95 The initial velocities can never be pushed as far as Use of the we have calculated for in this table; but we mean it last table. for a table of more extensive use than appears at first sight. Recollect, that while the proportion of the velocity at the vertex to the terminal velocity remains the same, the curves will be similar: therefore, if the initial velocities are as the square-roots of the diameters of the balls, they will describe similar curves, and the ranges will be as the diameters of the balls. Therefore, to have the range of a 12 pound shot, if projected at an elevation of 45, with the velocity 1500; suppose the diameter of the 12-pounder to be d, and that of the 24-pounder D; and let the velocities be v and V: Then say, ✅d: ✅D=1500, to a fourth proportional V. If the 24-pounder be projected with the velocity V, it will describe a curve similar to that described by the 12-pounder, having the initial velocity 1500. Therefore find (by interpolation) the range of the 24-pounder, having the initial velocity V. Call this R. Then D: d=R:r, the range of the 12-pounder which was wanted, and which is nearly 3380 yards. We see by this table the immense difference between the motions through the air and in a void. We see that the ranges through the air, instead of increasing in the duplicate ratio of the initial velocities, really increase slower than those velocities in all cases of military service; and in the most usual cases, viz. from Soo to 1600, they increase nearly as the square-roots of the velocities. A set of similar tables, made for different elevations, would almost complete what can be done by theory, and would be much more expeditious in their use than Mr Euler's Trajectories, computed with great labour by his English translator. The same table may also serve for computing the ranges of bomb-shells. We have only to find what must be the initial velocity of the 24 pound shot which corresponds to the proposed velocity of the shell. This must be deduced from the diameter and weight of the shell, by making the velocity of the 24-pounder such, that the ratio of its weight to the resistance may be the same as in the shell. Relation of That the reader may see with one glance the relation of those different quantities, we have given this table, expressed in a figure (fig. 10.). The abscissa, Fig. 10. or axis DA, is the scale of the initial velocities in feet 96 per second, measured in a scale of 400 equal parts in the diffean inch. The ordinates to the curve ACG express the rent quanyards of the range on a scale containing 800 yards in tities in it. an inch. The ordinates to the curve Axy express (by the same scale) the height to which the ball rises in the air. The ordinate BC (drawn through the point of the abscissa which corresponds to the initial velocity 2000) is divided on the points 4, 9, 12, 18, 24, 32, 42, in the ratio of the diameters of cannon-shot of different weights; and the same ordinate is produced on the other side of the axis, till BO be equal to BA; and then BO is divided in the subduplicate ratio of the same diameters. Lines are drawn from the point A, and from any point D of the abscissa, to these divisions. We see distinctly by this figure how the effect of the initial velocity gradually diminishes, and that in very great velocities the range is very little increased by its augmentation. The dotted curve APQR, shows what the ranges in vacuo would be. By this figure may the problems be solved. Thus, to find the range of the 12-pounder, with the initial velocity 1500. Set off 1500 from B to F; draw FH parallel to the axis, meeting the line 12 A in H; draw the ordinate HK, draw HL parallel to the axis, meeting 24 B in L; draw the ordinate LM, cutting 12 B in N. MN is the range required. If curves, such as ACG, were laid down in the same manner for other elevations, all the problems might be solved with great dispatch, and with much more accuracy than the theory by which the curves are drawn can pretend to. Note, that fig. 10. as given on Plate CCCCXLII, is one-half less than the scale according to which it is described; but the practical mathematician will find no difficulty in drawing the figure on the enlarged scale to correspond to the description. tereogra phic Pro PROJECTION OF THE SPHERE. THE PROJECTION of the SPHERE is a perspective representation of the circles on the surface of the ection of sphere; and is variously denominated according to the he Sphere different positions of the eye and plane of projection. There are three principal kinds of projection; the stereographic, the orthographic, and gnomic. In the tereographic projection the eye is supposed to be placed on the surface of the sphere; in the orthographic it is supposed to be at an infinite distance; and in the gnomic projection the eye is placed at the centre of the sphere. Other kinds of projection are, the globular, Mercator's, scenographic, &c. for which see the articles GEOGRAPHY, NAVIGATION, PERSPECTIVE, &c. DEFINITIONS. 1. The plane upon which the circles of the sphere are described, is called the plane of projection, or the. primitive circle. The pole of this circle is the pole Stereograof projection, and the place of the eye is the projecting phie Projection of point. the Sphere. 