mon pole of the circles B CD, E F G (37. Cor.), 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 arc which can be drawn from the point P to the circle A L7 (18. Cor. 1.), and if LP be produced to meet the circle in 7, Pl is the greatest. But every point of the circle B C D is at the distance PL; and every point in the equal circle E F G 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.

Bisect the angle ABC with the arc BP, and the angle A C B with the arc CP which meets the former arc in P (35.); from P draw Pa perpendicular to B C (34.); make B c 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 A C in b. Therefore, from the pole P, with the distance Pa, describe the circle a bc; and it will be the circle required. Therefore, &c.

Scholium.

The constructions in this section have

little or no practical utility, and have, accordingly, been added rather with a view to illustrate the analogies of Plane 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 plane, their sides more and of their angles (the excess of which more nearly straight lines, and the sum 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 face of a sphere, the radius of which as a spherical triangle upon the suris 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

Oblique Cylinder.

PART I.-Of Projection by Lines di

verging and by Lines parallel.

Ir 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 PQ.

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 p q, and the figure PQR the original of the figure p qr.

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.

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 parallel to P Q 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', pq is parallel to p' q'.

(2.

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.

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 q r 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.

and let V N be joined and produced to Let N be any point in the arc M P, fore (def. 1.) the projection of the point meet the plane AB in n, which is thereN: 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.

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 Vn, 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 MP, 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 arcs 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 arc, 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,

لـ

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, p will 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.

Because PH touches the curve PQR, the points of the curve PQR upon both sides of P lie towards the same part of PH, and therefore also the straight lines drawn from V through those points lie towards the same part of the plane VPH or Vph. But these straight lines are the same which are drawn from V to the points of the curve p q r on both sides of p (def. 1.). Therefore, the latter also lie towards the same part of the plane 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 pqr 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 MGH be a straight line touching the curve P Q R in the point M; then, if the curve pqr be the projection of the curve M PQR, and the straight line gh of the tangent