Cor. 1. (Euc. v. 13.) If A have to B the same ratio as C to D, and C to Da greater or a less ratio than E to F, A shall have to B a greater or a less ratio than E has to F.

Cor. 2. If A have to B a greater or a less ratio than C to D, and C to D the same ratio as E to F, A shall have to B a greater or a less ratio than E has to F.

PROP. 13. (EUc. v. D and C.) If four magnitudes be proportionals, and if the first be any multiple or part of the second, the third shall be the same multiple or part of the fourth: and conversely, if one magnitude be the same multiple or part of another, that a third magnitude is of a fourth, the four magnitudes shall be proportionals.

The first is evident from def. 7.; for it is supposed in that definition that if any part of the second be contained a certain number of times in the first exactly, a like part of the fourth will be contained the same number of times in the third exactly. And, in the second place, if A be the same multiple or part of B that C is of D, A, B, C, D will be commensurable proportionals, and therefore ([9]) also proportionals by def. 7. Therefore, &c.

PROP. 14. (Euc. v. A.)

pro

If four magnitudes A, B, C, D be portionals, and if the first be greater than the second, the third shall be greater than the fourth; if equal, equal; and if less, less.

For, if A be greater than B, a part of the latter may be found which shall be less than their difference, and which shall therefore be contained a greater number of times exactly, or with a remainder, in A, than in B: therefore, because A, B, C, D are proportionals, the like part of D will also be contained a greater number of times exactly or with a remainder in C than in D, that is, C will be greater than D.

And, in the same manner it may be shown, that if A be equal to B, C will be equal to D; and if less, less. Therefore, &c.

2

PROP. 15. (Euc. v. B.)

If four magnitudes A, B, C, D be proportionals, they shall also be proportionals when taken inversely: that is, invertendo, B: A:D: C.

Of A, C take any equimultiples whatever m A, m C, and of B, D,

any equimultiples whatever n B, nD. Then, according as n B is equal to, or greater than, or less than m A, it is plain that m A will be equal to, or less than, or greater than n B; and, therefore, on account of the proportion, (9. Cor. 2.) m C will be likewise equal to, or less than, or greater than n D, that is, n D will be equal to, or greater than, or less than m C. And this will be the case, whatever be the numbers n and m. Therefore (9. Cor. 2.) B, A, D, C are proportionals. Therefore, &c.

PROP. 16. (Euc. v. 4.)

If four magnitudes A, B, C, D be proportionals, and if there be taken any equimultiples of the first and third; and also any equimultiples of the second and fourth these equimultiples shall likewise be proportionals.

Let m A, m C be any equimultiples whatever of A, C, and n B, n D any equimultiples whatever of B, D. Take K, M any equimultiples whatever of m A, m C, and therefore (2.) likewise equimultiples of A, C ; also L, N any equimultiples whatever of n B, n D, and therefore likewise equimultiples of B, D. Then, because A:B:: C:D, if K be greater than L, M will be greater than N; if equal, equal and if less, less (9. Cor. 2.). But K, M are any equimultiples whatever of m A, m C, and M, N any equimultiples whatever of n B, n D. Therefore (9. Cor. 2.) m A: n B::m C:n D. Therefore, &c.

Cor. If A, B, C, D be proportionals, and if any like parts of the first and third be taken, and also any like parts of the second and fourth, these like parts will likewise be proportionals.

PROP. 17. (Euc. v. 15.) Magnitudes have the same ratio to one another which their equimultiples have.

For, if m A, m B be any equimultiples whatever of A, B, it is evident, that any part P of B will be contained in A exactly, or with a remainder, as often as m times that part or m P is contained in mA with a remainder. Also, whatever part P is of B, m P is the like part of m B. Therefore, A:B::m A: m B (def. 7).

als, and if any equimultiples whatever be taken of the first and second, and also any equimultiples whatever of the third and fourth, these equimultiples will likewise be proportionals (12).

