M po H it is sup P Y Z n T it has been shown that it never meets the curve, to whatever extent it is produced. Therefore, it is an asymptote to the curve pq r (def. 6.). Therefore, &c. Cor. 1. In that part of the proposiB tion which relates to the tangent at a point M in the vertical plane; posed that the tangent MH does not lie in the vertical plane* ; for in this case it is evident that no point of the tangent T 9 k can be projected upon plane AB, and consequently there is no asymptote. MGH, the straight line g h shall be an Cor. 2. The perspective projection of asymptote to the curve pq?: a straight line which cuts à curve is a Because the point M lies in the plane straight line which cuts the projection EF, and therefore (1.) cannot be of the curve, if the first point of interprojected upon the plane A B, and be- section is without the vertical plane. cause the tangent M G H does not meet Scholium. 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 par 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 its projection g h in the same point T, and axiom. That such is the case in the same point of it, may be regarded as an let the plane of the curve M P Q R 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:-Čonceive a circle gT, 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 g T, ciently small to fall within and let it cut TZ in Z, and join MZ. the curve, as in the adjoined Then, because M Z falls within the tan figure. Then, since gent MHT, and that no straight line can be drawn through the point of contact through the point M so near straight line can be drawn between the curve and its tangent so as not to cut the curve, M Z must cut the the circumference of this to the tangent as not to cut curve PQ R in some point N. And, circle, and since the curve because VM and gT are sections of the lies between the circumferparallel planes EF and AB by the plane VMT, V M is parallel to g I (IV. 12.); tangent; much less can any straight ence of this circle and the but g T is parallel to Y Z; therefore line be drawn so near to the tangent as (IV.6.), VM is likewise parallel to Y Z not to intercept a part of the curve beBut the point N is in the plane V M2, tween itself and the tangent, and, conthat is, in the plane of VM and YZ. sequently, being produced, to cut the Therefore, if V N be joined, and pro çurve. duced, it will cut the straight line Y Z in PROP. 7. some point n; and, because the point in is also in the plane A B, it is the pro The direction C D and the plane of jection of N, and therefore a point in the projection A B being given; the ortho curve pqr, which is the projection of graphic projection of any point P whatMPQR. Also, because Y Z is parallel ever may be found upon the plane A B. to gh T, n is at the same distance from gh T (I. 16.) that Y is, that is, at a less * It may, perhaps, appear at first, that if the tan gent lies in the vertical plane, the curve must likedistance than the given distance D. wise lie in that plane; this, however, is not a necesTherefore, g h T being produced may sary consequence; the tangent MH may be the com. made to approach nearer to the curve mon section of the plane of the curve with the vertical p q r than by any given distance. And plane, and this is ihe case wbich is supposed in the corollary no NI H be B R' R B For, the direc Cor. 3. The orthographic projection tion CD not being of a straight line, which is parallel to parallel to the the plane of projection, is parallel to its plane AB (def. 2.), original (IV. 10.). the straight line A PROP. 9. which is drawn through P parallel The orthographic projections of pato CD is not pa rallel straight lines are parallel straight rallel to that plane; lines, and have the same ratio to their since, otherwise, respective originals. the line thus drawn would be parallel Let CD be the to the common section of a plane pass- direction, A B the ing through it with the plane AB plane of projection, (IV. 10.), and therefore, also (IV. 6.), PQ and P'Q' any CD would be parallel to the same com- twoparallel straight mon section, that is, to a straight line lines, and p.q, p'q' in the plane A B, for which reason CD their respective pro p7 would be parallel also to the plane A B jections: pq shall (IV.10.), which is contrary to the sup- be parallel to p' q'., position. Therefore, the straight line Because Q qand which is drawn through P parallel to CD Q'q' are each of may be produced to meet the plane them parallel to CD (def. 2.), they are A B in some point p; and the point p parallel to one another (IV. 6.); also thus found (def. 2.) is the orthographic PQ is parallel to P'Q'; therefore, the projection of the point P. plane P Q 9 is parallel to the plane Therefore, &c. P'Q' q' (IV. 15.). But pq and p q' are PROP. 