308 



THE POPULAE EDUCATOR. 



straight line p x of indefinite length towards x, and along p x 

 from P set off PC, CD, each equal to A. Through c draw c Y 

 of unlimited length towards T, and along it from c set off c E 

 equal to B. Through p and T) draw p F, D G parallel to c E and 

 equal to it, and through E draw 

 F G parallel and equal to p x. 

 Divide p c, c D into any number 

 of equal parts in the points a, b, 

 c, etc., and divide p F, D G each 

 into the same number of equal 

 '. parts in the points m, n, o, etc., 

 s, t, u, etc., as each of the straight 

 lines P c, c D have been divided 

 into. Draw straight lines through 

 the points a, b, c, etc., parallel to 

 1 C E, and from E draw straight 

 5" lines to the points m, n, o, etc., 

 x in P F, and the points s, t, u, 

 etc., in i> G. Then through the 

 points 1, 2, 3, etc., formed by the 



Fig. 91. 



intersection of E r with the parallel through /, B g, with the 

 parallel through e, E p, with the parallel through d, etc., trace 

 the curve E P above the axis E o, and the curve E D below it. 

 The curve p E D is the required parabola. 



PROBLEM LXIV. To describe an hyperbola by mechanical 

 means. 



The hyperbola, instead of being considered as a single curve, 

 is frequently represented as consisting of two equal and sym- 

 metrical curves, having their vertices opposite each other, and 

 their branches proceeding in contrary directions. The reason 

 of this may be understood from Fig. 92, in which two cones are 

 represented, the one having its apex against the apex of the 

 other, and its base turned in the contrary direction. Such a 

 double cone as this may be generated by the revolution of two 

 equal equiangular and similar right-angled triangles, having 

 their vertices contiguous, and their altitudes in the same 

 straight line as the triangles A B c, A D E, in the figure. We 

 may also conceive the double cone to be generated by the 

 revolution of a straight line, B A F, or D A G, 

 round its central point, A (which is fixed), and 

 inclined at any angle less than a right angle 

 to a perpendicular straight line, E A c, passing 

 through A, which perpendicular becomes the 

 axis of the cone thus generated. Now if we 

 suppose a plane H K M L to pass through the 

 axis E c of the double cone, and the double 

 cone to be cut by another plane parallel to 

 the plane H K M L, as the plane N o p Q, it is 

 manifest that it will cut each branch of the 

 cone in NOB, P s Q, which form two equal 

 symmetrical and opposite curves, and which 

 are considered as each forming a branch of 

 the complete hyperbola. 



Our readers will now more readily compre 



Fig. 92. 



hend the method of describing an hyperbola by mechanical 

 means, and when certain data are given ; and they will also 

 understand why an hyperbola is said to have two foci, like an 

 ellipse. 



Inx T (Fig. 93), which represents any straight line of indefinite 

 length, let two points, A and B, be selected as the foci of the 

 hyperbola to be described. Take a flat, narrow ruler, c D, with 

 a hole in it near one end, through which a pin may be inserted 

 to fasten the ruler to the paper or board on which the hyperbola 

 is to be traced, the ruler working freely round the pin. Suppose 

 JP, in A B, be selected as the vertex of the hyperbola that is to 

 be traced. Take another point, E in A B, so that A E is equal 

 to F B ; then E will be the vertex of the opposite branch of the 

 hyperbola, and E F the major axis of the curve. Let a string 

 be fastened at the end, D, of the ruler c B, and let the string 

 be of unlimited length, or, what is as well, of the same length 

 as the ruler. Set off along the ruler from the point A, in the 

 direction A D, a straight line A G, equal to E F, and holding the 

 string tightly to the edge of the ruler, mark it at the point 

 opposite to the point G in the ruler ; then thrust a pin through 

 the string at the point thus marked, and fasten it down at the 

 point B. Keeping the cord stretched to its utmost tension 

 with a pencil-point, and having the edge of the ruler applied 

 to the straight line x T, move it slowly upwards round the 



Fig. 93. 



pivot A. Before starting, when the edge of the ruler is con- 

 tiguous to x Y, the point G will be at H, the pencil-point at F, 

 and the string in the position B F, F K. As the ruler moves 

 upwards, the pencil-point traces out the curve F L N p, the point 

 G describing or moving in the path of the arc H M, and the 

 end of the ruler D in the path of the arc K o. The point 

 Q, where A B is bisected at right angles by the perpendicular 

 E S, is the centre of the hyperbola. By reversing the ruler, 

 and repeating the operation below x Y, the lower part, F T, of 

 the curve 

 p F T may be 

 traced ; and 

 by fixing the 

 ruler so that 

 the point re- 

 presented in 

 the figure at 

 A may be at 

 B, and the 

 end of the 

 string fast- 

 ened at A, 

 the opposite 

 branch of 

 the hyper- 

 bola passing 

 through E 

 may be de- 

 scribed. The 



straight line v B u, passing through the focus B, is called the 

 latus rectum of the hyperbola ; F B the abscissa, and B v the 

 ordinate of the point V ; F z the abscissa, and p z, T z the 

 ordinates of the points p and T. The chief peculiarity of 

 the parabola is, that the distance of every point in the curve, as 

 the ruler passes from one position to another from the focus, is 

 equal to its distance from the point marked G in the ruler. 

 Thus, when the ruler is in the position A K, and G is at H, 

 F H is equal to F B ; in the position AD, L B is equal to L G ; 

 in the position A o, N M is equal to N B ; while in the position 

 A p, p w is equal to P B. The distances, A F, F B ; A L, L B ; 

 AN, N B ; A P, P B, are called the focal distances of the points 

 F, L, N, P respectively, and the difference between the greater 

 and the lesser of any of these pairs of distances from the foci 

 of the hyperbola is equal to E F, the major axis of the hyper- 

 bola ; and this is true for every point in the curve. For this 

 reason, in the commencement of the problem, A G was made 

 equal to E F. 



PROBLEM LXV. To describe an hyperbola by fixing a number 

 of points through which the curve may be traced, the major axis, 

 and the abscissa and ordinate of any point in the curve being 

 given. 



In Fig. 94 let any indefinite straight line, x Y, be the axis of the 

 required hy- 

 perbola; the 

 portion in- 

 tercepted be- 

 tween the 

 points A and 

 B being set 

 off equal to 

 p, the given 

 major axis ; 

 and Q, E be- 

 ing the given 

 abscissa and 

 ordinate of 

 a point in 

 the curve. 

 From B, set 

 off along x Y, 

 in the direc- 

 tion of Y, 



Fig. 94. 



B c equal to Q, and through c draw the straight line D E of 

 indefinite length, at right angles to x Y ; and from c along D E, 

 in the directions of D and E, set off c F, c G, each equal to E. 

 The points F and G are points in the required curve. Through 

 F and G draw F H, G K parallel to x Y, and through B draw H K 

 parallel to DE. Divide c F, co each into five equal parts in 



