PROBLEM III.

(63) To make an angle of any proposed number of degrees upon a given straight line, by the scale of chords.

Upon the line DB to make an angle of 30°. tent of 60° from the line of chords, with which and the centre D, describe the arc e f Take 30° from the same scale of chords and set them off from e to c; through c draw the line DC, then CDB is the angle required.

To make an angle of 150°. Produce the line BD to e, with the centre D and the chord of 60° describe a semicircle,

take the given obtuse angle from 180° e

Take the ex

f...

e

B

-B

and set off the remainder, viz. 30° from e to c, through c draw CD, then CDB is the angle required.

PROBLEM IV

(64) An angle being given, to find how many degrees it contains, by a scale of chords.

With the chord of 60° in your compasses and centre D describe an arc e f, cutting DB and DC in e and f. Then take the distance e f in your compasses, and setting one foot on the brass pin at the beginning of the chords on your scale, observe how many degrees the other foot reaches to, and that will be g the number of degrees contained in the arc ef, or angle CDB.

B

B

If the extent ef reach beyond the scale, which will alway be the case when the angle is obtuse, extend the line BD from D towards 9, and measure the arc gf in the same manner; the degrees it contains deducted from 180°, will give the measure of the angle CDB.

DEFINITIONS, ETC. OF TRIANGLES.

(65) A plane triangle is a space included by three straight lines, and contains three angles.

(66) Triangles are of various kinds, but in trigonometry only two kinds are considered, viz. right-angled triangles, and oblique-angled triangles; which indeed include the whole.

(67) Aright-angled triangle is that which has one right angle in it, as P. The longest side AC, or that opposite to the right angle B, is called the hypothenuse; the other two, AB and BC, are called the legs, or sides, or the base and perpendicular.

(68) The sum of the three angles of every plane triangle is equal to 180°; hence in a right-angled plane triangle, if either acute angle be taken from 90°, the remainder will be the other acute angle.

(69) The complement of an arc, or angle less than 90°, is what that angle wants of a quadrant, or 90°.

(70) If one acute angle of a right-angled triangle be 45°, or half a right angle, the other acute angle will also be 45°, or half a right angle; and the base and perpendicular will be equal to each other.

(71) If two right-angled triangles ABC and Abc, have the angle a common, they are equiangular and similar; that is, the sides about the equal angles are proportional, viz.

AB: BC:: Ab: bc, and AB BC::AC: AC, &c.

A

b B

(72) If a straight line be drawn parallel to one of the sides of a plane triangle, it will cut the other two sides proportionally.

(73) An oblique-angled triangle is that which has not a right angle in it; hence two of its angles must necessarily be acute, or each less than a right angle, but the remaining angle may be either greater or less than a right angle; as A and B.

A

B

(74) Any one angle of an oblique plane triangle subtracted from 180°, leaves the sum of the other two angles. Or the sum of any two angles subtracted from 180°, leaves the third angle.

(75) The supplement of any angle is what that angle wants of 180°. Hence the supplement of any one angle is always equal to the sum of the other two.

(76) Any two sides of a triangle added together are greater than the third side. The greatest side of any triangle is opposite to the greatest angle; and the contrary, the greatest angle is opposite to the greatest side.

An equilateral triangle has three equal sides; an isosceles triangle has two of its sides equal; an acute-angled triangle has three acute angles; and an obtuseangled triangle has two acute angles, and one obtuse.

(77) Every triangle has two of its angles acute, and if the third angle be either a right angle, or an obtuse angle, it is opposite to the greatest side.

(78) If a perpendicular BD be drawn upon the longest side of any triangle, from the opposite angle, it will fall within the triangle; and the greater segment AD, will meet the greater (AB) of the other two sides and the less segment DC, will meet the less of these sides (BC). (79) In an equilateral or isosceles triangle, a perpendicular BD drawn from the vertical angle, will bisect both the base and the vertical angle.

A

ED

(80) If any one side of a plane triangle be produced or extended beyond the angular point, the outward angle will be equal to the two inward angles,

C

opposite to the angular point where

64

the side is extended. Thus in the

triangle ABC, if the measure of the

angles be as expressed in the tri- /55 angle, and the side AB be produced A

to D; then will the angle CBD be equal to the angles BAC and BCA together.

(81) In a right-angled plane triangle, the hypothenuse is equal to the square-root of the sum of the squares of the base and perpendicular: and the square-root of the difference of the squares of the hypothenuse and either of the other sides, is equal to the remaining side.

(82) Each of the angles of an equilateral triangle is 60 degrees.

