LESSONS IN GEOMETRY. 



ints a, 6, etc., e, f, etc., and divide H r, K a also into 

 ! uU parts in the poL.ta k, I, etc., o, p, etc. Of courw, 

 . -i large, th > greater the number of port* into 

 ublo ordinate, FU, and tho parallel*, ir K, K a, aro 

 more accurately the curve can bo traced, care 

 to iliviil.i tin' jiarullcls into tho same number of 

 equal i>uru a-t ciu-h half of the double ordinate, FO, ia divided 

 into. From tho point A draw straight linen through tho points 

 r, a, b, etc., and from B draw straight lines through tho points 

 fc, I, etc., o, p, etc., and through tho points of intersection of 



Fig. 95. 



i lines AO, sn, Afo, BTH, etc., numbered 1, 2, etc., and the points 

 r, B, and G, trace the curve p B o. This curve is the hyperbola 

 ijuired. 



Lest some of our readers may bo tempted to inquire of whai 

 practical use it may be to be acquainted with the method ol 

 tracing parabolas and hyperbolas of different degrees of cur- 

 vature, we may remind them that the parabola sometimes is 

 used in forming an arch, while such articles as tazzas and wine- 

 glasses, and other pieces of useful and ornamental china-ware, 

 may be formed by the revolution of an hyperbola about its 

 axis, as may be seen by copying the curve in Fig. 94, BO that the 

 vertex, B, points downwards, and then adding a slender stem 

 and foot to form a wine-glass. 



PROBLEM LXVI. To describe the curve called the cycloid. 

 The term cycloid, derived from the Greek KVK\oet5t)s (ku-klo- 

 i'-dees), like a circle, is a name given to the curve traced by any 

 point in the circumference of a circle during the complete revolu- 

 tion of the circle while rolling along a straight line. For example, 

 as a carriage is drawn along on a road or railroad, tho end of 



spoke in one of its wheels, or a nail in the tire, describes 

 locossion of curves, similar to tho curve resembling half of an 

 lipse in Fig. 95. That the reader may understand how the curve 

 [ traced, let A B c D represent a circle, having two diameters, A c, 

 B D, intersecting each other at right angles, and let tho circle bo 

 standing on a straight line, XY, of indefinite length, so that 

 the diameter A c is at right angles to x Y, which is a tangent 

 to the circle A B c D, the circle touching it only in tho point A. 

 Suppose the circle to roll slowly along tho straight lino x Y, in 

 the direction of x, and pass into the position A' B' c' D'. It 

 has now performed a quarter of a complete revolution, and tho 

 point A in ascending into the position A' has traced a path 

 represented by the curve A A'. In the next quarter of a revo- 

 lution the point A is brought to the top in tho position A", 

 and when a complete revolution of the circle has been made it 

 has passed from A" to A'" and A"", having traced in its passage 

 from A to A"" the curve A A' A" A"' A"". Practically, the cycloid 

 may be traced by causing a thin disc of metal, ivory, or even 

 cardboard, having a slight nick in its circumference to receive a 

 pencil point, to travel slowly along the edge of a ruler until a 



D 



fig. 96. 



plete revolution has been made. At tho commencement of 

 the revolution the pencil-point must be on the line along which 

 the disc is to revolve, as A, in tho straight line x Y, in Fig. 95 

 above. 



lake tho ellipse, parabola, and hyperbola, the cycloid has 

 certain properties peculiar to itself. Suppose the circle B K D 

 (Fig. 96) revolving along the straight line A c, to have traced on. 

 " ie cycloid A B c ly tho passage of tho point B from A to c 



fig. 97. 



daring the revolution. It w evident that, M ovary point of th* 

 circumference of tho circlo in snooessioo touehM the straight 

 line A c during the revolution, A c, which wo may 5HJI the base 

 of the cycloid, in equal in length to the etroamforenoo of the 

 circle B K D. If tho circle be catuod to return to it* position 

 in tho centre of the cycloid when B ia at its highest position, M 

 in tho figure, and straight lines, such as ic if, x r, be drawn 

 through tho circle parallel to tho baso <md terminating both 

 ways in tho curve of tho cycloid, theno straight I in** pass 

 through opposite points in the circumference of tho circle B K D, 

 at equal distance* from the diameter B D, whioh is perpendi- 

 cular to the base, o L, being equal to o o, and B X to B P. It 

 will bo found that L H in equal to tho aro B L, and that tho aro 

 B ii ia equal to twice the chord B L, and so on for the other 

 points, H, N, F, in the curve of tho cycloid, through which 

 ) straight lines have been drawn parallel to the ba*e. Tho are 

 B o will therefore bo equal to twice B D, the diameter of the 

 generating circle, and tho wholo curve ABC consequently 

 equal to four times B n. Thw curve is Raid to have been 

 discovered and ita properties first investigated by Galileo. 

 PEOBLEM LXVII. To describe a spiral. 

 Take any point, A (Fig. 97), as the centre of tho spiral to be 

 drawn. Draw a horizontal straight line, x Y, of indefinite length 

 through A, and from A as centre, with any distance, A B, 

 describe tho semicircle B D c. 

 Then from the point B as 

 centre, with the distance B c, 

 describe the semicircle c E F 

 on the opposite side of A B. 

 Next, from A as centre, with 

 tho distance A F, describe 

 tho semicircle F a H, and 

 then from B and A, in alter- 

 nation as centres, at tho 

 distances B H, A L, etc. etc., 

 describe as many semicircles 

 in succession as may bo 

 required. A spiral of any 



given number of turns may be described on a given straight 

 line by dividing the given straight lino into as many equal parts 

 as there are turns required, and bisecting the central division 

 if the number of turns bo odd, or tho division on the right or 

 left of the centre of tho lino if the number of turns be even, 

 Tho centres to be fixed in describing the semicircles mufrt be 

 the point of bisection, and cither of tho points of division imme- 

 diately contiguous to it if tho number of turns be odd, or the 

 point of bisection and the centre of the divided line if the 

 number of turns be even. Thus, in Fig. 97, if it be required, 

 to describe a spiral of eight turns or semicircles on the given 

 straight lino B T, divide R T into eight equal parts, in the 

 points N, H, c, B, F, L, P, and bisect B c, or B F, in A for the 

 centre of the spiral. Then from the points A and B, in alterna- 

 tion, describe the semicircles B D c, c K F, etc. etc. 



PROBLEM LXVI1I. Any two straight lines being given, to 

 determine a curve 

 by which they shall 

 be connected. 



Let A B, CD (Fig. 

 98) be any two 

 straightlineswhich 

 t is required to 

 connect by a curve. 

 Produce A B, c D 

 H the direction of 

 B and c, until they 

 meet in K. Bisect 

 die angle B E c by 

 :ho straight lino 

 E F. From the 

 extremities B and 

 c of the straight 

 lines A B, c D, draw B F, c F perpendicular to A B and c l> 

 respectively, and intersecting each other and tho straight line 

 E F in the point F. From F as centre, with tho distance F B or 

 F c, describe the aro B c. This aro connects tho straight lines 

 A B, CD. The same process ia followed when the given straight 

 ines are at right angles to each other, as A B, a H, which are 

 connected by a curve, B o, struck from K as centre, tho point of 



