Imágenes de páginas
PDF
EPUB

b 2. def. found, let this be the ratio of the given magnitudes DE, EF: And because the given magnitude AB has to BC the given ratio of DE to EF, if to

c 2. dat. d 4. dat.

8.

A

D

C

B

F E

DE, EF, AB a fourth propor-
tional can be found, this, which is
BC, is given; and because AB is given, the other part AC
is given d.

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 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 2; let this be the ratio of the given magnitudes

АВСДЕН
F G

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 D to H: Therefore
the ratio of A to C is given, be-
cause the ratio of the given magni-
tudes 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 ra-
tios of A to B, and B to C, that is, of the given ratios of D
to E, and F to G.

[ocr errors]

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 is given2: but the ratio of B to E is given, therefore the ratio of D to E is given, and because the ratio of B to

A

D

B

E

F

C

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 F is 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.

A

9.

a 9. dat.

22.

a 9. dat.

B

C

b 7. dat.

Because AB, BC have each of them a given ratio to D, the ratio of AB to BC is given 2: And, by composition, the ratio of AC to CB is given: But the ratio of BC to D is given ; therefore the ratio of AC to D is given.

D

c hyp.

[blocks in formation]

PROP. XII.

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.

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.

A

E

[ocr errors]

C F G D

B

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 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, wherefore the ratio of FD to FG is given; and, by conversion, the ratio of FD to DG is given: And because AB has to each of the magnitudes CD, CG a given ratio, the ratio of CD to CG is given; 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 consequently d 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; wherefore the ratio of AE to EB is given; as also the ratio of AB to each of them f: The ratio therefore of every one to every one is given.

See N.

24.

a 2. def.

PROP. XIII.

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 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

b

e

d 1. def.

e 2. cor. 20. 6. f 11. 5.

1

proportional F; therefore the rectangle contained by D and c 17. 6. 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 square of A is to the square of B, as the square of D to the square of F: As therefore & the straight line 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 because the ratio of the given straight lines D, F, which is the same with it, has been found.

a

PROP. XIV.

A

A

B

C

FE

g 22. 6.

D

a 2. def.

A.

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

[blocks in formation]

A

B

E

a 2. dat.

[blocks in formation]

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 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

nitude 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 ratio

of BE to FD is given, and BE is A

a 2. dat. given, wherefore

c 12. 5.

See N.

a 2. dat.

B.

FD is given: And because as AE to CD, so is BE to FD; AB is to CF, as AE to CD: But the

C

E

B

DF

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.

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

B

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, and AE is given, wherefore a FD is given And because as BE to b Cor. 19.5. CD, so is AE to FD: AB is to FC, as BE to CD: And the ratio

10.

of BE to CD is given, wherefore

E

F

C D

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.

PROP. XVI.

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.

« AnteriorContinuar »