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of e b, that mp is of fd. And because a b is to
be as cd is to df, and that of ab and d, gk
and In are equimultiples, and of eb and fd, hx
and m p are equimultiples; if gk be greater than
hx, then 1n is greater than mp; and if equal,
equal; and if less, less (v def. 5); but if g h be
greater than kx, by adding the common part hk
to both, gk is greater than hx; wherefore also n
is greater than mp; and by taking away mn from h
both, 1m is greater than np: therefore, if g h be
greater than kx, 1m is greater than np. In like
manner it may be demonstrated, that if g h be equal
to kx, 1m likewise is equal to n p; and if less, less :
and gh, lm are any equimultiples whatever of
a e, cf, and kx, np are any whatever of eb, fd.
Therefore (v. def. 5), as a e is to e b, so is cf to fd.
If then magnitudes, &c. Q. E. D.

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PROPOSITION XVIII. THEOREM.

If magnitudes, taken separately, be proportionals, they shall also be proportionals when taken jointly, that is, if the first be to the second as the third to the fourth, the first and second together shall be to the second as the third and fourth together to the fourth.

LET ae, eb, cf, fd be proportionals; that is, as ae to eb, so is cf to fd; they shall also be proportionals when taken jointly; that is, as ab to be, so cd to df.

Take of a b, be, cd, df any equimultiples whatever gh, hk, 1m, mn and again, of be, df, take any equimultiples whatever ko, np: and because ko, np are equimultiples of be, df; and that k h, n m are equimultiples likewise of be, df, if ko, the multiple of be, be greater than kh, which is a multiple of the same be, np likewise the multiple of df, shall be greater than mn, the multiple of the same df; and if ko be equal to kh, np shall be equal to nm; and if less, less.

First, let ko not be greater than kh, therefore np is not greater than nm: and because gh, hk, are equimultiples of ab, be, and that ab is greater than be, therefore gh is greater (v. ax. 3) than hk; but ko is not greater than kh, wherefore gh is greater than ko. In like manner it may be shewn, that Im is greater than np. Therefore, if ko be not greater than kh, then gh, the multiple of a b, is always greater than o, the multiple of be; and likewise 1m, the multiple of cd, greater than np, the multiple of df.

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Next, let ko be greater than kh: therefore, as has been shewn, np

is greater than nm the whole a b, that

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and because the whole gh is the same multiple of hk is of be, the remainder gk is the same multiple of the remainder ae that gh is of a b (v. 5): which is the same that 1m is of cd. In like manner, because Im is the same multiple of cd, p that m n is of df, the remainder In is the same multiple of the remainder cf, that the whole 1m is of the whole cd (v. 5): but it was shewn that 1m is the same multiple of cd, that gk is of ae; therefore gk is the same multiple of a ẹ, that In is of cf; that is, gk, ln are equimultiples of a e, cf: and because ko, np, are equimultiples of be, df, if from ko, np, there be taken kh, nm, which are likewise equimultiples of be, df, the remainders ho, mp are either equal to be, df, or equimultiples of them (v. 6). First, let ho, mp be equal to be, df; and because ae is to eb, as cf to fd, and that gk In are equimultiples of ae, cf; gk shall be to eb, as In to fd (v. 4 cor.): but ho is equal to eb, and mp to fd; wherefore gk is to ho, as In to mp. If therefore gk be greater than ho, ln is greater than mp; and if equal, equal; and if less, less (5 ax.).

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But let ho, mp be equimultiples of eb, fd; and because ae is to eb, as cf to fd, and that of e, cf are taken equimultiples gk, ln; and of eb, fd, the equimultiples ho, mp; if gk be greater than ho, 1n is

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greater than mp; and if equal, equal; and if less, less (v. def. 5); which was likewise shewn in the preceding case. If therefore gh be greater than ko, taking k h from both, gk is pgreater than ho; wherefore also In is greater than mp; and consequently adding nm to both, im is greater than np: therefore, if g h be greater than ko, 1m is greater than np. In like manner it may be shewn, that if gh be equal to ko, lm is equal to np; and if less, less. And in the case in which ko is not greater than kh, it has been shewn that gh is always greater than k o, and likewise 1m than np: but gh, lm are any equimultiples of a b, cd, and ko, np are any whatever of be, df; therefore (v. def. 5), as a b is to be, so is cd to df. If then magnitudes, &c. Q. E. D.

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PROPOSITION XIX.-THEOREM.

If a whole magnitude be to a whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder shall be to the remainder as the whole to the whole.

