The English Works of Thomas Hobbes of Malmesbury (1 of 11)

PART II. THE FIRST GROUNDS OF PHILOSOPHY. — CHAPTER VII. OF PLACE AND TIME. — CHAPTER VIII. OF BODY AND ACCIDENT. — CHAPTER IX. OF CAUSE AND EFFECT. — CHAPTER X. OF POWER AND ACT. — CHAPTER XI. OF IDENTITY AND DIFFERENCE. — CHAPTER XII. OF QUANTITY. — CHAPTER XIII. OF ANALOGISM, OR THE SAME PROPORTION. — CHAP. XIV. OF STRAIT AND CROOKED, ANGLE AND FIGURE.

1. Things that have no existence, may nevertheless be understood and computed.—2. What is Space.—3. Time.—4. Part.—5. Division.—6. One.—7. Number.—8. Composition.—9. The whole.—10. Spaces and times contiguous, and continual.—11. Beginning, end, way, finite, infinite.—12. What is infinite in power. Nothing infinite can be truly said to be either whole, or one; nor infinite spaces or times, many.—13. Division proceeds not to the least.

1. In the teaching of natural philosophy, I cannot begin better (as I have already shewn) than from privation; that is, from feigning the world to be annihilated. But, if such annihilation of all things be supposed, it may perhaps be asked, what would remain for any man (whom only I except from this universal annihilation of things) to consider as the subject of philosophy, or at all to reason upon; or what to give names unto for ratiocination's sake.

92I say, therefore, there would remain to that man ideas of the world, and of all such bodies as he had, before their annihilation, seen with his eyes, or perceived by any other sense; that is to say, the memory and imagination of magnitudes, motions, sounds, colours, &c. as also of their order and parts. All which things, though they be nothing but ideas and phantasms, happening internally to him that imagineth; yet they will appear as if they were external, and not at all depending upon any power of the mind. And these are the things to which he would give names, and subtract them from, and compound them with one another. For seeing, that after the destruction of all other things, I suppose man still remaining, and namely that he thinks, imagines, and remembers, there can be nothing for him to think of but what is past; nay, if we do but observe diligently what it is we do when we consider and reason, we shall find, that though all things be still remaining in the world, yet we compute nothing but our own phantasms. For when we calculate the magnitude and motions of heaven or earth, we do not ascend into heaven that we may divide it into parts, or measure the motions thereof, but we do it sitting still in our closets or in the dark. Now things may be considered, that is, be brought into account, either as internal accidents of our mind, in which manner we consider them when the question is about some faculty of the mind; or as species of external things, not as really existing, but appearing only to exist, or to have a being without us. And in this manner we are now to consider them.

932. If therefore we remember, or have a phantasm of any thing that was in the world before the supposed annihilation of the same; and consider, not that the thing was such or such, but only that it had a being without the mind, we have presently a conception of that we call space: an imaginary space indeed, because a mere phantasm, yet that very thing which all men call so. For no man calls it space for being already filled, but because it may be filled; nor does any man think bodies carry their places away with them, but that the same space contains sometimes one, sometimes another body; which could not be if space should always accompany the body which is once in it. And this is of itself so manifest, that I should not think it needed any explaining at all, but that I find space to be falsely defined by certain philosophers, who infer from thence, one, that the world is infinite (for taking space to be the extension of bodies, and thinking extension may encrease continually, he infers that bodies may be infinitely extended); and, another, from the same definition, concludes rashly, that it is impossible even to God himself to create more worlds than one; for, if another world were to be created, he says, that seeing there is nothing without this world, and therefore (according to his definition) no space, that new world must be placed in nothing; but in nothing nothing can be placed; which he affirms only, without showing any reason for the same; whereas the contrary is the truth: for more cannot be put into a place already filled, so much is empty space fitter than that, which is full, for the receiving of new bodies. 94Having therefore spoken thus much for these men's sakes, and for theirs that assent to them, I return to my purpose, and define space thus: SPACE is the phantasm of a thing existing without the mind simply; that is to say, that phantasm, in which we consider no other accident, but only that it appears without us.

3. As a body leaves a phantasm of its magnitude in the mind, so also a moved body leaves a phantasm of its motion, namely, an idea of that body passing out of one space into another by continual succession. And this idea, or phantasm, is that, which (without receding much from the common opinion, or from Aristotle's definition) I call Time. For seeing all men confess a year to be time, and yet do not think a year to be the accident or affection of any body, they must needs confess it to be, not in the things without us, but only in the thought of the mind. So when they speak of the times of their predecessors, they do not think after their predecessors are gone, that their times can be any where else than in the memory of those that remember them. And as for those that say, days, years, and months are the motions of the sun and moon, seeing it is all one to say, motion past and motion destroyed, and that future motion is the same with motion which is not yet begun, they say that, which they do not mean, that there neither is, nor has been, nor shall be any time: for of whatsoever it may be said, it has been or it shall be, of the same also it might have been said heretofore, or may be said hereafter, it is. What then can days, months, and years, be, but the names of such 95computations made in our mind? Time therefore is a phantasm, but a phantasm of motion, for if we would know by what moments time passes away, we make use of some motion or other, as of the sun, of a clock, of the sand in an hour-glass, or we mark some line upon which we imagine something to be moved, there being no other means by which we can take notice of any time at all. And yet, when I say time is a phantasm of motion, I do not say this is sufficient to define it by; for this word time comprehends the notion of former and latter, or of succession in the motion of a body, in as much as it is first here then there. Wherefore a complete definition of time is such as this, TIME is the phantasm of before and after in motion; which agrees with this definition of Aristotle, time is the number of motion according to former and latter; for that numbering is an act of the mind; and therefore it is all one to say, time is the number of motion according to former and latter; and time is a phantasm of motion numbered. But that other definition, time is the measure of motion, is not so exact, for we measure time by motion and not motion by time.

4. One space is called part of another space, and one time part of another time, when this contains that and something besides. From whence it may be collected, that nothing can rightly be called a PART, but that which is compared with something that contains it.

5. And therefore to make parts, or to part or DIVIDE space or time, is nothing else but to consider one and another within the same; so that 96if any man divide space or time, the diverse conceptions he has are more, by one, than the parts he makes; for his first conception is of that which is to be divided, then of some part of it, and again of some other part of it, and so forwards as long as he goes on in dividing.

But it is to be noted, that here, by division, I do not mean the severing or pulling asunder of one space or time from another (for does any man think that one hemisphere may be separated from the other hemisphere, or the first hour from the second?) but diversity of consideration; so that division is not made by the operation of the hands but of the mind.

6. When space or time is considered among other spaces or times, it is said to be ONE, namely one of them; for except one space might be added to another, and subtracted from another space, and so of time, it would be sufficient to say space or time simply, and superfluous to say one space or one time, if it could not be conceived that there were another. The common definition of one, namely, that one is that which is undivided, is obnoxious to an absurd consequence; for it may thence be inferred, that whatsoever is divided is many things, that is, that every divided thing, is divided things, which is insignificant.

7. NUMBER is one and one, or one one and one, and so forwards; namely, one and one make the number two, and one one and one the number three; so are all other numbers made; which is all one as if we should say, number is unities.

8. To COMPOUND space of spaces, or time of times, is first to consider them one after another, 97and then altogether as one; as if one should reckon first the head, the feet, the arms, and the body, severally, and then for the account of them all together put man. And that which is so put for all the severals of which it consists, is called the WHOLE; and those severals, when by the division of the whole they come again to be considered singly, are parts thereof; and therefore the whole and all the parts taken together are the same thing. And as I noted above, that in division it is not necessary to pull the parts asunder; so in composition, it is to be understood, that for the making up of a whole there is no need of putting the parts together, so as to make them touch one another, but only of collecting them into one sum in the mind. For thus all men, being considered together, make up the whole of mankind, though never so much dispersed by time and place; and twelve hours, though the hours of several days, may be compounded into one number of twelve.

9. This being well understood, it is manifest, that nothing can rightly be called a whole, that is not conceived to be compounded of parts, and that it may be divided into parts; so that if we deny that a thing has parts, we deny the same to be a whole. For example, if we say the soul can have no parts, we affirm that no soul can be a whole soul. Also it is manifest, that nothing has parts till it be divided; and when a thing is divided, the parts are only so many as the division makes them. Again, that a part of a part is a part of the whole; and thus any part of the number four, as two, is a part of the number eight; for four is 98made of two and two; but eight is compounded of two, two, and four, and therefore two, which is a part of the part four, is also a part of the whole eight.

10. Two spaces are said to be CONTIGUOUS, when there is no other space betwixt them. But two times, betwixt which there is no other time, are called immediate, as A B, B C.|A      B     C| And any two spaces, as well as times, are said to be CONTINUAL, when they have one common part, as A C, B D, where the part B C is common; |A   B   C   D| and more spaces and times are continual, when every two which are next one another are continual.

11. That part which is between two other parts, is called a MEAN; and that which is not between two other parts, an EXTREME. And of extremes, that which is first reckoned is the BEGINNING, and that which last, the END; and all the means together taken are the WAY. Also, extreme parts and limits are the same thing. And from hence it is manifest, that beginning and end depend upon the order in which we number them; and that to terminate or limit space and time, is the same thing with imagining their beginning and end; as also that every thing is FINITE or INFINITE, according as we imagine or not imagine it limited or terminated every way; and that the limits of any number are unities, and of these, that which is the first in our numbering is the beginning, and that which we number last, is the end. When we say number is infinite, we mean only that no number is expressed; for when we 99speak of the numbers two, three, a thousand, &c. they are always finite. But when no more is said but this, number is infinite, it is to be understood as if it were said, this name number is an indefinite name.

12. Space or time is said to finite in power, or terminable, when there may be assigned a number of finite spaces or times, as of paces or hours, than which there can be no greater number of the same measure in that space or time; and infinite in power is that space or time, in which a greater number of the said paces or hours may be assigned, than any number that can be given. But we must note, that, although in that space or time which is infinite in power, there may be numbered more paces or hours than any number that can be assigned, yet their number will always be finite; for every number is finite. And therefore his ratiocination was not good, that undertaking to prove the world to be finite, reasoned thus; If the world be infinite, then there may be taken in it some part which is distant from us an infinite number of paces: but no such part can be taken; wherefore the world is not infinite; because that consequence of the major proposition is false; for in an infinite space, whatsoever we take or design in our mind, the distance of the same from us is a finite space; for in the very designing of the place thereof, we put an end to that space, of which we ourselves are the beginning; and whatsoever any man with his mind cuts off both ways from infinite, he determines the same, that is, he makes it finite.

Of infinite space or time, it cannot be said that 100it is a whole or one: not a whole, because not compounded of parts; for seeing parts, how many soever they be, are severally finite, they will also, when they are all put together, make a whole finite: nor one, because nothing can be said to be one, except there be another to compare it with; but it cannot be conceived that there are two spaces, or two times, infinite. Lastly, when we make question whether the world be finite or infinite, we have nothing in our mind answering to the name world; for whatsoever we imagine, is therefore finite, though our computation reach the fixed stars, or the ninth or tenth, nay, the thousandth sphere. The meaning of the question is this only, whether God has actually made so great an addition of body to body, as we are able to make of space to space.

13. And, therefore, that which is commonly said, that space and time may be divided infinitely, is not to be so understood, as if there might be any infinite or eternal division; but rather to be taken in this sense, whatsoever is divided, is divided into such parts as may again be divided; or thus, the least divisible thing is not to be given; or, as geometricians have it, no quantity is so small, but a less may be taken; which may easily be demonstrated in this manner. Let any space or time, that which was thought to be the least divisible, be divided into two equal parts, A and B. I say either of them, as A, may be divided again. For suppose the part A to be contiguous to the part B of one side, and of the other side to some other space equal to B. This whole space, therefore, being greater than the 101space given, is divisible. Wherefore, if it be divided into two equal parts, the part in the middle, which is A, will be also divided into two equal parts; and therefore A was divisible.

1. Body defined.—2. Accident defined.—3. How an accident may be understood to be in its subject.—4. Magnitude, what it is.—5. Place, what it is, and that it is immovable.—6. What is full and empty.—7. Here, there, somewhere, what they signify.—8. Many bodies cannot be in one place, nor one body in many places.—9. Contiguous and continual, what they are.—10. The definition of motion. No motion intelligible but with time.—11. What it is to be at rest, to have been moved, and to be moved. No motion to be conceived, without the conception of past and future.—12. A point, a line, superficies and solid, what they are.—13. Equal, greater, and less in bodies and magnitudes, what they are.—14. One and the same body has always one and the same magnitude.—15. Velocity, what it is.—16. Equal, greater, and less in times, what they are.—17. Equal, greater, and less, in velocity, what.—18. Equal, greater, and less, in motion, what.—19. That which is at rest, will always be at rest, except it be moved by some external thing; and that which is moved, will always be moved, unless it be hindered by some external thing.—20. Accidents are generated and destroyed, but bodies not so.—21. An accident cannot depart from its subject.—22. Nor be moved.—23. Essence, form, and matter, what they are.—24. First matter, what.—25. That the whole is greater than any part thereof, why demonstrated.

1. Having understood what imaginary space is, in which we supposed nothing remaining without us, but all those things to be destroyed, that, by 102existing heretofore, left images of themselves in our minds; let us now suppose some one of those things to be placed again in the world, or created anew. It is necessary, therefore, that this new-created or replaced thing do not only fill some part of the space above mentioned, or be coincident and coextended with it, but also that it have no dependance upon our thought. And this is that which, for the extension of it, we commonly call body; and because it depends not upon our thought, we say is a thing subsisting of itself; as also existing, because without us; and, lastly, it is called the subject, because it is so placed in and subjected to imaginary space, that it may be understood by reason, as well as perceived by sense. The definition, therefore, of body may be this, a body is that, which having no dependance upon our thought, is coincident or coextended with some part of space.

2. But what an accident is cannot so easily be explained by any definition, as by examples. Let us imagine, therefore, that a body fills any space, or is coextended with it; that coextension is not the coextended body: and, in like manner, let us imagine that the same body is removed out of its place; that removing is not the removed body: or let us think the same not removed; that not removing or rest is not the resting body. What, then, are these things? They are accidents of that body. But the thing in question is, what is an accident? which is an enquiry after that which we know already, and not that which we should enquire after. For who does not always and in the same manner understand him that says any 103thing is extended, or moved, or not moved? But most men will have it be said that an accident is something, namely, some part of a natural thing, when, indeed, it is no part of the same. To satisfy these men, as well as may be, they answer best that define an accident to be the manner by which any body is conceived; which is all one as if they should say, an accident is that faculty of any body, by which it works in us a conception of itself. Which definition, though it be not an answer to the question propounded, yet it is an answer to that question which should have been propounded, namely, whence does it happen that one part of any body appears here, another there? For this is well answered thus: it happens from the extension of that body. Or, how comes it to pass that the whole body, by succession, is seen now here, now there? and the answer will be, by reason of its motion. Or, lastly, whence is it that any body possesseth the same space for sometime? and the answer will be, because it is not moved. For if concerning the name of a body, that is, concerning a concrete name, it be asked, what is it? the answer must be made by definition; for the question is concerning the signification of the name. But if it be asked concerning an abstract name, what is it? the cause is demanded why a thing appears so or so. As if it be asked, what is hard? The answer will be, hard is that, whereof no part gives place, but when the whole gives place. But if it be demanded, what is hardness? a cause must be shewn why a part does not give place, except the 104whole give place. Wherefore, I define an accident to be the manner of our conception of body.

3. When an accident is said to be in a body, it is not so to be understood, as if any thing were contained in that body; as if, for example, redness were in blood, in the same manner, as blood is in a bloody cloth, that is, as a part in the whole; for so, an accident would be a body also. But, as magnitude, or rest, or motion, is in that which is great, or which resteth, or which is moved, (which, how it is to be understood, every man understands) so also, it is to be understood, that every other accident is in its subject. And this, also, is explicated by Aristotle no otherwise than negatively, namely, that an accident is in its subject, not as any part thereof, but so as that it may be away, the subject still remaining; which is right, saving that there are certain accidents which can never perish except the body perish also; for no body can be conceived to be without extension, or without figure. All other accidents, which are not common to all bodies, but peculiar to some only, as to be at rest, to be moved, colour, hardness, and the like, do perish continually, and are succeeded by others; yet so, as that the body never perisheth. And as for the opinion that some may have, that all other accidents are not in their bodies in the same manner that extension, motion, rest, or figure, are in the same; for example, that colour, heat, odour, virtue, vice, and the like, are otherwise in them, and, as they say, inherent; I desire they would suspend their judgment for the present, and expect a little, till it be found out 105by ratiocination, whether these very accidents are not also certain motions either of the mind of the perceiver, or of the bodies themselves which are perceived; for in the search of this, a great part of natural philosophy consists.

4. The extension of a body, is the same thing with the magnitude of it, or that which some call real space. But this magnitude does not depend upon our cogitation, as imaginary space doth; for this is an effect of our imagination, but magnitude is the cause of it; this is an accident of the mind, that of a body existing out of the mind.

5. That space, by which word I here understand imaginary space, which is coincident with the magnitude of any body, is called the place of that body; and the body itself is that which we call the thing placed. Now place, and the magnitude of the thing placed, differ. First in this, that a body keeps always the same magnitude, both when it is at rest, and when it is moved; but when it is moved, it does not keep the same place. Secondly in this, that place is a phantasm of any body of such and such quantity and figure; but magnitude is the peculiar accident of every body; for one body may at several times have several places, but has always one and the same magnitude. Thirdly in this, that place is nothing out of the mind, nor magnitude any thing within it. And lastly, place is feigned extension, but magnitude true extension; and a placed body is not extension, but a thing extended. Besides, place is immovable; for, seeing that which is moved, is understood to be carried from place to place, if place were moved, it would also be carried from place to 106place, so that one place must have another place, and that place another place, and so on infinitely, which is ridiculous. And as for those, that, by making place to be of the same nature with real space, would from thence maintain it to be immovable, they also make place, though they do not perceive they make it so, to be a mere phantasm. For whilst one affirms that place is therefore said to be immovable, because space in general is considered there; if he had remembered that nothing is general or universal besides names or signs, he would easily have seen that that space, which he says is considered in general, is nothing but a phantasm, in the mind or the memory, of a body of such magnitude and such figure. And whilst another says: real space is made immovable by the understanding; as when, under the superficies of running water, we imagine other and other water to come by continual succession, that superficies fixed there by the understanding, is the immovable place of the river: what else does he make it to be but a phantasm, though he do it obscurely and in perplexed words? Lastly, the nature of place does not consist in the superficies of the ambient, but in solid space; for the whole placed body is coextended with its whole place, and every part of it with every answering part of the same place; but seeing every placed body is a solid thing, it cannot be understood to be coextended with superficies. Besides, how can any whole body be moved, unless all its parts be moved together with it? Or how can the internal parts of it be moved, but by leaving their place? But the internal parts of a body cannot leave the 107superficies of an external part contiguous to it; and, therefore, it follows, that if place be the superficies of the ambient, then the parts of a body moved, that is, bodies moved, are not moved.

6. Space, or place, that is possessed by a body, is called full, and that which is not so possessed, is called empty.

7. Here, there, in the country, in the city, and other the like names, by which answer is made to the question where is it? are not properly names of place, nor do they of themselves bring into the mind the place that is sought; for here and there signify nothing, unless the thing be shewn at the same time with the finger or something else; but when the eye of him that seeks, is, by pointing or some other sign, directed to the thing sought, the place of it is not hereby defined by him that answers, but found out by him that asks the question. Now such shewings as are made by words only, as when we say, in the country, or in the city, are some of greater latitude than others, as when we say, in the country, in the city, in such a street, in a house, in the chamber, in bed, &c. For these do, by little and little, direct the seeker nearer to the proper place; and yet they do not determine the same, but only restrain it to a lesser space, and signify no more, than that the place of the thing is within a certain space designed by those words, as a part is in the whole. And all such names, by which answer is made to the question where? have, for their highest genus, the name somewhere. From whence it may be understood, that whatsoever is somewhere, is in some place properly so called, which place is part of 108that greater space that is signified by some of these names, in the country, in the city, or the like.

8. A body, and the magnitude, and the place thereof, are divided by one and the same act of the mind; for, to divide an extended body, and the extension thereof, and the idea of that extension, which is place, is the same with dividing any one of them; because they are coincident, and it cannot be done but by the mind, that is by the division of space. From whence it is manifest, that neither two bodies can be together in the same place, nor one body be in two places at the same time. Not two bodies in the same place; because when a body that fills its whole place is divided into two, the place itself is divided into two also, so that there will be two places. Not one body in two places; for the place that a body fills being divided into two, the placed body will be also divided into two; for, as I said, a place and the body that fills that place, are divided both together; and so there will be two bodies.

9. Two bodies are said to be contiguous to one another, and continual, in the same manner as spaces are; namely, those are contiguous, between which there is no space. Now, by space I understand, here as formerly, an idea or phantasm of a body. Wherefore, though between two bodies there be put no other body, and consequently no magnitude, or, as they call it, real space, yet if another body may be put between them, that is, if there intercede any imagined space which may receive another body, then those bodies are not contiguous. And this is so easy to be understood, that I should wonder at some men, who being 109otherwise skilful enough in philosophy, are of a different opinion, but that I find that most of those that affect metaphysical subtleties wander from truth, as if they were led out of their way by an ignis fatuus. For can any man that has his natural senses, think that two bodies must therefore necessarily touch one another, because no other body is between them? Or that there can be no vacuum, because vacuum is nothing, or as they call it, non ens? Which is as childish, as if one should reason thus; no man can fast, because to fast is to eat nothing; but nothing cannot be eaten. Continual, are any two bodies that have a common part; and more than two are continual, when every two, that are next to one another, are continual.

10. MOTION is a continual relinquishing of one place, and acquiring of another; and that place which is relinquished is commonly called the terminus a quo, as that which is acquired is called the terminus ad quem; I say a continual relinquishing, because no body, how little soever, can totally and at once go out of its former place into another, so, but that some part of it will be in a part of a place which is common to both, namely, to the relinquished and the acquired places. | A G B I E
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C H D K F| For example, let any body be in the place A C B D; the same body cannot come into the place B D E F, but it must first be in G H I K, whose part G H B D is common to both the places A C B D, and G H I K, and whose part B D I K, is common to both the places G H I K, and B D E F. Now it cannot be conceived 110that any thing can be moved without time; for time is, by the definition of it, a phantasm, that is, a conception of motion; and, therefore, to conceive that any thing may be moved without time, were to conceive motion without motion, which is impossible.

11. That is said to be at rest, which, during any time, is in one place; and that to be moved, or to have been moved, which, whether it be now at rest or moved, was formerly in another place than that which it is now in. From which definitions it may be inferred, first, that whatsoever is moved, has been moved; for if it be still in the same place in which it was formerly, it is at rest, that is, it is not moved, by the definition of rest; but if it be in another place, it has been moved, by the definition of moved. Secondly, that what is moved, will yet be moved; for that which is moved, leaveth the place where it is, and therefore will be in another place, and consequently will be moved still. Thirdly, that whatsoever is moved, is not in one place during any time, how little soever that time be; for by the definition of rest, that which is in one place during any time, is at rest.

There is a certain sophism against motion, which seems to spring from the not understanding of this last proposition. For they say, that, if any body be moved, it is moved either in the place where it is, or in the place where it is not; both which are false; and therefore nothing is moved. But the falsity lies in the major proposition; for that which is moved, is neither moved in the place where it is, nor in the place where is not; but 111from the place where it is, to the place where it is not. Indeed it cannot be denied but that whatsoever is moved, is moved somewhere, that is, within some space; but then the place of that body is not that whole space, but a part of it, as is said above in the seventh article. From what is above demonstrated, namely, that whatsoever is moved, has also been moved, and will be moved, this also may be collected, that there can be no conception of motion, without conceiving past and future time.

12. Though there be no body which has not some magnitude, yet if, when any body is moved, the magnitude of it be not at all considered, the way it makes is called a line, or one single dimension; and the space, through which it passeth, is called length; and the body itself, a point; in which sense the earth is called a point, and the way of its yearly revolution, the ecliptic line. But if a body, which is moved, be considered as long, and be supposed to be so moved, as that all the several parts of it be understood to make several lines, then the way of every part of that body is called breadth, and the space which is made is called superficies, consisting of two dimensions, one whereof to every several part of the other is applied whole. Again, if a body be considered as having superficies, and be understood to be so moved, that all the several parts of it describe several lines, then the way of every part of that body is called thickness or depth, and the space which is made is called solid, consisting of three dimensions, any two whereof are applied whole to every several part of the third.

112But if a body be considered as solid, then it is not possible that all the several parts of it should describe several lines; for what way soever it be moved, the way of the following part will fall into the way of the part before it, so that the same solid will still be made which the foremost superficies would have made by itself. And therefore there can be no other dimension in any body, as it is a body, than the three which I have now described; though, as it shall be shewed hereafter, velocity, which is motion according to length, may, by being applied to all the parts of a solid, make a magnitude of motion, consisting of four dimensions; as the goodness of gold, computed in all the parts of it, makes the price and value thereof.

13. Bodies, how many soever they be, that can fill every one the place of every one, are said to be equal every one to every other. Now, one body may fill the same place which another body filleth, though it be not of the same figure with that other body, if so be that it may be understood to be reducible to the same figure, either by flexion or transposition of the parts. And one body is greater than another body, when a part of that is equal to all this; and less, when all that is equal to a part of this. Also, magnitudes are equal, or greater, or lesser, than one another, for the same consideration, namely, when the bodies, of which they are the magnitudes, are either equal, or greater, or less, &c.

14. One and the same body is always of one and the same magnitude. For seeing a body and the magnitude and place thereof cannot be comprehended in the mind otherwise than as they 113are coincident, if any body be understood to be at rest, that is, to remain in the same place during some time, and the magnitude thereof be in one part of that time greater, and in another part less, that body's place, which is one and the same, will be coincident sometimes with greater, sometimes with less magnitude, that is, the same place will be greater and less than itself, which is impossible. But there would be no need at all of demonstrating a thing that is in itself so manifest, if there were not some, whose opinion concerning bodies and their magnitudes is, that a body may exist separated from its magnitude, and have greater or less magnitude bestowed upon it, making use of this principle for the explication of the nature of rarum and densum.

15. Motion, in as much as a certain length may in a certain time be transmitted by it, is called VELOCITY or swiftness: &c. For though swift be very often understood with relation to slower or less swift, as great is in respect of less, yet nevertheless, as magnitude is by philosophers taken absolutely for extension, so also velocity or swiftness may be put absolutely for motion according to length.

16. Many motions are said to be made in equal times, when every one of them begins and ends together with some other motion, or if it had begun together, would also have ended together with the same. For time, which is a phantasm of motion, cannot be reckoned but by some exposed motion; as in dials by the motion of the sun or of the hand; and if two or more motions begin and end with this motion, they are said to be made in 114equal times; from whence also it is easy to understand what it is to be moved in greater or longer time, and in less time or not so long; namely, that that is longer moved, which beginning with another, ends later; or ending together, began sooner.

17. Motions are said to be equally swift, when equal lengths are transmitted in equal times; and greater swiftness is that, wherein greater length is passed in equal time, or equal length in less time. Also that swiftness by which equal lengths are passed in equal parts of time, is called uniform swiftness or motion; and of motions not uniform, such as become swifter or slower by equal increasings or decreasings in equal parts of time, are said to be accelerated or retarded uniformly.

18. But motion is said to be greater, less, and equal, not only in regard of the length which is transmitted in a certain time, that is, in regard of swiftness only, but of swiftness applied to every smallest particle of magnitude; for when any body is moved, every part of it is also moved; and supposing the parts to be halves, the motions of those halves have their swiftness equal to one another, and severally equal to that of the whole; but the motion of the whole is equal to those two motions, either of which is of equal swiftness with it; and therefore it is one thing for two motions to be equal to one another, and another thing for them to be equally swift. And this is manifest in two horses that draw abreast, where the motion of both the horses together is of equal swiftness with the motion of either of them singly; but the motion of both is greater than the motion of one 115of them, namely, double. Wherefore motions are said to be simply equal to one another, when the swiftness of one, computed in every part of its magnitude, is equal to the swiftness of the other computed also in every part of its magnitude: and greater than one another, when the swiftness of one computed as above, is greater than the swiftness of the other so computed; and less, when less. Besides, the magnitude of motion computed in this manner is that which is commonly called FORCE.

19. Whatsoever is at rest, will always be at rest, unless there be some other body besides it, which, by endeavouring to get into its place by motion, suffers it no longer to remain at rest. For suppose that some finite body exist and be at rest, and that all space besides be empty; if now this body begin to be moved, it will certainly be moved some way; seeing therefore there was nothing in that body which did not dispose it to rest, the reason why it is moved this way is in something out of it; and in like manner, if it had been moved any other way, the reason of motion that way had also been in something out of it; but seeing it was supposed that nothing is out of it, the reason of its motion one way would be the same with the reason of its motion every other way, wherefore it would be moved alike all ways at once; which is impossible.

In like manner, whatsoever is moved, will always be moved, except there be some other body besides it, which causeth it to rest. For if we suppose nothing to be without it, there will be no reason why it should rest now, rather than at 116another time; wherefore its motion would cease in every particle of time alike; which is not intelligible.

20. When we say a living creature, a tree, or any other specified body is generated or destroyed, it is not to be so understood as if there were made a body of that which is not-body, or not a body of a body, but of a living creature not a living creature, of a tree not a tree, &c. that is, that those accidents for which we call one thing a living creature, another thing a tree, and another by some other name, are generated and destroyed; and that therefore the same names are not to be given to them now, which were given them before. But that magnitude for which we give to any thing the name of body is neither generated nor destroyed. For though we may feign in our mind that a point may swell to a huge bulk, and that this may again contract itself to a point; that is, though we may imagine something to arise where before was nothing, and nothing to be there where before was something, yet we cannot comprehend in our mind how this may possibly be done in nature. And therefore philosophers, who tie themselves to natural reason, suppose that a body can neither be generated nor destroyed, but only that it may appear otherwise than it did to us, that is, under different species, and consequently be called by other and other names; so that that which is now called man, may at another time have the name of not-man; but that which is once called body, can never be called not-body. But it is manifest, that all other accidents besides magnitude or extension may be generated and destroyed; 117as when a white thing is made black, the whiteness that was in it perisheth, and the blackness that was not in it is now generated; and therefore bodies, and the accidents under which they appear diversely, have this difference, that bodies are things, and not generated; accidents are generated, and not things.

21. And therefore, when any thing appears otherwise than it did by reason of other and other accidents, it is not to be thought that an accident goes out of one subject into another, (for they are not, as I said above, in their subjects as a part in the whole, or as a contained thing in that which contains it, or as a master of a family in his house,) but that one accident perisheth, and another is generated. For example, when the hand, being moved, moves the pen, motion does not go out of the hand into the pen; for so the writing might be continued though the hand stood still; but a new motion is generated in the pen, and is the pen's motion.

22. And therefore also it is improper to say, an accident is moved; as when, instead of saying, figure is an accident of a body carried away, we say, a body carries away its figure.

23. Now that accident for which we give a certain name to any body, or the accident which denominates its subject, is commonly called the ESSENCE thereof; as rationality is the essence of a man; whiteness, of any white thing, and extension the essence of a body. And the same essence, in as much as it is generated, is called the FORM. Again, a body, in respect of any accident, is called the SUBJECT, and in respect of the form it is called the MATTER.

118Also, the production or perishing of any accident makes its subject be said to be changed; only the production or perishing of form makes it be said it is generated or destroyed; but in all generation and mutation, the name of matter still remains. For a table made of wood is not only wooden, but wood; and a statue of brass is brass as well as brazen; though Aristotle, in his Metaphysics, says, that whatsoever is made of any thing ought not to be called ἐκεινὸ, but ἐκέινινον; as that which is made of wood, not ξύλον, but ξύλινον, that is, not wood, but wooden.

24. And as for that matter which is common to all things, and which philosophers, following Aristotle, usually call materia prima, that is, first matter, it is not any body distinct from all other bodies, nor is it one of them. What then is it? A mere name; yet a name which is not of vain use; for it signifies a conception of body without the consideration of any form or other accident except only magnitude or extension, and aptness to receive form and other accident. So that whensoever we have use of the name body in general, if we use that of materia prima, we do well. For as when a man not knowing which was first, water or ice, would find out which of the two were the matter of both, he would be fain to suppose some third matter which were neither of these two; so he that would find out what is the matter of all things, ought to suppose such as is not the matter of anything that exists. Wherefore materia prima is nothing; and therefore they do not attribute to it either form or any other accident besides quantity; whereas all singular things have their forms and accidents certain.

119Materia prima, therefore, is body in general, that is, body considered universally, not as having neither form nor any accident, but in which no form nor any other accident but quantity are at all considered, that is, they are not drawn into argumentation.

25. From what has been said, those axioms may be demonstrated, which are assumed by Euclid in the beginning of his first element, about the equality and inequality of magnitudes; of which, omitting the rest, I will here demonstrate only this one, the whole is greater than any part thereof; to the end that the reader may know that those axioms are not indemonstrable, and therefore not principles of demonstration; and from hence learn to be wary how he admits any thing for a principle, which is not at least as evident as these are. Greater is defined to be that, whose part is equal to the whole of another. Now if we suppose any whole to be A, and a part of it to be B; seeing the whole B is equal to itself, and the same B is a part of A; therefore a part of A will be equal to the whole B. Wherefore, by the definition above, A is greater than B; which was to be proved.

1. Action and passion, what they are.—2. Action and passion mediate and immediate.—3. Cause simply taken. Cause without which no effect follows, or cause necessary by supposition.—4. Cause efficient and material.—5. An entire cause is always sufficient to produce its effect. At the same instant that the cause is entire, the effect is produced. Every effect has a necessary cause.—6. The generation of effects is continual. What is the beginning in causation.—7. No cause of motion but in a body contiguous and moved.—8. The same agents and patients, if alike disposed, produce like effects though at different times.—9. All mutation is motion.—10. Contingent accidents, what they are.

1. A body is said to work upon or act, that is to say, do something to another body, when it either generates or destroys some accident in it: and the body in which an accident is generated or destroyed is said to suffer, that is, to have something done to it by another body; as when one body by putting forwards another body generates motion in it, it is called the AGENT; and the body in which motion is so generated, is called the PATIENT; so fire that warms the hand is the agent, and the hand, which is warmed, is the patient. That accident, which is generated in the patient, is called the EFFECT.

2. When an agent and patient are contiguous to one another, their action and passion are then said to be immediate, otherwise, mediate; and when another body, lying betwixt the agent and patient, is contiguous to them both, it is then itself both an 121agent and a patient; an agent in respect of the body next after it, upon which it works, and a patient in respect of the body next before it, from which it suffers. Also, if many bodies be so ordered that every two which are next to one another be contiguous, then all those that are betwixt the first and the last are both agents and patients, and the first is an agent only, and the last a patient only.