8. the respective sections of these parallel The orthographic projection of a planes made by the plane of projection straight line is a straight line. A B (8. Cor. 2.). Therefore, pq is paLet CD be the rallel to pl a' (IV. 12.). direction, A B the Also, the projections pq and p' q' have plane of projec the same ratio to the original raight tion, P Q any lines P Q and P' Q' respectively. straight line, and For, if P Q is parallel to pq, then, p. its orthogra because P'Q' and pq are each of them phic projection : parallel to P Q, P Q is parallel to pa g pq shall likewise (IV.6.); and because pi q' is likewise be a straight line. parallel to pq by the former part of the Because P Q is a straight line, and proposition, 'Q' is parallel to p q'. that the parallels to CD, which are Also, because Pp and Q q are each of drawn through the several points of PQ, them parallel to C D (def. 2.), Pp is are parallel to one another (IV. 6.), these parallel to Qq; and, for the like reason, parallels lie in one and the same plane P'p' is parallel to Q'q'. Therefore, the P Qq (IV. 1. Cor. 2.): but (def. 2.) the figures PpqQ, P'p'q'Q' are, in this case, points of p q lie in these parallels respec- parallelograms; and, because (I. 22.) tively; therefore, the points of p q lie in the opposite sides of parallelograms are the plane P Q q. But they lie also in equal to one another, pq and p'q' are the plane AB. Therefore they lie in the equal to P Q and P Q respectively ; common section of the planes PQ q and that is, the projections have the same A B, that is, in a straight line (IV. 2.). ratio to their respective originals, viz. Therefore, &c. the ratio of equality. But, if P Q is not Cor. 1. It is supposed in the proposi- parallel to pq, draw P R parallel to pa tion that the original straight line PQ to meet Q q in R, and P'R' parallel to is not parallel to CD; for, then, it is p'q' to meet Q' q' in R'. Then, because evident that all its points have for their Ppq R and P'p'q'R' are parallelograms, projections the single point in which it PR and P'R' are equal to p q and p' q' cuts the plane of projection. respectively (I. 22.). But, because PR Cor. 2. The orthographic projection and P'R' are parallel to p q and p' q' reof any given straight line is a part of the spectively, and that pq is parallel to p'a', common section of two planes, viz. a PR is parallel to P'R' (IV.6.). Thereplane which passes through the given fore, the triangles PQR, P'Q'R' have straight line parallel to the direction CD the three sides of the one parallel to the and the plane of projection A B. three sides of the other, each to each, P B R H a B and (IV. 15.) are equiangular. Conse- is not parallel to the direction CD; for, quently, P R is to PQ as P'R' to P Q' then, it is evident that the projection of (11. 31.), that is, since pq is equal to the curve is a straight line, and that the P R and p' q' to P'R', pq is to P Q as projection of the tangent is confounded p'q' to P'Q. with (or, if parallel to the direction CD Therefore, &c. (8. Cor. 1.) is only a point in the pro[ Cor. If the original straight lines are jection of the curve. parallel to the plane of projection, the Cor. 2. The orthographic projection of orthographic projections are equal to a straight line which cuts any curve is a their respective originals. straight line which cuts the projection of that curve. PROP, 10. The orthographic projection of a curved PART II.- Of the Plane Sections of the line* is a curved line ; and, if a straight Right Cone, or Conic Sections. line touches the original curve, the pro- It is easy to perceive that every section jection of that straight line shall like- of a right cone which is made by a plane wise touch the projection of the curve. passing through the vertex is rectilineal, Let CD be the and, again, that every section which is direction, A B the made by a plane parallel to the base is a plane of projec circle. The former follows from the tion; PQR any definition (V. def. 4.) of a cone; the curved line, and latter will be demonstrated at large por its projec hereafter (in prop. 11.). But, if a right tion: por shall cone be cut by a plane which neither likewise be passes through the vertex nor is parallel curved line. to the base, the section will be neither For, if any part of por, as p q, be rectilineal nor circular; but will, aca straight line, then, because P Q is cording to the position of the cutting the orthographic projection of pq(def.2.) plane, take one of the three forms menupon the plane PQR, P Q must like- tioned in the following definitions, wise be a straight line (8.), which is Def. 7. If the slant contrary to the supposition. There, sides of a right cone fore, no part of pqr is a straight line, are produced upthat is, (I. def. 6.) pqr is a