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tude 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, fo make BE to FD; therefore the ratio of BE to

А

E B FD is given, and BE is given, wherefore a. 2. Dat. FD is given a. and because as AE to CD, C. 12. 5. fo is BE to FD, AB is to CF, as c AE to

DF 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 DFhas a given ratio to AB.

B.

See N.

PROP. XV.
F a magnitude together with that to which another

magnitude has a given ratio, be given ; this other is given together with that to which the first magnitude has a given ratio.

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, so make

AE to FD; therefore the ratio of AE to FD is given, and AE is a. 2. Dat. given, wherefore a FD is given. and be- A

B E cause as BE to CD, fo is AE to FD; b.Cor.19.5. AB is b to FC, as BE to CD. and the F C D

ratio of BE to CD is given, wherefore
the ratio of AB to FC is given. and FD is given, that is CD to-
gether with FC to which AB has a given ratio is given.

IO.

See N.

PROP. XVI.
F the excess of a magnitude above a given magni.

tude, 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 a 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 to BC.

A DB C and because DB has a given ratio to BC, the ratio of DC to CB is given a, and AD is given; therefore DC, the excess of AC above a. 7. Data 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 one of

A them BC a given ratio ; either the

D BEC ëxcess of the other of them AB above a 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, the excess b. Cor. 6.

Dat. 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 e 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.

if.

PROP. XVII. F the excess of a magnitude above a given magnitude See N.

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 magnie tude 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.

Аа

Let AD be the given magnitude; and because DB, the excess of

AB above AD, has a given ratio to BC; the ratio of DC to DB is a. 7. Dat. given a. make the ratio of AD to DE the same with this ratio ;

therefore the ratio of AD to DE is

given. and AD is given, wherefore A E DB C b. 2. Dat. b DE, and the remainder AE are C. 12. So 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, so make d. 6. Dat. AD to DE; therefore the ratio of AD to DE is given, as also d

the ratio of AD to AE. and AE is given, wherefore 6 AD is given. and because as the whole, AC, to the 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 given, wherefore the ratio of DC f. Cor. 6. to DB is given, as also f the ratio of DB to BC. and AD is gi

ven, therefore DB, the excess of AB above the given magnitude AD, has a given ratio to BC.

c. 19. 5.

Dat.

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I

F to each of two magnitudes, which have a given

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 other.

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

ai' 1. Dat.

b. 12. 5.

and if this ratio be the same with A B TC
the ratio of AB to CD, the ratio of
AE to CF, which is the fame with C D F
the given ratio of AB to CD, shall be
given.

But if the ratio of BE to DF be not the fame with the ratio of
AB to CD; either it is greater than the ratio of AB to CD, or, bý
inversion, the ratio of DF to BE is greater than the ratio of CD to
AB. first, let the ratio of BE to DF
be greater than the ratio of AB to CD;

А в G E and as AB to CD, fo 'make BG to

C DF; therefore the ratio of BG to DF is given; and DF is given, therefore c

c. 2. Data BG is given. and because BE has a greater ratio to DF than (AB to CD, that is than) BG to DF, BE is greater d than BG. and d. 10. 5 because as AB to CD, so is BG to DF, therefore AG is b to CF, as AB to CD. but the ratio of AB to CD is given, wherefore the ratio of AG to CF is given; and because BE, BG are each of them given, GE is given. therefore AG, the excess of AE above the given magnitude GE has a given ratio to CF. the other case is demonstrated in the same manner.

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F from each of two magnitudes, which have a given

ratio to one another, a given nagnitude be taken; the remainders 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 magnitudes AB, CD have a given ratio to one another, and from AB let the given magnitude AE be taken, and from CD, the given magnitude CF. the remainders EB, FD 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.

A E B Because AE, CF are each of them given, their ratio is given a;

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a. I. Dat. and if this ratio be the same with the ratio of AB to CD, the ratio of the remainder EB to the re

b. 19. 5. mainder FD, which is the same with the given ratio of AB to

CD, shall be given.

But if the ratio of AB to CD be not the same with the ratio of AE to CF, either it is greater than the ratio of AE to CF, or, by inversion, the ratio of CI) to AB is greater than the ratio of CF to AE. first, let the ratio of AB to CD be greater than the ratio of AE to CF, and as AB to CD, fo make AG to CF; therefore the

ratio of AG to CF is given, and CF C. 2. Dat.

A

B is given, wherefore AG is given. and because the ratio of AB to CD, C F D that is the ratio of AG to CF, is

greater than the ratio of AE to CF; d. 10. 5. AG is greater than AE. and AG, AE are given, therefore the

remainder EG is given. and as AB to CD, fo is AG to CF, and fo is b the remainder GB to the remainder FD; and the ratio of AB to CD is given, wherefore the ratio of GB to FD is given; therefore GB, the excess of EB above the given magnitude EG, has a given ratio to FD. in the same manner the other case is demonstrated.

EG

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IF to one of two magnitudes which have a given ratio

to one another, a given magnitude be added, and from the other a given magnitude be taken; the excess of the sum above a given magnitude shall have a given ratio to the remainder.

Let the two magnitudes AB, CD have a given ratio to one another, and to AB let the given magnitude E A be added, and from CD let the given magnitude CF be taken; the excess of the sum EB above a given magnitude has a given ratio to the remainder FD.

Because the ratio of AB to CD is given, make as AB to CD, so

AG to CF. therefore the ratio of AG to CF is given, and CF is 8. 2. Dat. given, wherefore a AG is given; E A GB

and EA is given, therefore the
whole EG is given. and because

C

F D as AB to CD, so is AG to CF, and so is b the remainder GB to the remainder FD; the ratio of GB to FD is given. and EG is given,,

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