MATHEMATIC AL GEOGRAPHY.
Tun fundamental principle of all mathematical geogra phy, and what of course naturally claims our first attention, is the spherical figure of the earth. The proof of this, how eve•, is neither elaborate nor abstruse, the various pheno mena from which it is inferred being so obvious and so conclusive, as to require only to be mentioned. The first, and perhaps the most simple of these which we shall no tice, is the appearance of a ship at sea, either approaching to, or receding from, an observer on the shore. In the former case the vessel seems to rise out of the water, and in the latter to sink beneath it, a phenomenon that can only be accounted for from the convexity of the earth's surface ; and as the same appearance is observed at all times and in all situations, this convexity must also hold in every direc tion, that is, the earth must be spherical. The sante con elusion may also be drawn from other phenomena ; as the change which takes place in the visible part of the earth's surface, as well as of the heavens, to an observer who changes his situation—from the circular form of the earth's shadow, as observed in eclipses of the moon—and, finally, from the actual circumnavigation of the globe. Our readers will find these appearances illustrated at greater length under the article As•noxomv, in the second volume of our work. Without, therefore, attempting any farther proof of the fact, we shall proceed on the sup position that the earth is a perfect sphere. This, indeed, is not exactly the case, the globe being flattened or com pressed at two opposite points, forming what mathema ticians call an oblate spheroid, and at the same time having its surface diversified with numerous elevations and de pressions. But to the geographer, these inequalities are of no importance, as they are too inconsiderable to affect any of the problems that he may have occasion to solve. The longest diameter of the earth is to the shortest nearly as 1 to .9968, or as 301 to SOO. and the highest mountain on
the earth, it represented on a sphere of six feet nine inches diameter, would not project from its surface farther than of an inch. In a system of geography, therefore, we may safely omit the consideration of such minute irregula rities, and regard the globe as really a perfect sphere.
As our chief object in the present article is to render the principles of geography intelligible to our readers in gene ral, we shall endeavour, as much as possible, to exhibit a popular view of the subject, referring the scientific reader to those articles of our work, where the propositions that we may assume, and the phenomena that we shall have oc casion to explain, arc examined and illustrated on the most rigid principles. Agreeably to this plan, we shall here throw into the form of definitions, some of the properties of the sphere in general, referring for a demonstration of these properties to the article TRIGONOMETRY.
A sphere is an uniformly round body, every point of whose surface is equally distant from a point within the body, called the centre. I fence, If a circle is made to revolve about its diameter, which remains fixed, its circumference will describe or trace out the surface of a sphere. The circle thus revolving is call ed the generating circle.
The diameter of a sphere is a straight line passing through the centre, and terminated both ways by the sur face.
The axis of a sphere is that diameter about which the generating circle, or sphere itself, is supposed to revolve.
11 an indefinitely thin plane or flat surface cut or pass through a sphere, the part of the plane that lies within the sphere will be a circle, whose circumference appears on the surface, and is called a circle of the sphere.
The pole of a circle of the sphere, is a point on the sur face, from which every point in the circle is equally distant. Hence Every circle of the sphere has two poles, diametrically opposite to one another.