3. An agent is understood to produce its determined or certain effect in the patient, according to some certain accident or accidents, with which both it and the patient are affected; that is to say, the agent hath its effect precisely such, not because it is a body, but because such a body, or so moved. For otherwise all agents, seeing they are all bodies alike, would produce like effects in all patients. And therefore the fire, for example, does not warm, because it is a body, but because it is hot; nor does one body put forward another body because it is a body, but because it is moved into the place of that other body. |Cause without which no effect follows, or cause necessary by supposition.|The cause, therefore, of all effects consists in certain accidents both in the agents and in the patients; which when they are all present, the effect is produced; but if any one of them be wanting, it is not produced; and that accident either of the agent or patient, without which the effect cannot be produced, is called causa sine qua non, or cause necessary by supposition, as also the cause requisite for the production of the effect. But a CAUSE simply, or an entire cause, is the aggregate of all the accidents both of the agents how many soever they be, and of the patient, put together; which when they 122are all supposed to be present, it cannot be understood but that the effect is produced at the same instant; and if any one of them be wanting, it cannot be understood but that the effect is not produced.

4. The aggregate of accidents in the agent or agents, requisite for the production of the effect, the effect being produced, is called the efficient cause thereof; and the aggregate of accidents in the patient, the effect being produced, is usually called the material cause; I say the effect being produced; for where there is no effect, there can be no cause; for nothing can be called a cause, where there is nothing that can be called an effect. But the efficient and material causes are both but partial causes, or parts of that cause, which in the next precedent article I called an entire cause. And from hence it is manifest, that the effect we expect, though the agents be not defective on their part, may nevertheless be frustrated by a defect in the patient; and when the patient is sufficient, by a defect in the agents.

5. An entire cause is always sufficient for the production of its effect, if the effect be at all possible. For let any effect whatsoever be propounded to be produced; if the same be produced, it is manifest that the cause which produced it was a sufficient cause; but if it be not produced, and yet be possible, it is evident that something was wanting either in some agent, or in the patient, without which it could not be produced; that is, that some accident was wanting which was requisite for its production; and therefore, that cause was not entire, which is contrary to what was supposed.

123It follows also from hence, that in whatsoever instant the cause is entire, in the same instant the effect is produced. For if it be not produced, something is still wanting, which is requisite for the production of it; and therefore the cause was not entire, as was supposed.

And seeing a necessary cause is defined to be that, which being supposed, the effect cannot but follow; this also may be collected, that whatsoever effect is produced at any time, the same is produced by a necessary cause. For whatsoever is produced, in as much as it is produced, had an entire cause, that is, had all those things, which being supposed, it cannot be understood but that the effect follows; that is, it had a necessary cause. And in the same manner it may be shewn, that whatsoever effects are hereafter to be produced, shall have a necessary cause; so that all the effects that have been, or shall be produced, have their necessity in things antecedent.

6. And from this, that whensoever the cause is entire, the effect is produced in the same instant, it is manifest that causation and the production of effects consist in a certain continual progress; so that as there is a continual mutation in the agent or agents, by the working of other agents upon them, so also the patient, upon which they work, is continually altered and changed. For example: as the heat of the fire increases more and more, so also the effects thereof, namely, the heat of such bodies as are next to it, and again, of such other bodies as are next to them, increase more and more accordingly; which is already no little argument that all mutation consists in motion 124only; the truth whereof shall be further demonstrated in the ninth article. But in this progress of causation, that is, of action and passion, if any man comprehend in his imagination a part thereof, and divide the same into parts, the first part or beginning of it cannot be considered otherwise than as action or cause; for, if it should be considered as effect or passion, then it would be necessary to consider something before it, for its cause or action; which cannot be, for nothing can be before the beginning. And in like manner, the last part is considered only as effect; for it cannot be called cause, if nothing follow it; but after the last, nothing follows. And from hence it is, that in all action the beginning and cause are taken for the same thing. But every one of the intermediate parts are both action and passion, and cause and effect, according as they are compared with the antecedent or subsequent part.

7. There can be no cause of motion, except in a body contiguous and moved. For let there be any two bodies which are not contiguous, and betwixt which the intermediate space is empty, or, if filled, filled with another body which is at rest; and let one of the propounded bodies be supposed to be at rest; I say it shall always be at rest. For if it shall be moved, the cause of that motion, by the 8th chapter, article 19, will be some external body; and, therefore, if between it and that external body there be nothing but empty space, then whatsoever the disposition be of that external body or of the patient itself, yet if it be supposed to be now at rest, we may conceive it will continue so till it be touched by some other body. 125But seeing cause, by the definition, is the aggregate of all such accidents, which being supposed to be present, it cannot be conceived but that the effect will follow, those accidents, which are either in external bodies, or in the patient itself, cannot be the cause of future motion. And in like manner, seeing we may conceive that whatsoever is at rest will still be at rest, though it be touched by some other body, except that other body be moved; therefore in a contiguous body, which is at rest, there can be no cause of motion. Wherefore there is no cause of motion in any body, except it be contiguous and moved.

The same reason may serve to prove that whatsoever is moved, will always be moved on in the same way and with the same velocity, except it be hindered by some other contiguous and moved body; and consequently that no bodies, either when they are at rest, or when there is an interposition of vacuum, can generate or extinguish or lessen motion in other bodies. There is one that has written that things moved are more resisted by things at rest, than by things contrarily moved; for this reason, that he conceived motion not to be so contrary to motion as rest. That which deceived him was, that the words rest and motion are but contradictory names; whereas motion, indeed, is not resisted by rest, but by contrary motion.

8. But if a body work upon another body at one time, and afterwards the same body work upon the same body at another time, so that both the agent and patient, and all their parts, be in all things as they were; and there be no difference, except only in time, that is, that one action be former, the 126other later in time; it is manifest of itself, that the effects will be equal and like, as not differing in anything besides time. And as effects themselves proceed from their causes, so the diversity of them depends upon the diversity of their causes also.

9. This being true, it is necessary that mutation can be nothing else but motion of the parts of that body which is changed. For first, we do not say anything is changed, but that which appears to our senses otherwise than it appeared formerly. Secondly, both those appearances are effects produced in the sentient; and, therefore, if they be different, it is necessary, by the preceding article, that either some part of the agent, which was formerly at rest, is now moved, and so the mutation consists in this motion; or some part, which was formerly moved, is now otherwise moved, and so also the mutation consists in this new motion; or which, being formerly moved, is now at rest, which, as I have shewn above, cannot come to pass without motion; and so again, mutation is motion; or lastly, it happens in some of these manners to the patient, or some of its parts; so that mutation, howsoever it be made, will consist in the motion of the parts, either of the body which is perceived, or of the sentient body, or of both. Mutation therefore is motion, namely, of the parts either of the agent or of the patient; which was to be demonstrated. And to this it is consequent, that rest cannot be the cause of anything, nor can any action proceed from it; seeing neither motion nor mutation can be caused by it.

10. Accidents, in respect of other accidents which precede them, or are before them in time, 127and upon which they do not depend as upon their causes, are called contingent accidents; I say, in respect of those accidents by which they are not generated; for, in respect of their causes, all things come to pass with equal necessity; for otherwise they would have no causes at all; which, of things generated, is not intelligible.

1. Power and cause are the same thing.—2. An act is produced at the same instant in which the power is plenary.—3. Active and passive power are parts only of plenary power.—4. An act, when said to be possible.—5. An act necessary and contingent, what.—6. Active power consists in motion.—7. Cause, formal and final, what they are.

1. Correspondent to cause and effect, are POWER and ACT; nay, those and these are the same things; though, for divers considerations, they have divers names. For whensoever any agent has all those accidents which are necessarily requisite for the production of some effect in the patient, then we say that agent has power to produce that effect, if it be applied to a patient. But, as I have shewn in the precedent chapter, those accidents constitute the efficient cause; and therefore the same accidents, which constitute the efficient cause, constitute also the power of the agent. Wherefore the power of the agent and the efficient cause are the same thing. But they are considered with this difference, that cause is 128so called in respect of the effect already produced, and power in respect of the same effect to be produced hereafter; so that cause respects the past, power the future time. Also, the power of the agent is that which is commonly called active power.

In like manner, whensoever any patient has all those accidents which it is requisite it should have, for the production of some effect in it, we say it is in the power of that patient to produce that effect, if it be applied to a fitting agent. But those accidents, as is defined in the precedent chapter, constitute the material cause; and therefore the power of the patient, commonly called passive power, and material cause, are the same thing; but with this different consideration, that in cause the past time, and in power the future, is respected. Wherefore the power of the agent and patient together, which may be called entire or plenary power, is the same thing with entire cause; for they both consist in the sum or aggregate of all the accidents, as well in the agent as in the patient, which are requisite for the production of the effect. Lastly, as the accident produced is, in respect of the cause, called an effect, so in respect of the power, it is called an act.

2. As therefore the effect is produced in the same instant in which the cause is entire, so also every act that may be produced, is produced in the same instant in which the power is plenary. And as there can be no effect but from a sufficient and necessary cause, so also no act can be produced but by sufficient power, or that power by which it could not but be produced.

1293. And as it is manifest, as I have shewn, that the efficient and material causes are severally and by themselves parts only of an entire cause, and cannot produce any effect but by being joined together, so also power, active and passive, are parts only of plenary and entire power; nor, except they be joined, can any act proceed from them; and therefore these powers, as I said in the first article, are but conditional, namely, the agent has power, if it be applied to a patient; and the patient has power, if it be applied to an agent; otherwise neither of them have power, nor can the accidents, which are in them severally, be properly called powers; nor any action be said to be possible for the power of the agent alone or of the patient alone.

4. For that is an impossible act, for the production of which there is no power plenary. For seeing plenary power is that in which all things concur, which are requisite for the production of an act, if the power shall never be plenary, there will always be wanting some of those things, without which the act cannot be produced; wherefore that act shall never be produced; that is, that act is IMPOSSIBLE: and every act, which is not impossible, is POSSIBLE. Every act, therefore, which is possible, shall at some time be produced; for if it shall never be produced, then those things shall never concur which are requisite for the production of it; wherefore that act is impossible, by the definition; which is contrary to what was supposed.

5. A necessary act is that, the production whereof it is impossible to hinder; and therefore 130every act, that shall be produced, shall necessarily be produced; for, that it shall not be produced, is impossible; because, as is already demonstrated, every possible act shall at some time be produced; nay, this proposition, what shall be, shall be, is as necessary a proposition as this, a man is a man.

But here, perhaps, some man may ask whether those future things, which are commonly called contingents, are necessary. I say, therefore, that generally all contingents have their necessary causes, as is shewn in the preceding chapter; but are called contingents in respect of other events, upon which they do not depend; as the rain, which shall be tomorrow, shall be necessary, that is, from necessary causes; but we think and say it happens by chance, because we do not yet perceive the causes thereof, though they exist now; for men commonly call that casual or contingent, whereof they do not perceive the necessary cause; and in the same manner they used to speak of things past, when not knowing whether a thing be done or no, they say it is possible it never was done.

Wherefore, all propositions concerning future things, contingent or not contingent, as this, it will rain tomorrow, or this, tomorrow the sun will rise, are either necessarily true, or necessarily false; but we call them contingent, because we do not yet know whether they be true or false; whereas their verity depends not upon our knowledge, but upon the foregoing of their causes. But there are some, who though they confess this whole proposition, tomorrow it will either rain, or not rain, to be true, yet they will not acknowledge the parts of it, as, tomorrow it will rain, or, tomorrow 131it will not rain, to be either of them true by itself; because they say neither this nor that is true determinately. But what is this determinately true, but true upon our knowledge, or evidently true? And therefore they say no more but that it is not yet known whether it be true or no; but they say it more obscurely, and darken the evidence of the truth with the same words, with which they endeavour to hide their own ignorance.

6. In the 9th article of the preceding chapter, I have shewn that the efficient cause of all motion and mutation consists in the motion of the agent, or agents; and in the first article of this chapter, that the power of the agent is the same thing with the efficient cause. From whence it may be understood, that all active power consists in motion also; and that power is not a certain accident, which differs from all acts, but is, indeed, an act, namely, motion, which is therefore called power, because another act shall be produced by it afterwards. For example, if of three bodies the first put forward the second, and this the third, the motion of the second, in respect of the first which produceth it, is the act of the second body; but, in respect of the third, it is the active power of the same second body.

7. The writers of metaphysics reckon up two other causes besides the efficient and material, namely, the ESSENCE, which some call the formal cause, and the END, or final cause; both which are nevertheless efficient causes. For when it is said the essence of a thing is the cause thereof, as to be rational is the cause of man, it is not intelligible; for it is all one, as if it were said, to be a 132man is the cause of man; which is not well said. And yet the knowledge of the essence of anything, is the cause of the knowledge of the thing itself; for, if I first know that a thing is rational, I know from thence, that the same is man; but this is no other than an efficient cause. A final cause has no place but in such things as have sense and will; and this also I shall prove hereafter to be an efficient cause.

1. What it is for one thing to differ from another.—2. To differ in number, magnitude, species, and genus, what.—3. What is relation, proportion, and relatives.—4. Proportionals, what.—5. The proportion of magnitudes to one another, wherein it consists.—6. Relation is no new accident, but one of those that were in the relative before the relation or comparison was made. Also the causes of accidents in the correlatives, are the cause of relation.—7. Of the beginning of individuation.

1. Hitherto I have spoken of body simply, and accidents common to all bodies, as magnitude, motion, rest, action, passion, power, possible, &c.; and I should now descend to those accidents by which one body is distinguished from another, but that it is first to be declared what it is to be distinct and not distinct, namely, what are the SAME and DIFFERENT; for this also is common to all bodies, that they may be distinguished and differenced from one another. Now, two bodies are said to differ from one another, when something may be said of one of them, which cannot be said of the other at the same time.

1332. And, first of all, it is manifest that no two bodies are the same; for seeing they are two, they are in two places at the same time; as that, which is the same, is at the same time in one and the same place. All bodies therefore differ from one another in number, namely, as one and another; so that the same and different in number, are names opposed to one another by contradiction.

In magnitude bodies differ when one is greater than another, as a cubit long, and two cubits long, of two pound weight, and of three pound weight. And to these, equals are opposed.

Bodies, which differ more than in magnitude, are called unlike; and those, which differ only in magnitude, like. Also, of unlike bodies, some are said to differ in the species, others in the genus; in the species, when their difference is perceived by one and the same sense, as white and black; and in the genus, when their difference is not perceived but by divers senses, as white and hot.

3. And the likeness, or unlikeness, equality, or inequality of one body to another, is called their RELATION; and the bodies themselves relatives or correlatives; Aristotle calls them τὰ πρὸς τί; the first whereof is usually named the antecedent, and the second the consequent; and the relation of the antecedent to the consequent, according to magnitude, namely, the equality, the excess or defect thereof, is called the PROPORTION of the antecedent to the consequent; so that proportion is nothing but the equality or inequality of the magnitude of the antecedent compared to the magnitude of the consequent by their difference only, or compared also with their difference. For example, 134the proportion of three to two consists only in this, that three exceeds two by unity; and the proportion of two to five in this, that two, compared with five, is deficient of it by three, either simply, or compared with the numbers different; and therefore in the proportion of unequals, the proportion of the less to the greater, is called DEFECT; and that of the greater to the less, EXCESS.

4. Besides, of unequals, some are more, some less, and some equally unequal; so that there is proportion of proportions, as well as of magnitudes; namely, where two unequals have relation to two other unequals; as, when the inequality which is between 2 and 3, is compared with the inequality which is between 4 and 5. In which comparison there are always four magnitudes; or, which is all one, if there be but three, the middlemost is twice numbered; and if the proportion of the first to the second, be equal to the proportion of the third to the fourth, then the four are said to be proportionals; otherwise they are not proportionals.

5. The proportion of the antecedent to the consequent consists in their difference, not only simply taken, but also as compared with one of the relatives; that is, either in that part of the greater, by which it exceeds the less, or in the remainder, after the less is taken out of the greater; as the proportion of two to five consists in the three by which five exceeds two, not in three simply only, but also as compared with five or two. For though there be the same difference between two and five, which is between nine and twelve, namely three, yet there is not the same inequality; 135and therefore the proportion of two to five is not in all relation the same with that of nine to twelve, but only in that which is called arithmetical.

6. But we must not so think of relation, as if it were an accident differing from all the other accidents of the relative; but one of them, namely, that by which the comparison is made. For example, the likeness of one white to another white, or its unlikeness to black, is the same accident with its whiteness; and equality and inequality, the same accident with the magnitude of the thing compared, though under another name: for that which is called white or great, when it is not compared with something else, the same when it is compared, is called like or unlike, equal or unequal. And from this it follows that the causes of the accidents, which are in relatives, are the causes also of likeness, unlikeness, equality and inequality; namely, that he, that makes two unequal bodies, makes also their inequality; and he, that makes a rule and an action, makes also, if the action be congruous to the rule, their congruity; if incongruous, their incongruity. And thus much concerning comparison of one body with another.

7. But the same body may at different times be compared with itself. And from hence springs a great controversy among philosophers about the beginning of individuation, namely, in what sense it may be conceived that a body is at one time the same, at another time not the same it was formerly. For example, whether a man grown old be the same man he was whilst he was young, or another man; or whether a city be in different ages the same, or another city. Some place individuity in 136the unity of matter; others, in the unity of form; and one says it consists in the unity of the aggregate of all the accidents together. For matter, it is pleaded that a lump of wax, whether it be spherical or cubical, is the same wax, because the same matter. For form, that when a man is grown from an infant to be an old man, though his matter be changed, yet he is still the same numerical man; for that identity which cannot be attributed to the matter, ought probably to be ascribed to the form. For the aggregate of accidents, no instance can be made; but because, when any new accident is generated, a new name is commonly imposed on the thing, therefore he, that assigned this cause of individuity, thought the thing itself also was become another thing. According to the first opinion, he that sins, and he that is punished, should not be the same man, by reason of the perpetual flux and change of man's body; nor should the city, which makes laws in one age and abrogates them in another, be the same city; which were to confound all civil rights. According to the second opinion, two bodies existing both at once, would be one and the same numerical body. For if, for example, that ship of Theseus, concerning the difference whereof made by continual reparation in taking out the old planks and putting in new, the sophisters of Athens were wont to dispute, were, after all the planks were changed, the same numerical ship it was at the beginning; and if some man had kept the old planks as they were taken out, and by putting them afterwards together in the same order, had again made a ship of them, this, without doubt, had also been the same numerical 137ship with that which was at the beginning; and so there would have been two ships numerically the same, which is absurd. But, according to the third opinion, nothing would be the same it was; so that a man standing would not be the same he was sitting; nor the water, which is in the vessel, the same with that which is poured out of it. Wherefore the beginning of individuation is not always to be taken either from matter alone, or from form alone.

But we must consider by what name anything is called, when we inquire concerning the identity of it. For it is one thing to ask concerning Socrates, whether he be the same man, and another to ask whether he be the same body; for his body, when he is old, cannot be the same it was when he was an infant, by reason of the difference of magnitude; for one body has always one and the same magnitude; yet, nevertheless, he may be the same man. And therefore, whensoever the name, by which it is asked whether a thing be the same it was, is given it for the matter only, then, if the matter be the same, the thing also is individually the same; as the water, which was in the sea, is the same which is afterwards in the cloud; and any body is the same, whether the parts of it be put together, or dispersed; or whether it be congealed, or dissolved. Also, if the name be given for such form as is the beginning of motion, then, as long as that motion remains, it will be the same individual thing; as that man will be always the same, whose actions and thoughts proceed all from the same beginning of motion, namely, that which was in his generation; and that will be the same river 138which flows from one and the same fountain, whether the same water, or other water, or something else than water, flow from thence; and that the same city, whose acts proceed continually from the same institution, whether the men be the same or no. Lastly, if the name be given for some accident, then the identity of the thing will depend upon the matter; for, by the taking away and supplying of matter, the accidents that were, are destroyed, and other new ones are generated, which cannot be the same numerically; so that a ship, which signifies matter so figured, will be the same as long as the matter remains the same; but if no part of the matter be the same, then it is numerically another ship; and if part of the matter remain and part be changed, then the ship will be partly the same, and partly not the same.

1. The definition of quantity.—2. The exposition of quantity, what it is.—3. How line, superficies, and solid, are exposed.—4. How time is exposed.—5. How number is exposed.—6. How velocity is exposed.—7. How weight is exposed.—8. How the proportion of magnitudes is exposed.—9. How the proportion of times and velocities is exposed.

1. What and how manifold dimension is, has been said in the 8th chapter, namely, that there are three dimensions, line or length, superficies, and solid; every one of which, if it be determined, that is, if the limits of it be made known, is commonly called quantity; for by quantity all men understand 139that which is signified by that word, by which answer is made to the question, How much is it? Whensoever, therefore, it is asked, for example, How long is the journey? it is not answered indefinitely, length; nor, when it is asked, How big is the field? is it answered indefinitely, superficies; nor, if a man ask, How great is the bulk? indefinitely, solid: but it is answered determinately, the journey is a hundred miles; the field is a hundred acres; the bulk is a hundred cubical feet; or at least in some such manner, that the magnitude of the thing enquired after may by certain limits be comprehended in the mind. QUANTITY, therefore, cannot otherwise be defined, than to be a dimension determined, or a dimension, whose limits are set out, either by their place, or by some comparison.

2. And quantity is determined two ways; one, by the sense, when some sensible object is set before it; as when a line, a superficies or solid, of a foot or cubit, marked out in some matter, is objected to the eyes; which way of determining, is called exposition, and the quantity so known is called exposed quantity; the other by memory, that is, by comparison with some exposed quantity. In the first manner, when it is asked of what quantity a thing is, it is answered, of such quantity as you see exposed. In the second manner, answer cannot be made but by comparison with some exposed quantity; for if it be asked, how long is the way? the answer is, so many thousand paces; that is, by comparing the way with a pace, or some other measure, determined and known by exposition; or the quantity of it is to some other quantity 140known by exposition, as the diameter of a square is to the side of the same, or by some other the like means. But it is to be understood, that the quantity exposed must be some standing or permanent thing, such as is marked out in consistent or durable matter; or at least something which is revocable to sense; for otherwise no comparison can be made by it. Seeing, therefore, by what has been said in the next preceding chapter, comparison of one magnitude with another is the same thing with proportion; it is manifest, that quantity determined in the second manner is nothing else but the proportion of a dimension not exposed to another which is exposed; that is, the comparison of the equality or inequality thereof with an exposed quantity.

3. Lines, superficies, and solids, are exposed, first, by motion, in such manner as in the 8th chapter I have said they are generated; but so as that the marks of such motion be permanent; as when they are designed upon some matter, as a line upon paper; or graven in some durable matter. Secondly, by apposition; as when one line or length is applied to another line or length, one breadth to another breadth, and one thickness to another thickness; which is as much as to describe a line by points, a superficies by lines, and a solid by superficies; saving that by points in this place are to be understood very short lines; and, by superficies, very thin solids. Thirdly, lines and superficies may be exposed by section, namely, a line may be made by cutting an exposed superficies; and a superficies, by the cutting of an exposed solid.

1414. Time is exposed, not only by the exposition of a line, but also of some moveable thing, which is moved uniformly upon that line, or at least is supposed so to be moved. For, seeing time is an idea of motion, in which we consider former and latter, that is succession, it is not sufficient for the exposition of time that a line be described; but we must also have in our mind an imagination of some moveable thing passing over that line; and the motion of it must be uniform, that time may be divided and compounded as often as there shall be need. And, therefore, when philosophers, in their demonstrations, draw a line, and say, Let that line be time, it is to be understood as if they said, Let the conception of uniform motion upon that line, be time. For though the circles in dials be lines, yet they are not of themselves sufficient to note time by, except also there be, or be supposed to be, a motion of the shadow or the hand.

5. Number is exposed, either by the exposition of points, or of the names of number, one, two, three, &c.; and those points must not be contiguous, so as that they cannot be distinguished by notes, but they must be so placed that they may be discerned one from another; for, from this it is, that number is called discreet quantity, whereas all quantity, which is designed by motion, is called continual quantity. But that number may be exposed by the names of number, it is necessary that they be recited by heart and in order, as one, two, three, &c.; for by saying one, one, one, and so forward, we know not what number we are at beyond two or three; which also appear to us in this manner, not as number, but as figure.

1426. For the exposition of velocity, which, by the definition thereof, is a motion which, in a certain time, passeth over a certain space, it is requisite, not only that time be exposed, but that there be also exposed that space which is transmitted by the body, whose velocity we would determine; and that a body be understood to be moved in that space also; so that there must be exposed two lines, upon one of which uniform motion must be understood to be made, that the time may be determined; and, upon the other, the velocity is to be computed. |A     B
C    D| As if we would expose the velocity of the we would expose the velocity of the body A, we draw two lines A B and C D, and place a body in C also; which done, we say the velocity of the body A is so great, that it passeth over the line A B in the same time in which the body C passeth over the line C D with uniform motion.

7. Weight is exposed by any heavy body, of what matter soever, so it be always alike heavy.

8. The proportion of two magnitudes is then exposed, when the magnitudes themselves are exposed, namely, the proportion of equality, when the magnitudes are equal; and of inequality, when they are unequal. For seeing, by the 5th article of the preceding chapter, the proportion of two unequal magnitudes consists in their difference, compared with either of them; and when two unequal magnitudes are exposed, their difference is also exposed; it follows, that when magnitudes, which have proportion to one another, are exposed, their proportion also is exposed with them; and, in like manner, the proportion of equals, 143which consists in this, that there is no difference of magnitude betwixt them, is exposed at the same time when the equal magnitudes themselves are exposed. For example, if the exposed lines A B and C D be equal, the proportion of equality is exposed in them; |A      B
C      D
E  G  F| and if the exposed lines, E F and E G be unequal, the proportion which E F has to E G, and that which E G has to E F are also exposed in them; for not only the lines themselves, but also their difference, G F, is exposed. The proportion of unequals is quantity; for the difference, G F, in which it consists, is quantity. But the proportion of equality is not quantity; because, between equals, there is no difference; nor is one equality greater than another, as one inequality is greater than another inequality.

9. The proportion of two times, or of two uniform velocities, is then exposed, when two lines are exposed by which two bodies are understood to be moved uniformly; and therefore the same two lines serve to exhibit both their own proportion, and that of the times and velocities, according as they are considered to be exposed for the magnitudes themselves, or for the times or velocities. For let the two lines A and B be exposed; | A 
 B | their proportion therefore (by the last foregoing article) is exposed; and if they be considered as drawn with equal and uniform velocity, then, seeing their times are greater, or equal, or less, according as the same spaces are transmitted in greater, or equal, or less time, the lines A and B will exhibit the equality or inequality, that is, the proportion 144of the times. To conclude, if the same lines, A and B, be considered as drawn in the same time, then, seeing their velocities are greater, or equal, or less, according as they pass over in the same time longer, or equal, or shorter lines, the same lines, A and B, will exhibit the equality, or inequality, that is, the proportion of their velocities.

1, 2, 3, 4. The nature and definition of proportion, arithmetical and geometrical.—5. The definition, and some properties of the same arithmetical proportion.—6, 7. The definition and transmutations of analogism, or the same geometrical proportion.—8, 9.. The definitions of hyperlogism and hypologism, that is, of greater and less proportion, and their transmutations.—10, 11, 12. Comparison of analogical quantities, according to magnitude.—13, 14, 15. Composition of proportions.—16, 17, 18, 19, 20, 21, 22, 23, 24, 25. The definition and properties of continual proportion.—26, 27, 28, 29. Comparison of arithmetical and geometrical proportions.

1. Great and little are not intelligible, but by comparison. Now that, to which they are compared, is something exposed; that is, some magnitude either perceived by sense, or so defined by words, that it may be comprehended by the mind. Also that, to which any magnitude is compared, is either 145greater or less, or equal to it. And therefore proportion (which, as I have shewn, is the estimation or comprehension of magnitudes by comparison,) is threefold, namely, proportion of equality, that is, of equal to equal; or of excess, which is of the greater to the less; or of defect, which is the proportion of the less to the greater.

Again, every one of these proportions is two-fold; for if it be asked concerning any magnitude given, how great it is, the answer may be made by comparing it two ways; first, by saying it is greater or less than another magnitude, by so much; as seven is less than ten, by three unities; and this is called arithmetical proportion. Secondly, by saying it is greater or less than another magnitude, by such a part or parts thereof; as seven is less than ten, by three tenth parts of the same ten. And though this proportion be not always explicable by number, yet it is a determinate proportion, and of a different kind from the former, and called geometrical proportion, and most commonly proportion simply.

2. Proportion, whether it be arithmetical or geometrical, cannot be exposed but in two magnitudes, (of which the former is commonly called the antecedent, and the latter the consequent of the proportion) as I have shewn in the 8th article of the preceding chapter. And, therefore, if two proportions be to be compared, there must be four magnitudes exposed, namely, two antecedents and two consequents; for though it happen sometimes that the consequent of the former proportion be the same with the antecedent of the latter, yet in that double comparison it must of necessity be 146twice numbered; so that there will be always four terms.

3. Of two proportions, whether they be arithmetical or geometrical, when the magnitudes compared in both (which Euclid, in the fifth definition of his sixth book, calls the quantities of proportions,) are equal, then one of the proportions cannot be either greater or less than the other; for one equality is neither greater nor less than another equality. But of two proportions of inequality, whether they be proportions of excess or of defect, one of them may be either greater or less than the other, or they may both be equal; for though there be propounded two magnitudes that are unequal to one another, yet there may be other two more, unequal, and other two equally unequal, and other two less unequal than the two which were propounded. And from hence it may be understood, that the proportions of excess and defect are quantity, being capable of more and less; but the proportion of equality is not quantity, because not capable neither of more, nor of less. And therefore proportions of inequality may be added together, or subtracted from one another, or be multiplied or divided by one another, or by number; but proportions of equality not so.

4. Two equal proportions are commonly called the same proportion; and, it is said, that the proportion of the first antecedent to the first consequent is the same with that of the second antecedent to the second consequent. And when four magnitudes are thus to one another in geometrical proportion, they are called proportionals; and by some, more briefly, analogism. And greater 147proportion is the proportion of a greater antecedent to the same consequent, or of the same antecedent to a less consequent; and when the proportion of the first antecedent to the first consequent is greater than that of the second antecedent to the second consequent, the four magnitudes, which are so to one another, may be called hyperlogism.

Less proportion is the proportion of a less antecedent to the same consequent, or of the same antecedent to a greater consequent; and when the proportion of the first antecedent to the first consequent is less than that of the second to the second, the four magnitudes may be called hypologism.

5. One arithmetical proportion is the same with another arithmetical proportion, when one of the antecedents exceeds its consequent, or is exceeded by it, as much as the other antecedent exceeds its consequent, or is exceeded by it. And therefore, in four magnitudes, arithmetically proportional, the sum of the extremes is equal to the sum of the means. For if A. B :: C. D be arithmetically proportional, and the difference on both sides be the same excess, or the same defect, E, then B + C (if A be greater than B) will be equal to A - E + C; and A + D will be equal to A + C - E; but A - E + C and A + C - E are equal. Or if A be less than B, then B + C will be equal to A + E + C; and A + D will be equal to A + C + E; but A + E + C and A + C + E are equal.

Also, if there be never so many magnitudes, arithmetically proportional, the sum of them all will be equal to the product of half the number of the terms multiplied by the sum of the extremes.

148For if A. B :: C. D :: E. F be arithmetically proportional, the couples A + F, B + E, C + D will be equal to one another; and their sum will be equal to A + F, multiplied by the number of their combinations, that is, by half the number of the terms.

If, of four unequal magnitudes, any two, together taken, be equal to the other two together taken, then the greatest and the least of them will be in the same combination. Let the unequal magnitudes be A, B, C, D; and let A + B be equal to C + D; and let A be the greatest of them all; I say B will be the least. For, if it may be, let any of the rest, as D, be the least. Seeing therefore A is greater than C, and B than D, A+B will be greater than C + D; which is contrary to what was supposed.

If there be any four magnitudes, the sum of the greatest and least, the sum of the means, the difference of the two greatest, and the difference of the two least, will be arithmetically proportional. For, let there be four magnitudes, whereof A is the greatest, D the least, and B and C the means; I say A + D. B + C :: A - B. C - D are arithmetically proportional. For the difference between the first antecedent and its consequent is this, A + D - B - C; and the difference between the second antecedent and its consequent this, A - B - C + D; but these two differences are equal; and therefore, by this 5th article, A + D. B + C :: A - B. C - D are arithmetically proportional.

If, of four magnitudes, two be equal to the other two, they will be in reciprocal arithmetical proportion. For let A + B be equal to C + D, I say A. C :: D. B are arithmetically proportional. For 149if they be not, let A. C :: D. E (supposing E to be greater or less than B) be arithmetically proportional, and then A + E will be equal to C + D; wherefore A + B and C + D are not equal; which is contrary to what was supposed.

6. One geometrical proportion is the same with another geometrical proportion; when the same cause, producing equal effects in equal times, determines both the proportions.

If a point uniformly moved describe two lines, either with the same, or different velocity, all the parts of them which are contemporary, that is, which are described in the same time, will be two to two, in geometrical proportion, whether the antecedents be taken in the same line, or not. For, from the point A (in the 10th figure at the end of the 14th chapter) let the two lines, A D, A G, be described with uniform motion; and let there be taken in them two parts A B, A E, and again, two other parts, A C, A F; in such manner, that A B, A E, be contemporary, and likewise A C, A F contemporary. I say first (taking the antecedents A B, A C in the line A D, and the conquents A E, A F in the line A G) that A B. A C :: A E. A F are proportionals. For seeing (by the 8th chap, and the 15th art.) velocity is motion considered as determined by a certain length or line, in a certain time transmitted by it, the quantity of the line A B will be determined by the velocity and time by which the same A B is described; and for the same reason, the quantity of the line A C will be determined by the velocity and time, by which the same A C is described; and therefore the proportion of A B to A C, whether 150it be proportion of equality, or of excess or defect, is determined by the velocities and times by which A B, A C are described; but seeing the motion of the point A upon A B and A C is uniform, they are both described with equal velocity; and therefore whether one of them have to the other the proportion of majority or of minority, the sole cause of that proportion is the difference of their times; and by the same reason it is evident, that the proportion of A E to A F is determined by the difference of their times only. Seeing therefore A B, A E, as also A C, A F are contemporary, the difference of the times in which A B and A C are described, is the same with that in which A E and A F are described. Wherefore the proportion of A B to A C, and the proportion of A E to A F are both determined by the same cause. But the cause, which so determines the proportion of both, works equally in equal times, for it is uniform motion; and therefore (by the last precedent definition) the proportion of A B to A C is the same with that of A E to A F; and consequently A B. A C :: A E. A F are proportionals; which is the first.

