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12.

PROP. V. See n. If of three magnitudes, the first together with the

second be given, and also the second together with the third ; either the first is equal to the third, or one of their is greater than the other by a given magnitude.

Let AB, BC, CD, be three magnitudes, of which AB together with: BC, that is, AC, is given ; and also BC together with CD, that is, BD, is given. Either AB is equal to CD, or one of them is greater than the other by a given magnitude.

Because AC, BD, are each of them given, they are either equal to one another, or not A B

C D equal. First, let them be equal, and because AC is equal to BD, take away the common part BC; therefore the remainder AB is equal to the remainder CD.

But if they be unequal, let AC be greater than BD, and make CE equal to BD. Therefore CE is given, because

BD is given. And the whole A E B C D. * 4 Dat. AC is given ; therefore a AE =

the remainder is given. And because EC is equal to BD, by taking BC from both, the remainder EB is equal to the remainder CD. And AE is given ; wherefore AB exceeds EB, that is, CD, by the given magnitude AE.

5.

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PROP. VI. See N. IF a magnitude has a given ratio to a part of it, it

shall also have a given ratio to the remaining part of it.

Let the magnitude AB have a given ratio to AC a part of it; it has also a given ratio to ihe remainder BC.

Because the ratio of AB to AC is given, a ratio may be * 2 Def. founda which is the same to it: Let this be the ratio of • DE, a given magnitude to the given. A

magnitude DF. And because DE, O 4 Dat. DF, are given, the remainder FE is

given : Ånd because AB is to AC, as Dan © E. 5. DE to DF, by conversion C AB is to BC, as DE to EF.

Therefore the ratio of AB to BC is given, because the ratio of the given magnitudes DE, EF, which is the same with it, has been found,

Cor. From this it follows, that the parts AC, CB, have a mi 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.

PROP. VII.

6. 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 founda which is the same with it; let this be the ratio of a 2 Det. the given magnitudes DE, EF: $ And because DE, EF, are given,

B_C. the whole DF is given b: And

63 Dat. because as AB to BC, so is DE D

E F . to EF; by compositione AC is to CB as DF to FE; and, - 18. 5. by conversion", AC is to AB, as DF to DE: Wherefore d 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 givena.

PROP. VIII. If 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 magnitude 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 I and CB are each of them given.

Because the ratio of AC to CB is given, the ratio of AB to BC is givená, therefore a ratio, which is the same with a 7 Dat. 52 Def. it can be found); 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 D F E :

fourth proportional can be € 2 Dat. found, this which is BC is givenc; 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.

PROP. IX. .. 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 founda:
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 É H
D to H: Therefore the ratio of
A to C is givena, because the ra-
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 ra-
tios of D to E, and F to G.

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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 a; but the ratio p.

Eof B to Eis given: there. B forea the ratio of D to E 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 given a : 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.

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 D.

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

в с: of AB to BC is givena : And by composition, the ratio of AC to D. CB is given b: But the ratio of BC to D is given : therefore a the ratio of AC to D is given.'

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107 Dat.

23.

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 re* 19. 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

В 09 Dat. the ratio of FD to FG is given";

and, by conversion, the ratio of c. c6 Dat. FD to DG is givenc: 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 c 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

dat. tio of CF to AE is given, as also the ratio of FD to EB; • 10 Dat. wherefore e the ratio of AE to EB is given; as also the ra77 Dat. tio of AB to each of themf. The ratio therefore of every

one to every one is given.

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 113. 6. given straight lines D, E; and between D and E find ab

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