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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 given; Make the ratio of AD to DE the same with this ratio; therefore

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the ratio of AD to DE is given;

A

EDB

C

2 Dat. and AD is given, wherefore DE and the remainder AE are given. And because as DC to DB, so is AD to DE,

12. 5. AC is 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 also the ratio of AD to AE: And AE is given, wherefore AD is given: And because, as the whole AC, 19. 5. to the whole EB, so is AD to DE, the remainder DC is 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 f 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.

dat.

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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 excess of one of them above a given magnitude has a given ⚫ 1 Dat. ratio to the other a.

Because BE, DF, are each of them given, their ratio is

given, and if this ratio be the same with the ratio of AB to CD, the ratio of AE to CF, which is the same b with the given ratio of AB to CD, shall be given.

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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

er than the ratio of AB to CD; C

D

E

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
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,
so is BG to DF; therefore AG is 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.

2 Dat.

10. 5.

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 A given-ratio to the other.

Because AE, CF are each of

them given, their ratio is given"; and if this ratio be the same

CF

B

15.

E

D

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with the ratio of AB to CD, the ratio of the remainder EB

b

19. 5. to 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 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, 2 Dat. wherefore AG is given :

A

F

GB

F

D

than AE: and EG is given ; the remainder

And because the ratio of AB to CD, that is, the ra- C tio of AG to CF, is greater 10. 5. than the ratio of AE to CF; AG is greater AG, AE are given, therefore the remainder and as AB to CD, so is AG to CF, and so is GB to the remainder FD; and the ratio of AB to CD is given: Wherefore 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 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 A

G

Ģ B

F

D

C

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

above the given magnitude EG, has a given ratio to the remainder FD.

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Ir 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 GB to FD: Therefore the ratio of GB to FD is given, and FD is given, where

fore GB is given; and BE is G

A

BE

2 Def.

given, the whole GE is there

fore given and because as AB F

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to CD, so is GB to FD, and so

is b GA to FC; the ratio of GA to FC is given: And AE ↳ 19. 5. 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.

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If two magnitudes have a given ratio to one another, See N. 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 AG to CF: The ratio of AG to CF is therefore 2 Dat. given, and CF is given, wherefore AG is given; and AE is given, and therefore the remainder EG is given: And A because as AB to CD, so is

b 19. 5. AG to CF: And so is the re- C

mainder BG to the remainder

E B Ꮐ

DF

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 magnitudes 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 be the same with the ratio

A

C

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F D

19. 5. of AE to CF, then the remainder EB has 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; therefore the ratio of AG to CD is given, because the b 2 Dat. ratio of AE to CF is given; and CD is given; wherefore b

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