upon them others. If it be not so, or that the Doctor could not write it right when the copy was right before him, let him tell me where he did ever read in Plautus, adducta res in fastidium. I find the whole sentence in Pliny’s preface to Vespasian (out of whom in the precedent paragraph he cites it) about the middle: alia vero ita multis prodita, ut in factidium sint adducta, which is the very example Stephanus useth, although he doth premise his adducta res in fastidium. Let the Doctor tell where he ever did read in Horace, Ova noctuæ, &c. tædium vini adducunt. Did he, or any else, with the interposition of an &c. make Trochaics? I say, and Stephanus says so, too, that it is in Pliny, lib. xiii. cap. 15, near the end; the whole sentence runs thus: Ebriosis Ova noctuæ per triduum data in vino, tædium ejus adducunt. I doubt not but these are the places he aimed at, although he disguised and minced the quotations; if they be not, I should be glad to augment my Stephanus with his additions.

These things premised, I come to consider the Doctor’s proofs: Res eo adducta est: adducta vita in extremum: adducta res in fastidium: rem ad mucrones et manus adducere: contractares et adducta in augustum: res ad concordiam adduci potest: in ordinem adducerem: adducere febres, 409sitim, tedium vini (all in Robert Stephen) betwixt which and adducere malleum, what a vast difference there is, I leave them to umpire qui terretes et religiosas nacti sunt aures, who are the competent judges of elegancy, and only cast in the verdict of one or two, who are in any place (where the purity of the Latin tongue flourisheth) of great esteem. Losæus, in his Scopæ Linguæ Latinæ, ad purgandam Linguam a barbarie, &c. (would any think that the Doctor’s elegant expression, frequent in all authors, which none but the malicious or ignorant can deny, should suffer so contumelious an expurgation?) Losæus, I say, hath these words: Adferre plerique minus attenti utuntur pro adducere. Quod Plautus, in Pseudolo, insigni exemplo notat.

Satis igitur admonet discriminis inter ducere, reducere, adducere, et abducere, quæ de persona; et ferre, adferre, &c. quæ de re dicuntur. Idem, Demetrium, quem ego novi, adduce: argentum non moror quin feras. Cavendum igitur est ne vulgi more, (let the Doctor mark this, and know that this author is authentic amongst the Ciceronians), adferre de persona, dicamus, sed adducere; licet et hoc de certis quibusdam rebus non inepte dicatur. In this last clause he saith as much as Mr. Hobbes saith, and what the Doctor proves; but, that ever the Doctor brought an example which might resemble adducis malleum, is denied; for I have mentioned already his allegations, 410every one, of adduco. Another author, (a fit antagonist for the elegant Doctor), is the Farrago sordidorum Verborum, joined with the Epitome of L. Valla’s Elegancies. He saith: Accerse, adhuc Petrum, Latine dicitur, pro eo quod pueri dicunt, adfer Petrum. And this may suffice to justify Mr. Hobbes’s exception who proceeded no further than this author to tell the Doctor that adduco was used of animals. But the Doctor replies, this signification is true, but so may the other be also. I say if it never have been used so, it cannot be so, for we cannot coin new Latin words, no more than French or Spanish who are foreigners. Mr. Hobbes was upon the negative, and not to disprove the contrary opinion. If the Doctor would be believed, he must prove it by some example, (which is all the proof of elegancy), and till he do so, not to believe him, it is sufficient not to have cause. But, Doctor Wallis, why not adduco for a hammer as well as a tree? I answer yes, equally for either, and yet for neither. Did ever anybody go about to mock his readers thus solemnly? I do not find, to my best remembrance, any example of it in Stephen, and the Doctor is not wiser than his book; if there be, it is strange the Doctor should omit the only pertinent example, and trouble us with such impertinences for three or four pages. In Stephen there are adducere habenas and adducere lorum, but in a different sense. It is not impossible I may guess at the Doctor’s aim. In Tully de Nat. Deor. as I remember, there is this passage: Quum autem ille respondisset, in agro ambulanti ramulum adductum ut remissus esset, in oculum suum recidisse, where it signifies nothing 411else but to be bent, bowed, pulled back, and in that sense, the hammer of a clock, or that of a smith, when he fetcheth his stroke, may be said adduci. And this, I conceive, the Doctor would have us in the close think to have been his meaning; else, what doth he drive at in these words? “When you have done the best you can, you will not be able to find better words than adducere malleum and reducere, to signify the two contrary motions of the hammer, the one when you strike with it (excellently trivial!) the other when you take it back (better and better), What to do? to fetch another stroke. If any can believe that this was his meaning, I shall justify his Latin, but must leave it to him to prove its sense. If he intended no more, why did he go about to defend the other meaning, and never meddle with this? Which yet might have been proved by this one example of mine? May not, therefore, his own saying be justly retorted upon him in this case, Adducis malleum, ut occidas muscam?

Another exception is, Falsæ sunt, et multa istiusmodi (propositiones). I wish the Doctor could bring so good parallels, and so many, out of any author, for his Adducis malleum, as Tully affords in this case. Take one for all, out of the beginning of his Paradoxes: Animadverti sæpe Catonem, cum in senatu sententiam diceret, Locos graves ex Philosophia tractare, abhorrentes ab hoc usu forensi, et publico, sed dicendo consequi tamen, ut illa etiam populo probabilia viderentur. This is but a Solæcophanes, and hath many precedents more, as in the second book of his Academical Questions, &c.

412I cannot now stay upon each particular passage; I do not see any necessity of tracing the Doctor in all his vagaries. Now, he disallows tanquam diceremus, as if we should say. But why is that less tolerable than tanquam feceris, as if you had done? “It should be quasi, (forsooth!) or ac si, or tanquam si, which is Tully’s own word.” What is tanquam si become but one word? Tanquam si tua res agatur, &c. Good Doctor, leave out Tully and all Ciceronians, or you will for ever suffer for this, and your Adducis malleum. Is not this to put yourself on their verdict when you oppose Mr. Hobbes with Tully? But the Doctor gives his reason. And though he hath had the luck in his Adducis malleum, to follow the first part of that saying, Loquendum cum vulgo, yet now it is, sentiendum cum sapientibus. For tanquam without si signifies but as, not as if. It is pity the Doctor could not argue in symbols too, that so we might not understand him; but suppose all his papers to carry evidence with them, because they are mathematically scratched. How does he construe this:--

So Cœlius, one much esteemed by Cicero, who hath inserted his Epistles into his works, saith, in his fifth Epistle (Tul. Epist. Fam. lib. viii. ep. 5), Omnia desiderantur ab eo tanquam nihil denegatum sit ei quo minus paratissimus esset qui publico negotio præpositus est. But it was not possible the Doctor should know this, it not being in Stephen, where his examples for tanquam si are.

But, the Doctor having pitched upon this criticism, and penned it, somebody, I believe, put him 413in mind of the absurdity thereof; and yet the generous Professor, (who writes running hand and never transcribed his papers, if I am not misinformed), presumed nobody else could be more intelligent than he, who had perused Stephen. He would not retract anything, but subjoins, “That he will allow it as passable, because other modern writers, and some of the ancients, have so used it, as Mr. Hobbes hath done.” I know not what authors the Doctor meant, for, if I am not much mistaken, I do not find any in Stephen. His citation of Columella is not right, (lib. v. cap. 5), nor can I deduce anything thence till I have read the passage, but, if he take Juvenal and Cœlius for modern authors, I hope he will admit of Accius, Nævius, and Carmenta, for the only ancients. Let him think upon this criticism, and never hope pardon for his Adducis malleum, which is not half so well justified, and yet none but madmen or fools reject it.

But certainly the Doctor should not have made it his business to object Anglicisms, in whose Elenchus I doubt not but there may be found such phrases as may serve to convince him that he is an Englishman, however Scottified in his principles. If the Doctor doubt of it, or but desire a catalogue, let him but signify his mind, and he shall be furnished with a Florilegium. But I am now come to the main controversy about Empusa. The Doctor saith nothing in defence of his quibble, nor gives any reason why he jumbled languages to make a silly clinch, which will not pass for wit either at Oxford or at Cambridge; no, nor at Westminster.

It seems he had derived Empusa from ἓν and ποῦς, 414and said it was a kind of Hobgoblin that hopped upon one leg: and hence it was that the boys’ play (Fox come out of thy hole) came to be called Empusa. I suppose he means Ludus Empusæ. This derivation he would have to be good, and that we may know his reading, (though he hath scarce consulted any of the authors), he saith Mr. Hobbes did laugh at it, until somebody told him that it was in the Scholiast of Aristophanes (as good a critic as Mr. Hobbes), Eustathius, Erasmus, Cœlius Rhodiginus, Stephanus, Scapula, and Calepine. But sure he doth not think to scape so. To begin with the last; Calepine doth indeed say, uno incedit pede, unde et nomen. But he is a Modern, and I do not see why his authority should outweigh mine if his author’s reasons do not. He refers to Erasmus and Rhodiginus. Erasmus in the adage, Proteo mutabilior hath these words of Empusa: Narrant autem uno videri pedi--this is not to hop--unde et nomen inditum putant, Ἔμπουσαν ὁιονεὶ ἑνίποδα. He doth not testify his approbation of the derivation at all, only lets you know what etymologies some have given before him. And doth anybody think that Dr. Harmar was the first which began to show his wit, (or folly), in etymologizing words? Cœlius Rhodiginus doth not own the derivation, only saith, Nominis ratio est, ut placet Eustathio, quia uno incedit pede;--is this to hop?--sed nec desunt qui alterum interpretentur habere æneum pedem, et inde appellatam Empusam; quod in Batrachis Aristophanes expressit. And then he recites the interpretation that Aristophanes’s Scholiast doth give upon the text, of which by and by. If any credit be to be attributed to this allegation, his last 415thoughts are opposite to Dr. Wallis; and Empusa must be so called, not because she hopped upon one leg, but because she had but one, the other being brass. But for the former derivation he refers to Eustathius.