2. The line of measures of any circle of the sphere is that diameter of the primitive, produced indefinitely, which passes through the centre of the projected circle. AXIOM. The projection, or representation of any point, is where the straight line drawn from it to the projecting point intersects the plane of projection. SECTION I.. Of the Stereographic Projection of the Sphere. IN the stereographic projection of the sphere, the eye Stereogra- eye is placed on the surface of the sphere in the pole of phic Pro- the great circle upon which the sphere is to be projectjection of ed. The projection of the hemisphere opposite to the the Sphere. eye falls within the primitive, to which the projection is generally limited; it, however, may be extended to the other hemisphere, or that wherein the eye is placed, the projection of which falls without the primitive. Plate As all circles in this projection are projected either into circles or straight lines, which are easily described, it is therefore more generally understood, and by many preferred to the other projections. PROPOSITION I. THEOREM I. Every great circle which passes through the projecting point is projected into a straight line passing through the centre of the primitive; and every arch of it, reckoned from the other pole of the primitive, is projected into its semitangent. Let ABCD (fig. 1.) be a great circle passing through CCCCXLIII. A, C, the poles of the primitive, and intersecting it in fig. 1. the line of common section BED, E being the centre of the sphere. From A, the projecting point, let there be drawn straight lines AP, AM, AN, AQ, to any number of points P, M, N, Q, in the circle ABCD: these lines will intersect BED, which is in the same plane with them. Let them meet it in the points p, m, n, q; then P, m, n, q, are the projections of P, M, N, Q: hence the whole circle ABCD is projected into the straight line BED, passing through the centre of the primitive. Again, because the pole C is projected into E, and the point M into m; therefore the arch CM is projected into the straight line Em, which is the semitangent of the arch CM to the radius AE. In like manner, the arch CP is projected into its semitangent, E P, &c. COROLLARIES. 1. Each of the quadrants contiguous to the projecting point is projected into an indefinite straight line, and each of those that are remote into a radius of the primitive. 2. Every small circle which passes through the projecting point is projected into that straight line which is its common section with the primitive. 3. Every straight line in the plane of the primitive, and produced indefinitely, is the projection of some circle on the sphere passing through the projecting point. 4. The projection of any point in the surface of the sphere, is distant from the centre of the primitive, by the semitangent of the distance of that point from the pole opposite to the projecting point. PROPOSITION II. THEOREM II. Every circle on the sphere which does not pass through the projecting point is projected into a circle. If the given circle be parallel to the primitive, then a straight line drawn from the projecting point to any point in the circumference, and made to revolve about the circle, will describe the surface of a cone; which being cut by the plane of projection parallel to the base, the section will be a circle. See CONIC-Sections. Stereogra But if the circle MN (fig. 2.) be not parallel to the primitive circle BD, let the great circle ABCD, pas- phic Prosing through the projecting point, cut it at right angles jection of in the diameter MN, and the primitive in the diameter BD. Through M, in the plane of the great circle, let Fig. 2. MF be drawn parallel to BD; let AM, AN be joined, and meet BD in m, n. Then, because AB, AD are quadrants, and BD, MF parallel, the arch AM is equal to AF, and the angle AMF or Amn is equal to ANM. Hence the conic surface described by the revolution of AM about the circle MN is cut by the primitive in a subcontrary position; therefore the section is in this case likewise a circle. COROLLARIES. 1. The centres and poles of all circles parallel to the primitive have their projection in its centre. 2. The centre and poles of every circle inclined to the primitive have their projections in the line of mea sures. 3. All projected great circles cut the primitive in two points diametrically opposite; and every circle in the plane of projection, which passes through the extremities of a diameter of the primitive, or through the projections of two points that are diametrically opposite on the sphere, is the projection of some great circle. 4. A tangent to any circle of the sphere, which does not pass through the projecting point, is projected into a tangent to that circle's projection; also, the circular projections of tangent circles touch one another. 5. The extremities of the diameter, on the line of measures of any projected circle, are distant from the centre of the primitive by the semitangents of the least and greatest distances of the circle on the sphere, from the pole opposite to the projecting point. 