THE HYPERBOLA AWD CYCLOID.] MATHEMATICS. PR ACTICAL GEOMETRY. 



611 



dicular to A B, or parallel to C D, cutting the former 

 lines ; then the curve drawn through the points of in- 

 tersection will be the parabola required. 



3RD METHOD. Place a ruler G H (Fig. 75) at any 

 convenient distance from C, parallel to the base A B, 

 Fig. 75. 



and take a piece of wood (called a square or set square), 

 made in the form of a right-angled triangle I O K, placing 

 the base I O against the ruler, and the other edge O K 

 to coincide with the line C D ; having found the focus 

 F (as in the 1st method), fasten one end of a string at F, 

 place a pencil at the point C, passing the string round 

 it, and bringing the string back to K, fasten it to the 

 end or point of the triangle ; move the triangle or square 

 along the ruler, keeping the pencil always against the 

 edge of the square (as at E), and with the string stretched 

 the pencil will describe one-half of the curve. By re- 

 versing the square and proceeding in a similar manner, 

 the other half may be drawn, and the parabola required 

 completed. 



THE HYPERBOLA. 



If the reasoning employed in the case of the ellipse be 

 carefully gone over, attending only to the difference 

 that will result from the circumstance that e 7 1. the 

 student will readily deduce that the following expressions 

 will be found to hold good of the hyperbola, which are 

 entirely analogous to the corresponding expressions in 

 the case of the ellipse. 



(1.) SC = AC. 



(2.) BC must be so taken that BC 1 = CS I C A= 

 (e a 1) C A". 



(3.) SP=e. ON AC. 



,, , PN_C_N , 



1 ' CB 2 CA 



HP = e. CN+AC. 

 HP SP-2AC. 



Fig. 14. 



It is on this (6th) property that the following practical 

 construction is founded. 



The terms major axis, transverse axis, <kc., in the 

 hyperbola, are entirely analogous to the same terms in 

 the ellipse. 



To detcribe a hyperbola, the transverse (A B) and conjugate 

 (C D) axes being given. 



Through B (Fig. 77), at one end of the transverse 

 axis A B, draw G H 

 parallel to the conjugate 

 axis C D, and make GH 

 equal to C D ; with the 

 centre E and radius, 

 equal to EH, describe 

 a circle cutting A B, 

 produced both ways, in 

 the points F and/, which 

 will be the foci of op- 

 posite hyperbolas ; take 

 any number of points, 

 1, 2,3, 4 (<fec.), in AB 

 produced, and with the 

 centres F and /, and 

 radii Bl, B2, B3, B4, 

 fB Ac.), and Al, A2, 



A3, A4 (A <tc.), describe arcs cutting each other ; the 

 curve drawn through the points of intersection will be 

 the hyperbola required. 



The following consideration materially simplifies the 

 construction of a hyperbola. 



Through A (Fig. 78) draw A D parallel and equal to 

 B C, join C D and produce it indefinitely, take P any 

 point in the curve, draw pP N, p; g- 73, 



meeting CD produced in p. Then 



DA pN PS 



/% 



CN' ' A 



But (4) 



CA 



A subtracting * 



Now it is plain that PN and pN increase as P is far- 

 ther from A, or N from A, and therefore, the farther P 

 is from the vertex the nearer it approaches C D produced. 

 C D is called an asymptote after a short distance the 

 curve sensibly coincides with the asymptote. Hence, if 

 in the practical construction above given E H be pro- 

 duced, it will serve as a guide to the curve, which can be 

 drawn very accurately after a very few points have been 

 determined by the construction. 



THE CYCLOID. 



Besides the conic sections, which we have briefly dis- 

 cussed above, there are several curves possessing curious 

 or useful properties. Amongst the chief of these is the 

 cycloid. The construction of this curve is sometimes 

 useful to the artist. The following will suffice to explain 

 the nature of the curve, and the method of its con- 

 struction. 



The cycloid Is a curve formed by a point in the 

 circumference of a circle (called the generating circle), 

 revolving on a straight or level line ; it may be best 

 described as the curve traced out by a point in the wheel 

 of a carriage when in motion along a level road. When 

 the generating circle revolves round another circle, the 

 curve described by a point in the circle is then called an 

 epicycloid, and is constructed in a similar manner to the 

 cycloid. 



To describe a cycloid, the diameter or width at the base 

 (AB), and height (CD), of the curve being given. 



The most common method of describing the cycloid is 

 by placing a ruler along the line A B, and taking a 

 circle, such as a shilling, &c., according to the size or 

 height of the curve required, and having fixed a point in 

 that circle, to move it slowly along the ruler, marking 

 different points in the curve, or by keeping a pencil fixed 

 at the point chosen, and thus describing the curve. 

 This, however, is liable to error, as the circle used very 

 often slips, and cannot then revolve accurately ; by con- 

 struction, however, the curve may be correctly formed 

 thus : 



Let A C B (Fig. 79) represent the curve in question, 

 A B its base, C D its height. Bisect C D in O, and 

 Fig. 79. 



through O draw E F parallel to A- B ; with centre O and 

 radius OC describe a circle, and'divide the circumference 

 into any number of equal parts, as Cp lt Cj) 2 , . . . join 



