If the angle C is acute, and the side B less than A, then the arc described from the centre E with the radius EF = B will cut the side DF in two points, F and G, lying on 1 B D G the same side of D: hence, there will be two triangles, DEF, and DEG, either of which will satisfy all the condi tions of the problem. XI. The adjacent sides of a parallelogram, with the angle which they contain, being given, to describe the paral lelogram. 58. Let A and B be the given sides, and C the given angle. Draw the line DH, and lay off DE equal to A; at the point D, make the angle EDF= 0; take DF=B: describe two arcs, the one from = D A B F E H F, as a centre, with a radius FG DE, the other from E, as a centre, with a_radius_EG = DF; through the point G, where these arcs intersect each other, draw FG, EG; DEGF will be the parallelogram required. XII. To find the centre of a given circle or arc. 59. Take three points, A, B, C, any where in the cir cumference, or in the arc: draw AB, BC; bisect these two lines by the perpendiculars, DE, FG: the point 0, where these perpendiculars meet, will be the centre sought. The same construction serves for making a circumference pass through three given points A, B, B C, and also for describing a circumference, about a given PLANE TRIGONOMETRY. SECTION III. DEFINITIONS.-APPLICATION TO HEIGHTS AND DISTANCES. 1. In every plane triangle there are six parts: three sides and three angles. These parts are so related to each other, that when one side and any two other parts are given, the remaining ones can be obtained, either by geometrical construction or by trigonometrical computation. 2. Plane Trigonometry explains the methods of computing the unknown parts of a plane triangle, when a sufficient number of the six parts is given, 3. For the purpose of trigonometrical calculation, the circumference of the circle is supposed to be divided into 360 equal parts, called degrees; each degree is supposed to be divided into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are designated respectively, by the characters '". For example, ten degrees, eighteen minutes, and fourteen seconds, would be written 10° 18' 14". 4. If two lines be drawn through the centre of the circle, at right angles to each other, they will divide the circumference into four equal parts, of 90° each. Every right angle then, as EOA, is measured by an arc of 90°; every acute angle, as BOA, by an are less than 90°; and every obtuse angle, as FOA, by an arc greater than 90°. 5. The complement of an arc is what remains after subtracting the arc from 90°. Thus, the arc EB is the complement of AB. The sum of an arc and its complement is equal to 90°. 6. The supplement of an arc is what remains after subtracting the L G N E I F B H D 14 supplement of the arc AEF. The sum of an arc and its supplement is equal to 180°. 7. The sine of an arc is the perpendicular let fall from one extremity of the arc on the diameter which passes through the other extremity. Thus, BD is the sine of the arc AB. 8. The cosine of an arc is the part of the diameter intercepted between the foot of the sine and centre. Thus, OD is the cosine of the arc AB. 9. The tangent of an arc is the line which touches it at one extremity, and is limited by a line drawn through the other extremity and the centre of the circle. Thus, AC is the tangent of the arc AB. 10. The secant of an arc is the line drawn from the centre of the circle through one extremity of the arc, and limited by the tangent passing through the other extremity. Thus, OC is the secant of the arc AB. 11. The four lines, BD, OD, AC, OC, depend for their values on the arc AB and the radius OA; they are thus designated : sin AB for BD cos AB for OD tan AB for AC sec AB for OC Let the lines ET and IB 12. If ABE be equal to a quadrant, or 90°, then EB will be the complement of AB. be drawn perpendicular to OE. Then, ET, the tangent of EB, is called the cotangent of AB; In general, if A is any are or angle, we have, arc is equal to the cosine of its supplement.* Furthermore, AQ is the tangent of the arc AF, and OQ is its secant: GL is the tangent, and OL the secant But since AQ is equal to of the supplemental arc GF. GL, and OQ to OL, it follows that, the tangent of an are is equal to the tangent of its supplement; and the secant of an arc is equal to the secant of its supplement.* TABLE OF NATURAL SINES. 14. Let us suppose, that in a circle of a given radius, the lengths of the sine, cosine, tangent, and cotangent, have been calculated for every minute or second of the quadrant, and arranged in a table; such a table is called a table of sines and tangents. If the radius of the circle is 1, the table is called a table of natural sines. A table of natural sines, therefore, shows the values of the sines, cosines, tangents, and cotangents of all the arcs of a quadrant, which is divided to minutes or seconds. If the sines, cosines, tangents, and secants are known for arcs less than 90°, those for arcs which are greater can be found from them. For if an arc is less than 90°, its supplement will be greater than 90°, and the numerical values of these lines are the same for an arc and its supplement. Thus, if we know the sine of 20°, we also know the sine of its supplement 160°; for the two are equal to each other. The Table of Natural Sines, beginning at page 63, of the tables shows the values of the sines and cosines only. *These relations are between the numerical values of the trigonometrical lines; TABLE OF LOGARITHMIC SINES. 15. In this table are numerical values of the tangents of all the arcs of a quadrant, calculated to a radius of 10,000,000,000. The logarithm of this radius is 10. In the first and last horizontal lines of each page, are written the degrees whose sines, cosines, &c., are expressed on the page. The vertical columns on the left and right, are arranged the logarithms of the sines, cosines, tangents, and co columns of minutes. CASE I. To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given arc or angle. 16. If the angle is less than 45°, look for the degrees in the first horizontal line of the different pages: when the degrees are found, descend along the column of minutes, on the left of the page, till you reach the number showing the minutes then pass along a horizontal line till you come into the column designated, sine, cosine, tangent, or cotangent, as the case may be: the number so indicated is the logarithm sought. Thus, on page 37, for 19° 55', we find, 17. If the angle is greater than 45°, search for the degrees along the bottom line of the different pages: when the number is found, ascend along the column of minutes on the right hand side of the page, till you reach the number express. ing the minutes: then pass along a horizontal line into the column designated tang, cot, sine, or cosine, as the case may be: the number so pointed out is the logarithm required. 18. The column designated sine, at the top of the page, is designated by cosine at the bottom; the one designated tang, by cotang, and the one designated cotang, by tang. The angle found by taking the degrees at the top of the page, and the minutes from the left hand vertical column, |