As to Eustathius, I do easily conjecture that the reader doth believe that Rhodiginus doth mean Eustathius upon Homer, for that is the book of most repute and fame, his other piece being no way considerable for bulk or repute. But it is not that book, nor yet his History of Ismenias, but his notes upon the 725th verse of Dionysius Περῖηγητής. The poet had said of the stone Jaspis, that it was

Upon which Eustathius thus remarks: Δοκεῖ γαρ ἀλεξίκακος εἶναι ἡ λίθος ἅυτη, καὶ ἀποτροπιαςτικὴ φασμἀτων, ὧν ἕν ἐςτι καὶκαὶ ἡ Ἔμπουσα, δαιμονιόν τι τερί τἰὼ Ἑκάτην, ἑνὶ ποδὶ δοκοῦν δἰήκεσθαι· (fortè διερείδεσθαι Steph.) ὄθεν καὶ παρονομάζεται, ὡς ἔι τις ἔιπη μονόπους ποδι ζωοῦ· ὡς τοῦ ἑτέρου ποδος χαλκοῦ ὄντος, κατὰ τὸν μῦθον. This testimony doth not prove anything of hopping, and, as to the derivation, I cannot but say that Eustathius had too much of the grammarian in him, and this is not the first time, neither in this book, nor elsewhere, wherein he hath trifled. It is observable out of the place, that there were more Empusas than one, as, indeed, the name is applied by several men to any kind of frightful phantasm. And so it is used by several authors, and for as much as phantasms are various, according as the persons affrighted have been severally educated, &c. every man did impose this name upon his own apprehensions. This gave men occasion to fain Empusa as such--for who will believe that she was not apprehended as having four legs, when she 416appeared in the form of a cow, dog, &c.--but, as apprehended by Bacchus and his man at that time. I do not find that she appeared in any shape but such as made use of legs in going, whence I imagine that Empusæ might be opposite to the θεοὶ νεποδες, which appellation was anciently fixed upon the gods, (propitious) upon a two-fold account; first, for that they were usually effigiated as having no feet, which is evident from ancient sculpture, and secondly, for that they are all said not to walk, but rather swim, if I may so express that non gradiuntur, sed fluunt, which is the assertion of all the commentators I have ever seen upon that verse of Virgil:--

This whole discourse may be much illustrated from a passage in Heliodorus, Æthiop. lib. iii. sec. 12, 13. Calasiris told Cnemon that the Gods Apollo and Diana did appear unto him; Cnemon replied, Ἀλλὰ τίνα δὴ τρόπον ἒφασκες ἐνδεδεῖχθαἱ σοι τοῦς θεοῦς ὅτι μὴ ἐνύπνιον ἦλθον, ἀλλ’ ἐναργῶς ἐφᾶνησαν; upon this the old priest answered, that both gods and demons, when they appear to men, may be discovered by the curious observer, both in that they never shut their eyes, καὶ τῳ βαδίσματι πλέον, οὐ κατὰ διάστησιν τῶν ποδῶν οὐδέ μετάθεσιν ἀνυομένω, ἀλλὰ κατὰ τινα ρὕμην ἀέριον, καὶ ὁρμὴν ἀπαραπόδιστον, τεμνόντων μᾶλλον τὸ περιεχον ἢ διαπορευομένων. Δὶο δὴ καὶ τὰ ἀγάλματα τῶν θεῶν ἈιγύπτιοιἈιγύπτιοι τὼ πὸδέ ζευγνύντες καὶ ὥσπερ ἑνοῦντες ἵστᾶσιν. ἅ δὴ καὶ Ὅμηρος ἐιδῶς, ἅτε Ἀιγύπτιος, καὶκαὶ τὴν ἱερὰν πάιδευσιν ἐκδιδαχθείς, συμβολικῶς τοῖς ἔπεσιν ἐναπεθετο, τοῖς δυναμένοις συνιέναι γνωριζειν καταλιπών, ἐπι τοῦ ποσειδῶνος, το

οἴον ῥέοντος ἐν τῆ πορεία, τοῦτο γάρ εστι τό ῥεῖ ἀπιόντος, καὶ οῦχ ὥς 417τινες ἠπάτηνται, ῥᾳδίως ἔγνωνῥᾳδίως ἔγνων ὑπολαμβάνοντες. Farnaby, upon the place in Virgil, observes, that Deorum incessus est continuus et æqualis, non dimotis pedibus, neque transpositis, ἀλλὰ κατὰ ῥύμην ἀέριονἀέριον. Cornelius Schrevelius in the new Leyden notes saith, Antiquissima quæque Deorum simulachra, quod observarunt viri magni, erant τοῦς πόδας συμβεβηκότα, diique ipsi non gradiuntur sed fluunt. Their statues were said to stand rather upon columns than upon legs, for they seem to have been nothing but columns shaped out into this or that figure, the base whereof carrying little of the representation of a foot. These things being premised, I suppose it easy for the intelligent reader to find out the true etymology of Empusa, quasi ἐν ποσιν οῦσα, or βάινουσα, from going on her feet, whereas the other gods and demons had a different gait. If any can dislike this deduction, and think her so named from ἑνιπους, whereas she always went upon two legs, (if her shape permitted it) though she might draw the one after her, as a man doth a wooden leg: I say, if any, notwithstanding what hath been said, can join issue with the Doctor, my reply shall be Σοὶ μὲν ταῦτα δοκοῦντ’ ἐστὶν, ἐμὸι δὲ τάδε.

Now, as to the words of Aristophanes upon which the Scholiast descants, they are these:--speaking of an apparition strangely shaped, sometimes like a camel, sometimes like an ox, a beautiful woman, a dog, &c. Bacchus replies:

The Scholiast hereupon tells us that Empusa, 418was Φαντασμα δαιμονιῶδες ὑπὸ Ἑκάτης ἐπιπεμπόμενον καὶ φαινόμενον τοἴς δυστυχοῦσιν, ὅ δοκεῖ πολλὰς μορφας αλλασσεω καὶ ὁι μεν φασιν ἀυτην μονοποδα εῖναι, καὶ ἐτυμολογοῦσιν’ ὁιονεὶ ἑνιποδα, διὰ το ἑνὶ ποδι κεχρῆσθαι. And this is all that is material in the Scholiast, except that he adds by and by, that βολιτινον σκελος is all one with the leg of an ass. And this very text and Scholiast is that to which all the authors he names, and more, do refer.

