54.) since Fis the pole of BAC, AC measures the least angle AFC, and AB measures the greatest angle AFB.

In the same manner, also, on the other side of AF, it may be shewn, that the angle AFB is greater than AGB, and that AGB is greater than AHB.

(142.) Cor. 2. If a point be taken, in the surface of a sphere, from which there fall more than two equal arches of great circles to the circumference of a given circle in the sphere, that point is the pole of the given circle.

For, if it be not the pole, then from a point in a sphere's surface, which is not the pole of a given circle, there may be drawn more than two arches of great circles that are equal to one another: which (Art. 140.) is absurd.

PROP. XIV. (143.) Theorem. The base of an oblique-angled scalene spherical triangle, and the opposite or vertical angle, are of the same species, if one of the remaining sides, compared with the base, and with the arch which, together with the base, makes two quadrants, is greater than the one, and less than the other : the base and the vertical angle, are also, of the same species, if one of the angles at the base, compared with the vertical angle, and with the angle which, together with the vertical angle makes up two right angles, is greater than the one, and less than the other.

2861 19B

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

[merged small][merged small][merged small][merged small][ocr errors][ocr errors][ocr errors][merged small]

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 hy. pothesis, 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


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.



Spherical Geometry.




(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.

« ForrigeFortsett »