EF, and join On, VT. Then, because VO is perpendicular to the plane AB, which is parallel to the plane EF, VO is also perpendicular to the plane E F (IV. 11.); but N T is perpendicular to the same plane; therefore NT is parallel to VO (IV. 5.). And, because the plane of the parallels TN, VO (I. def. 12.) cuts the planes E F, AB in the lines TV, On respectively, TV is parallel to On (IV. 12.). Therefore, because the triangles VOn, TNV have the sides VO, On of the first parallel to the sides NT, TV of the other, each to each, and their sides V n, NV in the same straight line, they are equiangular (I. 15.), and, consequently (II. 31.), On is to OV as TV to TN. Therefore (II. 9.), if TV contains TN any number of times exactly or with a remainder, On will contain OV the same number of times exactly or with a remainder. But, if the point N be made to approach to the point M, TV will approach in magnitude, as well as position, to MV, and TN, which is the distance of the point N from the plane EF, will be diminished without limit: consequently, there is no limit to the number of times TV may be made to contain TN. Therefore, also, there is no limit to the number of times On may be made to contain OV, that is, the line On may be increased without limit, and the point n will describe an arc of unlimited extent corresponding to the arc PNM or M P, which is terminated in M. Therefore, &c.

Cor. 1. In the demonstration of this proposition it is supposed that the plane of the original curve does not pass through the vertex V; for, in this case, it is evident that its projection upon the plane A B is a straight line (IV. 2.).

Cor. 2. If the original curve cuts the vertical plane in the point M, so that one part, as KLM, lies upon one side of that plane, and the other part, as MPQ, upon the other side of it, the projection shall have two arcs which are extended without limit in opposite directions, corresponding to the two ares K L M, M PQ, which are termiminated in the point M.

Def. 6. If a curve has an arc of unlimited extent, and if a straight line is drawn which never meets that are, but which, being produced, may be made to approach nearer to it than by any given distance, such straight line is called an asymptote to the arc.

lying without the vertical plane E F, and PH a straight line which touches the curve PQR in the point P; then, if the point p be the projection of the point P (def. 1.), because P is a point both in the curve PQR and the tangent PH, pwill be a point in the projection of each; let, therefore, the curve p q r be the projection of the curve PQR, and the straight lineph the projection of the tangent PH: the straight line p h shall likewise touch the curve p q r in the point p.

the points of the curve P Q R upon both Because PH touches the curve PQR, sides of P lie towards the same part of drawn from V through those points lie PH, and therefore also the straight lines VPH or Vph. But these straight lines towards the same part of the plane to the points of the curve p q r on both are the same which are drawn from V ter also lie towards the same part of the sides of p (def. 1.). Therefore, the latplane Vph; and, consequently, the points of the curve pqr on both sides of p lie towards the same part of the straight line ph, that is, ph touches the curve p q r in the point p.

But, in the next place, let M be a point of the curve PQR lying in the vertical plane E F, and let M G H be a straight line touching the curve PQR in the point M; then, if the curve p q r be the projection of the curve MPQR, and the straight line g h of the tangent

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.

stration of the foregoing proposition,
The assumption made in the demon-
viz. that "no straight line can be drawn
through the point of contact between a
the curve," or, in other words, that a
curve and its tangent so as not to cut
curve admits of only one tangent at the
same point of it, may be regarded as an
axiom. That such is the case in the
circle has been shown in Book iii.
other curves, generally, it may be illus-
Prop. 2.; and hence, with regard to
trated as follows:-Conceive a circle
having the same tangent with the curve
at the point M, and suffi-
the curve, as in the adjoined
ciently small to fall within
figure. Then, since no
through the point M so near
straight line can be drawn
the circumference of this
to the tangent as not to cut
circle, and since the curve
lies between the circumfer-
ence of this circle and the

M GH, the straight line g h shall be an asymptote to the curve pqr. 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 P Q R in any other point, no point can be found in which the projection g h of the tangent meets the projection pqr of the curve, to whatever extent both of them may be produced. Again, let D be any given distance: produce the tangent MH to meet the plane A B in T, and therefore also to meet its projection g h in the same point T, and let the plane of the curve MPQR be produced to meet the plane AB in the straight line TZ (IV. 2.); from T, in the plane AB, draw T Y perpendicular to g T, and let TY be taken of any length, so that it be less than the given distance D; through Y draw Y Z parallel to g T, and let it cut TZ in Z, and join MZ. Then, because M Z falls within the tangent 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 V MT, V M 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 V M 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 pqr than by any given distance. And corollary.

M

H

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

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 plane, and this is the case which is supposed in the

Because 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 PQ 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

Let CD be the direction, A B the plane of projection; PQR any curved line, and Pqr its projection: pqr shall likewise be a curved line.

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. Therefore, 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 H Pp. 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 HPp, 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 curve pqr.

Therefore, &c.

Cor. 1. It is supposed in the proposition that the plane of the original curve * See note, page 210,

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 CD (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 up-
wards through the
vertex, the produced
parts will, it is evi-
dent, lie in the sur-
face of another right

cone which has the
same vertex, and its
axis lying in the
same straight line

C

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 called conic sections, inasmuch as they likewise are plane sections of a cone: the term is, however, usually appropriated to the other plane sections, viz. the ellipse (def. 10,), the parabola (def. 11.), and the hyperbola (def. 12.).

Let the plane a be 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 oa, ob, O A, OB. Then, be