252 



THE POPULAE EDUCATOE. 



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to display these curves, the reader must endeavour to under- 

 stand what a cone is, mathematically speaking. The word cone 

 is derived from the Greek KUVOS (ko'-nos), or the Latin conus, 

 terms applied by the Greeks and Romans respectively to objects 

 rising to a point, and by the ancient mathematicians to a solid 

 figure having a circular base and pointed apex, and so formed 

 that straight lines drawn from any points in the circumference of 

 the base to the apex, lie wholly in 

 the superficies or surface of the solid. 

 The ordinary cone, which is usually 

 called a right cone, is formed by the 

 revolution of a right-angled triangle 

 round its altitude. Thus, in Fig. 82, 

 if the right-angled triangle ABC, hav- 

 ing the altitude A B, the base B c, and 

 the hypothenuse A c, were caused to 

 revolve around its altitude A B in a 

 direction from left to right, the 

 fxL -'V" % \B triangle would occupy the positions 



x -^ //* "\ \ _./ A B D, ABE, A B F, A B G, A B H, SUC- 



~"tf '^6 cessively, in its revolution round A B, 



Fig. 82. and when it had returned to its original 



position the movement of the hypo- 

 thenuse would have described a figure which is called a cone. 

 The base of this figure is a circle, c D E F G H, described 

 virtually from the point B as centre, with a radius B c. 

 The path traced by every point in the hypothenuse, as the 

 triangle revolves around its altitude, is a circle ; as, for example, 

 the circle K L M N, traced by the revolution of the point K in 

 the hypothenuse round A B. As any figure which can be formed 

 by cutting a cone in any particular direction is a conic section, 

 it is obvious that the circle is also a conic section, as well as the 

 ellipse, parabola, and hyperbola ; and in a cone circles of any 

 size, varying from nothing at the apex of the cone, to the extent 

 of the circle forming the base, may be obtained by cutting the 

 cone at successive points in the altitude from top to bottom, 

 or from the point A to the point B in Fig. 82, in an hori- 

 zontal direction at right angles to the altitude. Any section, 

 therefore, of a cone made by cutting the cone transversely in 

 a direction perpendicular to the altitude is a circle. 



In Fig. 83 let Abe the apes, and B c D E the circular base of a 

 right cone, the axis or altitude of the cone being the straight 

 line, A F, drawn from the apex, A, to F, the centre of the circular 

 base. It has been already shown that if the cone be cut trans- 

 versely, in a plane at right angles to its axis or altitude, the sec- 

 tion will be a circle. Now, instead of cutting the axis at right 

 angles, let the plane of section be inclined to the axis, or a plane 

 passing through the axis, as the plane A B D, at an angle that is 

 not a right angle, as the plane of section G H K L is inclined to 

 the plane of section A B D, at an angle, A M G, which is less than 

 a right angle. The plane of section, G H K L, is called an ellipse. 

 The line K G is called the major axis, or greater axis of the ellipse. 

 Next, suppose the straight line G K to be produced both ways to 

 an indefinite length, and to turn on M as on a pivot, moving in 

 the plane shown by the triangle ACE, until it assumed the posi- 

 tion N o. On examination of the figure it will be found that N o 

 is parallel to A c, a line drawn from 

 apex to base on the side of the cone. 

 If, then, a plane be drawn passing 

 through the points N and o, and the 

 points H L, as well as the plane of 

 section N H P Q L, this plane of sec- 

 tion will be parallel to the side of the 

 cone, as well as its axis N o. The 

 plane of section N H P Q L is called a 

 parabola. If, however, the plane of 

 section be parallel to a plane passing 

 through the axis of the cone, as the 

 plane B s T, it is called an hyperbola. 

 To recapitulate what has been said 

 of the various curves produced by cut- 

 ting the cone in different directions 



1. If the cone be cut by a plane parallel to the base, or at right 

 angles to the axis or altitude of the cone, the plane of section is 

 a circle. 



