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GA, to which the remainder FC has a given ratio, is given. The second part is manifest from Prop. 15.

D.

PROP. XXII.

See N.

* 2 Dat.

19. 5.

If two magnitudes have a given ratio to one another, 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 given, and CF is given, wherefore * AG is given ; and AE is given, and therefore the remainder

А

G
EG is given: and because

Е В
as AB to CD, so is AG to
.CF: and so is * the re-

с. D F
mainder BG to the re-
mainder 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.

PROP. XXIII.
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.

20.

See N.

Because AB, CD, are each of them given, the A E

B ratio of AB to CD is given: and if this ratio be the same

с with the ratio of AE to CF,

F D then the remainder EB has

* 19.5. 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 ratio of AE to CF is given; and CD is given ; wherefore * AG is given; and because * 2 Dat. the ratio of AB to CD is greater than the ratio of (AE to CF that is, than the ratio of) AG to CD; AB is

A E G B greater * than AG: and

* 10. 5. AB, AG, are given; there

C fore the remainder BG is

FD given : and because as AE to CF, so is AG to CD, and so is * EG to FD; the ratio * 19.5. 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 shewn in the same way.

PROP. XXIV. If there be three magnitudes, the first of which has a given See N.

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. 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 AB to CD, so make AG to CF; therefore the ratio of AG to CF, is given: and CF is given, wherefore * AG is given : * 2 Dat.

13.

A A

19.5.

* 9 Dat.

17.

and because as AB to CD, so is AG
to CF, and so is * GB to FD; the

А
ratio of GB to FD is given. And the
ratio of FD to E is given, wherefore *
the ratio of GB to E is given, and AG GH.

С
is given ; therefore GB the excess of

FH
AB above a given magnitude AG has
a given ratio to E.

Cor. 1. And if the first have a given
ratio to the second, and the excess of B D E
the first above a given magnitude have
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.

Cor. 2. Also, if the first have a given ratio to the second, and the excess of the third above a given magnitude have also a given ratio to the second, the same excess shall have a given ratio to the first; as is evident from the 9th Dat.

PROP. XXV.
If there be three magnitudes, the excess of the first where-

of above a given magnitude has a given ratio to the
second ; and the excess of the third above a given mag-
nitude has a given ratio to the same second : the first
shall either have a given ratio to the third, or the ex-
cess of one of třem above a given magnitude shall have
a given ratio to the other.

Let AB, C, DE, be three magnitudes, and let the excesses of each of the two AB, DE, above given magnitudes, have given ratios to C; AB, DE, 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.

Let FB the excess of AB above the given magnitude AF have a given ratio to C; and let GE, the excess of DE

A above the given magnitude DG have a

DI

1
given ratio to C; and because FB, GE
have each of them a given ratio to C,

GH
they have a given * ratio to one an-
other. But to FB, GE the given mag-
nitudes AF, DG are added; there-

CL EI fore* the whole magnitudes AB, DE,

* 9 Dat.

BI

* 18 Dat.

have either a given ratio to one another, or the excess of one of them above a given magnitude, has a given ratio to the other.

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If there be three magnitudes, the excesses of one of which

above given magnitudes have given ratios to the other two magnitudes; these two 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, EF, be three magnitudes, and let GD, the excess of one of them CD above the given magnitude CG, have a given ratio to AB: and also let KD, the excess of the same CD above the given magnitude CK, have a given ratio to EF, either AB has a given ratio to EF, or the excess of one of them above a given magnitude has a given ratio to the other.

Because GD has a given ratio to AB, as GD to AB, so make CG to HA; therefore the ratio of CG to HA is given; and CG is given, wherefore * HA is given : * 2 Dat. and because as GD to AB, so is CG to HA, and so is * CD to HB: the ratio of CD to HB is given : also * 12. 5. because KD has a given ratio to EF, as KD to EF, so make CK to LE: therefore the ratio of CK to LE is given ; and CK H is given, wherefore LE* is given ; and because as KD to EF, so is CK to

A LE, and so* is CD to LF; the ratio of

* 12. 5.

GL
CD to LF is given : but the ratio of
CD to HB is given; wherefore * the

KÉ EL * 9 Dat. ratio of HB to LF is given; and from HB, LF the given magnitudes HA, BD LE being taken, the remainders AB, EF shall 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 *.

* 2 Dat.

D! FL

19 Dat.

Another Demonstration. Let AB, C, DE, be three magnitudes, and let the excesses of one of them C above given magnitudes have given ratios to AB, and DE; either AB, DE, 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.

* 14 Dat.

G

* 14 Dat.

Because the excess of C above a given magnitude has a given ratio to AB: therefore* AB together with a given magnitude has a given ratio to C: let this given magnitude be AF,

FI wherefore FB has a given ratio to C:

A

Di also because the excess of C above a given magnitude has a given ratio to DE; therefore * DE together with a given magnitude has a given ratio to BI CE C: let this given magnitude be DG, wherefore GË has a given ratio to C: and FB has a given ratio to C, therefore * the ratio of FB to GE is given: and from FB, GE, the given magnitudes AF, DG being taken, the remainders AB, DE, 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*.

* 9 Dat.

* 19 Dat.

19.

PROP. XXVII.

If there be three magnitudes, the excess of the first of

which above a given magnitude has a given ratio to the second ; and the excess of the second above a given magnitude has also a given ratio to the third : the excess of the first above a given magnitude shall have a given ratio to the third.

Let AB, CD, E, be three magnitudes, the excess of the first of which AB above the given magnitude, AG, viz. GB, has a given ratio to CD: and FD the excess of CD above the given magnitude CF, has a given ratio to E: the excess of AB above a given magnitude has a given ratio to E.

Because the ratio of GB to CD is given, as GB to
CD, so make GH, to CF; therefore
the ratio of GH to CF is given ; and

ΑΙ
CF is given, wherefore * GH is given;
and AG is given, wherefore the whole

GH
AH is given : and because as GB to

CI
CD, so is GH to CF, and so is * the
remainder HB to the remainder FD;
the ratio of HB to FD is given : and
the ratio of FD to E is given, where- Bl

BI DI E
fore* the ratio of HB to E is given :
and AH is given; therefore HB the excess of AB
above a given magnitude AH has a given ratio to E.

2 Dat.

19. 5.

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9 Dat.

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