I come now to Stephen, who, in his index, and in the word ποδίζω, gives the derivation of Empusa. Ποδιζω, gradior, incedo, (not to hop) sic Suidas Ἔμπουσαν dictam ait παρὰ το ἑνὶ ποδιζειν. In the index thus: sunt qui dictam putent παρὰ τὸ ἑνὶ ποδὶζειν, quod uno incedat pedi, quasi Ἔμπουσαν, alterum enim pedem æneum habet. But neither Stephen, nor any else, except Suidas, whom the hypercritical Doctor had not seen, no, not the Scholiast of Aristophanes (a better critic than Mr. Hobbes) doth relate the etymology as their own. Nay, there is not one that saith Empusa hopped on one leg, which is to be proved out of them. The great Etymological Dictionary deriveth it παρὰπαρὰ τὸ ἐμποδιζειν, to hinder, let, &c. its apparition being a token of ill luck. But, as to the Doctor’s deduction, it saith, ἜμπουσαἜμπουσα Ψιλοῦπαι, εἰ καὶ δοκεῖ παρὰ τὸ ἕνα συγκεῖσθαι. It doth only seem so. And it is strange that ἑν should not alter only its aspiration, but change its ν into μ, which I can hardly believe admittable in Greek, least there should be no difference betwixt its derivatives and those of ἐν. When I consider the several μορμόνες which the Grecians had, some whereof did fly, some had no legs, &c.&c., I can think that the origin of this name may have been thus: some amazed person 419saw a spectrum, and, giving another notice of it, his companion might answer, it is Βριμὼ, Μορμὼ Ἡκὰτη, but he, meeting with a new phantasm, cries, ἐν ποσὶ βαίνει or βαδίζει, for which apprehension of his, somebody coined this expression of Ἔμποῦσα. It may also be possibly deduced from Ἐμποδὶζω, so that τύχη ἐμποδιζουσαἐμποδιζουσα might afterwards be reduced to the single term of Empusa. Nor do I much doubt but that those who are conversant in languages, and know how that several expressions are often jumbled together to make up one word upon such like cases, will think this a probable origination. I believe, then, that Mr. Hobbes’s friend did never tell him it was in Eustathius, or that Empusa was an hopping phantasm. It had two legs and went upon both, as a man may upon a wooden leg. ἜμποῦσαἜμποῦσα is also a name for Lamia, and such was that which Menippus might have married, which, I suppose, did neither hop nor go upon one leg, for he might have discovered it. But Mr. Hobbes did not except against the derivation, (although he might justly, derivations made afterwards carrying more of fancy than of truth, and the Doctor is not excused for asserting what others barely relate, none approve), but asked him where that is, in what authors he read that boys’ play to be so called. To which question, the Doctor, to show his reading and the good authors he is conversant in, replies, in Junius’s Nomenclator, Rider and Thomas’s Dictionary, sufficient authors in such a business, which, methinks, no man should say that were near to so copious a library. It is to be remembered that the trial now is in Westminster School, and amongst Ciceronians, neither whereof 420will allow those to be sufficient authors of any Latin word. Alas, they are but Vocabularies; and, if they bring no author for their allegation, all that may be allowed them is, that, by way of allusion, our modern play may be called Ludus Empusæ. But that it is so called we must expect, till some author do give it the name. These are so good authors, that I have not either of them in my library. But I have taken the pains to consult, first, Rider; I looked in him, (who was only author of the English Dictionary) and I could not find any such thing. It is true, in the Latin Dictionary, which is joined with Rider, but made by Holyoke; (O that the Doctor would but mark!) in the index of obsolete words, there is Ascoliasmus, Ludus Empusæ, Fox to thy hole, for which word, not signification, he quoteth Junius. The same is in Thomasius, who refers to Junius in like manner. But could the Doctor think the word obsolete, when the play is still in fashion? Or, doth he think that this play is so ancient as to have had a name so long ago, that it should now be grown obsolete? As for Junius’s interpretation of Empusa, it is this: Empusa, spectrum, quod se infelicibus ingerit, uno pede ingrediens. Had the Doctor ever read him, he would have quoted him for his derivation of Empusa, I suppose. In Ascoliasmus, he saith, Ascoliasmus, Empusæ Ludus, fit ubi, altero pede in aere librato, unico subsiliunt pede: ἀσκολιασμὸς Pollux; Almanicè, Hinckelen; Belgicè, Op een been springhen; Hinckepincken, Flandris. But what is it in English he doth not tell, although he doth so in other places often. What the Doctor can pick out of the Dutch I 421know not; but, if that do not justify him, as I think it doth not, he hath wronged Junius, and greatly imposed upon his readers.

But, to illustrate this controversy further, I cannot be persuaded the Doctor ever looked into Junius, for, if he had, I am confident, according to his wonted accurateness, he would have cited Pollux’s Onomasticon into the bargain, for Junius refers to him, and I shall set down his words, that so the reader may see what Ascoliasmus was, and all the Doctor’s authors say Ludus Empusæ and Ascoliasmus were one and the same thing. Julius Pollux (lib. ix. cap. 7): Ὁ δε Ἀσκολὶασμὸς, (old editions read it, Ἀ’σκολιασμὸς et ασκολιάζω) τοῦ ἑτέρου ποδὸς αἰωρουμένου, κατὰ μόνου τοῦ ἑτέρου πηδᾶν ἔπόιει; ὅπερ Ἀσκωλιάζὲιν ὠνόμαζον· ἤτοι εἰς μἢκος ἐνήλλαντο, ἢ ὁ μὲν ἐδίωκεν οὕτως, οἱοἱ δὲ ὑπέφευγον ἐπ’ ἀμφοῖν θὲοντες, ἕως τινὸς τῳ φερομένῳ ποδὶ ὁ διὼκων δυνηθῇ τυχεῖν· ἤ καὶ στάντες ἐπήδων, ἀριθμοῦντες τὰ πηδήματα· προσέκειτο γὰρ τῷ πλήθει τὸ νικᾶν. Ἀσκωλιάζειν δὲ ἐκαλεῖτο καὶ τὸ ἐπιπηδᾶνἐπιπηδᾶν ἀσκῷ κενῷ καὶ ὑποπλέω πνευματος, ἠλείμμένω, ἵναπερ ὀλισθάνοιεν περὶ τὴν ἀλοιφὴν. “So that Ascoliasmus, and consequently, Ludus Empusæ, was a certain sport which consisted in hopping, whether it were by striving who could hop furthest, or whether only one did pursue the rest hopping, and they fled before him on both legs, which game he was to continue till he had caught one of his fellows, or whether it did consist in the boys’ striving who could hop longest. Or, lastly, whether it did consist in hopping upon a certain bladder, which, being blown up and well oiled over, was placed upon the ground for them to hop upon, that so the unctuous bladder might slip from under them and give them a fall.” And this is all that Pollux holds forth. Now, of all these ways, there is none that hath any resemblance with our Fox to thy 422hole; but the second: and yet, in its description, there is no mention of beating him with gloves, as they do now-a-days, and wherein the play consists as well as in hopping. It might, notwithstanding, be called Ludus Empusæ, but not in any sort our Fox to thy hole; so that the Doctor and his authors are out, imposing that upon Junius and Pollux which they never said. And thus much may suffice as to this point. I shall only add out of Meursius’s Ludi Græci, that Ascolia were not Ludus Empusæ but Bacchisacra, and he quotes Aristophanes’s Scholiast in Plutus, Ἀσκώλια ἑορτὴ Διονύσου ἀσκὸν γαρ οἵνουοἵνου πληροῦντες, ἑνὶ ποδὶ τοῦτον ἐπεπήδον, καὶ ὁ πηδήσας ἆθλον εἶχε τὸν οἵνουοἵνου. As also Hesychius, ἈσκωλιάζεινἈσκωλιάζειν, κυρίως τὸ ἐπὶ τοῦς ἀσκοὺς ἅλλεσθαι.

But I could have told the Doctor where he might have read of Empusa as being the name of a certain sport or game, and that is, in Turnebus Adversaria, lib. xxvii. cap. 33. There he speaks of several games mentioned by Justinian in his Code, at the latter end of the third book, one of which he takes to be named Empusa; adding withal, that the other are games, it is indisputable, only Empusa in lite et causa erit, quod nemo nobis facile assensurus sit Ludum esse, cum constet spectrum quoddam fuisse formas, varie mutans. Sed quid vetat eo nomine Ludum fuisse? Certe ad vestigia vitiatæ Scripturæ quam proximo accedit. Yet he only is satisfied in this conjecture, till somebody else shall produce a better. And now what shall I say? Was not Turnebus as good a critic, and of as great reading as Dr. Wallis, who had read over Pollux, and yet is afraid that nobody will believe Empusa to have been a 423game, and all he allegeth for it is, quid vetat? Truly, all I shall say, and so conclude this business, is, that he had read over an infinity of books, yet, had not had the happiness, which the Doctor had, to consult with Junius’s Nomenclator, Thomasius and Rider’s Dictionary, authors sufficient in such a case.

I now come to the Doctor’s last and greatest triumph, at which I cannot but stand in admiration, when I consider he hath not got the victory. Had the Doctor been pleased to have conversed with some of the fifth form in Westminster School, (for he needed not to have troubled the learned master), he might have been better informed than to have exposed himself thus.

Mr. Hobbes had said that στιγμὴ signified a mark with a hot iron; upon which saying the Doctor is pleased to play the droll thus: “Prithee tell me, good Thomas, before we leave this point, (O the wit of a divinity doctor!) who it was told thee that στιγμὴ was a mark with an hot iron, for it is a notion I never heard till now, and do not believe it yet. Never believe him again that told thee that lie, for as sure as can be, he did it to abuse thee; ϛιγμὴ signifies a distinctive point in writing, made with a pen or quill, not a mark made with a hot iron, such as they brand rogues withal; and, accordingly, ϛιζω δῖαϛιζω, distinguo, interstinguo, are often so used. It is also used of a mathematical point, or somewhat else that is very small, στιγμὴ χρὸνου, a moment, or the like. What should come in your cap, to make you think that ϛῖγμὴ signifies a mark or brand with a hot iron? I perceive where the business lies; it was ϛίγμα ran in 424your mind when you talked of ϛιγμὴ, and, because the words are somewhat alike, you jumbled them both together, according to your usual care and accurateness, as if they had been the same.”

