206 the triangles PDA, PDB have two sides of the one equal to two sides of the other, each to each, and the included angles PDA, PDB equal to one another (13.). And, for the same reason, if PA and PC are joined, PA will be equal to PC. But PÅ and P A' are together equal to a semicircumference, because AA' is a diameter of the sphere. Therefore, PA and PC are together equal to a semicircumference; and, consequently (3. Cor.), the parallel circles AGH, CK L are equal to one another. Also, because PA is equal to PB, the circle AG H passes through the point B. Therefore, &c. To describe a triangle which shall be equal to a given spherical polygon, and shall have a side and adjacent angle the sume with a given side AB and adjacent angle B of the polygon. First, let the given polygon be a quadrilateral A B C D. Through the two points A, C, and the B E third point D, describe two equal and parallel small circles (38.), and let the arc B C, which cuts one of them in C, be produced to cut the other in E (22. Cor. 2.), and join AE, AC (31.). Then, because the triangles A CD, A CE lie between the same equal and parallel small circles, they are equal to one another; and, therefore, the triangle ABC being added to each, the triangle ABE is equal to the quadrilateral A B C D. F Next, let ABCDEF be the given polygon. Join AC, A D. Make, as in the former case, the triangle ADG equal to the quadrilateral ADEF, the triangle ACH equal to the quadrilateral A ACDG, and the triangle ABK equal to the quadrilateral ABCH. It is evi B Ꮐ E H K dent that the triangle A B K is equal to the polygon ABCDEF. And, in each case, the triangle described has the same side AB and angle B with the given polygon. Therefore, &c. PROP. 40. Prob. 12. Given two spherical arcs A B and Q, which are together less than a semicircumference; to place them so, that, with a third not given, they may contain the greatest surface possible. Bisect AB in H (32.), and produce HA to P, so that HP may be equal to a quadrant ; from the pole P, with the distance PB, describe the small circle BCK, and from the pole B, with the distance Q, describe a circle cutting the circle BCK in C. Join AC, BC. The triangle A B C shall be the triangle required. K C For, if, with the distances PA, PH, there are described from the pole P the small circle ALM and the great circle HGN, the latter cutting the arc AC in G, the circles A L M and B C K will be equal to one another, because the distances PA, PB are together equal to twice the quadrant PH (I. ax. 9.), that is, to a semicircumference (3. Cor.); and they are parallel because they have the same poles; and HGN is the great circle to which they are parallel; therefore, AG is equal to GC (22.). But, because P is the pole of the great circle HGN, PH is at right angles to HG (5.); and, because the triangles GHA, GHB have two sides of the one equal to two sides of the other, each to each, and the included angles G HA, G H B equal to one another, G B is equal to GA or G C (13.). Therefore, because in the isosceles triangles GA B, GBC, the angles GBA, GBC are equal to the angles GAB, GCB respectively (11.), the whole angle ABC of the triangle A B C is equal to the sum of the two angles CAB, ACB; wherefore the triangle ABC is the greatest that can be formed with the two sides A B, B C, or the greatest that can be formed with the given sides A B and Q (26. Cor.). Therefore, &c. PROP. 41. Prob. 13. Through a given point A to describe a great circle, which shall touch two given equal and parallel small circles BCD, EFG. Find the point P which is the com mon pole of the circles B CD, E F G (37. Ĉor.), and join PA (31.): from the pole P, with the distance P A, describe a circle, and from the point H, in which PA cuts the circle B CD, draw H K at right angles to PA (33.), and let it cut the circle, which was described through A, in K: join PK (31.), and let it cut the circle B C D in L: the great circle A L which passes through the points A and L shall be the great circle required. For, because in the triangles PH K, PLA, the two sides PH, PK are equal to the two sides PL, PA, and the included angle LPH common to both triangles, the angle PLA is equal to the angle PHK, (13.) that is, to a right angle; and the arc P L is less than a quadrant; therefore, P L is the least are which can be drawn from the point P to the circle ALI (18. Cor. 1.), and if LP be produced to meet the circle in, P is the greatest. But every point of the circle B C D is at the distance PL; and every point in the equal circle EFG at the distance Pl, because P7 and PL are together equal to a semicircumference (3. Cor.). Therefore, the circle ALM, which has been described through the given point A, touches the given circles B CD and E F G in the points L and l. Therefore, &c. PROP. 42. Prop. 14. To inscribe a circle in a given spherical triangle AB C. A Bisect the angle ABC with the arc BP, and the angle ACB with the arc CP which meets the former arc in P (35.); from P draw Pa perpendicular to B C (34.); make Bc equal to Ba, and Cb equal to Ca, and join P c, Pb. Then, because the triangles P Ba, PB c have two sides of the one equal to two sides B of the other, each to each, and the included angles PBa, PB c equal to one another, Pc is equal to Pa (13.), and the angle PCB to the angle Pa B, that is, to a right angle: