PROP. X.

If two or more magnitudes have given ratios to one another, and if they have given ratios, though they be not the same, to some other magnitudes: these other magnitudes shall also have given ratios to one another.

Let two or more magnitudes A, B, C, have given ratios to one another; and let them have given ratios, though they be not the same, to some other magnitudes D, E, F : The magnitudes D, E, F, have given ratios to one another. Because the ratio of A to B is given, and likewise the ratio of A to D; therefore the ratio of D to B A is given; but the ratio of B to E is given; thereforea the ratio of D to E is given: And because

B

C

D

9.

a 9 Dat.

E

F

the ratio of B to C is given, and also the ratio of B to E; the ratio of E to C is given: And the ratio of C to Fis given; wherefore the ratio of E to F is given; D, E, F, have therefore given ratios to one another.

PROP. XI.

IF two magnitudes have each of them a given ratio to another magnitude, both of them together shall have a given ratio to that other.

Let the magnitudes AB, BC, have a given ratio to the magnitude D, AC has a given ratio to the same D.

Because AB, BC, have each of

22.

them a given ratio to D, the ratio A

B

C

of AB to BC is givena: And by

a 9 Dat.

composition, the ratio of AC to D

CB is given: But the ratio of

b 7 Dat.

BC to D is given: therefore the ratio of AC to D is given.

See N. IF the whole have to the whole a given ratio, and the parts have to the parts given, but not the same, ratios: every one of them, whole or part, shall have to every one a given ratio.

Let the whole AB have a given ratio to the whole CD, and the parts AE, EB, have given, but not the same, ratios to the parts CF, FD: every one shall have to every one, whole or part, a given ratio.

Because the ratio of AE to CF is given; as AE to CF, so make AB to CG; the ratio therefore of AB to CG is given wherefore the ratio of the remainder EB to the re19. 5. mainder FG is given, because it is the same with the ratio

of AB to CG: And the ratio of
EB to FD is given, wherefore A

9 Dat. the ratio of FD to FG is given";
and, by conversion, the ratio of

c 6 Dat. FD to DG is given: And be

E

B

GD

cause AB has to each of the magnitudes CD, CG, a given ratio, the ratio of CD to CG is given b, and therefore the ratio of CD to DG is given: But the ratio of GD to DF is given, wherefore the ratio of CD to DF is given, and Cor. 6. consequently the ratio of CF to FD is given; but the ratio of CF to AE is given, as also the ratio of FD to EB; 10 Dat. wherefore the ratio of AE to EB is given; as also the raf7 Dat. tio of AB to each of themf. The ratio therefore of every one to every one is given.

dat.

See N. IF the first of three proportional straight lines has a given ratio to the third, the first shall also have a given ratio to the second.

Let A, B, C, be three proportional straight lines, that is, as A to B, so is B to C; if A has to C a given ratio, A shall also have to B a given ratio.

Because the ratio of A to C is given, a ratio which is the • 2 Def. same with it may be founda: let this be the ratio of the 13. 6. given straight lines D, E; and between D and E find a

mean proportional F; therefore the rectangle contained by
D and E is equal to the square of F, and the
rectangle D, E, is given, because its sides D,
E, are given; wherefore the square of F, and
the straight line F is given: And because, as
A is to C, so is D to E; but as A to C, so
is the square of A to the square of B; and
as D to E, so is the square of D to the
square of F: Therefore the squared of A is to
the square of B, as the square of D to the
square of F: As therefore e the straight line D
A to the straight line B, so is the straight line
D to the straight line F; therefore the ratio
of A to B is given a, because the ratio of the
given straight lines D, F, which is the same
with it, has been found.

PROP. XIV.

F

E

• 2 Cor. 20. 6.

d 11. 5.

€ 22.6.

a 2 Def.

A.

Ir a magnitude, together with a given magnitude, See N. has a given ratio to another magnitude; the excess of this other magnitude above a given magnitude, has a given ratio to the first magnitude: And if the excess of a magnitude above a given magnitude has a given ratio to another magnitude; this other magnitude, together with a given magnitude, has a given ratio to the first magnitude.