The same, it is evident, may be stated with regard to any like parts taken of the first and second, and also of the third and fourth.

PROP. 18. (Euc. v. 14.)

If four magnitudes of the same kind be proportionals, and if the first be greater than the third, the second shall be greater than the fourth; if equal, equal; and if less, less.

Let A, B, C, D, be proportionals, and first, let A be greater than C; B shall be greater than D.

For A, being greater than C, has to B a greater ratio (11.) than C has to B: therefore (12. Cor. 1.) C has also to D a greater ratio than the same C has to B: therefore, (11. Cor. 3.) B is greater

than D.

And, in like manner it may be shown, that if A be equal to C, B must be equal to D; and if less, less.

Therefore, &c.

Cor. Hence, also, if four magnitudes of the same kind be proportionals, and if the second be greater than the fourth, the first will be greater than the third; if equal, equal; and if less, less.

PROP. 19. (Euc. v. 16.)

If four magnitudes A, B, C, D of the same kind be proportionals, they shall also be proportionals, when taken alternately; that is, alternando A: C:: B: D.

Of A, B take any equimultiples whatever m A, m B, and of C, D any equimultiples whatever n C, n D: then (17. Cor. 2.) m A, m B, n C, n D are proportionals; and, therefore, (18.) if m A be greater than n C, m B will also be greater than n D; if equal, equal; if less, less. But m A, m B are any equimultiples whatever of A, B, and n C, n D any equimultiples whatever of C, D. Therefore, (9. Cor. 2.) A: C::B: D. Therefore, &c.

PROP. 20. (EUc. v. 17.)

If four magnitudes A, B, C, D, be proportionals, they shall also be proportionals, when taken dividedly: that is, the difference of the first and second shall be to the second as the difference of the third and fourth to the fourth ;— dividendo A-B: B::C~D: D.

-or

For, if any like parts whatever, as the nth, of B and D be contained in A and C the same number of times, m, exactly, or with corresponding remainders, they will be contained in A-B and C-D, the same number of times, men, exactly, or with the same remainders. Therefore, (def. 7.) A~B : B :: C~D : D. Therefore, &c.

Cor. 1. (Euc. v. E.) If four magnitudes A, B, C, D be proportionals, then convertendo, A: A~B:: C: C-D (See [20] Cor. 1.)

Čor. 2. (Euc. v. 25.) If four magnitudes of the same kind be proportionals, the greatest and least of them together shall be greater than the other two together. (See [20] Cor. 2.)

Cor. 3. If three magnitudes be proportionals, half the sum of the extremes shall be greater than the mean: in other words, the arithmetical mean between two magnitudes is greater than the geometrical mean between the same two. (See [20] Cor. 3.)

PROP. 21. (EUc. v. 18.)

If four magnitudes A, B, C, D, be proportionals, they shall also be proportionals when taken conjointly; that is, the sum of the first and second shall be to the second, as the sum of the third and fourth to the fourth: or, componendo, A+B:B::C+D: D.

For, if any like parts whatever, as the nth, of B and D be contained in A and C the same number of times, m, exactly, or with corresponding remainders, they will be contained in A+B and C+Ď the same number of times, m+n, exactly, or with the same remainders. Therefore, (def. 7.) A+B:B::C+D:D. Therefore, &c.

Cor. If four magnitudes A, B, C, D be proportionals, A: A+B:: C:C+D. (See [21] Cor.)

PROP. 22. (Euc. v. 19.)

If one magnitude be to another as a magnitude taken from the first to a magnitude taken from the other, the remainder shall be to the remainder in the same ratio.

See the demonstration of [22].

Cor. 1. If there be any number of magnitudes A, B, C, D, &c. in geometrical progression, the differences A~B, B-C, C-D, &c. will form a geometrical progression in which the successive terms have the same ratio with the successive terms of the former. (See [22] Cor. 1.)