Secondly, (taking the antecedents in different lines) I say, A B. A E :: A C. A F are proportionals; for seeing A B, A E are described in the same time, the difference of the velocities in which they are described is the sole cause of the proportion they have to one another. And the same may be said of the proportion of A C to A F. But seeing both the lines A D and A G are passed over by uniform motion, the difference of the velocities in which A B, A E are described, will be the same 151with the difference of the velocities, in which A C, A F are described. Wherefore the cause which determines the proportion of A B to A E, is the same with that which determines the proportion of A C to A F; and therefore A B. A E :: A C. A F, are proportionals; which remained to be proved.

Coroll. I. If four magnitudes be in geometrical proportion, they will also be proportionals by permutation, that is, by transposing the middle terms. For I have shown, that not only A B. A C :: A E. A F, but also that, by permutation, A B. A E :: A C. A F are proportionals.

Coroll. II. If there be four proportionals, they will also be proportionals by inversion or conversion, that is, by turning the antecedents into consequents. For if in the last analogism, I had for A B, A C, put by inversion A C, A B, and in like manner converted A E, A F into A F, A E, yet the same demonstration had served. For as well A C, A B, as A B, A C are of equal velocity; and A C, A F, as well as A F, A C are contemporary.

Coroll. III. If proportionals be added to proportionals, or taken from them, the aggregates, or remainders, will be proportionals. For contemporaries, whether they be added to contemporaries, or taken from them, make the aggregates or remainders contemporary, though the addition or subtraction be of all the terms, or of the antecedents alone, or of the consequents alone.

Coroll. IV. If both the antecedents of four proportionals, or both the consequents, or all the terms, be multiplied or divided by the same number or quantity, the products or quotients will be proportionals. For the multiplication and division 152of proportionals, is the same with the addition and subtraction of them.

Coroll. V. If there be four proportionals, they will also be proportionals by composition, that is, by compounding an antecedent of the antecedent and consequent put together, and by taking for consequent either the consequent singly, or the antecedent singly. For this composition is nothing but addition of proportionals, namely, of consequents to their own antecedents, which by supposition are proportionals.

Coroll. VI. In like manner, if the antecedent singly, or consequent singly, be put for antecedent, and the consequent be made of both put together, these also will be proportionals. For it is the inversion of proportion by composition.

Coroll. VII. If there be four proportionals, they will also be proportionals by division, that is, by taking the remainder after the consequent is subtracted from the antecedent, or the difference between the antecedent and consequent for antecedent, and either the whole or the subtracted for consequent; as if A. B :: C. D be proportionals, they will by division be A - B. B :: C - D. D, and A - B. A :: C - D. C; and when the consequent is greater than the antecedent, B - A. A :: D - C. C, and B - A. B :: D - C. D. For in all these divisions, proportionals are, by the very supposition of the analogism A. B :: C. D, taken from A and B, and from C and D.

Coroll. VIII. If there be four proportionals, they will also be proportionals by the conversion of proportion, that is, by inverting the divided proportion, or by taking the whole for antecedent, and the difference or remainder for consequent.

153As, if A. B :: C. D be proportionals, then A. A - B :: C. C - D, as also B. A - B :: D. C - D will be proportionals. For seeing these inverted be proportionals, they are also themselves proportionals.

Coroll. IX. If there be two analogisms which have their quantities equal, the second to the second, and the fourth to the fourth, then either the sum or difference of the first quantities will be to the second, as the sum or difference of the third quantities is to the fourth. Let A. B :: C. D and E. B :: F. D be analogisms; I say A + E. B :: C + F. D are proportionals. For the said analogisms will by permutation be A. C :: B. D, and E. F :: B. D; and therefore A. C :: E. F will be proportionals, for they have both the proportion of B to D common. Wherefore, if in the permutation of the first analogism, there be added E and F to A and C, which E and F are proportional to A and C, then (by the third coroll.) A + E. B :: C + F. D will be proportionals; which was to be proved.

Also in the same manner it may be shown, that A - E. B :: C - F. D are proportionals.

7. If there be two analogisms, where four antecedents make an analogism, their consequents also shall make an analogism; as also the sums of their antecedents will be proportional to the sums of their consequents. For if A. B :: C. D and E. F :: G. H be two analogisms, and A. E :: C. G be proportionals, then by permutation A. C :: E. G, and E. G :: F. H, and A. C :: B. D will be proportionals; wherefore B. D :: E. G, that is, B. D :: F. H, and by permutation B. F :: D. H are proportionals; which is the first. Secondly, I say A + E. B + F :: C + G. D + H will be proportionals. For seeing 154A. E :: C. G are proportionals, A + E. E :: C + G. G will also by composition be proportionals, and by permutation A + E. C + G :: E. G will be proportionals; wherefore, also A + E. C + G :: B + F. D + H will be proportionals. Again, seeing, as is shown above, B. F :: D. H are proportionals, B + F. F :: D + H. H will also by composition be proportionals; and by permutation B + F. D + H :: F. H will also be proportionals; wherefore A + E. C + G :: B + F. D + H are proportionals; which remained to be proved.

Coroll. By the same reason, if there be never so many analogisms, and the antecedents be proportional to the antecedents, it may be demonstrated also that the consequents will be proportional to the consequents, as also the sum of the antecedents to the sum of the consequents.

8. In an hyperlogism, that is, where the proportion of the first antecedent to its consequent is greater than the proportion of the second antecedent to its consequent, the permutation of the proportionals, and the addition of proportionals to proportionals, and substraction of them from one another, as also their composition and division, and their multiplication and division by the same number, produce always an hyperlogism. For suppose A. B :: C . D and A. C :: E. F be analogisms, A + E. B :: C + F . D will also be an analogism; but A + E. B :: C. D will be an hyperlogism; wherefore by permutation, A + E. C :: B. D is an hyperlogism, because A. B :: C. D is an analogism. Secondly, if to the hyperlogism A + E. B :: C. D the proportionals G and H be added, A + E + G. B :: C + H. D will be an hyperlogism, by reason A + E + G. B :: C + F + H. D is an analogism. Also, if G and 155H be taken away, A + E - G. B :: C - H. D will be an hyperlogism; for A + E - G. B :: C + F - H. D is an analogism. Thirdly, by composition A + E + B. B :: C + D. D will be an hyperlogism, because A + E + B. B :: C + F + D. D is ah analogism, and so it will be in all the varieties of composition. Fourthly, by division, A + E - B. B :: C - D. D will by an hyperlogism, by reason A E - B. B :: C + F - D. D is an analogism. Also A + E - B. A + E :: C - D. C is an hyperlogism; for A + E - B. A + E :: C + F - D. C is an analogism. Fifthly, by multiplication 4 A + 4 E. B :: 4 C. D is an hyperlogism, because 4 A. B :: 4 C. D is an analogism; and by division ¼ A + ¼ E. B :: ¼ C.D is an hyperlogism, because ¼ A. B :: ¼ C. D is an analogism.

9. But if A + E. B :: C. D be an hyperlogism, then by inversion B. A + E :: D. C will be an hypologism, because B. A :: D. C being an analogism, the first consequent will be too great. Also, by conversion of proportion, A + E. A + E - B :: C. C - D is an hypologism, because the inversion of it, namely A + E - B. A + E :: C - D. C is an hyperlogism, as I have shown but now. So also B. A + E - B :: D. C - D is an hypologism, because, as I have newly shown, the inversion of it, namely A + E - B. B :: C - D. D is an hyperlogism. Note that this hypologism A + E. A + E - B :: C. C - D is commonly thus expressed; if the proportion of the whole, (A + E) to that which is taken out of it (B), be greater than the proportion of the whole (C) to that which is taken out of it (D), then the proportion of the whole (A + E) to the remainder (A + E - B) will be less than the proportion of the whole (C) to the remainder (C - D).

15610. If there be four proportionals, the difference of the two first, to the difference of the two last, will be as the first antecedent is to the second antecedent, or as the first consequent to the second consequent. For if A. B :: C. D be proportionals, then by division A - B. B :: C - D. D will be proportionals; and by permutation A - B. C - D :: B. D; that is, the differences are proportional to the consequents, and therefore they are so also to the antecedents.

11. Of four proportionals, if the first be greater than the second, the third also shall be greater than the fourth. For seeing the first is greater than the second, the proportion of the first to the second is the proportion of excess; but the proportion of the third to the fourth is the same with that of the first to the second; and therefore also the proportion of the third to the fourth is the proportion of excess; wherefore the third is greater than the fourth. In the same manner it may be proved, that whensoever the first is less than the second, the third also is less than the fourth; and when those are equal, that these also are equal.

12. If there be four proportionals whatsoever, A. B :: C.D, and the first and third be multiplied by any one number, as by 2; and again the second and fourth be multiplied by any one number, as by 3; and the product of the first 2 A, be greater than the product of the second 3 B; the product also of the third 2 C, will be greater than the product of the fourth 3 D. But if the product of the first be less than the product of the second, then the product of the third will be less than that of the fourth. And lastly, if the products of the first 157and second be equal, the products of the third and fourth shall also be equal. Now this theorem is all one with Euclid's definition of the same proportion; and it may be demonstrated thus. Seeing A. B :: C. D are proportionals, by permutation also (art. 6, coroll. I.) A. C :: B . D will be proportionals; wherefore (by coroll. IV. art. 6) 2 A. 2 C :: 3 B. 3 D will be proportionals; and again, by permutation, 2 A. 3 B :: 2 C. 3 D will be proportionals; and therefore, by the last article, if 2 A be greater than 3 B, then 2 C will be greater than 3 D; if less, less; and if equal, equal; which was to be demonstrated.

13. If any three magnitudes be propounded, or three things whatsoever that have any proportion one to another, as three numbers, three times, three degrees, &c.; the proportions of the first to the second, and of the second to the third, together taken, are equal to the proportion of the first to the third. Let there be three lines, for any proportion may be reduced to the proportion of lines, A B, A C, A D; and in the first place, let the proportion as well of the first A B to the second A C, as of the second A C to the|A  B  C  D| third AD, be the proportion of defect, or of less to greater; I say the proportions together taken of A B to A C, and of A C to A D, are equal to the proportion of A B to A D. Suppose the point A to be moved over the whole line A D with uniform motion; then the proportions as well of A B to A C, as of A C to A D, are determined by the difference of the times in which they are described; that is, A B has to A C such proportion as is determined by the different times of their description; and A C to AD such proportion 158as is determined by their times. But the proportion of A B to A D is such as is determined by the difference of the times in which A B and A D are described; and the difference of the times in which A B and A C are described, together with the difference of the times in which A C and A D are described, is the same with the difference of the times in which A B and A D are described. And therefore, 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. Wherefore, by the definition of the same proportion, delivered above in the 6th article, 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. In the second place, let A D be the first, A C the second, and A B the third, and let their proportion be the proportion of excess, or the greater to less; then, as before, the proportions of A D to A C, and of A C to A B, and of A D to A B, will be determined by the difference of their times; which in the description of A D and A C, and of A C and A B together taken, is the same with the difference of the times in the description of A D and A B. Wherefore the proportion of A D to A B is equal to the two proportions of A D to A C and of A C to A B.

In the last place. If one of the proportions, namely of A D to A B, be the proportion of excess, and another of them, as of A B to A C be the proportion of defect, thus also the proportion of A D to A C will be equal to the two proportions together taken of A D to A B, and of A B to A C. For the difference of the times in which AD and AB 159are described, is excess of time; for there goes more time to the description of A D than of A B; and the difference of the times in which A B and A C are described, is defect of time, for less time goes to the description of A B than of A C; but this excess and defect being added together, make D B - B C, which is equal to D C, by which the first A D exceeds the third A C; and therefore the proportions of the first A D to the second A B, and of the second A B to the third A C, are determined by the same cause which determines the proportion of the first A D to the third A C. Wherefore, if any three magnitudes, &c.

Coroll. I. If there be never so many magnitudes having proportion to one another, the proportion of the first to the last is compounded of the proportions of the first to the second, of the second to the third, and so on till you come to the last; or, the proportion of the first to the last is the same with the sum of all the intermediate proportions. For any number of magnitudes having proportion to one another, as A, B, C, D, E being propounded, the proportion of A to E, as is newly shown, is compounded of the proportions of A to D and of D to E; and again, the proportion of A to D, of the proportions of A to C, and of C to D; and lastly, the proportion of A to C, of the proportions of A to B, and of B to C.

Coroll. II. From hence it may be understood how any two proportions may be compounded. For if the proportions of A to B, and of C to D, be propounded to be added together, let B have to something else, as to E, the same proportion which C has to D, and let them be set in this order, 160A, B, E; for so the proportion of A to E will evidently be the sum of the two proportions of A to B, and of B to E, that is, of C to D. Or let it be as D to C, so A to something else, as to E, and let them be ordered thus, E, A, B; for the proportion of E to B will be compounded of the proportions of E to A, that is, of C to D, and of A to B. Also, it may be understood how one proportion may be taken out of another. For if the proportion of C to D be to be subtracted out of the proportion of A to B, let it be as C to D, so A to something else, as E, and setting them in this order, A, E, B, and taking away the proportion of A to E, that is, of C to D, there will remain the proportion of E to B.

Coroll. III. If there be two orders of magnitudes which have proportion to one another, and the several proportions of the first order be the same and equal in number with the proportions of the second order; then, whether the proportions in both orders be successively answerable to one another, which is called ordinate proportion, or not successively answerable, which is called perturbed proportion, the first and the last in both will be proportionals. For the proportion of the first to the last is equal to all the intermediate proportions; which being in both orders the same, and equal in number, the aggregates of those proportions will also be equal to one another; but to their aggregates, the proportions of the first to the last are equal; and therefore the proportion of the first to the last in one order, is the same with the proportion of the first to the last in the other order. Wherefore the first and the last in both are proportionals.

16114. If any two quantities be made of the mutual multiplication of many quantities, which have proportion to one another, and the efficient quantities on both sides be equal in number, the proportion of the products will be compounded of the several proportions, which the efficient quantities have to one another.

First, let the two products be A B and C D, whereof one is made of the multiplication of A into B, and the other of the multiplication of C into D. I say the proportion of A B to C D is compounded of the proportions of the efficient A to the efficient C, and of the efficient B to the efficient D. For let A B, C B and C D be set in order; and as B is to D, so let C be to another quantity as E; and let A, C, E be set also in order. |A B.   A.
C B.   C
C D.   E| Then (by coroll. IV. of the 6th art.) it will be as A B the first quantity to C B the second quantity in the first order, so A to C in the second order; and again, as C B to C D in the first order, so B to D, that is, by construction, so C to E in the second order; and therefore (by the last corollary) A B. C D :: A. E will be proportionals. But the proportion of A to E is compounded of the proportions of A to C, and of B to D; wherefore also the proportion of A B to C D is compounded of the same.

Secondly, let the two products be A B F, and C D G, each of them made of three efficients, the first of A, B and F, and the second of C, D and G; I say, the proportion of A B F to C D G is compounded of the proportions of A to C, of B to D, and of F to G. For let them be set in order as 162before; and as B is to D, so let C be to another quantity E; and again, as F is to G, so let E be to another, H; and let the first order stand thus, ABF, CBF, CDF and CDG; |A B F.   A.
C B F.   C.
C D F.   E.
C D G.  H.| and the second order thus, A, C, E, H. Then the proportion of A B F to C B F in the first order, will be as A to C in the second; and the proportion of CBF to CDF in the first order, as B to D, that is, as C to E (by construction) in the second order; and the proportion of CDF to CDG in the first, as F to G, that is, as E to H (by construction) in the second order; and therefore A B F. C D G:: A. H will be proportionals. But the proportion of A to H is compounded of the proportions of A to C, B to D, and F to G. Wherefore the proportion of the product A B F to C D G is also compounded of the same. And this operation serves, how many soever the efficients be that make the quantities given.

From hence ariseth another way of compounding many proportions into one, namely, that which is supposed in the 5th definition of the 6th book of Euclid; which is, by multiplying all the antecedents of the proportions into one another, and in like manner all the consequents into one another. And from hence also it is evident, in the first place, that the cause why parallelograms, which are made by the duction of two straight lines into one another, and all solids which are equal to figures so made, have their proportions compounded of the proportions of the efficients; and in the second place, why the multiplication of two or more fractions into one another is the same thing 163with the composition of the proportions of their several numerators to their several denominators. For example, if these fractions ¹⁄₂, ²⁄₃, ³⁄₄ be to be multiplied into one another, the numerators 1, 2, 3, are first to be multiplied into one another, which make 6; and next the denominators 2, 3, 4, which make 24; and these two products make the fraction ⁶⁄₂₄. In like manner, if the proportions of 1 to 2, of 2 to 3, and of 3 to 4, be to be compounded, by working as I have shown above, the same proportion of 6 to 24 will be produced.

15. If any proportion be compounded with itself inverted, the compound will be the proportion of equality. For let any proportion be given, as of A to B, and let the inverse of it be that of C to D; and as C to D, so let B be to another quantity; for thus they will be compounded (by the second coroll. of the 12th art.) Now seeing the proportion of C to D is the inverse of the proportion of A to B, it will be as C to D, so B to A; and therefore if they be placed in order, A, B, A, the proportion compounded of the proportions of A to B, and of C to D, will be the proportion of A to A, that is, the proportion of equality. And from hence the cause is evident why two equal products have their efficients reciprocally proportional. For, for the making of two products equal, the proportions of their efficients must be such, as being compounded may make the proportion of equality, which cannot be except one be the inverse of the other; for if betwixt A and A any other quantity, as C, be interposed, their order will be A, C, A, and the later proportion of C to A will be the inverse of the former proportion of A to C.

16416. A proportion is said to be multiplied by a number, when it is so often taken as there be unities in that number; and if the proportion be of the greater to the less, then shall also the quantity of the proportion be increased by the multiplication; but when the proportion is of the less to the greater, then as the number increaseth, the quantity of the proportion diminisheth; as in these three numbers, 4, 2, 1, the proportion of 4 to 1 is not only the duplicate of 4 to 2, but also twice as great; but inverting the order of those numbers thus, 1, 2, 4, the proportion of 1 to 2 is greater than that of 1 to 4; and therefore though the proportion of 1 to 4 be the duplicate of 1 to 2, yet it is not twice so great as that of 1 to 2, but contrarily the half of it. In like manner, a proportion is said to be divided, when between two quantities are interposed one or more means in continual proportion, and then the proportion of the first to the second is said to be subduplicate of that of the first to the third, and subtriplicate of that of the first to the fourth, &c.

This mixture of proportions, where some are proportions of excess, others of defect, as in a merchant's account of debtor and creditor, is not so easily reckoned as some think; but maketh the composition of proportions sometimes to be addition, sometimes substraction; which soundeth absurdly to such as have always by composition understood addition, and by diminution substraction. Therefore to make this account a little clearer, we are to consider (that which is commonly assumed, and truly) that if there be never so many quantities, the proportion of the first to 165the last is compounded of the proportions of the first to the second, and of the second to the third, and so on to the last, without regarding their equality, excess, or defect; so that if two proportions, one of inequality, the other of equality, be added together, the proportion is not thereby made greater nor less; as for example, if the proportions of A to B and of B to B be compounded, the proportion of the first to the second is as much as the sum of both, because proportion of equality, being not quantity, neither augmenteth quantity nor lesseneth it. But if there be three quantities, A, B, C, unequal, and the first be the greatest, the last least, then the proportion of B to C is an addition to that of A to B, and makes it greater; and on the contrary, if A be the least, and C the greatest quantity, then doth the addition of the proportion of B to C make the compounded proportion of A to C less than the proportion of A to B, that is, the whole less than the part. The composition therefore of proportions is not in this case the augmentation of them, but the diminution; for the same quantity (Euclid V. 8) compared with two other quantities, hath a greater proportion to the lesser of them than to the greater. Likewise, when the proportions compounded are one of excess, the other of defect, if the first be of excess, as in these numbers, 8, 6, 9, the proportion compounded, namely, of 8 to 9, is less than the proportion of one of the parts of it, namely, of 8 to 6; but if the proportion of the first to the second be of defect, and that of the second to the third be of excess, as in these numbers, 6, 8, 4, then shall the proportion of the first to the third 166be greater than that of the first to the second, as 6 hath a greater proportion to 4 than to 8; the reason whereof is manifestly this, that the less any quantity is deficient of another, or the more one exceedeth another, the proportion of it to that other is the greater. Suppose now three quantities in continual proportion, A B 4, A C 6, A D 9. Because therefore A D is greater than A C, but not greater than A D, the proportion of A D to AC will be (by Euclid, V. 8) greater than that of A D to A D; and likewise, because the proportions of A D to A C, and of A C to A B are the same, the proportions of A D to A C and of A C to A B, being both proportions of excess, make the whole proportion of A D to A B, or of 9 to 4, not only the duplicate of A D to A C, that is, of 9 to 6, but also the double, or twice so great. On the other side, because the proportion of A D to A D, or 9 to 9, being proportion of equality, is no quantity, and yet greater than that of A C to A D, or 6 to 9, it will be as 0 - 9 to 0 - 6, so A C to A D, and again, as 0 - 9 to 0 - 6, so 0 - 6 to 0 - 4; but 0 - 4, 0 - 6, 0 - 9 are in continual proportion; and because 0 - 4 is greater than 0 - 6, the proportion of 0 - 4 to 0 - 6 will be double to the proportion of 0 - 4 to 0 - 9, double I say, and yet not duplicate, but subduplicate.

If any be unsatisfied with this ratiocination, let him first consider that (by Euclid V. 8) the proportion of A B to A C is greater than that of A B to A D, wheresoever D be |  Ḅ  C̣   Ḍ
A——————E| placed in the line A C prolonged; and the further off the point 167D is from C, so much the greater is the proportion of A B to A C than that of A B to A D. There is therefore some point (which suppose be E) in such distance from C, as that the proportion of A B to A C will be twice as great as that of A B to A E. That considered, let him determine the length of the line A E, and demonstrate, if he can, that A E is greater or less than A D.

By the same method, if there be more quantities than three, as A, B, C, D, in continual proportion, and A be the least, it may be made appear that the proportion of A to B is triple magnitude, though subtriple in multitude, to the proportion of A to D.

17. If there be never so many quantities, the number whereof is odd, and their order such, that from the middlemost quantity both ways they proceed in continual proportion, the proportion of the two which are next on either side to the middlemost is subduplicate to the proportion of the two which are next to these on both sides, and subtriplicate of the proportion of the two which are yet one place more remote, &c. For let the magnitudes be C, B, A, D, E, and let A, B, C, as also A, D, E be in continual proportion; I say the proportion of D to B is subduplicate of the proportion of E to C. For the proportion of D to B is compounded of the proportions of D to A, and of A to B once taken; but the proportion of E to C is compounded of the same twice taken; and therefore the proportion of D to B is subduplicate of the proportion of E to C. And in the same manner, if there were three terms on either side, it might be demonstrated that the proportion of 168D to B would be subtriplicate of that of the extremes, &c.

18. If there be never so many continual proportionals, as the first, second, third, &c. their differences will be proportional to them. For the second, third, &c. are severally consequents of the preceding, and antecedents of the following proportion. But (by art. 10) the difference of the first antecedent and consequent, to difference of the second antecedent and consequent, is as the first antecedent to the second antecedent, that is, as the first term to the second, or as the second to the third, &c. in continual proportionals.

19. If there be three continual proportionals, the sum of the extremes, together with the mean twice taken, the sum of the mean and either of the extremes, and the same extreme, are continual proportionals. For let A. B. C be continual proportionals. Seeing, therefore, A. B :: B. C are proportionals, by composition also A + B. B :: B + C. C will be proportionals; and by permutation A + B. B + C :: B. C will also be proportionals; and again, by composition A + 2B + C. B + C :: B + C. C; which was to be proved.

20. In four continual proportionals, the greatest and the least put together is a greater quantity than the other two put together. Let A. B :: C. D be continual proportionals; whereof let the greatest be A, and the least be D; I say A + D is greater than B + C. For by art. 10, A - B. C - D :: A. C are proportionals; and therefore A - B is, by art. 11, greater than C - D. Add B on both sides, and A will be greater than C + B - D. And again, add D on both sides, and A + D will be greater than B + C; which was to be proved.

16921. If there be four proportionals, the extremes multiplied into one another, and the means multiplied into one another, will make equal products. Let A. B :: C. D be proportionals; I say A D is equal to B C. For the proportion of A D to B C is compounded, by art. 13, of the proportions of A to B, and D to C, that is, its inverse B to A; and therefore, by art. 14, this compounded proportion is the proportion of equality; and therefore also, the proportion of A D to B C is the proportion of equality. Wherefore they are equal.

22. If there be four quantities, and the proportion of the first to the second be duplicate of the proportion of the third to the fourth, the product of the extremes to the product of the means, will be as the third to the fourth. Let the four quantities be A, B, C and D; and let the proportion of A to B be duplicate of the proportion of C to D, I say A D, that is, the product of A into D is to B C, that is, to the product of the means, as C to D. For seeing the proportion of A to B is duplicate of the proportion of C to D, if it be as C to D, so D to another, E, then A. B :: C. E will be proportionals; for the proportion of A to B is by supposition duplicate of the proportion of C to D; and C to E duplicate also of that of C to D by the definition, art. 15. Wherefore, by the last article, A E or A into E is equal to B C or B into C; but, by coroll. IV. art. 6, A D is to A E as D to E, that is, as C to D; and therefore A D is to B C, which as I have shown is equal to A E, as C to D; which was to be proved.

Moreover, if the proportion of the first A to the second B be triplicate of the proportion of 170the third C to the fourth D, the product of the extremes to the product of the means will be duplicate of the proportion of the third to the fourth. For if it be as C to D so D to E, and again, as D to E so E to another, F, then the proportion of C to F will be triplicate of the proportion of C to D; and consequently, A. B :: C. F will be proportionals, and A F equal to B C. But as A D to A F, so is D to F; and therefore, also, as A D to B C, so D to F, that is, so C to E; but the proportion of C to E is duplicate of the proportion of C to D; wherefore, also, the proportion of A D to B C is duplicate of that of C to D, as was propounded.

23. If there be four proportionals, and a mean be interposed betwixt the first and second, and another betwixt the third and fourth, the first of these means will be to the second, as the first of the proportionals is to the third, or as the second of them is to the fourth. For let A. B :: C. D be proportionals, and let E be a mean betwixt A and B, and F a mean betwixt C and D; I say A. C :: E. F are proportionals. For the proportion of A to E is subduplicate of the proportion of A to B, or of C to D. Also, the proportion of C to F is subduplicate of that of C to D; and therefore A. E :: C. F are proportionals; and by permutation A. C :: E. F are also proportionals; which was to be proved.

24. Any thing is said to be divided into extreme and mean proportion, when the whole and the parts are in continual proportion. As for example, when A + B. A. B are continual proportionals; or when the straight line A C is so divided in B, that 171A C. A B. B C are in continual proportion. And if the same line A C be again divided |A       B   C
——+——+——
    D| in D, so as that A C. C D. A D be continual proportionals; then also A C. A B. A D will be continual proportionals; and in like manner, though in contrary order, C A. C D. C B will be continual proportionals; which cannot happen in any line otherwise divided.

25. If there be three continual proportionals, and again, three other continual proportions, which have the same middle term, their extremes will be in reciprocal proportion. For let A. B. C and D. B. E be continual proportionals, I say A. D :: E. C shall be proportionals. For the proportion of A to D is compounded of the proportions of A to B, and of B to D; and the proportion of E to C is compounded of those of E to B, that is, of B to D, and of B to C, that is, of A to B. Wherefore, by equality, A. D :: E. C are proportionals.

26. If any two unequal quantities be made extremes, and there be interposed betwixt them any number of means in geometrical proportion, and the same number of means in arithmetical proportion, the several means in geometrical proportion will be less than the several means in arithmetical proportion. For betwixt A the lesser, and E the greater extreme, let there be interposed three means, B, C, D, in geometrical proportion, and as many more, F, G, H, in arithmetical proportion; I say B will be less than F, C than G, and D than H. For first, the difference between A and F is the same with that between F and G, and with that between G and H, by the definition of arithmetical 172proportion; and therefore, the difference of the proportionals which stand next to one another, to the difference of the extremes, is, when there is but one mean, half their difference; when two, a third part of it; when three, a quarter, &c.; so that in this example it is a quarter. But the difference between D and E, by art. 17, is more than a |   A      A
   -       -
   B      F
   —     —
   C      G
  –—    –—
   D     H
  ——   ——
   E      E
 ——— ———| quarter of the difference between the extremes, because the proportion is geometrical, and therefore the difference between A and D is less than three quarters of the same difference of the extremes. In like manner, if the difference between A and D be understood to be divided into three equal parts, it may be proved, that the difference between A and C is less than two quarters of the difference of the extremes A and E. And lastly, if the difference between A and C be divided into two equal parts, that the difference between A and B is less than a quarter of the difference of the extremes A and E.

From the consideration hereof, it is manifest, that B, that is A together with something else which is less than a fourth part of the difference of the extremes A and E, is less than F, that is, than the same A with something else which is equal to the said fourth part. Also, that C, that is A with something else which is less than two fourth parts of the said difference, is less than G, that is, than A together with the said two-fourths. And lastly, that D, which exceeds A by less than three-fourths of the said difference, is less than H, which exceeds 173the same A by three entire fourths of the said difference. And in the same manner it would be if there were four means, saving that instead of fourths of the difference of the extremes we are to take fifth parts; and so on.

27. Lemma. If a quantity being given, first one quantity be both added to it and subtracted from it, and then another greater or less, the proportion of the remainder to the aggregate, is greater where the less quantity is added and substracted, than where the greater quantity is added and substracted. Let B be added to and substracted from the quantity A; so that A - B be the remainder, and A + B the aggregate; and again, let C, a greater quantity than B, be added to and substracted from the same A, so that A - C be the remainder and A + C the aggregate; I say A - B. A + B :: A - C. A + C will be an hyperlogism. For A - B. A :: A - C. A is an hyperlogism of a greater antecedent to the same consequent; and therefore A - B. A + B :: A - C. A + C is a much greater hyperlogism, being made of a greater antecedent to a less consequent.

28. If unequal parts be taken from two equal quantities, and betwixt the whole and the part of each there be interposed two means, one in geometrical, the other in arithmetical proportion; the difference betwixt the two means will be greatest, where the difference betwixt the whole and its part is greatest. |A   E   G   H   B
───┼──┼──┼───
────┼──┼──┼──
A    F    I   K  B|For let A B and A B be two equal quantities, from which let two unequal parts be taken, namely, A E the less, and A F the greater; and betwixt A B and A E let A G be a mean in geometrical proportion, and A H a mean in arithmetical 174proportion. Also betwixt A B and A F let A I be a mean in geometrical proportion, and A K a mean in arithmetical proportion; I say H G is greater than K I.

Now seeing the proportion of A B to A E is greater than that of A B to A F, the proportion of A B to A G, which is half the greater proportion, is greater than the proportion of A B to A I the half of the less proportion; and therefore A I is greater than A G. Wherefore the proportion of A B - A G to A B + A G, by the precedent lemma, will be greater than the proportion of A B - A I to A B + A I; and therefore also the proportion of H G to B H will be greater than that of K I to B K, and much greater than the proportion of K I to B H, which is greater than B K; for B H is the half of B E, as B K is the half of B F, which, by 175supposition, is less than B E. Wherefore H G is greater than K I; which was to be proved.

Coroll. It is manifest from hence, that if any quantity be supposed to be divided into equal parts infinite in number, the difference between the arithmetical and geometrical means will be infinitely little, that is, none at all. And upon this foundation, chiefly, the art of making those numbers, which are called Logarithms, seems to have been built.

29. If any number of quantities be propounded, whether they be unequal, or equal to one another; and there be another quantity, which multiplied by the number of the propounded quantities, is equal to them all; that other quantity is a mean in arithmetical proportion to all those propounded quantities.

1. The definition and properties of a strait line.—2. The definition and properties of a plane superficies.—3. Several sorts of crooked lines.—4. The definition and properties of a circular line.—5. The properties of a strait line taken in a plane.—6.. The definition of tangent lines.—7. The definition of an angle, and the kinds thereof.—8. In concentric circles, arches of the same angle are to one another, as the whole circumferences are.—9. The quantity of an angle, in what it consists. —10. The distinction of angles, simply so called.—11. Of strait lines from the centre of a circle to a tangent of the same. —12. The general definition of parallels, and the properties of strait parallels.—13. The circumferences of circles are to one another, as their diameters are.—14. In triangles, strait lines parallel to the bases are to one another, as the parts of the sides which they cut off from the vertex.—15. By what fraction of a strait line the circumference of a circle is made.—16. That an angle of contingence is quantity, but of a different kind from that of an angle simply so called; and that it can neither add nor take away any thing from the same.—17. That the inclination of planes is angle simply so called.—18. A solid angle what it is.—19. What is the nature of asymptotes.—20. Situation, by what it is determined.—21. What is like situation; what is figure; and what are like figures.

1. Between two points given, the shortest line is that, whose extreme points cannot be drawn farther asunder without altering the quantity, that is, without altering the proportion of that line to any other line given. For the magnitude of a line is computed by the greatest distance which may be 177between its extreme points; so that any one line, whether it be extended or bowed, has always one and the same length, because it can have but one greatest distance between its extreme points.

And seeing the action, by which a strait line is made crooked, or contrarily a crooked line is made strait, is nothing but the bringing of its extreme points nearer to one another, or the setting of them further asunder, a crooked line may rightly be defined to be that, whose extreme points may be understood to be drawn further asunder; and a strait line to be that, whose extreme points cannot be drawn further asunder; and comparatively, a more crooked, to be that line whose extreme points are nearer to one another than those of the other, supposing both the lines to be of equal length. Now, howsoever a line be bowed, it makes always a sinus or cavity, sometimes on one side, sometimes on another; so that the same crooked line may either have its whole cavity on one side only, or it may have it part on one side and part on the other side. Which being well understood, it will be easy to understand the following comparisons of strait and crooked lines.

First, if a strait and a crooked line have their extreme points common, the crooked line is longer than the strait line. For if the extreme points of the crooked line be drawn out to their greatest distance, it will be made a strait line, of which that, which was a strait line from the beginning, will be but a part; and therefore the strait line was shorter than the crooked line, which had the same extreme points. And for the same reason, 178if two crooked lines have their extreme points common, and both of them have all their cavity on one and the same side, the outermost of the two will be the longest line.

Secondly, a strait line and a perpetually crooked line cannot be coincident, no, not in the least part. For if they should, then not only some strait line would have its extreme points common with some crooked line, but also they would, by reason of their coincidence, be equal to one another; which, as I have newly shown, cannot be.

