MGH, the straight line g h shall be an asymptote to the curve pqr. it has been shown that it never meets the curve, to whatever extent it is produced. Therefore, it is an asymptote to the curve p q r (def. 6.). Therefore, &c. Cor. 1. In that part of the proposiBtion which relates to the tangent at a point M in the vertical plane; it is supposed that the tangent M H does not lie in the vertical plane*; for in this case it is evident that no point of the tangent can be projected upon plane AB, and consequently there is no asymptote. Cor. 2. The perspective projection of a straight line which cuts a curve is a straight line which cuts the projection of the curve, if the first point of intersection is without the vertical plane. Scholium. Because the point M lies in the plane E F, and therefore (1.) cannot be projected upon the plane A B, and because the tangent M G H does not meet the curve M PQR in any other point, The assumption made in the demonno point can be found in which the pro- stration of the foregoing proposition, jection g h of the tangent meets the pro- viz. that "no straight line can be drawn jection pqr of the curve, to whatever through the point of contact between a extent both of them may be produced. curve and its tangent so as not to cut Again, let D be any given distance: the curve," or, in other words, that a produce the tangent MH to meet the curve admits of only one tangent at the plane A B in T, and therefore also to meet same point of it, may be regarded as an its projection g h in the same point T, and axiom. That such is the case in the let the plane of the curve MPQR be circle has been shown in Book iii. produced to meet the plane A B in the Prop. 2.; and hence, with regard to straight line TZ (IV. 2.); from T, in the other curves, generally, it may be illusplane AB, draw T Y perpendicular to trated as follows:-Conceive a circle g T, and let TY be taken of any length, having the same tangent with the curve so that it be less than the given distance at the point M, and suffiD; through Y draw Y Z parallel to gTciently small to fall within and let it cut TZ in Z, and join M Z. the curve, as in the adjoined Then, because M Z falls within the tanfigure. Then, since straight line can be drawn through the point M so near the circumference of this to the tangent as not to cut circle, and since the curve lies between the circumference of this circle and the tangent; much less can any straight line be drawn so near to the tangent as not to intercept a part of the curve between itself and the tangent, and, consequently, being produced, to cut the no M H gent M HT, and that no straight line can be drawn through the point of contact between the curve and its tangent so as not to cut the curve, M Z must cut the curve PQR in some point N. And, because VM and g T are sections of the parallel planes EF and AB by the plane VMT, VM is parallel to g T (IV. 12.); but g T is parallel to YZ; therefore (IV. 6.), VM is likewise parallel to Y Z But the point N is in the plane V MZ, that is, in the plane of VM and Y Z. Therefore, if VN be joined, and produced, it will cut the straight line YZ in some point n; and, because the point n is also in the plane A B, it is the projection of N, and therefore a point in the curve pqr, which is the projection of MPQR. Also, because Y Z is parallel to gh T, n is at the same distance from ghT (I. 16.) that Y is, that is, at a less distance than the given distance D. Therefore, g h T being produced may be made to approach nearer to the curve p q r than by any given distance. And plane, and this is the case which is supposed in the curve. PROP. 7. The direction CD and the plane of projection A B being given; the orthographic projection of any point P whatever may be found upon the plane A B. *It may, perhaps, appear at first, that if the tangent lies in the vertical plane, the curve must likewise lie in that plane; this, however, is not a necessary consequence; the tangent MH may be the common section of the plane of the curve with the vertical corollary. plane AB (def. 2.), the straight line A which is drawn through P parallel to CD is not parallel to that plane; since, otherwise, B the line thus drawn would be parallel to the common section of a plane passing through it with the plane A B (IV. 10.), and therefore, also (IV. 6.), CD would be parallel to the same common section, that is, to a straight line in the plane A B, for which reason CD would be parallel also to the plane AB (IV.10.), which is contrary to the supposition. Therefore, the straight line which is drawn through Pparallel to CD may be produced to meet the plane AB in some point p; and the point p thus found (def. 2.) is the orthographic projection of the point P. Therefore, &c. PROP. 8. Cor. 3. The orthographic projection of a straight line, which is parallel to the plane of projection, is parallel to its original (IV. 10.). PROP. 9. The orthographic projections of parallel straight lines are parallel straight lines, and have the same ratio to their respective originals. Let CD be the direction, AB the plane of projection, PQ and P'Q' any twoparallel straight lines, and pq, p' q' their respective projections: pq shall be parallel to p' q'. B Becausé Qq and Q'q' are each of them parallel to CD (def. 2.), they are parallel to one another (IV. 6.); also PQ is parallel to P'Q'; therefore, the plane P Q q is parallel to the plane P'Q' q' (IV. 15.). But pq and p' q' are the respective sections of these parallel The orthographic projection of a planes made by the plane of projection straight line is a straight line. Let CD be the direction, A B the plane of projection, P Q any straight line, and Pq its orthographic projection : pq shall likewise be a straight line. A P B Because PQ is a straight line, and that the parallels to CD, which are drawn through the several points of PQ, are parallel to one another (IV. 6.), these parallels lie in one and the same plane PQq (IV. 1. Cor. 2.): but (def. 2.) the points of p q lie in these parallels