mon pole of the circles BCD, EFG Scholium. (37. Cor.), and join P A (31.): from the The constructions in this section have pole P, with the distance P A, describe little or no practical utility, and have, a circle, and from the point H, in which accordingly, been added rather with a P A cuts the circle BCD, draw H K at view to illustrate the analogies of Plane right angles to PA (33.), and let it cut the and Spherical Geometry, than to furnish circle, which was described through A, rules for practice. Some of these we in K: join P K (31.), and let it cut the have already had occasion to notice, circle B C D in L: the great circle ALI and others will have offered themselves which passes through the points A and to the reader; who will readily perceive L shall be the great circle required. that these striking points of resemblance For, because in the triangles PHK, (or, as he may be disposed to call them, PLA, the two sides PH, PK are equal to the two sides PL, PA, and cumstance, that spherical triangles, of identity) are to be ascribed to the cirthe included angle LPH common to when their sides are but small portions both triangles, the angle PL A is equal of great circles, and consequently their to the angle PHK, (13.) that is, to a surfaces small in comparison with the right angle; and the arc P L is less than surface of the sphere, become more and a quadrant; therefore, PL is the least more nearly plane, their sides more and arc which can be drawn from the point P to the circle A LI (18. Cor. 1.), and if of their angles (the excess of which more nearly straight lines, and the sum L P be produced to meet the circle in l, Pl is the greatest. But every point of ratio to eight right angles (21 Cor. 1.) above two right angles bears the same the circle B C D is at the distance PL; as the surface of the triangle to the and every point in the equal circle E FG surface of the sphere) more and more at the distance Pl, because Pl and P I nearly equal to two right angles. Thus are together equal to a semicircum. ference (3. Cor.). Therefore, the circle every plane triangle may be regarded as a spherical triangle upon the surALM, which has been described through face of a sphere, the radius of which the given point A, touches the given is indefinitely great; and in this way circles B C D and E F G in the points I of viewing the subject, the properties of and l. plane triangles resemble those of spheTherefore, &c. rical triangles, only as a particular case Prop. 42. Prop. 14. the general one in which it is included. But it may be asked, has the term To inscribe a circle in a given spheri- similar, which introduces us to so wide cal triangle A B C. a field in Plane Geometry, any place Bisect the angle ABC with the arc BP, in Spherics ? Not in propositions and the angle A C B with the arc CP which have reference only to the surwhich meets the former arc in P (35.); face of one and the same sphere. from P draw Pa per Similar figures upon the surface of the pendicular to B C A same sphere are likewise equal to one (34.); make B c equal another, and may be made to coincide. to Ba, and Cb equal But, when we consider the surfaces of to C a, and join Pc, different spheres, and compare the fiPb. Then, because the gures which are formed upon them, triangles P Ba, PB C here again we shall find room for the have two sides of the application of the term in its full and one equal to two sides peculiar sense. Thus, similar spherical of the other, each to each, and the in- triangles are such as are contained by cluded angles P B a, PB c equal to one similar arcs upon the surfaces of difanother, Pc is equal to Pa (13.), and the ferent spheres. It is easy to perceive angle PcB to the angle Pa B, that is, that such triangles are equiangular, and to a right angle: therefore, the circle have their sides about the equal angles which is described from the pole P, with proportionals; and that their surfaces the distance Pa, will touch A B in the bear the same ratio to one another as point c (18. Cor. 1.). And, in the same the surfaces of their respective spheres, manner, it may be shown that the same and, therefore, are to one another as the circle will touch AC in b. Therefore, squares of the radii of the spheres, or from the pole P, with the distance Pa, as the squares of the arcs which are describe the circle ubc; and it will be homologous sides of the triangles, the circle required. Therefore, &c. B 208 APPENDIX. Part I.-Of Projection by Lines diverging and by Lines parallel. Oblique Cylinder. a A Part I.