lying on the same side of D: hence, there will be two triangles, DEF and DEG, either of which will satisfy all the conditions of the problem.

XI. The adjacent sides of a parallelogram, with the angle which they contain, being given, to describe the parallelogram.

F

43. 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 C; take DF = B: describe two arcs, the one from F, as a centre, with a radius FG

=

D

AH

A, the

B

-H

other from E, as a centre, with a radius EG = BF; through the point G, where these arcs intersect each other, draw FG, EG: then, DEGF will be the parallelogram required.

XII. To find the centre of a given circle, or arc.

44. Take three points, A, B, C, anywhere in the circumference, 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.

A similar construction serves for

making a circumference pass through three given points, A, B, C, and also for describing a circumference, about a given triangle.

For, if we join the points by the straight lines AB, BC, and AC, and bisect either two of them, by perpendiculars, their point of intersection, O, will be the centre of the required circle.

SECTION III.

PLANE TRIGONOMETRY.

45. PLANE TRIGONOMETRY is that branch of Mathematics which treats of the solution of plane triangles.

In every plane triangle there are six parts: three sides and three angles. When three of these parts are given, one being a side, the remaining parts may be found by computation.

46. An angle is measured by the arc of a circle included between its sides, the centre of the circle being at the vertex, and its radius being equal to 1.

47. If two lines be drawn through the centre of a circle, at right angles to each other, they will divide the circumference into four equal parts, each of which is called a quadrant.

For convenience, the quadrant is divided into 90 equal parts, each of which is called a degree; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. Degrees, minutes, and seconds, are denoted by the symbols, ',". Thus, the expression 7° 22' 33", is read, 7 degrees, 22 minutes, and 33 seconds.

A quadrant contains 324,000 seconds, and an arc of 7° 22′ 33′′ contains 26,553 seconds; and any arc of a quadrant may be expressed in seconds.

48. The complement of an arc is what remains after subtracting the arc from 90°. Thus, the arc EB is the comple

49. The supplement of an arc is what remains after subtracting the arc from 180°. Thus, GF is the supplement of the arc AEF.

50. 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.

51. 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.

52. The tangent of an arc is the line which touches it at one extremity, and limited by a line drawn through the other extremity and the centre of the circle. Thus, AC is the tangent of the arc AB.

43 53. 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.

54. The four lines, BD, OD, AC, OC, depend for their values on the arc AB and the radius OA; they are thus written:

55. If ABE be a quadrant, or 90°, then EB will be the

complement of AB. Let the lines ET and IB be drawn perpendicular to OE. Then,

ET, the tangent of EB, is called the cotangent of AB;

IB, the sine of EB, is the cosine of AB;

OT, the secant of EB, is the cosecant of AB.

In general, if A is any arc or angle, we have,

cosine of an 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 of the supplemental arc GF. But since AQ is equal to GL, and OQ to OL, it follows that, the tangent of an arc 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.

57. A NATURAL SINE, COSINE, TANGENT, or COTANGENt, is the sine, cosine, tangent, or cotangent of an arc whose radius is 1.

A TABLE OF NATURAL SINES is, therefore, a table showing the values of the sines, cosines, tangents, and cotangents of all

* These relations are between the numerical values of the trigonometrical lines; the

the arcs of a quadrant, divided either to minutes or seconds. The Table of Natural Sines, beginning at page 63, of the tables, shows the values of the sines and cosines only.

TABLE OF LOGARITHMIC SINES.

58. In this table are arranged the logarithms of the numerical values of the sines, cosines, tangents, and cotangents 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 columns of minutes.

To find, in the table, the logarithmic sine, cosine, tangent, or cotangent of any given angle.

59. 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,

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 expressing the minutes: then pass along a horizontal line into the column.