Page:The American Cyclopædia (1879) Volume VII.djvu/711

 GEOMETRY 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 as- sumes 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 oc- cupies 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 po- sition only. A point may be considered inde- pendently 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 philoso- phers, 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 in- direct measure of magnitude. To measure a magnitude is to find how many times it con- tains 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 measure- ment 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 num- ber 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 inter- esting questions immediately presented them- selves. What was its diameter ? How many square miles did its surface contain ? Were all its diameters equal ? To answer these ques- tions 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 ele- ments of which each magnitude is composed were related to each other according to cer- tain fixed and definite laws. The number of different forms in space is infinite, and unless their relations to each other were fixed and definite, and they were susceptible of classifi- cation and comparison, there could be no sci- ence of geometry. The same would be the case if the different magnitudes which are the elements of every form were not connected by definite relations. Geometry shows that they are so related, and explains the nature of those relations. According to the different points of view from which it is regarded, geometry is variously divided. Its primary division is into elementary and higher geometry, mentary geometry treats of angles, straight lines, planes bounded by straight lines, solids bounded by planes, circles, cylinders, cones, and spheres. The treatment of all curves ex- cept the circle, and of all surfaces and soli< which involve the consideration of any curve other than the circle, belongs to higher geom- etry The only instruments necessary for the construction of the figures employed in treat- ing of elementary geometry are the rule and