When I read this I cannot but be astonished at the Doctor’s confidence, and applaud him who said, ἀμάθεια θάρσὸς φέρει. That the Doctor should never hear that ϛιγμὴ signifies a mark with a hot iron, is a manifest argument of his ignorance. But, that he should advise Mr. Hobbes not to believe his own readings, or any man’s else that should tell him it did signify any such thing, is a piece of notorious impudence. That ϛιγμὴ signifies a distinctive point in writing made with a pen or quill, (is a pen one thing and a quill another to write with?) nobody denies. But, it must be withal acknowledged it signifies many things else. I know the Doctor is a good historian, else he should not presume to object the want of history to another; let him tell us how long ago it is since men have made use of pens or quills in writing; for, if that invention be of no long standing, this signification must also be such, and so it could not be that from any allusion thereunto the mathematicians used it for a point. Another thing I would fain know of this great historian, how long ago ϛίζω and διαϛίζω began to signify interpungo? For, if the mathematics were studied before the mystery of printing was found out, (as shall be proved whenever it shall please the Doctor, out of his no reading, to maintain the contrary), then the mathematical use thereof should have been named before the grammatical. And, if this word be translatitious, and that sciences were the effect of long contemplation, 425the names used wherein are borrowed from talk, Mr. Hobbes did well to say, that στιγμὴ precedaneously to that indivisible signification which it afterwards had, did signify a visible mark made by a hot iron, or the like. And, in this procedure, he did no more than any man would have done, who considers that all our knowledge proceeds from our senses; as also that words do, primarily, signify things obvious to sense, and only secondarily, such as men call incorporeal. This leads me to a further consideration of this word. Hesychius, (of whom it is said that he is Legendus non tanquam Lexicographus, sed tanquam justus author), interprets στιγμὴ, νυγμή, which is a point of a greater or lesser size, made with any thing. So ϛίζω signifies to prick or mark with anything in any manner, and hath no impropriated signification in itself, but according to the writer that useth it. Thus, in a grammarian ϛίζω signifies to distinguish, by pointing often; sometimes, even in them, it is the same with ὀβελίζω; sometimes it signifies to set a mark that something is wanting in that place, which marks were called ϛιγμαί. In matters of policy, ϛίζω signifies to disallow, because they used to put a ϛιγμὴ (not ϛίγμα) before his name who was either disapproved or to be mulcted. In punishment it signifies to mark or brand, whereof I cannot at present remember any other ways than that of an hot iron, which is most usual in authors, because most practised by the ancients. But, that the mark which the Turks and others do imprint without burning may be said ϛίζεσθαι, I do not doubt, no more than that Herodian did to give that term to the ancient Britons, of whom he says, τὰ σώματα 426ἐϛίζοντο γραφαῖς ποικίλαις, καὶ ζώων παντοδαπῶν εἰκόσιεἰκόσι. Thus, horses that were branded with κάππα and σαν (κοππἀτιαι and σαμφοραι) were said ϛίζεσθαι. Thus, in its origin, ϛιγμὴ doth signify a brand or mark with an hot iron, or the like; and that must be the proper signification of στιγμὴ, which is proper to ϛίζω, none but such as Dr. Wallis can doubt. In its descendants it is no less evident, for, from στιγμὴ comes stigmosus, which signifies to be branded; Vitelliana cicatrice stigmosus, not stigmatosus. So Pliny in his Epistles, as Robert Stephen cites it. And στιγματιας (the derivative of στιγμὴ, which signifies any mark, as well as a brand, even such as remain after stripes, being black and blue), was a nickname imposed upon the grammarian Nicanor, ὅτι περὶ στιγμῶν ἐπολυλόγησε. And, though we had not any examples of στῖγμὴ being used in this sense, yet, from thence, for any man to argue against it, (but he who knows no more than Stephen tells him) is madness, unless he will deny that any word hath lost its right signification, and is used only, by the authors we have, although neither the Doctor nor I have read all them, in its analogical signification. I have always been of opinion, that στιγμὴ signified a single point, big or little, it matters not; and στίγμα, a composure of many; as γραμμὴ signifies a line, and γράμμα a letter, made of several lines. For στίγμα signified the owl, the sæmæna, the letter K, yea, whole words, lines, epigrams engraven in men’s faces; and στιγμὴ, I doubt not, had signified a single point, had such been used, and so it became translatitiously used by grammarians and mathematicians. I could give grounds for this conjecture, 427and not be so impertinent as the Doctor in his sermon, where he told men that σοφός was not in Homer; that from ἄφρων came ebrius; that sobrietas was not bad Latin, and that sobrius was once, as I remember, in Tully. Is this to speak suitably to the oracles of God, or rather to lash out into idle words? Hath the Doctor any ground to think these are not impertinences? Or, are we, poor mortals, accountable for such idle words as fall from us in private discourses, whilst these ambassadors from heaven droll in the pulpit without any danger of an after-reckoning?

But I proceed to a further survey of the Doctor’s intolerable ignorance. His charge in the end of the school-master’s rant is, that he should remember στίγμα and στιγμὴ are not all one. I complained before that he hath not cited Robert Stephen aright; now I must tell him he hath been negligent in the reading of Henry Stephen: for in him he might have found that στίγμα was sometimes all one with στιγμὴ, though there be no example in him wherein στιγμὴ is used for στίγμα. Hath not Hesiod, (as Stephen rightly citeth it), in his Scutum, 166-67.

ubi scholiastes ὥσπερ δὲ στιγμαὶ ἦσαν ἐπάνω, τῶν ῥάχεων τῶν δρακόντων, κατάστίκτοι γὰρ καὶ ποικίλοι ὁι ὄφεις. So Johannes Diaconus upon the place, a man who (if I may use the Doctor’s phrase) was as good a critic as the Geometry Professor.

Thus much for the Doctor. To the understanding reader, I say that στιγμὴ is used for burning with a 428hot iron: 2 Macchab. ix. 11, where speaking of Antiochus’s lamentable death, his body putrefying and breeding worms, he is said, ἐις ετίγνωσιν τοῦ θεοῦ ἔρχεθαι θείᾳ μάστιγι, κατα στιγμὴν ἐπιτεινόμενος ταῖς ἀλγηδόσι; being pained as if he had been pricked or burned with hot irons. And that this is the meaning of that elegant writer, shall be made good against the Doctor, when he shall please to defend the vulgar interpretation. Pausanias, in Bœoticis, speaking of Epaminondas, who had taken a town belonging to the Sicyonians, called Phœbia (Φουβία) wherein were many Bœotian fugitives, who ought, by law, to have been put to death, saith he dismissed them under other names, giving them only a brand or mark. Πόλισμα ἑλὼν Σικυωνἰων Φουβίαν, ἔνθὰ ἦσαν το πολὺ οἱ Βοιώτιοι φυγάδες, στιγμήν ἀφίησι τοῦς ἐγκαταληφθέντας ἄλλην σφίσιν ἣν ετυχε πατρίδα ἐπονομάζων ἐκάστω. It is true στιγμὴν is here put adverbially, but that doth not alter the case. Again, Zonaras, in the third tome of his History, in the life of the Emperor Theophilus, saith, that when Theophanes and another monk had reproved the said emperor for demolishing images, he took and stigmatized each of them with twelve iambics in their faces: εἶτα καὶ τὰς ὄψεις ἀυτῶν κάτεστιξε καὶ ταῖς στιγμαῖς μέλαν ἐπέχεε γράμματα δὲ ἐτύπουν τὰ στιγματα, τὰ δὲ ἦσαν ἴαμβοι οὗτοι. A place so evident, that I know not what the Doctor can reply. This place is just parallel to what the same author saith in the life of Irene, τἀς ὄψέις σφών καταστιξας ἐν γράμμασι, μέλανος εγχεομένου τοῖς στίγμασι. If the Doctor object that he is a modern author, he will never be able to render him as inconsiderable as Adrianus Junius’s Nomenclator, Thomasius and Rider. If any will 429deny that he writes good Greek, Hieronymus Wolfius will tell them, his only fault is περισσολογια, redundancy in words, and not the use of bad ones.

Another example of στιγμὴ used in this sense, is in the collections out of Diodorus Siculus, lib. xxxiv. as they are to be found at the end of his works, and as Photius hath transcribed them into his Bibliotheca. He saith that the Romans did buy multitudes of servants and employ them in Sicily: Οἷς, ἐκ τῶν σωματοτροφείων ἀγεληδὸν απαχθεῖσιν, ἐυθύς χαρακτῆρα ἐπέβαλλον, καὶ στιγμὰς τοἴς σώμασιν. These are the words but of one author, but ought to pass for the judgment of two, seeing Photius, by inserting them, hath made them his own.

Besides, it is the judgment of a great master of the Greek tongue, that stigmata non tam puncta ipsa quam punctis variatam superficiem Græci vocaverunt. I need not, I suppose, name him, so great a critic as the Doctor cannot be ignorant of him.

Nor, were στίγματα commonly, but upon extraordinary occasions, imprinted with an hot iron. The letters were first made by incision, then the blood pressed, and the place filled up with ink, the composition whereof is to be seen in Aetius. And thus they did use to matriculate soldiers also in the hand. Thus, did the Grecian emperor, in the precedent example of Zonaras. And if the Doctor would more, let him repair to Vinetus’s comment upon the fifteenth Epigram of Ausonius.

And now I conceive enough hath been said to vindicate Mr. Hobbes, and to show the insufferable ignorance of the puny professor, and unlearned 430critic. If any more shall be thought necessary, I shall take the pains to collect more examples and authorities, though I confess I had rather spend time otherwise, than in matter of so little moment. As for some other passages in his book, I am no competent judge of symbolic stenography. The Doctor (Sir Reverence) might have used a cleanlier expression than that of a shitten piece, when he censures Mr. Hobbes’s book.

