PRO P. VII. THE OR.

QUAL magnitudes have the fame proportion to the fame

proportion to equal magnitudes.

Book V.

Let A, B, be two equal magnitudes, and C any third magnitude, then A, B, will have the fame proportion to C. For, let D, E, be any equimultiples of A, B, and F any equimultiple of C; then, if D is equal to F, E is likewife equal to F; if greater, greater; and, if lefs, lefs; therefore A is to C as B is to C. Again, C is to A as C is to B; for, the fame construction re- a def. s. maining, if F is equal to D, it is likewife equal to E; wherefore Cis to B as C is to A. Wherefore, &c.

PRO P. VIII. THE O R.

THE greater of two unequal magnitudes has a greater proportion to fome third magnitude than the lefs has, and that third magnitude has a greater proportion to the leffer magnitude than it has to the greater.

Let AB, C, be two unequal magnitudes, of which AB is the greater; and let D be any third magnitude; then AB will have a greater proportion to D than Chas to D; but D has a greater proportion to C than it has to AB.

b Az; tà

For, because AB is greater than C, make BE equal to C, then AB will exceed C by AE; if AE is not greater than D, let. it be multiplied till it exceed D; and let this multiple be FG; let GH be the fame multiple of EB that FG is of AE; then FH will be the fame multiple of AB that GH is of EB; and make a r. K the fame multiple of C that GH is of EB; but EB is equal to C; therefore K is equal to GH; wherefore FH is the fame multiple of AB that K is of C. Now, of D let L be taken a double, M triple, and so on, till a multiple of D is found next greater than K, which let be N; and let M be the next multiple of D, lefs than N; then M will not be greater than K, that is, Kwill not be less than M; but M and D are equal to N ; and becaufe FG exceeds D, and GH is equal to K, FH will be greater than N'; that is, FH, the multiple of AB, exceeds N, the multiple of D; but K, the multiple of C, does not exceed N, he multiple of D; therefore AB has to D a greater proportion han Chas to Dc; but likewife D has to Ca greater proportion han it has to AB; for, the fame conftruction remaining, N, he multiple of D, exceeds K, the multiple of C; but does not Exceed FH, the multiple of AB. Wherefore, &c. PROP.

с

Book. V.

PROP. IX. THEOR.

MAGNITUDES which have the fame proportion to one and the fame magnitude are equal to one another; and if a magnitude has the fame proportion to other magnitudes, these magnitudes are equal to one another.

Let the magnitudes A, B, have the fame proportion, to C, then A is equal to B. If not, let A be greater or less than B; if greater, then A has a greater proportion to C than B has to Ca; but it has not; therefore A is not greater than B ; if lefs, then B has a greater proportion to C than it has to C; but it has not; therefore A is not lefs than B; and, fince neither greater nor lefs, it must be equal.

Again, if C have the fame proportion to A that it has to B, A is equal to B; if not, C will have a greater proportion to the leffer magnitude, than it has to the greater; but it has not; therefore A is not greater than B, or B is not greater than A; therefore A is equal to B. Wherefore, &c.

Ο

PROP. X. THE OR.

F magnitudes having proportion to the fame magnitude, that which has the greater proportion to fome third is the greater magnitude; and that magnitude to which the fame has a greater proportion is the leffer magnitude.

Of the magnitudes A, B, if A have a greater proportion to a third magnitude C, than B has to C, A is greater than B; for, if not, it will be either equal or lefs. If A is equal to B, then they would have the fame proportion to Ca; but they have not; therefore A is not equal to B; neither is it lefs; for then B would have a greater proportion to C than it has to Cb; but it has not: Therefore, fince A is neither equal nor less than B, it must be greater.

Again, if C have a greater proportion to B than it has to A, then B is lefs than A; if not, let it be equal or greater; if e qual, then C has the fame proportion to B that it has to A ; but it has not; therefore B is not equal to A. If greater, then C will have a greater proportion to A than it has to B; but it has not; therefore, fince B is not equal or greater than A, it muft be lefs. Wherefore, &c.

1

BOOK V.

PROP. XI. THEOR.

ROPORTIONS that are the fame to any third, are the fame to one another.

PR

Let A be to B, as C is to D, and C to D as E to F; then A will be to B as E to F.

For, let G, H, K, be any equimultiples of A, C, E; and L, M, N, any other equimultiples of B, D, F; now, because A is to B as C is to D; and G, H are equimultiples of A, C ; and L, M any other equimultiples of B, D; if G is equal to L, H will be equal to Ma; if greater, greater; and, if lefs, lefs: Like- a Def. 5. wife, because C is to D as E is to F; if H be equal to M, K will be equal to N2; if greater, greater; and, if lefs, lefs. Wherefore, if G be equal to L, K will be equal to N; for they are e

qual to H, Mb; wherefore A is to B as E to Fa. Where- b Ax. 1. 1. fore, &c.

