SECANT

, in Geometry, a line that cuts another, whether right or curved; Thus the line PA or PB, &c, is a Secant of the circle ABD, because cutting it in the point F, or G, &c. Properties of such Secants to the circle are as follow:

1. Of several Secants PA, PB, PD, &c, drawn from the same point P, that which passes through the centre C is the greatest; and from thence they decrease more and more as they recede farther from the centre; viz. PB less than PA, and PD less than PB, and so on, till they arrive at the tangent at E, which is the limit of all the Secants.

2. Of these Secants, the external parts PF, PG, PH, &c, are in the reverse order, increasing continually from F to E, the greater Secant having the less external part, and in such sort, that any Secant and its external part are in reciprocal proportion, or the whole is reciprocally as its external part, and consequently that the rectangle of every Secant and its external part is equal to a constant quantity, viz, the square of the tangent. That is, .

3. The tangent PE is a mean proportional between any Secant and its external part; as between PA and PF, or PB and PG, or PD and PH, &c.

4. The angle DPB, formed by two Secants, is measured by half the difference of its intercepted arcs DB and GH.

, in Trigonometry, denotes a right line drawn from the centre of a circle, and, cutting the circumference, proceeds till it meets with a tangent to the same circle. Thus, the line CD, drawn from the centre C, till it meets the tangent BD, is called a Secant; and particularly the Secant of the arc BE, to which BD is a tangent. In like manner, by producing DC to meet the tangent Ad in d, then Cd, equal to CD, is the Secant of the arch AE which is the supplement of the arch BE. | So that an arch and its supplement have their Secants equal, only the latter one is negative to the former, being drawn the contrary way. And thus the Secants in the 2d and 3d quadrant are negative, while those in the 1st and 4th quadrants are positive.

The Secant CI of the arc EF, which is the complement of the former arch BE, is called the cosecant of BE, or the Secant of its complement. The cosecants in the 1st and 2d quadrants are affirmative, but in the 3d and 4th negative.

The Secant of an arc is reciprocally as the cosine, and the cosecant reciprocally as the sine; or the rectangle of the Secant and cosine, and the rectangle of the cosecant and sine, are each equal to the square of the radius. For CD : CE :: CB : CH, or s : r :: r : c, and CI : CE :: CF : CK, or s : r :: r : s; and consequently r² = cs = ss; where r denotes the radius, s the sine, c the cosine, s the Secant, and s the cosecant.

An arc a, to the radius r, being given, the Secant s, and cosecant s, and their logarithms, or the logarithmic Secant and cosecant, may be expressed in infinite series, as follows, viz, where m is the modulus of the system of logarithms.

, Figure of. See Figure of Secants.

, Line of. See Sector, and Scale.