284 



THE POPULAR EDUCATOR 



LESSONS IN GEOMETRY. XXII. 



THE OVAL THE PARABOLA. 



THE next two problems will be found Useful by the practical 

 draughtsman, as the first enables him to draw an oval, a figure 

 approaching very nearly to the form of the ellipse, by a few turns 

 pf his compasses ; while the second shows how an ovoid or egg- 

 shaped figure, of which one end is more pointed than the other, 

 may be formed. The oval is as elegant in form as the ellipse, 

 and quite as useful for all practical purposes. It may be drawn 

 far more readily than the ellipse when it is necessary to trace the 

 curve by hand and determine points through which it must pass, 

 as in the last problem. 



PROBLEM LX. To describe an oval on any given straight 

 line as its greater diameter. 



Let A B (Fig. 87) be the given straight line about which, 



as its greater diameter, it is 

 required to describe an oval. 

 Trisect the straight line A B 

 in the points c and D. From 

 the centre c at the distance 

 c D or c A describe the circle 

 A E F, and from the centre D 

 at the distance D c or D B de- 

 scribe the circle B E F, and let 

 the circles A E F, B E F inter- 

 sect each other in the points 

 E and F. From the point E, 

 through D, draw the straight 

 line E G, meeting the circumference of the circle B E F in G, 

 and from the point F, through c, draw the straight line F H, 

 meeting the circumference of the circle A E F in H. From 

 the point E as centre, with E G as radius, describe the arc G K 

 meeting the circumference of the circle A E F in K ; and from 

 F as centre, with the radius F H, describe the arc H L, meeting the 

 circumference of the circle B E F in L. The figure A H L B u K 

 is an oval, and it is described on or about the given straight 

 line A B as its greater diameter. The straight line N o drawn 

 through the points of intersection E and F of the circles A E F, 

 B E F, is the lesser diameter of the oval, and M, the point in 

 which its diameters intersect each other, is its centre. 



There is another method of constructing an oval, the prin- 

 ciples of which may be readily applied to the mode of con- 

 struction just described, inasmuch as in both cases the oval is 

 described by arcs of circles drawn from four centres, which are 

 the angular points of a rhombus, a figure formed by placing to- 

 gether two equal equilateral triangles, base to base. 



In x T (Fig. 88), a straight line of indefinite length, take any 

 two points A and B, and on the straight line A B describe the 

 equal and opposite equilateral triangles ACB, ADB. Produce the 

 sides c A, c B of the triangle ACB indefinitely towards E and F, 

 and the sides D A, D B of the triangle ADB also indefinitely 

 towards G and H ; and through c and D, the opposite vertices of 

 the equilateral triangles A c D, ADB, draw the straight line K L 

 of indefinite length, intersecting the straight line x Y in the point 



z. In cz take any 

 point M, and from z 

 set off z N along z D, 

 equal to z M. From 

 D as centre at the 

 distance D M, de- 

 scribe the arc o M p, 

 Y meeting the straight 

 lines D G, D H in o 

 and P ; and from c 

 as centre, at the dis- 

 tance c N, describe 

 the arc Q N R, meet- 

 ing the straight lines 

 C E, C F in Q and 



-,. R. Then from the 



x if?, ob. . . 



point A as centre 



at the distance A o or A Q, and from the point B as centre at 

 the distance B p or B R, describe the arcs o Q, p R. The figure 

 o M P R N Q, composed of the four arcs o P, P R, R Q, Q o, described 

 from the four angular points of the rhombus D B c A as centres, 

 is an oval. By taking points beyond M and N at equal distances 

 from z, a series of similar ovals may be drawn, as shown by the 



dotted line surrounding the oval o M p R N Q. This shows us an , 

 easy and practical method of forming an oval grass-plot or 

 flower-bed, surrounded by a gravel walk of uniform width. 



PROBLEM LXI. To describe an ovoid or egg-shaped oval or 

 any given straight line taken as its lesser diameter. 



Let A B (Fig. 89) be the straight line that is given on or about 

 which to describe an ovoid or egg-shaped oval. Bisect A B in c, 

 and from c as centre, at the distance c A or c B, describe the circle 

 A D B E, and through c draw the straight line D z of indefinite length 

 towards z, at right angles to A B. Through the point E, from 

 the points A and B, draw the straight lines A T, B x of unlimited 

 length. Then from A as centre,, with A B as radius, describe the 

 arc B G meeting A Y in G ; and from B as centre, with B A as radius, 

 describe the arc A F, meeting B x in F. From the point E, at 

 the distance E F or EG, describe the arc F L G. The figure 

 DALE is an ovoid, and it is described about A B as its lesser 

 diameter as required. If it be required to make the ovoid, 

 longer or shorter than the ovoid D A L B, it is manifest that the 

 points from which the arcs forming the sides of the figure are 

 described must be without the points A and B in the straight 

 line A B, produced both ways in the first case, and within the 

 points A and B in the straight line A B itself in the second. Sup- 

 posing that it be required to make it longer than the ovoid DALE, 

 produce A B both ways to Q and R ; in A Q take any point H, and 

 make c K equal to c H. Take any point L in D z, and through L 

 from the points H and K draw the straight lines HP, K o of un- 

 limited length towards p and o. Then from H and K as centres, 

 at the distances H B, K A respectively, describe the arcs B N, A M, 

 meeting H p K o in M and W, and complete the ovoid DAMNS as 

 before, by drawing the arc M N from L as centre with the radius 

 L M or L N. The student may work out the remaining case for 

 himself, bearing in mind that the radius with which the arcs 

 forming the sides 

 of the ovoid are 

 described must be 

 necessarily greater 

 than the radius of 

 the circle described 

 about the given 

 lesser diameter, or, 

 in other words, 

 greater than one- 

 half of the straight 

 line given as the 

 lesser diameter, as, 

 when the centres of 

 the side arcs ap- 

 proach so closely 

 together as to coin- 

 cide with each other 

 and the centre of 

 the circle, there can be no elongation of the lower part of the 

 ovoid, which, in fact, then becomes identical with the circle. 



PROBLEM LXII. To describe a parabola by mechanical 

 means. 



Fix a long flat ruler by means of two brass pins on the piece 

 of wood or paper on which it is required to describe a parabola, 

 and from a point A (Fig. 90), taken as nearly as possible in the 

 middle of the ruler, draw a straight line A B at right angles to 

 the ruler, or the direction in which the ruler is fixed. The 

 straight line A B is called the axis of the parabola, while the 

 line c D which represents one side of the fixed ruler is called the 

 directrix of the parabola. Take a ruler made in the form of a 

 right-angled triangle (see Vol. I., page 96), and at the extremity 

 G of the longer of the two sides that contain the right angle G F E, 

 fasten a piece of thread or string, and let the thread have a knot 

 tied in it so that the length of the thread from G to the knot may 

 be exactly equal to the side G F of the triangular ruler. Thrust 

 a pin through the knot, and fix the pin through any point H, in 

 the straight line A B, which has been selected as the focus of the 

 parabola to be described. Place the edge F G of the triangular 

 ruler along the straight line A B, keeping the string tight with a 

 poncil-point, which, when the edge F G of the ruler is lying along 

 the straight line A B, will manifestly be at a point K, the point of 

 bisection of A H, the distance between the fixed ruler and the 

 focus of the required parabola. Slide the edge F E of the 

 triangular ruler slowly along the edge c D of the fixed ruler in 

 the direction of c, keeping the pencil-point against the edge F a 



