CHAPTER XIII.. Of Analogism, or the Same Proportion.

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.

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 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.

A B C 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.

A B. A. C B. C. C D. E.

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. 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; 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.

ABF. A. CBF. C. CDF. E. CDG. H.

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.

16. 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 the 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 be 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 placed in the line A C prolonged; and the further off the point

D 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.

Ḅ C̣ Ḍ A—————E

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 D 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.

21. 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 the 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 A C. A B. B C are in continual proportion. And if the same line A C be again divided 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.

A B C ----|----|---- D

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.

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 proportion; 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 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.

A A - - B F --- --- C G ----- ----- D H ------- ------- E 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 the 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. 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 proportion. 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.

A E G H B ---+---+---+---- ----+---+---+--- A F I K B

For in the first place we have this A B. A G :: B G. G E, by analogism article 18.

Then by composition we have this A B + A G. A B :: B G + G E that is, B E. B G.

And by taking the halves of the ½A B + ½A G. A B :: ½B G + ½G E, antecedents this third that is, B H. B G.

And by conversion a fourth A B. ½A B + ½A G :: B G. B H.

And by division this fifth ½A B - ½A G. ½A B + ½A G :: H G. B H.

And by doubling the first antecedent and the first consequent A B - A G. A B + A G :: H G. B H.

Also by the same method may be found out this analogism A B - A I. A B + AI :: K I. B K.

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 supposition, 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.

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CHAP. XIV.

OF STRAIT AND CROOKED, ANGLE AND FIGURE.

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 between 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, if 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 line, 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.

Fourthly, 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 was 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 will 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 be 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 and 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, those, 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 B 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 although 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 made 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.

12. 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.

Coroll. 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 away 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. For 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 points 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 subtense 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 C 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 the 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 circular 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 taking 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 that 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 will 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 point 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 one 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.

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

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