I

PRO P. XII. THE OR.

F any number of magnitudes be proportional, as one of the antecedents is to one of the confequents, fo are all the antecedents to all the confequents.

Let the magnitudes be A, B, C, D, E, F; and, as A is to B, fo is C to D, and E to F; then as A is to B, fo are A, C, E, all the antecedents, to B, D, F, all the confequents. For, let let G, H, K be equimultiples of A, C, E; and L, M, N be any other equimultiples of B, D, F; then, because A is to B as C is to D, and C to D as E to F, and G, H, K equimultiples of A, C, E, and L, M, N any other equimultiples of B, D, F, if G be equal to L, H will be equal to M, and K to N; if a Def, s. greater, greater; and, if lefs, lefs; wherefore, if G be equal to L; G, H, K, will be equal to L, M, N; if greater, greater; and, if lefs, lefs ; but G; G, H, K, are equimultiples of A; A, b 1. C, E, together; and L; L, M, N, equimultiples of B; B, D, F, together; fuch, that, if G be equal to L, G, H, K will be equal to L, M, N, together; wherefore A, C, E, together, are to B, D, F, together, as A is to B. Wherefore, &c. K

PROP.

PROP. XIII. THE OR.

TF the first is in the fame proportion to the fecond, as the third to the fourth; and if the third has a greater proportion to the fourth, than the fifth to the fixth; then shall also the first have a greater proportion to the fecond, than the fifth to the fixth.

Let the firft A have the fame proportion to the fecond B, that the third C has to the fourth D; but let the third C have a greater proportion to the fourth D, than the fifth E to the fixth F; then the firft A will have a greater proportion to the second B, than the fifth E has to the fixth F.

For, because C has a greater proportion to D, then E to F; let G, H be equimultiples of C, E; and K, L equimultiples of D, F; fuch, that, if G exceeds K, but H does not exceed L'; and let M be the fame multiple of A that G is of C; and N the fame multiple of B that K is of D; then, because G exceeds K, M will exceed N; but G exceeds K, and H does not exceed L; and M exceeds N, and H does not exceed L; and M, H are equimultiples of A, E; and N, L, of B, F; wherefore A has a greater proportion to B, than E has to F. Wherefore, &c.

a 8.

b 13.

€ 10.

PRO P. XIV. THE OR.

IF F the firft has the fame proportion to the fecond, that the third has to the fourth; if the first be greater than the third, the Jecond will be greater than the fourth; but, if the first be equal to the third, the fecond will be equal to the fourth; if the first is less than the third, the fecond will be less than the fourth.

Let the first A have the fame proportion to the fecond B, that the third C has to the fourth D; if A is greater than C, then B is greater than D.

For, if A is greater than C, and B any third magnitude, A has a greater proportion to B than C has to B; but A is to B as C is to D; therefore C has to D a greater proportion than C has to Bb; therefore D is lefs than B; that is, B is greater than D. In the fame manner it is proved, that, if A is equal to

BOOK V.

~

P

PRO P. XV. THEOR.

ARTS have the fame proportion as their like multiples, if
taken correfpondently.

Let AB be the fame multiple of C that DE is of F; then C will be to F as AB is to DE.

For, let AG, GH, HB be each equal to C; and DK, KL, LE, each equal to F; then AG, GH, HB are equal to one another; and likewife DK, KL, LE equal to one another ; Ax. I. I. therefore AG is to DK as GH is to KL, and as HB is to LE b; b 11. therefore AB is to DE as AG is to DK; that is, as C to c 12. F. Wherefore, &c.

B

PRO P. XVI. THE OR.

F four magnitudes of the fame kind are proportional, they shall alfo be alternately proportional.

Let the four magnitudes A, B, C, D, be proportional, viz. as A is to B, fo is C to D; they will likewife be proportional when taken alternately, that is, as A is to C, fo is B to D; for, take E, F equimultiples of A, B, and G, H any equimultiples of C and D; then, because E is the fame multiple of A that F is of B, and G the fame multiple of C that H is of D, A is to as E is to Fa; but A is to B, as C is to D; therefore C is to a 15. Das E is to F; and as C is to D, fo is G to H; therefore b 11. E is to F as G is to Hb; therefore, fince E, F, G, H are four magnitudes proportional, equimultiples of other four, A, B, C, D; therefore, if E is equal to G, Fis equal to H; if greater, greater; and, if lefs, lefs; wherefore A is to C as B is to Dd; c 14. Wherefore, &c.

d Def. 5.

PROP. XVII. THEOR.

IF magnitudes compounded are proportional, they shall also be proportional when divided..

Let