respectively; therefore, the points of p q lie in the plane PQ q. But they lie also in the plane AB. Therefore they lie in the common section of the planes PQ q and A B, that is, in a straight line (IV. 2.). Therefore, &c. Cor. 1. It is supposed in the proposition that the original straight line PQ is not parallel to CD; for, then, it is evident that all its points have for their projections the single point in which it cuts the plane of projection. Cor. 2. The orthographic projection of any given straight line is a part of the common section of two planes, viz. a plane which passes through the given straight line parallel to the direction CD and the plane of projection A. B. A B (8. Cor. 2.). Therefore, pq is parallel to p' q' (IV. 12.). Also, the projections p q and p' q' have the same ratio to the original straight lines PQ and P' Q' respectively. For, if PQ is parallel to pq, then, because P'Q' and pq are each of them parallel to P Q, P'Q' is parallel to p q (IV.6.); and because p' q' is likewise parallel to pq by the former part of the proposition, P'Q' is parallel to p'q'. Also, because Pp and Q q are each of them parallel to CD (def. 2.), Pp is parallel to Qq; and, for the like reason, P'p' is parallel to Q'q'. Therefore, the figures PpqQ, P'p'q'Q' are, in this case, parallelograms; and, because (I. 22.) the opposite sides of parallelograms are equal to one another, pq and p'q' are equal to P Q and PQ' respectively; that is, the projections have the same ratio to their respective originals, viz. the ratio of equality. But, if P Q is not parallel to pq, draw PR parallel to p q to meet Qq in R, and P' R' parallel to p'q' to meet Q'q' in R'. Then, because PpqR and P' p' q'R' are parallelograms, PR and P'R' are equal to p q and p' q' respectively (I. 22.). But, because PR and P' R' are parallel to p q and p' q' respectively, and that p q is parallel to p' q', PR is parallel to P'R' (IV. 6.). Therefore, the triangles P Q R, P'Q' R' have the three sides of the one parallel to the three sides of the other, each to each, The orthographic projection of a curved line* is a curved line; and, if a straight line touches the original curve, the projection of that straight line shall like wise touch the projection of the curve. Let CD be the direction, A B the plane of projection; PQR any curved line, and pqr its projection: p q r shall likewise be a curved line. H R B For, if any part of p q r, as p q, be a straight line, then, because PQ is the orthographic projection of p q (def. 2.) upon the plane PQR, PQ must likewise be a straight line (8.), which is contrary to the supposition. There fore, no part of pqr is a straight line, that is, (1. def. 6.) pqr is a curved line. Next, let the straight line PH touch the curve PQR in the point P, and let ph be the orthographic projection of PH:ph shall touch the curve p q r in p. For, CD being parallel to Pp (def. 2.), straight lines which are parallel to CD are parallel to Pp (IV. 6.), and therefore (IV. 10.) parallel to the plane HPp. Also, the points of the curve PQR on both sides of P fall, all of them, without and to the same part of the tangent PH. Therefore, the parallels to CD or Pp, which pass through these points likewise fall without and to the same part of the plane HP p. But these parallels pass through the corresponding points of the projection pqr (def. 2.). Therefore the points of p qr, on both sides of p, lie without and to the same part of the plane H Pp, and consequently also without and to the same part of the straight line ph which is in that plane (8. Cor. 2). Therefore, ph meets the curve pqr in p, but does not cut it, that is, ph touches the is not parallel to the direction CD; for, then, it is evident that the projection of the curve is a straight line, and that the projection of the tangent is confounded with (or, if parallel to the direction C D (8. Cor. 1.) is only a point in) the projection of the curve. Cor. 2. The orthographic projection of a straight line which cuts any curve is a straight line which cuts the projection of that curve. PART II.-Of the Plane Sections of the Right Cone, or Conic Sections. Ir is easy to perceive that every section of a right cone which is made by a plane passing through the vertex is rectilineal, and, again, that every section which is made by a plane parallel to the base is a circle. The former follows from the definition (V. def. 4.) of a cone; the latter will be demonstrated at large hereafter (in prop. 11.). But, if a right cone be cut by a plane which neither passes through the vertex nor is parallel to the base, the section will be neither rectilineal nor circular; but will, according to the position of the cutting plane, take one of the three forms mentioned in the following definitions. Def. 7. If the slant sides of a right cone are produced upwards through the vertex, the produced parts will, it is evident, lie in the surface of another right cone which has the same vertex, and its axis lying in the same straight line with the axis of the first. This cone, with reference to the first, is called the opposite cone, and its surface the opposite surface. The two opposite surfaces, infinitely produced, are to be considered as constituting one complete conical surface; which may be conceived to be generated by the revolution of a slant side infinitely produced both ways about the axis of the cone. 8. If a complete conical surface is cut by a plane which neither passes through the vertex nor is parallel to the base, the curved line in which such plane cuts the surface is called a conic section*. *The plane sections which are here excepted, viz. the straight line and circle, are likewise sometimes plane sections of a cone: the term is, however, usually called conic sections, inasmuch as they likewise are appropriated to the other plane sections, viz. the ellipse (def. 10,), the parabola (def. 11.), and the hyperbola (def. 12.). 