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.

THEOREM 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.

Proof. Let A B C D.

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

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

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

THEOREM 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.

Proof. Let A B :: C: D, then A : C :: B : D.

:

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 n,

A: C: B: D.

THEOREM: 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.

:

Proof. 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. :

THEOREM IO.

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.

Proof. Let AB:: 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 = 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 = (m + n)B,

or =

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 second has to their difference or sum.

THEOREM 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.

Proof. Let A
For pm. A>

and therefore p. mA>

B

P : Q, then mA : nB :: mP : nQ. or <qn. B, aš pm. P >= or <qn. Q,

or <q.nB, as p. mP> or <q.nQ,

whence, p, q being any numbers whatever,

mA nB :: mP : nQ.

THEOREM 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.

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

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 <C, so is mP> or <nR, and therefore, m and n being any numbers whatever,

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 A: C :: P: R; but C: D :: R: S,

and therefore A D :: P: S.

COR.

Q. E. D.

If A B QR and B: C: P: Q, then

A CP: R.

Proof. Let S be a fourth proportional to Q, R, P,

then QR: P: S,

therefore Q P :: R: S,

and P Q S : R.

Hence A B :: P: S and B: C: S: R,

therefore A: C: P: R.

(Th. 8.)

(Th. 3.)

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 set is equal to that compounded of the ratios of the other

set.

Also that the ratio compounded of a given ratio and its reciprocal is the ratio of equality.

Def. When two ratios are equal, the ratio compounded of them is called the duplicate ratio of either of the original ratios.

Def. When three ratios are equal, the ratio compounded of them is called the triplicate ratio of any one of the original ratios.

SECTION II.

FUNDAMENTAL GEOMETRICAL PROPOSITIONS.

LEMMA.

If on two straight lines AB, CD cut by two parallel straight lines AC, BD equimultiples of the intercepts respectively are taken; then the line joining the points of division will be parallel to AC or BD.