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BOOK V.

E

PROP. VII. THEOR.

QUAL magnitudes have the fame ratio to the fame magnitude; and the fame has the fame ratio to equal magnitudes.

Let A and B be equal magnitudes, and C any other; A has the fame ratio to C that B has to C; and C has the fame ratio to A that it has to B

2

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Take MA, MB any equimultiples of A. B: and because A is equal to B, and MA, MB are equimultiples of them, MA is 24. Ax. 5. equal to MB and equal magnitudes contain b1. Ax. 5. the fame equally b; therefore MA, MB contain C the fame number of times: and MA, MC are any equimultiples of A, B; as many times therefore as any multiple of A contains C, fo many times does the fame multiple of B conC € 5.Def. 5. tain C. Therefore, as A is to C, fo is B to C. Likewife, C has the fame ratio to A that it has to B. Take NC any multiple of C greater than A: and because A is equal to B, NC contains each of them the fame number of times b; that is, any multiple of C contains A the fame number of times that it contains B; therefore C is to A, as C is to B. Wherefore, &c. Q. E. D.

PROP. VIII,

c

THEOR.

C B

F unequal magnitudes, the greater has a greater

OF ratio to the fame than the lefs has; and the

fame has a greater ratio to the lefs, than it has to the greater.

Let AB, BC be unequal magnitudes, of which AB is the greater, and let D be any magnitude whatever: AB has a greater ratio to D than BC to D: and D has a greater ratio to BC than to AB.

Take EF, FG fuch equimultiples of AC, CB, that each of them may be greater than D: and because EF is E the fame multiple of AC that FG is of CB;

21. 5. the whole EG is the fame multiple a of the F whole AB that FG is of CB: but, because EF is greater than D, EG contains D a greater number of times than FG contains it. Wherefore a multiple of AB is found, viz. EG, which contains D a greater number of times than FG,

the

C

M

the fame multiple of BC contains D; therefore AB has to D Book V. a greater ratio than BC has to D.

Alfo D has to BC a greater ratio than it has to AB. For, b7. def. 5. having made the same construction, take also MD the least multiple of D, that is not lefs than FG: therefore MD is either equal to FG, or exceeds it by a magnitude lefs than D: but EF is greater than D; therefore MD is lefs than EG: but EG, FG contain AB, BC the fame number of times exactly; therefore MD does not contain AB fo many times as it contains BC; that is, a multiple of D is taken, which contains BC a greater number of times than it contains BA: therefore D has to BC a greater ratio than it has to AB. Wherefore, &c. Q. E. D.

M

PROP. IX. THEOR.

AGNITUDES which have the fame ratio to the fame magnitude, are equal to one another; and those to which the fame magnitude has the fame ratio, are equal to one another.

M

M

25. Def. 5.

B

Let A have the fame ratio to C that B has to C; A is equal to B: For if not, let A be the greater; then, by what was shown in the preceding propofition, there is fome multiple of A which contains C a greater number of times than the fanie multiple of B does. Let MA be that multiple of A which contains C a greater number of times than MB, the fame multiple of B contains C. But, because A is to C as B is to C, and MA, MB are equimultiples of A, B; MA, MB contain C the fame number of times a; but MA contains C a greater number of times than MB does; which is impoffible; A therefore and B are not unequal; that is, they are equal. Next, let C have the fame ratio to A that it has to B; A is equal to B. For, if not, let A be the greater; therefore, as was fhewn in Prop. VIII. there is fome multiple NC of C, which contains B a greater number of times than it contains A: but because C is to B, as C is to A, and that NC is a multiple of C; NC contains B and A the fame number of times : but it contains B oftener; which is impoffible. Therefore A is equal to B. Wherefore, &c. Q. E. D.

A

PROP.

Book V.

THA

PROP. X. THEOR.

HAT magnitude which has a greater ratio than another has to the fame magnitude, is the greater of the two: and that magnitude to which the fame has a greater ratio than it has to another, is the leffer of the two.

M

M

Let A have to C a greater ratio than B has to C; A is greater than B: For, because A has a greater ratio to C, than B has to C, there is fome multiple of A which contains Ca greater number of times than the fame a7.Dcf. 5. multiple of B2: Let MA be that multiple of A which contains C a greater number of times than MB, the fame multiple of B contains C; b2.Ax. 5. therefore MA is greater than MB and A, B are the fame parts of them; therefore A is

5. Ax. 5. greater than B..

b

; B is

Next, let C have a greater ratio to B than it has to A lefs than A: For, there is fome multiple NC of C, which contains B a greater number of times than it contains A; therefore

d3. Ax. 5. B is lefs than A. Wherefore, &c. Q. E. D.

R

PROP. XI. THEOR.

