95 a the : The initial velocities can never be pushed as far as That the reader may see with one glance the relaUse of the we have calculated for in this table ; but we mean it tion of those different quantities, we have given this last table. for a table of more extensive use than appears at first table, expressed in a figure (fig. 10.). The abscissa, Fig. 10. sight. Recollect, that while the proportion of the ve- or axis DA, is the scale of the initial velocities in feet 96 Relation of locity at the vertex to the terminal velocity remains the per second, measured in a scale of 400 equal parts in the diffesame, the curves will be similar: therefore, if the initial an inch. The ordinates to the curve ACG express rent quanvelocities are as the square-roots of the diameters of the yards of the range on a scale containing 800 yards in tilies in it. balls, they will describe similar curves, and the ranges an inch. The ordinates to the curve A x y express (by the same scale) the height to which the ball rises in the We see by this table the immense difference between great velocities the range is very little increased by its the ordinate HK, draw HL parallel to the axis, meet- solved with great dispatch, and with much more accu- Note, that fig. 10. as given on Plate CCCCXLII, is PROJECTION OF THE SPHERE, THE PROJECTION of the SPHERE is a perspective primitive circle. The pole of this circle is the pole Stereogratereographic Pro representation of the circles on the surface of the of projection, and the place of the eye is the projecting phie Pro jection of ection of sphere; and is variously denominated according to the point. the Spheres he Sphere. different positions of the eye and plane of projection. 2. The line of measures of any circle of the sphere is There are three principal kinds of projection ; the that diameter of the primitive, produced indefinitely, tereographic projection the eye is supposed to be placed AXIOM. The projection, or representation of any point, is where the straight line drawn from it to the proMercator's, scenographic, &c. for which see the artieles jecting point intersects the plane of projectiou. SECTION I. Of the Stereographic Projection of the Sphere. eye sures. a Stereogra- eye is placed on the surface of the sphere in the pole of But if the circle MN (fig. 2.) be not parallel to the Stereographic Pro- the great circle upon which the sphere is to be project- primitive circle BD, let the great circle ABCD, pas- phic to jection of ed. The projection of the bemisphere opposite to the sing through the projecting point, cut it at right angles jection of the Sphere. eye falls within the primitive, to which the projection in the diameter MN, and the primitive in the diameter the Sphere is generally limited; it, however, may be extended to BD. Through M, in the plane of the great circle, let Fig . 1. the other hemisphere, or that wherein the eye is placed, MF be drawn parallel to BD; let AM, AN be joined, the projection of which falls without the primitive. and meet BD in m, n. Then, because AB, AD are As all circles in this projection are projected either quadrants, and BD, MF parallel, the arch AM is equal into circles or straight lines, which are easily described, to AF, and the angle AMF or Amn is equal to ANM, it is therefore more generally understood, and by many Hence the conic surface described by the revolution of preferred to the other projections. AM about the circle MN is cut by the primitive in a subcontrary position ; therefore the section is in this PROPOSITION I. THEOREM I. case likewise a circle. COROLLARIES. 1. The centres and poles of all circles parallel to the 2. The centre and poles of every circle inclined to Plate Let ABCD (fig. 1.) be a great circle passing through the primitive have their projections in the line of meaccecxlii. A, C, the poles of the primitive, and intersecting it in fig. 1. the line of common section BED, E being the centre 3. All projected great circles cut the primitive in a tangent to that circle's projection ; also, the circular the pole opposite to the projecting point. 6. The extremities of the diameter, on the line of COROLLARIES. measures of any projected great circle, are distant from 1. Each of the quadrants contiguous to the projects the centre of the primitive by the tangent and cotaning point is projected into an indefinite straight line, gent of half the great circle's inclination to the primiand each of those that are remote into a radius of the tive. primitive. 7. The radius of any projected circle is equal to balf 2. Every small circle which passes through the pro the sum, or half the difference of the semitangents of jecting point is projected into that straight line which is the least and greatest distances of the circle from the its common section with the primitive. pole opposite to the projecting point, according as that 3. Every straight line in the plane of the primitive, pole is within or without the given circle. and produced indefinitely, is the projection of some circle on the sphere passing through the projecting PROPOSITION III. THEOREM III. point. An angle formed by two tangents at the same point 4. The projection of any point in the surface of the in the surface of the sphere, is equal to the angle sphere, is distant from the centre of the primitive, by formed by their projections. Let FGI and GH (fig. 3.) be the two tangents, Fig. 3• and A the projecting point; let the plane AGF cut PROPOSITION II. THEOREM II. the sphere in the circle AGL, and the primitive in the Every circle on the sphere which does not pass through tion of the plane AGH with the primitive : then the line BML. Also, let MN be the line of common secthe projecting point is projected into a circle. angle FGH=LMN. If the plane FGH be parallel If the given circle be parallel to the primitive, then to the primitive BLD, the proposition is manifest. If a straight line drawn from the projecting point to any not, through any point K in AG produced, let the point in the circumference, and made to revolve about plane FKH, parallel to the primitive, be extended to the circle, will describe the surface of a cone ; which meet FGH in the line FH." Then, because the plane being cut by the plane of projection parallel to the base, AGF meets the two parallel planes BLD, FKH, the the section will be a circle. See Conic-Sections. lines of common section LM, FK are parallel ; there fore a |