Hitherto the letter.^[1] By which you may see what came into my (not square) cap to call στιγμὴ a mark with a hot iron, and that they who told me that, did no more tell me a lie than they told you a lie that said the same of στίγμα; and, if στιγμὴ be not right as I use it now, then call these notes not στιγρας, but στίγματα. I will not contend with you for a trifle. For, howsoever you call them, you are like to be known by them. Sir, the calling of a divine hath justly taken from you some time that might have been employed in geometry. The study of algebra hath taken from you another part, for algebra and geometry are not all one; and you have cast away much time in practising and trusting to symbolical writings; and for the authors of geometry you have read, you have not examined their demonstrations to the bottom. Therefore, you perhaps may be, but are not yet, a geometrician, much less a good divine. I would you had but so much ethics as to be civil. But you are a notable critic; so fare you well, and consider 431what honour you do, either to the University where you are received for professor, or to the University from whence you came thither, by your geometry; and what honour you do to Emanuel College by your divinity; and what honour you do to the degree of Doctor, with the manner of your language. And take the counsel which you publish out of your encomiast his letter; think me no more worthy of your pains, you see how I have fouled your fingers.

THREE PAPERS PRESENTED TO THE ROYAL SOCIETY AGAINST DR. WALLIS.

TO THE RIGHT HONOURABLE AND OTHERS, THE LEARNED MEMBERS OF THE ROYAL SOCIETY, FOR THE ADVANCEMENT OF SCIENCES.

Presenteth to your consideration, your most humble servant, Thomas Hobbes, (who hath spent much time upon the same subject), two propositions, whereof the one is lately published by Dr. Wallis, a member of your Society, and Professor of Geometry; which if it should be false, and pass for truth, would be a great obstruction in the way to the design you have undertaken. The other is a problem, which, if well demonstrated, will be a considerable advancement of geometry; and though it should prove false, will in no wise be an impediment to the growth of any other part of philosophy.

Presenteth to your consideration, your most humble servant Thomas Hobbes, a confutation of a theorem which hath a long time passed for truth; to the great hinderance of Geometry, and also of Natural Philosophy, which thereon dependeth.

The four sides of a square being divided into any number of equal parts, for example into 10; and straight lines drawn through the opposite points, which will divide the square into 100 lesser squares; the received opinion, and which Dr. Wallis commonly useth, is, that the root of those 100, namely 10, is the side of the whole square.

The root 10 is a number of those squares, whereof the whole containeth 100, whereof one square is an unity; therefore the root 10, is 10 squares: Therefore the root of 100 squares is 10 squares, and not the side of any square; because the side of a square is not a superficies, but a 438line. For as the root of 100 unities is 10 unities, or of 100 soldiers 10 soldiers: so the root of 100 squares is 10 of those squares. Therefore the theorem is false; and more false, when the root is augmented by multiplying it by other greater numbers.

Hence it followeth, that no proposition can either be demonstrated or confuted from this false theorem. Upon which, and upon the numeration of infinites, is grounded all the geometry which Dr. Wallis hath hitherto published.

And your said servant humbly prayeth to have your judgment hereupon: and that if you find it to be false, you will be pleased to correct the same: and not to suffer so necessary a science as geometry to be stifled, to save the credit of a professor.

Your most humble servant Thomas Hobbes presenteth, that the quantity of a line calculated by extraction of roots is not to be truly found. And further presenteth to you the invention of a straight line equal to the arc of a circle.

A square root is a number which multiplied into itself produced a number.

And the number so produced is called a square number. For example: Because 10 multiplied by 10 makes 100; the root is 10, and the square number 100.

In the natural row of numbers, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, &c. every one is the square of some number in the same row. But square numbers (beginning at 1) intermit first two numbers, then four, then six, &c. So that none of the intermitted numbers is a square number, nor has any square root.

A square root (speaking of quantity) is not a line, such as Euclid defines, without latitude, but a rectangle.

Suppose A B C D be the square, and A B, B C, C D, D A, be the sides, and every side divided into 10 equal parts, and lines drawn through the opposite points of division; there will then be made 100 lesser squares, which taken altogether are equal to the square A B C D. Therefore the whole square is 100, whereof one square is an unit; therefore 10 units, which is the root, is ten of the lesser squares, and consequently has latitude; and therefore it cannot be the side of a square, which, according to Euclid, is a line without latitude.

It follows hence, that whosoever taketh for a principle, that a side of a square is a mere line without latitude, and that the root of a square is such a line (as Dr. Wallis continually does) demonstrates nothing. But if a line be divided into what number of equal parts soever, so the line have breadth allowed it (as all lines must, if they be drawn), and the length be to the breadth as number to an unit; the side and the roof will be all of one length.

Any number given is produced by the greatest root multiplied into itself, and into the remaining fraction. Let the number given be two hundred squares, the greatest root is 144
14 squares. I say that 200 is equal to the product of 14 into itself, together with 14 multiplied into 4
14. For 14 multiplied into itself makes 196. And 14 into 4
14 makes 56
14 which is equal to 4. And 4 added to 196 maketh 200; as was to be proved. Or take any other number 8, the greatest root is 2; which multiplied into itself is 4, and the remainder 2
4 multiplied into 2, is 4, and both together 8.

But the same square calculated geometrically by the like parts, consisteth (by Euclid II. 4) of the same numeral great square 196, and of the two rectangles under the greatest side 14, and the remainder of the side, or (which is all one) of one rectangle under the greatest side, and double the remainder of the side; and further of the square of the less segment; which altogether make 200, and moreover 1
49 of those 200 squares, as by the operation itself appeareth thus:

442The product of 14, the greatest segment, into the two fractions 4
14, that is, into 4
14 (or into twice 2
14) is 56
14 (that is 4); and that 4 added to 196 makes 200.

Lastly, the product of 2
14 into 2
14 or 1
7 into 1
7 is 1
49. And so the same square calculated by roots is less by 1
49 of one of those two hundred squares, than by the true and geometrical calculation; as was to be demonstrated.

It is hence manifest, that whosoever calculates the length of an arc or other line by the extraction of roots, must necessarily make it shorter than the truth, unless the square have a true root.

443The Radius of a Circle is a Mean Proportion between the Arc of a Quadrant and two-fifths of the same.

Describe a square A B C D, and in it a quadrant D C A. In the side D C take D T two-fifths of D C, and between D C and D T a mean proportional D R, and describe the quadrantal arcs R S, T V. I say the arc R S is equal to the straight line D C. For seeing the proportion of D C to D T is duplicate of the proportion of D C to D R, it will be also duplicate of the proportion of the arc C A to the arc R S, and likewise duplicate of the proportion of the arc R S to the arc T V.

Suppose some other arc, less or greater than the arc R S, to be equal to D C, as for example r s: then the proportion of the arc r s to the straight line D T will be duplicate of the proportion of R S to T V, or D R to D T. Which is absurd; because D r is by construction greater or less than D R. Therefore the arc R S is equal to the side D C, which was to be demonstrated.

Hence it follows that D R is equal to two-fifths of the arc C A. For R S, T V, D T, being continually proportional, and the arc T V being described by D T, the arc R S will be described by a straight line equal to T V. But R S is described by the straight line D R. Therefore D R is equal to T V, that is to two-fifths of C A.

444And your said servant most humbly prayeth you to consider, if the demonstration be true and evident, whether the way of objecting against it by square root, used by Dr. Wallis; and whether all his geometry, as being built upon it, and upon his supposition of an infinite number, be not false.

CONSIDERATIONS UPON THE ANSWER OF DOCTOR WALLIS TO THE THREE PAPERS OF MR. HOBBES. — TO THE LATTER PART OF THE FIRST PAPER. — TO THE SECOND PAPER. — TO THE THIRD PAPER.

Dr. Wallis says, all that is affirmed, is but if we SUPPOSE that, this will follow.

But it seemeth to me, that if the supposition be impossible, then that which follows will either be false, or at least undemonstrated.

First, this proposition being founded upon his Arithmetica Infinitorum, if there he affirm an absolute infiniteness, he must here also be understood to affirm the same. But in his thirty-ninth proposition he saith thus: “Seeing that the number of terms increasing, the excess above sub-quadruple is perpetually diminished, so at last it becomes less than any proportion that can be assigned; if it proceed in infinitum it must utterly vanish. And therefore if there be propounded an infinite row of quantities in triplicate proportion of quantities arithmetically proportioned (that is, according to the row of cubical numbers) beginning from a point or 0; that row shall be to a row of as many, equal to the greater, as 1 to 4.” It is therefore manifest that he affirms, that in an infinite row of quantities the last is given; and he knows well enough that this is but a shift.

Secondly, he says, that usually in Euclid, and all after him, by infinite is meant but, more than 446any assignable finite, or the greatest possible. I am content it be so interpreted. But then from thence he must demonstrate those his conclusions, which he hath not yet done. And when he shall have done it, not only the conclusions, but also the demonstration, will be the same with mine in Cap. XIV. Art. 2, 3, &c. of my book De Corpore. And so he steals what he once condemned. A fine quality.