6. The extremities of the diameter, on the line of measures of any projected great circle, are distant from the centre of the primitive by the tangent and cotangent of half the great circle's inclination to the primi tive. 7. The radius of any projected circle is equal to half the sum, or half the difference of the semitangents of the least and greatest distances of the circle from the pole opposite to the projecting point, according as that pole is within or without the given circle. PROPOSITION III. THEOREM III. An angle formed by two tangents at the same point in the surface of the sphere, is equal to the angle 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 the sphere in the circle AGL, and the primitive in the line BML. Also, let MN be the line of common section of the plane AGH with the primitive: then the angle FGH=LMN. If the plane FGH be parallel to the primitive BLD, the proposition is manifest. If not, through any point K in AG produced, let the plane FKH, parallel to the primitive, be extended to meet FGH in the line FH. Then, because the plane AGF meets the two parallel planes BLD, FKH, the lines of common section LM, FK are parallel; there fore Stereogra- for the angle AML AKF. But since A is the pole phic Pro. BLD, the chords, and consequently the arches AB jection of AL, are equal, and the arch ABG is the sum of the the Sphere. arches AL, BG; hence the angle AML is equal to an angle at the circumference standing upon AG, and therefore equal to AGI or FGK; consequently the angle FGK FKG, and the side FG-FK. In like manner HG=HK: hence the triangles GHF, KHF are equal, and the angle FGH FKH=LMN. 2. An angle contained by any two circles of the sphere is equal to the angle formed by the radii of their projections at the point of intersection. PROPOSITION IV. THEOREM IV. The centre of a projected great circle is distant from the centre of the primitive; the tangent of the inclination of the great circle to the primitive, and its radius, is the secant of its inclination. Let MNG (fig. 4.) be the projection of a great circle, meeting the primitive in the extremities of the diameter MN, and let the diameter BD, perpendicular to MN, meet the projection in F, G. Bisect FG in H, and join NH. Then, because any angle contained by two circles of the sphere is equal to the angle formed by the radii of their projections at the point of intersection; therefore the angle contained by the proposed great circle and the primitive is equal to the angle ENH, of which EH is the tangent, and NH the secant, to the radius of the primitive. COROLLARIES. 1. All circles which pass through the points M, N, are the projections of great circles, and have their centres in the line BG; and all circles which pass through the points F, G, are the projections of great circles, and have their centres in the line HI, perpendicular to BG. 2. If NF, NH be continued to meet the primitive in L, F; then BL is the measure of the great circle's inclination to the primitive; and MT=2BL. PROPOSITION V. THEOREM V. The centre of projection of a less circle perpendicular to the primitive, is distant from the centre of the primitive, the secant of the distance of the less circle from its nearest pole; and the radius of projection is the tangent of that distance. Let MN (fig. 5.) be the given less circle perpendicular to the primitive, and A the projecting point. Draw AM, AN to meet the diameter BD produced in G and H; then GH is the projected diameter of the less circle: bisect GH in C, and C will be its centre; join NE, NC. Then because AE, NI are parallel, the angle INE=NEA; but NEA=2NMA VOL. XVII. Part II. =2NHG NCG: hence ENC-INE+INC NCG Stereogra +INC a right angle; and therefore NC is a tan- phic Progent to the primitive at N; but the arch ND is the Jection of distance of the less circle from its nearest pole D: the Sphere. hence NC is the tangent, and EC the secant of the distance of the less circle from its pole to the radius of the primitive. PROPOSITION VI. THEOREM VI. The projection of the poles of any circle, inclined to the primitive, are, in the line of measures, distant from the centre of the primitive, the tangent, and cotangent, of half its inclination. Let MN (fig. 6.) be a great circle perpendicular to Fig. 6. the primitive ABCD, and A the projecting point; then P, p are the poles of MN, and of all its parallels mn, &c. Let AP, Ap meet the diameter BD in Ff, which will therefore be the projected poles of MN and its parallels. The angle BEM is the inclination of the circle MEN, and its parallels, to the primitive: and because BC and MP are quadrants, and MC common to both; therefore PC-BM: and hence PEC is also the inclination of MN and its parallels. Now EF is the tangent of EAF, or of half the angle PEC the inclination; and Efis the tangent of the angle EAƒ; but EAƒ is the complement of EAF, and Eƒ is the cotangent of half the inclination. COROLLARIES. 1. The projection of that pole which is nearest to the projecting point is without the primitive, and the projection of the other within. 2. The projected centre of any circle is always between the projection of its nearest pole and the centre of the primitive; and the projected centres of all circles are contained between the projected poles. PROPOSITION VII. THEOREM VII. Equal arches of any two great circles of the sphere will be intercepted between two other circles drawn on the sphere through the remote poles of those great circles. Let AGB, CFD (fig. 7.) be two great circles of the Fig. 7. sphere, whose remote poles are E, P; through which draw the great circle PBEC, and less circle PGE, intersecting the great circles AGB, CFD in the points B, G, and D, F; then the arch BG is equal to the arch DF. Because E is the pole of the circle AGB, and P the pole of CFD, therefore the arches EB, PD are equal; is equal to the arch PB. For the same reason, the and since BD is common to both, hence the arch ED arches EF, PG are equal; but the angle DEF is equal to the angle BPG: hence these triangles are equal, and therefore the arch DF is equal to the arch BG. PROPOSITION