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PRO P. VII. THE O R.

Book V.

QUAL magnitudes have the same proportion to the same
magnitude, and

one and the same magnitude has the same proportion to equal magnitudes.

E

Let A, B, be two equal magnitudes, and C any third magnitude, then A, B, will have the same 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 likewise equal to F; if greater, greater; and, if less, less; therefore A is to C as B is to C. Again, C is to A as C is to B; for, the same construction re. a def. S. maining, if F is equal to D, it is likewise equal to E ; wherefore C is to B as C is to A. Wherefore, &c.

PRO P. VIII. THE O R.

THE greater of two unequal magnitudes has a greater propor,

tion to some third magnitude than the less has, and that third magnitude has a greater proportion to the leser 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 C has to D; but D has a greater proportion to C than it has to AB.

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 same multiple of EB that FG is of AE ; then FH will be the same multiple of AB that GH is of EB?; and make a r. K the same multiple of C that GH is of EB; but EB is equal to C; therefore K is equal to GHb; wherefore FH is the same

b Ax re 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 ler be N; and let M be the next multiple of D, less than N; then M will not be greater than K, that is, K will not be less than M; but M and D are equal to N; and because 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, the multiple of D; therefore AB has to D a greater proportion Def. 7. than C has to DC; but likewise D has to Ca greater proportion than it has to AB; for, the same construction remaining, N, the multipie of D, exceeds K, the multiple of C; but does not exceed FH, the multiple of AB. Wherefore, &c.

PROP.

Book. V.

PRO P. IX. THE OR.

AGNITUDES which have the same proportion to one

and the same magnitude are equal to one another; and if a magnitude has the same proportion to other magnitudes, these magnitudes are equal to one another.

a 8.)

Let the magnitudes A, B, have the same 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 less, then B has a greater proportion to C than it has to C; but it has not“; therefore A is not less than B ; and, since neither greater nor less, it must be equal.

Again, if C have the same proportion to A that it has to B, A is equal to B; if not, C will have a greater proportion to the lefser 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.

PRO P. X. THE O R.

F magnitudes having proportion to the same magnitude,

that which has the greater proportion to some third is the greater magnitude ; and that magnitude to which the same has a greater proportion is the lefser magnitude.

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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 less. If A is equal to B, then they would have the same proportion to Ca; but they have not; therefore A is not equal to B; neither is it,less; for then B would have a greater proportion to C than it has to Cb; but it has not: Therefore, since 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 less than A; if not, let it be equal or greater ; if equal, then C has the fame proportion to B that it has to AC 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, since B is not equal or greater than A, it must be less. Wherefore, &c.

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Book V.

PRO P. XI. THEOR.

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

Let A be to B, as C is to D, and C to Das 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 Mo; if greater, greater; and, if less, less : Like- a Def. 5. wise, because C is to D as E is to F; if H be equal to M, K will be equal to N°; if greater, greater; and, if less, less. Wherefore, if G be equal to L, K will be equal to N; for they are e. qual to H, Mó; wherefore A is to B as E to F? Where. b Ax. 1. 1. fore, &c.

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PRO P. XII. T H E O R.

F any number of magnitudes be proportional, as one of the an

tecedents is to one of the consequents, so are all the antecedents to all the consequents.

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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, so are A, C, E, all the antecedents, to B, D, F, all the consequents. 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, so greater, greater ; and, if less, less ; wherefore, if G be equal to L; G, H, K, will be equal to L, M, N; if greater, greater ; and, if less, less b; but G; G, H, K, are equimultiples of A; A, b . C, E, together; and L; L, M, N, equimultiples of B; B, D, F, together ; such, that, if G be equal to L; G, H, K will be equal to L, M, N, together ; wherefore

A, C, E, together, are 10 B, D, F, together, as A is to Bo Wherefore, &c.

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PROP,

Book V.

PRO P. XIII. THEO R.

TF the first is in the same proportion to the second, as the third

to the fourth; and if the third has a greater proportion to the fourth, than the fifth to the sixth ; then shall also the first have a greater proportion to the second, than the fifth to the sixth.

7.

Let the first A have the same 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 sixth F; then the first 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; such, that, if Ġ exceeds K, but H does not exceed L"; and let M be the same 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 Nb; 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.

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PRO P. XIV. THEO R.

IF the first has the same proportion to the second, that the third

has to the fourth; if the first be greater than the third, the second will be greater than the fourth ; but, if the first be equal to the third, the second will be equal to the fourth; if the first is less than the third, the second will be less than the fourth.

a 8.

Let the first A have the same proportion to the second 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 Bd; therefore D is less than Bo; that is, B is greater than D. In the same manner it is proved, that, if A is equal to

b 13

€ IO.

Book V.

PRO P. XV. THEOR.

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ARTS have the same proportion as their like multiples, if taken correspondently.

Let AB be the same 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 a. nother ; and likewise DK, KL, LE equal to one another ; a Ax. 1, 5. therefore AG is to DK as GH is to KL, and as HB is to LEb; b 11. therefore AB is to DE as AG is to DK °; that is, as C to c 12. F. Wherefore, &c.

PRO P. XVI. THE O R.

F four magnitudes of the same kind are proportional, theyßball

they also be alternately proportional.

Let the four magnitudes A, B, C, D, be proportional, viz. as A is to B, so is C to D; they will likewise be proportional when taken alternately; that is, as A is to C, fo is B to D; for, take E, F equimultiples 6f A, B; and G, H any equimultiples of C and D; then, because E is the same multiple of A that F is of B, and G the same multiple of C that H is of D, A is to B 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 Fb; and as C is to D, fo is G to Hà; therefore b 11. E is to F as G is to Hb; therefore, since E, F, G, H are four magnitudes proportional, equimultiples of other four, A, B, C, D; therefore, if E is equal to G, F is equal to H; if greater, greater; and, if less, less; wherefore A is to C as B is to D d; c 14. Wherefore, &c.

d Def. s.

PROP. XVII. THEO R.

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

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