Thirdly, between two points given, there can be understood but one strait line; because there cannot be more than one least interval or length between the same points. For if there may be two, they will either be coincident, and so both of them will be one strait line; or if they be not coincident, then the application of one to the other by extension will make the extended line have its extreme points at greater distance than the other; and consequently, it was crooked from the beginning.

Fourthly, from this last it follows, that two strait lines cannot include a superficies. For if they have both their extreme points common, they are coincident; and if they have but one or neither of them common, then at one or both ends the extreme points will be disjoined, and include no superficies, but leave all open and undetermined.

Fifthly, every part of a strait line is a strait line. For seeing every part of a strait line is the least that can be drawn between its own extreme points, if all the parts should not constitute a strait 179line, they would altogether be longer than the whole line.

2. A plane or a plane superficies, is that which is described by a strait line so moved, that all the several points thereof describe several strait lines. A strait line, therefore, is necessarily all of it in the same plane which it describes. Also the strait lines, which are made by the points that describe a plane, are all of them in the same plane. Moreover, if any line whatsoever be moved in a plane, the lines, which are described by it, are all of them in the same plane.

All other superficies, which are not plane, are crooked, that is, are either concave or convex. And the same comparisons, which were made of strait and crooked lines, may also be made of plane and crooked superficies.

For, first, if a plane and crooked superficies be terminated with the same lines, the crooked superficies is greater than the plane superficies. For if the lines, of which the crooked superficies consists, be extended, they will be found to be longer than those of which the plane superficies consists, which cannot be extended, because they are strait.

Secondly, two superficies, whereof the one is plane, and the other continually crooked, cannot be coincident, no, not in the least part. For if they were coincident, they would be equal; nay, the same superficies would be both plane and crooked, which is impossible.

Thirdly, within the same terminating lines there can be no more than one plane superficies; because there can be but one least superficies within the same.

180Fourthly, no number of plane superficies can include a solid, unless more than two of them end in a common vertex. For if two planes have both the same terminating lines, they are coincident, that is, they are but one superficies; and if their terminating lines be not the same, they leave one or more sides open.

Fifthly, every part of a plane superficies is a plane superficies. For seeing the whole plane superficies is the least of all those, that have the same terminating lines; and also every part of the same superficies is the least of all those, that are terminated with the same lines; if every part should not constitute a plane superficies, all the parts put together would not be equal to the whole.

3. Of straitness, whether it be in lines or in superficies, there is but one kind; but of crookedness there are many kinds; for of crooked magnitudes, some are congruous, that is, are coincident when they are applied to one other; others are incongruous. Again, some are ὁμοιομερεῖς or uniform, that is, have their parts, howsoever taken, congruous to one another; others are ἀνομοιομερεῖς or of several forms. Moreover, of such as are crooked, some are continually crooked, others have parts which are not crooked.

4. If a strait line be moved in a plane, in such manner, that while one end of it stands still, the whole line be carried round about till it come again into the same place from whence it was first moved, it will describe a plane superficies, which will be terminated every way by that crooked line, which is made by that end of the strait line which 181was carried round. Now this superficies is called a CIRCLE; and of this circle, the unmoved point is the centre; the crooked line which terminates it, the perimeter; and every part of that crooked line, a circumference or arch; the strait line, which generated the circle, is the semidiameter or radius; and any strait line, which passeth through the centre and is terminated on both sides in the circumference, is called the diameter. Moreover, every point of the radius, which describes the circle, describes in the same time its own perimeter, terminating its own circle, which is said to be concentric to all the other circles, because this and all those have one common centre.

Wherefore in every circle, all strait lines from the centre to the circumference are equal. For they are all coincident with the radius which generates the circle.

Also the diameter divides both the perimeter and the circle itself into two equal parts. For if those two parts be applied to one another, and the semiperimeters be coincident, then, seeing they have one common diameter, they will be equal; and the semicircles will be equal also; for these also will be coincident. But if the semiperimeters be not coincident, then some one strait line, which passes through the centre, which centre is in the diameter, will be cut by them in two points. Wherefore, seeing all the strait lines from the centre to the circumference are equal, a part of the same strait line will be equal to the whole; which is impossible.

For the same reason the perimeter of a circle 182will be uniform, that is, any one part of it will be coincident with any other equal part of the same.

5. From hence may be collected this property of a strait line, namely, that it is all contained in that plane which contains both its extreme points. For seeing both its extreme points are in the plane, that strait line, which describes the plane, will pass through them both; and if one of them be made a centre, and at the distance between both a circumference be described, whose radius is the strait line which describes the plane, that circumference will pass through the other point. Wherefore between the two propounded points, there is one strait line, by the definition of a circle, contained wholly in the propounded plane; and therefore if another strait line might be drawn between the same points, and yet not be contained in the same plane, it would follow, that between two points two strait lines may be drawn; which has been demonstrated to be impossible.

It may also be collected, that if two planes cut one another, their common section will be a strait line. For the two extreme points of the intersection are in both the intersecting planes; and between those points a strait line may be drawn; but a strait line between any two points is in the same plane, in which the points are; and seeing these are in both the planes, the strait line which connects them will also be in both the same planes, and therefore it is the common section of both. And every other line, that can be drawn between those points, will be either coincident with that line, that is, it will be the same line; or it will not 183be coincident, and then it will be in neither, or but in one of those planes.

As a strait line may be understood to be moved round about whilst one end thereof remains fixed, as the centre; so in like manner it is easy to understand, that a plane may be circumduced about a strait line, whilst the strait line remains still in one and the same place, as the axis of that motion. Now from hence it is manifest, that any three points are in some one plane. For as any two points, if they be connected by a strait line, are understood to be in the same plane in which the strait line is; so, if that plane be circumduced about the same strait line, it will in its revolution take in any third point, howsoever it be situate; and then the three points will be all in that plane; and consequently the three strait lines which connect those points, will also be in the same plane.

6. Two lines are said to touch one another, which being both drawn to one and the same point, will not cut one another, though they be produced, produced, I say, in the same manner in which they were generated. And therefore if two strait lines touch one another in any one point, they will be contiguous through their whole length. Also two lines continually crooked will do the same, if they be congruous and be applied to one another according to their congruity; otherwise, if they be incongruously applied, they will, as all other crooked lines, touch one another, where they touch, but in one point only. Which is manifest from this, that there can be no congruity between a strait line and a line that is continually crooked; for otherwise the same line might be both strait 184and crooked. Besides, when a strait line touches a crooked line, if the strait line be never so little moved about upon the point of contact, it will cut the crooked line; for seeing it touches it but in one point, if it incline any way, it will do more than touch it; that is, it will either be congruous to it, or it will cut it; but it cannot be congruous to it; and therefore it will cut it.

7. An angle, according to the most general acceptation of the word, may be thus defined; when two lines, or many superficies, concur in one sole point, and diverge every where else, the quantity of that divergence is an ANGLE. And an angle is of two sorts; for, first, it may be made by the concurrence of lines, and then it is a superficial angle; or by the concurrence of superficies, and then it is called a solid angle.

Again, from the two ways by which two lines may diverge from one another, superficial angles are divided into two kinds. For two strait lines, which are applied to one another, and are contiguous in their whole length, may be separated or pulled open in such manner, that their concurrence in one point will still remain; and this separation or opening may be either by circular motion, the centre whereof is their point of concurrence, and the lines will still retain their straitness, the quantity of which separation or divergence is an angle simply so called; or they may be separated by continual flexion or curvation in every imaginable point; and the quantity of this separation is that, which is called an angle of contingence.

Besides, of superficial angles simply so called, 185those, which are in a plane superficies, are plane; and those, which are not plane, are denominated from the superficies in which they are.

Lastly, those are strait-lined angles, which are made by strait lines; as those which are made by crooked lines are crooked-lined; and those which are made both of strait and crooked lines, are mixed angles.

8. Two arches intercepted between two radii of concentric circles, have the same proportion to one another, which their whole perimeters have to one another. For let the point A (in the first figure) be the centre of the two circles B C D and E F G, in which the radii A E B and A F C intercept the arches B C and E F; I say the proportion of the arch B C to the arch E F is the same with that of the perimeter B C D to the perimeter E F G. For if the radius A F C be understood to be moved about the centre A with circular and uniform motion, that is, with equal swiftness everywhere, the point C will in a certain time describe the perimeter B C D, and in a part of that time the arch B C; and because the velocities are equal by which both the arch and the whole perimeter are described, the proportion of the magnitude of the perimeter B C D to the magnitude of the arch BC is determined by nothing but the difference of the times in which the perimeter and the arch are described. But both the perimeters are described in one and the same time, and both the arches in one and the same time; and therefore the proportions of the perimeter B C D to the arch B C, and of the perimeter E F G to the arch E F, are both determined by the same cause. Wherefore 186B C D. B C :: E F G. E F are proportionals (by the 6th art. of the last chapter), and by permutation B C D. E F G :: B C. E F will also be proportionals; which was to be demonstrated.

9. Nothing is contributed towards the quantity of an angle, neither by the length, nor by the equality, nor by the inequality of the lines which comprehend it. For the lines A B and A C comprehend the same angle which is comprehended by the lines A E and A F, or A B and A F. Nor is an angle either increased or diminished by the absolute quantity of the arch, which subtends the same; for both the greater arch B C and the lesser arch E F are subtended to the same angle. But the quantity of an angle is estimated by the quantity of the subtending arch compared with the quantity of the whole perimeter. And therefore the quantity of an angle simply so called may be thus defined: the quantity of an angle is an arch or circumference of a circle, determined by its proportion to the whole perimeter. So that when an arch is intercepted between two strait lines drawn from the centre, look how great a portion that arch is of the whole perimeter, so great is the angle. From whence it may be understood, that when the lines which contain an angle are strait lines, the quantity of that angle may be taken at any distance from the centre. But if one or both of the containing lines be crooked, then the quantity of the angle is to be taken in the least distance from the centre, or from their concurrence; for the least distance is to be considered as a strait line, seeing no crooked line can be imagined so little, but that there may be a less strait line. And 187although the least strait line cannot be given, because the least given line may still be divided, yet we may come to a part so small, as is not at all considerable; which we call a point. And this point may be understood to be in a strait line which touches a crooked line; for an angle is generated by separating, by circular motion, one strait line from another which touches it, as has been said above in the 7th article. Wherefore an angle, which two crooked lines make, is the same with that which is made by two strait lines which touch them.

10. From hence it follows, that vertical angles, such as are A B C, D B F in the second figure, are equal to one another. For if, from the two semiperimeters D A C, F D A, which are equal to one another, the common arch D A be taken away, the remaining arches A C, D F will be equal to one another.

Another distinction of angles is into right and oblique. A right angle is that, whose quantity is the fourth part of the perimeter. And the lines, which make a right angle, are said to be perpendicular to one another. Also, of oblique angles, that which is greater than a right, is called an obtuse angle; and that which is less, an acute angle. From whence it follows, that all the angles that can possibly be made at one and the same point, together taken, are equal to four right angles; because the quantities of them all put together make the whole perimeter. Also, that all the angles, which are made on one side of a strait line, from any one point taken in the same, are equal to two right angles; for if that point be 188made the centre, that strait line will be the diameter of a circle, by whose circumference the quantity of an angle is determined; and that diameter will divide the perimeter into two equal parts.

11. If a tangent be made the diameter of a circle, whose centre is the point of contact, a strait line drawn from the centre of the former circle to the centre of the latter circle, will make two angles with the tangent, that is, with the diameter of the latter circle, equal to two right angles, by the last article. And because, by the 6th article, the tangent has on both sides equal inclination to the circle, each of them will be a right angle; as also the semidiameter will be perpendicular to the same tangent. Moreover, the semidiameter, inasmuch as it is the semidiameter, is the least strait line which can be drawn from the centre to the tangent; and every other strait line, that reaches the tangent, will pass out of the circle, and will therefore be greater than the semidiameter. In like manner, of all the strait lines, which may be drawn from the centre to the tangent, that is the greatest which makes the greatest angle with the perpendicular; which will be manifest, if about the same centre another circle be described, whose semidiameter is a strait line taken nearer to the perpendicular, and there be drawn a perpendicular, that is, a tangent, to the same.

From whence it is also manifest, that if two strait lines, which make equal angles on either side of the perpendicular, be produced to the tangent, they will be equal.

18912. There is in Euclid a definition of strait-lined parallels; but I do not find that parallels in general are anywhere defined; and therefore for an universal definition of them, I say that any two lines whatsoever, strait or crooked, as also any two superficies, are PARALLEL; when two equal strait lines, wheresoever they fall upon them, make always equal angles with each of them.

From which definition it follows; first, that any two strait lines, not inclined opposite ways, falling upon two other strait lines, which are parallel, and intercepting equal parts in both of them, are themselves also equal and parallel. As if A B and C D (in the third figure), inclined both the same way, fall upon the parallels A C and B D, and A C and B D be equal, A B and C D will also be equal and parallel. For the perpendiculars B E and D F being drawn, the right angles E B D and F D H will be equal. Wherefore, seeing E F and B D are parallel, the angles E B A and F D C will be equal. Now if D C be not equal to B A, let any other strait line equal to B A be drawn from the point D; which, seeing it cannot fall upon the point C, let it fall upon G. Wherefore A G will be either greater or less than B D; and therefore the angles E B A and F D C are not equal, as was supposed. Wherefore A B and C D are equal; which is the first.

Again, because they make equal angles with the perpendiculars B E and D F; therefore the angle C D H will be equal to the angle A B D, and, by the definition of parallels, A B and C D will be parallel; which is the second.

That plane, which is included both ways within parallel lines, is called a PARALLELOGRAM.

190Coroll. I. From this last it follows, that the angles A B D and C D H are equal, that is, that a strait line, as B H, falling upon two parallels, as A B and C D, makes the internal angle A B D equal to the external and opposite angle C D H.

Coroll. II. And from hence again it follows, that a strait line falling upon two parallels, makes the alternate angles equal, that is, the angle A G F, in the fourth figure, equal to the angle G F D. For seeing G F D is equal to the external opposite angle E G B, it will be also equal to its vertical angle A G F, which is alternate to G F D.

Coroll. III. That the internal angles on the same side of the line F G are equal to two right angles. For the angles at F, namely, G F C and G F D, are equal to two right angles. But G F D is equal to its alternate angle A G F. Wherefore both the angles G F C and A G F, which are internal on the same side of the line F G, are equal to two right angles.

Coroll. IV. That the three angles of a strait-lined plain triangle are equal to two right angles; and any side being produced, the external angle will be equal to the two opposite internal angles. For if there be drawn by the vertex of the plain triangle A B C (fig. 5) a parallel to any of the sides, as to A B, the angles A and B will be equal to their alternate angles E and F, and the angle C is common. But, by the 10th article, the three angles E, C and F, are equal to two right angles; and therefore the three angles of the triangle are equal to the same; which is the first. Again, the two angles B and D are equal to two right angles, by the 10th article. Wherefore taking 191away B, there will remain the angles A and C, equal to the angle D; which is the second.

Coroll. V. If the angles A and B be equal, the sides A C and C B will also be equal, because A B and E F are parallel; and, on the contrary, if the sides A C and C B be equal, the angles A and B will also be equal. For if they be not equal, let the angles B and G be equal. Wherefore, seeing G B and E F are parallels, and the angles G and B equal, the sides G C and C B will also be equal; and because C B and A C are equal by supposition, C G and C A will also be equal; which cannot be, by the 11th article.

Coroll. VI. From hence it is manifest, that if two radii of a circle be connected by a strait line, the angles they make with that connecting line will be equal to one another; and if there be added that segment of the circle, which is subtended by the same line which connects the radii, then the angles, which those radii make with the circumference, will also be equal to one another. For a strait line, which subtends any arch, makes equal angles with the same; because, if the arch and the subtense be divided in the middle, the two halves of the segment will be congruous to one another, by reason of the uniformity both of the circumference of the circle, and of the strait line.

13. Perimeters of circles are to one another, as their semidiameters are. For let there be any two circles, as, in the first figure, B C D the greater, and E F G the lesser, having their common centre at A; and let their semidiameters be A C and A E. I say, A C has the same proportion to A E, which the perimeter B C D has to the perimeter E F G. 192For the magnitude of the semidiameters A C and A E is determined by the distance of the points C and E from the centre A; and the same distances are acquired by the uniform motion of a point from A to C, in such manner, that in equal times the distances acquired be equal. But the perimeters B C D and E F G are also determined by the same distances of the points C and E from the centre A; and therefore the perimeters B C D and E F G, as well as the semidiameters A C and A E, have their magnitudes determined by the same cause, which cause makes, in equal times, equal spaces. Wherefore, by the 13th chapter and 6th article, the perimeters of circles and their semidiameters are proportionals; which was to be proved.

14. If two strait lines, which constitute an angle, be cut by strait-lined parallels, the intercepted parallels will be to one another, as the parts which they cut off from the vertex. Let the strait lines A B and A C, in the 6th figure, make an angle at A, and be cut by the two strait-lined parallels B C and D E, so that the parts cut off from the vertex in either of those lines, as in A B, may be A B and A D. I say, the parallels B C and D E are to one another, as the parts A B and A D. For let A B be divided into any number of equal parts, as into A F, F D, D B; and by the points F and D, let F G and D E be drawn parallel to the base B C, and cut A C in G and E; and again, by the points G and E, let other strait lines be drawn parallel to A B, and cut B C in H and I. If now the point A be understood to be moved uniformly over A B, and in the same time B be moved to C, and all the 193points F, D, and B be moved uniformly and with equal swiftness over F G, D E, and B C; then shall B pass over B H, equal to F G, in the same time that A passes over A F; and A F and F G will be to one another, as their velocities are; and when A is in F, D will be in K; when A is in D, D will be in E; and in what manner the point A passes by the points F, D, and B, in the same manner the point B will pass by the points H, I, and C; and the strait lines F G, D K, K E, B H, H I, and I C, are equal, by reason of their parallelism; and therefore, as the velocity in A B is to the velocity in B C, so is A D to D E; but as the velocity in A B is to the velocity in B C, so is A B to B C; that is to say, all the parallels will be severally to all the parts cut off from the vertex, as A F is to F G. Wherefore, A F. G F :: A D. D E :: A B. B C are proportionals.

The subtenses of equal angles in different circles, as the strait lines B C and F E (in fig. 1), are to one another as the arches which they subtend. For (by art. 8) the arches of equal angles are to one another as their perimeters are; and (by art. 13) the perimeters as their semidiameters; but the subtenses B C and F E are parallel to one another by reason of the equality of the angles which they make with the semidiameters; and therefore the same subtenses, by the last precedent article, will be proportional to the semidiameters, that is, to the perimeters, that is, to the arches which they subtend.

15. If in a circle any number of equal subtenses be placed immediately after one another, and strait lines be drawn from the extreme point of the first 194subtense to the extreme points of all the rest, the first subtense being produced will make with the second subtense an external angle double to that, which is made by the same first subtense, and a tangent to the circle touching it in the extreme points thereof; and if a strait line which subtends two of those arches be produced, it will make an external angle with the third subtense, triple to the angle which is made by the tangent with the first subtense; and so continually. For with the radius A B (in fig. 7) let a circle be described, and in it let any number of equal subtenses, B C, C D, and D E, be placed; also let B D and B E be drawn; and by producing B C, B D and B E to any distance in G, H and I, let them make angles with the subtenses which succeed one another, namely, the external angles G C D, and H D E. Lastly, let the tangent K B be drawn, making with the first subtense the angle K B C. I say the angle G C D is double to the angle K B C, and the angle H D E triple to the same angle K B C. For if A C be drawn cutting B D in M, and from the point C there be drawn L C perpendicular to the same A C, then C L and M D will be parallel, by reason of the right angles at C and M; and therefore the alterne angles L C D and B D C will be equal: as also the angles B D C and C B D will be equal, because of the equality of the strait lines B C and C D. Wherefore the angle G C D is double to either of the angles C B D or C D B; and therefore also the angle G C D is double to the angle L C D, that is, to the angle K B C. Again, C D is parallel to B E, by reason of the equality of the angles C B E and D E B, and of the strait lines 195C B and D E; and therefore the angles G C D and G B E are equal; and consequently G B E, as also D E B is double to the angle K B C. But the external angle H D E is equal to the two internal D E B and D B E; and therefore the angle H D E is triple to the angle K B C, &c.; which was to be proved.

Coroll. I. From hence it is manifest, that the angles K B C and C B D, as also, that all the angles that are comprehended by two strait lines meeting in the circumference of a circle and insisting upon equal arches, are equal to one another.

Coroll. II. If the tangent B K be moved in the circumference with uniform motion about the centre B, it will in equal times cut off equal arches; and will pass over the whole perimeter in the same time in which itself describes a semiperimeter about the centre B.

Coroll. III. From hence also we may understand, what it is that determines the bending or curvation of a strait line into the circumference of a circle; namely, that it is fraction continually increasing in the same manner, as numbers, from one upwards, increase by the continual addition of unity. For the indefinite strait line K B being broken in B according to any angle, as that of K B C, and again in C according to a double angle, and in D according to an angle which is triple, and in E according to an angle which is quadruple to the first angle, and so continually, there will be described a figure which will indeed be rectilineal, if the broken parts be considered as having magnitude; but if they be understood to be the least that can be, that is, as so many points, then 196the figure described will not be rectilineal, but a circle, whose circumference will be the broken line.

Coroll. IV. From what has been said in this present article, it may also be demonstrated, that an angle in the centre is double to an angle in the circumference of the same circle, if the intercepted arches be equal. For seeing that strait line, by whose motion an angle is determined, passes over equal arches in equal times, as well from the centre as from the circumference; and while that, which is from the circumference, is passing over half its own perimeter, it passes in the same time over the whole perimeter of that which is from the centre, the arches, which it cuts off in the perimeter whose centre is A, will be double to those, which it makes in its own semiperimeter, whose centre is B. But in equal circles, as arches are to one another, so also are angles.

It may also be demonstrated, that the external angle made by a subtense produced and the next equal subtense is equal to an angle from the centre insisting upon the same arch; as in the last diagram, the angle G C D is equal to the angle C A D; for the external angle G C D is double to the angle C B D; and the angle C A D insisting upon the same arch C D is also double to the same angle C B D or K B C.

16. An angle of contingence, if it be compared with an angle simply so called, how little soever, has such proportion to it as a point has to a line; that is, no proportion at all, nor any quantity. For first, an angle of contingence is made by continual flexion; so that in the generation of it there is no 197circular motion at all, in which consists the nature of an angle simply so called; and therefore it cannot be compared with it according to quantity. Secondly, seeing the external angle made by a subtense produced and the next subtense is equal to an angle from the centre insisting upon the same arch, as in the last figure the angle G C D is equal to the angle C A D, the angle of contingence will be equal to that angle from the centre, which is made by A B and the same A B; for no part of a tangent can subtend any arch; but as the point of contact is to be taken for the subtense, so the angle of contingence is to be accounted for the external angle, and equal to that angle whose arch is the same point B.

Now, seeing an angle in general is defined to be the opening or divergence of two lines, which concur in one sole point; and seeing one opening is greater than another, it cannot be denied, but that by the very generation of it, an angle of contingence is quantity; for wheresoever there is greater and less, there is also quantity; but this quantity consists in greater and less flexion; for how much the greater a circle is, so much the nearer comes the circumference of it to the nature of a strait line; for the circumference of a circle being made by the curvation of a strait line, the less that strait line is, the greater is the curvation; and therefore, when one strait line is a tangent to many circles, the angle of contingence, which it makes with a less circle, is greater than that which it makes with a greater circle.

Nothing therefore is added to or taken from an angle simply so called, by the addition to it or 198taking from it of never so many angles of contingence. And as an angle of one sort can never be equal to an angle of the other sort, so they cannot be either greater or less than one another.

From whence it follows, that an angle of a segment, that is, the angle, which any strait line makes with any arch, is equal to the angle which is made by the same strait line, and another which touches the circle in the point of their concurrence; as in the last figure, the angle which is made between G B and B K is equal to that which is made between G B and the arch B C.

17. An angle, which is made by two planes, is commonly called the inclination of those planes; and because planes have equal inclination in all their parts, instead of their inclination an angle is taken, which is made by two strait lines, one of which is in one, the other in the other of those planes, but both perpendicular to the common section.

18. A solid angle may be conceived two ways. First, for the aggregate of all the angles, which are made by the motion of a strait line, while one extreme point thereof remaining fixed, it is carried about any plain figure, in which the fixed point of the strait line is not contained. And in this sense, it seems to be understood by Euclid. Now it is manifest, that the quantity of a solid angle so conceived is no other, than the aggregate of all the angles in a superficies so described, that is, in the superficies of a pyramidal solid. Secondly, when a pyramis or cone has its vertex in the centre of a sphere, a solid angle may be understood to be the proportion of a spherical superficies subtending 199that vertex to the whole superficies of the sphere. In which sense, solid angles are to one another as the spherical bases of solids, which have their vertex in the centre of the same sphere.

19. All the ways, by which two lines respect one another, or all the variety of their position, may be comprehended under four kinds; for any two lines whatsoever are either parallels, or being produced, if need be, or moved one of them to the other parallelly to itself, they make an angle; or else, by the like production and motion, they touch one another; or lastly, they are asymptotes. The nature of parallels, angles, and tangents, has been already declared. It remains that I speak briefly of the nature of asymptotes.

Asymptosy depends upon this, that quantity is infinitely divisible. And from hence it follows, that any line being given, and a body supposed to be moved from one extreme thereof towards the other, it is possible, by taking degrees of velocity always less and less, in such proportion as the parts of the line are made less by continual division, that the same body may be always moved forwards in that line, and yet never reach the end of it. For it is manifest, that if any strait line, as A F, (in the 8th figure) be cut anywhere in B, and again B F be cut in C, and C F in D, and D F in E, and so eternally, and there be drawn from the point F, the strait line F F at any angle A F F; and lastly, if the strait lines A F, B F, C F, D F, E F, &c., having the same proportion to one another with the segments of the line A F, be set in order and parallel to the same A F, the crooked line A B C D E, and the strait line F F, will be asymptotes, that is, they 200will always come nearer and nearer together, but never touch one another. Now, because any line may be cut eternally according to the proportions which the segments have to one another, therefore the divers kinds of asymptotes are infinite in number, and not necessary to be further spoken of in this place. In the nature of asymptotes in general there is no more, than that they come still nearer and nearer, but never touch. But in special in the asymptosy of hyperbolic lines, it is understood they should approach to a distance less than any given quantity.

20. Situation is the relation of one place to another; and where there are many places, their situation is determined by four things; by their distances from one another; by several distances from a place assigned; by the order of strait lines drawn from a place assigned to the places of them all; and by the angles which are made by the lines so drawn. For if their distances, order, and angles, be given, that is, be certainly known, their several places will also be so certainly known, as that they can be no other.

21. Points, how many soever they be, have like situation with an equal number of other points, when all the strait lines, that are drawn from some one point to all these, have severally the same proportion to those, that are drawn in the same order and at equal angles from some one point to all those. For let there be any number of points as A, B, and C, (in the 9th figure) to which from some one point D let the strait lines D A, D B, and D C be drawn; and let there be an equal number of other points, as E, F, and G, and from some 201point H let the strait lines H E, H F, and H G be drawn, so that the angles A D B and B D C be severally and in the same order equal to the angles E H F and F H G, and the strait lines D A, D B, and D C proportional to the strait lines H E, H F, and H G; I say, the three points A, B, and C, have like situation with the three points E, F, and G, or are placed alike. For if H E be understood to be laid upon D A, so that the point H be in D, the point F will be in the strait line D B, by reason of the equality of the angles A D B and E H F; and the point G will be in the strait line D C, by reason of the equality of the angles B D C and F H G; and the strait lines A B and E F, as also B C and F G, will be parallel, because A D. E H :: B D. F H :: C D. G H are proportionals by construction; and therefore the distances between the points A and B, and the points B and C, will be proportional to the distances between the points E and F, and the points F and G. Wherefore, in the situation of the points A, B, and C, and the situation of the points E, F and G, the angles in the same order are equal; so that their situations differ in nothing but the inequality of their distances from one another, and of their distances from the points D and H. Now, in both the orders of points, those inequalities are equal; for A B. B C :: E F. F G, which are their distances from one another, as also D A. D B. D C :: H E. H F. H G, which are their distances from the assumed points D and H, are proportionals. Their difference, therefore, consists solely in the magnitude of their distances. But, by the definition of like, (chapter I. article 2) those things, which differ only in magnitude, are like. Wherefore the points A, B, and C, have to 202one another like situation with the points E, F, and G, or are placed alike; which was to be proved.

Figure is quantity, determined by the situation or placing of all its extreme points. Now I call those points extreme, which are contiguous to the place which is without the figure. In lines therefore and superficies, all points may be called extreme; but in solids only those which are in the superficies that includes them.

Like figures are those, whose extreme points in one of them are all placed like all the extreme points in the other; for such figures differ in nothing but magnitude.

And like figures are alike placed, when in both of them the homologal strait lines, that is, the strait lines which connect the points which answer one another, are parallel, and have their proportional sides inclined the same way.

And seeing every strait line is like every other strait line, and every plane like every other plane, when nothing but planeness is considered; if the lines, which include planes, or the superficies, which include solids, have their proportions known, it will not be hard to know whether any figure be like or unlike to another propounded figure.

And thus much concerning the first grounds of philosophy. The next place belongs to geometry; in which the quantities of figures are sought out from the proportions of lines and angles. Wherefore it is necessary for him, that would study geometry, to know first what is the nature of quantity, proportion, angle and figure. Having therefore explained these in the three last chapters, I thought fit to add them to this part; and so pass to the next.

Vol. 1. Lat. & Eng.
C. XIV.
Fig. 1-10

PART III PROPORTIONS OF MOTIONS AND MAGNITUDES. — CHAPTER XV. OF THE NATURE, PROPERTIES, AND DIVERS CONSIDERATIONS OF MOTION AND ENDEAVOUR. — CHAPTER XVI. OF MOTION ACCELERATED AND UNIFORM, AND OF MOTION BY CONCOURSE. — CHAP. XVII. OF FIGURES DEFICIENT. — CHAPTER XVIII. OF THE EQUATION OF STRAIT LINES WITH THE CROOKED LINES OF PARABOLAS AND OTHER FIGURES MADE IN IMITATION OF PARABOLAS. — CHAPTER XIX. OF ANGLES OF INCIDENCE AND REFLECTION, EQUAL BY SUPPOSITION. — CHAPTER XX. OF THE DIMENSION OF A CIRCLE, AND THE DIVISION OF ANGLES OR ARCHES.

1.. Repetition of some principles of the doctrine of motion formerly set down.—2. Other principles added to them.—3. Certain theorems concerning the nature of motion.—4. Divers considerations of motion.—5. The way by which the first endeavour of bodies moved tendeth.—6. In motion which is made by concourse, one of the movents ceasing, the endeavour is made by the way by which the rest tend.—7. All endeavour is propagated in infinitum.—8. How much greater the velocity or magnitude is of a movent, so much the greater is the efficacy thereof upon any other body in its way.

1. The next things in order to be treated of are MOTION and MAGNITUDE, which are the most common accidents of all bodies. This place therefore most properly belongs to the elements of geometry. But because this part of philosophy, having been improved by the best wits of all ages, has afforded greater plenty of matter than can well 204be thrust together within the narrow limits of this discourse, I thought fit to admonish the reader, that before he proceed further, he take into his hands the works of Euclid, Archimedes, Apollonius, and other as well ancient as modern writers. For to what end is it, to do over again that which is already done? The little therefore that I shall say concerning geometry in some of the following chapters, shall be such only as is new, and conducing to natural philosophy.

I have already delivered some of the principles of this doctrine in the eighth and ninth chapters; which I shall briefly put together here, that the reader in going on may have their light nearer at hand.

First, therefore, in chap. VIII. art. 10, motion is defined to be the continual privation of one place, and acquisition of another.

Secondly, it is there shown, that whatsoever is moved is moved in time.

Thirdly, in the same chapter, art. 11, I have defined rest to be when a body remains for some time in one place.

Fourthly, it is there shown, that whatsoever is moved is not in any determined place; as also that the same has been moved, is still moved, and will yet be moved; so that in every part of that space, in which motion is made, we may consider three times, namely, the past, the present, and future time.

Fifthly, in art. 15 of the same chapter, I have defined velocity or swiftness to be motion considered as power, namely, that power by which a body moved may in a certain time transmit a 205certain length; which also may more briefly be enunciated thus, velocity is the quantity of motion determined by time and line.

Sixthly, in the same chapter, art. 16, I have shown that motion is the measure of time.

Seventhly, in the same chapter, art. 17, I have defined motions to be equally swift, when in equal times equal lengths are transmitted by them.

Eighthly, in art. 18 of the same chapter, motions are defined to be equal, when the swiftness of one moved body, computed in every part of its magnitude, is equal to the swiftness of another, computed also in every part of its magnitude. From whence it is to be noted, that motions equal to one another, and motions equally swift, do not signify the same thing; for when two horses draw abreast, the motion of both is greater than the motion of either of them singly; but the swiftness of both together is but equal to that of either.

Ninthly, in art. 19 of the same chapter, I have shown, that whatsoever is at rest will always be at rest, unless there be some other body besides it, which by getting into its place suffers it no longer to remain at rest. And that whatsoever is moved, will always be moved, unless there be some other body besides it, which hinders its motion.

Tenthly, in chap. IX. art. 7, I have demonstrated, that when any body is moved which was formerly at rest, the immediate efficient cause of that motion is in some other moved and contiguous body.

Eleventhly, I have shown in the same place, that whatsoever is moved, will always be moved in the same way, and with the same swiftness, if it be 206not hindered by some other moved and contiguous body.

2. To which principles I shall here add those that follow. First, I define ENDEAVOUR to be motion made in less space and time than can be given; that is, less than can be determined or assigned by exposition or number; that is, motion made through the length of a point, and in an instant or point of time. For the explaining of which definition it must be remembered, that by a point is not to be understood that which has no quantity, or which cannot by any means be divided; for there is no such thing in nature; but that, whose quantity is not at all considered, that is, whereof neither quantity nor any part is computed in demonstration; so that a point is not to be taken for an indivisible, but for an undivided thing; as also an instant is to be taken for an undivided, and not for an indivisible time.

In like manner, endeavour is to be conceived as motion; but so as that neither the quantity of the time in which, nor of the line in which it is made, may in demonstration be at all brought into comparison with the quantity of that time, or of that line of which it is a part. And yet, as a point may be compared with a point, so one endeavour may be compared with another endeavour, and one may be found to be greater or less than another. For if the vertical points of two angles be compared, they will be equal or unequal in the same proportion which the angles themselves have to one another. Or if a strait line cut many circumferences of concentric circles, the inequality of the points of intersection will be in the same proportion 207which the perimeters have to one another. And in the same manner, if two motions begin and end both together, their endeavours will be equal or unequal, according to the proportion of their velocities; as we see a bullet of lead descend with greater endeavour than a ball of wool.