Cor. 2. And conversely, any number of magnitudes A, B, C, D, &c. in geome trical progression may be considered as the differences of other magnitudes A', B', C', D', E', &c. forming a geometrical progression, in which the first term A' is to A, as A to A-B, and the successive terms have the same ratio with the successive terms of the former. (See [22]. Cor. 2.)

PROP. 23.

If one magnitude be to another as a third magnitude of the same kind to a fourth, the sum of the first and third "shall be to the sum of the second and I fourth in the same ratio.

See the demonstration of [23].

Cor. 1. (Euc. v. 12.) Hence, if there be any number of magnitudes of the same kind antecedents, and as many consequents, and if every antecedent have the same ratio to its consequent, the sum of all the antecedents shall have the same ratio to the sum of all the consequents.

Cor. 2. If the ratio of A' to B' be not the same with the ratio of A to B, the ratio of A+ A' to B+B' will not be the same with the ratio of A to B ; but less, if the magnitudes added be to one another in a less ratio, or greater, if the magnitudes added be to one another in a greater ratio.* (See [23]. Cor. 2.)

Cor. 3. Hence, if the ratio of A' to B' be not the same with the ratio of A to B, the ratio of A+ A' to B + B' will lie between the ratios of A to B and of A' to B'; that is, it will be greater than the lesser of the two, and less than the greater of the two.

PROP. 24.

If there be three magnitudes of the same kind, A, B, C, and other three A', B', C', which, taken two and two in order, have the same ratio; viz. A to B the same ratio as A' to B', and B to C the same ratio as B' to C'; then ex

*It is required in the demonstration of this Corollary, by help of the proposition, as in [23]. Cor. 2., that to two given magnitudes of the same kind, and a third, there may be found a fourth proportional. The case in which the two first magnitudes are commensurable has been already noticed in the note upon [23]. Cor, 2. in the other case, i. e. when they are incommensurable, we can only approximate to the fourth proportional as we approximate to the ratio of the two magnitudes numerically, as in the opening of this section. Since, however, such approximation may be continued without limit, and magnitudes can be obtained which are in a greater and less ratio to the given magnitude, according to def. 8., we presume that there is some magnitude between them which is to the given third magnitude in the same ratio which the second has to the first-that is, some magnitude which is a fourth proportional to the three.

æquali in proportione directâ (or ex æquo) the first shall be to the third of the first magnitudes, as the first to the third of the others; or, as it may be more briefly stated,

if A: B:: A': B' and B: C:: B': C'

then, ex æquali A: C :: A' : C'.

Of A, A' take any equimultiples mA, m A', of B, B' any equimultiples n B, n B', and of C, C' any equimultiples p C, pc': then, because A, B, A', Bare proportionals, m A, n B, m A', n B' are also proportionals, (16.): and, in like manner, because B, C, B', C' are proportionals, n B, p C, n B', p C' are also proportionals.

Now, if m A be greater than p C, it will have to n B a greater ratio than p C has (11.); and therefore, (12. Cor. 1.) m A will also have to n B' a greater ratio than p C has to n B; that is, on account of the proportionals p C, n B, p C', n B' (15.), a greater ratio than p C' has to n B (12. Cor. 2.); therefore, also, m A' will be greater than p C' (11. Cor. 3.). In the same manner it may be shown, that if m A be equal to p C, m A' will also be equal to p C'; and, if less, less. And the numbers m, p may be any whatever. Therefore (9. Cor. 2.) A, C, A', C are proportionals, and A: C:: A': C'.

Therefore, &c.

Cor. 1. (Euc. v. 22.) Hence, also, if A: B:: A': B' and BC: B': C'

and CD:: C': D'

ex æquali A: D:: A': D'; and the same is true whatever be the number of magnitudes A, B, C, D, &c. A', B', C', D', &c.

This may be stated in the following words: “Ratios, which are compounded of any number of equal ratios in the same order, are equal to one another."

Cor. 2. If four magnitudes A, B, C, D be proportionals, then miscendo A + B :AB::C+D: C~D. (See [24] Cor. 2.)