(83) Each of the angles at the base of an isosceles triangle is equal to half the supplement of the vertical angle, or to the complement of half the vertical angle.

(84) If two straight lines intersect each other, the vertical or opposite angles will be equal.

(85) If a straight line intersect two parallel straight lines, it makes the alternate angles equal to each other, and the exterior angle equal to the interior and opposite on the same side of the line.

(86) All angles in the same segment of a circle are equal to each other.

(87) An angle at the centre of a circle is double to an angle at the circumference when they both stand on the same arc. (88) Every angle in a semicircle is a right angle.

BOOK II.

CHAPTER I.

DEFINITIONS OF PLANE TRIGONOMETRY, WITH THE INVESTIGATION OF GENERAL RULES FOR CALCULATING THE SIDES AND ANGLES OF PLANE TRIANGLES.

(89) PLANE TRIGONOMETRY is the art of measuring and calculating the sides and angles of triangles described on a plane surface, or of such triangles as are composed of straight lines. It likewise includes the relation between the radius of a circle and certain other straight lines described in and about a circle.

(9) The theory of triangles is the very foundation of all geometrical knowledge, for all straight-lined figures may be reduced to triangles. The angles of a triangle determine only its relative species, and are measured in degrees, minutes, and seconds (56); but the sides determine its absolute magnitude, and are expressed in yards, feet, chains, or any other lineal

measure.

T

(91) A circle is a plain figure contained under one line, called the circumference, to which all lines drawn from the centre are equal. Thus ABDBHA is the circumference; c the centre, and CA, CD, cb, CB, CF, are all equal to each other. (92) The distance from the centre of a circle to the circumference is called the radius: thus CA, CB, CD, &c. are radii.

(93) A straight line drawn through the centre of a circle to touch the circumference in two points, is called a diameter, and is always double the radius. Thus AD and вb are diameters, and are each of them double of AC or of bc.

(94) The exact ratio between

A Co Tangk

H

ECo-Sin

Chord

Radius

Sine

Tangent

Co-Sin ver. B

GSin

the radius and the circumference of a circle being unknown, mathematicians were at a loss to form a comparison between the sides and angles of a triangle, since they could not compare a straight line with any part of the circumference of a circle. They were therefore under the necessity of determining the relation between the radius of a circle, and certain other straight lines, described in and about a circle, called chords, sines, tangents, &c.

(95) The chord of an arc is a straight line drawn from one extremity of the arc to the other. Thus bн is the chord of the arc bн, or of the arc Hafbdb.

The chord of an arc of 60° is equal to the radius of the circle.* (96) The complement of any arc is the difference between that arc and a quadrant. Or it is the difference between any angle and 90°. Thus the arc AF is the complement of the arc BF, or the angle ACF is the complement of the angle FCB.

(97) The supplement of any arc is the difference between that arc and a semi-circle. Or it is the number of degrees which any angle wants of 180°. Thus the arc BF is the supplement of the arc FAHb, or the angle FCB is the supplement of the angle Fcb.

(98) The sine of an arc is a straight line drawn from one end of that arc, perpendicular to a diameter passing through the other end of the same arc. Thus FG is the sine of the arc BF, or it is the sine of the supplemental arc FAHb.

The sine of an arc of 90° is equal to the radius, for AC is the sine of the arc Ba. The sine of an arc of 30° is equal to half the

radius.+

(99) The tangent of an arc is a straight line drawn from one extremity of the arc, perpendicular to the diameter, and is terminated by a straight line drawn through the centre of the circle and the other extremity of the arc. Thus BT is the tangent of the arc BF, or of the angle BCF.

The tangent of any arc is equal to the tangent of the supplement

* For (Euclid 15. of IV.) the side of a hexagon, which is the chord of 60°, is equal to the radius of the circumscribing circle.

The sine of any arc is equal to half the chord of double that arc; thus let BF and BH (Plate I. Fig. 1.) be equal arcs, then FGH is the chord of the double arc FBH; and FGH is bisected in G (Euclid 3. of III.). Therefore if FB be an arc of 30°, FG its sine will be half the chord of 60°, and the chord of 60° has been shown to be equal to the radius, therefore the sine of 30° is equal to half the radius.

The semi-tangent of an arc is the tangent of half that arc. Let BF be any are (Plate I. fig. 1.) then mc is the semi-tangent of that arc. For the angle FCB is double the angle гbв (Euclid 20. of III.), consequently the arc oc, of which cm is the tangent, is the half of the arc BF.