Let the magnitude AB, together with the given magnitude BE, that is, AE, have a given ratio to the magnitude CD: the excess of CD above a given magnitude has a given ratio to AB.

Because the ratio of AE to CD is given; as AE to CD, so make BE to FD; therefore the ratio of BE to FD is given, and BE is given; wherefore FD is given: And because as AE to CD, so is BE to FD, the remainder AB is to the remainder CF, as AE to CD: But the ratio

B E

a 2 Dat.

F D

b 19.5.

of AE to CD is given; therefore the ratio of AB to CF is given; that is, CF, the excess of CD above the given magnitude FD, has a given ratio to AB.

Next, Let the excess of the magnitude AB above the given magnitude BE, that is, let AE, have a given ratio to

the magnitude CD; CD together with a given magnitude has a given ratio to AB.

Because the ratio of AE to CD is given; as AE to CD so make BE to FD; therefore the ra

tio of BE to FD is given, and BE is A

* 2 Dat. given; wherefore FD is given a: And because, as AE to CD, so is BE to

12. 5. FD, AB is to CF, as AE to CD :

B.

C

ER

D F

But the ratio of AE to CD is given, therefore the ratio of AB to CF is given: that is, CF, which is equal to CD together with the given magnitude DF, has a given ratio to AB.

PROP. XV.

See N. IF a magnitude, together with that to which another magnitude has a given ratio, be given; the sum of this other, and that to which the first magnitude has a given ratio, is given.

Let AB, CD, be two magnitudes, of which AB, together with BE, to which CD has a given ratio, is given; CD is given, together with that magnitude to which AB has a given ratio.

A

BE

Because the ratio of CD to BE is given; as BE to CD, so make AE to FD; therefore the ratio of AE to FD is given, • 2 Dat. and AE is given, wherefore a FD is given: And because as BE to Cor. 19. 5. CD, so is AE to FD: AB is to FC, as BE to CD: And the ratio F of BE to CD is given: wherefore the ratio of AB to FC is given: And FD is given, that is, CD together with FC, to which AB, has a given ratio, is given.

C D

See N. IF the excess of a magnitude above a given magnitude has a given ratio to another magnitude; the excess of both together above a given magnitude shall have to that other a given ratio: And if the excess of two magnitudes together above a given magnitude, has to one of them a given ratio; either the excess of the other above a given magnitude has to that one a given ratio; or the other is given together with the magnitude to which that one has a given ratio.

1

Let the excess of the magnitude AB above a given magnitude, have a given ratio to the magnitude BC; the excess of AC, both of them together, above the given magnitude, has a given ratio to BC.

Let AD be the given magnitude, the excess of AB above which, viz. DB has a given

ratio to BC: And because

A

D B

C

DB has a given ratio to BC, the ratio to DC to CB is givena, 7 Dat. and AD is given; therefore DC, the excess of AC above

the given magnitude AD, has a given ratio to BC.

Next, Let the excess of two magnitudes AB, BC, together, above a given magni

A D BEC

tude, have to one of them
BC a given ratio; either the excess of the other of them
AB above the given magnitude shall have to BC a given
ratio; or AB is given, together with the magnitude to
which BC has a given ratio.

Let AD be the given magnitude, and first let it be less than AB; and because DC the excess of AC above AD has a given ratio to BC, DB has a given ratio to BC; that is, Cor. 6DB the excess of AB above the given magnitude AD has a given ratio to BC.

dat.

But let the given magnitude be greater than AB, and make AE equal to it; and because EC, the excess of AC above AE has to BC a given ratio, BC has a given ratio to 6 Dat. BE; and because AE is given, AB together with BE to which BC has a given ratio is given.

If the excess of a magnitude above a given magni- See N. tude has a given ratio to another magnitude; the excess of the same first magnitude above a given magnitude, shall have a given ratio to both the magnitudes together. And if the excess of either of two magnitudes above a given magnitude has a given ratio to both magnitudes together; the excess of the same above a given magnitude shall have a given ratio to the other.

Let the excess of the magnitude AB above a given magnitude have a given ratio to the magnitude BC; the excess of AB above a given magnitude has a given ratio to AC.