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For the angle AFE is to a right angle, as the arch on which it stands, to the fourth part of the circumference (by Prop. 33, B. 6); and therefore to four right angles, as the arch to the whole circumference (by Prop. 30, B. 5).

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COR. 2. Of unequal circles, the arches which subtend the equal angles, are similar.

For it appears that they have the same ratio to the whole circumference (by preceding Cor.)

COR. 3.-Hence it appears that arches, by which similar segments are contained, are similar.

END OF SIXTE BOOK.

TRIGONOMETRY.

TRIGONOMETRY is of two kinds, Plane and Spherical, it treats of the measurement of heights, distances, &c., by means of triangles. Plane Trigonometry, on which we shall comment, is divided into two parts, scilicet, right angled, and oblique, since all triangles must be either rectangular or otherwise. A triangle consists of six parts; id est, three angles and three sides. If any three of these parts be given, provided one be a side, the remaining three can be found; by means of the triangle undergoing the following change; thus if the vertex of either of the acute angles of a right-angled triangle be taken as a centre, and any of the sides as radius, a circle being described, the sides of the triangle receive the following names; if the radius be the longest side sine, cosine, and radius; and the following, if the radius be either of the other sides: tangent, secant, and radius. It must be remembered that they retain

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their names with regard to the angle which has been taken as a centre, therefore let ABC be a triangle, and taking AB as radius, and describing the circle BDLGF,

we have BC, the sine of the angle BAC, and AC the cosine of the same angle, but it must be also understood that the arc intercepted between the legs of an angle is the measure of that angle, so that if a circle be divided into 360 degrees, the arc intercepted between the legs of a right angle will be 90 degrees, and so every angle is said to be an angle of as many degrees as are contained in an arc intercepted between its legs. Again, if we take a minor side of the triangle, as radius, and the same angle as centre, and describe the circle CEH, we have the tangent BC of the angle BAC, and AB the secant of the same angle BAC. The circle is usually divided in ENGLAND into 360 degrees, but in FRANCE into 400 degrees; this mode of division has an advantage over that of England, since when degrees, minutes, seconds, and thirds are expressed, which are thus written, 20° 8′ 7′′ 6", we could also express them in the decimal notation by the very same numbers thus 20.876.

The complement of an angle is what it wants of a right angle, and its sine, tangent, &c., are called the co-sine or sine of the complement, &c. The supplement is what an angle wants of two right angles or a semicircle-and the sine, &c. of the supplement is the same as the sine, &c. of that arc or angle.

Now taking the angle BAF of the triangle BAC, or its measure which is either the arc BF or PC, we shall proceed to explain all the lines which can be drawn to the circle DLGF with regard to that angle, that is, all the lines used in Trigonometry, as a Cord, derived from Corda, the string of a bow; sine, from sinus, the bosom ; versed sine, from sagitta, an arrow; tangent, being the line touching the circle; secant, the line cutting the circle, and the complements of these, which are called co-sine, co-cord, &c. being merely the sine, cord, &c. of the complement of the angle in question.

Thus the sine BC of the angle BAC, or the arc BF, is a line perpendicular to the radius, and meeting an

extremity of the arc BF. The sine of BD, or the co-sine of BF, is a line BT, drawn from the same extremity of the arc, parallel and equal to AC. The tangent is a line RF drawn perpendicularly from the extremity of the radius, meeting the other leg of the angle in R, which is called the secant or cutting line, since it cuts the circle in B when produced. The line DI is the tangent of the angle EAB or the arc DB, but it is also the co-tangent of the angle BAF. The line AI meeting DI in I is called the secant of EAB, or the co-secant of BAF. The line CF, between the sign and tangent, is called the versed line. It must be now remembered that the sine, tangent, &c. of any arc is also the sine, &c. of its supplement, and that the sine of an arc is half the cord of double that arc, which is evident since BFK is double of BF, and BC is half of BK.

Next, in treating analytically of Trigonometry, it will be requisite to alternately consider the above lines as plus or minus, according as they become nothing or infinite, when removed from the commencement of the first quadrant round the circle.

In the first quadrant the lines are all considered as +, then if we take the sine and move it on through the semicircle in which it commences we find it will not become nothing until at the end of the semicircle, therefore the sine is + in the two first quadrants, and in the two second; and likewise + in the fifth quadrant, and similarly any number of times round the circle. Again it we take the co-sine, which is the radius at the commencement of the first quadrant, and move on towards the centre, we find it nothing on arriving at that point, therefore it becomes in the second quadrant, and also in the third, because it does not again become nothing till it returns to the centre, therefore it will be

in the fourth quadrant, and similarly any number of times round the circle. Next let us take the tangent, which is nothing at the beginning of the first quadrant, but increases until the are becomes a quadrant, when it

is evident that it changes its sign, since it becomes infinite, because it will be parallel to the line which limits it (the secant) and therefore the tangent will be - in the second quadrant, and similarly in each alternate one. The secant is radius at the commencement of the first quadrant, it increases by limiting the tangent, till it becomes infinite, when the angle becomes right, and therefore changes its sine to in the second quadrant, and retains the same in the third, because it neither becomes infinite nor nothing at the end of the second; but since it becomes infinite at the end of the third, it will also change its sign; therefore the fourth quadrant will be +, and so on in the same order, any number of times round the circle. The co-tangent undergoes the same change as the tangent, and the co-secant the same with the sine, therefore as often as those lines become nothing or infinite, they also change their sines. Lastly, the versed sine being either the diameter or radius, it always retains the sine plus.

On the next page we give, at one view, the values at the beginning and ending of each quadrant—but it must be particularly observed, that the lines never change their signs, when moved round the circle, unless they become nothing or infinite.