CHAPTER XII.

OF QUANTITY.

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 that 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 known 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.

4. 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.

6. 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, which 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 of 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.

CHAPTER XIII.

OF ANALOGISM, OR THE SAME PROPORTION.

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 greater 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 twice 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 proportion 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.

For 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 if 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 it 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 with 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 of 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.

As, 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 A. 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 H 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).

10. 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 and 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 as 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 are 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, A, 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.

14. 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 before; 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 with 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.