Secondly, I define IMPETUS, or quickness of motion, to be the swiftness or velocity of the body moved, but considered in the several points of that time in which it is moved. In which sense impetus is nothing else but the quantity or velocity of endeavour. But considered with the whole time, it is the whole velocity of the body moved taken together throughout all the time, and equal to the product of a line representing the time, multiplied into a line representing the arithmetically mean impetus or quickness. Which arithmetical mean, what it is, is defined in the 29th article of chapter XIII.

And because in equal times the ways that are passed are as the velocities, and the impetus is the velocity they go withal, reckoned in all the several points of the times, it followeth that during any time whatsoever, howsoever the impetus be increased or decreased, the length of the way passed over shall be increased or decreased in the same proportion; and the same line shall represent both the way of the body moved, and the several impetus or degrees of swiftness wherewith the way is passed over.

And if the body moved be not a point, but a strait line moved so as that every point thereof make a several strait line, the plane described by its motion, whether uniform, accelerated, or retarded, 208shall be greater or less, the time being the same, in the same proportion with that of the impetus reckoned in one motion to the impetus reckoned in the other. For the reason is the same in parallelograms and their sides.

For the same cause also, if the body moved be a plane, the solid described shall be still greater or less in the proportions of the several impetus or quicknesses reckoned through one line, to the several impetus reckoned through another.

This understood, let A B C D, (in figure 1, chap. XVII.) be a parallelogram; in which suppose the side A B to be moved parallelly to the opposite side C D, decreasing all the way till it vanish in the point C, and so describing the figure A B E F C; the point B, as A B decreaseth, will therefore describe the line B E F C; and suppose the time of this motion designed by the line C D; and in the same time C D, suppose the side A C to be moved parallel and uniformly to B D. From the point O taken at adventure in the line C D, draw O R parallel to B D, cutting the line B E F C in E, and the side A B in R. And again, from the point Q taken also at adventure in the line C D, draw Q S parallel to B D, cutting the line B E F C in F, and the side A B in S; and draw E G and F H parallel to C D, cutting A C in G and H. Lastly, suppose the same construction done in all the points possible of the line B E F C. I say, that as the proportions of the swiftness wherewith Q F, O E, D B, and all the rest supposed to be drawn parallel to D B and terminated in the line B E F C, are to the proportions of their several times designed by the several parallels H F, G E, A B, and all the 209rest supposed to be drawn parallel to the line of time C D and terminated in the line B E F C, the aggregate to the aggregate, so is the area or plane D B E F C to the area or plane A C F E B. For as A B decreasing continually by the line B E F C vanisheth in the time C D into the point C, so in the same time the line D C continually decreasing vanisheth by the same line C F E B into the point B; and the point D describeth in that decreasing motion the line D B equal to the line A C described by the point A in the decreasing motion of A B; and their swiftnesses are therefore equal. Again, because in the time G E the point O describeth the line OE, and in the same time the point S describeth the line S E, the line O E shall be to the line SE, as the swiftness wherewith OE is described to the swiftness wherewith SE is described. In like manner, because in the same time H F the point Q describeth the line Q F, and the point R the line R F, it shall be as the swiftness by which Q F is described to the swiftness by which R F is described, so the line itself Q F to the line itself R F; and so in all the lines that can possibly be drawn parallel to B D in the points where they cut the line B E F C. But all the parallels to B D, as S E, R F, A C, and the rest that can possibly be drawn from the line A B to the line B E F C, make the area of the plane A B E F C; and all the parallels to the same B D, as Q F, O E, D B and the rest drawn to the points where they cut the same line B E F C, make the area of the plane B E F C D. As therefore the aggregate of the swiftnesses wherewith the plane B E F C D is described, is to the aggregate of the swiftnesses wherewith 210the plane A C F E B is described, so is the plane itself B E F C D to the plane itself A C F E B. But the aggregate of the times represented by the parallels A B, G E, H F and the rest, maketh also the area A C F E B. And therefore, as the aggregate of all the lines Q F, O E, D B and all the rest of the lines parallel to B D and terminated in the line B E F C, is to the aggregate of all the lines H F, G E, A B and all the rest of the lines parallel to C D and terminated in the same line B E F C; that is, as the aggregate of the lines of swiftness to the aggregate of the lines of time, or as the whole swiftness in the parallels to D B to the whole time in the parallels to C D, so is the plane B E F C D to the plane A C F E B. And the proportions of Q F to F H, and of O E to E G, and of D B to B A, and so of all the rest taken together, are the proportions of the plane D B E F C to the plane A B E F C. But the lines Q F, O E, D B and the rest are the lines that design the swiftness; and the lines H F, G E, A B and the rest are the lines that design the times of the motions; and therefore the proportion of the plane D B E F C to the plane A B E F C is the proportion of all the velocities taken together to all the times taken together. Wherefore, as the proportions of the swiftnesses, &c.; which was to be demonstrated.

The same holds also in the diminution of the circles, whereof the lines of time are the semidiameters, as may easily be conceived by imagining the whole plane A B C D turned round upon the axis B D; for the line B E F C will be everywhere in the superficies so made, and the lines H F, G E, A B, which are here parallelograms, will be there 211cylinders, the diameters of whose bases are the lines H F, G E, A B, &c. and the altitude a point, that is to say, a quantity less than any quantity that can possibly be named; and the lines Q F, O E, D B, &c. small solids whose lengths and breadths are less than any quantity that can be named.

But this is to be noted, that unless the proportion of the sum of the swiftnesses to the proportion of the sum of the times be determined, the proportion of the figure D B E F C to the figure A B E F C cannot be determined.

Thirdly, I define RESISTANCE to be the endeavour of one moved body either wholly or in part contrary to the endeavour of another moved body, which toucheth the same. I say, wholly contrary, when the endeavour of two bodies proceeds in the same strait line from the opposite extremes, and contrary in part, when two bodies have their endeavour in two lines, which, proceeding from the extreme points of a strait line, meet without the same.

Fourthly, that I may define what it is to PRESS, I say, that of two moved bodies one presses the other, when with its endeavour it makes either all or part of the other body to go out of its place.

Fifthly, a body, which is pressed and not wholly removed, is said to RESTORE itself, when, the pressing body being taken away, the parts which were moved do, by reason of the internal constitution of the pressed body, return every one into its own place. And this we may observe in springs, in blown bladders, and in many other bodies, whose parts yield more or less to the endeavour which the pressing body makes at the 212first arrival; but afterwards, when the pressing body is removed, they do, by some force within them, restore themselves, and give their whole body the same figure it had before.

Sixthly, I define FORCE to be the impetus or quickness of motion multiplied either into itself, or into the magnitude of the movent, by means whereof the said movent works more or less upon the body that resists it.

3. Having premised thus much, I shall now demonstrate, first, that if a point moved come to touch another point which is at rest, how little soever the impetus or quickness of its motion be, it shall move that other point. For if by that impetus it do not at all move it out of its place, neither shall it move it with double the same impetus. For nothing doubled is still nothing; and for the same reason it shall never move it with that impetus, how many times soever it be multiplied, because nothing, however it be multiplied, will for ever be nothing. Wherefore, when a point is at rest, if it do not yield to the least impetus, it will yield to none; and consequently it will be impossible that that, which is at rest, should ever be moved.

Secondly, that when a point moved, how little soever the impetus thereof be, falls upon a point of any body at rest, how hard soever that body be, it will at the first touch make it yield a little. For if it do not yield to the impetus which is in that point, neither will it yield to the impetus of never so many points, which have all their impetus severally equal to the impetus of that point. For seeing all those points together work equally, if any one 213of them have no effect, the aggregate of them all together shall have no effect as many times told as there are points in the whole body, that is, still no effect at all; and by consequent there would be some bodies so hard that it would be impossible to break them; that is, a finite hardness, or a finite force, would not yield to that which is infinite; which is absurd.

Coroll. It is therefore manifest, that rest does nothing at all, nor is of any efficacy; and that nothing but motion gives motion to such things as be at rest, and takes it from things moved.

Thirdly, that cessation in the movent does not cause cessation in that which was moved by it. For (by number 11 of art. 1 of this chapter) whatsoever is moved perseveres in the same way and with the same swiftness, as long as it is not hindered by something that is moved against it. Now it is manifest, that cessation is not contrary motion; and therefore it follows that the standing still of the movent does not make it necessary that the thing moved should also stand still.

Coroll. They are therefore deceived, that reckon the taking away of the impediment or resistance for one of the causes of motion.

4. Motion is brought into account for divers respects; first, as in a body undivided, that is, considered as a point; or, as in a divided body. In an undivided body, when we suppose the way, by which the motion is made, to be a line; and in a divided body, when we compute the motion of the several parts of that body, as of parts.

Secondly, from the diversity of the regulation of motion, it is in body, considered as undivided. 214sometimes uniform and sometimes multiform. Uniform is that by which equal lines are always transmitted in equal times; and multiform, when in one time more, in another time less space is transmitted. Again, of multiform motions, there are some in which the degrees of acceleration and retardation proceed in the same proportions, which the spaces transmitted have, whether duplicate, or triplicate, or by whatsoever number multiplied; and others in which it is otherwise.

Thirdly, from the number of the movents; that is, one motion is made by one movent only, and another by the concourse of many movents.

Fourthly, from the position of that line in which a body is moved, in respect of some other line; and from hence one motion is called perpendicular, another oblique, another parallel.

Fifthly, from the position of the movent in respect of the moved body; from whence one motion is pulsion or driving, another traction or drawing. Pulsion, when the movent makes the moved body go before it; and traction, when it makes it follow. Again, there are two sorts of pulsion; one, when the motions of the movent and moved body begin both together, which may be called trusion or thrusting and vection; the other, when the movent is first moved, and afterwards the moved body, which motion is called percussion or stroke.

Sixthly, motion is considered sometimes from the effect only which the movent works in the moved body, which is usually called moment. Now moment is the excess of motion which the movent has above the motion or endeavour of the resisting body.

215Seventhly, it may be considered from the diversity of the medium; as one motion may be made in vacuity or empty place; another in a fluid; another in a consistent medium, that is, a medium whose parts are by some power so consistent and cohering, that no part of the same will yield to the movent, unless the whole yield also.

Eighthly, when a moved body is considered as having parts, there arises another distinction of motion into simple and compound. Simple, when all the several parts describe several equal lines; compounded, when the lines described are unequal.

5. All endeavour tends towards that part, that is to say, in that way which is determined by the motion of the movent, if the movent be but one; or, if there be many movents, in that way which their concourse determines. For example, if a moved body have direct motion, its first endeavour will be in a strait line; if it have circular motion, its first endeavour will be in the circumference of a circle.

6. And whatsoever the line be, in which a body has its motion from the concourse of two movents, as soon as in any point thereof the force of one of the movents ceases, there immediately the former endeavour of that body will be changed into an endeavour in the line of the other movent.

Wherefore, when any body is carried on by the concourse of two winds, one of those winds ceasing, the endeavour and motion of that body will be in that line, in which it would have been carried by that wind alone which blows still. And in the describing of a circle, where that which is moved has its motion determined by a movent in a 216tangent, and by the radius which keeps it in a certain distance from the centre, if the retention of the radius cease, that endeavour, which was in the circumference of the circle, will now be in the tangent, that is, in a strait line. For, seeing endeavour is computed in a less part of the circumference than can be given, that is, in a point, the way by which a body is moved in the circumference is compounded of innumerable strait lines, of which every one is less than can be given; which are therefore called points. Wherefore when any body, which is moved in the circumference of a circle, is freed from the retention of the radius, it will proceed in one of those strait lines, that is, in a tangent.

7. All endeavour, whether strong or weak, is propagated to infinite distance; for it is motion. If therefore the first endeavour of a body be made in space which is empty, it will always proceed with the same velocity; for it cannot be supposed that it can receive any resistance at all from empty space; and therefore, (by art. 7, chap, IX) it will always proceed in the same way and with the same swiftness. And if its endeavour be in space which is filled, yet, seeing endeavour is motion, that which stands next in its way shall be removed, and endeavour further, and again remove that which stands next, and so infinitely. Wherefore the propagation of endeavour, from one part of full space to another, proceeds infinitely. Besides, it reaches in any instant to any distance, how great soever. For in the same instant in which the first part of the full medium removes that which is next it, the second also removes that part which is next 217to it; and therefore all endeavour, whether it be in empty or in full space, proceeds not only to any distance, how great soever, but also in any time, how little soever, that is, in an instant. Nor makes it any matter, that endeavour, by proceeding, grows weaker and weaker, till at last it can no longer be perceived by sense; for motion may be insensible; and I do not here examine things by sense and experience, but by reason.

8. When two movents are of equal magnitude, the swifter of them works with greater force than the slower, upon a body that resists their motion. Also, if two movents have equal velocity, the greater of them works with more force than the less. For where the magnitude is equal, the movent of greater velocity makes the greater impression upon that body upon which it falls; and where the velocity is equal, the movent of greater magnitude falling upon the same point, or an equal part of another body, loses less of its velocity, because the resisting body works only upon that part of the movent which it touches, and therefore abates the impetus of that part only; whereas in the mean time the parts, which are not touched, proceed, and retain their whole force, till they also come to be touched; and their force has some effect. Wherefore, for example, in batteries a longer than a shorter piece of timber of the same thickness and velocity, and a thicker than a slenderer piece of the same length and velocity, work a greater effect upon the wall.

1. The velocity of any body, in what time soever it be computed, is that which is made of the multiplication of the impetus, or quickness of its motion into the time.—2-5. In all motion, the lengths which are passed through are to one another, as the products made by the impetus multiplied into the time.—6. If two bodies be moved with uniform motion through two lengths, the proportion of those lengths to one another will be compounded of the proportions of time to time, and impetus to impetus, directly taken.—7. If two bodies pass through two lengths with uniform motion, the proportion of their times to one another will be compounded of the proportions of length to length, and impetus to impetus reciprocally taken; also the proportion of their impetus to one another will be compounded of the proportions of length to length, and time to time reciprocally taken.—8. If a body be carried on with uniform motion by two movents together, which meet in an angle, the line by which it passes will be a strait line, subtending the complement of that angle to two right angles.—9, &c. If a body be carried by two movents together, one of them being moved with uniform, the other with accelerated motion, and the proportion of their lengths to their times being explicable in numbers, how to find out what line that body describes.

1. The velocity of any body, in whatsoever time it be moved, has its quantity determined by the sum of all the several quicknesses or impetus, which it hath in the several points of the time of the body's motion. For seeing velocity, (by the definition of it, chap, VIII, art. 15) is that power by which a body can in a certain time pass through a certain length; and quickness of motion or impetus, (by 219chap. XV, art. 2, num. 2) is velocity taken in one point of time only, all the impetus, together taken in all the points of time, will be the same thing with the mean impetus multiplied into the whole time, or which is all one, will be the velocity of the whole motion.

Coroll. If the impetus be the same in every point, any strait line representing it may be taken for the measure of time: and the quicknesses or impetus applied ordinately to any strait line making an angle with it, and representing the way of the body's motion, will design a parallelogram which shall represent the velocity of the whole motion. But if the impetus or quickness of motion begin from rest and increase uniformly, that is, in the same proportion continually with the times which are passed, the whole velocity of the motion shall be represented by a triangle, one side whereof is the whole time, and the other the greatest impetus acquired in that time; or else by a parallelogram, one of whose sides is the whole time of motion, and the other, half the greatest impetus; or lastly, by a parallelogram having for one side a mean proportional between the whole time and the half of that time, and for the other side the half of the greatest impetus. For both these parallelograms are equal to one another, and severally equal to the triangle which is made of the whole line of time, and of the greatest acquired impetus; as is demonstrated in the elements of geometry.

2. In all uniform motions the lengths which are transmitted are to one another, as the product of the mean impetus multiplied into its time, to the 220product of the mean impetus multiplied also into its time.

For let A B (in fig. 1) be the time, and A C the impetus by which any body passes with uniform motion through the length D E; and in any part of the time A B, as in the time A F, let another body be moved with uniform motion, first, with the same impetus A C. This body, therefore, in the time A F with the impetus A C will pass through the length A F. Seeing, therefore, when bodies are moved in the same time, and with the same velocity and impetus in every part of their motion, the proportion of one length transmitted to another length transmitted, is the same with that of time to time, it followeth, that the length transmitted in the time A B with the impetus A C will be to the length transmitted in the time A F with the same impetus A C, as A B itself is to A F, that is, as the parallelogram A I is to the parallelogram A H, that is, as the product of the time A B into the mean impetus A C is to the product of the time A F into the same impetus A C. Again, let it be supposed that a body be moved in the time A F, not with the same but with some other uniform impetus, as A L. Seeing therefore, one of the bodies has in all the parts of its motion the impetus A C, and the other in like manner the impetus A L, the length transmitted by the body moved with the impetus A C will be to the length transmitted by the body moved with the impetus A L, as A C itself is to A L, that is, as the parallelogram A H is to the parallelogram F L. Wherefore, by ordinate proportion it will be, as the parallelogram A I to the parallelogram F L, that is, as the product 221of the mean impetus into the time is to the product of the mean impetus into the time, so the length transmitted in the time A B with the impetus A C, to the length transmitted in the time A F with the impetus A L; which was to be demonstrated.

Coroll. Seeing, therefore, in uniform motion, as has been shown, the lengths transmitted are to one another as the parallelograms which are made by the multiplication of the mean impetus into the times, that is, by reason of the equality of the impetus all the way, as the times themselves, it will also be, by permutation, as time to length, so time to length; and in general, to this place are applicable all the properties and transmutations of analogisms, which I have set down and demonstrated in chapter XIII.

3. In motion begun from rest and uniformly accelerated, that is, where the impetus increaseth continually according to the proportion of the times, it will also be, as one product made by the mean impetus multiplied into the time, to another product made likewise by the mean impetus multiplied into the time, so the length transmitted in the one time to the length transmitted in the other time.

For let A B (in fig. 1) represent a time; in the beginning of which time A, let the impetus be as the point A; but as the time goes on, so let the impetus increase uniformly, till in the last point of that time A B, namely in B, the impetus acquired be B I. Again, let A F represent another time, in whose beginning A, let the impetus be as the point itself A; but as the time proceeds, so let the impetus 222increase uniformly, till in the last point F of the time A F the impetus acquired be F K; and let D E be the length passed through in the time A B with impetus uniformly increased. I say, the length D E is to the length transmitted in the time A F, as the time A B multiplied into the mean of the impetus increasing through the time A B, is to the time A F multiplied into the mean of the impetus increasing through the time A F.

For seeing the triangle A B I is the whole velocity of the body moved in the time A B, till the impetus acquired be B I; and the triangle A F K the whole velocity of the body moved in the time A F with impetus increasing till there be acquired the impetus F K; the length D E to the length acquired in the time A F with impetus increasing from rest in A till there be acquired the impetus F K, will be as the triangle A B I to the triangle A F K, that is, if the triangles A B I and A F K be like, in duplicate proportion of the time A B to the time A F; but if unlike, in the proportion compounded of the proportions of A B to A F and of B I to F K. Wherefore, as A B I is to A F K, so let D E be to D P; for so, the length transmitted in the time A B with impetus increasing to B I, will be to the length transmitted in the time A F with impetus increasing to F K, as the triangle A B I is to the triangle A F K; but the triangle A B I is made by the multiplication of the time A B into the mean of the impetus increasing to B I; and the triangle A F K is made by the multiplication of the time A F into the mean of the impetus increasing to F K; and therefore the length D E which is transmitted in the time A B 223with impetus increasing to B I, to the length D P which is transmitted in the time A F with impetus increasing to F K, is as the product which is made of the time A B multiplied into its mean impetus, to the product of the time A F multiplied also into its mean impetus; which was to be proved.

Coroll. I. In motion uniformly accelerated, the proportion of the lengths transmitted to that of their times, is compounded of the proportions of their times to their times, and impetus to impetus.

Coroll. II. In motion uniformly accelerated, the lengths transmitted in equal times, taken in continual succession from the beginning of motion, are as the differences of square numbers beginning from unity, namely, as 3, 5, 7, &c. For if in the first time the length transmitted be as 1, in the first and second times the length transmitted will be as 4, which is the square of 2, and in the three first times it will be as 9, which is the square of 3, and in the four first times as 16, and so on. Now the differences of these squares are 3, 5, 7, &c.

Coroll. III. In motion uniformly accelerated from rest, the length transmitted is to another length transmitted uniformly in the same time, but with such impetus as was acquired by the accelerated motion in the last point of that time, as a triangle to a parallelogram, which have their altitude and base common. For seeing the length D E (in fig. 1) is passed through with velocity as the triangle A B I, it is necessary that for the passing through of a length which is double to D E, the velocity be as the parallelogram A I; for the parallelogram A I is double to the triangle A B I.

4. In motion, which beginning from rest is so aclerated, 224that the impetus thereof increases continually in proportion duplicate to the proportion of the times in which it is made, a length transmitted in one time will be to a length transmitted in another time, as the product made by the mean impetus multiplied into the time of one of those motions, to the product of the mean impetus multiplied into the time of the other motion.

For let A B (in fig. 2) represent a time, in whose first instant A let the impetus be as the point A; but as the time proceeds, so let the impetus increase continually in duplicate proportion to that of the times, till in the last point of time B the impetus acquired be B I; then taking the point F anywhere in the time A B, let the impetus F K acquired in the time A F be ordinately applied to that point F. Seeing therefore the proportion of F K to B I is supposed to be duplicate to that of A F to A B, the proportion of A F to A B will be subduplicate to that of F K to B I; and that of A B to A F will be (by chap. XIII. art. 16) duplicate to that of B I to F K; and consequently the point K will be in a parabolical line, whose diameter is A B and base B I; and for the same reason, to what point soever of the time A B the impetus acquired in that time be ordinately applied, the strait line designing that impetus will be in the same parabolical line A K I. Wherefore the mean impetus multiplied into the whole time A B will be the parabola A K I B, equal to the parallelogram A M, which parallelogram has for one side the line of time A B and for the other the line of the impetus A L, which is two-thirds of the impetus B I; for every parabola is equal to two-thirds 225of that parallelogram with which it has its altitude and base common. Wherefore the whole velocity in A B will be the parallelogram A M, as being made by the multiplication of the impetus A L into the time A B. And in like manner, if F N be taken, which is two-thirds of the impetus F K, and the parallelogram F O be completed, F O will be the whole velocity in the time A F, as being made by the uniform impetus A O or F N multiplied into the time A F. Let now the length transmitted in the time A B and with the velocity A M be the strait line D E; and lastly, let the length transmitted in the time A F with the velocity A N be D P; I say that as A M is to A N, or as the parabola A K I B to the parabola A K F, so is D E to D P. For as A M is to F L, that is, as A B is to A F, so let D E be to D G. Now the proportion of A M to A N is compounded of the proportions of A M to F L, and of F L to A N. But as A M to F L, so by construction is D E to D G; and as F L is to A N (seeing the time in both is the same, namely, A F), so is the length D G to the length D P; for lengths transmitted in the same time are to one another as their velocities are. Wherefore by ordinate proportion, as A M is to A N, that is, as the mean impetus A L multiplied into its time A B, is to the mean impetus A O multiplied into A F, so is D E to D P; which was to be proved.

Coroll. I. Lengths transmitted with motion so accelerated, that the impetus increase continually in duplicate proportion to that of their times, if the base represent the impetus, are in triplicate proportion of their impetus acquired in the last 226point of their times. For as the length D E is to the length D P, so is the parallelogram A M to the parallelogram A N, and so the parabola A K I B to the parabola A K F. But the proportion of the parabola A K I B to the parabola A K F is triplicate to the proportion which the base B I has to the base F K. Wherefore also the proportion of D E to D P is triplicate to that of B I to F K.

Coroll. II. Lengths transmitted in equal times succeeding one another from the beginning, by motion so accelerated, that the proportion of the impetus be duplicate to the proportion of the times, are to one another as the differences of cubic numbers beginning at unity, that is as 7, 19, 37, &c. For if in the first time the length transmitted be as 1, the length at the end of the second time will be as 8, at the end of the third time as 27, and at the end of the fourth time as 64, &c.; which are cubic numbers, whose differences are 7, 19, 37, &c.

Coroll. III. In motion so accelerated, as that the length transmitted be always to the length transmitted in duplicate proportion to their times, the length uniformly transmitted in the whole time, and with impetus all the way equal to that which is last acquired, is as a parabola to a parallelogram of the same altitude and base, that is, as 2 to 3. For the parabola A K I B is the impetus increasing in the time A B; and the parallelogram A I is the greatest uniform impetus multiplied into the same time A B. Wherefore the lengths transmitted will be as a parabola to a parallelogram, &c., that is, as 2 to 3.

5. If I should proceed to the explication of such motions as are made by impetus increasing in proportion 227triplicate, quadruplicate, quintuplicate, &c., to that of their times, it would be a labour infinite and unnecessary. For by the same method by which I have computed such lengths, as are transmitted with impetus increasing in single and duplicate proportion, any man may compute such as are transmitted with impetus increasing in triplicate, quadruplicate, or what other proportion he pleases.

In making which computation he shall find, that where the impetus increase in proportion triplicate to that of the times, there the whole velocity will be designed by the first parabolaster (of which see the next chapter); and the lengths transmitted will be in proportion quadruplicate to that of the times. And in like manner, where the impetus increase in quadruplicate proportion to that of the times, that there the whole velocity will be designed by the second parabolaster, and the lengths transmitted will be in quintuplicate proportion to that of the times; and so on continually.

6. If two bodies with uniform motion transmit two lengths, each with its own impetus and time, the proportion of the lengths transmitted will be compounded of the proportions of time to time, and impetus to impetus, directly taken.

Let two bodies be moved uniformly (as in fig. 3), one in the time A B with the impetus A C, the other in the time A D with the impetus A E. I say the lengths transmitted have their proportion to one another compounded of the proportions of A B to A D, and of A C to A E. For let any length whatsoever, as Z, be transmitted by one of the bodies in the time A B with the impetus A C; and any other length, as X, be transmitted by the 228other body in the time A D with the impetus A E; and let the parallelograms A F and A G be completed. Seeing now Z is to X (by art. 2) as the impetus A C multiplied into the time A B is to the impetus A E multiplied into the time A D, that is, as A F to A G; the proportion of Z to X will be compounded of the same proportions, of which the proportion of A F to A G is compounded; but the proportion of A F to A G is compounded of the proportions of the side A B to the side A D, and of the side A C to the side A E (as is evident by the Elements of Euclid), that is, of the proportions of the time A B to the time A D, and of the impetus A C to the impetus A E. Wherefore also the proportion of Z to X is compounded of the same proportions of the time A B to the time A D, and of the impetus A C to the impetus A E; which was to be demonstrated.

Coroll. I. When two bodies are moved with uniform motion, if the times and impetus be in reciprocal proportion, the lengths transmitted shall be equal. For if it were as A B to A D (in the same fig. 3) so reciprocally A E to A C, the proportion of A F to A G would be compounded of the proportions of A B to A D, and of A C to A E, that is, of the proportions of A B to A D, and of A D to A B. Wherefore, A F would be to A G as A B to A B, that is, equal; and so the two products made by the multiplication of impetus into time would be equal; and by consequent, Z would be equal to X.

Coroll. II. If two bodies be moved in the same time, but with different impetus, the lengths transmitted will be as impetus to impetus. For if the 229time of both of them be A D, and their different impetus be A E and A C, the proportion of A G to D C will be compounded of the proportions of A E to A C and of A D to A D, that is, of the proportions of A E to A C and of A C to A C; and so the proportion of A G to D C, that is, the proportion of length to length, will be as A E to A C, that is, as that of impetus to impetus. In like manner, if two bodies be moved uniformly, and both of them with the same impetus, but in different times, the proportion of the lengths transmitted by them will be as that of their times. For if they have both the same impetus A C, and their different times be A B and A D, the proportion of A F to D C will be compounded of the proportions of A B to A D and of A C to A C; that is, of the proportions of A B to A D and of A D to A D; and therefore the proportion of A F to D C, that is, of length to length, will be the same with that of A B to A D, which is the proportion of time to time.

7. If two bodies pass through two lengths with uniform motion, the proportion of the times in which they are moved will be compounded of the proportions of length to length and impetus to impetus reciprocally taken.

For let any two lengths be given, as (in the same fig. 3) Z and X, and let one of them be transmitted with the impetus A C, the other with the impetus A E. I say the proportion of the times in which they are transmitted, will be compounded of the proportions of Z to X, and of A E, which is the impetus with which X is transmitted, to A C, the impetus with which Z is transmitted. For seeing A F is the product of the impetus A C multiplied 230into the time A B, the time of motion through Z will be a line, which is made by the application of the parallelogram A F to the strait line A C, which line is A B; and therefore A B is the time of motion through Z. In like manner, seeing A G is the product of the impetus A E multiplied into the time A D, the time of motion through X will be a line which is made by the application of A G to the strait line A D; but A D is the time of motion through X. Now the proportion of A B to A D is compounded of the proportions of the parallelogram A F to the parallelogram A G, and of the impetus A E to the impetus A C; which may be demonstrated thus. Put the parallelograms in order A F, A G, D C, and it will be manifest that the proportion of A F to D C is compounded of the proportions of A F to A G and of A G to D C; but A F is to D C as A B to A D; wherefore also the proportion of A B to A D is compounded of the proportions of A F to A G and of A G to D C. And because the length Z is to the length X as A F is to A G, and the impetus A E to the impetus A C as A G to D C, therefore the proportion of A B to A D will be compounded of the proportions of the length Z to the length X, and of the impetus A E to the impetus A C; which was to be demonstrated.

In the same manner it may be proved, that in two uniform motions the proportion of the impetus is compounded of the proportions of length to length and of time to time reciprocally taken.

For if we suppose A C (in the same fig. 3) to be the time, and A B the impetus with which the length Z is passed through; and A E to be the 231time, and A D the impetus with which the length X is passed through, the demonstration will proceed as in the last article.

8. If a body be carried by two movents together, which move with strait and uniform motion, and concur in any given angle, the line by which that body passes will be a strait line.

Let the movent A B (in fig. 4) have strait and uniform motion, and be moved till it come into the place C D; and let another movent A C, having likewise strait and uniform motion, and making with the movent A B any given angle C A B, be understood to be moved in the same time to D B; and let the body be placed in the point of their concourse, A. I say the line which that body describes with its motion is a strait line. For let the parallelogram A B D C be completed, and its diagonal A D be drawn; and in the strait line A B let any point E be taken; and from it let E F be drawn parallel to the strait lines A C and B D, cutting A D in G; and through the point G let H I be drawn parallel to the strait lines A B and C D; and lastly, let the measure of the time be A C. Seeing therefore both the motions are made in the same time, when A B is in C D, the body also will be in C D; and in like manner, when A C is in B D, the body will be in B D. But A B is in C D at the same time when A C is in B D; and therefore the body will be in C D and B D at the same time; wherefore it will be in the common point D. Again, seeing the motion from A C to B D is uniform, that is, the spaces transmitted by it are in proportion to one another as the times in which they are transmitted, when A C is in E F, 232the proportion of A B to A E will be the same with that of E F to E G, that is, of the time A C to the time A H. Wherefore A B will be in H I in the same time in which A C is in E F, so that the body will at the same time be in E F and H I, and therefore in their common point G. And in the same manner it will be, wheresoever the point E be taken between A and B. Wherefore the body will always be in the diagonal A D; which was to be demonstrated.

Coroll. From hence it is manifest, that the body will be carried through the same strait line A D, though the motion be not uniform, provided it have like acceleration; for the proportion of A B to A E will always be the same with that of A C to A H.

9. If a body be carried by two movents together, which meet in any given angle, and are moved, the one uniformly, the other with motion uniformly accelerated from rest, that is, that the proportion of their impetus be as that of their times, that is, that the proportion of their lengths be duplicate to that of the lines of their times, till the line of greatest impetus acquired by acceleration be equal to that of the line of time of the uniform motion; the line in which the body is carried will be the crooked line of a semiparabola, whose base is the impetus last acquired, and vertex the point of rest.

Let the straight line A B (in fig. 5) be understood to be moved with uniform motion to C D; and let another movent in the strait line A C be supposed to be moved in the same time to B D, but with motion uniformly accelerated, that is, 233with such motion, that the proportion of the spaces which are transmitted be always duplicate to that of the times, till the impetus acquired be B D equal to the strait line A C; and let the semiparabola A G D B be described. I say that by the concourse of those two movents, the body will be carried through the semiparabolical crooked line A G D. For let the parallelogram A B D C be completed; and from the point E, taken anywhere in the strait line A B, let E F be drawn parallel to A C and cutting the crooked line in G; and lastly, through the point G let H I be drawn parallel to the strait lines A B and C D. Seeing therefore the proportion of A B to A E is by supposition duplicate to the proportion of E F to E G, that is, of the time A C to the time A H, at the same time when A C is in E F, A B will be in H I; and therefore the moved body will be in the common point G. And so it will always be, in what part soever of A B the point E be taken. Wherefore the moved body will always be found in the parabolical line A G D; which was to be demonstrated.

10. If a body be carried by two movents together, which meet in any given angle, and are moved the one uniformly, the other with impetus increasing from rest, till it be equal to that of the uniform motion, and with such acceleration, that the proportion of the lengths transmitted be every where triplicate to that of the times in which they are transmitted; the line, in which that body is moved, will be the crooked line of the first semiparabolaster of two means, whose base is the impetus last acquired.

Let the strait line A B (in the 6th figure) be moved 234uniformly to C D; and let another movent A C be moved at the same time to B D with motion so accelerated, that the proportion of the lengths transmitted be everywhere triplicate to the proportion of their times; and let the impetus acquired in the end of that motion be B D, equal to the strait line A C; and lastly, let A G D be the crooked line of the first semiparabolaster of two means. I say, that by the concourse of the two movents together, the body will be always in that crooked line A G D. For let the parallelogram A B D C be completed; and from the point E, taken anywhere in the strait line A B, let E F be drawn parallel to A C, and cutting the crooked line in G; and through the point G let H I be drawn parallel to the strait lines A B and C D. Seeing therefore the proportion of A B to A E is, by supposition, triplicate to the proportion of E F to E G, that is, of the time A C to the time A H, at the same time when A C is in E F, A B will be in H I; and therefore the moved body will be in the common point G. And so it will always be, in what part soever of A B the point E be taken; and by consequent, the body will always be in the crooked line A G D; which was to be demonstrated.

