and receive particular names, according to the number of their sides and angles, thus:

A pentagon has 5 sides, hexagon 6 sides, heptagon 7 sides, octagon 8 sides, nonagon 9 sides, decagon 10 sides, undecagon 11 sides, duodecagon 12 sides.

A regular polygon has all its sides and angles equal to one another.

An equilateral triangle is also a regular polygon of three sides, a square is one of four sides, the former being called a trigon, the latter a tetragon.

The sum of three angles of every triangle is equal to two right angles, or 180 degrees.

The sum of two angles in any triangle taken from 180 degrees leaves the third angle.

In a right angled triangle, if either acute angle be taken from 90 degrees, the remainder will give the other acute angle.

When the sine of an acute angle is required, subtract such obtuse angle from 180 degrees, and take the sine of the remainder.

The circle is a plane figure bound by a curve line, called the circumference (it is often called the periphery), and is divided into 360 equal parts, called degrees; each degree is divided into 60 equal parts, called minutes, each minute into 60 equal parts, called seconds, the next thirds, and so on. Degrees are expressed by a small over the number, minutes by one dash', seconds by two dashes", thirds by three dashes "", and so on in progression, as thus: 45° 14' 10" 4" reads forty-five degrees, fourteen minutes, ten seconds, and four thirds.

The diameter is a line drawn through the centre of a circle to the circumference on both sides.

An arc of a circle is any part of the circumference.

A chord is a right line joining the extremes of an arc.

A segment is any part of a circle bounded by an arc and its chord.

A semicircle is half the circle cut off by the diameter, equal to 180 degrees.

A sector is any part of the circle bounded by an arc and two radii, drawn from the centre to its extremities.

A quadrant, or quarter of a circle, is a sector having a quarter of the circumference for its arc, and its two radii perpendicular to each other, equal to 90 degrees.

The height or altitude of a figure is a perpendicular let fall from an angle, or its vortex, to the opposite side of the base.

The complement of an arc, or an angle, is its difference from a quadrant, or what it wants of 90 degrees.

The supplement of an arc, or an angle, is its difference from a semicircle, or what it wants of 180 degrees.

The sine of an arc is a line drawn from one extremity of an arc, perpendicular to the diameter.

The versed sine of an arc is that part of the diameter intercepted between the arc and its sine.

The tangent of an arc is a line drawn perpendicular to the diameter, touching the circumference.

The secant of an arc is the line proceeding from the centre, and limiting the tangent of the same arc.

arc.

The chord of an arc is a line touching the extremities of the

The chord of 60° is the sine of 90°;

The versed sine of 90°; the tangent of 45°;

The secant of 0°; are all equal to radius.

Note.-The co-sine, &c., is the abbreviation of complement.-See Trigonometrical Cannon, Fig. 1, Plate 27.

LINES.

Problem 1.

To draw a line parallel to another, Plate 1, Fig. 1.

Take the distance required in the compass, from two or three points on the line A B, describe arcs as a b c, draw a line touching those arcs will be the parallel line required.

Problem 2.

Another method when the line is to pass through a given point, C, Fig. 2.

From C draw a line at pleasure to B, with any radius describe the arc A a, with the same radius describe the arc bc; then take the distance A a, with the same distance from b, intersect the arc at c, and draw the line C c, the parallel line required.

Problem 3.

To draw lines parallel to the curve A B C, the centre being given, Fig. 3.

Bisect A B and B C, and draw the perpendiculars DE and FG; then draw the lines A D, DG, GC; with the radius A D describe the arcs A B and B C; then with the radius DE describe arc E, with the same radius and centre G describe the arc F, and with the radius DH describe the arc H, and also from G the arc I, will complete the three parallel lines required.

Problem 4.

To bisect or divide a line into two equal parts, Fig. 4.

From A and B as centres take in the compass any distance. greater than half, and describe arcs intersecting each other, as at a and b; draw a line through those points, cutting A B at c, the point of division required.

N.B.-A perpendicular on any part of a line may be found by the same rule.

Problem 5.

To raise a perpendicular from a given point, C, on the line A B, Fig 5.

At C with any radius describe the arc abc; with the same radius from a intersect the arc at b; from b with the same radius draw the arc dc; also with the same radius from c intersect those arcs at bd; draw the line d C, will be the perpendicular required.

Problem 6.

Another method when the point is at the end of the line A B, Fig. 6.

Take in the compass any distance, and describe the arc a b c; then draw a line through a and the centre of the arc at d, intersecting the same arc at c; then through c, and the point of intersection at b, draw the perpendicular b c.

Problem 7.

Another method to let fall a perpendicular from a given point near the middle of a line, Fig. 7.

From A with any radius describe the arc bc; with the same radius describe the arcs intersecting each other at d; draw the line A d, the perpendicular required.

Note.-Perpendiculars are more readily drawn by a protractor.-See Instruments, or a Right Angle Square.

Problem 8.

To find a mean proportion between two lines, as A B and B C, Fig. 8.

Draw a line, on which set off A B and B C; bisect A C in a; with the radius A a or a C describe the arc Abc; from the point B raise a perpendicular, intersecting the arc at b; draw the line b B, the length of the mean proportion required.

Problem 9.

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To draw a line or scale equal in proportion to another, A B the given scale, B C the given line to be divided, Fig. 9.

From B draw the line B C at pleasure, and set off the given distance; divide A B into the required number of parts, draw the line A C, parallel to which draw lines through each division on A B, cutting CB, which will be the divided proportion required.

Problem 10.

To divide a line or scale into any number of equal parts, Fig. 10.

1

Given the length of the line A B, to be divided into five equal parts. Draw the line C D, on which mark off the given number of parts; with the radius C D describe arcs intersecting at a; from a, draw the lines a C and a D; then with the radius of the given length A B from a, set off A and B; draw lines from a, to the division, on CD, cutting the line A B into the required number of parts.

Problem 11.

To divide a line or scale A B into eight equal parts, Fig. 11. From A draw the line A b at pleasure, and from B draw the line B c parallel to Ab; from A set off seven equal divisions, and the same number on the line B c; draw lines from the corresponding numbers, and the intersections on A B will be the required divisions.

ANGLES.

Angles are usually expressed by three letters; that placed at the angular point is always in the middle, as E is the angle of AED or BED, Fig. 16a.

An angle is the inclination of two lines meeting in a point called the vertex, and is of a certain number of degrees, or part of the circumference of a circle. Angles are of three kinds.

A right angle is when one line is perpendicular to another equal to 90 degrees, or the fourth part of a circle, as A B C, Fig. 12, Plate 1.

An acute angle is that which is less than a right angle, as A B D.

An obtuse angle is that which is greater than a right angle, or more than 90 degrees, as D BE.

Problem 12.

To draw an angle equal to another, Fig. 13.

With any radius in the compass describe the arc B C; set off the length A B; then with the compass take the distance B C;

D