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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 cvery 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 fame, 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, lo make AB to CG; the ratio therefore of AB to CG is given;

wherefore the ratio of the remainder EB to the remainder a 39. 5. FG is given, because it is the fame with the ratio of AB to CG: And the ratio of EB to FD is

A E

B given, wherefore the ratio of FD to FG is given b; and by conver

fion, the fatio of FD to DG is C F GD € 6. dat. 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 d cor. 6. ratio of CD to DF is given, and consequently d the ratio of CF dat.

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

as also the ratio of AB to each of themf: The ratio therefore of every one to every one is given.

b . dat.

F

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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, fo 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. fame with it may be found; let this be the ratio of the gib 13. 6. ven straight lines D, E; and between D and E find a mean

proportional

Ć 2. cor.

20. 6.

proportional F; therefore the rectangle contained by D and
E is equal to the square of F, and the rect-
angle D, E is given, because its fides D, E are
given ; wherefore the square of F, and the
Araight 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 squared of A is to the square of

A B C
B, as the square of D to the square of F:
As therefore the straight line A to the D F E

C 22. 6. 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.

d Il. S.

a 2. def.

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IF a magnitude together with a given magnitude has a See N.

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, fo is BE A B E

a 2. dat. to FD, the remainder AB iso to the

b 19. 5. remainder CF, as AE to CD: But the C F D 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 the mag

nitude

Aa4

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

Because the ratio of AE to CD is given, as AE to CD, f make BE to FD; therefore the ratio of A

E B BE to FD is given, and BE is given, wherefore FD is given : And because as AE to CD, fo is BE to FD, AB is C DF to CF, asc 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.

a 2. dat.

C 12. 3

B.

PRO P. XV.

See N.

IF a magnitude together with that to which another

magnitude has a given ratio, be given ; the fum of this other, and that to which the first magnitode 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.

Because the ratio of CD to BE is given, as BE to CD, fo make A E to FD; therefore the ratio of AE to FD is given, and AE is given, wherefore • FD

A B E is given : And because as BE to h Cor. 19. CD, so is AE to FD: AB is b to FC, as BE to CD: And the ratio

F 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.

a 2. dat.

.

C D

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See N. F 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 magnicude, 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 bas a given ratio.

Let

Let the excess of the magnitude AB above a given magni. tude, 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 ratio A D B C to BC: And because DB has a given ratio to BC, the ratio of DC to

-H CB is given , and AD is given ; therefore DC, the excess of a 7. dat. 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 to

A D BEC 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 b a given ratio to BC; that is, DB, the ex- b Cor. 6. cess of AB above the given magnitude AD, has a given ratio to BC.

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; and be

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

dat.

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IF the excess of a magnitude above a given magnitude Sce N.

has a given ratio to another magnitude ; the excess of the saine 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 fame above a given magnitude thall have a given ratio to the other.

Let the excess of the magnitude AB above a given magni. tude have a given ratio to the magnitude BC, the excess of AB above a given magnitude has a given ratio to AC.

Lec

c 12. 5.

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

the ratio e a y. dat. DC to DB is given : Make the ratio of AD to DE the fame

with this ratio ; therefore the ratio
of AD to DE is given : And AD

A E_DB_C b 2. dat. is given, wherefore b DE, and the remainder AĖ are given: And

because as DC to DB, so is AD to DE, AC is 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, fo

make AD to DE ; therefore the ratio of AD to DE is given, d 6. dat. as also a the ratio of AD to AE : And AE is given, where

fore 0 AD is given : And because, as the whole AC, to the € 19. 5. whole EB, fo is AD to DE; the remainder DC ise 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 ex. cess of AB above a given magnitude AD, has a given ratio to BC.

dat.

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IF to each of two magnitudes, which bave a given ratio

to one another, a given magnitude be added; the whole shall either have a given ratio to one another, or the excess of one of them above a given magnitude fhall have a given ratio to the other.

1

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 a 1. dat. above a given magnitude has a given ratio to the other. Because BE, DF are each of them given, their ratio is given,

and

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