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the magnitude 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 FD is given, and BE is

B. • 2 Dat. given; wherefore FD is given a : And because, as AE to CD, so is BE to

с

F * 12. 5. FD, AB is to CF, as c 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.

B.

See N. If a magnitude, together with that to which an

other magnitude has a given ratio, be given ; the sum of this other, and that to which the first 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 magnituile to which AB bas a given ratio.

Because the ratio of CD to BE is given; as BE to CD, 50

make AE to FD; therefore the ratio of AE to FD is given, • 2 Dat. and AE is given, wherefore a FD is given : And because as BE to

А

E *Cor. 19. 5. CD, so is AE to FD: AB is b to

FC, as BE to CD: And the ratio F C D
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.

10.

PROP. XVI.
Soe N. If the excess of a magnitude above a given magni-

tude has a given ratio to another magnitude; the ex-
cess of both together above a given 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.

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 the 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

D B

C ratio to BC: And because DB has a given ratio to BC, the ratio to DC to CB is given *; a 7 Dat. and AD is given; therefore DC, the excess of 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 magni- A tude, have to one of them BC a given ratio; either the excess of the other of them AB above the 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, Cor. 6. DB the excess of AB above 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 bas to BC a given ratio, BC has a given ratio to o 6 Dat. BE; and because AE is given, AB together with BE to which BC 'has a given ratio is given.

dat.

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If the excess of a magnitude above a given magni. See N. tude has a given ratio to another magnitude; the excess of the same first magnitude above a given magnitude, shall bave 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 shall have a given ratio to 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 AC.

b

Let AD be the given magnitude ; and because DB, the

excess of AB above AD, has a given ratio to BC; the ratio * 7 Dat. of DC to DB is givena; Make the ratio of AD to DE the same with this ratio ; therefore

A the ratio of AD to DE is given;

Ę DB_C 2 Dat. and AD is given, wherefore b DE and the remainder AE

are given. And because as DC to DB, so is AD to DE, * 12. 5. AC ise 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 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 so make AD to DE; therefore the ratio of AD to DE is 6 Dat. given, as alsod the ratio of AD to AE: And AE is given,

wherefore 6 AD is given : And because, as the whole AC, * 19. 5. to the whole EB, so is AD to DE, the remainder DC ise 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 given, as * Cor. 6. alsof the ratio of DB to BC: And AD is given; therefore

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

1

dat,

PROP. XVIII. Ir 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 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 ex

cess of one of them above a given magnitude has a given * i Dat. ratio to the othera.

Because BE, DF, are each of them given, their ratio is given, and if this ratio be the same

A B

E with the ratio of AB to CD, the tatio of AE to CF, which is the same b

b 12. 5. with the given ratio of AB to CD, CD shall be given.

But if the ratio of BE to DF be not the same with the ratio of AB to CD, either it is greater than the ratio of AB to CD, or, by inversion, the ratio of DF to BE is greater than the ratio of CD to AB: First, A B let the ratio of BE to DF be great

Ģ er than the ratio of AB to CD; C D

F and as AB to CD, so make BG to DF; therefore the ratio of BG to DF is given ; and DF is given, therefore « BG is given : And because BE has a cg Dat. greater ratio to DF than (AB to CD, that is, than) BG to DF, BE is greater than BG: And because as AB to CD, 10.5. so is BG to DF; therefore AG isb to CF, as AB to CD: But the ratio of AB to CD is given, wherefore the ratio of AG to CF is given; and because BE, BG are each of them given, GE is given : Therefore AG, the excess of AE above a given magnitude GE, has a given ratio to CF. The other case is demonstrated in the same manner.

15.

PROP. XIX. If from each of two magnitudes, which have a given ratio to one another, a given magnitude be taken, the remainders shall 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 magnitudes AB, CD have a given ratio to one another, and from AB let the given magnitude AE be taken, and from CD the given magnitudes CF: The remainders EB, FD shall either have a given ratio to one another, or the excess of one of them above a given magnitude shall have a

E

B given-ratio to the other.

Because AE, CF are each of them given, their ratio is givena;

CF -D

* 1 Dat, and if this ratio be the same with the ratio of AB to CD, the ratio of the remainder EB

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19. 5. to the remainder FD, which is the same b with the given

ratio of AB to CD, shall be given.

But if the ratio of AB to CD be not the same with the ratio of AE to CF, either it is greater than the ratio of AE to CF, or, by inversion, the ratio of CD to AB is greater than the ratio of CF to AE: First, let the ratio of AB to CD be greater than the ratio of AE to CF, and as AB to CD, so make AG to CF: therefore the ratio of AG to CF

is given, and CF is given,
ca Dat.whereforec AG is given: A E GB

And because the ratio of
AB to CD, that is, the ra- C F D
tio of AG to CF, is

greater
d 10. 5. than the ratio of AE to CF; AG is greaterd than AE: and

AG, AE are given, therefore the remainder EG is given;
and as AB to CD, so is AG to CF, and so is b the remainder
GB to the remainder FD; and the ratio of AB to CD is
given : Wherefore the ratio of GB to FD is given; there-
fore GB; the excess of EB above a given magnitude EG,
has a given ratio to FD. In the same manner the other
case is demonstrated,

16.

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PROP. XX.
IF to one of two magnitudes which have a given
ratio to one another, a given magnitude be added

,
and from the other a given magnitude be taken;
the excess of the sum above a given magnitude
shall have a given ratio to the remainder.

Let the two magnitudes AB, CD have a given ratio to one another, and to AB let the given magnitude EA be added, and from CD let the given magnitude CF be taken; the excess of the sum EB above a given magnitude has a given ratio to the remainder FD.

Because the ratio of AB to CD is given, make as AB to CD, so AG to CF: Therefore the ratio of AG to CF is given, and CF is given, where

E А

B a 2 Dat. forea AG is given: and EA is given, therefore the whole EG

C

F D
is given: And because as AB
19. 5. to CD, so is AG to CF, and so is the remainder GB to

the remainder FD; the ratio of GB to FD is given. And
EG is given, therefore GB, the excess of the sum EB

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