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PROPERTIES OF THE

TRACT 13.

figures, leads us to, and forms only a particular case of, aproperty in some other figure, less simple ; afterwards thisagain turns out to be no more than a particular case of an-other still more general ; and so on, till at last we often tracethe tendency to end in a general property of all figureswhatever.

The few properties which make a part of this paper, con-stitute a small specimen of the analogy, and even identity,of some of the more remarkable properties of the circle, withthose of the sphere. To wdiich are added some propertiesof the lines of section, and of contact, between the sphereand cone. Both which may be further extended as occasionsmay offer : like as all of these properties have occurred Iromthe circumstance, mentioned near the end of the paper, ofconsidering the inner surface of a hollow spherical vessel,as viewed by an eye, or as illuminated by rays, from a givenpoint.

proposition 1.

All the tangents are equal, which arc drawn, from a givenpoint without a sphere, to the surface of the sphere, quitearound.

Demons. —For, let pt be any tangentfrom the given point p ; and draw pc tothe centre c, and join tc. Also let ctabe a great circle of the sphere in theplane of the triangle tpc. Then, cpand ct, as well as the angle T, which isright (Eucl. iii. 18), being Constant, inevery position of the tangent, or of thepoint of contact T ; the square of pt will be every whereequal to the difference of the squares of the, constant linescp, ct, and therefore constant; and consequently the line ortangent pt itself of a constant length, in every position, quiteround the surface of the sphere.

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