curved line, wards through the Next, let the straight line PH touch vertex, the produced the curve P Q R in the point P, and let parts will, it is eviph be the orthographic projection of dent, lie in the surPH:ph shall touch the curve pqrin p. face of another right For, CD being parallel to Pp (def , 2.), cone which has the straight lines which are parallel to CD same vertex, and its are parallel to Pp (IV. 6.), and there- axis lying in the fore (IV, 10.) parallel to the plane H Pp. same straight line Also, the points of the curve PQR on with the axis of the both sides of P fall, all of them, without first. This cone, with and to the same part of the tangent PH. reference to the first, is called the opposite Therefore, the parallels to CD or Pp, cone, and its surface the opposite surface. which pass through these points likewise The two opposite surfaces, infinitely fall without and to the same part of the produced, are to be considered as conplane H Pp. But these parallels pass stituting one complete conical surface; through the corresponding points of the which may be conceived to be geneprojection par (def. 2.). Therefore the rated by the revolution of a slant side points of par, on both sides of p, lie with- infinitely produced both ways about the out and to the same part of the plane axis of the cone. HPp, and consequently also without and 8. If a complete conical surface is cut to the same part of the straight line ph by a plane which neither passes through which is in that plane (8. Cor.2). There- the vertex nor is parallel to the base, fore, ph meets the curve parin p, but the curved line in which such plane cuts does not cut it, that is, ph touches the the surface is called a conic section* * The plane sections which are here excepted, viz. Therefore, &c. the straight line and circle, are likewise sometimes Cor. 1. It is supposed in the proposi- plane sections of a cone : the term is, however, usually called conic sections, inasmuch as they likewise are tion that the plane of the original curve appropriated to the other plane sections, viz. the ellipse (def. 10.), the parabola (def. 11.), and the hyperbola * See note, page 210, (def, 12.), ourve par M case A 9. The vertical plane of a conic sec- because a part of each is intercepted betion is a plane which passes through the tween the vertical plane and the plane of vertex of the cone parallel to the plane the conic section; and because there are of the conic section. two slant sides in each surface which lie in 10. If the ver the vertical plane, and therefore cannot tical plane of be produced to meet the plane of the coa conic section nic section, the section has two infinite falls entirely with arcs in each surface corresponding to out the surface, them. the conic section These curves, or the conic sections surrounds the properly so called, different as they are lower cone, and is in form, the first a complete figure incalled an ellipse. closing an area, the second having two The reason why infinite arcs, the third four, are neverthe section in this theless very nearly related to one anosurrounds ther in their properties, many of which the lower cone is, bear a striking analogy to the properthat the lower surface only is intercepted ties of the circle. Thus, “if, in any between the planes; and because every conic section two chords are drawn slant side of this surface cuts the vertical which cut (or are produced to cut) one plane in V, it will, if produced, cut also another, and other two chords parallel the plane of the conic section which is to the former respectively which likeparallel to the vertical plane (1. Cor.). wise cut one another, the rectangles 11. If the verti contained under the segments of the cal plane touches former two shall have the same ratio the conical surface to one another as the rectangles which in a slant side, the are contained under the segments of the conic section has latter two;" a property which we have two infinite arcs, seen (III. 20.) obtains in the circle, the and is called a pa ratio in this case being always that of rabola. equality. It is proposed in the present The section has part of the Appendix to demonstrate in this case two in few of these properties, among them the finite arcs, because one just stated; and it will be found one slant side of that the demonstrations are considerthe cone lies in the vertical plane, and ably aided and abridged by help of the therefore can never be produced to meet principles laid down in the preceding the plane of the conic section; it is still, part. however, only the lower surface which is PROP. 11. intercepted between the planes, and be Every section of a right cone which cause every other is maile by a plane parallel to its base, slant side of this is a circle having