23. Sce N.

[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, fhall 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 fame, ratios to the part CF, FD; every one fhall have to every one, whole or part, a given ratio.

Because the ratio of AE to CF is given, as AE to CF, fo 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, be a. 19. 5. caufe it is the fame with the ratio of AB to CG, and the ratio of El to FD is given, wherefore the ratio of b. 9. Dat. FD to FG is given b; and by convere. 6. Dat. fion, the ratio of FD to DG is given.

and becaufe AB has to each of the

A
F

E

B

Ꮐ Ꭰ

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 CF to FD is given; but the ratio CF to AE is given, as alfo the ratio of FD to EB; wherefore t ratio of AE to EB is given; as alfo the ratio of AB to each them f. the ratio therefore of every one to every one is given.

Dat.

•. 10. Dat.

f. 7. Dat.

24. See N.

IF

PRO P. XIII.

F the first of three proportional straight lines has a § ven ratio to the third, the first fhall also have a giv ratio to the fecond.

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

Because the ratio of A to C is given, a ratio which is the 2.2. Def. with it may be found 2; let this be the ratio of the given strai b. 13. 6. lines D, E; and between D and E find ab mean proportional

the

therefore the rectangle contained by D and E is equal to the
fquare of F, and the rectangle D, E is given be-
cause its fides D, E are given; wherefore the
fquare 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, fo is the fquare of A to the fquare of B; and
as D to E, fo is the fquare of D to the fquare of
F; therefore the fquare of A is to the fquare of A B

e

d

B, as the fquare of D to the fquare of F. as therefore the straight line A to the straight line B, fo 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 ftraight lines D, F which is the fame with it has been found.

DFE

c. 2. Cor. 20.6.

d. 11. 5.

e. 22. 6.

a. 2. Def.

IF a

a magnitude together with a given magnitude has a see N. given ratio to another magnitude; the excefs of this other magnitude above a given magnitude has a given ratio to the first magnitude. and if the excefs 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, fo 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 to FD,

the remainder AB is der CF, as AE to CD.

of AE to CD is given,

BE
F D

therefore the ratio of AB to CF is given; that is, CF the excefs of CD above the given magnitude FD has a given ratio to AB.

Next, Let the excefs of the magnitude AB above the given magnitude BE, that is, let AE have a given ratio to the magni

tude

a. 2. Dat.

b. 19.5.

tude CD; CD together with a given magnitude has a given ratio to AB.

A

E B

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 a. 2. Dat. FD is given. and because as AE to ć. ia. 5. CD, fo is BE to FD, AB is to CF, as AE C 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 ra

B. Sce N.

tio to AB.

IF

PROP. XV.

D 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

2.2. Dat. given, wherefore FD is given. and

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

10.

See N.

and the ratio of BE to CD is given,

A

F

BE

Ꮯ Ꭰ

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.

PRO P. XVI.

IF the excess of a magnitude above a given magnitude, has a given ratio to another magnitude; the excefs of both together above a given magnitude shall have to thať other a given ratio. and if the excefs 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

1

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 excefs of AB above which, viz. DB, has a given ratio to BC.

D B

BC, the ratio of DC to CB is gi

ven, and AD is given; therefore DC, the excefs of AC above the a. 7. Dat.

given magnitude AD, has a given ratio to BC.

Next, let the excefs of two magnitudes AB, BC together a

bove a given magnitude have to

one of them BC a given ratio; ei

ther the excess of the other of them

D BEC

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 firft let it be less than AB; and because DC the excefs of AC above AD has a given ratio to BC, DB has a given ratio to BC; that is DB, the excefs 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 becaufe EC, the excefs of AC above AE, has to

C

Dat.

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

IF

F the excefs of a magnitude above a given magnitude See N. has a given ratio to another magnitude; the excefs of the same first magnitude above a given magnitude, fhall 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 fhall have a given ratio to the other.

Let the excefs of the magnitude AB above a given magnitude" have a given ratio to the magnitude BC; the excefs 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 DC to DB is

a. 7. Dat. given2. make the ratio of AD to DE the fame with this ratio; therefore the ratio of AD to DE is

b. 2. Dat

c. 12. 5.

given. and AD is given, where- A

foreb DE, and the remainder A E

EDB C

are given. and becaufe as DC to DB, fo 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 excefs of AB above the given magnitude AE, has a given ratio to AC.

Next, let the excefs 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 becaufe EB the excess of AB above AE has to AC a given ratio, as AC to EB, fo make AD d. 5. Dat. to DE; therefore the ratio of AD to DE is given, as alfo d the ratio of AD to AE. and AE is given, wherefore ↳ 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 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 gi ven, as alfof the ratio of DB to BC. and AD is given, therefore DB, the exccfs of AB above the given magnitude AD, has a given ratio to BC.

f. Cor. 6

Dat.

e

I

PRO P. XVIII.

F to cach of two magnitudes, which have a given ratio to one another, a given magnitude be added; the wholes fhall either have a given ratio to one another, or the excefs of one of them above a given magnitude fhall 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 rafio to one another, or the excefs 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‘.

and