See N.

a. 19. 5.

23.

b. 9. Dat.

c. 6. Dat.

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, 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 parts CF, FD; every one fhall have to every one, whole or part, a gi

ven ratio.

Becaufe 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, becaufe it is the fame with the ratio of AB to CG. and the ratio of EB to FD is given, wherefore the ratio of A FD to FG is given b; and by converfion, the ratio of FD to DG is given. C and becaufe AB has to each of the mag

E

B

F

G D

nitudes 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 d. Cor. 6. confequently the ratio of CF to FD is given; but the ratio of CF to AE is given, as alfo the ratio of FD to EB; wherefore the e. 1c. Dat. ratio of AE to EB is given; as alfo the ratio of AB to each of f. 7. Dat. them f. the ratio therefore of every one to every one is given.

Dat.

Sce N.

24.

IF

F the first of three proportional straight lines has a given ratio to the third, the firft fhall alfo have a given ratio to the fecond.

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.

a

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

b. 13. 6.

C

c

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

IF

d

PRO P. XIV.

c. z. Cor. 20. 6.

B

Ĉ d. 11. 5.

D

FE

e. 22. 6.

a. 2. Def.

A.

F 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.

A

B

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 to the C remainder CF, as AE to CD. but

E

a. 2. Dat.

F D

b. 19. 5.

the ratio of AE to CD is given, 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 CD; CD together with a given magnitude has a given ratio to AB.

A

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 a. 2. Dat. FD is given". and becaufe as AE to CD, fo is BE to FD, AB is to CF, as AE to C CD. but the ratio of AE to CD is given,

C. 12. 5.

B. See N.

c

E

B

D F

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.

IF

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 firft 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, fo make AE to FD; therefore the ratio of AE to FD is given, and AE is

a. a. Dat. given, wherefore a FD is given. and be A

caufe as BE to CD, fo is AE to FD;

B

C D

E

b.Cor. 19.5. AB is b to FC, as BE to CD. and the F 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 given.

See N.

10.

PROP. XVI.

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 fhall 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 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 excefs 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. and becaufe DB has a given ratio

A

to BC, the ratio of DC to CB is

D B

C

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

them BC a given ratio; either the

excess of the other of them AB a

bove a given magnitude hall 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 lefs 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 excefs b. Cor. of AB above the given magnitude AD, has a given ratio to BC.

Dat.

But let the given magnitude be greater than AB, and make AE equal to it; and becaufe EC, the excefs of AC above AZ, has to BC a given ratio, BC has a given ratio to BE; and be- c. 6. Dats caufe AE is given, AB together with BE to which BC has a given ratio, is given.

IF the excefs of a magnitude above a given magnitude See N. has a given ratio to another magnitude; the exce's of the fame first magnitude above a given magnitude, fhall have a given ratio to both the magnitudes together. and if the excefs of either of two magnitudes above a given magnitude has a given ratio to both magnitades together; the excess of the fame above a given magnitude shall 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.

A a

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 PL 3

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

given. and AD is given, wherefore A

b. 2. Dat. b DE, and the remainder AE are gi

O. 12. 5.

ED B

C

ven. and becaufe as DC to DB, fo is AD to DE, AC is to E3, 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 2bove the given magnitude AE, has a given ratio to AC.

Next, let the excefs of AB above a given magnitu 'e have a given ratio to AB and BC together, that is to AC; the excus of AB above a given magnitude has a given ratio to BC.

Let AE be the given magnitude; and becaufe EB the excl of AB above AE has to AC a given ratio, as AC to EB, fo mile d. 6. Dat. AD to DE; therefore the ratio of AD to DE is given, as al the ratio of AD to AE. and AE is given, wherefore AD is gi ven. and because as the whole, AC, to the whole, EB, f› is A) to DE; the remainder DC is to the remainder DB, as AC to 12; and the ratio of AC to EB is given, wherefore the ratio of DC to DB is given, as alfof the ratio of DB to BC. and AD is gven, therefore DB, the excefs of AB above the given magnade AD, has a given ratio to BC.

C. 19. S.

f. Cor. 6.

Dat.

e

14.

2. 1. Dat.

IF to each 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 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 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 ".