SUBCONTRARY. This word is particularly applied to the sections of a cone, in a manner which, without interfering with that application, would allow of its definition being generalised as follows :—When figure or solid is symmetrical, so that equal lines or polygons can be drawn on two different sides, those equal lines or polygons may be called subcontrary. Thus, in Euclid, i. 5, the equal lines which are obliquely deflected from the two ends of the base of the isosceles triangle, are subcontrary. In a right cone every section has its sub contrary, except only the circle which generates the cone, and its parallels. Let v be the vertex of an oblique circular cone, and ABCD the circle on which it is described. Let the plane v A c be that which passes through the centre of the circle perpendicularly to its plane. Then the cone is exactly the same on one side of the plane v A o as on the other; and if a plane A o r be drawn through A perpendicular to the line which bisects the angle AVE, the section AOF is such that either luilf would take the place of the other, if it were to make a half revolution about • F. It is then an ellipse of which A F is one of the principal axes ; and the middle point of A r, falling in the line which bisects A V F, is the centre. Consequently every section of this cone
has a aubcontrary section, except only those which are parallel to AO F. Hence the generating circle A a c D has a subcontrary circle E B F D, made by taking the line E F subcoutmry to A c, and drawing through E F a plane perpendicular to the plane A v F. The angles v E F and v 0 • are equal, 34 MAO A C and V F E.
In the limited use of the word subcontrary, no sections are considered in this light except the two circular sections of an oblique cone. Con sequently, when subcontrary sections are mentioned, these circular sections are understood. The proofs given of the existence of these subcontrary sections usually conceal the fact of all cones described upon a circle being symmetrical when produced in every direction, and seem to make the existence of a second circular section a sort of accident of the circle, as if no other section had its aubcontrary.
Since all parallel sections of a cone are similar, it follows that through every point of the surface two subcontrary circles can be drawn. The surfaces of the second order generally have the same property. [Sun