Geometry (Gr. fromthe earth, andto measure), the science of relations in space. As its name indicates, it originally denoted the measurement of land, and was equivalent to what is known in modern times as surveying. As at present understood, surveying is but a subordinate application of the science, and although geometry retains its ancient name, it has by the labors of many successive generations grown to be a vast and comprehensive system, forming the basis of many of the most important arts and sciences. It has been defined as "the science which treats of forms in space;" and if we give a sufficiently extended meaning to the word "form,'1 the definition is perhaps as good as any other. It regards material objects only in so far as they occupy space. With their other physical qualities, their color, weight, hardness, etc, geometry has nothing to do. Assuming that a billiard ball and the sun are each a perfect sphere, then the only geometrical difference between them is the difference in size. Neither has geometry anything to do with the nature of space abstractly considered. It assumes the notion of space as it is assumed by all men in practical life, and leaves to philosophy the discussion of its nature.

It assumes that space is infinite in extent; that is, it assumes as undeniable, and therefore as requiring no proof, that we can neither in fact nor in thought set any boundary to space and rightfully say there is no space beyond. It assumes that space is infinitely divisible; that is, that no portion of space is so small that we cannot conceive it as being divided. Finally, it assumes that space is continuous; that is, that which separates any two definite portions of space is itself space. Any definite portion of space, whether occupied by a body or not, is in geometry called a solid or volume, and the property of a body by virtue of which it occupies space is called extension. Extension is said to have three dimensions, length, breadth, and thickness. The limits of a solid are called surfaces, and are said to have length and breadth without thickness. The limits of a surface are called lines, and are said to have length without breadth or thickness. The limits of a line are called points, and are said to have neither length, breadth, nor thickness, but position only. A point may be considered independently of any line, a line independently of any surface, and a surface independently of any solid.

The definitions of these fundamental notions of geometry have always been matters of controversy among geometers and philosophers, but practically all men are agreed as to its nature. The idea of space involves three notions which are indissolubly connected, viz.: position, direction, and magnitude. Starting from any given point, we can suppose lines to be drawn in an infinity of different directions. The difference in the direction of any two of these lines is called an angle. A line whose direction is everywhere the same is called a straight or right line; a line which changes its direction at every point is called a curved line. When the word line is used alone, and there is nothing to indicate the contrary, a straight line is always meant, and a curved line is usually called simply a curve. In treating of forms in space, straight lines, angles, and curves, and their mutual relations, are the principal things which the geometer has to consider. The object of geometry is the indirect measure of magnitude. To measure a magnitude is to find how many times it contains a known magnitude of like nature with itself, which is assumed as a unit.

Thus, to measure a line is to find how many times it contains a line of known length, as an inch, a foot, a yard, a metre; to measure a surface is to find how many times it contains a known surface, as a square inch, a square foot, a square yard, a square metre, an acre, a square mile; to measure a solid is to find how many times it contains a known solid or volume, as a cubic inch, a cubic foot, a cubic yard, a cubic metre, a cubic mile. To measure a straight line, the most obvious method is to apply to it the assumed unit, for example, a foot, and count the number of times the line to be measured contains it. This method of measurement is purely mechanical, and geometry has nothing to do with it; it is a question, not of geometry, but of physics and arithmetic. In many cases, as in measuring the height of a mountain, this method is impracticable; in many others, as the distance of the moon from the earth, it is impossible. And when we pass from the measurement of straight lines to the measurement of curves, surfaces, and solids, we find that in almost all cases the mechanical method is either impracticable or impossible. Thus the every-day problem, to find how many acres there are in a farm, would, in the absence of all geometrical knowledge, remain for ever insoluble.

It is evidently necessary to find some method of measuring indirectly that which we cannot measure directly. Thus in the case of a farm we can measure by me-chanical means the length and directions of its boundary lines, and then geometry teaches how, knowing these, we can find the number of acres it contains. Let us take as another example a problem of a higher kind. From the observation of certain physical facts men long ago concluded that the earth was a spherical body. A great number of interesting questions immediately presented themselves. What was its diameter? How many square miles did its surface contain? Were all its diameters equal ? To answer these questions by direct measurement was impossible; all that could be done was to measure here and there a line upon its surface. Yet with the aid of a few direct measurements and of the principles of geometry all these questions have been answered. It is evident that the attainment of these results would be hopeless, and that geometry would be impossible, unless the different magnitudes of space and the elements of which each magnitude is composed were related to each other according to certain fixed and definite laws.