11. By the same method it may be shown, what line it is that is made by the motion of a body carried by the concourse of any two movents, which are moved one of them uniformly, the other with acceleration, but in such proportions of spaces and times as are explicable by numbers, as duplicate, triplicate, &c., or such as may be designed by any broken number whatsoever. For which this is the rule. Let the two numbers of the length 235and time be added together; and let their sum be the denominator of a fraction, whose numerator must be the number of the length. Seek this fraction in the table of the third article of the XVIIth chapter; and the line sought will be that, which denominates the three-sided figure noted on the left hand; and the kind of it will be that, which is numbered above over the fraction. For example, let there be a concourse of two movents, whereof one is moved uniformly, the other with motion so accelerated, that the spaces are to the times as 5 to 3. Let a fraction be made whose denominator is the sum of 5 and 3, and the numerator 5, namely the fraction ⁵⁄₈. Seek in the table, and you will find ⁵⁄₈ to be the third in that row, which belongs to the three-sided figure of four means. Wherefore the line of motion made by the concourse of two such movents, as are last of all described, will be the crooked line of the third parabolaster of four means.

12. If motion be made by the concourse of two movents, whereof one is moved uniformly, the other beginning from rest in the angle of concourse with any acceleration whatsoever; the movent, which is moved uniformly, shall put forward the moved body in the several parallel spaces, less than if both the movents had uniform motion; and still less and less, as the motion of the other movent is more and more accelerated.

Let the body be placed in A, (in the 7th figure) and be moved by two movents, by one with uniform motion from the strait line A B to the strait line C D parallel to it; and by the other with any acceleration, from the strait line A C to the strait 236line B D parallel to it; and in the parallelogram A B D C let a space be taken between any two parallels E F and G H. I say, that whilst the movent A C passes through the latitude which is between E F and G H, the body is less moved forwards from A B towards C D, than it would have been, if the motion from A C to B D had been uniform.

For suppose that whilst the body is made to descend to the parallel E F by the power of the movent from A C towards B D, the same body in the same time is moved forwards to any point F in the line E F, by the power of the movent from A B towards C D; and let the strait line A F be drawn and produced indeterminately, cutting G H in H. Seeing therefore, it is as A E to A G, so E F to G H; if A C should descend towards B D with uniform motion, the body in the time G H, (for I make A C and its parallels the measure of time,) would be found in the point H. But because A C is supposed to be moved towards B D with motion continually accelerated, that is, in greater proportion of space to space, than of time to time, in the time G H the body will be in some parallel beyond it, as between G H and B D. Suppose now that in the end of the time G H it be in the parallel I K, and in I K let I L be taken equal to G H. When therefore the body is in the parallel I K, it will be in the point L. Wherefore when it was in the parallel G H, it was in some point between G and H, as in the point M; but if both the motions had been uniform, it had been in the point H; and therefore whilst the movent A C passes over the latitude which is between E F and G H, the body is less moved forwards from A B towards C D, than 237it would have been, if both the motions had been uniform; which was to be demonstrated.

13. Any length being given, which is passed through in a given time with uniform motion, to find out what length shall be passed through in the same time with motion uniformly accelerated, that is, with such motion that the proportion of the lengths passed through be continually duplicate to that of their times, and that the line of the impetus last acquired be equal to the line of the whole time of the motion.

Let A B (in the 8th figure) be a length, transmitted with uniform motion in the time A C; and let it be required to find another length, which shall be transmitted in the same time with motion uniformly accelerated, so that the line of the impetus last acquired be equal to the strait line A C.

Let the parallelogram A B D C be completed; and let B D be divided in the middle at E; and between B E and B D let B F be a mean proportional; and let A F be drawn and produced till it meet with C D produced in G; and lastly, let the parallelogram A C G H be completed. I say, A H is the length required.

For as duplicate proportion is to single proportion, so let A H be to A I, that is, let A I be the half of A H; and let I K be drawn parallel to the strait line A C, and cutting the diagonal A D in K, and the strait line A G in L. Seeing therefore A I is the half of A H, I L will also be the half of B D, that is, equal to B E; and I K equal to B F; for B D, that is, G H, B F, and B E, that is, I L, being continual proportionals, A H, A B and A I will also be continual proportionals. But as A B is to 238A I, that is, as A H is to A B, so is B D to I K, and so also is G H, that is, B D to B F; and therefore B F and I K are equal. Now the proportion of A H to A I is duplicate to the proportion of A B to A I, that is, to that of B D to I K, or of G H to I K. Wherefore the point K will be in a parabola, whose diameter is A H, and base G H, which G H is equal to A C. The body therefore proceeding from rest in A, with motion uniformly accelerated in the time A C, when it has passed through the length A H, will acquire the impetus G H equal to the time A C, that is, such impetus, as that with it the body will pass through the length A C in the time A C. Wherefore any length being given, &c., which was propounded to be done.

14. Any length being given, which in a given time is transmitted with uniform motion, to find out what length shall be transmitted in the same time with motion so accelerated, that the lengths transmitted be continually in triplicate proportion to that of their times, and the line of the impetus last of all acquired be equal to the line of time given.

Let the given length A B (in the 9th figure) be transmitted with uniform motion in the time A C; and let it be required to find what length shall be transmitted in the same time with motion so accelerated, that the lengths transmitted be continually in triplicate proportion to that of their times, and the impetus last acquired be equal to the time given.

Let the parallelogram A B D C be completed; and let B D be so divided in E, that B E be a third part of the whole B D; and let B F be a mean proportional 239between B D and B E; and let A F be drawn and produced till it meet the strait line C D in G; and lastly, let the parallelogram A C G H be completed. I say, A H is the length required.

For as triplicate proportion is to single proportion, so let A H be to another line, A I, that is, make A I a third part of the whole A H; and let I K be drawn parallel to the strait line A C, cutting the diagonal A D in K, and the strait line A G in L; then, as A B is to A I, so let A I be to another, A N; and from the point N let N Q be drawn parallel to A C, cutting A G, A D, and F K produced in P, M, and O; and last of all, let F O and L M be drawn, which will be equal and parallel to the strait lines B N and I N. By this construction, the lengths transmitted A H, A B, A I, and A N, will be continual proportionals; and, in like manner, the times G H, B F, I L and N P, that is, N Q, N O, N M and N P, will be continual proportionals, and in the same proportion with A H, A B, A I and A N. Wherefore the proportion of A H, A B, A I and A N. Wherefore the proportion of A H to A N is the same with that of B D, that is, of N Q to N P; and the proportion of N Q to N P triplicate to that of N Q to N O, that is, triplicate to that of B D to I K; wherefore also the length A H is to the length A N in triplicate proportion to that of the time B D, to the time I K; and therefore the crooked line of the first three-sided figure of two means whose diameter is A H, and base G H equal to A C, shall pass through the point O; and consequently, A H shall be transmitted in the time A C, and shall have its last acquired impetus G H equal to A C, and the proportions of the lengths acquired in any of the times triplicate to the proportions 240of the times themselves. Wherefore A H is the length required to be found out.

By the same method, if a length be given which is transmitted with uniform motion in any given time, another length may be found out which shall be transmitted in the same time with motion so accelerated, that the lengths transmitted shall be to the times in which they are transmitted, in proportion quadruplicate, quintuplicate, and so on infinitely. For if B D be divided in E, so that B D be to B E as 4 to 1; and there be taken between B D and B E a mean proportional F B; and as A H is to A B, so A B be made to a third, and again so that third to a fourth, and that fourth to a fifth, A N, so that the proportion of A H to A N be quadruplicate to that of A H to A B, and the parallelogram N B F O be completed, the crooked line of the first three-sided figure of three means will pass through the point O; and consequently, the body moved will acquire the impetus G H equal to A C in the time A C. And so of the rest.

15. Also, if the proportion of the lengths transmitted be to that of their times, as any number to any number, the same method serves for the finding out of the length transmitted with such impetus, and in such time.

For let A C (in the 10th figure) be the time in which a body is transmitted with uniform motion from A to B; and the parallelogram A B D C being completed, let it be required to find out a length in which that body may be moved in the same time A C from A, with motion so accelerated, that the proportion of the lengths transmitted to that of the times be continually as 3 to 2.

Let B D be so divided in E, that B D be to B E 241as 3 to 2; and between B D and B E let B F be a mean proportional; and let A F be drawn and produced till it meet with C D produced in G; and making A M a mean proportional between A H and A B, let it be as A M to A B, so A B to A I; and so the proportion of A H to A I will be to that of A H to A B as 3 to 2; for of the proportions, of which that of A H to A M is one, that of A H to A B is two, and that of A H to A I is three; and consequently, as 3 to 2 to that of G H to B F, and (F K being drawn parallel to B I and cutting A D in K) so likewise to that of G H or B D to I K. Wherefore the proportion of the length A H to A I is to the proportion of the time B D to I K as 3 to 2; and therefore if in the time A C the body be moved with accelerated motion, as was propounded, till it acquire the impetus H G equal to A C, the length transmitted in the same time will be A H.

16. But if the proportion of the lengths to that of the times had been as 4 to 3, there should then have been taken two mean proportionals between A H and A B, and their proportion should have been continued one term further, so that A H to A B might have three of the same proportions, of which A H to A I has four; and all things else should have been done as is already shown. Now the way how to interpose any number of means between two lines given, is not yet found out. Nevertheless this may stand for a general rule; if there be a time given, and a length be transmitted in that time with uniform motion; as for example, if the time be A C, and the length A B, the strait line A G, which determines the length C G or A H, 242transmitted in the same time A C with any accelerated motion, shall so cut B D in F, that B F shall be a mean proportional between B D and B E, B E being so taken in B D, that the proportion of length to length be everywhere to the proportion of time to time, as the whole B D is to its part B E.

17. If in a given time two lengths be transmitted, one with uniform motion, the other with motion accelerated in any proportion of the lengths to the times; and again, in part of the same time, parts of the same lengths be transmitted with the same motions, the whole length will exceed the other length in the same proportion in which one part exceeds the other part.

For example, let A B (in the 8th figure) be a length transmitted in the time A C, with uniform motion; and let A H be another length transmitted in the same time with motion uniformly accelerated, so that the impetus last acquired be G H equal to A C; and in A H let any part A I be taken, and transmitted in part of the time A C with uniform motion; and let another part A B be taken and transmitted in the same part of the time A C with motion uniformly accelerated; I say, that as A H is to A B, so will A B be to A I.

Let B D be drawn parallel and equal to H G, and divided in the midst at E, and between B D and B E let a mean proportional be taken as B F; and the strait line A G, by the demonstration of art. 13, shall pass through F. And dividing A H in the midst at I, A B shall be a mean proportional between A H and A I. Again, because A I and A B are described by the same motions, if I K be 243drawn parallel and equal to B F or A M, and divided in the midst at N, and between I K and I N be taken the mean proportional I L, the strait line A F will, by the demonstration of the same art. 13, pass through L. And dividing A B in the midst at O, the line A I will be a mean proportional between A B and A O. Where A B is divided in I and O, in like manner as A H is divided in B and I; and as A H to A B, so is A B to A I. Which was to be proved.

Coroll. Also as A H to A B, so is H B to B I; and so also B I to I O.

And as this, where one of the motions is uniformly accelerated, is proved out of the demonstration of art. 13; so, when the accelerations are in double proportion to the times, the same may be proved by the demonstration of art. 14; and by the same method in all other accelerations, whose proportions to the times are explicable in numbers.

18. If two sides, which contain an angle in any parallelogram, be moved in the same time to the sides opposite to them, one of them with uniform motion, the other with motion uniformly accelerated; that side, which is moved uniformly, will affect as much with its concourse through the whole length transmitted, as it would do if the other motion were also uniform, and the length transmitted by it in the same time were a mean proportional between the whole and the half.

Let the side A B of the parallelogram A B D C, (in the 11th figure) be understood to be moved with uniform motion till it be coincident with C D; and let the time of that motion be A C or B D. Also in the same time let the side A C be understood to 244be moved with motion uniformly accelerated, till it be coincident with B D; then dividing A B in the middle in E, let A F be made a mean proportional between A B and A E; and drawing F G parallel to A C, let the side A C be understood to be moved in the same time A C with uniform motion till it be coincident with F G. I say, the whole A B confers as much to the velocity of the body placed in A, when the motion of A C is uniformly accelerated till it comes to B D, as the part A F confers to the same, when the side A C is moved uniformly and in the same time to F G.

For seeing A F is a mean proportional between the whole A B and its half A E, B D will (by the 13th article) be the last impetus acquired by A C, with motion uniformly accelerated till it come to the same B D; and consequently, the strait line F B will be the excess, by which the length, transmitted by A C with motion uniformly accelerated, will exceed the length transmitted by the same A C in the same time with uniform motion, and with impetus every where equal to B D. Wherefore, if the whole A B be moved uniformly to C D in the same time in which A C is moved uniformly to F G, the part F B, seeing it concurs not at all with the motion of the side A C which is supposed to be moved only to F G, will confer nothing to its motion. Again, supposing the side A C to be moved to B D with motion uniformly accelerated, the side A B with its uniform motion to C D will less put forwards the body when it is accelerated in all the parallels, than when it is not at all accelerated; and by how much the greater the acceleration is, by so much the less it will put it forwards, 245as is shown in the 12th article. When therefore A C is in F G with accelerated motion, the body will not be in the side C D at the point G, but at the point D; so that G D will be the excess, by which the length transmitted with accelerated motion to B D exceeds the length transmitted with uniform motion to F G; so that the body by its acceleration avoids the action of the part A F, and comes to the side C D in the time A C, and makes the length C D, which is equal to the length A B. Wherefore uniform motion from A B to C D in the time A C, works no more in the whole length A B upon the body uniformly accelerated from A C to B D, than if A C were moved in the same time with uniform motion to F G; the difference consisting only in this, that when A B works upon the body uniformly moved from A C to F G, that, by which the accelerated motion exceeds the uniform motion, is altogether in F B or G D; but when the same A B works upon the body accelerated, that, by which the accelerated motion exceeds the uniform motion, is dispersed through the whole length A B or C D, yet, so that if it were collected and put together, it would be equal to the same F B or G D. Wherefore, if two sides which contain an angle, &c.; which was to be demonstrated.

19. If two transmitted lengths have to their times any other proportion explicable by number, and the side A B be so divided in E, that A B be to A E in the same proportion which the lengths transmitted have to the times in which they are transmitted, and between A B and A E there be taken a mean proportional A F; it may be shown by the same method, that the side, which is moved 246with uniform motion, works as much with its concourse through the whole length A B, as it would do if the other motion were also uniform, and the length transmitted in the same time A C were that mean proportional A F.

And thus much concerning motion by concourse.

Vol. 1. Lat. & Eng.
C. XVI.
Fig. 1-11

1. Definitions of a deficient figure; of a complete figure; of the complement of a deficient figure; and of proportions which are proportional and commensurable to one another.—2. The proportion of a deficient figure to its complement.—3. The proportions of deficient figures to the parallelograms in which they are described, set forth in a table.—4. The description and production of the same figures.—5. The drawing of tangents to them.—6. In what proportion the same figures exceed a strait-lined triangle of the same altitude and base.—7. A table of solid deficient figures described in a cylinder.—8. In what proportion the same figures exceed a cone of the same altitude and base.—9. How a plain deficient figure may be described in a parallelogram, so that it be to a triangle of the same base and altitude, as another deficient figure, plain or solid, twice taken, is to the same deficient figure, together with the complete figure in which it is described.—10. The transferring of certain properties of deficient figures described in a parallelogram to the proportions of the spaces transmitted with several degrees of velocity.—11. Of deficient figures described in a circle.—12. The proposition demonstrated in art. 2 confirmed from the elements of philosophy.—13. An unusual way of reasoning concerning the equality between the superficies of a portion of a sphere and a circle.—14. How from the description of deficient figures in a parallelogram, any number of mean proportionals may be found out between two given strait lines.

1. I call those deficient figures which may be understood to be generated by the uniform motion 247of some quantity, which decreases continually, till at last it have no magnitude at all.

And I call that a complete figure, answering to a deficient figure, which is generated with the same motion and in the same time, by a quantity which retains always its whole magnitude.

The complement of a deficient figure is that which being added to the deficient figure makes it complete.

Four proportions are said to be proportional, when the first of them is to the second as the third is to the fourth. For example, if the first proportion be duplicate to the second, and again, the third be duplicate to the fourth, those proportions are said to be proportional.

And commensurable proportions are those, which are to one another as number to number. As when to a proportion given, one proportion is duplicate, another triplicate, the duplicate proportion will be to the triplicate proportion as 2 to 3; but to the given proportion it will be as 2 to 1; and therefore I call those three proportions commensurable.

2. A deficient figure, which is made by a quantity continually decreasing to nothing by proportions everywhere proportional and commensurable, is to its complement, as the proportion of the whole altitude to an altitude diminished in any time is to the proportion of the whole quantity, which describes the figure, to the same quantity diminished in the same time.

Let the quantity A B (in fig. 1), by its motion through the altitude A C, describe the complete figure A D; and again, let the same quantity, by 248decreasing continually to nothing in C, describe the deficient figure A B E F C, whose complement will be the figure B D C F E. Now let A B be supposed to be moved till it lie in G K, so that the altitude diminished be G C, and A B diminished be G E; and let the proportion of the whole altitude A C to the diminished altitude G C, be, for example, triplicate to the proportion of the whole quantity A B or G K to the diminished quantity G E. And in like manner, let H I be taken equal to G E, and let it be diminished to H F; and let the proportion of G C to H C be triplicate to that of H I to H F; and let the same be done in as many parts of the strait line A C as is possible; and a line be drawn through the points B, E, F and C. I say the deficient figure A B E F C is to its complement B D C F E as 3 to 1, or as the proportion of A C to G C is to the proportion of A B, that is, of G K to G E.

For (by art. 2, chapter XV.) the proportion of the complement B E F C D to the deficient figure A B E F C is all the proportions of D B to B A, O E to E G, Q F to F H, and of all the lines parallel to D B terminated in the line B E F C, to all the parallels to A B terminated in the same points of the line B E F C. And seeing the proportions of D B to O E, and of D B to Q F &c. are everywhere triplicate of the proportions of A B to G E, and of A B to H F &c. the proportions of H F to A B, and of G E to A B &c. (by art. 16, chap. XIII.), are triplicate of the proportions of Q F to D B, and of O E to D B &c. and therefore the deficient figure A B E F C, which is the aggregate 249of all the lines H F, G E, A B, &c. is triple to the complement B E F C D made of all the lines Q F, O E, D B, &c.; which was to be proved.

It follows from hence, that the same complement B E F C D is ¹⁄₄ of the whole parallelogram. And by the same method may be calculated in all other deficient figures, generated as above declared, the proportion of the parallelogram to either of its parts; as that when the parallels increase from a point in the same proportion, the parallelogram will be divided into two equal triangles; when one increase is double to the other, it will be divided into a semiparabola and its complement, or into 2 and 1.

The same construction standing, the same conclusion may otherwise be demonstrated thus.

Let the strait line C B be drawn cutting G K in L, and through L let M N be drawn parallel to the strait line A C; wherefore the parallelograms G M and L D will be equal. Then let L K be divided into three equal parts, so that it may be to one of those parts in the same proportion which the proportion of A C to G C, or of G K to G L, hath to the proportion of G K to G E. Therefore L K will be to one of those three parts as the arithmetical proportion between G K and G L is to the arithmetical proportion between G K and the same G K wanting the third part of L K; and K E will be somewhat greater than a third of L K. Seeing now the altitude A G or M L is, by reason of the continual decrease, to be supposed less than any quantity that can be given; L K, which is intercepted between the diagonal B C and the side B D, 250will be also less than any quantity that can be given; and consequently, if G be put so near to A in g, as that the difference between C g and C A be less than any quantity that can be assigned, the difference also between C l (removing L to l) and C B, will be less than any quantity that can be assigned; and the line g l being drawn and produced to the line B D in k, cutting the crooked line in e, the proportion of G k to G l will still be triplicate to the proportion of G k to G e, and the difference between k and e, the third part of k l, will be less than any quantity that can be given; and therefore the parallelogram e D will differ from a third part of the parallelogram A e by a less difference than any quantity that can be assigned. Again, let H I be drawn parallel and equal to G E, cutting C B in P, the crooked line in F, and O E in I, and the proportion of C g to C H will be triplicate to the proportion of H F to H P, and I F will be greater than the third part of P I. But again, setting H in h so near to g, as that the difference between C h and C g may be but as a point, the point P will also in p be so near to l, as that the difference between C p and C l will be but as a point; and drawing h p till it meet with B D in i, cutting the crooked line in f and having drawn e o parallel to B D, cutting D C in o, the parallelogram f o will differ less from the third part of the parallelogram g f, than by any quantity that can be given. And so it will be in all other spaces generated in the same manner. Wherefore the differences of the arithmetical and geometrical means, which are but as so many points B, e, f, &c. 251(seeing the whole figure is made up of so many indivisible spaces) will constitute a certain line, such as is the line B E F C, which will divide the complete figure A D into two parts, whereof one, namely, A B E F C, which I call a deficient figure, is triple to the other, namely, B D C F E, which I call the complement thereof. And whereas the proportion of the altitudes to one another is in this case everywhere triplicate to that of the decreasing quantities to one another; in the same manner, if the proportion of the altitudes had been everywhere quadruplicate to that of the decreasing quantities, it might have been demonstrated that the deficient figure had been quadruple to its complement; and so in any other proportion. Wherefore, a deficient figure, which is made, &c. which was to be demonstrated.

The same rule holdeth also in the diminution of the bases of cylinders, as is demonstrated in the second article of chapter XV.

3. By this proposition, the magnitudes of all deficient figures, when the proportions by which their bases decrease continually are proportional to those by which their altitudes decrease, may be compared with the magnitudes of their complements; and consequently, with the magnitudes of their complete figures. And they will be found to be, as I have set them down in the following tables; in which I compare a parallelogram with three-sided figures; and first, with a strait-lined triangle, made by the base of the parallelogram continually decreasing in such manner, that the altitudes be always in proportion to one another 252as the bases are, and so the triangle will be equal to its complement; or the proportions of the altitudes and bases will be as 1 to 1, and then the triangle will be half the parallelogram. Secondly, with that three-sided figure which is made by the continual decreasing of the bases in subduplicate proportion to that of the altitudes; and so the deficient figure will be double to its complement, and to the parallelogram as 2 to 3. Then, with that where the proportion of the altitudes is triplicate to that of the bases; and then the deficient figure will be triple to its complement, and to the parallelogram as 3 to 4. Also the proportion of the altitudes to that of the bases may be as 3 to 2; and then the deficient figure will be to its complement as 3 to 2, and to the parallelogram as 3 to 5; and so forwards, according as more mean proportionals are taken, or as the proportions are more multiplied, as may be seen in the following table. For example, if the bases decrease so, that the proportion of the altitudes to that of the bases be always as 5 to 2, and it be demanded what proportion the figure made has to the parallelogram, which is supposed to be unity; then, seeing that where the proportion is taken five times, there must be four means; look in the table amongst the three-sided figures of four means, and seeing the proportion was as 5 to 2, look in the uppermost row for the number 2, and descending in the second column till you meet with that three-sided figure, you will find ⁵⁄₇; which shows that the deficient figure is to the parallelogram as ⁵⁄₇ to 1, or as 5 to 7. 253

4. Now for the better understanding of the nature of these three-sided figures, I will show how they may be described by points; and first, those which are in the first column of the table. Any parallelogram being described, as A B C D (in figure 2) let the diagonal B D be drawn; and the strait-lined triangle B C D will be half the parallelogram; then let any number of lines, as E F, be drawn parallel to the side B C, and cutting the diagonal B D in G; and let it be everywhere, as E F to E G, so E G to another, E H; and through all the points H let the line B H H D be drawn; and the figure B H H D C will be that which I call a three-sided figure of one mean, because in three proportionals, as E F, E G and E H, there is but one mean, namely, E G; and this three-sided figure will be ²⁄₃ of the parallelogram, and is called a parabola. Again, let it be as E G to E H, so E H to another, E I, and let the line B I I D be drawn, making the three-sided figure B I I D C; and this will be ³⁄₄ of the parallelogram, and is by many called a cubic parabola. In like manner, if the 254proportions be further continued in E F, there will be made the rest of the three-sided figures of the first column; which I thus demonstrate. Let there be drawn strait lines, as H K and G L, parallel to the base D C. Seeing therefore the proportion of E F to E H is duplicate to that of E F to E G, or of B C to B L, that is, of C D to L G, or of K M (producing K H to A D in M) to K H, the proportion of B C to B K will be duplicate to that of K M to K H; but as B C is to B K, so is D C or K M to K N, and therefore the proportion of K M to K N is duplicate to that of K M to K H; and so it will be wheresoever the parallel K M be placed. Wherefore the figure B H H D C is double to its complement B H H D A, and consequently ²⁄₃ of the whole parallelogram. In the same manner, if through I be drawn O P I Q parallel and equal to C D, it may be demonstrated that the proportion of O Q to O P, that is, of B C to B O, is triplicate that of O Q to O I, and therefore that the figure B I I D C is triple to its complement B I I D A, and consequently ¾ of the whole parallelogram, &c.

Secondly, such three-sided figures as are in any of the transverse rows, may be thus described. Let A B C D (in fig. 3) be a parallelogram, whose diagonal is B D. I would describe in it such figures, as in the preceding table I call three-sided figures of three means. Parallel to D C, I draw E F as often as is necessary, cutting B D in G; and between E F and E G, I take three proportionals E H, E I and E K. If now there be drawn lines through all the points H, I and K, that through all the points H will make the figure B H D C, which is the first of those three-sided figures; and that 255through all the points I, will make the figure B I D C, which is the second; and that which is drawn through all the points K, will make the figure B K D C the third of those three-sided figures. The first of these, seeing the proportion of E F to E G is quadruplicate of that E F to E H, will be to its complement as 4 to 1, and to the parallelogram as 4 to 5. The second, seeing the proportion of E F to E G is to that of E F to E I as 4 to 2, will be double to its complement, and ⁴⁄₆ or ²⁄₃ of the parallelogram. The third, seeing the proportionproportion of E F to E G is that of E F to E K as 4 to 3, will be to its complement as 4 to 3, and to the parallelogram as 4 to 7.

Any of these figures being described may be produced at pleasure, thus; let A B C D (in fig. 4) be a parallelogram, and in it let the figure B K D C be described, namely, the third three-sided figure of three means. Let B D be produced indefinitely to E, and let E F be made parallel to the base D C, cutting A D produced in G, and B C produced in F; and in G E let the point H be so taken, that the proportion of F E to F G may be quadruplicate to that of F E to F H, which may be done by making F H the greatest of three proportionals between F E and F G; the crooked line B K D produced, will pass through the point H. For if the strait line B H be drawn, cutting C D in I, and H L be drawn parallel to G D, and meeting C D produced in L; it will be as F E to F G, so C L to C I, that is, in quadruplicate proportion to that of F E to F H, or of C D to C I. Wherefore if the line B K D be produced according to its generation, it will fall upon the point H.

2565. A strait line may be drawn so as to touch the crooked line of the said figure in any point, in this manner. Let it be required to draw a tangent to the line B K D H (in fig. 4) in the point D. Let the points B and D be connected, and drawing D A equal and parallel to B C, let B and A be connected; and because this figure is by construction the third of three means, let there be taken in A B three points, so, that by them the same A B be divided into four equal parts; of which take three, namely, A M, so that A B may be to A M, as the figure B K D C is to its complement. I say, the strait line M D will touch the figure in the point given D. For let there be drawn anywhere between A B and D C a parallel, as R Q, cutting the strait line B D, the crooked line B K D, the strait line M D, and the strait line A D, in the points P, K, O and Q. R K will therefore, by construction, be the least of three means in geometrical proportion between R Q and R P. Wherefore (by coroll. of art. 28, chapter XIII.) R K will be less than R O; and therefore M D will fall without the figure. Now if M D be produced to N, F N will be the greatest of three means in arithmetical proportion between F E and F G; and F H will be the greatest of three means in geometrical proportion between the same F E and F G. Wherefore (by the same coroll. of art. 28, chapter XIII.) F H will be less than F N; and therefore D N will fall without the figure, and the strait line M N will touch the same figure only in the point D.

6. The proportion of a deficient figure to its complement being known, it may also be known 257what proportion a strait-lined triangle has to the excess of the deficient figure above the same triangle; and these proportions I have set down in the following table; where if you seek, for example, how much the fourth three-sided figure of five means exceeds a triangle of the same altitude and base, you will find in the concourse of the fourth column with the three-sided figures of five means ²⁄₁₀; by which is signified, that that three-sided figure exceeds the triangle by two-tenths or by one-fifth part of the same triangle.

			1	2	3	4	5	6	7
		The triangle	1	:	:	:	:	:	:
		A three-sided fig. of 1 mean	1⁄3	:	:	:	:	:	:
The ex- cess of		A three-sided fig. of 2 means	2⁄4	1⁄5	:	:	:	:	:
		A three-sided fig. of 3 means	3⁄5	2⁄6	1⁄7	:	:	:	:
		A three-sided fig. of 4 means	4⁄6	3⁄7	2⁄8	1⁄9	:	:	:
		A three-sided fig. of 5 means	5⁄7	4⁄8	3⁄9	2⁄10	1⁄11	:	:
		A three-sided fig. of 6 means	6⁄8	5⁄9	4⁄10	3⁄11	2⁄12	1⁄13	:
		A three-sided fig. of 7 means	7⁄9	6⁄10	5⁄11	4⁄12	3⁄13	2⁄14	1⁄15

7. In the next table are set down the proportion of a cone and the solids of the said three-sided figures, namely, the proportions between them and a cylinder. As for example, in the concourse of the second column with the three-sided figures of four means, you have ⁵⁄₉; which gives you to understand, that the solid of the second three-sided figure of four means is to the cylinder as ⁵⁄₉ to 1, or as 5 to 9. 258

			1	2	3	4	5	6	7
			1	2	3	4	5	6	7
		A cylinder	1	:	:	:	:	:	:
		A cone	1⁄3	:	:	:	:	:	:
		A three-sided fig. of 1 mean	2⁄4	:	:	:	:	:	:
		A three-sided fig. of 2 means	3⁄5	3⁄7	:	:	:	:	:
The sol- ids of		A three-sided fig. of 3 means	4⁄6	4⁄8	4⁄10	:	:	:	:
		A three-sided fig. of 4 means	5⁄7	5⁄9	5⁄11	5⁄13	:	:	:
		A three-sided fig. of 5 means	6⁄8	6⁄10	6⁄12	6⁄14	6⁄16	:	:
		A three-sided fig. of 6 means	7⁄9	7⁄11	7⁄13	7⁄15	7⁄17	7⁄19	:
		A three-sided fig. of 7 means	8⁄10	8⁄12	8⁄14	8⁄16	8⁄18	8⁄20	8⁄22

8. Lastly, the excess of the solids of the said three-sided figures above a cone of the same altitude and base, are set down in the table which follows:

				1	2	3	4	5	6	7
		The Cone		1	:	:	:	:	:	:
The exces-ses of the sol-ids of these three- sided fig- ures above a cone.		Of the solid of a three-sided figure of 1 mean		6⁄12	:	:	:	:	:	:
		Ditto ditto, 2 means		12⁄15	6⁄21	:	:	:	:	:
		Ditto ditto, 3 means		18⁄18	12⁄24	6⁄30	:	:	:	:
		Ditto ditto, 4 means		24⁄21	18⁄27	12⁄33	6⁄39	:	:	:
		Ditto ditto, 5 means		30⁄24	24⁄30	18⁄36	12⁄42	6⁄48	:	:
		Ditto ditto, 6 means		36⁄27	30⁄33	24⁄39	18⁄45	12⁄51	6⁄57	:
		Ditto ditto, 7 means		42⁄30	36⁄36	30⁄42	24⁄48	18⁄54	12⁄60	6⁄66

9. If any of these deficient figures, of which I have now spoken, as A B C D (in the 5th figure) be inscribed within the complete figure B E, having A D C E for its complement; and there be made upon C B produced the triangle A B I; and the parallelogram A B I K be completed; and there be drawn parallel to the strait line C I, any number of lines, as M F, cutting every one of them the 259crooked line of the deficient figure in D, and the strait lines A C, A B and A I in H, G, and L; and as G F is to G D, so G L be made to another, G N; and through all the points N there be drawn the line A N I: there will be a deficient figure A N I B, whose complement will be A N I K. I say, the figure A N I B is to the triangle A B I, as the deficient figure A B C D twice taken is to the same deficient figure together with the complete figure B E.

For as the proportion of A B to A G, that is, of G M to G L, is to the proportion of G M to G N, so is the magnitude of the figure A N I B to that of its complement A N I K, by the second article of this chapter.

But, by the same article, as the proportion of A B to A G, that is, of G M to G L, is to the proportion of G F to G D, that is, by construction, of G L to G N, so is the figure A B C D to its complement A D C E.

And by composition, as the proportion of G M to G L, together with that of G L to G N, is to the proportion of G M to G L, so is the complete figure B E to the deficient figure A B C D.

And by conversion, as the proportion of G M to G L is to both the proportions of G M to G L and of G L to G N, that is, to the proportion of G M to G N, which is the proportion compounded of both, so is the deficient figure A B C D to the complete figure B E.

But it was, as the proportion of G M to G L to that of G M to G N, so the figure A N I B to its complement A N I K. And therefore, A B C D. B E :: A N I B. A N I K are proportionals. And by composition, 260A B C D + B E. A B C D :: B K. A N I B are proportionals.

And by doubling the consequents, A B C D + B E. 2 A B C D :: B K. 2 A N I B are proportionals.

And by taking the halves of the third and the fourth, A B C D + B E. 2 A B C D :: A B I. A N I B are also proportionals; which was to be proved.

10. From what has been said of deficient figures described in a parallelogram, may be found out what proportions spaces, transmitted with accelerated motion in determined times, have to the times themselves, according as the moved body is accelerated in the several times with one or more degrees of velocity.

For let the parallelogram A B C D, in the 6th figure, and in it the three-sided figure D E B C be described; and let F G be drawn anywhere parallel to the base, cutting the diagonal B D in H, and the crooked line B E D in E; and let the proportion of B C to B F be, for example, triplicate to that of F G to F E; whereupon the figure D E B C will be triple to its complement B E D A; and in like manner, I E being drawn parallel to B C, the three-sided figure E K B F will be triple to its complement B K E I. Wherefore the parts of the deficient figure cut off from the vertex by strait lines parallel to the base, namely, D E B C and E K B F, will be to one another as the parallelograms A C and I F; that is, in proportion compounded of the proportions of the altitudes and bases. Seeing therefore the proportion of the altitude B C to the altitude B F is triplicate to the proportion of the base D C to the base F E, the figure D E B C to the 261figure E K B F will be quadruplicate to the proportion of the same D C to F E. And by the same method, may be found out what proportion any of the said three-sided figures has to any part of the same, cut off from the vertex by a strait line parallel to the base.

Now as the said figures are understood to be described by the continual decreasing of the base, as of C D, for example, till it end in a point, as in B; so also they may be understood to be described by the continual increasing of a point, as of B, till it acquire any magnitude, as that of C D.