- Of Projection by Lines di- either perspectively or orthographically, verging and by Lines parallel. upon a given plane A B, when all its points are so projected; and the line It is not here intended to enter at large p q which contains the projections of the upon the subject of perspective, or to latter is called the perspective (fig. of anticipate in any manner the rules by def. 1.) or orthographic (fig. of def. 2.) which it affords such material assistance projection of the line P Q. to the draughtsman and artist. We 4. A figure P Q R is said to be propropose, on the contrary, no more than jected, either perspectively or orthograthe explanation of a few terms, and the phically, upon a given plane A B, when statement of a few theorems, occasion- all its containing lines are so projected ; ally serving to simplify the consideration and the figure p qr, which is contained of lines in different planes, and which by the projections of the latter is called will be of immediate service in the ac- the perspective (fig. of def. 1.) or orthocount which will be subsequently given graphic (fig. of def. 2.) projection of the of the general properties of the conic figure P Q R. sections. 5. Any point, line, or figure is called Def. 1. Let AB an original point, line, or figure with be a plane given reference to its perspective or orthograin position, and V phic projection. a given point with Thus, in the figures of def. 1. and out it; then, if def. 2., the point P is called the original through any point of the point p, the line P Q the original P a straight line of the line pq, and the figure PQ R the is drawn from V original of the figure pqr. to meet the plane It is almost needless to observe that AB in p, the point in these definitions the planes E F and p is called the per A B, although they necessarily appear spective projec circumscribed in the figures, are consition of the point P upon the plane A B. dered to be of unlimited extent; and P The plane A B is called the plane of the same is to be understood in the folprojection, the point V the vertex, and lowing propositions. the plane E F, which is drawn through PROP. 1. the point V parallel to A B, the vertical plane. In perspective projection, no point in 2. Let A B be a plane given in posi- the vertical plane É F can be projected tion, and C D a straight line given in from the vertex V upon the plane of position, not parallel to the plane A B: projection AB; but, of every point Then, if through which is not in that plane, the projection any point P a may be found upon the plane AB. straight line is For, the straight drawn parallel to line which is drawn CD to meet the from the point V plane AB in p. through any point the point pis called in the plane EF, the orthographic can never meet the projection of the plane A B ; be q point P upon the cause it lies enplane AB. tirely in the plane The straight line C D is called the EF, which is paraldirection, and the plane A B, as before, lel to AB. Therethe plane of projection. fore, no point in the plane E F can be 3. A line P Q is said to be projected, projected from V upon the plane A B. B R IC E P M P A B N B But, if P be a point which is not in the vertex V cuts Q PM in some point the plane E F, draw V O perpendicular between Q and M. to the plane A B (IV.36.), and let the Therefore, &c. plane PVO cut the parallel planes EF Cor. 1. It is supposed in the propoand A B in the straight lines VM and sition that the original straight line PQ ON respectively. Then, because the does not pass through the vertex V; sections of parallel planes by the same for, in this case, it is evident that all its plane are parallel straight lines (IV. 12.), points have for their projections the sinVM is parallel to ON; and, because VM gle point in which it cuts the plane of is parallel to ON, and that V P cuts projection. V M in V, VP may be produced to cut Cor. 2, The perspective projection ON in some point p (I. 14. Cor. 3.); but of any given straight line is a part of the if it cuts O N in any point, it must cut common section of two planes, viz. the the plane AB in the same point, because plane which passes through the vertex O N lies in that plane: therefore p, the and given straight line, and the plane of projection of the point P (def. 1.), may be projection. found. Cor. 3. The perspective projection of Therefore, &c. a straight line which is parallel to the Cor. It is shewn in this proposition, plane of projection, is parallel to its orithat if a straight line cuts one of two ginal (IV, 10.) parallel planes, it may be produced to Cor. 4. The perspective projection of cut the other likewise. a straight line which is not parallel to the plane of projection, shall pass, if PROP. 2. produced, through the point in which a The perspective projection of a straight vertex cuts the plane of projection. For parallel to the original drawn through the line is a straight line; and if any point such parallel is in the plane which passes of the original straight line be in the through the vertex and the original vertical plane, the straight line which straight line, and