THE ELLIPSE.] 



MATHEMATICS. PRACTICAL GEOMETRY. 



609 



Now, Op 2 = ^ B. and G^ = ^ O D. 



Also, 



OD-PN OD- fl B 



ON OB 0~^~ n' 



Again, draw P M perpendicular D O. Then 



CD 

 OB. 



Op 2 

 OC 



MP 



ON 



MC' or OC + OM 



p OB 



Now OC = OD 



OD+PN 



OD 



ON 



But we have already seen that 



OD PN 



ON 

 OD* 



OT). 

 OB 



O.V 



PN* 0X2 

 OD 1 + OB T 



= i 



Hence by article (4), P is a point in an ellipse, whose 

 major and minor axes are O B and O D respectively. 



On this principle the second practical method is 

 founded. 



(7.) If A B, C D (Fig 63) are two lines at right angles 

 to each other, and Fig. 63. 



bisecting each 

 other in the point 

 O, then if Pot be 

 a straight line, so 

 fiuceii that P6 = 

 O C and Pa = __ 

 O A, then the 

 point P it in an 

 ellipse, whose axes 

 an AS and C D. 



For draw P N, PM at right angles to O C and A. 

 Now by similar triangles, 



06 



ab' 



Oa 



I'M 



Pa' 



PN 



_PN^ON Ob + Oa* 



"~" 6 



_ON 



PN 



oo' 



ab* 

 ~ ^jj"~ ! Since 60a is 



a right angle. 



Hence (art. 4) P is a point in the specified ellipse. 

 Q. E. D. 



This is the principle of the trammel described in the 

 fourth practical method. 



To describe an Ellipse, the transverse (A B) and conjugate 

 ' (C D) diameters being given. 



IST METHOD. Draw A B and C D, bisecting each 

 other at right angles, in the centre O (Fig. 64) ; with 

 the centre D or C 

 and radius O A or 

 O B, describe area 

 cutting A B in F 

 and F', which points 

 will be the foci of 

 the ellipse ; take 

 any other point P 

 in the transverse 

 diameter A B, and 

 with the centres F, 

 F' and radii equal to A P and P B, respectively, describe 

 arcs intersecting in the point E, which will be a point 

 in the curve. By taking several points in the transverse 

 diameter, and proceeding in a similar manner, as many 

 points in the curve may be found as may be required, 

 through which the ellipse may be drawn. 



2uu MKTHOD. Through the extremities of the given 

 diameters draw the rectangle E F G H (Fig. 65) ; divide 

 VOL. I. 



Fig. 65. 



A E, A F and 

 BH, BG into 

 any number of 

 equal parts (say 

 3) ; also divide 

 A O and OB 

 into the same 

 number of equal 

 parts ; join D 

 and the points 

 of division in 

 AF and B G, 

 also C and those 

 in A E, and B H. Again, from D draw lines through 

 the points of division in A O and O B, intersecting the ! 

 lines drawn from C to the points in A E and B H ; and 

 in the same manner from C draw lines through the 

 points of division in A O and O B, intersecting the lines 

 drawn from D to the points in A F and B G ; the in- 

 tersections thus derived are points in the curve through 

 which the required ellipse may be drawn. 



SKD METHOD. An ellipse may easily be drawn by 

 means of a thread, in the following manner : 



Find the foci F and F' (Fig. 66) as in the first method ; 

 take a piece of thread equal in length to the distance 

 between the foci add- 

 ed to the transverse 

 diameter, and fasten 

 the two ends ; having 

 fixed two pins in the 

 foci, pass the thread 

 round them and 

 stretch it, place a pen- 

 cil in the angle made 

 by the thread when 

 stretched, and com- 

 mencing at one end (D) of the conjugate diameter, the 

 thread being in the position F D F (as shown in dotted 

 lines), move the pencil-point round, keeping the thread 

 always stretched until the point again meets at D ; the 

 ellipse will be drawn. 



4TH METHOD. An instrument called a trammel is 

 sometimes used for describing an ellipse, and is generally 

 formed of two pieces of wood fixed at right angles to 

 each other, and having a groove running through the 

 centre of each, the groove being made a little wider at 

 the bottom than at the top, in order to keep the guides 

 6, c, (Fig. 67), attached to the bar, abc, in the groove ; 

 the bar, abc, should have a pencil fixed at the end a, 

 the two guides, 6 and c, being movable or sliding along 

 the bar, but when set ready for use, secured by small 

 screws in the proper positions. The method of using 

 the trammel is as follows : 



1st. Set the 

 grooved pieces of 

 wood on the 

 given diameters 

 A B and C O of 

 the ellipse (as in 

 the figure) ; the 

 centre, where 

 the two pieces 

 cross, being on 

 the centre O of 

 the ellipse. 



2nd. Make the 



distance from the pencil at a to the first sliding-guide at 

 6 equal to half the conjugate diameter, (or ab equal to 

 O D or O C) ; and from the same point a to the second 

 sliding-guide at c equal to half the transverse diameter 

 (or ac equal to O A or O B). 



3rd. Move the end a round, commencing at the point 

 D, allowing the guides to move freely in the grooves, 

 and the pencil will be found to describe the ellipse 

 required. 



The OVAL, although not an ellipse, so nearly resembles 

 it, that very frequently, from the simplicity of drawing 

 an oval, it is used in place of an ellipse. It is formed of 

 parts of circles by the following methods : 



4l 