Thirdly, he says, (by Euclid’s tenth proposition, but he tells not of what book), that a line may be bisected, and the halves of it may again be bisected, and so onwards infinitely; and that upon such supposed section infinitely continued, the parts must be supposed infinitely many.

I deny that; for Euclid, if he says a line may be divisible into parts perpetually divisible, he means that all the divisions, and all the parts arising from those divisions, are perpetually finite in number.

Fourthly, he says, that there may be supposed a row of quantities infinitely many, and continually increasing, whereof the last is given.

It is true, a man may say, (if that be supposing) that white is black: but, if supposing be thinking, he cannot suppose an infinite row of quantities whereof the last is given. And if he say it, he can demonstrate nothing from it.

Fifthly, he says (for one absurdity begets another) that a superficies or solid may be supposed so constituted as to be infinitely long, but finitely great, (the breadth continually decreasing in greater proportion than the length increaseth), and so as to have no centre of gravity. Such is 447Toricellio’s Solidum Hyperbolicum acutum, and others innumerable, discovered by Dr. Wallis, Monsieur Fermat, and others. But, to determine this, requires more of geometry and logic, (whatsoever it do of the Latin tongue), than Mr. Hobbes is master of.

I do not remember this of Toricellio, and I doubt Dr. Wallis does him wrong and Monsieur Fermat too. For, to understand this for sense, it is not required that a man should be a geometrician or a logician, but that he should be mad.

In the next place, he puts to me a question as absurd as his answers are to mine. Let him ask himself, saith he, if he be still of opinion, that there is no argument in natural philosophy to prove that the world had a beginning. First, whether, in case it had no beginning, there must not have passed an infinite number of years before Mr. Hobbes was born. Secondly, whether, at this time, there have not passed more, that is, more than that infinite number. Thirdly, whether, in that infinite (or more than infinite) number of years, there have not been a greater number of days and hours, and of which, hitherto, the last is given. Fourthly, whether, if this be an absurdity, we have not then, (contrary to what Mr. Hobbes would persuade us), an argument in nature to prove the world had a beginning.

To this I answer, not willingly, but in service to the truth, that, by the same argument, he might as well prove that God had a beginning. Thus, in case he had not, there must have passed an infinite length of time before Mr. Hobbes was born; but there hath passed at this day more than that infinite 448length, by eighty-four years. And this day, which is the last, is given. If this be an absurdity, have we not then an argument in nature to prove that God had a beginning? Thus it is when men entangle themselves in a dispute of that which they cannot comprehend. But, perhaps, he looks for a solution of his argument to prove that there is somewhat greater than infinite; which I shall do so far as to show it is not concluding. If from this day backwards to eternity be more than infinite, and from Mr. Hobbes his birth backwards to the same eternity be infinite, then take away from this day backwards to the time of Adam, which is more than from this day to Mr. Hobbes his birth, then that which remains backwards must be less than infinite. All this arguing of infinites is but the ambition of school-boys.

There is no doubt if we give what proportion we will of the radius to the arc, but that the arc upon that arc will have the same proportion. But that is nothing to my demonstration. He knows it, and wrongs the Royal Society in presuming they cannot find the impertinence of it.

My proof is this: that if the arc on T V, and the arc R S, and the straight line C D, be not equal, then the arc on T V, the arc on R S, and the arc on C A, cannot be proportional; which is manifest by supposing in D C a less than the said D C, but equal to R S, and another straight line, less than R S, equal to the arc on T V; and anybody may examine it by himself.

I have been asked by some that think themselves 449logicians, why I proceeded upon ⅖ rather than any other part of the radius. The reason I had for it was, that, long ago, some Arabians had determined, that a straight line, whose square is equal to 10 squares of half the radius, is equal to a quarter of the perimeter; but their demonstrations are lost. From that equality it follows, that the third proportional to the quadrant and radius, must be a mean proportional between the radius and ⅖ of the same. But, my answer to the logicians was, that, though I took any part of the radius to proceed on, and lighted on the truth by chance, the truth itself would appear by the absurdity arising from the denial of it. And this is it that Aristotle means, where he distinguishes between a direct demonstration and a demonstration leading to an absurdity. Hence it appears that Dr. Wallis’s objections to my Rosetum are invalid as built upon roots.

First, he says that it concerns him no more than other men, which is true. I meant it against the whole herd of them who apply their algebra to geometry. Secondly, he says that a bare number cannot be the side of a square figure. I would know what he means by a bare number. Ten lines may be the side of a square figure. Is there any number so bare, as by it we are not to conceive or consider anything numbered? Or, by 10 nothings understands he bare 10? He struggles in vain, his conscience puzzles him. Thirdly, he says 10 squares is the root of 100 square squares. To 450which I answer, first, that there is no such figure as a square square. Secondly, that it follows hence, that a root is a superficies, for such is 10 squares. Lastly, he says that, neither the number 10, nor 10 soldiers, is the root of 100 soldiers; because 100 soldiers is not the product of 10 soldiers into 10 soldiers. This last I grant, because nothing but numbers can be multiplied into one another. A soldier cannot be multiplied by a soldier. But no more can a square figure by a square figure, though a square number may. Again, if a captain will place his 100 men in a square form, must he not take the root of 100 to make a rank or file? And are not those 10 men?

He objects nothing here, but that the side of a square is not a superficies, but a line, and that a square root (speaking of quantity) is not a line, but a rectangle, is a contradiction. The reader is to judge of that.

To his scoffings I say no more, but that they may be retorted in the same words, and are therefore childish.

And now I submit the whole to the Royal Society, with confidence that they will never engage themselves in the maintenance of these unintelligible doctrines of Dr. Wallis, that tend to the suppression of the sciences which they endeavour to advance.

LETTERS AND OTHER PIECES. — I. A LETTER FROM MR. HOBBS TO MY MR. [2] — II. TO A FRIEND IN ENGLAND. — III. TO MY WORTHY FRIEND MR. GLEN. [3] — IV. LETTER TO SIR CHARLES CAVENDISH. [5] — V. LETTER TO MR. BEALE. [6] — VI. LETTER TO MR. OLDENBURG. [7] — VII. TO THE RIGHT HONOURABLE THE MARQUIS OF NEWCASTLE. [8] — VIII. TO THE KING’S MOST EXCELLENT MAJESTY THE HUMBLE PETITION OF THOMAS HOBBES [9]

Honorable Sir,

Though I may goe whither and when I will for anie necessity you have of my service, yet there is a necessity of good manners that obliges me as yo^r servant to lett you knowe att all times where to find me. Wee goe out of Paris 3 weekes hence, or sooner, towards Venice, but by what way I knowe not, because the ordinary high way through the territory of Milan is encumbered with the warre betweene the French and the Spaniards. Howsoever, wee have to be there in October next. If you require anie service that I can doe there, it may please you to convey your command by Devonshire house. But if you command me nothing, I have forbidden my letters to look for answer: their busines being only to informe and to lett you knowe that the image of your noblenes decayes not in my memory, but abides fresh to keepe me eternally

Worthy Sir,

I have been behind hand with you a long time for a letter I received of yours at Angers, that place affording nothing wherewith to pay a debt of that kind, all matter of news being sooner known in England than here: and the news you writ me was of that kind, that none from England could be more welcome, because it concerned the honour of Welbeck and Clifton, two houses in which I am very much obliged.

Monsieur having given the slip to the Spaniards at Bruxelles, came to the King about ten days ago at St. Germains, where he was received with great joy. The next day the Cardinal entertained him at Ruelle: and the day after that he went to Limours, where he is now, and from thence he goes away shortly to Bloys, to stay there this winter. The Cardinal of Lyons is going to Rome to treat about the annulling of Monsieur’s marriage, which is here by Parliament declared void, but yet they require the sentence of the Pope. There goes somebody thither on the part of his wife, to get the marriage approved: but who that is, I ’know not. The Swedish party in Germany is in low estate, but the French prepare a great army for those parts, pretending to defend the places which the Swedes have put into the King of France his protection, whereof Philipsbourgh is one; a place of importance for the Lower Palatinate. This is all the French news.

For your question, why a man remembers less 455his own face, which he sees often in a glass, than the face of a friend that he has not seen of a great time, my opinion in general is, that a man remembers best those faces whereof he has had the greatest impressions, and that the impressions are the greater for the oftener seeing them, and the longer staying upon the sight of them. Now you know men look upon their own faces but for short fits, but upon their friends’ faces long time together, whilst they discourse or converse together; so that a man may receive a greater impression from his friend’s face in a day, than from his own in a year; and according to this impression, the image will be fresher in his mind. Besides, the sight of one’s friend’s face two hours together, is of greater force to imprint the image of it, than the same quantity of time by intermissions. For the intermissions do easily deface that which is but lightly imprinted. In general, I think that lasteth longer in the memory which hath been stronglier received by the sense.