consequently the point is its projection shall be of unlimited in which it cuts the plane of projection extent. is in the conimon section of the two Let AB be the planes. plane of projec Cor. 5. If the original straight line tion, V the ver cuts the vertical plane, in the point M, tex, P Q any so that one part, as K M, lies upon straight line, and one side of that plane, and the other p. 9 its projec part, as M PQ, upon the other side of tion: p q shall it, the projections of the two parts shall likewise be together make up the whole of a straight straight line. line infinitely produced both ways, exBecause the cept only the finite interval kq between points of p q lie the projections of its extreme points K in straight lines and Q. drawn through Cor. 6. And if such original finite V and the cor straight line KM P Q be infinitely proresponding points of P Q, and that duced both ways, the projections of the PQ is a straight line, the points of pq produced parts shall together make up lie in the plane V P Q. But they lie the finite interval kq between the proalso in the plane A B. Therefore they jections of its extreme points K and Q. lie in the common section of the planes For, if Vi be drawn parallel to KQ V P Q and A B, that is (IV. 2.), in a to meet the plane A B in i, the projecstraight line. tion of every point in the part produced Also, if any point M of the original beyond K will be found between k and straight line QPM lie in the verticali, and the projection of every point in plane E F; the straight line qp, which the part produced beyond Q between is its projection, shall be of unlimited 9 and i. extent towards p. For the projection PROP. 3. of the point M cannot be found upon The perspective projections of parallel the plane A B (1.); and every point in straight lines, which are likewise paralap produced is the perspective projec- lel to the plane of projection, are parallel tion of some point of QPM, because the straight lines. straight line which is drawn from it to Let A B be the plane of projection, M а P A B V P P B L F M T N V the vertex, P Q and P'Q' any two projection. Therefore, because V C is parallel straight lines, which are like- parallel both to P Q and P'Q', the prowise parallel to the plane AB, and pqjections pq and p' q' pass both of them and p'q' their pro through the point Č. And in the same jections: p q shall manner, it may be shown that the probe parallel to p'q'. jection of any other straight line which Because P Q is is parallel to PQ, passes through the parallel to the e' same point C; for such straight line plane A B, the pro is likewise parallel to VC (IV. 6.). jection p q is pa pr Therefore, &c. rallel to P Q (2. PROP. 5. Cor. 3.) and, for The perspective projection of a curved the like reason, line* is a curved line; and, if any point p'd' is parallel to P'Q'.' Again, be- of the original curve be in the vertical cause p 9 and P Q are each of them plane, the curve which is its projection parallel to PQ, pq is parallel to P'Q' shall have an arc of unlimited extent (IV. 6.); and, because p q and p' q' are each of them parallel to P'Q', pq is corresponding to the arc of the original curve which is terminated in that parallel to pla. And in the same manner it may be point. shewn that if there are any number of v the vertex, PQR any curved line, Let A B be the plane of projection, parallel straight lines which are likewise and pqr its projection: pqr shall likeparallel to the plane A B, their perspec- wise be a curved line. tive projections shall be parallel to one another. Therefore, &c. PROP. 4. ब The perspective projections of parallel straight lines, which cut the plane of projection, are straight lines, which are not parallels, but which pass, when produced, all of them, through one and the same point, the point, namely, in which 9 a straight line drawn through the vertex parallel to the original straight lines, cuts the plane of projection. For, if any part of par, as p.q, be Let A B be the a straight line, then, since P Q is the plane of projec perspective projection of pq upon the tion, V the vertex, P Q and P'Q' plane P Q R (def . 