PROP. 25. (Euc. v. 24.)

If two proportions have the same consequents, the sum of the first antecedents shall be to their consequent, as the sum of the second antecedents to their consequent; that is, if A: B::C: D, and if A': B:: C': D, then A + A': B:: C+ C': D..

See the demonstration of [25].

Cor, 1. The same may be stated of

any number of proportions having the same consequents; that is, the sum of all the first antecedents shall be to their consequent as the sum of all the second antecedents to their consequent.

Cor. 2. In like manner, also, it may be shown (by" dividendo" instead of " componendo") that if two proportions have the same consequents, the difference of the first antecedents shall be to their consequent as the difference of the second antecedents to their consequent.

Cor. 3. Hence if there be any number of magnitudes of the same kind A, B, C, D, E, F, and as many others A', B', C', D', E', F', and if the ratios of the first to the second, of the second to the third, of the third to the fourth, and so on, be respectively the same in the two series; any two combinations by sum and difference of the magnitudes of the first series, e. g. A + C – E and B-C+D shall be to one another as two similar combinations of the corresponding magnitudes of the second series, viz. A' + C'E' and B' C' + D'.

PROP. 26.

If there be three magnitudes of the same kind A, B, C, and other three A', B', C', which taken two and two, but in a cross order, have the same ratio, viz. A to B the same ratio us B' to C', and B to C the same ratio as A' to B'; then, ex æquali in proportione perturbata (or ex æquo perturbato), the first shall be to the third of the first magnitudes as the first to the third of the others; or, as it may be more briefly stated,

if A: B:: B': C' and B: C:: A': B'

and

ex æquo perturbato A: C:: A' : C'. Of A, B and A' take any equimultiples m A, m B and m A', and of C, B' and C' any equimultiples n C, n B' n C': then, because A, B, B′, C', are proportionals, m A, m B, n B', n C' are also proportionals (17. Cor. 2.); and, because B, C, A', B' are proportionals, m B, n C, m A', n B' are also proportionals (16).

Now, if m A be greater than n C, it will have to m B a greater ratio than nC has to m B (11.); and therefore, (12. Cor.1.)nB' will also have to n C' a greater ratio than n C has to m B, that is, on account of the proportionals n C, m B, n B', m A' (15.), a greater ratio than n B' has to mA (12. Cor. 2.): therefore, also, m A' will be greater than n C' (11. Cor.3.). In the same manner, it may be shown

that if m A be equal to n C, m A' will also be equal to `n C', and if less, less. And the numbers m, n may be any whatever. Therefore (9. Cor. 2.), A, C, A', C' are proportionals, and A: C :: A': C'.

Therefore, &c.

Cor. 1. (Euc. v. 23.) Hence, also, if A: B: C': D' and B: C: B': C' and CD: A': B' ex æquo perturbato A: D:: A': D'; and the same is true whatever be the number of magnitudes A, B, C, D, &c. A', B', C', D', &c.

This may be stated in the following words: "ratios which are compounded of any number of equal ratios, but in a reverse order, are equal to one another."

PROP. 27. (Euc. v. F.)

Ratios which are compounded of the same ratios, in whatsoever orders, are the same with one another.

The case of ratios which are compounded of the same ratios in the same order is that of 24. Cor. 1.

The case, again, of ratios which are compounded of the same ratios in a reverse order, is that of 26. Cor. 1.

Let the ratio of A to D, therefore, be compounded of the ratios of A to B, of B to C, and of C to D: and let these ratios be the same respectively with the ratios of C' to D', of A' to B' and of B' to C', of which the ratio of A' to D' is compounded, but in a different order: the ratio of A to D shall be the same with the ratio of A' to D'.

For, because B: C: A': B'
and C:D :: B': C'

ex æquali, B: D
but again, A: B::

Therefore ex æquo
perturbato

A': C

C': D'

A:D:: A': D'.

