Book V.

PROP. D. THEOR.

IF the first be to the second as the third to the See N. fourth, and if the first be a multiple or part of the second; the third is the same multiple, or the same part of the fourth.

Let A be to B, as C is to D; and first let A be a multiple of B C is the same multiple of D.

Take E equal to A, and whatever multiple A or E is of B, make F the same multiple of D: Then, because A is to B, as C is to D; and of B the second, and D the fourth, equimultiples have been taken E and F; A is to E, as C to Fa: But A is equal to E, therefore C is equal to Fb: and F is the same multiple of D, A that A is of B. Wherefore C is the same multiple of D, that A is of B.

Next, Let the first A be a part of the second B; C the third is the same part of the fourth D.

Because A is to B, as C is to D; then inversely, B is to A, as D to C: But A is a part of B, therefore B is a multiple of A; and, by the preceding case, D is the

a Cor. 4. 5.

B C D'A. 5.

E

F

same multiple of C; that is, C is the same part of D, that A is of B: Therefore, if the first, &c. Q. E.D.

See the fi

gure at the foot of the preceding page. c B. 5.

PROP. VII. THEOR.

EQUAL magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes.

Let A and B be equal magnitudes, and C any other. A and B have each of them the same ratio to C, and C has the same ratio to each of the magnitudes A and B.

Take of A and B any equimultiples whatever D and E, and of C any multiple whatever F: Then because D is the

2

Book V. same multiple of A, that E is of B, and
that A is equal to B; D is equal to E:
* 1 Ax. 5. Therefore, if D be greater than F, E is
greater than F; and if equal, equal; if
less, less: And D, E are any equimultiples
of A, B, and F is any multiple of C.
5 Def. 5. Therefore, as A is to C, so is B to C.

Likewise C has the same ratio to A, D
that it has to B: For, having made the
same construction, D may in like manner E
be shown equal to E: Therefore, if F be
greater than D, it is likewise greater than
E; and if equal, equal; if less, less: And
F is any multiple whatever of C, and D, E,
are any equimultiples whatever of A, B.
Therefore, C is to A as C is to Bb. There-
fore, equal magnitudes, &c. Q. E. D.

PROP. VIII. THEOR.

See N. OF unequal magnitudes the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less, than it has to the greater.

Fig. 1.

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 unto AB.

If the magnitude which is not the
greater of the two AC, CB, be not less
than D, take EF, FG, the doubles of
AC, CB, as in Fig. 1. But if that which
is not the greater of the two AC, CB,
be less than D (as in Fig. 2. and 3.) this G
magnitude can be multiplied, so as to
become greater than D, whether it be Ļ
AC or CB. Let it be multiplied, until
it become greater than D, and let the
other be multiplied as often; and let EF
be the multiple thus taken of AC, and
FG the same multiple of CB: There-

C

A

B

1

KH D

fore EF and FG are each of them greater than D: and in Book V. every one of the cases, take H the double of D, K its triple, and so on, till the multiple of D be that which first becomes greater than FG: Let L be that multiple of D which is first greater than FG, and K the multiple of D which is next less than L.

Fig. 2.

Fig. 3.
ΕΙ

2

Then, because L is the multiple of D, which is the first that becomes greater than FG, the next preceding multiple K is not greater than FG; that is, FG is not less than K: And since EF is the same multiple of AC, that FG is of CB; FG is the same multiple of CB, that EG is of AB; wherefore EG and FG are equimultiples of AB and × 1. 5. CB And it was shown, that FG was not less than K, and, by the construction, EF is greater than D; therefore the whole EG is greater than K and D together: But K, together with D, is equal to L; therefore EG is greater

A

F

A

than L; but FG is not

greater than L; and

C

EG, FG are equi- L KH D

has to D a greater ratio than BC has to D. Also D has to BC a greater ratio than it has to AB: For, having made the

same construction, it may be shown, in like manner, that
L is greater than FG, but that it is not greater than EG: and
L is a multiple of D; and FG, EG are equimultiples of CB,
AB; therefore D has to CB a greater ratiob than it has to
AB. Wherefore, of unequal magnitudes, &c. Q. E.D.

Book V.

PROP. IX. THEOR.

See N. MAGNITUDES which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another.

Let A, B have each of them the same ratio to C; A is equal to B. For, if they are not equal, one of them is greater than the other: Let A be the greater; then, by what was shown in the preceding proposition, there are some equimultiples of A and B, and some multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C. Let such multiples be taken, and let D, E, be the equimul-' tiples of A, B, and F the multiple of C, so that D may be greater than F, and E not greater than F: But, because A is to C as B is to C, and of A, B, are taken equimultiples D, E, and of C is taken a multiple F; and that D is greater than a5 Def. 5. F; E shall also be greater than Fa; but E is not greater than F; which is im- A possible; A therefore and B are not unequal; that is, they are equal.

B

E

F

Next, let C have the same ratio to each of the magnitudes A and B; A is equal to B: For, if they are not, one of them is greater than the other; let A be the greater; therefore, as was shown in Prop. 8th, there is some multiple F of C, and some equimultiples E and D, of B and A such, that F is greater than E, and not greater than D; but because C is to B, as C is to A, and that F, the multiple of the first, is greater than E, the multiple of the second; F the multiple of the third, is greater than D, the multiple of the fourtha: But F is not greater than D, which is impossible. Therefore A is equal to B. Wherefore, magnitudes which, &c. Q.E. D.

Book V.

PROP. X. THEOR.

THAT magnitude which has a greater ratio than See N. another has unto the same magnitude, is the greater of the two: And that magnitude to which the same has a greater ratio than it has unto another magnitude, is the lesser of the two.

a

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 are some equimultiples of A and Def. 5. B, and some multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than it: Let them be taken, and let D, E be equimultiples of A, B, and F a multiple of C such, that D is greater than F, but E is not greater than F: Therefore D is greater than E: And, A because D and E are equimultiples of A and B, and D is greater than E; therefore A is b greater than B.

Next, let C have a greater ratio to B than it has to A; B is less than A: Fora B there is some multiple F of C, and some equimultiples E and D of B and A such, that F is greater than E, but is not great

C

D

E

b

F

b 4 Ax. 5.

er than D: E therefore is less than D; and because E and D are equimultiples of B and A, therefore B is less than A. The magnitude, therefore, &c. Q.E. D.

PROP. XI. THEOR.

RATIOS that are the same to the same ratio, are the same to one another.

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

Take of A, C, E, any equimultiples whatever G, H, K; and of B, D, F, any equimultiples whatever L, M, N. Therefore, since A is to B, as C to D, and G, H are taken equimultiples of A, C, and L, M, of B, D; if G be greater