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and if this ratio be the same with A B

E
the ratio of AB to CD, the ratio
of AE to CF, which is the sameb C

D
F

b 12. 5. with the given ratio of AB to 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, Let the ratio of BE to DF be greater than A B G E the ratio of AB to CD; and as

+ AB to CD, so make BG to DF; D

F therefore the ratio of BG to DF is given; and DF is given, therefore · BG is given: And because BE has a greater ratio to DF c 2. dat. than (AB to CD, that is, than) BG to DF, BE is greater d d 10. 5. than BG: And because as AB to CD, so is BG to DF; therefore AG is b 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.

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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 A E be taken, and from CD the given magnitude 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 given A E

B ratio to the other.

Because AE, CF are each of C F D them given, their ratio is given"; and if this ratio be the same with the ratio of AB to CD, the ratio of the remainder EB to

a 1. dat.

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b 19. 5. the remainder FD, which is the same 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 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 A EG B e 2. dat. CF is given, wherefore C AG is

given: And because the ratio of C F D
AB to CD, that is, the ratio of

AG to CF, is greater than the d 10. 5.

ratio of A E to ČF; AG is greater d 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 the remainder GB to the
remainder FD, and the ratio of AB to CD is given : Where-
fore the ratio of GB to FD is given; therefore 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.

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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 E A 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 is AG to CF: Therefore the ratio of AG to CF is

given, and CF is given, where-
a 2. dat. fore a AG is given; and EA is E A G B

given, therefore the whole EG
is given. And because as AB to C

F D b 19. 5. CD, so is AG to CF, and so is 6

the remainder GB to the remain-
der FD; the ratio of GB to FD is given. And EG is given,

therefore GB the excess of the sum EB above the given magnitude EG, has a given ratio to the remainder FD.

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If two magnitudes have a given ratio to one an- See N. other, if a given magnitude be added to one of them, and the other be taken from a given magnitude ; the sum, together with the magnitude to which the remainder has a given ratio, is given : And the remainder is given, together with the magnitude to which the sum has a given ratio.

Let the two magnitudes AB, CD have a given ratio to one another : and to AB let the given magnitude BE be added, and let CD be taken from the given magnitude FD: The sum AE is given together with the magnitude to which the remainder FC has a given ratio.

Because the ratio of AB to CD is given, make as AB to CD, so is GB to FD: Therefore the ratio of GB to FD is given, and FD is given, wherefore GB is given a ; and BE is

G A В Е given, the whole GE is therefore given : and because as AB

E c

D to CD, so is GB to FD, and so ist GA to FC; the ratio of GA to FC is given: And AE together with GA is given, because GE is given; therefore the sum AE together with GA, to which the remainder FC has a given ratio, is given.

The second part is manifest from Prop. 15.

a 2. dat.

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b 19. 5.

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If two magnitudes have a given ratio to one an- See N. other, if from one of them a given magnitude be taken, and the other be taken from a given magnitude ; each of the remainders is given, together with the magnitude to which the other remainder has a given ratio.

Let the two magnitudes, AB, CD have a given ratio to one another, and from AB let the given magnitude AE be taken,

and let CD be taken from the given magnitude CF: The remainder EB is given, together with the magnitude to which the other remainder DF has a given ratio.

Because the ratio of AB to CD is given, make as AB to CD, so is AG to CF: The ratio of AG to CF is therefore

given, and CF is given, wherea 2. dat. fore a AG is given; and AE

Α. E B G is given, and therefore the remainder EG is given: And be

C D cause as AB to CD, so is AG

F b 19. 5. to CF: And so is b the remain

der BG to the remainder DF; the ratio of BG to DF is given :
And EB together with BG is given, because EG is given :
Therefore the remainder EB, together with BG, to which
DF, the other remainder, has a given ratio, is given.

The second part is plain from this and Prop. 15.

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See N. If from two given magnitudes there be taken mag.

nitudes which have a given ratio to one another, 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 AB, CD be two given magnitudes, and from them let the magnitudes AE, CF, which have a given ratio to one another, be taken: the remainders EB, FD either have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other.

Because AB, CD are each of them given, the ratio of AB to CD is given: And if this ratio

A

E B be the same with the ratio of AE

to CF, then the remainder EB C F D a 19. 5.

has a the same given ratio to the
remainder FD.

But if the ratio of AB to CD be not the same with the ratio of AE to CF, it is either greater than it, 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 AE to CF, so make AG to CD; there

fore the ratio of AG to CD is given, because the ratio of b 2. dat. AE to CF is given; and CD is given, wherefore 6 AG is

c 10. 5.

given; and because the ratio of AB to CD is greater than
the ratio of (AE to CF, that
is, than the ratio of) AG to A E

GB
CD; AB is greater C than
AG: And AB, AG are given; C F D
therefore the remainder BG
is given : And because as AE
to CF, so is AG to CD, and so is a EG to FD; the ratio of
EG to FD is given: and GB is given; therefore EG, the
excess of EB above a given magnitude GB, has a given ratio
to FD.

The other case is shown in the same way.

a 19. 5.

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If there be three magnitudes, the first of which has See N. a given ratio to the second, and the excess of the second above a given magnitude has a given ratio to the third ; the excess of the first above a given magnitude shall also have a given ratio to the third.

a 2. dat.

b 19. 5.

Let AB, CD, E be the three magnitudes of which AB has a given ratio to CD; and the excess of CD above a given magnitude has a given ratio to E; the excess of AB above a given magnitude has a given ratio to E.

Let CF be the given magnitude, the excess of CD above which, viz. FD has a given ratio to E: And because the ratio of AB to CD is given, as А AB to CD, so make AG to CF; therefore the ratio of AG to CF is given : and CF is given, wherefore a AG is given: And because as AB to CD, so is AG to CF, and so is 6 GB to FD; the ratio of GB to FD F is given. And the ratio of FD to E is given, wherefore c the ratio of GB to E is given, and AG is given; therefore GB the excess of AB above a given magnitude AG has a

B'D' E given ratio to E.

Cor. 1. And, if the first has a given ratio to the second, and the excess of the first above a given magnitude has a given ratio to the third ; the excess of the second above a given magnitude shall have a given ratio to the third. For, if the second be called the first, and the first the second, this corollary will be the same with the proposition.

c 9. dat.

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