PROPOSITION 14.

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 if the line be given in both length and sense, there is in each case one and only one such point of division.

Let A B be the given straight line; it may be divided, as at E, in a given ratio P : Q.

Then CE,

G

For, on the straight line AG making any convenient. angle with AB lay off AC=P, CD=Q. drawn parallel to DB to meet AB in E, will divide A B at E in the given ratio. (Prop. 13.)

Since CE and D B are parallel, C and E lie on the same side of D and B, and hence the A division will be internal if A and D are on opposite sides of C, but external if A and D are on the same side of C.

A

FE B

B

F E

Fig. 5

If the line to be divided be estimated in a given sense, as from A to B, there is in each case only one point of division in the given ratio. as F, be joined to C and BG be drawn parallel to FC, AF: FB: AC: CG,

For if any other point,

then (Prop. 13.) so that F divides A B in the ratio AC: CG, different from

the given ratio.

If the given ratio be a ratio of equality, the construction in the case of external division fails.

PROPOSITION 15.

"A straight line which divides the sides of a triangle proportionally is parallel to the base of the triangle."

Let DE divide the sides A B, AC of the triangle ABC proportionally, so that

then

AD: DB:: AE: EC,

DE is parallel to BC.

If possible, let DF be parallel to BC, F some other point than E; then

AD: DB::AF: FC.

But by hypothesis

(Prop. 13.)

AD:DB:: AE: EC

... AF: FC:: AE: EC

which is only possible when F coincides

with E.

FROPOSITION 16.

Fig. 6.

E

Rectangles of equal altitude are to one another in the same ratio as their bases."

Let KA, KB be two rectangles having the common altitude OK and their bases OA, OB extending in the same line from O to the right; then

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

KN.

Whatever multiples OM and ON are of OA and

O B, the rectangles K M and KN are the same respective multiples of the rectangles K A and KB; that is,

and according as

OM (or m.OA) >= or < ON (or 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

On the two circumferences respectively take

OM = m.OA, O'N = n.O'B,

B

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 O KM 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 OKA 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 18.

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'. (Prop. 13, Cor. iii.)

Therefore, m being any integer,

mCm C':: R : R'.

(Prop. 8.)

Let m be the number of equal parts into which the angle at O is divided; then mC 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

S: S': R: R'.

Q. E. D. (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.)