Sidebilder
PDF
ePub

PROP. A. THEOR.

Book V.

IF 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 second, the third is alfo 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 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 lefs than it. Therefore if the firft, &c. Q. E. D.

IF

PROP. B. THEOR.

F four magnitudes are proportionals, they are propor- See N. tionals alfo 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 second and fourth, E and F are equimultiples; and that G is lefs than E, H is alfo lefs than F; that is, Fis 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 fhewn to be equal to H; and if lefs, lefs. and E, F are any equimultiples whatever of B and D, and G, H

GABE

HC D F

a. 5. Def. 5.

any

b.

Book V. any whatever of A and C. Therefore as B is to A, fo is D to C. If then four magnitudes, &c. Q. E. D.

See N.

IF

PROP. C. THEOR.

F 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 first 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 a. 3. s. multiple of B, that F is of D*; therefore E and

5.

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 inanner, 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, D. Therefore A is to B, Def. 5. as C is to Db.

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 preceeding cafe B is to c. B. 5. 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.

C

A B C D
E GFH

A B C D

PROP.

PROP. D. THEOR.

Book V.

IF the first be to the second 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 bc 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 fecond 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 F. 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 second 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 preceeding case, D is the fame multiple of

A B

C D

E

F

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

PR Q P. VII. THEOR.

EQUAL magnitudes have the fame ratio to the same

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

Let A and B be equal magnitudes, and C any other. A and B have 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 of

a. Cor. 4. S b. A. s.

See the Fi

gure at the foot of the preceeding

page.

c. B. 5.

C any

Book V. C any multiple whatever F. then because D is the fame multiple

of A, that E is of B, and that A is equal to

a. 1. Ax. 5. B ; D is equal to E. therefore if D be greater

[blocks in formation]

multiples of A, B,

and F is any multiple of

5. Def. 5. C. Therefore bas A is to C, fo is B to C.

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 fhewn 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 equimultiples whatever of A, B. Therefore C is to A, as C is to Bb. Therefore equal magnitudes, &c. Q. E. D.

DA
E B

Sce N.

[blocks in formation]

C F

OF

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.

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 lefs
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.

E

F

A

G B
L

KH D

therefore

therefore 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 fo on, till the multiple of D be that which first becomes greater than FG. let L be that multiple of D which is firft greater than FG, and K the multiple of D which is next lefs than L.

Then because L is the multiple of D which is the first that becomes greater than FG, the next preceeding 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 FG are equimultiples of AB a. 1. 5. and CB. and it was shewn

E

that FG was not less than
K, and, by the conftructi- F
on, EF is greater than D;

E

[blocks in formation]

the fame construction, it may be fhewn, 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 than it has to AB. Wherefore of unequal magnitudes, &c. Q. E. D.

PROP.

« ForrigeFortsett »