6. From the longer of two unequal right lines to cut off a part equal to the shorter.-Let A B (fig. 6) be the longer of the two lines, from which it

is required to cut off a part equal to C the shorter. From the point A draw in any direction the line A D equal to C (5). From A as a centre, and with radius A D, describe a cir

Fig. 6.

cle cutting A B in E; then AE is equal in length to C.

ANGLES.

Fig. 7.

7. To construct an angle equal to a given angle.-Let BAC (fig. 7) be the given angle made by the two lines AB and AC meeting at A. From A as a centre and with any radius. A B, describe a circle cutting AB and AC in B and C; join BC. Draw any right line DE, and from D as a centre, with a radius equal to A B, describe a circle cutting DE in the point E. From E as a centre, with a radius equal to BC, describe an arc cutting the circle in the point F. then the angle FDE is equal to the angle BA C. parallel (12) to A C, then D F will be also parallel to A B.

Join FD;

If DE is

8. To divide a given rectilinear angle into two or more equal angles.-Let BAC (fig. 8) be the given angle made by the lines A B and A C. From A as a centre, with any radius A B, describe a circle cutting the lines in B and C; join B and C. From B and C as centres, and with BC as a radius, describe circles intersecting at the point D.

Join AD; then the angle BAC is bisected or divided into two equal angles by the line A D.

In practice it is sufficient to bisect the line BC (2) in the

these again can be bisected, the angle will be divided into eight equal angles; and in this way we can subdivide an angle into any number of equal angles, that number being a power of two, as 4, 8, 16, 32, and so forth.

We cannot divide an angle mathematically into any odd number of equal parts, but practically this can be done as nearly as we please by dividing the arc BC by means of dividers, or by the use of an instrument called a protractor.

Fig. 9.

9. To divide a right angle into three or more equal angles.-Let A O B (fig. 9) be the given right angle. From O as a centre, and with any radius OA, describe a circle; and from A and B as centres, with the same radius, draw arcs, cutting the circle in the points C and D. Join CO and DO; then the angles AOC, COD, DOB are all equal.

Also, since we can bisect each of the angles AOC, COD, DOB (8), and divide them into any number of equal parts being a power of 2; we are enabled to divide a right angle into 3, 6, 12, 24, 48 equal

parts, or any number being a product of 3 and a power of 2.

The half, quarter, eighth, and so on, of a right angle can be subdivided in the same way, for one-sixth of a right angle is one-third of half a right angle; and one-twelfth of a right angle is one-third of a quarter of a right angle; and so forth.

10. To divide a right angle into five or fifteen equal parts. Let the line A E (fig. 10) be at right angles to A B. Take any length AB,

and divide it at C in extreme and mean ratio

(19). From A as a centre, with radius A B, describe a circle, and from B as a centre, with AC for radius, draw an arc cutting the circle in D. Join DA, then the

Fig. 10.

angle DAB is one-fifth of two right angles, or two-fifths of one right angle; bisect the angle D A B (8) by the line AF, and the angle BAF is one-fifth of the right angle EA B. Divide the angle FA E into four equal parts by the lines DA, JA, HA, and the right angle will be then divided into five parts, each equal to FA B.

To divide the angle B A E into fifteen equal parts: draw the circle BDE from A as a centre; then from E as a centre, with EA for radius, draw an arc cutting the circle at G; join GA. Then G AB is one-third or five-fifteenths of a right angle, and DAB being two-fifths or six-fifteenths of a right angle, consequently the angle GAD is onefifteenth of the right angle. By measuring DG along the

quadrant BE, we can divide it into fifteen equal parts, and hence the right angle can also be divided into fifteen equal parts. Since each of the five or fifteen parts can be divided by two, four, eight, &c., it is possible to divide the right angle into five, ten, fifteen, twenty, thirty, and any other number of equal parts, being a product of five or fifteen into a power of two.

11. From two given points to draw two right lines meeting a given right line in the same point and making equal angles

angles to CE (4); produce A C to D, making CD equal to CA. Draw DB, cutting the given line at E, and join AE; then the lines AE and BE make equal angles at E, with the line CE.

If the given points, as B and D, are on opposite sides of the given line, draw DB cutting the line at E, then DE and BE make equal angles at E with the given line.

PARALLELS.

12. DEFINITIONS.-Two right lines which lie in the same plane are said to be parallel to one another when their distance apart is everywhere the same, so that however far they may be produced in either direction they will never meet.

If two lines are each parallel to a third line, they are

parallel to one another. If two lines which are parallel are also of equal length, the right lines which join the adjacent extremities of those lines will be themselves equal and parallel. If the lines A B and EF (fig. 12) are parallel and any line CD is drawn across them, then the angles ECD, CDB are equal, and the angles ADC, DCF are also equal; these angles are called alternate angles. The sum of the two angles FCD, CDB is equal to two right angles, and the sum of the angles ECD, CDA is also equal to two right angles.

13. To draw a right line through a given point and parallel to a given line.-Let AB (fig. 12) be the given line, C the given point. Draw

any right line CD, cutting A B in D; and from D and C as centres, with DC as radius, describe circles CH and DE. Let the arc CH cut AB in H, and join CH. Then from D as a centre, and with

Fig. 12.

HC as radius, describe an arc cutting the circle D E in E. Draw the right line CE, which will be parallel to the given line A B.

Another method is to draw CB perpendicular to AB (4), and then to draw EC perpendicular to CB; EC will be parallel to A B. This is a rule that is readily available for setting out parallel lines upon the ground.

When an instrument for setting out angles is used, we have only to make the angle ECD equal to the angle CDH, and the lines E C and A B will be parallel.

The line CB perpendicular to both the parallels represents the shortest distance between them, which is always of the same length for the same pair of parallels.