therefore, the circle which is described from the pole P, with the distance Pa, will touch AB in the point c (18. Cor. 1.). And, in the same manner, it may be shown that the same circle will touch AC in b. Therefore, from the pole P, with the distance P a, describe the circle abc; and it will be the circle required. Therefore, &c. Scholium. The constructions in this section have little or no practical utility, and have, view to illustrate the analogies of Plane accordingly, been added rather with a and Spherical Geometry, than to furnish rules for practice. Some of these we have already had occasion to notice, and others will have offered themselves to the reader; who will readily perceive that these striking points of resemblance (or, as he may be disposed to call them, of identity) are to be ascribed to the circumstance, that spherical triangles, when their sides are but small portions of great circles, and consequently their surfaces small in comparison with the surface of the sphere, become more and more nearly straight lines, and the sum more nearly plane, their sides more and of their angles (the excess of which above two right angles bears the same ratio to eight right angles (21 Cor. 1.) as the surface of the triangle to the surface of the sphere) more and more nearly equal to two right angles. Thus every plane triangle may be regarded as a spherical triangle upon the surface of a sphere, the radius of which is indefinitely great; and in this way of viewing the subject, the properties of plane triangles resemble those of spherical triangles, only as a particular case the general one in which it is included. But it may be asked, has the term similar, which introduces us to so wide a field in Plane Geometry, any place in Spherics? Not in propositions which have reference only to the surface of one and the same sphere. Similar figures upon the surface of the same sphere are likewise equal to one another, and may be made to coincide. But, when we consider the surfaces of different spheres, and compare the figures which are formed upon them, here again we shall find room for the application of the term in its full and peculiar sense. Thus, similar spherical triangles are such as are contained by similar arcs upon the surfaces of different spheres. It is easy to perceive that such triangles are equiangular, and have their sides about the equal angles proportionals; and that their surfaces bear the same ratio to one another as the surfaces of their respective spheres, and, therefore, are to one another as the squares of the radii of the spheres, or as the squares of the arcs which are homologous sides of the triangles, 208 APPENDIX. PART I.-Of Projection by Lines diverging and by Lines parallel. PART II.-Of the Plane Sections of the Right Cone, or Conic Sections. PART III-Plane Sections of the Oblique Cone, of the Right Cylinder, and of the Oblique Cylinder. PART I.-Of Projection by Lines di verging and by Lines parallel. It is not here intended to enter at large upon the subject of perspective, or to anticipate in any manner the rules by which it affords such material assistance to the draughtsman and artist. We propose, on the contrary, no more than the explanation of a few terms, and the statement of a few theorems, occasionally serving to simplify the consideration of lines in different planes, and which will be of immediate service in the account which will be subsequently given of the general properties of the conic sections. either perspectively or orthographically, upon a given plane A B, when all its points are so projected; and the line pq which contains the projections of the latter is called the perspective (fig. of def. 1.) or orthographic (fig. of def. 2.) projection of the line P Q. 4. A figure PQR is said to be projected, either perspectively or orthographically, upon a given plane A B, when all its containing lines are so projected; and the figure p qr, which is contained by the projections of the latter is called the perspective (fig. of def. 1.) or orthographic (fig. of def. 2.) projection of the figure PQR. 5. Any point, line, or figure is called an original point, line, or figure with reference to its perspective or orthographic projection. Thus, in the figures of def. 1. and def. 2., the point P is called the original of the point p, the line PQ the original of the line pq, and the figure P Q R the original of the figure pqr. It is almost needless to observe that in these definitions the planes E F and A B, (although they necessarily appear circumscribed in the figures, are considered to be of unlimited extent; and the same is to be understood in the following propositions. But, if P be a point which is not in the plane E F, draw V O perpendicular to the plane A B (IV. 36.), and let the plane PVO cut the parallel planes E F and AB in the straight lines V M and ON respectively. Then, because the sections of parallel planes by the same plane are parallel straight lines (IV. 12.), VM is parallel to ON; and, because VM is parallel to ON, and that V P cuts VM in V, V P may be produced to cut ON in some point p (I. 14. Cor. 3.); but if it cuts ON in any point, it must cut the plane AB in the same point, because ON lies in that plane: therefore p, the projection of the point P (def. 1.), may be found. Because the points of pq lie in