and according as

OM (or m.OA) >= or < ON (or m.OB) so is KM (or m.KA)> or < KN (or n.KB)

=

.. rect. KA : rect. KB :: OA: OB.

(Def. 5.)

(i). COROLLARY: "Parallelograms or triangles of the same altitude are to one another as their bases."

PROPOSITION 17.

"In the same circle, or in equal circles, angles at the centre and sectors are to one another as the arcs on which they stand."

Let there be two equal circles with centres at K and K', and on their circumferences any two arcs OA, O'B; then angle OKA: angle O'K'B :: OA: O'B, sector OKA: sector O'K'B :: OA: O'B.

and

m and n being integers. Whatever multiples OM and O'N are of OA and O'B, the same multiples respectively are the angles or sectors OK M and O'K'N of the angles or sectors OK A and O'K'B; that is,

OKM = m.OKA, O'K'N

=

n. O'K'B,

and according as

so is

OM (or m.OA) >= or < O'N (or n. O'B)

OKM (or m.OKA)> or < O'K'N (or n. O'K'B).

=

.'. OKA: O'K’B :: OA : O'B,

(Def. 5.) wherein OK A and O'K'B represent either angles or sectors.

(i). COROLLARY: In any two given concentric circles, corresponding arcs intercepted by common radii bear always the same ratio to one another.

That is, if u, u', u'', . . . be arcs on one of the circles determined by a series of radii, and the same radii intercept on the other circle the corresponding arcs v, v', v'', . . . then u : v :: u' : v' :: u'' : v'' . . .

PROPOSITION IS.

Arcs of circles that subtend the same angle or equal angles at their centres are to one another as their radii.

their angles AOD and A'O D' either equal and distinct or common, and let R, R' be their respective radii; then

S: S': R: R'.

If the two arcs be not concentric, let them be made so, and let their bounding radii be made to coincide. Then

the proposition proved for the concentric will also be true for the non-concentric arcs.

Conceive the angle at O to be divided into m equal parts, m being any integer, by radii setting off the arcs S and S' into the same number of equal parts, and draw the equal chords of the submultiple arcs of S and the like equal chords of the submultiple arcs of S'. Let C and C' be the respective lengths of these chords.

Then, since the chords C, C' cut off equal segments on the lines O A', OB' they are parallel (Prop. 12), and

C: C': R: R'.

Therefore, m being any integer,

mCm C':: R: R'.

(Prop. 13, Cor. iii.)

(Prop. 8.)

Let m be the number of equal parts into which the angle at O is divided; then m C and m C' are the lengths of the polygonal lines formed by the equal chords of S and S' respectively.

If now m be increased indefinitely, the chords decrease in length but increase in number, and the two polygonal lines which they form approach coincidence with the arcs S and S' respectively; and by increasing m sufficiently the aggregate of all the spaces between the arcs and their chords may be made smaller than any previously assigned arbitrarily small magnitude. Under these circumstances it is assumed as axiomatic that the relation existing between the polygonal lines exists also between the arcs, which are called limits. Under this assumption it follows that

(i). COROLLARY: Circumferences are to one another as their radii.

(ii). COROLLARY: Of two arcs of circles that subtend the same angle or equal angles at their centres, that is the longer which has the longer radius. (By Prop. 2.)

E. AGENDA: SUPPLEMENTARY PROPOSITIONS.

(1). If two geometrical magnitudes A, B, have the same ratio as two integers m, n, prove that

nAm B.

(2). If A, B be two geometrical magnitudes and m, n two integers such that n A = mB, prove that AB: m : n.

Hence infer the statement in the first part of Definition 4, page 4, concerning commensurable magnitudes.

(3). Given A: B :: P: Q and ʼn A η

=

m B, prove that

(4). It is a corollary of (3), that if A: B :: P: Q and A be a multiple, part, or multiple of a part of B, then P is the same multiple, part, or multiple of a part of Q.

(5). Given A: B :: P: Q and B: C:: Q: R, prove that P> = or < R according as A > = or < C.

(6). Given A: B :: P: Q and B: C:: Q: R, prove that A: C: P: R. (Ex aequali.)

::

(7). Given A: B :: P: Q and B: C :: Q: R and C : D :: RS and D: E:: S: T, prove that

AE P T.

State and prove the general theorem of which this is a particular case.

(8). Given A : B :: Q: R and B: C:: P: Q, prove that A: C: PR.

1. Quantities in General. Quantities, whatever their nature, may be expressed in terms of geometrical magnitudes; in particular they may be thought of as straight lines of definite fixed or variable length. Such mag

nitudes, in so far as they represent the quantities of ordinary algebra, are of three kinds: real, imaginary, and complex; real if, when considered by themselves (laid off upon the real axis), they are supposed to involve only the idea of length, positive or negative, without regard to direction, imaginary when they involve not only length, but also turning or rotation through a right angle, that is, length and direction at right angles to the axis of real quantities, finally complex if they embody length and rotation through any angle, that is, length and unrestricted direction in the plane.

If we think of the straight line as generated by the motion of a point, we may translate length positive or negative into motion forwards or backwards; and it will sometimes be convenient to use the latter terminology in place of the former.

It is at once evident that both reals and imaginaries are particular forms of complex quantities, reals involving motion forwards or backwards and rotation through a zero-angle, imaginaries involving motion forwards or