[Let A B :: P: Q, then as A> or < B, so is P > = or <Q.

:

This is contained in Def. 4, if the multiples taken be the magnitudes themselves.]

THEOR. 3. If two ratios are equal, their reciprocal ratios are

For, since the multiples of A are distributed among those of B

as the multiples of P among those of Q, the multiples of B are distributed among those of A as the multiples of Q among those of P.]

THEOR. 4. If the ratios of each of two magnitudes to a third magnitude be taken, the first ratio will be greater than, equal to, or less than the other as the first magnitude is greater than, equal to, or less than the other and if the ratios of one magnitude to each of two others be taken, the first ratio will be greater than, equal to, or less than the other as the first of the two magnitudes is less than, equal to, or greater than the other.

[Let A, B, C be three magnitudes of the same kind, then

A: C>= or < B: C, as A> or <B

=

If A=B, it follows directly from Def. 4 that A: C:: B: C and
CA: B: A.

If A >B, m can be found such that mB is less than mA by a
greater magnitude than C.

Hence if mA be between nC and (n+1)C, or if mA=nC, mB will be less than C, whence (Def. 6) A : C>B: C;

Also, since C>mB while nC is not> mA (Def. 6) C: B > C: A or C: A<C: B.

If A <B, then B>A and therefore B: C>A : C, that is A: C <B: C. and so also C: A> C: B.

Hence the proposition is proved.]

COR. The converses of both parts of the proposition are true, since the "Rule of Conversion" is applicable.

THEOR. 5. The ratio of equimultiples of two magnitudes is equal to that of the magnitudes themselves.

[Let A, B be two magnitudes, then mA : mB :: A : B.

For as pA> or <qB, so is m.pA>=or<m.qB; but m.pA= p.mA and m.qB=q.mB, therefore as pA>=or<qB, so is p.mA > or <q.mB, whatever be the values of p and q, and hence mA mB:: A: B.]

:

THEOR. 6. If two magnitudes (A, B) have the same ratio as two whole numbers (m, n), then nA = mB: and conversely

if nA = mB, A has to B the same ratio as m to n.
[Of A and m take the equimultiples nA and n.m, and of B and
n take the equimultiples mB and m.n, then since n.m=m.n, it
follows (Def. 4) that nA=mB.

Again since by Def. 4 mB : nB :: m : n we have, if nA = mB,
nAnB: mn; whence it follows (Theor. 5) that A : B ::
m: n.]

=

COR. If A: B:: P: Q and nA= mB, then nP mQ; whence if A be a multiple, part, or multiple of a part of B, P is the same multiple, part, or multiple of a part of Q.

THEOR. 7. If four magnitudes of the same kind be proportionals, the first will be greater than, equal to, or less than the third, according as the second is greater than, equal to, or less than the fourth.

Then if A=C, A: B:: C: B, and therefore C: D :: C: B, whence BD.

Also if A>C, A: B>C: B, and therefore C: D>C: B, whence B> D.

Again if A <C, A : B<C: B, and therefore C: D<C: B, whence B<D.]

THEOR. 8. If four magnitudes of the same kind be proportionals, the first will have to the third the same ratio as the second to the fourth.

For (Th. 6) mA : mB :: A: B and nC : nD :: C: D;
therefore mA : mB:: nC: nD,

whence (Th. 7) mA>=or <nC, as mB>= or <nD,
and this being true for all values of m and »,

THEOR. 9. If any number of magnitudes of the same kind be proportionals, as one of the antecedents is to its consequent, so shall the sum of the antecedents be to the sum of the consequents.

[Let A B : C: D :: E : F, then A : B :: A+C+E : B+D+F.

For as mA>=<nB, so is mC>=or<nD,

and so also is mE>=or <nF; whence it follows that so also is mA+mC+mE>=or<nB+nD+nF and therefore so is m (A+C+E)>=or<n (B+D+F), whence A B :: A+C+E : B+D+F.]

