being perpendicular to the plane of the circle. The poles of a small circle are unequally distant from its plane, the inequality of distance amounting to twice Cc; but in a great circle this inequality vanishes, and the poles are equidistant from the circle.

As the poles of any circle are at the extremities of a diameter of the sphere, an infinite number of great circles may be drawn through them; indeed, every circle passing through them will necessarily be a great circle, because the entire diameter joining them must be comprised in every plane drawn through them. The distance of any circle from either of its poles, measured upon any of these infinite number of great circles, is constantly the same, that is, the distances or arcs PB, PD', PD, PA, &c. are equal, because the constant line Pc is the common versed sine of all these arcs to the common radius CP; hence the other distances P'B, P'E, &c. must be equal. Every arc of a great circle is thus distant from either pole by a quadrant or 90°.

(36.) Two great circles always intersect in two points, at the distance of a semicircle from each other, that is, the circumferences bisect each other. For as the plane of each circle passes through the centre of the sphere their intersection must be a diameter common to both circles, and it is at the extremities of this diameter that the circumferences cross each other.

From this we learn that if from any point on the sphere two quadrantal arcs can be drawn to two points in any great circle, the distance between the points being less than 180°, then the first point must be the pole of this great circle; for it is necessarily the pole of some great circle passing through the proposed points, and as only one great circle can pass through two points, which are not 180° apart, the pole must belong to the circle in question.

In spherical trigonometry, the arcs of great circles only are concerned, and the angle included between two such arcs, that is to say, a spherical angle, is measured in a manner analogous to that in which a plane angle is measured. For the measure of a plane angle we take the intercepted arc of that circle whose centre is at the vertex, and whose radius is some assumed unit: in like manner for the measure of a spherical angle we take the intercepted arc of that circle whose pole is at the vertex, and whose radius is some fixed unit, viz. the radius of the sphere on whose surface the angle is: thus, in the foregoing figure the

spherical angle DPD' is measured by the intercepted arc QQ', of which the pole is P, and radius, CQ, that of the sphere.

It is as easy to justify the propriety of adopting this mode of measuring spherical angles as it is to justify the method of measuring plane angles, for in both cases the intercepted arc varies as the angle; this, by the by, is true of the intercepted arc DD' of any small circle whose pole is P, but we are compelled to refer the measure to a great circle, in order that all the trigonometrical lines concerned in the same inquiry may be related to a common radius, for, as we have before remarked, the sides of a spherical triangle are always arcs of great circles.

From what we have just said it appears that a spherical angle DPD has the same measure as either of the equal plane angles QCQ', DcD', &c. situated in the planes of the circles whose common pole is P, and whose sides are formed by the intersection of these planes with those of the two great circles, forming the sides of the spherical angle. If at P tangents were drawn to the two great circles PD, PD', and in their planes they would obviously include the same angle as the lines CQ, CQ', to which they are parallel; indeed if we conceive the plane of the circle HQQ' to move parallel to itself towards the pole, P, the path of C being along the line CP, the angle QCQ' will successively coincide with QCQ, DcD', &c. till C coincides with P, when the lines CQ, CQ, will become tangents to the circles at P, and will remain each in the plane along which it has moved; hence the measure of the angle included between these tangents is also the measure of the spherical angle.

(37.) If in the plane of HQI perpendiculars be drawn from C to each of the planes of the circles PQP', PQ'P', these will be perpendicular to the lines CQ, CQ', and will therefore, include the same angle, which angle will be measured by the arc of HQI, which the said perpendiculars intercept; but these perpendiculars will meet the surface at the poles of the circles to whose planes they are perpendicular; hence the great circle distance between the poles of two intersecting great circles measures their angle of intersection.

Every great circle which passes through the poles af another is at right angles to it. Thus the great circle PDQP', through the poles of HQQ'I, is at right angles to HQQ'I; for if a tangent were drawn to PQP' at the point Q it would be in the same plane with and parallel to CP, and

if a tangent were drawn to HQI at the point Q it would be in the same plane with and parallel to CH; hence if these two tangents were to move simultaneously parallel to themselves, the path of their point of concourse Q being along QC, they would necessarily coincide with the perpendiculars CP, CH, when Q arrived at C; these tangents, therefore, form a right angle; hence the great circles are perpendicular to each other, or the spherical angle at Q is a right angle.

(38.) Any one side of a spherical triangle is less than the sum of the other two.

