and draw the perpendiculars, BE, CD, meeting that semi-circumference in the points E, D. Join OE, OD, and with these lines as radii from the centre, O, describe circles; these circles will divide the given circle into the required number of equal parts.

A

B C

For join A E, AD; then the angle AD O, being in a semicircle, is a right angle (Prop. XVIII. Cor. 2, Bk. III.); hence the triangles DA O, D C O are similar, and consequently are to each other as the squares of their homologous sides; that is,

consequently, since circles are to each other as the squares of their radii (Prop. XIII.), it follows that the circle whose radius is OA, is to that whose radius is OD, as O A to OC; that is to say, the latter is one third of the former.

In the same manner, by means of the right-angled triangles E AO, E BO, it may be proved that the circle whose radius is O E, is two thirds that whose radius is OA. Hence, the smaller circle and the two surrounding annular spaces are all equal.

NOTE.

This useful problem was first solved by Dr. Hutton, the justly distinguished English mathematician.

BOOK VII.

PLANES. DIEDRAL AND POLYEDRAL ANGLES.

DEFINITIONS.

388. A STRAIGHT line is perpendicular to a plane, when it is perpendicular to every straight line which it meets in that plane. Conversely, the plane, in the

same case, is perpendicular to the line.

A

M

B

The foot of the perpendicular is the point in which it meets the plane.

Thus the straight line A B is perpendicular to the plane MN; the plane MN is perpendicular to the straight line AB; and B is the foot of the perpendicular A B.

389. A line is parallel to a plane when it cannot meet the plane, however far both of them may be produced. Conversely, the plane, in the same case, is parallel to the line.

390. Two planes are parallel to each other, when they cannot meet, however far both of them may be produced.

391. A DIEDRAL ANGLE is an

angle formed by the intersection of two planes, and is meas

ured by the inclination of two straight lines drawn from any A point in the line of intersection, perpendicular to that line, one being drawn in each plane.

B

M

N

The line of common section is called the edge, and the two planes are called the faces, of the diedral angle.

Thus the two planes A B M, ABN, whose line of intersection is AB, form a diedral angle, of which the line AB

is the edge, and the planes ABM, ABN are the faces.

392. A diedral angle may be acute, right, or obtuse. If the two faces are perpendicular to each other, the angle is right.

393. A POLYEDRAL ANGLE is an angle formed by the meeting at one point of more than two plane angles, which are not in the same plane.

The common point of meeting of the planes is called the vertex, each of the plane angles a face,

A

B

S

C

and the line of common section of any two of the planes an edge of the polyedral angle.

Thus the three plane angles ASB, BSC, CSA form a polyedral angle, whose vertex is S, whose faces are the plane angles, and whose edges are the sides, AS, BS, CS, of the same angles.

394. A polyedral angle formed by three faces is called a triedral angle; by four faces, a tetraedral; by five faces, a pentaedral, &c.

PROPOSITION I.THEOREM.

395. A straight line cannot be partly in a plane, and partly out of it.

For, by the definition of a plane (Art. 10), a straight

line which has two points in common with a plane lies wholly in that plane.

396. Scholium. To determine whether a surface is a plane, apply a straight line in different directions to that surface, and ascertain whether the line throughout its whole extent touches the surface.

PROPOSITION II.-THEOREM.

397. Two straight lines which intersect each other lie in the same plane and determine its position.

Let AB, AC be two straight lines which intersect each other in A; then these lines will be in the same plane.

B

A

Conceive a plane to pass through A B, and to be turned about AB, until it pass through the point C; then, the two points A and C being in this plane, the line AC lies wholly in it (Art. 10). Hence, the position of the plane is determined by the condition of its containing the two straight lines A B, A C.

398. Cor. 1. A triangle, ABC, or three points, A, B, C, not in a straight line, determine the position of a plane.

399. Cor. 2. Hence, also, two parallels, A B, CD, determine the position of a plane; for, drawing the secant EF, the plane of the two straight lines A B, EF is that of the parallels AB, CD.

A

E

B

C

D

PROPOSITION III. THEOREM.

400. If two planes cut each other, their common section is a straight line.

Let the two planes A B, CD cut each other, and let E, F be two points in their common section. Draw the straight line EF. Now, since the points E and F are in the plane A B, and also in the plane CD, the straight line EF, joining E and F,, must be wholly in each plane, or is common to both of them. Therefore, the common section of the two planes AB, A CD is a straight line.

PROPOSITION IV. - THEOREM.

401. If a straight line is perpendicular to each of two straight lines, at their point of intersection, it is perpendicular to the plane in which the two lines lie.

Let the straight line AB be. perpendicular to each of the straight lines CD, EF, at B, the point of their intersection, and MN the plane in which the lines CD, EF lie; then will A B be perpendicular to the plane MN.

A

M

F

B

E

N

Through the point B draw any straight line, BG, in the plane MN; and through any point G draw D G F, meeting the lines CD, EF in such a manner that DG shall be equal to GF (Prob. XXVIII. Bk. V.). Join AD, AG, AF.

The line D F being divided into two equal parts at the point G, the triangle D B F gives (Prop. XIV. Bk. IV.) B F2 + B D2 =2BG2+2 GF2.

The triangle D AF, in like manner, gives

A F2 + A D2 = 2 A G2 + 2 G F2.

Subtracting the first equation from the second, and ob