PROP. A.

THEOR.

Book V.

F the first of four magnitudes has to the fecond, the see N. fame ratio which the third has to the fourth; then, if the first be greater than the fecond, the third is alfo greater than the fourth; and, if equal, equal; if lefs, lefs.

Take any equimultiples of each of them, as the doubles of each; then, by def. 5th of this book, if the double of the first be greater than the double of the fecond, the double of the third is greater than the double of the fourth; but, if the first be greater than the fecond, the double of the first is greater than the double of the fecond; wherefore alfo the double of the third is greater than the double of the fourth; therefore the third is greater than the fourth: In like manner, if the first be equal to the fecond, 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 fout magnitudes are proportionals, they are propor- see N.

tionals alío when taken inverfely.

If the magnitude A be to B, as C is to D, then alfo 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 lefs 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 fecond and fourth, E and F are equimultiples; and that G is lefs than E, H is alfo a lefs than F; that is, F is greater than H; if therefore E be greater than G, F is greater than H: In like manner, if E be equal to G, F may be shown to be equal to H; and, if lefs, lefs; and E, F are any equimultiples whatever of B and D, and G, H any whatever of A and C; therefore, as B

GABE

a

HC D F 5. def. 5.

Book V. is to A, fo is D to C. If, then, four magnitudes, &c. in

Q. E. D.

Sce N.

a 3. 5.

IF

PRO P. C. THEOR.

the first be the fame multiple of the fecond, or the fame part of it, that the third is of the fourth; the firft is to the fecond, as the third is to the fourth.

Let the firft A, be the fame multiple of B the fecond, that C the third is of the fourth D: A is to B as C is to D.

Take of A and C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H: Then, because A is the fame multiple of B that C is of D; and that E is the fame multiple of A, that F is of C; E is the fame multiple of B, that F is of D; therefore E and F are the fame 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 lefs; F is equal to H, or lefs than it. But E, F are equimultiples, any whatever, of A, C, and G, H any equimultiples whatever of B, bs, def. 5. D. Therefore A is to B, as C is to D b.

* B. 5.

Next, Let the first A be the fame part of the fecond B, that the third C is of the fourth D: A is to B, as C is to D: For B is the fame multiple of A, that D is of C; wherefore, by the preceding cafe, B is to A, as D is to C; and inverfely A is to B, as C is to D. Therefore, if the first be the fame multiple, &c. Q. E. D.

A B C D
EGF H

A B C D

PROP.

Book V.

PROP. D. THE OR.

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

Take E equal to A, and whatever multiple A or E is of B, make F the fame 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 F: But A is equal to E, therefore C is equal to Fb: And F is the fame multiple of D, that A is of B. Wherefore C is the fame multiple of D, that A is of B.

Next, Let the first A be a part of the fecond B; C the third is the fame 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 cafe, D is the fame

A B

E

C D

F

multiple of C, that is, C is the fame part of D, that A is of B: Therefore, if the firft, &c. Q. E. D.

a Cor. 4. 5.

b A. 5.

See the fi

gure at the toot of the preceding page.

c B. 5.

QUAL magnitudes have the fame ratio to the fame
magnitude; and the fame has the fame ratio to e-

E fame

qual magnitudes.

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

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

Book V.

a 1. Ax. 5.

of C any multiple whatever F: Then, becaufe D is the fame multiple of A, that E is of B, and that A is equal to B; D is equal to E: Therefore, if D be greater than F, E is greater than F; and if equal, equal; if lefs, lefs: And D, E are any equimultiples of A, B, and F is any mulb 5. def. 5. tiple of C. Therefore, as A is to C, fo is B to C.

See N.

Likewife C has the fame ratio to A, that it has to B: For, having made the fame conftruction, D may in like manner be shown equal to E: Therefore, if F be greater than D, it is likewife greater than E; and if equal, equal, if lefs, lefs: 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 magnitudes, &c. Q. E. D.

O

PROP. VIII.

D

E B

THEOR.

CF

F unequal magnitudes, the greater has a greater ratio to the fame than the lefs has; and the fame magnitude has a greater ratio to the lefs, 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 ra tio to BC than unto AB.

E

If the magnitude which is not the
greater of the two AC, CB, be not lefs F
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 lefs than D (as in Fig. 2. and 3.) this
magnitude can be multiplied, fo as to
become greater than D, whether it be
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 fame multiple of CB: Therefore
EF and FG are each of them greater than

A

of

G B

LKH D

D:

D: And in every one of the cafes, take H the double of D, K Book V. its triple, and fo 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 lefs than L.

E

F

Fig. 2.

Fig. 3.

E

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 lefs than K: And fince EF is the fame multiple of AC, that FG is of CB; FG is the fame multiple of CB, that EG is of AB; wherefore EG and a 1. 5. FG are equimultiples of AB and CB: And it was shown, that FG was not less than K, and, by the conftruction, 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 than L; but FG is not greater than L; and EG, FG are equimultiples of AB, BC, and is a multiple of D; therefore AB has b to D a greater ratio than BC has to D.

Alfo D has to BC a greater ratio than it has to AB: For, having made the fame conftruction, it may be fhown, in like manner, that is greater than

A

F+

A

of

GB

b 7. def. 5.

LK D

G B
LKHD

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.