PROP. X.

IF two or more magnitudes have given ratios to one another, and if they have given ratios, though they be not the fame, to fome other magnitudes; thefe other magnitudes fhall alfo have given ratios to one an

other.

Let two or more magnitudes A, B, C have given ratios to one another; and let them have given ratios, though they be not the fame, to fome other magnitudes D, E, F: The magnitudes D, E, F have given ratios to one another.

A

Because the ratio of A to B is given, and likewife the ratio of A to D ; therefore the ratio of D to B is given a ; but the ratio of B to E is given, B therefore a the ratio of D to E is given: And because the C

D

E

F

ratio of B to C is given, and alfo the ratio of B to E; the ratio of E to C is given a: And the ratio of C to F is given; wherefore the ratio of E to F is given; D, E, F have therefore given ratios to one another.

PROP. XI.

IF two magnitudes have each of them a given ratio to another magnitude; both of them together fhall

I have a given ratio to that other.

Let the magnitudes AB, BC have a given ratio to the magnitude D; AC has a given ratio to the fame D. Because AB, BC have each of

A

them a given ratio to D, the ratio
of AB to BC is given a: And by com-
pofition, the ratio of AC to CB is D
given b: But the ratio of BC to D

is given; therefore a the ratio of AC to D is given.

a 9. dat.

B C

a 9. dat.

7. dat.

23:

See N.

PRO P. XII.

IF 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 given ratio.

Becaule 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 19. 5. FG is given, because it is the fame a with the ratio of AB to CG: And the ratio of EB to FD is given, wherefore the ratio of FD

b 9. dat. to FG is given b; and by conver

A

fion, the ratio of FD to DG is CF

e 6. dat. given c: And becaufe AB has to

E

B

G D

each of the magnitudes CD, CG a given ratio, the ratio of CD to CG is given b; and therefore e the ratio of CD to DG is given: But the ratio of GD to DF is given, wherefore b the d cor. 6ratio of CD to DF is given, and confequently d the ratio of CF to FD is given, but the ratio of CF to AE is given, as alfo the 10. dat. ratio of FD to EB; wherefore e the ratio of AE to EB is given; f7. dat. as alfo the ratio of AB to each of them f: The ratio therefore of every one to every one is given.

dat.

See N. IF 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 ftraight lines, that is, as A to B, to is B to C; if A has to C a given ratio, A fhall also have to B a given ratio.

Because the ratio of A to C is given, a ratio which is the def. fame with it may be found a; let this be the ratio of the gi 36. ven straight lines D, E; and between D and E find a b mean

proportional

proportional F; therefore the rectangle contained by D and
E is equal to the fquare of F, and the rect-
angle D, E is given becaufe its fides D, E are
given; wherefore the fquare of F, and the
ftraight line F is given: And because as A is
to C, fo is D to E; but as A to C, fo is C 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; ABC
therefore the fquared of A is to the fquare of
B, as the fquare of D to the fquare of F:
As therefore e the ftraight line A to the
ftraight line B, fo is the ftraight line D to the
traight line F: Therefore the ratio of A to B
is given a, becaufe the ratio of the given
ftraight lines D, F which is the fame with it
has been found.

a

PROP. XIV.

DFE

IF a magnitude together with a given magnitude has a 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 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 excefs of CD above a given magnitude has a given ratio to AB.

A

B

E

a 2. dat.

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 2: And because as AE to CD, fo is BE to FD, the remainder AB is b to the remainder CF, as AE to CD: But the C 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.

F D

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

b 19.5.

nitude 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 A
BE to FD given, and BE is given

♣ 2. dat. wherefore FD is given a : And becaufe
as AE to CD, fo is BE to FD, ABC
12. 5. to CF, as c AE to CD: But the ratio

B.

E B

D F

AB 10 CF is gi

of AE to CD is given, therefore the ratio of
yen; that is, CF which is equal to CD together with the given
magnitude DF has a given ratio to AB.

PROP. 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 magnitude has a given ratio is given.

5.

Let AB, CD be two magnitudes of which AB together with BE to which CD has a given ratio, is given; CD is given toge ther 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 o FD; therefore the ratio of AE to FD is given,

A

BE

C D

a 2. dat. and AE is given, wherefore a FD is given And becaufe as B to Cor. 19. CD, fo 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.

See N. IF the excefs 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 excefs of two magnitudes together above a given magnitude, has to one of them a given ratio; either the excels 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 ratie:

Let

Let the excefs 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 the given magnitude, has a given fatio 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 gi-A

ven ratio to BC, the ratio of DC 10.

DB

C

CB is given a, and AD is given; therefore DC, the excets 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; ei.
ther the excess of the other of them
AB above the given magnitude fhall have to BC a given ra-
tio; 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 becaufe DC the excefs of AC above AD has a given ratio to BC, DB has b a given ratio to BC; that is, DB, b Cor. 6. the excefs of AB above the given magnitude AD, has a given dat. 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 BC a given ratio, BC has e a given ratio to BE; and be-c 6. dat. caufe AE is given, AB together with BE to which BC has a given ratio, is given.

PROP. XVII.

II.

F the excess of a magnitude above a given magnitude See N. has a given ratio to another magnitude; the excefs 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 magnitudes together; the excefs 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 excels of AB above a given magnitude has a given ratio to AC.

Let