all their parts; hence the side QE=PB, and the angle FQE =СРВ.

Now, the triangles DFQ, ACP, which have their sides respectively equal, are at the same time isosceles, and capable of coinciding, when applied to each other; for having placed PA on its equal QF, the side PC will fall on its equal QD, and thus the two triangles will exactly coincide; hence they arc equal; and the surface DQF=APC. For a like reason, the surface FQE=CPB, and the surface DQE APB; hence we have DQF+FQE-DQE=APC+CPB-APB, or DFE= ABC; hence the two symmetrical triangles ABC, DEF are equal in surface.

Scholium. The poles P and Q might lie within the triangles. ABC, DEF: in which case it would be requisite to add the three triangles DQF, FQE, DQE together, in order to make up the triangle DEF; and in like manner to add the three triangles APC, CPB, APB together, in order to make up the triangle ABC in all other respects, the demonstration and the result would still be the same.

PROPOSITION XXI. THEOREM.

If the circumferences of two great circles intersect each other on the surface of a hemisphere, the sum of the opposite triangles thus formed, is equivalent to the surface of a lune whose angle is equal to the angle formed by the circles.

Let the circumferences AOB, COD, intersect on the hemisphere OACBD; then will the opposite triangles AOC, BOD be equal to the lune whose angle is BOD.

For, producing the arcs OB, OD on the other hemisphere, till they meet in N, the arc OBN will be a semi-circumference, and AOB one also ; and taking OB from both, we shall

A

D

B

C

have BN=AO. For a like reason, we have DN=CO, and BD=AC. Hence the two triangles AOC, BDN have their three sides respectively equal; besides, they are so placed as to be symmetrical; hence (Prop. XIX. Sch.) they are equal in surface, and the sum of the triangles AOC, BOD is equivalent to the lune OBNDO, whose angle is BOD.

Scholium. It is likewise evident that the two spherical pyramids, which have the triangles AOC, BOD for bases, are together equivalent to the spherical ungula whose angle is BOD.

PROPOSITION XXII. THEOREM.

The surface of a spherical triangle is measured by the excess of the sum of its three angles above two right angles, multiplied by the tri-rectangular triangle.

Let ABC be the proposed triangle: produce its sides till they meet the great circle DEFG, drawn at pleasure without the triangle. By the last Theorem, the two triangles ADE, AGH, are together equivalent to the lune whose angle is A, and which is measured by 2A.T (Prop. XIX. Cor. 2.) Hence we have ADE+AGH= 2A.T; and, for a like reason, BGF+ BID=2B.T, and CIH+CFE=2C.T.

H

G F

A

E

B

But the sum of these six triangles exceeds the hemisphere by twice the triangle ABC, and the hemisphere is represented by 4T; therefore, twice the triangle ABC is equal to 2A.T+ 2B.T+2C.T—4T; and consequently, once ABC=(A+B+C . -2)T; hence every spherical triangle is measured by the sum of all its angles minus two right angles, multiplied by the trirectangular triangle.

Cor. 1. However many right angles there may be in the sum of the three angles minus two right angles, just so many tri-rectangular triangles, or eighths of the sphere, will the proposed triangle contain. If the angles, for example, are each equal to of a right angle, the three angles will amount to four right angles, and the sum of the angles minus two right angles will be represented by 4-2, or 2; therefore the surface of the triangle will be equal to two tri-rectangular triangles, or to the fourth part of the whole surface of the sphere.

Scholium. While the spherical triangle ABC is compared with the tri-rectangular triangle, the spherical pyramid, which has ABC for its base, is compared with the tri-rectangular pyramid, and a similar proportion is found to subsist between them. The solid angle at the vertex of the pyramid, is in like manner compared with the solid angle at the vertex of the trirectangular pyramid. These comparisons are founded on the coincidence of the corresponding parts. If the bases of the pyramids coincide, the pyramids themselves will evidently coincide, and likewise the solid angles at their vertices. From this, some consequences are deduced.

First. Two triangular spherical pyramids are to each other as their bases; and, since a polygonal pyramid may always be divided into a certain number of triangular ones, it follows that any two spherical pyramids are to each other as the polygons which form their bases.

Second. The solid angles at the vertices of these pyramids are also as their bases: hence, for comparing any two solid angles, we have merely to place their vertices at the centres of two equal spheres, and the solid angles will be to each other as the spherical polygons intercepted between their planes or faces.

