856

THEORY AND PRACTICE OF ENGINEERING.

Rook II.

point on the horizontal limb half-way between all the readings will give the angle towhich the vernier is to be placed, in order that the telescope may point to the positionoccupied by the sun at noon.

On the Mensuration of Distances, Heights, §*c., by the calculation of Sines, Tangents , and

Secants -By the terms sines, tangents, and secants, is understood the knowledge of the

sides and angles of triangles, by means of which, and the assistance of tables calculated forthe purpose, we can obtain the length of the unknown sides and angles.

A Sine is the side of a right-angled triangle, tlffehypotenuse of which has served as a radius to describea circle comprising the right-angled triangle, as A 13 C,the hypotenuse of which, A B, is the radius of thecircle DBEFG, which encloses the triangle ABC,the side of which, B C, is the sine of the angle CAB;for the same reason the side A C is the sine of theangle ABC, and the hypotenuse A B that of the angleA B C. Every angle of a triangle is the sine of itsopposite side, as in that of ABC: the angle C A B isthe sine of the side B C, which is opposite to it; andthe angle ABC is the sine of the side A C; and theangle B C A is the sine of the hypotenuse A B.

Total sine, radius, sine of 90°, or entire sine, is thehypotenuse of a right-angled triangle, which servesas a radius to describe a circle enclosing a right-angled triangle. In the triangle ABC, the hypote-nuse A B is a total sine, and the two other sides A C and B C are only sines; so thatthe total sine A B, being commonly divided into 100,000 parts which are equal, thetwo other sides or sines being each smaller, must have less than that number.

Right Sine of an Angle is a line which falls perpendicular from the point where thehypotenuse cuts the circle, on to the extremity of another line, which forms an anglewith the hypotenuse, as the line B C is the right sine of the angle C A B.

Sine of an Arc is the right line drawn from one extremity of the arc perpendicularly tothe radius which answers to the other extremity, as the right line B C is the sine of thearc B E.

Versed Sine is the remaining portion of the radius which is comprised between the lineof the right sine of an angle and one of the same sine, as the line CE is a versed sine.

Sine of the Complement of an Arc is a right line drawn from the extremity of an arc,perpendicular to the radius, which does not touch the arc, but which together with the arcterminate a quarter circle: the line BII is the sine of the complement of the arc BE;because the right line BII is drawn from the extremity B of the arc BE, perpendicularto the radius A D, which does not touch the arc I) B, which bounds it.

'The reason why each side of a right-angled triangle inclosed within a circle is called asine is said to be from the word sinus, signifying the heart, the most inward part of man ;thus, sines, which are likewise enclosed or rather found inscribed in a circle, are called so ; andas the heart is the most important part of man, so are sines in a circle those which producethe most useful acquirements in mathematics.

Tangents and Secants .—A tangent is a line whichtouches the circumference of a circle, but does notcut it if prolonged, as B C : tangent of an angle is aline perpendicular to the extremity of a radius at thepoint on which it touches the circle, and this perpen-dicular terminates at the other line, which forms theangle of the tangent with the radius; the right lineB C is the tangent of the angle B A C , because it isperpendicular to the extremity of the radius A C, atthe point C, where it touches the circle DCEF ;and the perpendicular BC terminates at the'line cA B, which together with the radius A C forms theangle B A C of the tangent B C.

A Secant is a line which is drawn from the centreinto the circumference of a circle, as the line A B isa secant; for being drawn from the centre A, it cutsthe circumference of the circle DCEF in the pointG.

Secant of an Angle is a line drawn from the centreof a circle, which cutting the circumference extends to the tangent; as the right line A B isthe secant of the angle C A B, because it is a line drawn from the centre of a circle intoits circumference DCE F, in the point G, and extends beyond the circle to the tangent B C.

B

Fig. 1078.

F

Fig. 1077.