Def. A straight line divided as in vi. 30 (i. e. as in ii. 11) is said to be divided in extreme and mean ratio.

Proposition 31.

THEOREM-If similar rectilineal figures are similarly described on the three sides of a right-angled triangle, the figure on the hypotenuse is equal to the sum of the figures on the other sides.

Since simr. figs. are as sqs. on their homologous sides;
... X: Y = sq. on BC: sq. on AC.

.., componendo,

X + Y : Y = sq. on BC + sq. on AC : sq. on AC,
=sq. on AB : sq. on AC,

= Z: Y.

.. X + Y = Z.

Proposition 32.

THEOREM-If two triangles have two sides of the one proportional to two sides of the other, and are so placed at an angle that the homologous sides are parallel, the remaining sides of the triangles are in a straight line.

This proposition is omitted as quite useless.

Without some further limitation to the given conditions, it is not even necessarily true.

Then it is clear that either of the positions of ▲ BYC, given in fig., satisfies, the stated condns., but that only one of them gives the stated result.

Proposition 33.

THEOREMS-In equal circles (or the same circle) the ratio

of--

(a) any two angles at the centres;

or (B) any two angles at the circumferences ;

or (y) any two sectors;

is equal to the ratio of the respective arcs on which they stand.

S

B

R

W

Let PQ, XY be any arcs of equal Os whose centres are respecty.

A and B.

Along circumf. of O, centre A, set off any

number of arcs

QR, RS, each of which = PQ.

Along circumf. of O, centre B, set off any number of arcs

YU, UV, VW, each of which = XY.

Join A to each of the pts. R, S; and B to each of pts. U, V, W. Then (a) arc PQ = arc QR = arc RS,

.. PÂS and arc PS are equimults. of PÂQ and arc PQ. Similarly XÊW and arc XW are equimults. of XBY and arc XY. And PAS >,=, or < XBW,

according as arc PS >,=, or < arc XW.

But this is the criterion that

8

Also (8) an exactly similar process will serve for Ʌs which PQ, XY subtend at circumfs.

Or the proportionality of Ʌ at the circumfs. to the arcs on which they stand, may be deduced from that of the corresponding As at the centres by the consideration that they are equisubmults.—viz. halves of them.

Lastly (y) first prove, as a Lemma, iii. Addenda (8); and then prove the proportionality of sectors APQ, BXY to their arcs, exactly in the same way as (a) is done here.

Note-All the propositions of Euclid's first four, and sixth, Books have been enunciated in the preceding pages. Most of the more familiar intercalations of his various Editors will be found in the Addenda to the different Books.

COROLLARIES TO THE PROPS. IN Book vi.

Def. The distance between an opposite pair of sides of a parallelogram (measured by the length of a perpendicular dropped from any point in one on the other) is called an altitude of the parallelogram, with respect to either of those sides considered as base.

Note-Obviously a parallelogram has two altitudes.

vi. 1. (a) Parallelograms of the same altitude are in the same ratio as their bases this follows at once from the Prop. by considering that the parallelograms are equimultiples (viz. doubles) of triangles of the same altitude; or it can be proved by a precisely similar method to that used in the Prop.

(B) As in the Prop. it could be shown that triangles on equal bases are as their altitudes.

(7) So also parallelograms on equal bases are as their altitudes.

(8) Triangles of equal altitude are as their bases.

(e) Parallelograms of equal altitude are as their bases.

(5) The Converses of the Prop., and of all the preceding Corollaries,

follow easily by reductio ad absurdum.

vi. 4. (a) A line drawn across a triangle, parallel to a side, cuts off a similar triangle.

(B) In equiangular triangles the altitudes drawn to homologous sides are proportional to those sides.

(7) A line drawn from a corner of a triangle (considered as vertex) to meet the base, divides every parallel to the base (terminated by the sides, or sides produced) in the same ratio.

vi. 10.

The external section of a line, in a given ratio, can be done in exactly the same way.

vi. 13. By this Prop. 3, 7, 15, &c., means may be found between two given lines for after one mean is found, a mean can be found between it and each of the given lines, thus getting 3 means; then, again, finding means between each successive two of these we get 7 means; and continuing this process we can find 2-1 means, where n is any positive integer.

Note The Problem of finding 2 means between two given lines is insoluble by the ungraduated ruler and compasses alone. On p. 284 will be found one of the many ways of solving it by further mechanical aid.

vi. 16. The principle alternando follows at once from this Prop. in the case of straight lines. If therefore it can be shown that the ratio of any two magnitudes of the same kind can be represented by the ratio of two straight lines, the principle (for all such magnitudes) is an immediate deduction from this Prop.

Now in the geometry of the Point, Line, and Circle, the only magnitudes that can occur are

(1) Lines (including straight lines, and arcs of circles).

(2) Angles.

(3) Areas (including rectilineal figures, circles, and sectors of circles). But to any arc of a circle there is a straight line equivalent in length; and though we cannot (by the use of the ruler and compasses) find this line, it clearly has an existence, and might be hypothetically reasoned about. Hence arcs of circles are proportional to straight lines.

Angles again are proportional to arcs of any (the same) circle.

Rectilineal figures can be reduced to rectangles, having a common altitude, and therefore proportional to their bases.

Circles can be shown (Euclid xii. 2) to be proportional to the squares on their radii; and therefore come under the conditions of rectilineal figures.

Sectors of the same circle are proportional to their arcs; and sectors of different circles are proportional to the circles they are parts of; and therefore to the squares on their radii.

So that (assuming xii. 2, and the above hypothetical construction) alternando follows from vi. 16; and then (as on p. 244) ex æquali and componendo can be deduced in this way Book v. might be dispensed with.

Note-It is however to be carefully noted that Euclid does not permit the use of hypothetical constructions; and therefore that to introduce such, is to travel outside the limitations of geometrical reasoning which he has laid down-though not necessarily to be illogical.

vi. 20.

The perimeters of similar rectilineal figures are proportional to their homologous sides.

vi. 22.

If four lines are proportional, the squares on them are proportional; and, conversely, if four squares are proportional, their sides are proportional.

vi. 23. (a) Triangles which have an angle of the one equal, or supplementary, to an angle of the other, have to one another the ratio compounded of the ratios of the sides about these angles.

(B) Equiangular parallelograms are in the same ratio as the rectangles under the sides forming a pair of equal angles.

(7) Triangles which have an angle of the one equal, or supplementary, to an angle of the other, are in the same ratio as the rectangles under the sides forming these angles.

vi. 25. The shape of any given rectilineal figure may be changed (without altering its area) to the shape of any other given rectilineal figure.

vi. 30. The greater segment will be itself divided in extreme and mean ratio by setting off, from one end of it, a part equal to the lesser segment; and this process can be continued indefinitely. Cf. Cor. to ii. 11, and v. Addenda (1).