Note-Although any treatment of the ratios of Geometric Magnitudes (which vary continuously) based on their representation by Arithmetic Numbers (which vary discontinuously) is entirely fallacious—still (as in the two preceding cases) by making the unit of measurement small enough, we can find numbers whose ratio will represent any proposed geometric ratio, to any assigned degree of accuracy.

The following method of considering Proportion is simple, and may be found an instructive aid towards gaining a clearer view of the subject.

Def. When two variable quantities are so related to each other that they vanish together, and that equal increments of the one always involve equal increments of the other, they are said to be proportional.

C2 C3

For example-suppose the altitude of a ▲ ABC to be fixed, but the base BC to vary.

As the base shrinks to nothing the ▲ also shrinks to nothing; and, as the base increases, any two equal increments C, C, and CC, of the

base, involve equal increments AC, C2 and AC, C, of the A: hence the ▲ and the base are said to be proportional to each other.

If we select any two values of the base, and the two corresponding values of the area of the ▲, they are said to form a proportion; and the first base is said to have to the second base the same ratio as the first area has to the second area. (Cf. vi. 1.)

The two bases and the two areas are called the four terms of the proportion. A proportion always consists of four terms, and the first term has to the second the same ratio as the third has to the fourth.

This method might be substituted for Euclid's treatment of Proportion, without any loss of exactitude, and with considerable gain in simplicity. The Student should (as an Exercise) apply it to vi. 33.

* Communicated to the Editor by Professor Everett, F.R.S., of Queen's College, Belfast.

BOOK vi.

Proposition 1.

THEOREM-If two triangles have the same altitude, then the ratio which one triangle has to the other is equal to the ratio which the base of the first has to the base of the second.

A

N M B

C

E P

R

Let ABC, ADE be ▲s which have the same altitude—viz. the from A on their common line of base BCDE.

In the production of CB set off any number of parts,

BM, MN, each of which = BC.

In the production of DE set off any number of parts,
EP, PQ, QR, each of which = DE.

Join A to each of the pts. M, N, P, Q, R.

..A ANC and line NC are equimults. of A ABC and base BC.

Similarly

AARD and line RD are equimults. of ▲ ADE and base DE. And A ANC >, =, or < ▲ ARD,

according as NC >,=, or < RD.

But this is the criterion that

▲ ABC: A ADE BC: DE,

which is .. true.

Proposition 2.

THEOREMS-(a) A straight line parallel to one side of a triangle cuts the other two sides (or these produced) proportionally; so that the segments terminated at the point of concurrence of the latter two sides are homologous : (B) the converse of this is also true.

(a) Let PQ, || to BC, one of the sides of▲ ABC, cut sides AB, AC in P, Q respecty. Join BQ, CP.

..

Then ▲ PQB = ▲ PQC;

they are on same base, and between same ||s.

Δ Δ
▲ PQB: ▲ PQA =

But Δ Δ
▲ PQB: A PQA

And A PQC: A PQA

=

A PQC : ▲ PQA.
Δ

BP: PA;

.. BP: PA CQ: QA.

(B) Next let PQ cut AB, AC so that

BP: PACQ: QA.

Δ Δ

Then ▲ PQB: A PQA BP: PA;

=

Proposition 3.

THEOREMS (a) If the vertical angle of a triangle is bisected, internally or externally, by a straight line which also cuts the base, then the base is divided internally or externally in the ratio of the sides of the triangle; so that each segment and its conterminous side are homologous : (B) the converse of this is also true.

BAC in fig. (1),

or CÂE, external to BẤC, in fig. (2).

Draw CD | to AX, meeting BA, or BA produced, in D.

Then ADC

..

= BAX in fig. (1), or = EAX in fig. (2),

=

CÂX in both figs.

= AĈD.

AD =

AC.

BA: AC

BX: XC.

(8) Next let AX meet BC, fig. (1), or BC produced, fig. (2),

so that BA: AC = BX : XC.

Constructing as before, we have

BA: AD BX: XC.

.. CÂX BÂX in fig. (1), or EÂX in fig. (2) :

=

i. e. AX bisects BÂC, or CAE external to BAC.

Proposition 4.

THEOREM-If two triangles are equiangular to each other, the sides which contain any one of the angles of the one, are proportional to the sides which contain that angle which is equal to it in the other; and those sides which are opposite equal angles are homologous terms in the ratios.

Ад

A

A =

B

D

Let ABC, DEF be

As which have ▲ 8 at

E

A, B, C and at D, E, F, respectively equal.

we can place ▲ DEF on ▲ ABC,

so that D may be on A, DE on AB, and DF on AC.
Then E will take a position X in AB, or AB produced;
and F will take a position Y in AC, or AC produced.

And

AXY = E = B;

... XY is to BC.

.. AB: AC = AX: AY,

= DE: DF.

Similarly for sides about other pairs of equal Ʌ".