VIII. THEOREM VIII. If from either pole of a projected great circle, two straight lines be drawn to meet the primitive and the projection, they will intercept similar arches of these circles. On 3 H Stereogra- On the plane of projection AGB (fig. 7.) let the phic Pro- great circle CFD be projected into cfd, and its pole P jection of into p; through p draw the straight lines pd, pf, then the Sphere. are the arches GB, fd similar. Fig. 8. Fig. 9. Fig. 10. Fig. 11. Since pd lies both in the plane AGB and APBE, it is in their common section, and the point B is also in their common section; therefore pd passes through the point B. In like manner it may be shown that the line pf passes through G. Now the points D, F are projected into d, f: hence the arches FD, fd are similar; but GB is equal to FD, therefore the intercepted arch of the primitive GB is similar to the projected arch fd. To describe the projection of a great circle through two given points in the plane of the primitive. Let P and B be given points, and C the centre of the primitive. straight line EGH, meeting the diameter AB produ- Stereogra ced if necessary in H; then from the centre H, with phic Frathe radius HE, describe the oblique circle DIE, and it jection of will be the projection of the great circle required. the Spart Or, make DK equal to FA; join EK, which intersects the diameter AB in I; then through the three points, D, I, E, describe the oblique circle DIE. PROPOSITION XI. PROBLEM III. To find the poles of a great circle. 1. When the given great circle is the primitive, its centre is the pole. 2. To find the pole of the right circle ACB (fig. 11.). Draw the diameter PE perpendicular to the given circle AB; and its extremities P, E are the poles of the circle ACB. 3. To find the pole of the oblique circle DEF (fig. Fig. 13. 13.). Join DF, and perpendicular thereto draw the diameter AB, cutting the given oblique circle DEF in E. Draw the straight line FEG, meeting the cir-. cumference in G. Make GI, GH, each equal to AD; then FI being joined, cuts the diameter AB in P, the lower pole; through F and H draw the straight line FHp, meeting the diameter AB produced in p, which will be the opposite or exterior pole. PROPOSITION XII. PROBLEM IV. 1. When one point P (fig. 8.) is the centre of the primitive, a diameter drawn through the given points To describe a less circle about any given point as a pole, will be the great circle required. 2. When one point P (fig. 9.) is in the circumference of the primitive. Through P draw the diameter PD; and an oblique circle described through the three. points P, B, D, will be the projection of the required great circle. 3. When the given points are neither in the centre nor circumference of the primitive. Through either of the given points P (fig. 10.) draw the diameter ED, and at right angles thereto draw the diameter FG. From F through P draw the straight line FPH, meeting the circumference in H: draw the diameter HI, and draw the straight line FIK, meeting ED produced in K; then an arch, terminated by the circumference, being described through the three points, P, B, K, will be the great circle. PROPOSITION X. PROBLEM II. To describe the representation of a great circle about any given point as a pole. Let P be the given pole, and C the centre of the pri mitive. 1. When P (fig. 8.) is in the centre of the primitive, then the primitive will be the great circle required. 2. When the pole P (fig. 11.) is in the circumference of the primitive. Through P draw the diameter PE, and the diameter AB drawn 'at right angles to PE will be the projected great circle required. 3. When the given pole is neither in the centre nor circumference of the primitive. Through the pole P Plate (fig. 12.) draw the diameter AB, and draw the diameCCCCXLIV. ter DE perpendicular to AB; through E and P draw the straight line EPF, meeting the circumference in F. Make FG equal to FD; through E and G draw the 4 and at any given distance from that pole. 1. When the pole of the less circle is in the centre of the primitive; then from the centre of the primitive, with the semitangent of the distance of the given circle from its pole, describe a circle, and it will be the projection of the less circle required. 2. If the given pole is in the circumference of the primitive, from C (fig. 14.) the centre of the primitive, Fig. 14set off CE the secant of the distance of the less circle from its pole P; then from the centre E, with the tangent of the given distance, describe a circle, and it will be the less circle required. Or, make PG, PF each equal to the chord of the distance of the less circle from its pole. Through B, G draw the straight line BGD meeting CP produced in D: bisect GD in H, and draw HE perpendicular to GD, and meeting PD in E; then E is the centre of the less circle. 3. When the given pole is neither in the centre nor circumference of the primitive. Through P (fig. 15.), Fig. 19 the given pole, and C the centre of the primitive, draw the diameter AB, and draw the diameter DE perpendicular to AB; join EP, and produce it to meet the primitive in p; make pF, pG, each equal to the chord of the distance of the less circle from its pole; join EF which intersects the diameter AB in H; from E through G draw the straight line EGI, meeting the diameter AB produced in I; bisect HI in K: Then a circle described from the centre K, at the distance KH or KI, will be the projection of the less circle. PROPOSITION XIII. PROBLEM V. |