This is my opinion of the question you propounded in your letter. Other new truths I have none, at least they appear not new to me. Therefore if this resolution of your first question seems probable, you may propound another, wherein I will endeavour to satisfy you, as also in any thing of any other nature you shall command me, to my utmost power; taking it for an honour to be esteemed by you, as I am in effect,

Paris, Oct. 21/31, 1634.

My Lord Fielding and his Lady came to Paris on Saturday night last.

Worthy Sir,

I received here in Florence, two days since, a letter from you of the 19th of January. It was long by the way; but when it came it did thoroughly recompence that delay. For it was worth all the pacquets I had received a great while together. All that passeth in these parts is equally news, and therefore no news; else I would labour to requite your letter in that point, though in the handsome setting down of it, I should still be your inferior.

I long infinitely to see those books of the Sabbaoth^[4], and am of your mind they will put such thoughts into the heads of vulgar people, as will confer little to their good life. For when they see one of the ten commandments to be jus humanum merely, (as it must be if the Church can alter it), they will hope also that the other nine may be so too. For every man hitherto did believe that the ten commandments were the moral, that is, the eternal law.

I desire also to see Selden’s Mare Clausum, having already a great opinion of it.

You may perhaps, by some that go to Paris, send me those of the Sabbaoth, for the other being in Latin, I doubt not to find it in the Rue St. Jaques.

457We are now come hither from Rome, and hope to be in Paris by the end of June. I thank you for your letter, and desire you to believe that I can never grow strange to one, the goodness of whose acquaintance I have found by so much experience. But I have to write to so many, that I write to you seldomer than I desire; which I pray pardon, and esteem me

Florence, Apr. 6
16 1636.

My Lord and Mr. Nicholls, and all our company commend them to you.

Honorable Sir,

The last weeke I had the honor to receave two letters from you at once, one of the 30 of Dec., the other of the 7^th of Jan., w^ch I acknowledged, but could not answer in my last. In the first you begin with a difficulty on the principle of Mons^r de Cartes, that it is all one to move a weight two spaces, or the double of that waight one space, and so on in other proportions: to w^ch you object the difference of swiftnesse, w^ch is greater when a waight is moved two spaces thanthan when double waight is moved one space. Certenly de Cartes his meaning was by force the same that mine, 458namely, a multiplication of the weight of a body in to the swiftnesse wherew^th it is moved. So that when I move a pound two foote atat the rate of a mile an howre, I do the same as if of 2 poundes I moved one pound a foote at y^e rate of a mile an howrehowre, the other pound another foote at the same rate, not in directū, but parallell to the first pound. As if the wayt A B were moved to C D at the rate of a mile an howre, ’tis all one as if the waight A E were moved to F H at the same rate. Here is all the difference: this swiftnesse or rate of a mile an howre is, in the first case, layd out in the 2 spaces A G, G C, the latter, in the 2 spaces A G, E G. The first case, as like as if a footman should run w^th double swiftnesse endwayes, w^ch is y^e doubling of swiftnesse in one man: in the other, it is as if you doubled the swiftnesse by doubling the man: for every man has his owne swiftnesse. And so A H is the swiftnesse A G doubled, as well as A D. For that, that Mons^r de Cartes will not have just twice the force requisite to move the same weight twice as fast, I can say nothing. The papers I have of his touching that are in my trunk, w^ch hath bene taken by Dunkerkers, and taken againe from them by French, and at length recovered by frends I made: but I shall not have it yet this fortnight. In the meane time I am not in that opinion, but do assure myself, the patient being the same, double force in the agent shall worke upon it double effect.

In the same letter you require a better explication of y^e proportion I gather betweene wayght 459and swiftnesse: wherein, because you have not my figure, I imagine you have mistaken me very much. And first, you thinke, I suppose, D E equall to A B: w^ch I am sure is a mistake. For I put A B for any line you will to expresse a minutū secundum. I will, therefore, go over againe the demonstration I sent you before, and see if I can do it cleerer.

Let A B stand for the time knowne wherein the waight D descendeth to E. And let there bee a cylinder of the same matter the waight D consisteth of, and let the altitude of that cylinder be D C: w^ch I shew before was the swiftnesse wherew^th that cylinder presseth, not wherew^th it falleth. And wee are now to enquire how farre such matter as the cylinder is made of must descend from D, before it attayne a swiftnesse equall to this pressing swiftnesse D C. And I say it must fall to L. For in the time A B it is knowne that the waight in D will fall to E: and it is demonstrated by Gallileo, that when such waight comes to E, it shall be able to go twice the space it hath fallen in the same time. Therefore the waight D being in E, hath velocity to carry him the space D K (w^ch I put double to D E) in the same time A B. But I put B F equall to D K. Therefore, in the time A B, the waight’s velocity acquired in E shall be such as to go from B to F without decrease of velocity by the way. Hence I go on to finde in what point the waight in D comes to where it getteth a velocity equall to 460C D. Therefore, I apply D C to G H, parallel to B F: and then it is, as the time A B to the time A G, so the velocity acquired at the end of the time A B to the velocity acquired at the end of the time A G. For the swiftnesse acquired from time to time (I say, not from place to place, but from time to time) are proportionable to the times wherein they are acquired: w^ch is the postulate on w^ch Galileo builds all his doctrine. And as A B to A G, so the line B F to the line G H. But, at the end of the time A B, the waight D is by supposition in E, in that degree of velocity as to go B F or D K in the same time A B. The question therefore is, where the waight D shall be at the end of the time A G. For there it hath the velocity of going G H or D C in the same time, because the velocity G H is to the velocity B F as the line G H to the line B F, or as the time of descent A G to the time A B. But, because the spaces of the descent are in double the proportion of the times of descent, make it as B F to G H, that is, D K to D C, so D C to another, D L. The velocity, therefore, acquired in the point of descent E, namely the velocity D K or B F, is to the velocity acquired in the point L, namely, the velocity G H or D C, (w^ch is the velocity of the cylinder’s waight), as D K to D L. And therefore in L the waight D has acquired a velocity equal to the velocity of the waight of the cylinder.

In the same letter you desire to knowe, how any mediū, as water, retardeth the motion of a stone that falls into it. To w^ch I answer out of that you say afterwards, that nothing can hinder motion but contrary motion: that the motion of the water, 461when a stone falls into it, is point blanke contrary to the motion of the stone. For the stone by descent causeth so much water to ascend as the bignesse of the stone comes to. For imagine so much water taken out of the place w^ch the stone occupies, and layd upon the superficies of the water: it presseth downeward as the stone does, and maketh the water that is below to rise upwards, and this rising upwards is contrary to the descent, and is no other operation than we see in scales, when of two equal bullets in magnitude that w^ch is of heavier metal maketh the other to rise. And thus farre goes your letter of Dec. 30.

For the first quære in your second letter, concerning how we see in the time the lucide body contracts itselfe, I have no other solution but that w^ch your selfe hath given: w^ch is, that the reciprocation is so quicke, that the effect of the first motion lasteth till the next comes, and longer. For by experience we observe that the end of a firebrand swiftely moved about in circle, maketh a circle of fire: w^ch could not be, if the impression made at the beginning of the circulation did not last till the end of it. For if the same firebrand be moved slower, there will appear but a peece of a circle, bigger or lesser according to the swiftnesse or slownesse of y^e motion. For the cause of such reciprocation, it is hard to guess what it is. It may well be the reaction of the medium. For though the mediū yeld, yet it resisteth to: for there can be no passion w^thout reaction. And if a man could make an hypothesis to salve that contraction of y^e sun, yet such is the nature of naturall thinges, as a cause may be againe demanded of such hypothesis: and never should one come to an end w^thout assigning 462the immediate hand of God. Whereas in mathematicall sciences wee come at last to a definition, w^ch is a beginning or principle, made true by pact and consent amongst ourselves. Further, you conceave a difficulty how the medium can be continually driven on, if there be such an alternate contraction. To w^ch, first I answer, that the motion forward is propagated to the utmost distance in an instant, and the first push is therefore enough, and in another instant is made the returne back in y^e like manner. And though it were not done in an instant, yet we see by experience in rivers, as in y^e Thames, that the tide goes upward towards London pushed by the water below, and yet at the same instant the water below is going backe to the sea. For seeing it is high-water at Blackwall before ’tis so at Greenwich, the water goes backe from Blackwall when it goes on at Greenwich. And so it would happen, though Blackwall and Greenwich were nerer together then that any quantity given could come betweene.

In my letter from London, speaking of the refraction of a bullet, I thinke I delivered my opinion to be, that a bullet falling out of a thinner medium into a thicker, looseth in the entring nothing but motion perpendicular: but being entred, he looseth proportionably both of one and the other. For suppose a bullet, whose diameter is A B, be in the thiner mediū, and enter at C into the thicker medium. The thicker medium, at the first touch of B in the point C, worketh nothing upon the line A B. And when the diameter A B is entred, suppose halfe way, 463yet the thicker medium operates laterally but on one halfe of it. So that in the somme there is a losse of velocity perpendicular (to the quantity that the diameter A B requires) without any offence to the motion laterall, but so much of the diameter as is within the thicker mediū is retarded both wayes, and looses of his absolute motion, w^ch is compounded of perpendicular and laterall, and that proportionally. Suppose now that a bullet passe from A to D, and receave a peculiar losse of his perpendicular motion by entring at D, so great that he proceed in the perpendicular but halfe so farre, as for example from D to I: and then being in, the thicknesse of the medium take away more of his velocity both perpendicular and laterall, suppose halfe that w^ch was left of the perpendicular motion and halfe of his first laterall motion, so that the perpendicular motion is but D K, and the laterall motion D E. Then will the line of refraction be D G. As for that argumentation of Des Cartes, it is, in my opinion, as I have heretofore endeavored to shew you, a mere paralogism.

