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b 2. def. found “, let this be the ratio of the given magnitudes DE, EF: And because the given

A

С B. magnitude AB bas to BC the given ratio of DE to EF, if to DE, EF, AB a fourth propor

D F E tional can be found, this, which is c 2. dat. BC, is giveno; and because AB is given, the other part AC d 4. dat. is given a.

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 a 2. def. same to it may be found ; 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 B C D E H A to C, so is D to H: Therefore the ratio of A to C is given “, be

F G cause the ratio 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.

a 9. dat.

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 is given : A

D but the ratio of B to E is

B

E given, therefore a the ratio

C

F 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 4: 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,

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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.

a 9. dat.

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 of A B to BC is givena: And, by composition, the ratio A

B C of AC to CB is given b: But the ratio of BC to D is giveno;

D therefore the ratio a of AC to D is given.

b 7. dat.

c hyp.

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See N.

! If the whole has to the whole a given ratio, and the parts have to the parts given ratios, but not the same ratios : Every one of them, whole or part, shall have to every one a given ratio.

1

Let the whole AB have a given ratio to the whole CD, and the parts AE, EB have given ratios, 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 remainder a 19. 5. FG is given, because it is the same with the ratio of AB to

CG: And, by hypothesis, the
ratio of EB to FD is given, A E

B. wherefore the ratio of FD to b 9. dat. FG is given b; and, by con

C F G D version, the ratio of FD to DG c 6. dat. is given : And because 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, whered cor. 6. fore b the ratio of CD to DF is given, and consequently d the

ratio of CF to FD is given; but the ratio of CF to AE is e 10. dat. given, as also the ratio of FD to EB; wherefore the ratio of f 7. dat. AE to EB is given; as also the ratio of AB to each of them?:

The ratio therefore of every one to every one is given.

dat.

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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 a 2. def. same with it may be found a ; let this be the ratio of the given b 15. 6, straight lines D, E; and between D and E find a mean

e 2. cor.

proportional F; therefore the rectangle contained by D and c 17. 6.
E is equal to the square of F, and the rect-
angle D, E is given, because its sides D, E
are given, wherefore the square of F, and

dl. def.
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 ise the square of D to the
square of F: Therefore the square of A is À B Ć fll. 5.
to the square of B, as the square of D to the
square of F: As therefore & the straight line D F 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

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

20. 6.

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g 22. 6.

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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, A E, 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 A E to CD, so make BE to FD; therefore the ratio of BE to FD is given, and BE is given ; wherefore FD

A

B. E
is given a : And because as AE
to CD, so is BE to FD, the re-

с
mainder AB is b to the remainder

F D
CF, as A E 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 magni-
tude 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 mag-

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a 2. dat.

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b 19. 5.

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

Because the ratio of A E to CD is given, as A E to CD, so make BE to FD; therefore the ratio of BE to FD is given, and BE is A

E

B a 2. dat. given, wherefore FD is given a : And

С DF because as A E to CD, so is BE to FD; c 12. 5. AB is to CF, as c AE to CD: 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.

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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 A B bas a given ratio.

Because the ratio of CD to BE is given, as BE to CD, so

make A E to FD; therefore the ratio of AE to FD is given, a 2. dat. and AE is given, wherefore a FD

A

B E is given : And because as BE to b Cor.19.5. CD, so is AE to FD: AB is o to

F С D FC, as BE to CD: And the ratio 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.

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See N.

If the excess of a magnitude above a given mag. nitude 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.

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