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Cor. From this it follows, that the parts AC, CB, have a given ratio to one another: Because as AB to BC, so is DE to EF; by division", AC is to CB, as DF to FE; and DF, " 17.5. FE, are given; therefore a the ratio of AC to CB is given. - 2 Def.

6.

PROP. VII. If two magnitudes which have a given ratio to one See N. another be added together ; the whole magnitude shall have to each of them a given ratio.

Let the magnitudes AB, BC, which have a given ratio to one another, be added together: the whole AC bas to each of the magnitudes AB, BC, a given ratio. Because the ratio of AB to BC is given, a ratio

may

be found a which is the same with it; let this be the ratio of a 2 Det. the given magnitudes DE, EF:

С And because DE, EF, are given,

B the whole DF is given b: And

b 3 Dat.

1 because as AB to BC, so is DE to EF; by composition AC is to CB as DF to FE; and, o 18. 5. by conversion", AC is to AB, as DF to DE: Wherefore : E. 5. because AC is to each of the magnitudes AB, BC, as DF to each of the others DE, EF; the ratio of AC to each of the magnitudes AB, BC, is given a.

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PROP. VIII. Ir a given magnitude be divided into two parts See N. which have a given ratio to one another, and if a fourth proportional can be found to the sum of the two inagnitudes by which the given ratio is exhibited, one of them, and the given magnitude; each of the parts is given.

Let the given maguitude AB be divided into the parts AC, CB, which have a given ratio to one another; if a fourth pro

B portional can be found to the above-named magnitudes; AC D and CB are each of them given.

Because the ratio of AC to CB is given, the ratio of AB to BC is givena, therefore a ratio, which is the same with a 7 Dat.

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ç B

• 2 Def. it can be found b; let this be the ratio of the given magni

tudes, DE, EF: And because
the given magnitude AB has to A
BC the given ratio of DE to
EF, if unto DE, EF, AB, a
fourth proportional can be

F E € ? Dat. found, this which is BC is giveno; and because AB is 4 Dat. given, the other part AC is givend.

In the same manner, and with the like limitation, if the difference AC of two magnitudes AB, BC, which have a given ratio be given; each of the magnitudes AB, BC, is given.

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MAGNITUDES which have given ratios to the same magnitude, have also a given ratio to one another.

Let A, C, have each of them a given ratio to B; A has a given ratio to C.

Because the ratio of A to B is given, a ratio which is the 2 Def. same to it may be founda; let this be the ratio of the given

magnitudes D), E: And because the ratio of B to C is given, a ratio which is the same with it may

be found a : let this be the ratio of the given magnitudes F, G: To F, G, E, find a fourth proportional H, if it can be done ; and because as A is to B, so is D to E; and as B to C, so is (F to G, and so is) E to H; ex æquali

, as A to C, so is A B C D E H D to H: Therefore the ratio of A to C is givena, because the ra

F G tio of the given magnitudes D and H, which is the same with it, has been found : But if a fourth proportional to F, G, E, cannot be found, then it can only be said that the ratio of A to C is compounded of the ratios of A to B, and B to C, that is, of the given ratios of D to E, and F to G.

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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.-

D is given a; but the ratio

B В of B to E is given; there

E forea the ratio of D to E

C

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

a 9 Dat.

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

Because AB, BC, have each of them a given ratio to D, the ratio A

C of AB to BC is given a : And by

1 9 Dat composition, the ratio of AC to D CB is given : But the ratio of BC to D is given : therefore the ratio of AC to Dis given.

B

7 Dat.

+

23.

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PROP. XII.
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 a with the ratio

of AB to CG: And the ratio of

EB to FD is given, wherefore E B 9 Dat the ratio of FD to FG is given"; and, by conversion, the ratio of C

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

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 b the ratio of CD to DF is given, and Cor. 6. consequently d the ratio of CF to FD is given ; but the ra

tio of CF to AE is given, as also the ratio of FD to EB; e 10 Dat. wherefore e the ratio of AE to EB is given; as also the raf7 Dat. tio of AB to each of them f. The ratio therefore of every

one to every one is given.

*

dat.

24.

PROP. XIII.
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 strafght 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 tlie 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
isc the
square of A to the square of B; and

« ? Cor. as D to E,'so is the square of D to the

A B C 20.6. 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 Dr E 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

a 2 Def. given straight lines D, F, which is the same with it, has been found.

d 11. 5.

e 22. 6.

A

PROP. XIV. If 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 a : And because as AE A B_E to CD, so is BE to FD, the remainder AB is b to the remainder

C FD CF, as AE to CD: But the ratio 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

E a 2 Dat.

19. 5.

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