straight lines drawn through V and the corresponding points of P Q, and that PQ is a straight line, the points of p q lie in the plane VPQ. But they lie also in the plane A B. Therefore they lie in the common section of the planes VPQ and AB, that is (IV. 2.), in a straight line. the vertex V cuts Q PM in some point between Q and M. Therefore, &c. Cor. 1. It is supposed in the proposition that the original straight line P Q does not pass through the vertex V; for, in this case, it is evident that all its points have for their projections the single point in which it cuts the plane of projection. Cor. 12. The perspective projection of any given straight line is a part of the common section of two planes, viz. the plane which passes through the vertex and given straight line, and the plane of projection. Čor. 3. The perspective projection of a straight line which is parallel, to the plane of projection, is parallel to its original (IV. 10.) Cor. 4. The perspective projection of a straight line which is not parallel to the plane of projection, shall pass, if produced, through the point in which a parallel to the original drawn through the vertex cuts the plane of projection. For such parallel is in the plane which passes through the vertex and the original straight line, and consequently the point in which it cuts the plane of projection is in the common section of the two planes. Cor. 5. If the original straight line cuts the vertical plane, in the point M, so that one part, as K M, lies upon one side of that plane, and the other part, as MPQ, upon the other side of it, the projections of the two parts shall together make up the whole of a straight line infinitely produced both ways, except only the finite interval kq between the projections of its extreme points K and Q. Cor. 6. And if such original finite straight line K M P Q be infinitely produced both ways, the projections of the produced parts shall together make up the finite interval kq between the projections of its extreme points K and Q. For, if V i be drawn parallel to KQ to meet the plane A B in i, the projection 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 vertical, 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 of the point M cannot be found upon the plane AB (1.); and every point in qp produced is the perspective projection of some point of QPM, because the straight line which is drawn from it to PROP. 3. The perspective projections of parallel straight lines, which are likewise parallel to the plane of projection, are parallel straight lines. Let A B be the plane of projection 210 A P P V the vertex, PQ and P'Q' any two parallel straight lines, which are likewise parallel to the plane AB, and pq and p'q' their projections: p q shall be parallel to p'q'. Because PQ is parallel to the plane A B, the projection p q is rallel to P Q (2. Cor. 3.) and, for the like reason, p'q' is parallel to P'Q'. Again, because p q and P' Q' are each of them parallel to PQ, pq is parallel to P'Q' (IV. 6.); and, because p q and p' q' are each of them parallel to P' Q', p q is parallel to p' q'. ра p B And in the same manner it may be shewn that if there are any number of parallel straight lines which are likewise parallel to the plane A B, their perspective projections shall be parallel to one another. Therefore, &c. PROP. 4. 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 a straight line drawn through the vertex parallel to the original straight lines, cuts the plane of projection. Let A B be the plane of projection, V the vertex, PQ and P' Q' any two parallel straight lines which are not parallel to the plane AB, and pq and p'q' their projec A p B tions; also let VC be drawn through the point V parallel to PQ to meet the plane AB in C: the straight lines pq, p' q' produced, shall, each of them, pass through the point C. Because straight lines which are parallel to the same straight line are parallel to one another (IV. 6.) VC, which is parallel to PQ, is parallel also to P'Q' But it has been already shown (2 Cor. 4.) that the perspective projection of a straight line which is not parallel to the plane of projection, passes through the point in which a parallel to it drawn through the vertex cuts the plane of F For, if any part of pqr, as pq, be a straight line, then, since PQ is the perspective projection of pq upon the plane P Q R (def. 1.), PQ must likewise be a straight line (2.), which is contrary to the supposition. Therefore, no part of p qr is a straight line, that is, p q r is a curved line (I. def. 6.). Also, if any point M of the curve PQR lies in the vertical plane E F, the projection pqr shall have an arc of unlimited extent corresponding to the are MP, which is terminated in M. Let N be any point in the arc M P, and let V N be joined and produced to meet the plane AB in n, which is therefore (def. 1.) the projection of the point N: from V draw V O perpendicular to the plane AB (IV. 36.), and from N draw NT perpendicular to the plane The whole of which (it is also understood) lies in one plane. For, if the parts of a curve lie in different planes, of which one or more pass through the vertex of projection, the projections of the corresponding parts will be straight lines, (see Cor. 1 of this proposition). The demonstration given in the text applies only to a plane curve. B |