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23.

See N.

PROP. XI. IF (F the whole have to the whole a given ratio, and the

parts have to the parts given, but not the fame, 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, but not the same, ratios to the parti CF, FD; every one shall have to every one, whole or part, a giren ratio.

Because the ratio of AE to CF is given, as AE to CF, To make AB to CG; the ratio therefore of AB to CG is giveni; wherefore

the ratio of the remainder EB to the remainder FG is given, be a. 19. 5. cause it is the famea with the ratio of AB to CG, and the ratio of El to FD is given, wherefore the ratio of

A E B b. 9. Dat. FD to FG is given b; and by converc. 6. Dat. fion, the ratio of FD to DG is giveno.

С F G D 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 GI

to DF is given, wherefore b the ratio of CD to DF is given, an d. Cor. 6. confequently the ratio of C F to FD is given ; but the ratio

CF to AE is given, as also the ratio of FD to EB; wherefore th 6. 10. Dat. ratio of AE to EB is given ; as also the ratio of AB to each f. 7. Dat. them f. the ratio therefore of every one to every one is given.

Dat.

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Sce N.

F the first of three proportional straight lines has a g

ven ratio to the third, the first shall also have a giv ratio to the second.

Let A, B, C be three proportional straight lines, that is as A B, fo is B to C; if A has to C a given ratio, A shall also have to a given ratio.

Because the ratio of A to C is given, a ratio which is the £ a. 2. Def, with it may be found”; let this be the ratio of the given strai 1. 13. 6. lincs D, E; and between D and E find ab mean proportional

th

A

therefore the rectangle contained by D and E is equal to the
square of F, and the rectangle D, E is given be-
cause 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, fo 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, fo is the square of D to the square of
F; therefore the square d of A is to the square'of A B C
B, as the square of D to the square of F. as

D F E
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 givena, because
the ratio of the given straight lines D, F which
is the same with it has been found.

C. 2. Cor.

10.6.

d. 11. s.

e. 22. 6.

a. 2. Def.

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IF
[F 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.

a. 2. Dat,

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

A Β Ε because as AE to CD, so is BE to FD, the remainder AB is b to the remain

F D der 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 magnitude FD has a given ratio to AB.

Next, Let the excess of the magnitude A B above the given magnitude BE, that is, let A E have a given ratio to the magni

tude

b. 19.5.

C

See N.

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, so make
BE to FD; therefore the ratio of BE to

А
FD is given, and BE is given, wherefore

E B a. 2. Dat. FD is given", and because as AE to Ć. ir. s. CD, so is BE to FD, AB is to CF, as AE C D F

to CD. but the ratio of AE to CD is gi-
ven, 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 ra-

tio to AB.
B.

PRO P. XV.
IF
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 togetñer with that magnitnde 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 2. 2. Dat. given, wherefore a FD is given. and

A Β Ε because as BE to CD, fo is AE to b.Cor.19 5.FD; AB is b to FC, as BE to CD,

F C D 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 giver..

10.

See N.

PRO P. XVI.
IF
F the excess of a magnitude above a given magnitude,

has a given ratio to another magnitude; the excess of both together above a giren 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

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D BEC

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.
and because DB has a given ratio to

C BC, the ratio of DC to CB is given”, and AD is given; therefore DC, the excess of AC above the a. 7. Dat. 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 them BC a given ratio ; ei A ther the excess 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 b a given ratio to BC; that is DB, the excess of AB above b. Cor. 6. 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 because AE is c. 6. Dal! gren, AB together with BE to which BC has a given ratio, is given.

Dat.

PROP.
XVII.

ii. 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 magnitude hall have a given ratio co 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

Le:

Aa

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Ę DBC

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 2. 7. Dat. givena. make :he ratio of AD to DE the same with this ratio ;

therefore the ratio of AD to DE is

given. and AD is given, where. A E ED C 6. 2. Dat fore b DE, and the remainder A E 6.12. s. are given. and because as DC to DB, so is AD to DE, AC is to

IB, 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 d. 6. Dat. to DE; therefore the ratio of AD to DE is given, as also d the ra

tio of AD to AE, and AE is given, wherefore b AD is given, and

because as the whole, AC, to the whole, EB, fo is AD to DE; C. 19. 5. the remainder DC is e to the remainder DB, as AC to EB; and the

ratio of AC to EB is given, wherefore the ratio of DC to DB is gif. Cor. 6

ven, as also f the ratio of DB to BC. and AD is given, therefore Dat.

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

P R O P. XVIII.

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

and

6. 1. Dat.

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