Let ABC be any spherical triangle, and O the centre of the sphere; draw the radii OA, OB, OC, then there will be about O three angles in three distinct planes respectively, measured by the arcs AB, BC, CA. Let AB be the greatest of these arcs, then it will only be necessary to show that AB < AC + CB, or that AOBAOC + BOC. In the plane of

A

AOB draw any line A'B', and then draw OD, making an angle B'OD equal to BOC; make OC' equal to OD, and join CB", C'A'.

Then since by construction the two sides 'BO, OD', and the included angle, are respectively equal to the two sides B'O, OC', and the included angle, BD = B'C'. But in the plane triangle A'B'C', A'B' < A'C' + B'C' .. A'D < A'C'; hence the two sides OA', OD, of the triangle A ́OD, are equal to the two sides OA', OC', of the triangle A'OC', but the third side A'D of the former is less than the third side A'C' of the latter, and, consequently, A'OD < A'O C'; hence, since B'OD has been made equal to B'OC', it follows that

A'OD + B'OD = A'OB′ < A'OC' + B'OC'

... AB < AC + CB.

(39.) The sum of all the three sides of a spherical triangle is less than the circumference of a great circle.

Let ABC be any spherical triangle; produce the sides AB, AC, till they meet again in D, then the arcs ABD, ACD, will be semi-circumferences, since (36,) two great circles always bisect each other. But in the triangle BCD we have BC < BD+CD, and, consequently, by adding AB + AC to both, we shall have

AB+ AC+ BC < ABD + ACD;

B

that is to say, the sum of the three sides is less than a whole circumference.

By help of this theorem we may show that the sum of the sides of any spherical polygon whatever is less than the circumference of a great circle.

Take the spherical pentagon ABCDE for example. Produce the sides AB, DC, till they meet in F; then since BC < BF + CF, the perimeter of the pentagon will be less than the quadrilateral AEDF. Again, produce the sides DE, BA, till they meet in G; we shall have EAEG + AG; hence the perimeter of the quadrilateral AEDF is less than that of

B

F

the triangle DFG; which last is itself less than the circumference of a great circle; the perimeter of the original polygon is, therefore, less

still.

(40.) If from the three vertices of a spherical triangle, taken as poles, arcs be described, forming a new triangle, then the vertices of the new triangle will be the poles of the other triangle.

For let ABC be any spherical triangle, and with the pole A, and circular radius AG equal to a quadrant, describe the arc EF; in like manner with the pole B and same radius describe the arc FD, meeting the former in F; and, lastly, with the pole C and same radius describe the arc ED, E completing the spherical triangle DEF.

M

G

F

H

Then, because the arcs, whose poles are A and C, intersect at E, the

points A, C, are each 90° distant from E; and as the arc AC is less than 180°, E must be the pole of AC (36). In like manner is it shown that F is the pole of AB, and D the pole of BC.

The triangle DEF is sometimes, from the mode of its construction, called the polar triangle, and the original one ABC the primitive triangle.

(41.) Any angle of the primitive triangle is the supplement of the side opposite to it of the polar triangle, and any angle of the polar triangle is the supplement of the side opposite to it in the primitive triangle.

For EH being the radius of HL is = 90°, and FG being the radius of GK is also = 90°, and the sum of these radii, namely, EF+GH=180°, therefore, GH, which is the measure of the angle A, is the supplement of the side EF opposite to it. In like manner is it shown that B is the supplement of DF, and C the supplement of DE. Again, BI being the radius of ID, and CM the radius of MD, the sum of these MI + BC= 180°; therefore, BC is the supplement of MI, which measures the angle D.

On account of the property just demonstrated, the triangles ABC, DEF, are frequently called supplemental triangles.

It is proper to remark here, as Legendre has done, that besides the triangle DEF three others might be formed by the intersection of the three arcs DE, EF, DF. But the proposition immediately before us is applicable only to the central triangle, which is distinguished from the others by the circumstance that the two angles

A and D (see preceding fig.) be on the same side of BC, the two B and E on the same side of AC, and the two C and F on the same side of AB.

(42.) From the foregoing proposition it follows

E

that the three angles of every spherical triangle are together greater than two right angles, and less than six.

For the sides of the supplemental triangle DEF are together less than four right angles (39), and as these are supplements of the angles A, B, C, and therefore when added to them make six right angles, these last must together exceed two right angles. But they cannot amount to six right angles, for in that case the sum of the sides of the supplemental triangle would be 0, which is absurd. Hence, unlike plane triangles,