Sidebilder
PDF
ePub

44

A SYLLABUS OF

[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 11,

A:C :: B:D.]

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, mn being any whole number, n may be found such that
either mA is between nB and (12+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 (12+1) B or =nB,
so is mp between nQ and (n + 1) or =nQ;
whence as m (A+B) is between (m + n)B and (m +1+1)B or
= (1 + n)B,

so is m (P+Q) between (m+n) and (m+n+I)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 nB when A>B
and therefore m<n, and subtracting mA and nB from mB
when A<B and therefore m>n, it

may be proved that A~B :B :: P~Q : Q.]

or

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 difference of the antecedent and consequent of the

other has to their difference or sum. THEOR. 11. 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 <an.Q, and therefore p.mA >=or <q nB, as p.mP >=or <q.nl, whence, p, 2 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 A: C :: P: R.
Lemma.-As A>=or <C, so is P>=or <R.

46

A SYLLABUS OF

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 <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 A :C :: P:R; 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 Q : R :: P : S,
whence Q :P :: R: S and P :( :: S : R.

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

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. II. When two ratios are equal, the ratio compounded of

them is called the duplicate ratio of either of the

original ratios. DEF. 12. When three ratios are equal, the ratio compounded of

them is called the triplicate ratio of any one of the original ratios.

SECTION 2.

FUNDAMENTAL GEOMETRICAL PROPOSITIONS.

THEOR, I. If two straight lines are cut by three parallel straight

lines, the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other.

(Let the three parallel lines AA', BB', CC', cut other two lines in A, B, C, and A', B', C' respectively:

then AB : BC :: A'B' : B'C'. On the line ABC take BM=m.AB and BN=n.BC, M and N being taken on the same side of B. Also on the line A'B'C' take B M'=m. A'B' and B'N'= n.B'C', M', N' being on the same side of B' as M, N are of B. It is easy to prove that MM and NN’ are both parallel to BB'. Hence, whatever be the values of 11 and n, as BM (or m. AB) is greater than, equal to, or less than BN (or n.BC), so is B M' (or m. A'B') greater than, equal to, or less than B'N' (or n.B'C'), therefore AB : BC :: A'B' : B'C'. It will be observed that the reasoning holds good, whether B be between A and C or beyond A or beyond C.]

48

A SYLLABUS OF

COR. 1. If the sides of a triangle are cut by a straight line

parallel to the base, the segments of one side are to one another in the same ratio as the segments of the other side.

COR. 2. If two straight lines are cut by four parallel straight

lines the intercepts on the one are to one another in the same ratio as the corresponding intercepts on the other.

THEOR. 2. A given finite straight line can be divided internally

into segments having any given ratio, and also externally into segments having any given ratio except the ratio of equality: and in each case there is only one such point of division.

[Let AB be the given straight line and, since any given ratio
may be expressed as the ratio of two straight lines, let AC, CD
be two lines having the given ratio taken on an indefinite line
drawn from A making any angle with AB; then CE parallel to
DB and meeting AB in E will (Theor. 1) divide AB internally
in E in the given ratio.
If it could be divided internally at F in the same ratio, BG
being drawn parallel to CF to meet AD in G, AF would be to
FB as AC to CG, and therefore not as AC to CD. Hence E
is the only point which divides AB internally in the given ratio.
If CD be taken so that A and D are on the same side of C, the
like construction will determine the external point of division.
In this case the construction will fail, if CD = AC. A like
demonstration will shew that there can be only one point of
external division in the given ratio.]

THEOR. 3. A straight line which divides the sides of a triangle

proportionally is parallel to the base of the triangle.

[From Theor. 1 by the Rule of Identity, since by Theor. 2 there is only one point of division of a given line in a given ratio.]

« ForrigeFortsett »