Sidebilder
PDF
ePub

See N.

23.

a 19. 5.

b 9. dat.

c 6. dat.

d cor. 6.

dat.

e 10. dat. f 7. dat.

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

B

C F

G D

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.

[blocks in formation]

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; let this be the ratio of the given straight lines D, E; and between D and E find a mean

b 15. 6,

[ocr errors]

d 1. def.

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

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.

PROP. XIV.

A

B C

g 22. 6.

D FE

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 is given And because as AE to CD, so is BE to FD, the remainder AB is to the remainder

A

B

E

a 2. dat.

C

F D

b 19. 5.

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 2. dat. given, wherefore FD is given 2: And because as AE to CD, so is BE to FD; AB is to CF, as AE to CD: But the

c 12. 5.

See N.

a 2. dat.

B.

A

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 of BE to CD is given, wherefore

10.

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.

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 ra

tio to BC: And because DB A D B C has a given ratio to BC, the ra

tio of DC to CB is given 2, 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 magnitude, have

A DBE C

+

a 7. dat.

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, DB b Cor. 6. the excess of AB above the given magnitude AD has a given ratio to BC.

с

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 BE; c 6. dat. and because AE is given, AB together with BE to which BC has a given ratio, is given.

[blocks in formation]

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.

Let AD be the given magnitude; and because DB, the excess of AB above AD, has a given ratio to BC; the ratio of

a 7. dat. DC to DB is given: Make the ratio of AD to DE the same with this ratio; therefore the

ratio of AD to DE is given; A

b 2. dat.

and AD is given, wherefore b

c 12. 5.

DE and the remainder AE are

EDB

C

given: And because as DC to DB, so is AD to DE, AC is c to EB, as DC to DB; and the ratio of DC to DB is given; wherefore the ratio of AC to EB is given: And because the ratio of EB to AC is given, and that AE is given, therefore EB the excess of AB above the given magnitude AE, has a given ratio to AC.

Next, Let the excess of AB above a given magnitude have a given ratio to AB and BC together, that is, to AC; the excess of AB above a given magnitude has a given ratio to BC.

Let AE be the given magnitude; and because EB the excess of AB above AE has to AC a given ratio, as AC to EB, so make AD to DE; therefore the ratio of AD to DE is given, d 6. dat. as also the ratio of AD to AE: And AE is given, wherefore AD is given: And because, as the whole AC, to the whole EB, so is AD to DE, the remainder DC is to the remainder DB, as AC to EB; and the ratio of AC to EB is f Cor. 6. given; wherefore the ratio of DC to DB is given, as also f the ratio of DB to BC: And AD is given; therefore DB, the excess of AB above a given magnitude AD, has a given ratio to BC.

e 19.5.

dat.

[blocks in formation]

If to each of two magnitudes, which have a given ratio to one another, a given magnitude be added; the wholes shall either have a given ratio to one another, or the excess of one of them above a given magnitude shall have a given ratio to the other.

Let the two magnitudes AB, CD have a given ratio to one another, and to AB let the given magnitude BE be added, and the given magnitude DF to CD: The wholes AE, CF either have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the a 1. dat. other a.

Because BE, DF are each of them given, their ratio is given,

« ForrigeFortsett »