Suppose now the figure B E D C to be described by the increasing of the point B to the magnitude C D. Seeing therefore the proportion of B C to B F is triplicate to that of C D to F E, the proportion of F E to C D will, by conversion, as I shall presently demonstrate, be triplicate to that B F to B C. Wherefore if the strait line B C be taken for the measure of the time in which the point B is moved, the figure E K B F will represent the sum of all the increasing velocities in the time B F; and the figure D E B C will in like manner represent the sum of all the increasing velocities in the time B C. Seeing therefore the proportion of the figure E K B F to the figure D E B C is compounded of the proportions of altitude to altitude, and base to base; and seeing the proportion of F E to C D is triplicate to that of B F to B C; the proportion of the figure E K B F to the figure D E B C will be quadruplicate to that of B F to B C; that is, the proportion of the sum of the velocities in the time B F, to the sum of the velocities in the time B C, will be quadruplicate to the proportion of B F to 262B C. Wherefore if a body be moved from B with velocity so increasing, that the velocity acquired in the time B F be to the velocity acquired in the time B C in triplicate proportion to that of the times themselves B F to B C, and the body be carried to F in the time B F; the same body in the time B C will be carried through a line equal to the fifth proportional from B F in the continual proportion of B F to B C. And by the same manner of working, we may determine what spaces are transmitted by velocities increasing according to any other proportions.

It remains that I demonstrate the proportion of F E to C D to be triplicate to that of B F to B C. Seeing therefore the proportion of C D, that is, of F G to F E is subtriplicate to that of B C to B F; the proportion of F G to F E will also be subtriplicate to that of F G to F H. Wherefore the proportion of F G to F H is triplicate to that of F G, that is, of C D to F E. But in four continual proportionals, of which the least is the first, the proportion of the first to the fourth, (by the 16th article of chapter XIII.), is subtriplicate to the proportion of the third to the same fourth. Wherefore the proportion of F H to G F is subtriplicate to that of F E to C D; and therefore the proportion of F E to C D is triplicate to that of F H to F G, that is, of B F to B C; which was to be proved.

It may from hence be collected, that when the velocity of a body is increased in the same proportion with that of the times, the degrees of velocity above one another proceed as numbers do in immediate succession from unity, namely, as 1, 2, 3, 4, &c. And when the velocity is increased in proportion 263duplicate to that of the times, the degrees proceed as numbers from unity, skipping one, as 1, 3, 5, 7, &c. Lastly, when the proportions of the velocities are triplicate to those of the times, the progression of the degrees is as that of numbers from unity, skipping two in every place, as 1, 4, 7, 10, &c., and so of other proportions. For geometrical proportionals, when they are taken in every point, are the same with arithmetical proportionals.

11. Moreover, it is to be noted that as in quantities, which are made by any magnitudes decreasing, the proportions of the figures to one another are as the proportions of the altitudes to those of the bases; so also it is in those, which are made with motion decreasing, which motion is nothing else but that power by which the described figures are greater or less. And therefore in the description of Archimedes' spiral, which is done by the continual diminution of the semidiameter of a circle in the same proportion in which the circumference is diminished, the space, which is contained within the semidiameter and the spiral line, is a third part of the whole circle. For the semidiameters of circles, inasmuch as circles are understood to be made up of the aggregate of them, are so many sectors; and therefore in the description of a spiral, the sector which describes it is diminished in duplicate proportions to the diminutions of the circumference of the circle in which it is inscribed; so that the complement of the spiral, that is, that space in the circle which is without the spiral line, is double to the space within the spiral line. In the same manner, if 264there be taken a mean proportional everywhere between the semidiameter of the circle, which contains the spiral, and that part of the semidiameter which is within the same, there will be made another figure, which will be half the circle. And to conclude, this rule serves for all such spaces as may be described by a line or superficies decreasing either in magnitude of power; so that if the proportions, in which they decrease, be commensurable to the proportions of the times in which they decrease, the magnitudes of the figures they describe will be known.

12. The truth of that proposition, which I demonstrated in art. 2, which is the foundation of all that has been said concerning deficient figures, may be derived from the elements of philosophy, as having its original in this; that all equality and inequality between two effects, that is, all proportion, proceeds from, and is determined by, the equal and unequal causes of those effects, or from the proportion which the causes, concurring to one effect, have to the causes which concur to the producing of the other effect; and that therefore the proportions of quantities are the same with the proportions of their causes. Seeing, therefore, two deficient figures, of which one is the complement of the other, are made, one by motion decreasing in a certain time and proportion, the other by the loss of motion in the same time; the causes, which make and determine the quantities of both the figures, so that they can be no other than they are, differ only in this, that the proportions by which the quantity which generates the figure proceeds in describing of the same, that 265is, the proportions of the remainders of all the times and altitudes, may be other proportions than those by which the same generating quantity decreases in making the complement of that figure, that is, the proportions of the quantity which generates the figure continually diminished. Wherefore, as the proportion of the times in which motion is lost, is to that of the decreasing quantities by which the deficient figure is generated, so will the defect or complement be to the figure itself which is generated.

13. There are also other quantities which are determinable from the knowledge of their causes, namely, from the comparison of the motions by which they are made; and that more easily than from the common elements of geometry. For example, that the superficies of any portion of a sphere is equal to that circle, whose radius is a strait line drawn from the pole of the portion to the circumference of its base, I may demonstrate in this manner. Let B A C (in fig. 7) be a portion of a sphere, whose axis is A E, and whose base is B C; and let A B be the strait line drawn from the pole A to the base in B; and let A D, equal to A B, touch the great circle B A C in the pole A. It is to be proved that the circle made by the radius A D is equal to the superficies of the portion B A C. Let the plain A E B D be understood to make a revolution about the axis A E; and it is manifest that by the strait line A D a circle will be described; and by the arch A B the superficies of a portion of a sphere; and lastly, by the subtense A B the superficies of a right cone. Now seeing both the strait line A B and the arch A B make 266one and the same revolution, and both of them have the same extreme points A and B, the cause why the spherical superficies, which is made by the arch, is greater than the conical superficies, which is made by the subtense, is, that A B the arch is greater than A B the subtense; and the cause why it is greater consists in this, that although they be both drawn from A to B, yet the subtense is drawn strait, but the arch angularly, namely, according to that angle which the arch makes with the subtense, which angle is equal to the angle D A B (for an angle of contingence adds nothing to an angle of a segment, as has been shown in chapter XIV, article 16.) Wherefore the magnitude of the angle D A B is the cause why the superficies of the portion, described by the arch A B, is greater than the superficies of the right cone described by the subtense A B.

Again, the cause why the circle described by the tangent A D is greater than the superficies of the right cone described by the subtense A B (notwithstanding that the tangent and the subtense are equal, and both moved round in the same time) is this, that A D stands at right angles to the axis, but A B obliquely; which obliquity consists in the same angle D A B. Seeing therefore the quantity of the angle D A B is that which makes the excess both of the superficies of the portion, and of the circle made by the radius A D, above the superficies of the right cone described by the subtense A B; it follows, that both the superficies of the portion and that of the circle do equally exceed the superficies of the cone. Wherefore the circle made by A D or A B, and 267the spherical superficies made by the arch A B, are equal to one another; which was to be proved.

14. If these deficient figures, which I have described in a parallelogram, were capable of exact description, then any number of mean proportionals might be found out between two strait lines given. For example, in the parallelogram A B C D, (in figure 8) let the three-sided figure of two means be described (which many call a cubical parabola); and let R and S be two given strait lines; between which, if it be required to find two mean proportionals, it may be done thus. Let it be as R to S, so B C to B F; and let F E be drawn parallel to B A, and cut the crooked line in E; then through E let G H be drawn parallel and equal to the strait line A D, and cut the diagonal B D in I; for thus we have G I the greatest of two means between G H and G E, as appears by the description of the figure in article 4. Wherefore, if it be as G H to G I, so R to another line, T, that T will be the greatest of two means between R and S. And therefore if itit be again as R to T, so T to another line, X, that will be done which was required.

In the same manner, four mean proportionals may be found out, by the description of a three-sided figure of four means; and so any other number of means, &c.

Vol. 1. Lat. & Eng.
C. XVII.
Fig. 1-8

1. To find the strait line equal to the crooked line of a semiparabola.—2. To find a strait line equal to the crooked line of the first semiparabolaster, or to the crooked line of any other of the deficient figures of the table of the 3d article of the precedent chapter.

1. A parabola being given, to find a strait line equal to the crooked line of the semiparabola.

Let the parabolical line given be A B C (in figure 1), and the diameter found be A D, and the base drawn D C; and the parallelogram A D C E being completed, draw the strait line A C. Then dividing A D into two equal parts in F, draw F H equal and parallel to D C, cutting A C in K, and the parabolical line in O; and between F H and F O take a mean proportional F P, and draw A O, A P and P C. I say that the two lines A P and P C, taken together as one line, are equal to the parabolical line A B O C.

For the line A B O C being a parabolical line, is generated by the concourse of two motions, one uniform from A to E, the other in the same time uniformly accelerated from rest in A to D. And because the motion from A to E is uniform, A E may represent the times of both those motions from the beginning to the end. Let therefore A E be the time; and consequently the lines ordinately 269applied in the semiparabola will design the parts of time wherein the body, that describeth the line A B O C, is in every point of the same; so that as at the end of the time A E or D C it is in C, so at the end of the time F O it will be in O. And because the velocity in A D is increased uniformly, that is, in the same proportion with the time, the same lines ordinately applied in the semiparabola will design also the continual augmentation of the impetus, till it be at the greatest, designed by the base D C. Therefore supposing uniform motion in the line A F, in the time F K the body in A by the concourse of the two uniform motions in A F and F K will be moved uniformly in the line A K; and K O will be the increase of the impetus or swiftness gained in the time F K; and the line A O will be uniformly described by the concourse of the two uniform motions in A F and F O in the time F O. From O draw O L parallel to E C, cutting A C in L; and draw L N parallel to D C, cutting E C in N, and the parabolical line in M; and produce it on the other side to A D in I; and I N, I M and I L will be, by the construction of a parabola, in continual proportion, and equal to the three lines F H, F P and F O; and a strait line parallel to E C passing through M will fall on P; and therefore O P will be the increase of impetus gained in the time F O or I L. Lastly, produce P M to C D in Q; and Q C or M N or P H will be the increase of impetus proportional to the time F P or I M or D Q. Suppose now uniform motion from H to C in the time P H. Seeing therefore in the time F P with uniform motion and the impetus increased in proportion to the times, is described 270the straight line A P; and in the rest of the time and impetus, namely, P H, is described the line C P uniformly; it followeth that the whole line A P C is described with the whole impetus, and in the same time wherewith is described the parabolical line A B C; and therefore the line A P C, made of the two strait lines A P and P C, is equal to the parabolical line A B C; which was to be proved.

2. To find a strait line equal to the crooked line of the first semiparabolaster.

Let A B C be the crooked line of the first semiparabolaster; A D the diameter; D C the base; and let the parallelogram completed be A D C E, whose diagonal is A C. Divide the diameter into two equal parts in F, and draw F H equal and parallel to D C, cutting A C in K, the crooked line in O, and E C in H. Then draw O L parallel to E C, cutting A C in L; and draw L N parallel to the base D C, cutting the crooked line in M, and the strait line E C in N; and produce it on the other side to A D in I. Lastly, through the point M draw P M Q parallel and equal to H C, cutting F H in P; and join C P, A P and A O. I say, the two strait lines A P and P C are equal to the crooked line A B O C.

For the line A B O C, being the crooked line of the first semiparabolaster, is generated by the concourse of two motions, one uniform from A to E, the other in the same time accelerated from rest in A to D, so as that the impetus increaseth in proportion perpetually triplicate to that of the increase of the time, or which is all one, the lengths transmitted are in proportion triplicate to 271that of the times of their transmission; for as the impetus or quicknesses increase, so the lengths transmitted increase also. And because the motion from A to E is uniform, the line A E may serve to represent the time, and consequently the lines, ordinately drawn in the semiparabolaster, will design the parts of time wherein the body, beginning from rest in A, describeth by its motion the crooked line A B O C. And because D C, which represents the greatest acquired impetus, is equal to A E, the same ordinate lines will represent the several augmentations of the impetus increasing from rest in A. Therefore, supposing uniform motion from A to F, in the time F K there will be described, by the concourse of the two uniform motions A F and F K, the line A K uniformly, and K O will be the increase of impetus in the time F K; and by the concourse of the two uniform motions in A F and F O will be described the line A O uniformly. Through the point L draw the strait line L M N parallel to D C, cutting the strait line A D in I, the crooked line A B C in M, and the strait line E C in N; and through the point M the strait line P M Q parallel and equal to H C, cutting D C in Q and F H in P. By the concourse therefore of the two uniform motions in A F and F P in the time F P will be uniformly described the strait line A P; and L M or O P will be the increase of impetus to be added for the time F O. And because the proportion of I N to I L is triplicate to the proportion of I N to I M, the proportion of F H to F O will also be triplicate to the proportion of F H to F P; and the proportional impetus gained in the time F P is P H. 272So that F H being equal to D C, which designed the whole impetus acquired by the acceleration, there is no more increase of impetus to be computed. Now in the time P H suppose an uniform motion from H to C; and by the two uniform motions in C H and H P will be described uniformly the strait line P C. Seeing therefore the two strait lines A P and P C are described in the time A E with the same increase of impetus, wherewith the crooked line A B O C is described in the same time A E, that is, seeing the line A P C and the line A B O C are transmitted by the same body in the same time and with equal velocities, the lines themselves are equal; which was to be demonstrated.

By the same method (if any of the semiparabolasters in the table of art. 3 of the precedent chapter be exhibited) may be found a strait line equal to the crooked line thereof, namely, by dividing the diameter into two equal parts, and proceeding as before. Yet no man hitherto hath compared any crooked with any strait line, though many geometricians of every age have endeavoured it. But the cause, why they have not done it, may be this, that there being in Euclid no definition of equality, nor any mark by which to judge of it besides congruity (which is the 8th axiom of the first Book of his Elements) a thing of no use at all in the comparing of strait and crooked; and others after Euclid (except Archimedes and Apollonius, and in our time Bonaventura) thinking the industry of the ancients had reached to all that was to be done in geometry, thought also, that all that could be propounded was either to be 273deduced from what they had written, or else that it was not at all to be done: it was therefore disputed by some of those ancients themselves, whether there might be any equality at all between crooked and strait lines; which question Archimedes, who assumed that some strait line was equal to the circumference of a circle, seems to have despised, as he had reason. And there is a late writer that granteth that between a strait and a crooked line there is equality; but now, says he, since the fall of Adam, without the special assistance of Divine Grace it is not to be found.

Vol. 1. Lat. & Eng.
C. XVIII.
Fig. 1-2

1. If two strait lines falling upon another strait line be parallel, the lines reflected from them shall also be parallel.—2. If two strait lines drawn from one point fall upon another strait line, the lines reflected from them, if they be drawn out the other way, will meet in an angle equal to the angle made by the lines of incidence.—3. If two strait parallel lines, drawn not oppositely, but from the same parts, fall upon the circumference of a circle, the lines reflected from them, if produced they meet within the circle, will make an angle double to that which is made by two strait lines drawn from the centre to the points of incidence.—4. If two strait lines drawn from the same point without a circle fall upon the circumference, and the lines reflected from them being produced meet within the circle, they will make an angle equal to twice that angle, which is made by two strait lines drawn from the centre to the points of incidence, together with the angle which the incident lines themselves make.—5. If two strait lines drawn from one point 274fall upon the concave circumference of a circle, and the angle they make be less than twice the angle at the centre, the lines reflected from them and meeting within the circle will make an angle, which being added to the angle of the incident lines will be equal to twice the angle at the centre.—6. If through any one point two unequal chords be drawn cutting one another, and the centre of the circle be not placed between them, and the lines reflected from them concur wheresoever, there cannot through the point, through which the two former lines were drawn, be drawn any other strait line whose reflected line shall pass through the common point of the two former lines reflected.—7. In equal chords the same is not true.—8. Two points being given in the circumference of a circle, to draw two strait lines to them, so that their reflected lines may contain any angle given.—9. If a strait line falling upon the circumference of a circle be produced till it reach the semidiameter, and that part of it, which is intercepted between the circumference and the semidiameter, be equal to that part of the semidiameter which is between the point of concourse and the centre, the reflected line will be parallel to the semidiameter.—10. If from a point within a circle, two strait lines be drawn to the circumference, and their reflected lines meet in the circumference of the same circle, the angle made by the reflected lines will be a third part of the angle made by the incident lines.

Whether a body falling upon the superficies of another body and being reflected from it, do make equal angles at that superficies, it belongs not to this place to dispute, being a knowledge which depends upon the natural causes of reflection; of which hitherto nothing has been said, but shall be spoken of hereafter.

In this place, therefore, let it be supposed that the angle of incidence is equal to the angle of reflection; that our present search may be applied, not to the finding out of the causes, but some consequences of the same.

I call an angle of incidence, that which is made 275between a strait line and another line, strait or crooked, upon which it falls, and which I call the line reflecting; and an angle of reflection equal to it, that which is made at the same point between the strait line which is reflected and the line reflecting.

1. If two strait lines, which fall upon another strait line, be parallel, their reflected lines shall be also parallel.

Let the two strait lines A B and C D (in fig. 1), which fall upon the strait line E F, at the points B and D, be parallel; and let the lines reflected from them be B G and D H. I say, B G and D H are also parallel.

For the angles A B E and C D E are equal by reason of the parallelism of A B and C D; and the angles G B F and H D F are equal to them by supposition; for the lines B G and D H are reflected from the lines A B and C D. Wherefore B G and D H are parallel.

2. If two strait lines drawn from the same point fall upon another strait line, the lines reflected from them, if they be drawn out the other way, will meet in an angle equal to the angle of the incident lines.

From the point A (in fig. 2) let the two strait lines A B and A D be drawn; and let them fall upon the strait line E K at the points B and D; and let the lines B I and D G be reflected from them. I say, I B and G D do converge, and that if they be produced on the other side of the line E K, they shall meet, as in F; and that the angle B F D shall be equal to the angle B A D.

For the angle of reflection I B K is equal to the 276angle of incidence A B E; and to the angle I B K its vertical angle E B F is equal; and therefore the angle A B E is equal to the angle E B F. Again, the angle A D E is equal to the angle of reflection G D K, that is, to its vertical angle E D F; and therefore the two angles A B D and A D B of the triangle A B D are one by one equal to the two angles F B D and F D B of the triangle F B D; wherefore also the third angle B A D is equal to the third angle B F D; which was to be proved.

Coroll. I. If the strait line A F be drawn, it will be perpendicular to the strait line E K. For both the angles at E will be equal, by reason of the equality of the two angles A B E and F B E, and of the two sides A B and F B.

Coroll. II. If upon any point between B and D there fall a strait line, as A C, whose reflected line is C H, this also produced beyond C, will fall upon F; which is evident by the demonstration above.

3. If from two points taken without a circle, two strait parallel lines, drawn not oppositely, but from the same parts, fall upon the circumference; the lines reflected from them, if produced they meet within the circle, will make an angle double to that which is made by two strait lines drawn from the centre to the points of incidence.

Let the two strait parallels A B and D C (in fig. 3) fall upon the circumference B C at the points B and C; and let the centre of the circle be E; and let A B reflected be B F, and D C reflected be C G; and let the lines F B and G C produced meet within the circle in H; and let E B and E C 277be connected. I say the angle F H G is double to the angle B E C.

For seeing A B and D C are parallels, and E B cuts A B in B, the same E B produced will cut D C somewhere; let it cut it in D; and let D C be produced howsoever to I, and let the intersection of D C and B F be at K. The angle therefore I C H, being external to the triangle C K H, will be equal to the two opposite angles C K H and C H K. Again, I C E being external to the triangle C D E, is equal to the two angles at D and E. Wherefore the angle I C H, being double to the angle I C E, is equal to the angles at D and E twice taken; and therefore the two angles C K H and C H K are equal to the two angles at D and E twice taken. But the angle C K H is equal to the angles D and A B D, that is, D twice taken; for A B and D C being parallels, the altern angles D and A B D are equal. Wherefore C H K, that is the angle F H G is also equal to the angle at E twice taken; which was to be proved.

Coroll. If from two points taken within a circle two strait parallels fall upon the circumference, the lines reflected from them shall meet in an angle, double to that which is made by two strait lines drawn from the centre to the points of incidence. For the parallels A B and I C falling upon the points B and C, are reflected in the lines B H and C H, and make the angle at H double to the angle at E, as was but now demonstrated.

4. If two strait lines drawn from the same point without a circle fall upon the circumference, and the lines reflected from them being produced meet within the circle, they will make an angle equal to 278twice that angle, which is made by two strait lines drawn from the centre to the points of incidence, together with the angle which the incident lines themselves make.

Let the two strait lines A B and A C (in fig. 4) be drawn from the point A to the circumference of the circle, whose centre is D; and let the lines reflected from them be B E and C G, and, being produced, make within the circle the angle H; also let the two strait lines D B and D C be drawn from the centre D to the points of incidence B and C. I say, the angle H is equal to twice the angle at D together with the angle at A.

For let A C be produced howsoever to I. Therefore the angle I C H, which is external to the triangle C K H, will be equal to the two angles C K H and C H K. Again, the angle I C D, which is external to the triangle C L D, will be equal to the two angles C L D and C D L. But the angle I C H is double to the angle I C D, and is therefore equal to the angles C L D and C D L twice taken. Wherefore the angles C K H and C H K are equal to the angles C L D and C D L twice taken. But the angle C L D, being external to the triangle A L B, is equal to the two angles L A B and L B A; and consequently C L D twice taken is equal to L A B and L B A twice taken. Wherefore C K H and C H K are equal to the angle C D L together with L A B and L B A twice taken. Also the angle C K H is equal to the angle L A B once and A B K, that is, L B A twice taken. Wherefore the angle C H K is equal to the remaining angle C D L, that is, to the angle at D, twice taken, and 279the angle L A B, that is, the angle at A, once taken; which was to be proved.

Coroll. If two strait converging lines, as I C and M B, fall upon the concave circumference of a circle, their reflected lines, as C H and B H, will meet in the angle H, equal to twice the angle D, together with the angle at A made by the incident lines produced. Or, if the incident lines be H B and I C, whose reflected lines C H and B M meet in the point N, the angle C N B will be equal to twice the angle D, together with the angle C K H made by the lines of incidence. For the angle C N B is equal to the angle H, that is, to twice the angle D, together with the two angles A, and N B H, that is, K B A. But the angles K B A and A are equal to the angle C K H. Wherefore the angle C N B is equal to twice the angle D, together with the angle C K H made by the lines of incidence I C and H B produced to K.

5. If two strait lines drawn from one point fall upon the concave circumference of a circle, and the angle they make be less than twice the angle at the centre, the lines reflected from them and meeting within the circle will make an angle, which being added to the angle of the incident lines, will be equal to twice the angle at the centre.

Let the two lines A B and A C (in fig. 5), drawn from the point A, fall upon the concave circumference of the circle whose centre is D; and let their reflected lines B E and C E meet in the point E; also let the angle A be less than twice the angle D. I say, the angles A and E together taken are equal to twice the angle D.

For let the strait lines A B and E C cut the 280strait lines D C and D B in the points G and H; and the angle B H C will be equal to the two angles E B H and E; also the same angle B H C will be equal to the two angles D and D C H; and in like manner the angle B G C will be equal to the two angles A C D and A, and the same angle B G C will be also equal to the two angles D B G and D. Wherefore the four angles E B H, E, A C D and A, are equal to the four angles D, D C H, D B G and D. If, therefore, equals be taken away on both sides, namely, on one side A C D and E B H, and on the other side D C H and D B G, (for the angle E B H is equal to the angle D B G, and the angle A C D equal to the angle D C H), the remainders on both sides will be equal, namely, on one side the angles A and E, and on the other the angle D twice taken. Wherefore the angles A and E are equal to twice the angle D.

Coroll. If the angle A be greater than twice the angle D, their reflected lines will diverge. For, by the corollary of the third proposition, if the angle A be equal to twice the angle D, the reflected lines B E and C E will be parallel; and if it be less, they will concur, as has now been demonstrated. And therefore, if it be greater, the reflected lines B E and C E will diverge, and consequently, if they be produced the other way, they will concur and make an angle equal to the excess of the angle A above twice the angle D; as is evident by art. 4.

6. If through any one point two unequal chords be drawn cutting one another, either within the circle, or, if they be produced, without it, and the centre of the circle be not placed between them, and the lines reflected from them concur wheresoever; 281there cannot, through the point through which the former lines were drawn, be drawn another strait line, whose reflected line shall pass through the point where the two former reflected lines concur.

Let any two unequal chords, as B K and C H (in fig. 6), be drawn through the point A in the circle B C; and let their reflected lines B D and C E meet in F; and let the centre not be between A B and A C; and from the point A let any other strait line, as A G, be drawn to the circumference between B and C. I say, G N, which passes through the point F, where the reflected lines B D and C E meet, will not be the reflected line of A G.

For let the arch B L be taken equal to the arch B G, and the strait line B M equal to the strait line B A; and L M being drawn, let it be produced to the circumference in O. Seeing therefore B A and B M are equal, and the arch B L equal to the arch B G, and the angle M B L equal to the angle A B G, A G and M L will also be equal, and, producing G A to the circumference in I, the whole lines L O and G I will in like manner be equal. But L O is greater than G F N, as shall presently be demonstrated; and therefore also G I is greater than G N. Wherefore the angles N G C and I G B are not equal. Wherefore the line G F N is not reflected from the line of incidence A G, and consequently no other strait line, besides A B and A C, which is drawn through the point A, and falls upon the circumference B C, can be reflected to the point F; which was to be demonstrated.

It remains that I prove L O to be greater than G N; which I shall do in this manner. L O and 282G N cut one another in P; and P L is greater than P G. Seeing now L P. P G :: P N. P O are proportionals, therefore the two extremes L P and P O together taken, that is L O, are greater than P G and P N together taken, that is, G N; which remained to be proved.

7. But if two equal chords be drawn through one point within a circle, and the lines reflected from them meet in another point, then another strait line may be drawn between them through the former point, whose reflected line shall pass through the latter point.

Let the two equal chords B C and E D (in the 7th figure) cut one another in the point A within the circle B C D; and let their reflected lines C H and D I meet in the point F. Then dividing the arch C D equally in G, let the two chords G K and G L be drawn through the points A and F. I say, G L will be the line reflected from the chord K G. For the four chords B C, C H, E D and D I are by supposition all equal to one another; and therefore the arch B C H is equal to the arch E D I; as also the angle B C H to the angle E D I; and the angle A M C to its verticle angle F M D; and the strait line D M to the strait line G M; and, in like manner, the strait line A C to the strait line F D; and the chords C G and G D being drawn, will also be equal; and also the angles F D G and A C G, in the equal segments G D I and G C B. Wherefore the strait lines F G and A G are equal; and, therefore, the angle F G D is equal to the angle A G C, that is, the angle of incidence equal to the angle of reflection. Wherefore the line G L is reflected from the incident line C G; which was to be proved.

283Coroll. By the very sight of the figure it is manifest, that if G be not the middle point between C and D, the reflected line G L will not pass through the point F.

8. Two points in the circumference of a circle being given to draw two strait lines to them, so as that their reflected lines may be parallel, or contain any angle given.

In the circumference of the circle, whose centre is A, (in the 8th figure) let the two points B and C be given; and let it be required to draw to them from two points taken without the circle two incident lines, so that their reflected lines may, first, be parallel.

Let A B and A C be drawn; as also any incident line D C, with its reflected line C F; and let the angle E C D be made double to the angle A; and let H B be drawn parallel to E C, and produced till it meet with D C produced in I. Lastly, producing A B indefinitely to K, let G B be drawn so that the angle G B K may be equal to the angle H B K, and then G B will be the reflected line of the incident line H B. I say, D C and H B are two incident lines, whose reflected lines C F and B G are parallel.

For seeing the angle E C D is double to the angle B A C, the angle H I C is also, by reason of the parallels E C and H I, double to the same B A C; therefore also F C and G B, namely, the lines reflected from the incident lines D C and H B, are parallel. Wherefore the first thing required is done.

Secondly, let it be required to draw to the points B and C two strait lines of incidence, so that the 284lines reflected from them may contain the given angle Z.

To the angle E C D made at the point C, let there be added on one side the angle D C L equal to half Z, and on the other side the angle E C M equal to the angle D C L; and let the strait line B N be drawn parallel to the strait line C M; and let the angle K B O be made equal to the angle N B K; which being done, B O will be the line of reflection from the line of incidence N B. Lastly, from the incident line L C, let the reflected line C O be drawn, cutting B O at O, and making the angle C O B. I say, the angle C O B is equal to the angle Z.

Let N B be produced till it meet with the strait line L C produced in P. Seeing, therefore, the angle L C M is, by construction, equal to twice the angle B A C, together with the angle Z; the angle N P L, which is equal to L C M by reason of the parallels N P and M C, will also be equal to twice the same angle B A C, together with the angle Z. And seeing the two strait lines O C and O B fall from the point O upon the points C and B; and their reflected lines L C and N B meet in the point P; the angle N P L will be equal to twice the angle B A C together with the angle C O B. But I have already proved the angle N P L to be equal to twice the angle B A C together with the angle Z. Therefore the angle C O B is equal to the angle Z; wherefore, two points in the circumference of a circle being given, I have drawn, &c.; which was to be done.

But if it be required to draw the incident lines from a point within the circle, so that the lines reflected from them may contain an angle equal to 285the angle Z, the same method is to be used, saving that in this case the angle Z is not to be added to twice the angle B A C, but to be taken from it.

9. If a strait line, falling upon the circumference of a circle, be produced till it reach the semidiameter, and that part of it which is intercepted between the circumference and the semidiameter be equal to that part of the semidiameter which is between the point of concourse and the centre, the reflected line will be parallel to the semidiameter.

Let any line A B (in the 9th figure) be the semidiameter of the circle whose centre is A; and upon the circumference B D let the strait line C D fall, and be produced till it cut A B in E, so that E D and E A may be equal; and from the incident line C D let the line D F be reflected. I say, A B and D F will be parallel.

Let A G be drawn through the point D. Seeing, therefore, E D and E A are equal, the angles E D A and E A D will also be equal. But the angles F D G and E D A are equal; for each of them is half the angle E D H or F D C. Wherefore the angles F D G and E A D are equal; and consequently D F and A B are parallel; which was to be proved.

Coroll. If E A be greater then E D, then D F and A B being produced will concur; but if E A be less than E D, then B A and D H being produced will concur.

10. If from a point within a circle two strait lines be drawn to the circumference, and their reflected lines meet in the circumference of the same circle, the angle made by the lines of reflection will be a third part of the angle made by the lines of incidence.

286From the point B (in the 10th figure) taken within the circle whose centre is A, let the two strait lines B C and B D be drawn to the circumference; and let their reflected lines C E and D E meet in the circumference of the same circle at the point E. I say, the angle C E D will be a third part of the angle C B D.

Let A C and A D be drawn. Seeing, therefore, the angles C E D and C B D together taken are equal to twice the angle C A D (as has been demonstrated in the 5th article); and the angle C A D twice taken is quadruple to the angle C E D; the angles C E D and C B D together taken will also be equal to the angle C E D four times taken; and therefore if the angle C E D be taken away on both sides, there will remain the angle C B D on one side, equal to the angle C E D thrice taken on the other side; which was to be demonstrated.

Coroll. Therefore a point being given within a circle, there may be drawn two lines from it to the circumference, so as their reflected lines may meet in the circumference. For it is but trisecting the angle C B D, which how it may be done shall be shown in the following chapter.

Vol. 1. Lat. & Eng.
C. XIX.
Fig. 1-10

1. The dimension of a circle never determined in numbers by Archimedes and others.—2. The first attempt for the finding out of the dimension of a circle by lines.—3. The second attempt for the finding out of the dimension of a circle from the consideration of the nature of crookedness.—4. The third attempt; and some things propounded to be further searched into.—5. The equation of the spiral of Archimedes with a strait line.—6. Of the analysis of geometricians by the powers of lines.

1. In the comparing of an arch of a circle with a strait line, many and great geometricians, even from the most ancient times, have exercised their wits; and more had done the same, if they had not seen their pains, though undertaken for the common good, if not brought to perfection, vilified by those that envy the praises of other men. Amongst those ancient writers whose works are come to our hands, Archimedes was the first that brought the length of the perimeter of a circle within the limits of numbers very little differing from the truth; demonstrating the same to be less than three diameters and a seventh part, but greater than three diameters and ten seventy-one parts of the diameter. So that supposing the radius to consist of 10,000,000 equal parts, the arch of a quadrant will be between 15,714,285 and 15,704,225 of the same parts. In our times, Ludovicus Van Cullen and Willebrordus Snellius, 288with joint endeavour, have come yet nearer to the truth; and pronounced from true principles, that the arch of a quadrant, putting, as before, 10,000,000 for radius, differs not one whole unity from the number 15,707,963; which, if they had exhibited their arithmetical operations, and no man had discovered any error in that long work of theirs, had been demonstrated by them. This is the furthest progress that has been made by the way of numbers; and they that have proceeded thus far deserve the praise of industry. Nevertheless, if we consider the benefit, which is the scope at which all speculation should aim, the improvement they have made has been little or none. For any ordinary man may much sooner and more accurately find a strait line equal to the perimeter of a circle, and consequently square the circle, by winding a small thread about a given cylinder, than any geometrician shall do the same by dividing the radius into 10,000,000 equal parts. But though the length of the circumference were exactly set out, either by numbers, or mechanically, or only by chance, yet this would contribute no help at all towards the section of angles, unless happily these two problems, to divide a given angle according to any proportion assigned, and to find a strait line equal to the arch of a circle, were reciprocal, and followed one another. Seeing therefore the benefit proceeding from the knowledge of the length of the arch of a quadrant consists in this, that we may thereby divide an angle according to any proportion, either accurately, or at least accurately enough for common use; and seeing this cannot be done by arithmetic, I 289thought fit to attempt the same by geometry, and in this chapter to make trial whether it might not be performed by the drawing of strait and circular lines.

2. Let the square A B C D (in the first figure) be described; and with the radii A B, B C, and D C, the three arches B D, C A, and A C; of which let the two B D and C A cut one another in E, and the two B D and A C in F. The diagonals therefore B D and A C being drawn will cut one another in the centre of the square G, and the two arches B D and C A in two equal parts in H and Y; and the arch B H D will be trisected in F and E. Through the centre G let the two strait lines K G L and M G N be drawn parallel and equal to the sides of the square A B and A D, cutting the four sides of the same square in the points K, L, M, and N; which being done, K L will pass through F, and M N through E. Then let O P be drawn parallel and equal to the side B C, cutting the arch B F D in F, and the sides A B and D C in O and P. Therefore O F will be the sine of the arch B F, which is an arch of 30 degrees; and the same O F will be equal to half the radius. Lastly, dividing the arch B F in the middle in Q, let R Q, the sine of the arch B Q, be drawn and produced to S, so that Q S be equal to R Q, and consequently R S be equal to the chord of the arch B F; and let F S be drawn and produced to T in the side B C. I say, the strait line B T is equal to the arch B F; and consequently that B V, the triple of B T, is equal to the arch of the quadrant B F E D.