And, in a similar manner, the case of ratios which are compounded of the same three ratios in any other order, may be demonstrated. For, if K, L, M represent the three ratios in one order, in whatever other order they may be arranged, two of them will be found which are contiguous in both arrangements; commencing with which two, the demonstration will differ little from the above.

Hence, again, it may be shown, that ratios which are compounded of the same four ratios K, L, M, N in whatsoever orders, are the same with one another; as for instance, in the orders K, L, M, N, and M, K, N, L: for the

latter ratio is the same with the ratio which is compounded of the same ratios in the order M,K,L,N, because the ratio which is compounded of K, N, L, is the same with that which is compounded of K, L, N; and, for a similar reason, the ratio which is compounded of M, K, L, N, is the same with that which is compounded of K, L, M, N.

And the same reasoning may be extended to five, six, or any other number

of ratios.

Therefore, &c.

Cor. 1. (Euc. v. G.) If there be any number of ratios as those of A to B, of C to D, of E to F, &c. magnitudes which have to one another a ratio compounded of any two of these shall have the same ratio to one another with any other magnitudes which have to one another a ratio compounded of the same two; and, in like manner, magnitudes which have to one another a ratio compounded of any three of these shall have the same ratio to one another with any other magnitudes which have to one another a ratio compounded of the same three;

and so on.

Cor. 2. If the ratios of A to B, of B to D, of E to F, &c. be all equal to one another, magnitudes which have to one another a ratio compounded of any two of them will have to one another a ratio which is the same with the duplicate ratio of A to B and in like manner, magnitudes which have to one another a ratio compounded of any three of them will have to one another a ratio which is the same with the triplicate ratio of A to B; and so on.

Cor. 3. Ratios which are the duplicate, or triplicate, &c. of the same ratio are the same with one another.

Cor. 4. In the composition of ratios any two which are reciprocals of one another may be neglected, without affecting the resulting compound ratio. (See 10. Cor.)

Cor. 5. Hence, if two ratios be equal to one another (and therefore compounded of equal ratios, having any order in the composition of each), and if any of the equal ratios be subducted or taken away, the remaining ratios will be equal to one another. (See [27] Cor. 5.)

PROP. 28.

If there be two fixed magnitudes A and B, which are the limits of two others P and Q (that is, to which P and Q, by increasing together, or by diminishing together, may be made to approach more

nearly than by any the same given difference) and if P be to Q always in the same given ratio of C to D; A shall be to B in the same ratio.

See the demonstration of [28.] and the note at 23. Cor. 2.

It has been already observed, that this proposition is of extensive application in Geometry, and the uses have been mentioned to which it will be found applied in subsequent parts of this treatise. The present section, or rather, the passing from a property demonstrated of commensurable proportionals to the demonstration of the same property with regard to incommensurable proportionals, offers an immediate illustration of it; which seems also the rather not to be passed by in this place, as we here take leave of the abstract theory of proportion, to consider its application to the proper subjects of Geometry, viz. lines, surfaces and solids; and a theory of proportion would scarcely appear complete without some notice of the equality existing between the products of the extremes and means. In what sense this expression is to be interpreted with regard to a proportion of four magnitudes of any kind A, B, C, D, was pointed out in the General Scholium at the end of Section II.; viz. that it supposes the magnitudes to be commensurable, and is to be understood of the numbers which stand for them. But we have seen that the term product is sometimes also, for the sake of brevity, used synonymously for rectangle,-as whenever the product of two lines is spoken of, meaning the rectangle which they contain.

We propose then to demonstrate generally, that, if four straight lines A, B, C, D be proportionals, (whether commensurable or otherwise,) the rectangle under the extremes will be equal to the rectangle under the means; and that in such a manner as may serve to illustrate the use of Prop. 28.

c.n

First, let A and B be commensurable, and therefore also C and D: and let their common ratio be any whatever, as 7:5; that is, let there be common measures M, N, the first of A and B, and the second of C and D, which are contained in A and C respectively 7 times, and in B and D respectively times. Then because A contains M 7 times, and that D contains

M N