2. If the cone be cut by a plane making any angle other than 

 a right angle with the axis of the cone, the plane of section is an 

 ellipse. 



3. If the cone be cut by a plane parallel to the side of the 

 cone, the plane of section is a parabola. 



4. If the cone be cut by a plane parallel to the axis of the 

 cone, or parallel to any plane passing through its axis, the plane 

 of section is an hyperbola. Indeed, any plane of section which 

 makes with the base of the cone an angle greater than the incli- 

 nation of the side of the cone to the base, is an hyperbola. 



PROBLEM LVIII. To trace the curve of an ellipse by mecha- 

 nical contrivances. 



To students of practical geometry who are seeking a know- 

 ledge of the subject that will aid them in their daily avocations, 

 it will be far more acceptable to have a brief statement of the 

 practical means adopted for tracing the conic sections than it 

 j will to have an exposition of the properties of these curves, 

 which belongs to the higher branches of mathematics. Failing 

 a pair of compasses, every artisan knows that a circle may be 

 readily traced by tying one end of a piece of string round a 

 pencil, and fastening the other end to a board, or anything on 

 which the circle is to be traced, by a nail. A similar plan may 

 be adopted for tracing an ellipse, which may be briefly described 

 as follows : 



Take a piece of thread and two small fine pins, if you are 

 going to draw a small ellipse on paper, or a piece of twine and 

 two nails, if you wish to trace a larger ellipse on a piece of 

 board. Draw a straight line, x Y, of indefinite length, and 

 in it take any two points, A and B. Having tied two knots 

 in the thread or twine, distant from each other a little more 

 than the length of the line A B, thrust a pin through each, and 

 fasten the pins through the paper at the points A and B. Then, 

 stretching the thread as tightly as you can on the point of a lead 

 pencil, and keeping it at its utmost tension during the operation, 

 you trace the half of 

 the ellipse namely, the 

 curve c E D with the 

 pencil point. After 

 this, pulling the thread 

 across x Y, between 

 the pins at A and B, the 

 thread is again tight- 

 ened, and the ellipse 

 is completed by tracing 

 the lower half of the 

 ellipse, the curve c F D, 

 with the pencil point 

 as before. The points 

 A and B, in which the 



pins have been inserted, are called the foci of the ellipse. The 

 straight line, c D, passing through the points A and B, is called 

 the axis major, or transverse diameter of the ellipse. The 

 point G, in which c D is bisected, is called the centre of the 

 ellipse, while the straight line E F, drawn through the centre 

 G, at right angles to c D, is called the axis minor, or conjugate 

 diameter of the ellipse. As the length of the thread does not 

 vary during the process of tracing the curve, it must follow that 

 the sums of the distances of any points in the circumference of 

 the ellipse from the foci are equal to one another. Thus, in Fig. 

 84, the sum of A H -f- H B is equal to the sum of A K + K B, and 

 the sum of B F + F A. The straight line c D, the axis major 

 of the ellipse, is also equal to A H + H B, A K + K B, or A F + F B ; 

 so to describe an ellipse having a major axis of a certain length, 

 all that is necessary to be done is to make the distance between 

 the knots in the thread equal to the given length of the axis 

 major. It must be remembered that the nearer the foci A and B 

 are to the extremities c and D of the axis major, the narrower 

 the ellipse will become in its minor axis, or the distance from E to 

 F ; but, on the contrary, the nearer the foci approach to G, the 

 centre of the ellipse and point of bisection of the axis major, the 

 wider the ellipse will become in its minor axis, approximating 

 more and more closely to the form of a circle, until it becomes a 

 circle, if the foci approach so closely that they coincide with 

 each other and the centre of the ellipse, G. If the foci be re- 

 moved so far apart that they coincide with c, D, the extremities 

 of the axis major, the ellipse assumes the form of a straight line. 

 The straight line and circle are therefore the extremes between 

 which the forms of the ellipse vary, the minor axis being 

 longer or shorter (the length of the axis major remaining the 

 same in all cases) according as the foci are more remote from or 

 nearer to each other. 



