to the base, the section will be similar to the base, i.e., a circle. But if a cone be cut in some other way, the section has a distinctive name. Thus 2. An ellipse is a section of a cone, produced by a plane which is not parallel to the base. Ex. AB NOTE 1.-Such a figure has two diameters, unequal in length; viz., the long diameter A B, called the transverse or major axis, and the short diameter ab, called the conjugate or minor axis. NOTE 2.—There are two important fixed points in the transverse axis called foci (singular, focus) equally distant from the centre, and are such that the sum of two straight lines F1A, AF, drawn from them to any point A in the circumference, is equal to the length of the major axis. 3. A parabola is a section of a cone, produced by a plane which is parallel to one of the sides. Ex. ABC NOTE.--Its base AB is termed its double ordinate, AC or CB being its ordinate ; and its altitude CD is called its abscissa. 4. A hyperbola is a section of a cone, produced by a plane which is parallel to its axis. Ex. ABC NOTE 1.--AB is termed its double ordinate, AC its ordinate, CD its abscissa, and CE its diameter. NOTE 2.—The three foregoing sections are usually known as the “conic sections.” 5. An oval, as its name implies, is simply an egg-shaped figure, being wider at one end than at the other. Ex. A Problem 80. To describe an ellipse, its axes or transverse and conjugate diameters AB and CD being given. 1. Place the transverse diameter AB and the conjugate diameter CD perpendicular to each other at their centres E. 2. Through A and B draw the lines FG, HK parallel to CD (Pr. 8), and through C and D draw FH, GK parallel to AB, forming the rectangle GFHK. 3. Divide AE and AF into any number of equal parts, in this case four (Pr. 15). Draw lines 1 C, 2 C, 3 C; and from D, through points 1, 2, 3 in the transverse diameter, draw lines which will intersect the former lines. The points of intersection will be in the curve of the ellipse required. NOTE 1.-By repeating the process in the other divisions of the rectangle, the curve of the required ellipse will be completed. NOTE 2.—The ellipse must be carefully drawn by hand. Another Method. 1. Place the transverse diameter AB and the conjugate diameter CD perpendicular to each other at their centres. 2. From C or D, with half AB as radius, describe arcs, cutting AB in F1, F2. These points are the foci of the ellipse. 3. From F 1 to the centre of AB, mark off any number of parts, as 1, 2, 3, 4, &c., and it will be more convenient if the divisions lessen as they approach F 1. 4. From F 1, with radius A 1, A2, A 3, &c., describe arcs in the spaces AC and AD. 5. From F 2, with B 1 (the first division towards A beyond the centre of AB), B2, B3, dc., as radius, describe arcs cutting the arcs already described from' F 1; radius Bl cutting arc A 1, &c., in a, b, c, d, &c. The points of intersection will be in the curve of the ellipse required. NOTE 1.-By repeating the process in the spaces BC, BD, the curve of the required ellipse will be completed. NOTE 2.—The ellipse must be carefully drawn by hand. Another Method. 1. Place the diameters perpendicular to each other at their centres E, as before. 2. From E, with radii EC and EA, describe circles. 3. Divide the circumference of the larger circle into any number of equal parts, 1, 2, 3, 4, &c. 4. Draw radii from each point of division, cutting the cir cumference of the smaller circle also in 1, 2, 3, 4, &c. 5. From the divisions of the smaller circle, draw lines parallel to the transverse axis AB. 6. From the divisions of the larger circle, draw lines parallel to the conjugate axis CD. The points of intersection will be in the curve of the ellipse required. Another Method. 1. Place the given diameters AB, CD perpendicular to each other at their centres E. 2. From A, with CD as radius, mark the point F. 3. Divide FB into three equal parts. 4. From E, with two of these parts as radius, cut AB in G and H. 5. From G and H, with GH as radius, describe arcs in K and L. 6. From K and L, with radius KD, describe arcs MN, OP; and from G and H, with radius HB, describe arcs Mo, PN, which complete the required ellipse. NOTE.—Lines drawn from K and L, through G and H, will show where the four arcs unite, N.B.--This method is by means of arcs of circles. |