The vertical angle of the tri-rectangular pyramid is formed by three planes, at right angles to each other. This angle, which may be called a right solid angle, will serve as a very natural unit of measure for all other solid angles. If, for example, the area of the triangle is of the tri-rectangular triangle, then the corresponding solid angle will also be of the right solid angle.

PROPOSITION XXIII. THEOREM.

The surface of a spherical polygon is measured by the sum of all its angles, minus two right angles multiplied by the number of sides in the polygon less two, into the tri-rectangular triangle.

From one of the vertices A, let diagonals AC, AD, be drawn to all the other vertices; the polygon ABCDE will be divided into as many triangles, minus two, as it has sides. But the surface of each triangle is measured by the sum of all its angles minus two B right angles, into the tri-rectangular triangle; and the sum of the angles in all

A

E

the triangles is evidently the same as that of all the angles of the polygon: hence, the surface of the polygon is equal to the sum of all its angles, diminished by twice as many right angles as it has sides, less two, into the tri-rectangular triangle.

Scholium. Let s be the sum of all the angles in a spherical polygon, n the number of its sides, and T the tri-rectangular triangle; the right angle being taken for unity, the surface of the polygon will be measured by

(s—2 (n—2,)) T, or (s—2 n+4) T

ANALYTICAL PLANE TRIGONOMETRY.

CHAPTER I.

PLANE TRIGONOMETRY is the science which treats of the relations of the sides and angles of plane triangles.

In every triangle there are six parts: three sides and three angles; which have such relations to each other that the value of one depends on the value of the others; and if a sufficient number of these are known the others may thereby be determined.

The sides of triangles consist of absolute magnitude, but the angles are only the relations of those sides to each other in position or direction, without regard to their magnitudes.

Angles have no absolute measure in terms of the sides; but are, nevertheless, susceptible of measure; for if two lines meet each other the space included between them within a given distance from their point of contact is proportional to their mutual inclination, and hence (Prop. XVIII. Cor. B. III. El. Geom.) the arc of the circumference of a circle intercepted by two lines drawn from its centre, may be regarded as the measure of the angle or inclination of those lines, and therefore the arc of the circumference may be regarded as the measure of angular magnitude.

For this purpose the circumference of the circle is supposed to be divided into 360 equal parts, called degrees, and each of those degrees is divided into 60 equal parts called minutes, and each minute into 60 equal parts called seconds; and so on, to thirds, fourths, &c.

These divisions are designated by the following characters, • &c. Thus the expression 30° 20' 12" 22"", represents an arc or an angle of 30 degrees 20 minutes 12 seconds 22 thirds.

The circumference of any circle may in this manner be applied as the measure of angles, without regard to its magnitude or the length of its radius; hence a degree is not a mag

nitude of any definite length, but is a certain portion of the whole circumference of any circle, for it is evident that the 360th part of the circumference of a large circle is greater than the same part of a smaller one, but the number of degrees in the small circumference is the same as in the large one. The fourth part of the circumference of a circle is called a quadrant and contains 90 degrees: hence 90 degrees is the measure of the right angle.

Thus, if we draw two straight lines AD, BE, so as to cross each other at right angles, and from their point of intersection, C, we discribe a circle with any radius so as to cut those lines in any A points, as a, b, d, e, the circumference of the circle will thus be divided into four equal arcs, ab, bd, de, ea, each of which measures or subtends a right angle at the centre C, of the circle.

B

b

C

B

D

If a line CP be made to revolve round a fixed point Cas the centre of a circle, and so as to pass successively through every point of the circumference, commencing in the point a, then, while it is in the position Ca, or while it coincides with the line Ca, those two lines form but one, and intercept no arc on the circumference of the circle, and hence form no angle with each other; but when the line CaP comes into the position CP, it forms with AC an acute angle at C, which is measured by the arc aP, and when it comes into the position CbP, it then forms a right Aangle ACP with the line AC, which angle is measured by the quadrant ab. Now let it come into the position CP,, and the angle which it forms with CA, will be measured by the arc aP,, which is greater than a right angle, and hence is an obtuse angle.

P

P

P

-D

P

Let it now come into the position CdP; it then coincides with the right line Cd, which is a portion of the line AC produced, since the line CP, in this position, coincides with the line AD, it can be said to form with it no angle; yet the space passed over by the line CP, from the position CaP, is equal to two quadrants, or two right angles equal to 180 degrees, and for trigonometrical investigation the lines CbP and CA are said to subtend the angle measured by the arc abd.

After passing the point d, and coming into the position CP,, it forms with AC, and on the upper side of it the angle P,CA