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LET the whole a b be to the whole cd, as a e, a magnitude taken from ab, to cf, a magnitude taken from cd; the remainder eb a shall be to the remainder fd, as the whole ab to the whole c d. Because ab is to cd, as ae to cf; likewise, alternately (v. 16), ba is to ae, as dc to cf; and because if magnitudes, taken jointly, be proportionals, they are also proportionals e (v. 17) when taken separately; therefore, as be is to ea, so is df to fc, and alternately, as be is to d f, so is ea to fc; but, as ae to cf, so, by the hypothesis, is a b to c d therefore also be, the remainder, shall be to the remainder d f as the whole ab to the whole c d. Wherefore, if the whole, &c. Q. E. D. COR. If the whole be to the whole, as a magnitude taken from the first is to a magnitude taken from the other, the remainder likewise is to the remainder, as the magnitude taken from the first to that taken from the other. The demonstration is contained in the preceding.

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PROPOSITION E.-THEOREM.

If four magnitudes be proportionals, they are also proportionals by conversion: that is, the first is to its excess above the second, as the third to its excess above the fourth.

LET ab be to be, as cd to df; then ba is to ae, as dc to cf.

Because ab is to be, as cd to df, by division (v. 17), a e is to eb, as cfto fd; and by inversion (v. B.), be is to ea as df to fc. Wherefore, by composition (v. 18), ba is to ae as dc is to cf. If, therefore, four magnitudes, &c. Q. E. D.

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PROPOSITION XX.—THEOREM.

If there be three magnitudes, and other three, which, taken two and two, have the same ratio; if the first be greater than the third, the fourth shall be greater than the sixth, and if equal, equal; and if less, less.

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LET a, b, c be three magnitudes, and d, e, f other three, which, taken two and two, have the same ratio, viz. as a is to b, so is d to e; and as b to c, so is e to f. If a be greater than c, d shall be greater than f; and if equal, equal; and if less, less.

Because a is greater than c, and b is any other magnitude, and that the greater has to the same magnitude a greater ratio than the less has to it (v. 8); therefore a has to b a greater ratio than c has to b; but as d is to e, so is fa to b; therefore (v. 13) d has to e a greater ratio than c to b; and because b is to c, as e to f, by inversion, c is to b, as f is to e; and d was shewn to have to e a greater ratio than c to b; therefore d has to e a greater ratio than fto e (v. 13. cor.). But the magnitude which has a greater ratio than another to the same magnitude, is the greater of the two (v. 10); d is therefore greater than f.

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Secondly, let a be equal to c; d shall be equal to f. Because a and c are equal to one another, a is to b as c is to b (v. 7): but a is to bas d to e; and c is to bas f to e; wherefore d is to e as f to e (v. 11); and therefore d is equal to f (v. 9).

Next, let a be less than c; d shall be less f than f; for c is greater than a, and, as was shewn in the first case, c is to b, as f to e, and in like manner, b is to a, as e to d; therefore f is greater than d, by the first case; and therefore d is less than f. Therefore, if there

be three magnitudes, &c. Q. E. D.

PROPOSITION XXI.—THEOREM.

If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.

LET a, b, c be three magnitudes, and d, e, f other three, which have the same ratio, taken two and two, but in a cross order, viz. as a is to b, so is e to f, and as b is to c, so is d to e. If a be greater than c, d shall be greater than f; and if equal, equal; and if less, less.

Because a is greater than c, and b is any other magnitude, a has to b a greater ratio (v. 8) than c has to b: but as e is to f, so is a to b:

therefore (v. 13) e has to f a greater ratio than c to b. And because b is to c as d to e, by inversion, c is to b as e to d: and e was shewn to have to f a greater ratio than c to b; therefore e has to f a greater ratio than e to d (v. 13. cor.); but the magnitude to which the same has a greater ratio than it has to another, is the lesser of the two (v. 10): f therefore is less than d; that is, d is greater than f.

Secondly, let a be equal to c; d shall be equal to f. Because a and c are equal, a is (v. 7) to b, as c is to b: but a is to b, as e to f; and c is to b, as e to d; wherefore e is to f, as e to d (v. 11); and therefore d is equal to f (v. 9).

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Next, let a be less than c d shall be less than f: for c is greater than a, and, as was shewn, c is to b, as e to d, and in like manner b is to a, as fto e; therefore f is greater than d, by case first; and therefore, d is less than f. Therefore, if there be three magnitudes, &c. Q. E. D.

PROPOSITION XXII.-THEOREM.

If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N.B.-This is usually cited by the words " ex æquali," or ex æquo."

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FIRST, let there be three magnitudes a, b, c, and as many others d, e, f, gk m h 1 n

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