Let AED be an oblique-angled scalene spherical triangle: let any one of its sides, as AD, be considered as its base; and let DA, produced, meet DE, produced, in M: therefore, (Art. 7.) AM is the arch, which together with AD, makes up two quadrants: also, (Art. 42.) the angles AED and AEM are, together, equal to two right angles: and, first, if one of the other sides, as AE, of the triangle AED, be greater than AD, and less than AM; AD, and the opposite angle AED, are of the same species.

For, find (Art. 64.) the pole P of the circle ED; join (Art. 66.) P, A, and produce PA, both ways, to meet the

circumference of ED in B and C: also, at the point A, in CA, make (Art. 96.) the angle CAK equal to DAC, and produce KA, to meet the circumference of DE in G; so that (Art. 140.) AG is equal to AM, and AE is situated between AD and AG.

Then, it is manifest, that if AD be less than a quadrant, AM, and AG, are each of them greater than a quadrant; wherefore, (Art. 137.) the angle AGM is obtuse, and, consequently, the angle AGD acute : but (Art. 141.) the angle AED is less than AGD: wherefore the angle. AED is less than a right angle, when AD is less than a quadrant: that is, AD and the angle AED are of the same species: and, in the same manner, if AD be greater than a quadrant, the angle AED may be shewn to be obtuse, that is, to be of the same species as AD.

Again, in the spherical triangle AED, let the angle ADE, at the base, be greater than the vertical angle AED, and less than the angle AEM.

Then (Art. 129.) AE is greater than AD: and, because the angle ADE, or (Art. 56.) AME, is, by the hypothesis, less than the angle AEM, therefore, (Art. 129.) AE is less than AM: and, therefore, by the preceding case, AD and the opposite or vertical angle AED are of the same species.

(144.) COR. Whenever, by means of Art. 143, the species of two sides, and of the two opposite angles, of a spherical triangle, can be determined, it may be known, from Art. 134. whether the arch of a great circle, drawn perpendicular to the third side from the angle opposite to it, fall within, or without, that side.

PART I.

THE ELEMENTS OF

Spherical Geometry.

SECTION V.

ON THE MUTUAL CONTACT OF CIRCLES IN A SPHERE.

DEFINITION.

(145.) CIRCLES in a sphere are said to touch one

another, the circumferences of which meet, but do not cut one another.

(146.) COR. Two great circles of a sphere cannot touch one another: for (Art. 7.) they bisect each other.

PROP. I.

(147.) Theorem. If two circles in a sphere touch one another, their common section touches each of the

circles, in the point that is common to their two circumferences and conversely.

For (E. 3. 11.) the common section of the planes of the two circles is a straight line; and it is in the plane of each of the circles; unless, therefore, it touch each of them, there will be more than one point common to the circumferences of the two circles, and therefore, (Art. 145.) they will not touch one another: which is contrary to the supposition.

The converse proposition may be shewn to be true, by a similar proof, ex absurdo.

(148.) COR. 1. If two lesser circles, in a sphere, both touch the same great circle, their common sections with it are in the same plane.

For (Art. 147.) both the sections are in the plane of the great circle.

(149.) COR. 2. Two circles, in a sphere, which touch any the same circle, of the sphere, in the same point, also touch one another.

(150.) COR. 3. If two circles, in a sphere, touch each other, a straight line which touches either of them, in their point of contact, also touches the other.

For two planes cannot cut each other in more than one straight line; nor can more than one straight line touch a circle in the same point, of its circumference, the tangent straight line being supposed to be in the same plane with the circle.

PROP. II.

(151.) Theorem. If two given circles, in a sphere, meet each other in any the same point of the circumference of the great circle in which are their poles, they shall touch one another.

For (Art. 27.) their planes are both of them perpendicular to the great circle in which are their poles; wherefore, (E. 19. 11.) the common section of their planes is perpendicular to the plane of that great circle; and therefore, (E. 3. Def. 11.) it is perpendicular to the two diameters, drawn from their point of concourse, one in each of the given circles; for these diameters are in the plane (Art. 28.) of the great circle: and, since the common section of the two given circles is perpendicular to a diameter in each, it (E. 16. 3. Cor.) touches them both; wherefore, (Art. 147.) the two circles touch one another.

The proposition may, also, easily be proved ex absurdo.

(152.) COR. If two circles in a sphere meet each other, and if an arch of a great circle, drawn from the pole of either of them, to the point in which they meet, be perpendicular to the circumference of the other, the two circles shall touch one another.

For (Art. 50.) the poles of any circle are in the arch of a great circle, which cuts its circumference at right. angles. It is manifest, therefore, that the poles of both