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Book V,

PROP. A. THEOR.

If the first of four magnitudes have to the second Sec Note. the same ratio which the third has to the fourth; then, if the first be greater then the second, the third is also greater than the fourth ; and if. equal, equal; if less, less.

Take any equimultiples of each of them, as the doubles of each; then, by def. 5th of this book, if ile double of the first be greater than the double of the second, the double of the third is greater then the double of the fourth; but, if the first be greater then the second, the double of the first is greater then the double of the second; wherefore also the double of the third is greater then the double of the fourth ; therefore the third is greater than the fourth : in like manner, if the first be equal to the second, or less than it, the third can be proved to be equal to the fourth, or less than it. Therefore, if the first, &c. Q. E. D.

PROP. B. THEOR.

IF four magnitudes be proportionals, they are pro- See Note portionals also when taken inversely.

If the magnitude A be to B, as C is to D, then also inversely B is to A, as D to C.

Take of B and D any equimultiples whatever E and F; and of A and C any equimultiples whatever G and H. First, Let E be greater than G, then G is less than E; and, because A is to B, as C is to D, and of A and C, the first and third, G, and H are equimultiples; and of B and D, the second and fourth, E and Fare equimultiples; and that G is less than E, H is also G A B E a less than E ; that is, F is greater than H C D F a 5. def. 5. H; if therefnre E be greater than G, F is greater than H: in like manner, if Ete equal to G, F may be shown to be equal to H; and, if less, less; and E, F are any equimultiples whatever of B and-D, and G, H any whatever of A and C; therefore, as B is.

R

Book v. to A, so is D to C. If, then, four magnitudes, &c. Q. E. D.

PROP. C. THEOR.

See Note.

If the first be the same multiple of the second, or the same part of it, that the third is of the fourth ; the first is to the second, as the third is to the fourth.

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Let the first A be the same multiple of B the second, that the third is of the fourth D: A is to B as C is to D.

Take of A and C any equimultiples what ever E and F; and of B and D any equi. multiples whatever G and H: then, because A is the same multiple of B that C is of D; A B C D and that E is the same multiple of A, that E G F H.

F is of C; E is the same multiple of B, that 3. 5.

F is of D R; therefore E and F are the same multiples of B and D: but G and H are equimultiples of B and D; therefore, if E be a greater multiple of B, than G is, F is a greater multiple of D, than H is of D; that is, if E be greater than G, F is greater than H: in like manner, if E be equal to G, or less ; F is equal to H, or less than it. But E, F are any equimuhiples whatever, of A, C, and G, Hariy equimultiples whatever of B, D.

Therefore A is to B, as C is to D b. b 5. def. 5.

Next, Let the first A be the same part of the second B, that the third C is of the fourth D: A is to B, as C is to D: for B is the same multiple of A, that D is of C: wherefore, by the preceding case, B is to A, as D is to c; and in

versely c A is to B, as C is to D. ThereCB 5.

fore, if the first be the same multiple, &c.
Q. E. D.

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Book v.

PROP. D. THEOR.

If the first be to the second as the third to the See Note. fourth, and if the first be a multiple, or part of the se. cond; 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 mula aiple 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

a Cor. 4. 5. equal to E, therefore Cis equal to Fb: and

b A. 5. F is the same multiple of D, that A is of A B с B. Wherefore C is the same multiple of D, that A is of B.

E

F Next, Let the first A be a part of the

See the fisecond B; C the third is the same part of

gure at the

foot of the the fourth D.

preceding Because A is to B, as C is to D; then,

page. inversely, B is c to A, as D to C: but A is

c B. 5. a part of B, therefore B is a multiple of A; and, by the preceding case, D is thə 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.)

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 fatio to each of the magnitudes A and B.

Take of A and B any equimultiples whatever Dand E, and

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Book V. of C any multiple whatever F: 'then, because D is the same

Tempe multiple of A, that E is of B, and that A is a 1. Ax. 5. equal to B; D is a equal to E : 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 b 5. def. 5. multiple of C. Thereforeb, as A is to C, so

is B to C.
Likewise C has the same ratio to A, that

D
it has to B : for, having made the same con-
struction, D may in like manner be shown

E B
· 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 equimul-
tiples whatever of A, B. Therefore C is to
A, as C is to Bb. Therefore equal magni-
tudes, &c. Q. E. D.

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PROP. VIII. THEOR.

See note,

OF unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same mag. nitude bas 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 E to BC than to AB.

If the magnitude which is not the greater of the two AC, CB, be not less

A
than D, take EF, FG, the doubles of

F
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
magnitude can be multiplied, so as to
bccome greater than D, whether it be. G B
AC, or CB. Let it be multiplied, until

L
it become greater than D, and let the

K HD other be multiplied as often ;, and let EF be the multiple thus taken of AC, and FG the same multiple of CB : therefore EF and FG are each of them greater than

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Fig. 2.

.

D.: and in every one of the cases, take H the double of D, K, Book V. its triple, and so on, till the multiple of D be that which first

pe 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:

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

a 1. 5. is the same multiple of CB, that EG is of AB a; wherefore EG and FG are equimultiples of AB and CB: and it was shown, that FG was

Fig. 3. not less than K, and, by the construction,

E EF is greater than D;

E therefore the whole EG is greater than K F.

A and D together: but, K together with D, is

A equal to L; therefore EG is greater than L;

c

F but' FG is not greater than L; and EG, FG

C are equimultiples of AB, BC, and L is a G B

G B multiple of Ds there

L

L foreb AB has to D a

K. H D

K Db 7. def. 5. 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 struction, 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 ratio b than it has to AB. Wherefore, of unequal magnitudes, &c. Q. E. D.

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