1.), PQ must likewise be a straight line (2.), which is contrary any two paral to the supposition. Therefore, no part lel straight lines of p qr is a straight line, that is, paris which are not pa a curved line (I. def. 6.). rallel to the plane Also, if any point M of the curve A B, and pq and PQ R lies in the vertical plane E F, the p' q' their projections; also let VC be drawn through the projection par shall have an arc of un limited extent corresponding to the are point V parallel to P Q to meet the plane MP, which is terminated in M. y A B in C: the straight lines produced, shall , each of them, pass and let V N be joined and produced to Let N be any point in the arc M P, through the point C. Because straight lines which are pa. fore (def. 1.) the projection of the point meet the plane XB in n, which is thererallel to the same straight line are pa- N : from V draw VO perpendicular to rallel to one another (IV.6.) VC, which the plane. A B (IV. 36.), and from N is parallel to PQ, is parallel also to P'Q” : draw NT perpendicular to the plane But it has been already shown (2 Cor. 4.) that the perspective projection of a * The whole of which (it is also understood) lies in one plane. For, if the parts of a curve lie in different straight line which is not parallel to the planes, of which one or more pass through the vertex plane of projection, passes through the of projection, the projections of the corresponding point in which a parallel to it drawn parts will be straight lines, (see Cor. 1 of this propo sition). The demonstration given in the text applies through the vertex cuts the plane of only to a plane curye. P B 72 Q P P с B E V R P B E F, and join On, VT. Then, because This definition will be best illustrated VO is perpendicular to the plane AB, by the example contained in the followwhich is parallel to the plane EF, VO ing proposition. is also perpendicular to the plane E F PROP. 6. (IV. 11.); but N T is perpendicular to the same plane; therefore NT is pa- The perspective projection of a rallel to VO (IV. 3.). And, because straight line which touches any curve is the plane of the parallels TN, VO (I. a straight line which touches the prodef. 12.) cuts the planes E F, AB in the jection of that curve, if the point of lines TV, On respectively, TV is parallel contact is without the vertical plane; to On (IV. 12.). Therefore, because but, if the point of contact is in that the triangles V On, TNV have the plane, the projection of the tangent is an sides Vo, On of the first parallel to the asymptote to the projection of the curve. sides NT, TV of the other, each to each, Let A B be the plane of projection, V and their sides V n, NV in the same the vertex, and P Q K any curve; and, straight line, they are equiangular (I. first, let_Pbe a point of the curve PQ R 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 O V 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 magni Th tude, 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 lying without the vertical plane E F, and limit to the number of times TV may PH a straight line which touches the be made to contain TN. Therefore, curve P Q R in the point P; then, if the also, there is no limit to the number of point p be the projection of the point P times On may be made to contain OV, (def. 1.), because P is a point both in the that is, the line On may be increased curve PQ R and the tangent PH, p will without limit, and the point n will de- be a point in the projection of each ; let, scribe an arc of unlimited extent cor- therefore, the curve p q r be the projecresponding to the arc PNM or MP, tion of the curve P Q R, and the straight which is terminated in M. line ph the projection of the tangent PH: Therefore, &c. the straight line ph shall likewise touch the curve pqr in the point p. Cor. 1. In the demonstration of this Because PH touches the curve PQR, proposition it is supposed that the plane the points of the curve P Q R upon both of the original curve does not pass sides of P lie towards the same part of through the vertex V; for, in this case, PH, and therefore also the straight lines it is evident that its projection upon the drawn from V through those points lie plane A B is a straight line (IV. 2.). towards the same part of the plane Cor. 2. If the original curve cuts the VPH or Vph. But these straight lines vertical plane in the point M, are the same which are drawn from V one part, as KLM, lies upon one side to the points of the curve pqr on both of that plane, and the other part, as sides of p (def. 1.). Therefore, the latMPQ, upon the other side of it, the ter also lie towards the same part of the projection shall have two arcs which plane V ph; and, consequently, the are extended without limit in opposite points of the curve pqr on both sides directions, corresponding, to the two of p lie towards the same part of the arcs KLM, MP.Q, which are termi- straight line ph, that is, ph touches the minated in the point M. curve pgr in the point p. Def. 6. If a curve has an arc of unli- But, in the next place, let M be a mited extent, and if a straight line is point of the curve P Q R lying in the drawn which never meets that arc, but vertical plane E F, and let M G H be a which, being produced, may be made to straight line touching the curve PQR approach nearer to it than by any given in the point M; then, if the curve par distance, such straight line is called be the projection of the curve M PQR, an asymptote to the arc, and the straight line gh of the tangent so that |