Let T F be produced till it meet the side B A produced in X; and dividing O F in the middle 290in Z, let Q Z be drawn and produced till it meet with the side B A produced. Seeing therefore the strait lines R S and O F are parallel, and divided in the midst in Q and Z, Q Z produced will fall upon X, and X Z Q produced to the side B C will cut B T in the midst in α.

Upon the strait line F Z, the fourth part of the radius A B, let the equilateral triangle a Z F be constituted; and upon the centre a, with the radius a Z, let the arch Z F be drawn; which arch Z F will therefore be equal to the arch Q F, the half of the arch B F. Again, let the strait line Z O be cut in the midst in b, and the strait line b O in the midst in c; and let the bisection be continued in this manner till the last part O c be the least that can possibly be taken; and upon it, and all the rest of the parts equal to it into which the strait line O F may be cut, let so many equilateral triangles be understood to be constituted; of which let the last be d O c. If, therefore, upon the centre d, with the radius d O, be drawn the arch O c, and upon the rest of the equal parts of the strait line O F be drawn in like manner so many equal arches, all those arches together taken will be equal to the whole arch B F, and the half of them, namely, those that are comprehended between O and Z, or between Z and F, will be equal to the arch B Q or Q F, and in sum, what part soever the strait line O c be of the strait line O F, the same part will the arch O c be of the arch B F, though both the arch and the chord be infinitely bisected. Now seeing the arch O c is more crooked than that part of the arch B F which is equal to it; and seeing also 291that the more the strait line X c is produced, the more it diverges from the strait line X O, if the points O and c be understood to be moved forwards with strait motion in X O and X c, the arch O c will thereby be extended by little and little, till at the last it come somewhere to have the same crookedness with that part of the arch B F which is equal to it. In like manner, if the strait line X b be drawn, and the point b be understood to be moved forwards at the same time, the arch c b will also by little and little be extended, till its crookedness come to be equal to the crookedness of that part of the arch B F which is equal to it. And the same will happen in all those small equal arches which are described upon so many equal parts of the strait line O F. It is also manifest, that by strait motion in X O and X Z all those small arches will lie in the arch B F, in the points B, Q and F. And though the same small equal arches should not be coincident with the equal parts of the arch B F in all the other points thereof, yet certainly they will constitute two crooked lines, not only equal to the two arches B Q and Q F, and equally crooked, but also having their cavity towards the same parts; which how it should be, unless all those small arches should be coincident with the arch B F in all its points, is not imaginable. They are therefore coincident, and all the strait lines drawn from X, and passing through the points of division of the strait line O F, will also divide the arch B F into the same proportions into which O F is divided.

Now seeing X b cuts off from the point B the 292fourth part of the arch B F, let that fourth part be B e; and let the sine thereof, f e, be produced to F T in g, for so f e will be the fourth part of the strait line f g, because as O b is to O F, so is f e to f g. But B T is greater than f g; and therefore the same B T is greater than four sines of the fourth part of the arch B F. And in like manner, if the arch B F be subdivided into any number of equal parts whatsoever, it may be proved that the strait line B T is greater than the sine of one of those small arches, so many times taken as there be parts made of the whole arch B F. Wherefore the strait line B T is not less than the arch B F. But neither can it be greater, because if any strait line whatsoever, less than B T, be drawn below B T, parallel to it, and terminated in the strait lines X B and X T, it would cut the arch B F; and so the sine of some one of the parts of the arch B F, taken so often as that small arch is found in the whole arch B F, would be greater than so many of the same arches; which is absurd. Wherefore the strait line B T is equal to the arch B F; and the strait line B V equal to the arch of the quadrant B F D; and B V four times taken, equal to the perimeter of the circle described with the radius A B. Also the arch B F and the strait line B T are everywhere divided into the same proportions; and consequently any given angle, whether greater or less than B A F, may be divided into any proportion given.

But the strait line B V, though its magnitude fall within the terms assigned by Archimedes, is found, if computed by the canon of signs, to be somewhat greater than that which is exhibited by 293the Rudolphine numbers. Nevertheless, if in the place of B T, another strait line, though never so little less, be substituted, the division of angles is immediately lost, as may by any man be demonstrated by this very scheme.

Howsoever, if any man think this my strait line B V to be too great, yet, seeing the arch and all the parallels are everywhere so exactly divided, and B V comes so near to the truth, I desire he would search out the reason, why, granting B V to be precisely true, the arches cut off should not be equal.

But some man may yet ask the reason why the strait lines, drawn from X through the equal parts of the arch B F, should cut off in the tangent B V so many strait lines equal to them, seeing the connected straight line X V passes not through the point D, but cuts the strait line A D produced in l; and consequently require some determination of this problem. Concerning which, I will say what I think to be the reason, namely, that whilst the magnitude of the arch doth not exceed the magnitude of the radius, that is, the magnitude of the tangent B C, both the arch and the tangent are cut alike by the strait lines drawn from X; otherwise not. For A V being connected, cutting the arch B H D in I, if X C being drawn should cut the same arch in the same point I, it would be as true that the arch B I is equal to the radius B C, as it is true that the arch B F is equal to the strait line B T; and drawing X K it would cut the arch B I in the midst in i; also drawing A i and producing it to the tangent B C in k, the strait line B k will be the 294tangent of the arch B i, (which arch is equal to half the radius) and the same strait line B k will be equal to the strait line k I. I say all this is true, if the preceding demonstration be true; and consequently the proportional section of the arch and its tangent proceeds hitherto. But it is manifest by the golden rule, that taking B h double to B T, the line X h shall not cut off the arch B E, which is double to the arch B F, but a much greater. For the magnitude of the straight lines X M, X B, and M E, being known (in numbers), the magnitude of the strait line cut off in the tangent by the strait line X E produced to the tangent, may also be known; and it will be found to be less than B h; Wherefore the strait line X h being drawn, will cut off a part of the arch of the quadrant greater than the arch B E. But I shall speak more fully in the next article concerning the magnitude of the arch B I.

And let this be the first attempt for the finding out of the dimension of a circle by the section of the arch B F.

3. I shall now attempt the same by arguments drawn from the nature of the crookedness of the circle itself; but I shall first set down some premises necessary for this speculation; and

First, if a strait line be bowed into an arch of a circle equal to it, as when a stretched thread, which toucheth a right cylinder, is so bowed in every point, that it be everywhere coincident with the perimeter of the base of the cylinder, the flexion of that line will be equal, in all its points; and consequently the crookedness of the arch of a 295circle is everywhere uniform; which needs no other demonstration than this, that the perimeter of a circle is an uniform line.

Secondly, and consequently: if two unequal arches of the same circle be made by the bowing of two strait lines equal to them, the flexion of the longer line, whilst it is bowed into the greater arch, is greater than the flexion of the shorter line, whilst it is bowed into the lesser arch, according to the proportion of the arches themselves; and consequently, the crookedness of the greater arch is to the crookedness of the lesser arch, as the greater arch is to the lesser arch.

Thirdly: if two unequal circles and a strait line touch one another in the same point, the crookedness of any arch taken in the lesser circle, will be greater than the crookedness of an arch equal to it taken in the greater circle, in reciprocal proportion to that of the radii with which the circles are described; or, which is all one, any strait line being drawn from the point of contact till it cut both the circumferences, as the part of that strait line cut off by the circumference of the greater circle to that part which is cut off by the circumference of the lesser circle.

For let A B and A C (in the second figure) be two circles, touching one another, and the strait line A D in the point A; and let their centres be E and F; and let it be supposed, that as A E is to A F, so is the arch A B to the arch A H. I say the crookedness of the arch A C is to the crookedness of the arch A H, as A E is to A F. For let the strait line A D be supposed to be equal to the arch A B, and the strait line A G to the arch A C; and 296let A D, for example, be double to A G. Therefore, by reason of the likeness of the arches A B and A C, the strait line A B will be double to the strait line A C, and the radius A E double to the radius A F, and the arch A B double to the arch A H. And because the strait line A D is so bowed to be coincident with the arch A B equal to it, as the strait line A G is bowed to be coincident with the arch A C equal also to it, the flexion of the strait line A G into the crooked line A C will be equal to the flexion of the strait line A D into the crooked line A B. But the flexion of the strait line A D into the crooked line A B is double to the flexion of the strait line A G into the crooked line A H; and therefore the flexion of the strait line A G into the crooked line A C is double to the flexion of the same strait line A G into the crooked line A H. Wherefore, as the arch A B is to the arch A C or A H; or as the radius A E is to the radius A F; or as the chord A B is to the chord A C; so reciprocally is the flexion or uniform crookedness of the arch A C, to the flexion or uniform crookedness of the arch A H, namely, here double. And this may by the same method be demonstrated in circles whose perimeters are to one another triple, quadruple, or in whatsoever given proportion. The crookedness therefore of two equal arches taken in several circles are in proportion reciprocal to that of their radii, or like arches, or like chords; which was to be demonstrated.

Let the square A B C D be again described (in the third figure), and in it the quadrants A B D, B C A and D A C; and dividing each side of the square A B C D in the midst in E, F, G and H, let 297E G and F H be connected, which will cut one another in the centre of the square at I, and divide the arch of the quadrant A B D into three equal parts in K and L. Also the diagonals A C and B D being drawn will cut one another in I, and divide the arches B K D and C L A into two equal parts in M and N. Then with the radius B F let the arch F E be drawn, cutting the diagonal B D in O; and dividing the arch B M in the midst in P, let the strait line E a equal to the chord B P be set off from the point E in the arch E F, and let the arch a b be taken equal to the arch O a, and let B a and B b be drawn and produced to the arch A N in c and d; and lastly, let the strait line A d be drawn. I say the strait line A d is equal to the arch A N or B M.

I have proved in the preceding article, that the arch E O is twice as crooked as the arch B P, that is to say, that the arch E O is so much more crooked than the arch B P, as the arch B P is more crooked than the strait line E a. The crookedness therefore of the chord E a, of the arch B P, and of the arch E O, are as 0, 1, 2. Also the difference between the arches E O and E O, the difference between the arches E O and E a, and the difference between the arches E O and E b, are as 0, 1, 2. So also the difference between the arches A N and A N, the difference between the arches A N and A c, and the difference between the arches A N and A d, are as 0, 1, 2; and the strait line A c is double to the chord B P or E a, and the strait line A d double to the chord E b.

Again, let the strait line B F be divided in the midst in Q, and the arch B P in the midst in R; 298and describing the quadrant B Q S (whose arch Q S is a fourth part of the arch of the quadrant B M D, as the arch B R is a fourth part of the arch B M, which is the arch of the semiquadrant A B M) let the chord S e equal to the chord B R be set off from the point S in the arch S Q; and let B e be drawn and produced to the arch A N in f; which being done, the strait line A f will be quadruple to the chord B R or S e. And seeing the crookedness of the arch S e, or of the arch A c, is double to the crookedness of the arch B R, the excess of the crookedness of the arch A f above the crookedness of the arch A c will be subduple to the excess of the crookedness of the arch A c above the crookedness of the arch A N; and therefore the arch N c will be double to the arch c f. Wherefore the arch c d is divided in the midst in f, and the arch N f is ¾ of the arch N d. And in like manner if the arch B R be bisected in V, and the strait line B Q in X, and the quadrant B X Y be described, and the strait line Y g equal to the chord B V be set off from the point Y in the arch Y X, it may be demonstrated that the strait line B g being drawn and produced to the arch A N, will cut the arch f d into two equal parts, and that a strait line drawn from A to the point of that section, will be equal to eight chords of the arch B V, and so on perpetually; and consequently, that the strait line A d is equal to so many equal chords of equal parts of the arch B M, as may be made by infinite bisections. Wherefore the strait line A d is equal to the arch B M or A N, that is, to half the arch of the quadrant A B D or B C A.

Coroll. An arch being given not greater than 299the arch of a quadrant (for being made greater, it comes again towards the radius B A produced, from which it receded before) if a strait line double to the chord of half the given arch be adapted from the beginning of the arch, and by how much the arch that is subtended by it is greater than the given arch, by so much a greater arch be subtended by another strait line, this strait line shall be equal to the first given arch.

Supposing the strait line B V (in fig. 1) be equal to the arch of the quadrant B H D, and A V be connected cutting the arch B H D in I, it may be asked what proportion the arch B I has to the arch I D. Let therefore the arch A Y be divided in the midst in o, and in the strait line A D let A p be taken equal, and A q double to the drawn chord A o. Then upon the centre A, with the radius A q, let an arch of a circle be drawn cutting the arch A Y in r, and let the arch Y r be doubled at t; which being done, the drawn strait line A t (by what has been last demonstrated) will be equal to the arch A Y. Again, upon the centre A with the radius A t let the arch t u be drawn cutting A D in u; and the strait line A u will be equal to the arch A Y. From the point u let the strait line u s be drawn equal and parallel to the strait line A B, cutting M N in x, and bisected by M N in the same point x. Therefore the strait line A x being drawn and produced till it meet with B C produced in V, it will cut off B V double to B s, that is, equal to the arch B H D. Now let the point, where the strait line A V cuts the arch B H D, be I; and let the arch D I be divided in the midst in y; and in the strait line D C, let D z 300be taken equal, and D δ double to the drawn chord D y; and upon the centre D with the radius D δ let an arch of a circle be drawn cutting the arch B H D in the point n; and let the arch n m be taken equal to the arch I n; which being done, the strait line D m will (by the last foregoing corollary) be equal to the arch D I. If now the strait lines D m and C V be equal, the arch B I will be equal to the radius A B or B C; and consequently X C being drawn, will pass through the point I. Moreover, if the semicircle B H D ϐ being completed, the strait lines ϐ I and B I be drawn, making a right angle (in the semicircle) at I, and the arch B I be divided in the midst at i, it will follow that A i being connected will be parallel to the strait line ϐ I, and being produced to B C in k, will cut off the strait line B k equal to the strait line k I, and equal also to the strait line A γ cut off in A D by the strait line ϐ I. All which is manifest, supposing the arch B I and the radius B C to be equal.

But that the arch B I and the radius B C are precisely equal, cannot (how true soever it be) be demonstrated, unless that be first proved which is contained in art. 1, namely, that the strait lines drawn from X through the equal parts of O F (produced to a certain length) cut off so many parts also in the tangent B C severally equal to the several arches cut off; which they do most exactly as far as B C in the tangent, and B I in the arch B E; insomuch that no inequality between the arch B I and the radius B C can be discovered either by the hand or by ratiocination. It is therefore to be further enquired, whether the 301strait line A V cut the arch of the quadrant in I in the same proportion as the point C divides the strait line B V, which is equal to the arch of the quadrant. But however this be, it has been demonstrated that the strait line B V is equal to the arch B H D.

4. I shall now attempt the same dimension of a circle another way, assuming the two following lemmas.

Lemma I. If to the arch of a quadrant, and the radius, there be taken in continual proportion a third line Z; then the arch of the semiquadrant, half the chord of the quadrant, and Z, will also be in continual proportion.

For seeing the radius is a mean proportional between the chord of a quadrant and its semichord, and the same radius a mean proportional between the arch of the quadrant and Z, the square of the radius will be equal as well to the rectangle made of the chord and semichord of the quadrant, as to the rectangle made of the arch of the quadrant and Z; and these two rectangles will be equal to one another. Wherefore, as the arch of a quadrant is to its chord, so reciprocally is half the chord of the quadrant to Z. But as the arch of the quadrant is to its chord, so is half the arch of the quadrant to half the chord of the quadrant. Wherefore, as half the arch of the quadrant is to half the chord of the quadrant (or to the sine of 45 degrees), so is half the chord of the quadrant to Z; which was to be proved.

Lemma II. The radius, the arch of the semiquadrant, the sine of 45 degrees, and the semiradius, are proportional.

302For seeing the sine of 45 degrees is a mean proportional between the radius and the semiradius; and the same sine of 45 degrees is also a mean proportional (by the precedent lemma) between the arch of 45 degrees and Z; the square of the sine of 45 degrees will be equal as well to the rectangle made of the radius and semiradius, as to the rectangle made of the arch of 45 degrees and Z. Wherefore, as the radius is to the arch of 45 degrees, so reciprocally is Z to the semiradius; which was to be demonstrated.

Let now A B C D (in fig. 4) be a square; and with the radii A B, B C and D A, let the three quadrants A B D, B C A and D A C, be described; and let the strait lines E F and G H, drawn parallel to the sides B C and A B, divide the square A B C D into four equal squares. They will therefore cut the arch of the quadrant A B D into three equal parts in I and K, and the arch of the quadrant B C A into three equal parts in K and L. Also let the diagonals A C and B D be drawn, cutting the arches B I D and A L C in M and N. Then upon the centre H with the radius H F equal to half the chord of the arch B M D, or to the sine of 45 degrees, let the arch F O be drawn cutting the arch C K in O; and let A O be drawn and produced till it meet with B C produced in P; also let it cut the arch B M D in Q, and the strait line D C in R. If now the strait line H Q be equal to the strait line D R, and being produced to D C in S, cut off D S equal to half the strait line B P; I say then the strait line B P will be equal to the arch B M D.

For seeing P B A and A D R are like triangles, 303it will be as P B to the radius B A or A D, so A D to D R; and therefore as well P B, A D and D R, as P B, A D (or A Q) and Q H are in continual proportion; and producing H O to D C in T, D T will be equal to the sine of 45 degrees, as shall by and by be demonstrated. Now D S, D T and D R are in continual proportion by the first lemma; and by the second lemma D C. D S:: D R. D F are proportionals. And thus it will be, whether B P be equal or not equal to the arch of the quadrant B M D. But if they be equal, it will then be, as that part of the arch B M D which is equal to the radius, is to the remainder of the same arch B M D; so A Q to H Q, or so B C to C P. And then will B P and the arch B M D be equal. But it is not demonstrated that the strait lines H Q and D R are equal; though if from the point B there be drawn (by the construction of fig. 1) a strait line equal to the arch B M D, then D R to H Q, and also the half of the strait line B P to D S, will always be so equal, that no inequality can be discovered between them. I will therefore leave this to be further searched into. For though it be almost out of doubt, that the strait line B P and the arch B M D are equal, yet that may not be received without demonstration; and means of demonstration the circular line admitteth none that is not grounded upon the nature of flexion, or of angles. But by that way I have already exhibited a strait line equal to the arch of a quadrant in the first and second aggression.

It remains that I prove D T to be equal to the sine of 45 degrees.

304In B A produced let A V be taken equal to the sine of 45 degrees; and drawing and producing V H, it will cut the arch of the quadrant C N A in the midst in N, and the same arch again in O, and the strait line D C in T, so that D T will be equal to the sine of 45 degrees, or to the strait line A V; also the strait line V H will be equal to the strait line H I, or the sine of 60 degrees.

For the square of A V is equal to two squares of the semiradius; and consequently the square of V H is equal to three squares of the semiradius. But H I is a mean proportional between the semiradius and three semiradii; and, therefore, the square of H I is equal to three squares of the semiradius. Wherefore H I is equal to H V. But because A D is cut in the midst in H, therefore V H and H T are equal; and, therefore, also D T is equal to the sine of 45 degrees. In the radius B A let B X be taken equal to the sine of 45 degrees; for so V X will be equal to the radius; and it will be as V A to A H the semiradius, so V X the radius to X N the sine of 45 degrees. Wherefore V H produced passes through N. Lastly, upon the centre V with the radius V A let the arch of a circle be drawn cutting V H in Y; which being done, V Y will be equal to H O (for H O is, by construction, equal to the sine of 45 degrees) and Y H will be equal to O T; and, therefore, V T passes through O. All which was to be demonstrated.

I will here add certain problems, of which if any analyst can make the construction, he will thereby be able to judge clearly of what I have now said concerning the dimension of a circle. Now 305these problems are nothing else (at least to sense) but certain symptoms accompanying the construction of the first and third figure of this chapter.

Describing, therefore, again, the square A B C D (in fig. 5) and the three quadrants A B D, B C A and D A C, let the diagonals A C and B D be drawn, cutting the arches B H D and C I A in the middle in H and I; and the strait lines E F and G L, dividing the square A B C D into four equal squares, and trisecting the arches B H D and C I A, namely, B H D in K and M, and C I A in M and O. Then dividing the arch B K in the midst in P, let Q P the sine of the arch B P, be drawn and produced to R, so that Q R be double to Q P; and, connecting K R, let it be produced one way to B C in S, and the other way to B A produced in T. Also let B V be made triple to B S, and consequently, (by the second article of this chapter) equal to the arch B D. This construction is the same with that of the first figure, which I thought fit to renew discharged of all lines but such as are necessary for my present purpose.

In the first place, therefore, if A V be drawn, cutting the arch B H D in X, and the side D C in Z, I desire some analyst would, if he can, give a reason why the strait lines T E and T C should cut the arch B D, the one in Y, the other in X, so as to make the arch B Y equal to the arch Y X; or if they be not equal, that he would determine their difference.

Secondly, if in the side D A, the strait line D a be taken equal to D Z, and V a be drawn; why V a and V B should be equal; or if they be not equal, what is the difference.

306Thirdly, drawing Z b parallel and equal to the side C B, cutting the arch B H D in c, and drawing the strait line A c, and producing it to B V in d; why A d should be equal and parallel to the strait line a V, and consequently equal also to the arch B D.

Fourthly, drawing e K the sine of the arch B K, and taking (in e A produced) e f equal to the diagonal A C, and connecting f C; why f C should pass through a (which point being given, the length of the arch B H D is also given) and c; and why f e and f c should be equal; or if not, why unequal.

Fifthly, drawing f Z, I desire he would show, why it is equal to B V, or to the arch B D; or if they be not equal, what is their difference.

Sixthly, granting f Z to be equal to the arch B D, I desire he would determine whether it fall all without the arch B C A, or cut the same, or touch it, and in what point.

Seventhly, the semicircle B D g being completed, why g I being drawn and produced, should pass through X, by which point X the length of the arch B D is determined. And the same g I being yet further produced to D C in h, why A d, which is equal to the arch B D, should pass through that point h.

Eighthly, upon the centre of the square A B C D, which let be k, the arch of the quadrant E i L being drawn, cutting e K produced in i, why the drawn strait line i X should be parallel to the side C D.

Ninthly, in the sides B A and B C taking g l and B m severally equal to half B V, or to the arch B H, and drawing m n parallel and equal to the 307side B A, cutting the arch B D in o, why the strait line which connects V l should pass through the point o.

Tenthly, I would know of him why the strait line which connects a H should be equal to B m; or if not, how much it differs from it.

The analyst that can solve these problems without knowing first the length of the arch B D, or using any other known method than that which proceeds by perpetual bisection of an angle, or is drawn from the consideration of the nature of flexion, shall do more than ordinary geometry is able to perform. But if the dimension of a circle cannot be found by any other method, then I have either found it, or it is not at all to be found.

From the known length of the arch of a quadrant, and from the proportional division of the arch and of the tangent B C, may be deduced the section of an angle into any given proportion; as also the squaring of the circle, the squaring of a given sector, and many the like propositions, which it is not necessary here to demonstrate. I will, therefore, only exhibit a strait line equal to the spiral of ArchimedesArchimedes, and so dismiss this speculation.

5. The length of the perimeter of a circle being found, that strait line is also found, which touches a spiral at the end of its first conversion. For upon the centre A (in fig. 6) let the circle B C D E be described; and in it let Archimedes' spiral A F G H B be drawn, beginning at A and ending at B. Through the centre A let the strait line C E be drawn, cutting the diameter B D at right angles; and let it be produced to I, so that A I be equal to the perimeter B C D E B. Therefore I B being drawn will touch 308the spiral A F G H B in B; which is demonstrated by Archimedes in his book De Spiralibus.

And for a strait line equal to the given spiral A F G H B, it may be found thus.

Let the strait line A I, which is equal to the perimeter B C D E, be bisected in K; and taking K L equal to the radius A B, let the rectangle I L be completed. Let M L be understood to be the axis, and K L the base of a parabola, and let M K be the crooked line thereof. Now if the point M be conceived to be so moved by the concourse of two movents, the one from I M to K L with velocity encreasing continually in the same proportion with the times, the other from M L to I K uniformly, that both those motions begin together in M and end in K; Galilæus has demonstrated that by such motion of the point M, the crooked line of a parabola will be described. Again, if the point A be conceived to be moved uniformly in the strait line A B, and in the same time to be carried round upon the centre A by the circular motion of all the points between A and B; Archimedes has demonstrated that by such motion will be described a spiral line. And seeing the circles of all these motions are concentric in A; and the interior circle is always less than the exterior in the proportion of the times in which A B is passed over with uniform motion; the velocity also of the circular motion of the point A will continually increase proportionally to the times. And thus far the generations of the parabolical line M K, and of the spiral line A F G H B, are like. But the uniform motion in A B concurring with circular motion in the perimeters of all the concentric circles, describes that 309circle, whose centre is A, and perimeter B C D E; and, therefore, that circle is (by the coroll. of art. 1, chap, XVI) the aggregate of all the velocities together taken of the point A whilst it describes the spiral A F G H B. Also the rectangle I K L M is the aggregate of all the velocities together taken of the point M, whilst it describes the crooked line M K. And, therefore the whole velocity by which the parabolical line M K is described, is to the whole velocity with which the spiral line A F G H B is described in the same time, as the rectangle I K L M is to the circle B C D E, that is to the triangle A I B. But because A I is bisected in K, and the strait lines I M and A B are equal, therefore the rectangle I K L M and the triangle A I B are also equal. Wherefore the spiral line A F G H B, and the parabolical line M K, being described with equal velocity and in equal times, are equal to one another. Now, in the first article of chap. XVIII, a strait line is found out equal to any parabolical line. Wherefore also a strait line is found out equal to a given spiral line of the first revolution described by Archimedes; which was to be done.

6. In the sixth chapter, which is of Method, that which I should there have spoken of the analytics of geometricians I thought fit to defer, because I could not there have been understood, as not having then so much as named lines, superficies, solids, equal and unequal, &c. Wherefore I will in this place set down my thoughts concerning it.

Analysis is continual reasoning from the definitions of the terms of a proposition we suppose true, and again from the definitions of the terms of 310those definitions, and so on, till we come to some things known, the composition whereof is the demonstration of the truth or falsity of the first supposition; and this composition or demonstration is that we call Synthesis. Analytica, therefore, is that art, by which our reason proceeds from something supposed, to principles, that is, to prime propositions, or to such as are known by these, till we have so many known propositions as are sufficient for the demonstration of the truth or falsity of the thing supposed. Synthetica is the art itself of demonstration. Synthesis, therefore, and analysis, differ in nothing, but in proceeding forwards or backwards; and Logistica comprehends both. So that in the analysis or synthesis of any question, that is to say, of any problem, the terms of all the propositions ought to be convertible; or if they be enunciated hypothetically, the truth of the consequent ought not only to follow out of the truth of its antecedent, but contrarily also the truth of the antecedent must necessarily be inferred from the truth of the consequent. For otherwise, when by resolution we are arrived at principles, we cannot by composition return directly back to the thing sought for. For those terms which are the first in analysis, will be the last in synthesis; as for example, when in resolving, we say, these two rectangles are equal, and therefore their sides are reciprocally proportional, we must necessarily in compounding say, the sides of these rectangles are reciprocally proportional, and therefore the rectangles themselves are equal; which we could not say, unless rectangles have their sides reciprocally 311proportional, and rectangles are equal, were terms convertible.

Now in every analysis, that which is sought is the proportion of two quantities; by which proportion, a figure being described, the quantity sought for may be exposed to sense. And this exposition is the end and solution of the question, or the construction of the problem.

And seeing analysis is reasoning from something supposed, till we come to principles, that is, to definitions, or to theorems formerly known; and seeing the same reasoning tends in the last place to some equation, we can therefore make no end of resolving, till we come at last to the causes themselves of equality and inequality, or to theorems formerly demonstrated from those causes; and so have a sufficient number of those theorems for the demonstration of the thing sought for.

And seeing also, that the end of the analytics is either the construction of such a problem as is possible, or the detection of the impossibility thereof; whensoever the problem may be solved, the analyst must not stay, till he come to those things which contain the efficient cause of that whereof he is to make construction. But he must of necessity stay, when he comes to prime propositions; and these are definitions. These definitions therefore must contain the efficient cause of his construction; I say of his construction, not of the conclusion which he demonstrates; for the cause of the conclusion is contained in the premised propositions; that is to say, the truth of the proposition he proves is drawn from the propositions which prove the same. 312But the cause of his construction is in the things themselves, and consists in motion, or in the concourse of motions. Wherefore those propositions, in which analysis ends, are definitions, but such as signify in what manner the construction or generation of the thing proceeds. For otherwise, when he goes back by synthesis to the proof of his problem, he will come to no demonstration at all; there being no true demonstration but such as is scientifical; and no demonstration is scientifical, but that which proceeds from the knowledge of the causes from which the construction of the problem is drawn. To collect therefore what has been said into few words; ANALYSIS is ratiocination from the supposed construction or generation of a thing to the efficient cause or coefficient causes of that which is constructed or generated. And SYNTHESIS is ratiocination from the first causes of the construction, continued through all the middle causes till we come to the thing itself which is constructed or generated.

But because there are many means by which the same thing may be generated, or the same problem be constructed, therefore neither do all geometricians, nor doth the same geometrician always, use one and the same method. For, if to a certain quantity given, it be required to construct another quantity equal, there may be some that will inquire whether this may not be done by means of some motion. For there are quantities, whose equality and inequality may be argued from motion and time, as well as from congruence; and there is motion, by which two quantities, whether lines or superficies, though one of them be crooked, the 313other strait, may be made congruous or coincident. And this method Archimedes made use of in his book De Spiralibus. Also the equality or inequality of two quantities may be found out and demonstrated from the consideration of weight, as the same Archimedes did in his quadrature of the parabola. Besides, equality and inequality are found out often by the division of the two quantities into parts which are considered as indivisable; as Cavallerius Bonaventura has done in our time, and Archimedes often. Lastly, the same is performed by the consideration of the powers of lines, or the roots of those powers, and by the multiplication, division, addition, and subtraction, as also by the extraction of the roots of those powers, or by finding where strait lines of the same proportion terminate. For example, when any number of strait lines, how many soever, are drawn from a strait line and pass all through the same point, look what proportion they have, and if their parts continued from the point retain everywhere the same proportion, they shall all terminate in a strait line. And the same happens if the point be taken between two circles. So that the places of all their points of termination make either strait lines, or circumferences of circles, and are called plane places. So also when strait parallel lines are applied to one strait line, if the parts of the strait line to which they are applied be to one another in proportion duplicate to that of the contiguous applied lines, they will all terminate in a conical section; which section, being the place of their termination, is called a solid place, because it serves for the finding out of the quantity of any 314equation which consists of three dimensions. There are therefore three ways of finding out the cause of equality or inequality between two given quantities; namely, first, by the computation of motions; for by equal motion, and equal time, equal spaces are described; and ponderation is motion. Secondly, by indivisibles: because all the parts together taken are equal to the whole. And thirdly, by the powers: for when they are equal, their roots also are equal; and contrarily, the powers are equal, when their roots are equal. But if the question be much complicated, there cannot by any of these ways be constituted a certain rule, from the supposition of which of the unknown quantities the analysis may best begin; nor out of the variety of equations, that at first appear, which we were best to choose; but the success will depend upon dexterity, upon formerly acquired science, and many times upon fortune.

For no man can ever be a good analyst without being first a good geometrician; nor do the rules of analysis make a geometrician, as synthesis doth; which begins at the very elements, and proceeds by a logical use of the same. For the true teaching of geometry is by synthesis, according to Euclid's method; and he that hath Euclid for his master, may be a geometrician without Vieta, though Vieta was a most admirable geometrician; but he that has Vieta for his master, not so, without Euclid.

And as for that part of analysis which works by the powers, though it be esteemed by some geometricians, not the chiefest, to be the best way of solving all problems, yet it is a thing of no great extent; it being all contained in the doctrine of 315rectangles, and rectangled solids. So that although they come to an equation which determines the quantity sought, yet they cannot sometimes by art exhibit that quantity in a plane, but in some conic section; that is, as geometricians say, not geometrically, but mechanically. Now such problems as these, they call solid; and when they cannot exhibit the quantity sought for with the help of a conic section, they call it a lineary problem. And therefore in the quantities of angles, and of the arches of circles, there is no use at all of the analytics which proceed by the powers; so that the ancients pronounced it impossible to exhibit in a plane the division of angles, except bisection, and the bisection of the bisected parts, otherwise than mechanically. For Pappus, (before the 31st proposition of his fourth book) distinguishing and defining the several kinds of problems, says that "some are plane, others solid, and others lineary. Those, therefore, which may be solved by strait lines and the circumferences of circles, (that is, which may be described with the rule and compass, without any other instrument), are fitly called plane; for the lines, by which such problems are found out, have their generation in a plane. But those which are solved by the using of some one or more conic sections in their construction, are called solid, because their construction cannot be made without using the superficies of solid figures, namely, of cones. There remains the third kind, which is called lineary, because other lines besides those already mentioned are made use of in their construction, &c." And a 316little after he says, "of this kind are the spiral lines, the quadratrices, the conchoeides, and the cissoeides, And geometricians think it no small fault, when for the finding out of a plane problem any man makes use of conics, or new lines." Now he ranks the trisection of an angle among solid problems, and the quinquesection among lineary. But what! are the ancient geometricians to be blamed, who made use of the quadratrix for the finding out of a strait line equal to the arch of a circle? And Pappus himself, was he faulty, when he found out the trisection of an angle by the help of an hyperbole? Or am I in the wrong, who think I have found out the construction of both these problems by the rule and compass only? Neither they, nor I. For the ancients made use of this analysis which proceeds by the powers; and with them it was a fault to do that by a more remote power, which might be done by a nearer; as being an argument that they did not sufficiently understand the nature of the thing.

The virtue of this kind of analysis consists in the changing and turning and tossing of rectangles and analogisms; and the skill of analysts is mere logic, by which they are able methodically to find out whatsoever lies hid either in the subject or predicate of the conclusion sought for. But this doth not properly belong to algebra, or the analytics specious, symbolical, or cossick; which are, as I may say, the brachygraphy of the analytics, and an art neither of teaching nor learning geometry, but of registering with brevity and celerity the inventions of geometricians. For though it be easy to discourse 317by symbols of very remote propositions; yet whether such discourse deserve to be thought very profitable, when it is made without any ideas of the things themselves, I know not.

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