ATIOS that are equal to the fame ratio, are equal to one another.

Let A be to B, as C is to D, and as C to D, fo let E be to F; A is to B, as E to F.

M

:

M

M

T

I

Take MA, MC, ME any equimultiples of A, C, E and becaufe A is to B as C to D, and MA, MC are equimultiples of A, C; MA a 5. Def. 5. contains B the fame number of times a that MC contains D: and because C is to Das E is to F, and MC, ME are equimultiples of C, E; MC contains D the fame number of times, a that ME contains F: But MC contains D the fame number of times that MA contains B; therefore MA contains B the fame number of times that ME contains F: and MA, ME are any equimultiples of A. E; as many times, therefore, as any multiple of A contains B, fo many times does the fame multiple of E contain F: therefore, as A is to B, fo is E to F. Wherefore, &c. Q. E. D.

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A B C D

EF

PROP.

PROP. XII. THEOR.

F any number of magnitudes be proportionals, as

one of the antecedents is to its confequent, fo fhall all the antecedents taken together be to all the confequents.

Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, as A is to B, so C to D, and E to F ; as A is to B, fo fhall A, C, E together be to B, D, F together.

I

M

M

Take MA, MC, ME any equimultiples of A, C, E therefore MA is the fame multiple of A, that MA, MC, ME together is of A, C, E together : And be- M caufe A is to B, as C is to D, and as E to F; and that MA, MC, ME are equimultiples of A, C, E; MA contains B as many times b as MC contains D, and ME contains F: Wherefore, as many times as MÁ contains B, fo many times fhall MA, MC, ME together A B contain B, D, F together: But MA, and MA, MC, ME together, are any equimultiples of A, and of A, C, E together; as many times, therefore, as any multiple of A contains B, fo many times does the fame multiple of A, C, E together contain B, D, F together therefore, as A is to B, fo are A, C, E together to B, D, F together. Wherefore, &c. Q. E. D.

PROP. XIII. THEOR.

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C D E F

[F the firft has to the fecond the fame ratio that the

IF

third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth; the first fhall alfo have to the fecond a greater ratio than the fifth has to the fixth.

Let A the first have the fame ratio to B the fecond, which C the third has to D the fourth, but C the third, to D the fourth, a greater ratio than E the fifth, to F the fixth: Alfo the firft A hall have to the fecond B a greater ratio than the fifth E to the fixth F.

Because C has a greater ratio to D, than E to F, there is fome multiple of C which contains D, a greater number of times a than the fame multiple of E contains F: Let MC be that multiple of C, which contains D a greater number of times than

ME,

BOOK V.

a 1. 5.

b5. Def. 5.

a7. Def. 5.

BOOK V. ME, the fame multiple of E, contains F; and whatever multiple
MC is of C, take MA the fame multiple of A: then, because
A is to B, as C to D, and MA, MC M
are equimultiples of A, C; MA con-

b

b5. Def. 5. tains B the fame nnmber of times that MC contains D: but MC contains D a greater number of times than ME contains F; therefore MA contains B a greater number of times

M

M

I

than ME contains F; and MA, ME À ǹ C D
are equimultiples of A, E; there is

therefore fome multiple of A which contains B a greater number of times than the fame multiple of E contains F: therea 7. Def. 5. fore A has a greater ratio to B 2, than E has to F. Wherefore, &c. Q. E. D.

COR. And if the firft has a greater ratio to the second, than the third has to the fourth, but the third the fame ratio to the fourth, which the fifth has to the fixth; it may be demonftrated in like manner, that the firft has a greater ratio to the fecond, than the fifth has to the fixth.

IF

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F the firft has to the fecond the fame ratio which the third has to the fourth; then, if the first be greater than the third, the fecond fhall be greater than the fourth; and if equal, equal; and if lefs, lefs.

Let the first A have to the fecond B, the fame ratio which the third C has to the fourth D; if A be greater than C, B is greater than D.

Because A is greater than C, and B is any other magnitude, 23. 5. A has to В a greater ratio than C to B a : But, as A is to B, fo is C to D; therefore alfo C has to D a greater ratio than C has to B: But of two magnitudes, that to which the fame has the greater ratio is the leffer : Wherefore D is lefs than B; that is, B is greater than D.

b 13. 5. C10. 5.

I

A B C D Secondly, If A be equal to C, B is equal to D: For A is to 9. 5. B, as C, that is A, is to D; therefore B is equal to Da.

Thirdly, If A be lefs than C, B fhall be lefs than D: For C is greater than A; and becaufe C is to D, as A to B, and C greater than A, D is greater than B, by the firft cafe; therefore B is lefs than D. Wherefore, &c. Q. E. D.

PROP.

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