b 2. def.

2. dat.

which is the fame with it can be found b, let this be the ratio of
the given magnitudes DE, EF: And
A
because the given magnitude AB has
to BC the given ratio of DE to EF, if
unto DE, EF, AB a fourth propor-D

tional can be found, this which is BC

B

FE

is given; and becaufe AB is given, the other part AC is d 4. dat. given d.

3.

a 2. def.

In the fame manner, and with the like limitation, if the dif ference AC of two magnitudes AB, BC which have a given ratio be given; each of the magnitudes AB, BC is given.

PROP. IX.

AGNITUDES which have given ratios to the

Mfame magnitude, have alfo a given ratio to one

another.

Let A, C have each of them a given ratio to B; A has a gi ven ratio to C.

Because the ratio of A to B is given, a ratio which is the fame to it may be found; let this be the ratio of the given magnitudes D, E: And because the ratio of B to C is given, a ratio which is the fame with it may be found; let this be the ratio of the given magnitudes F, G: To F, G, E find a fourth proportional H, if it can be done; and becaufe as A is to B, to is D to E; and as B to C, fo is (F to G, and

fo is) E to H; ex aequali, as A to

C, fo is D to H: Therefore the ratio ABCDE H

EН

of A to C is given, because the

FG

ratio of the given magnitudes D and
H, which is the fame with it, has
been found: But if a fourth propor-
tional to F, G, E cannot be found,

then it can only be faid that the ratio of A to C is compounded
of the ratios of A to B, and B to C, that is, of the given ratios
of D to E, and F to G.

PROP

PROP. X.

F 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; these other magnitudes fhall also have given ratios to one another.

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.

Because the ratio of A to B is given, and likewife the ratio of A to D; therefore the ra

A

tio of D to B is given ; but
the ratio of B to E is given, B

therefore the ratio of D to
E is given And because the C

D

E

F

ratio of B to C is given, and also the ratio of B to E; the ratio of E to C is given: 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.

two magnitudes have each of them a given ratío to another magnitude, both of them together shall

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

them a given ratio to D, the ratio

A

of AB to BC is given: And by compofition, the ratio of AC to CB is D

given: But the ratio of BC to D

a

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

A a 3

9.

29. dat.

22.

B

IF

PROP. XII.

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 given ratio.

A

E

B

F

Ꮐ Ꭰ

GD

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, because it is the fame with the ratio of AB to CG: And the ratio of EB to FD is given, wherefore the ratio of FD to FG is given b; and by converfion, the ratio of FD to DG is given 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 GD to DF is given, wherefore the ratio of CD to DF is given, and 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 ratio of AE to EB is given; as alfo the ratio of AB to each of them f: The ratio therefore of every one to every one is given.

C

d

See N. IF the first of three proportional ftraight lines has a given ratio to the third, the firft fhall also have a given ratio to the fecond.

a 2. def.

b 13. 6.

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

Because the ratio of A to C is given, a ratio which is the fame with it may be found"; let this be the ratio of the given ftraight lines D, E; and between D and E find a 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, because its fides D, E are
given; wherefore the fquare of F, and the
traight 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 fquared of A is to the fquare of
B, as the fquare of D to the fquare of F:
As therefore the ftraight line A to the
ftraight line B, fo is the ftraight line D to the
ftraight 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.

A B C

DFE

Ć 2. cor.

20. 6.

d II. 5.

c 22. 6.

a 2. def.

IF 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 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 firft 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. 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 remainder CF, as AE to CD: But the C ratio of AE to CD is given, therefore

A

BE

FD

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 mag

A a 4

nitude

a 2. dat.

b 19. 5.

a 2. dat.

C 12, 5.

See N.

B.

nitude CD: CD together with a given magnitude has a given

ratio to AB.

E

B

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 is given, and BE is given," wherefore FD is given: And because as AE to CD, fo is BE to FD, AB is C to 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 which is equal to CD together with the given magnitude DF has a given ratio to AB.

PROP. XV.

D F

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 firft magnitude has a given ratio is given.

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,

a 2. dat. and AE is given, wherefore FD And becaufe as BE to

is given

IC.

of BE to CD is given, wherefore

A

F

BE

CD

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.

PROP. XVI.

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