its centre in the axis surface cuts the ver of the cone. tical plane in V, it Let V be the vertex may be produced to of a right cone, VO its cut the plane of the axis, and ABC its conic section (1. base; and let abc be Cor.). a section which is 12. If the vertical made by any plane plane cuts the coni parallel to the base cal surface in two A B C: the section slant sides, the conic abc shall be a circle section has four in having its centre in finite arcs, two lying the axis V O. in one and two in Let the plane abc cut the axis of the the other of the op cone in the point o: in the curve, or posite surfaces, and circumference, a be, take any two points is called an hyper a, b; join Va, V b, and produce them to bola. M meet the circumference of the circle The section in this ABC in the points A, B respectively, case cuts both of the opposite surfaces, and join o a, ob; 0 A, OB. Then, be a A B tion; A om cause o a and O A are sections of pa- rallel to the circular section, falls entirely rallel planes, by the plane V O A, oa is without the cone, so that no point of parallel to 0 A (IV. 12.), and conse- the conic section is found in that plane. quently (11. 30. Cor. 2.), o ais to 0 A as Cor. 3. And so, the projection of V o to V O. And, in the same manner, it every point in the circular section may may be shown that o b is to 0 B as Vo be found in the conic section; except, in to VO. Therefore (II. 12.) o a is to OA the case of the parabola, the projection as ob to O B; and, because OA is equal of the point in which the vertical plane to O B, oa is equal to o b (II. 18. Cor.). touches the circular section, and except, In like manner, if c be any other point in the case of the hyperbola, the projecin the circumference abc, and if oc be tions of the two points in which the verjoined, it may be shown that o cis equal tical plane cuts the circular section. to o a or o b. Therefore, every point in the circumference abc is at the same PROP. 13. distance from the point o; that is (I. def. Every conic section is symmetrically 24.), abc is a circle of which o is the divided by a straight line, which is the centre. common section of the cutting plane, Therefore, &c. and a plane which passes through the axis of the cone perpendicular to the PROP. 12. cutting plane. Let y be the Every conic section QPR is the per- vertex, and vo spective projection of a circular section the axis of the qpr, upon the plane of the conic section, by straight lines drawn from the ver be the conic sec cone, and let PQR tex V; and the vertical plane of such from V draw perspective projection is the vertical Vu, perpendicu VU plane of the conic section. lar to the plane igl For, every straight PQR, and let the a line which is drawn plane UVO, which from V through a passes through point of the circum VO, and (IV. 18.) ference par to meet is perpendicular the plane of the to the plane conic section, meets PQR, cut the latthat plane in some ter plane in the straight line AM: the point of the conic conic section PQR shall be divided section; and there is symmetrically by the straight line A M. no point of Q PR Through the point A let there be which is not in a drawn a plane perpendicular to the axis straight line with V V O, and let it cut the cone in the cirand some point of cular section pq?, having the centre o 9 pm ; therefore (11.), and the plane Q P R in the straight (def. 1.), QPR is line A F (IV. 2.); through V draw VD the perspective pro parallel to AF (1.48.). Take any point P jection of a pr by straight lines drawn in the conic section, join V P, and let the from V. And, because the vertical plane plane D V P cut the planes of the conic of the conic section Q PR is parallel to section and circle in the straight lines the plane Q PR (def. 9.), that vertical PQ, and p q respectively (IV. 2.); also, plane is also the vertical plane of pro- let these straight lines cut AM, A O in jection (def. 1.). the points M, m respectively. Then, beTherefore, &c. cause VD is parallel to AF, it is parallel Cor. 1. In like manner, also, every (IV. 10. Cor. 1.) to PQ and to pa, which circular section qpr may be considered are the common sections of planes passas the perspective projection of the conic ing through A F with the plane D VP section Q PR by straight lines drawn which passes through DV: therefore, from the vertex V. also, PQ and pq are parallel to one Cor. 2. The projection of every point another (IV. 6.). Now, because the in the conic section may be found in the plane par of the circle is perpendicular circular section, whether it be an ellipse, to the axis VO, it is perpendicular to or a parabola, or an hyperbola (1.): for the plane U V O, which passes through the plane which passes through V pa- VO (IV, 18.); and the plane PQR is F R M P P R |