Page images
PDF
EPUB

IF

PROP. D. THEOR.

Book V.

F 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 second; 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 same 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 Fa. 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 same 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, Dis the fame

ABCD
E

F

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

[blocks in formation]

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 same 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. 5. b. A. 5.

See the Fi

gure at the

foot of the

preceding

page.

с. В. 5.

Book V. Cany 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. A 5. B; Disa 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 multiple of C.

b. 5. Def. 5. Therefore b as A is to C, so is B to C.
Likewife C has the fame ratio to A that
it has to B. for, having made the same con-
DA
struction, D may in like manner be shewn E B
equal to E. therefore if F be greater than D,
it is likewife greater than E; and if equal,
equal; if less, less. and Fis any multiple what-
ever 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.

See N.

PROP. VIII. THEOR.

CF

F 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.

Let AB, BC be unequal magnitudes of which AB is the greater,

and let D be any magnitude whatever. AB E
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 great-
er 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 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.

F

A

C

GB

LKHD.

therefore EF and FG are each of them greater than D. and in Book V. every one of the cafes 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.

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 fince EF is the fame multiple of AC, that FG is of CB; FG is the fame multiple of CB, that EG is of AB2; wherefore EG and FG are equi- a. 1. 5.

[blocks in formation]

the same construction, it may be shewn, 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.

Book V.

See N.

a. 5. Def. 5.

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.

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 thewn in the preceding Propofition, there are some equimultiples of A and B, and fome 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 equimultiples 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 F;
E shall also be greater than Fa; but E is
not greater than F, which is impossible.
A therefore and B are not unequal; that
is, they are equal.

Next, Let C have the fame ratio to

A

D

F

[ocr errors]

E

each of the magnitudes A and B; A is B
equal to B. for if they are not, one of
them is greater than the other; let A be
the greater, therefore, as was shewn in
Prop. 8th, there is some multiple F of C, and fome equimultiples
E and D of B and A such, that F is greater than E, and not great-
er 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
fourth a. but F is not greater than D, which is impossible. There-
fore A is equal to B. Wherefore magnitudes which, &c. Q. E. D.

PROP. X. THEOR.

Book V.

T

CHAT magnitude which has a greater ratio than See N. another has unto the fame magnitude is the greater of the two. and that magnitude to which the fame has a greater ratio than it has unto another magnitude is the lesser of the two.

and D is greater
greater than B.

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 a some equimultiples of A and B, and some multiple of a. 7. Def. 5. C fuch, 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 because D and E are equi-
multiples of A and B,
than E; therefore A is

A A

D

C

F

b. 4. Ax. 5.

[blocks in formation]

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

R

PROP. XI. THEOR.

ATIOS that are the fame to the fame ratio, are
the fame 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 fince A is to B, as C to D, and of A, C are taken equimultiples G,

« PreviousContinue »