Lastly, you make this quære, why light hath not at severall inclinations severall swiftnesses as well as a bullet. The bullet itself passeth through the severall media: whereas in the motion of light, the body moved, w^ch is the mediū, entreth not into the other medium, but thrusteth it on: and so the parts of that medium thrust on one another, whereby the laterall motion of the thicker medium hath nothing to worke upon, because nothing enters, but stoppes onely and retardes, in oblique incidence, that end w^ch comes first to it, and thereby causes a refraction the contrary way to 464that of a bullet, in such manner as I set forth to you in one of my letters from hence concerning the cause of refraction. And this is all I can say for the present to the quæres of y^r two last letters.

I have enquired concerning perspectives after the manner of De Cartes. Mydorgius tells me there is none that goes about them, as a thing too hard to do. And I believe it. For here is one Mons^r de Bosne in towne, that dwells at Bloys, an excellent workman, but by profession a lawier, and is counsellor of Bloys, and a better philosopher in my opinion then De Cartes, and not inferior to him in the analytiques. I have his acquaintance by Pere Mersenne. He tells me he hath tryed De Cartes his way, but cannot do it: and now he workes upon a crooked line of his owne invention. He sayes he shall have made one w^thin a moneth after he shall returne to Bloys: after that he will see what he can discover in the heavens himselfe, and then if he discover any new thing he will let his way be publique together w^th the effects. This is all the hope I can give you yet. So w^th my prayers to God to keepe you in prosperity this troublesome time, I rest

Paris, Feb. 8, stile no. 1641.

To the Right Honorable

Sr Charles Cavendysshe

present these

at Wellinger.

Sir,

The young woman at Over-Haddon hath been visited by divers persons of this house. My Lord himself hunting the hare one day at the Town’s end, with other gentlemen and some of his servants, went to see her on purpose: and they all agree with the relation you say was made to yourself. They further say on their own knowledge, that part of her Belly touches her Back-bone. She began (as her Mother says) to loose her appetite in December last, and had lost it quite in March following: insomuch as that since that time she has not eaten nor drunk any thing at all, but only wetts her lips with a feather dipt in water. They were told also that her gutts (she alwayes keeps her bed) lye out by her at her fundament shrunken. Some of the neighbouring ministers visit her often: others that see her for curiosity give her mony, sixpence or a shilling, which she refuseth, and her mother taketh. But it does not appear they gain by it so much as to breed a suspition of a cheat. The woman is manifestly sick, and ’tis thought she cannot last much longer. Her talk (as the gentlewoman that went from this house told me) is most heavenly. To know the certainty, there bee many things necessary which cannot honestly be pryed into by a man. First, whether her gutts (as ’tis said) lye out. Secondly, whether any excrement 466pass that way, or none at all. For if it pass, though in small quantity, yet it argues food proportionable, which may, being little, bee given her secretly and pass through the shrunken intestine, which may easily be kept clean. Thirdly, whether no urine at all pass: for liquors also nourish as they go. I think it were somewhat inhumane to examin these things too nearly, when it so little concerneth the commonwealth: nor do I know of any law that authoriseth a Justice of peace, or other subject, to restrain the liberty of a sick person so farr as were needful for a discovery of this nature. I cannot therefore deliver any judgment in the case. The examining whether such a thing as this bee a miracle, belongs I think to the Church. Besides, I myself in a sickness have been without all manner of sustenance for more than six weeks together: which is enough to make mee think that six months would not have made it a miracle. Nor do I much wonder that a young woman of clear memory, hourely expecting death, should bee more devout then at other times. ’Twas my own case. That which I wonder at most, is how her piety without instruction should bee so eloquent as ’tis reported.

Chatsworth, Oct. 20. 68.

Worthy S^r

In the last Transactions for September and October I find a letter addressed to you from D^r Wallis, in answer to my LUX MATHEMATICA. I pray you tell me that are my old acquaintance, whether it be (his words and characters supposed to be interpreted) intelligible. I know very well you understand sense both in Latine, Greeke, and many other languages. He shows you no ill consequence in any of my arguments. Whereas I say there is no proportion of infinite to finite. He answers, he meant indefinite; but derives not his conclusion from any other notion than simply infinite. I said the root of a square number cannot be the length of the side of a square figure, because a root is part of a square number, but length is no part of a square figure. To which he answers nothing. In like manner, he shuffles off all my other objections, though he know well enough that whatsoever he has written in Geometry (except what he has taken from me and others) dependeth on the truth of my objections. I perceive by many of his former writings that I have reformed him somewhat as to the Principles of Geometry, though he thanke me not. He shuffles and struggles in vaine, he has the hooke in his guills, I will give him line enough: for (which I pray you tell him) I will no more teach him by 468replying to any thing he shall hereafter write, whatsoever they shall say that are confident of his Geometry. Qui volunt decipi, decipiuntur. He tells you that I bring but crambe sæpe cocta. For which I have a just excuse, and all men do the same; they repeat the same words often when they talk with them that cannot heare.

I desire also this reasonable favour from you: that, if hereafter I shall send you any paper tending to the advancement of physiques or mathematiques, and not too long, you will cause it to be printed by him that is printer to the Society, as you have done often for D^r Wallis: it will save me some charges.

November the 26th, 1672.

The passions of man’s mind, except onely one, may bee observed all in other living creatures. They have desires of all sorts, love, hatred, feare, hope, anger, pitie, æmulation, and y^e like: onely of curiositie, which is y^e desire to know y^e causes of thinges, I never saw signe in any other living creature but in man. And where it is in man, I find alwaies a defalcation or abatement for it of another passion, which in beastes is commonly predominant, namely, a ravenous qualitie, which in man is called avarice. The desire of knowledge and desire of needlesse riches are incompatible, and destructive one of another. And therefore as in the cognitive faculties reason, so in the motive curiositie, are the markes that part y^e bounds of man’s nature from that of beastes. Which makes mee, when I heare a man, upon the discovery of any new and ingenious knowledge or invention, aske gravely, that is to say, scornefully, what ’tis good for, meaning what monie it will bring in, (when he knows as little, to one that hath sufficient what that overplus of monie is good for), to esteeme that man not sufficiently removed bn 484.png 470from brutalitie. Which I thought fit to say by way of anticipation to y^e grave scorners of philosophie, and that your lordship, after having performed so noble and honourable acts for defence of your countrie, may thinke it no dishonour in this unfortunate leasure to have employed some thoughts in the speculation of the noblest of the senses, vision.

That which I have written of it is grounded especially upon that w^ch about 16 yeares since I affirmed to your Lo^PP at Welbeck, that light is a fancy in the minde, caused by motion in the braine, which motion againe is caused by the motion of y^e parts of such bodies as we call lucid: such as are the sunne and y^e fixed stars, and such as here on earth is fire. By putting you in mind hereof, I doe indeed call you to witnesse of it: because, the same doctrine having since been published by another, I might bee challenged for building on another man’s ground. Yett philosophical ground I take to be of such a nature, that any man may build upon it that will, especially if the owner himselfe will nott. But upon this ground, with the helpe of some other speculations drawne from the nature of motion and action, I have, I thinke, derived y^e reason of all the phænomena I have mett with concerning light and vision, both solidly enough nott to be confuted, and withall easilie enough to be understood by such as can give that attention thereto which the figures, whereby such motion as causeth vision is described, do require. All that I shall bee ever able to adde to it, is polishing: for, being the first draught, it could nott bee so perfect as I hope hereafter to make it in Latin. 471Butt as it is, it will sufficiently give your Lo^PP satisfaction in those quæres you were pleased to make concerning this subject. I am content that it passe, in respect of some drosse that yett cleaves to it, for ore: w^ch is much better than old ends raked out of the kennell of sophisters’ bookes. And for such I commend it to your Lo^p, and myselfe to your accustomed good opinion: which hath beene hitherto so greate honour to mee, as I am nott known to the world by any thing so much as by being,

The treatise ends with the following passage:--

Sheweth, that though your Majesty hath been pleased to take off the restraint of late years laid upon the pensions payable out of your privy purse, yet your Majesty’s Officers refuse to pay the pension of your petitioner without your Majesty’s express command.

And humbly beseaceth your Majesty, (considering his extreme age, perpetual infirmity, frequent and long sickness, and the aptness of his enemies to take any occasion to report that your petitioner by some ill behaviour hath forfeited your 474wonted favour), that you would be pleased to renew your order for the payment of it in such manner as to his great comfort he hath for many years enjoyed it.^[10]

And daily prayeth to God Almighty to bless your Majesty with long life, constant health, and happinesse.