THEOR. 10. If two ratios are equal, the sum or difference of the antecedent and consequent of the first has to the consequent the same ratio as the sum or difference of the antecedent and consequent of the other has to its consequent.

[Let A B :: P: Q, then A+B : B :: P+Q: Q and A B B P~Q: Q.

~

For, m being any whole number, n may be found such that either mA is between nB and (n+1) B or mA = nB

and therefore mA+mB is between mB+nB and mB+(n+1)B
or=mB+nB;

but mA+mB:
3=m (A+B) and mB+nB=(m+n)B,

therefore m (A+B) = is between (m+n)B and (m+n+1)B or=
(m+n)B.

But as mA is between nB and (n+1) B or =nB,

so is mP between nQ and (n+1)Q or =nQ;

whence as m (A+B) is between (m+n)B and (m+n+1)B or =(m+n)B,

so is m (P+Q) between (m+n)Q and (m+n+1)Q_or = (m + n)Q,

and therefore, since m is any whole number whatever,

A+B B: P+Q: Q.

By like reasoning subtracting mB from mA and B when A>B and therefore m<n, and subtracting mA and B from mB when AB and therefore m>n, it may be proved that

COR. If two ratios are equal, the sum or difference of the antecedent and consequent of the first has to their difference or sum the same ratio as the sum or difference of the antecedent and consequent of the other has to their difference or sum.

THEOR. II. If two ratios are equal, and equimultiples of the antecedents and also of the consequents are taken, the multiple of the first antecedent has to that of its consequent the same ratio as the multiple of the other antecedent has to that of its consequent.

[Let A B :: P: Q, then mA : nB :: mP : nQ.

For pm. A> or <qn.B, as pm. P>=or<qn.Q, and therefore p.mA>=or < q nB, as p.mP >=or <q.nQ, whence, p, q being any numbers whatever,

mA: nB :: mP : nQ.]

THEOR. 12. If there be two sets of magnitudes, such that the first is to the second of the first set as the first to the second of the other set, and the second to the third of the first set as the second to the third of the other, and so on to the last magnitude: then the first is to the last of the first set as the first to the last of the other.

[Let the two sets of three magnitudes be A, B, C and P, Q, R, and let A B: P Q and B: C:: Q: R,

then AC :: P: R.

Lemma.-As A >= or <C, so is P >=or <R.

For if A> C, A: B>C: B and C

therefore P: Q>R: Q, whence P > R.

B:: R: Q,

Similarly if A = C or if A < C. Hence the lemma is proved.
By Theor. 6, mA : mB :: mP : mQ, and by Theor. 11, mB : nC
:: mQ: nR, whence by the lemma as mA>=or<nC, so
is mP >= or <nR, and therefore, m and n being any numbers
whatever,

A C: P: R.

If there be more magnitudes than three in each set, as A, B, C,
D and P, Q, R, S;

then, since A: B :: P Q and B: C:: Q: R,

therefore AC: PR; but C: D :: R: S,

and therefore A: D:: P: S.]

COR. If A: B::Q:R and B: C:: P: Q, then A: C :: P: R.

[Let S be a fourth proportional to Q, R, P,

then QR: P: S,

whence Q P :: R S and P: Q :: S: R, Hence A B :: P S and BC :: S: R, therefore AC :: P: R.]

DEF. 9. If there are any number of magnitudes of the same kind, the first is said to have to the last the ratio compounded of the ratios of the first to the second, of the second to the third, and so on to the last magnitude.

DEF. 10. If there are any number of ratios, and a set of magnitudes is taken such that the ratio of the first to the second is equal to the first ratio, and the ratio of the second to the third is equal to the second ratio, and so on, then the first of the set is said to have to the last the ratio compounded of the original ratios.

OBS. From these Definitions it follows, by Theor. 12, that if there be two sets of ratios equal to one another, each to each, the ratio compounded of the ratios of the first