12. If the vertical plane cuts the conical surface in two slant sides, the conic section has four infinite arcs, two lying in one and two in the other of the opposite surfaces, and is called an hyper because a part of each is intercepted between the vertical plane and the plane of the conic section; and because there are two slant sides in each surface which lie in the vertical plane, and therefore cannot be produced to meet the plane of the conic section, the section has two infinite arcs in each surface corresponding to them. These curves, or the conic sections properly so called, different as they are in form, the first a complete figure inclosing an area, the second having two infinite arcs, the third four, are nevertheless very nearly related to one another in their properties, many of which bear a striking analogy to the properties of the circle. Thus, "if, in any conic section two chords are drawn which cut (or are produced to cut) one another, and other two chords parallel to the former respectively which likewise cut one another, the rectangles contained under the segments of the former two shall have the same ratio to one another as the rectangles which are contained under the segments of the latter two;" a property which we have seen (III. 20.) obtains in the circle, the ratio in this case being always that of equality. It is proposed in the present part of the Appendix to demonstrate a few of these properties, among them the one just stated; and it will be found that the demonstrations are considerably aided and abridged by help of the principles laid down in the preceding part. PROP. 11. Every section of a right cone which is made by a plane parallel to its base, is a circle having its centre in the axis of the cone. Let V be the vertex of a right cone, VO its axis, and ABC its base; and let abc be a section which is made by any plane parallel to the base A B C the section abc shall be a circle having its centre in the axis V O. A Let the plane abe cut the axis of the cone in the point o: in the curve, or circumference, a bc, take any two points a, b; join Va, V b, and produce them to meet the circumference of the circle ABC in the points A, B respectively, case cuts both of the opposite surfaces, and join o a, ob, O A, OB. Then, be dola. The section in this M. cause o a and OA are sections of parallel planes, by the plane V O A, oa is parallel to OA (IV. 12.), and consequently (II. 30. Cor. 2.), o a is to OA as Voto V O. And, in the same manner, it may be shown that o b is to O B as Vo to VO. Therefore (II. 12.) o a is to OA as ob to OB; and, because OA is equal to O B, oa is equal to o b (II. 18. Cor.). In like manner, if c be any other point in the circumference abc, and if oc be joined, it may be shown that o c is equal to o a or o b. Therefore, every point in the circumference abc is at the same distance from the point o; that is (I. def. 24.), abc is a circle of which o is the centre. Therefore, &c, PROP. 12. Every conic section QPR is the perspective projection of a circular section qpr, upon the plane of the conic section, by straight lines drawn from the vertex V; and the vertical plane of such perspective projection is the vertical plane of the conic section. P For, every straight line which is drawn from V through a point of the circumference pqr to meet the plane of the conic section, meets that plane in some point of the conic section; and there is no point of Q PR which is not in a straight line with V and some point of q pr; therefore (def. 1.), QPR is the perspective projection of qpr by straight lines drawn from V. And, because the vertical plane of the conic section QPR is parallel to the plane QPR (def. 9.), that vertical plane is also the vertical plane of projection (def. 1.). Therefore, &c. R Cor. 1. In like manner, also, every circular section qpr may be considered as the perspective projection of the conic section QPR by straight lines drawn from the vertex V. Cor. 2. The projection of every point in the conic section may be found in the circular section, whether it be an ellipse, or a parabola, or an hyperbola (1.): for the plane which passes through V pa rallel to the circular section, falls entirely without the cone, so that no point of the conic section is found in that plane. Cor. 3. And so, the projection of every point in the circular section may be found in the conic section; except, in the case of the parabola, the projection of the point in which the vertical plane touches the circular section, and except, in the case of the hyperbola, the projections of the two points in which the vertical plane cuts the circular section. PROP. 13. Every conic section is symmetrically divided by a straight line, which is the common section of the cutting plane, and a plane which passes through the axis of the cone perpendicular to the cutting plane. Let V be the vertex, and VO the axis of the cone, and let PQR be the conic sec tion; from V draw VU, perpendicular to the plane PQR, and let the a plane UVO, which passes through VO, and (IV. 18.) is perpendicular to the plane PQR, cut the lat ter plane in the straight line AM: the conic section PQR shall be divided symmetrically by the straight line A M. Through the point A let there be drawn a plane perpendicular to the axis V O, and let it cut the cone in the circular section pqr, having the centre O (11.), and the plane QPR in the straight line AF (IV. 2.); through V draw VD parallel to AF (I. 48.). Take any point P in the conic section, join V P, and let the plane DVP cut the planes of the conic section and circle in the straight lines PQ, and p q respectively (IV. 2.); also, let these straight lines cut A M, AO in the points M, m respectively. Then, because VD is parallel to AF, it is parallel (IV. 10. Cor. 1.) to PQ and to p q, which are the common sections of planes passing through AF with the plane DVP which passes through DV: therefore, also, PQ and pq are parallel to one another (IV. 6.). Now, because the plane pqr of the circle is perpendicular to the axis VO, it is perpendicular to the plane UVO, which passes through VO (IV, 18.); and the plane PQR is |