Fig. 15.

To measure any spherical angle.

1. If the angle is at the centre of the primitive, it is measured as a plane angle.

Stereogra- great circle. If therefore the parallel great circle be phic Progiven, then its poles being found by Prob. III. will be jection of those of the less circle. the Sphere. But if the parallel great circle be not given, let HMIN (fig. 15.) be the given less circle. Through its centre, and C the centre of the pri mitive, draw the line of measures IAHB; and draw the diameter DE perpendicular to it, also draw the straight line EHF meeting the primitive in F; make Fp equal to the chord of the distance of the less circle from its pole join Ep, and its intersection P with the diameter AB is the interior pole. Draw the diameter p CL through E and L, draw ELq meeting the diameter AB produced in q; then q is the external pole. Or thus: Join EI intersecting the primitive in G; join also EH, and produce it to meet the primitive in F; bisect the arch GH in p; from E top draw the straight line EP p, and P is the pole of the given less circle.

ig. 16.

'ig. 17.

ig. 18.

PROPOSITION XIV. PROBLEM. VI.

To measure any arch of a great circle.

1. Arches of the primitive are measured on the line of chords.

2. Right circles are measured on the line of semitangents, beginning at the centre of the primitive. Thus, the measure of the portion AC (fig. 16.) of the right circle DE, is found by applying it to the line of semitangents. The measure of the arch DB is found by subtracting that of BC from 90°: the measure of the arch AF, lying partly on each side of the centre, is obtained by adding the measures of AC and CF. Lastly, To measure the part AB, which is neither terminated at the centre or circumference of the primitive, apply CA to the line of semitangents; then CB, and the difference between the measures of these arches, will be that of

AB.

Or thus Draw the diameter GH perpendicular to DE; then from either extremity, as D, of this diameter, draw lines through the extremities of the arch intended to be measured; and the intercepted portion of the primitive applied to the line of chords will give the measure of the required arch. Thus IK applied to the line of chords will give the measure of AB.

3. To measure an arch of an oblique circle: draw lines from its pole through the extremities of the arch to meet the primitive, then the intercepted portion of the primitive applied to the line of chords will give the measure of the arch of the oblique circle. Thus, let AB (fig. 17.), be an arch of an oblique circle to be measured, and P its pole; from P draw the lines PAD, PBE meeting the primitive in B and E; then the arch DE applied to the line of chords will give the measure of the arch of the oblique circle AB.

PROPOSITION XV. PROBLEM VII.

To measure any arch of a less circle. Let DEG (fig. 18.) be the given less circle, and DE the arch to be measured: find its internal pole P; and describe the circle AFI parallel to the primitive, and whose distance from the projecting point may be equal to the distance of the given less circle from its pole P: then join PD, PE, which produce to meet the parallel circle in A and F. Now AF applied to a

phic Projection of the Sphere.

2. When the angular point is in the circumference of the primitive; let A (fig. 19.) be the angular point, Fig. 19. and ABE an oblique circle inclined to the primitive. Through P, the pole of ABE, draw the line AP p meeting the circumference in p: then the arch Ep is the measure of the angle BAD, and the arch AFp is the measure of its supplement BAF: also p F is the measure of the angle BAC, and p ED that of its supple

ment.

3. If the angular point is neither at the centre nor circumference of the primitive. Let A (fig. 20.) be Fig. 20. the angular point, and DAH, or GAF, the angle to be measured, P the pole of the oblique circle DAF, and p the pole of GAH: then from A, through the points Pp, draw the straight lines APM, A p Ñ, and the arch MN will be the measure of the angle DAH; and the supplement of MN will be the measure of the angle HAF or DAG.

PROPOSITION XVII. PROBLEM IX. To draw a great circle perpendicular to a projected great circle, and through a point given in it. Find the pole of the given circle, then a great circle described through that pole and the given point will be perpendicular to the given circle. Hence if the given circle be the primitive, then a diameter drawn through the given point will be the required perpendicular. If the given circle is a right one, draw a diameter at right angles to it; then through the extremities of this diameter and the given point describe an oblique circle, and it will be perpendicular to that given. If the given circle is inclined to the primitive, let it be represented by BAD (fig. 21.), whose pole is P, and let A be the point through which the perpendicular is to be drawn, then, by Prob. I. describe a great circle through the points P and A, and it will be perpendicular to the oblique circle BAD.

Through a point in a projected great circle, to describe another great circle to make a given angle with the former, provided the measure of the given angle is not less than the distance between the given point and circle.

Let the given circle he the primitive, and let A (fig. 19.) be the angular point. Draw the diameter AE, DF perpendicular to each other; and make the angle CAG equal to that given, or make CG equal to the tangent of the given angle; then from the centre G, with the distance GC, describe the oblique circle ABE, and it will make with the primitive an angle equal to that given.

Fig. 21.

If the given circle be a right one, let it be APB (fig. Fig. 22. 22.) and let P be the given point. Draw the diameter 3 H 2

GH

about n as a pole, describe the great circle EDF, cut- Stereogra ting the primitive and given circle in E and D, and it phic Prowill be the great circle required. jection of the Sphere

Stereogra- GH perpendicular to AB; join GP, and produce it to phic Pro- a; make Hb equal to twice A a: and Gb being joined jection of intersects AB in C. Draw CD perpendicular to AB, the Sphere and equal to the cotangent of the given angle to the radius PC; or make the angle CPD equal to the complement of that given: then from the centre D, with the radius DP, describe the great circle FPE, and the angle APF, or BPE, will be equal to that given.

Fig. 23.

Plate

If APB (fig. 23.) is an oblique circle. From the angular point P, draw the lines PG, PC through the centres of the primitive and given oblique circle. Through C, the centre of APB, draw GCD at right angles to PG; make the angle GPD equal to that given; and from the centre D, with the radius DP, describe the oblique circle FPE, and the angle APF, or BPE, will be equal to that proposed.

PROPOSITION XIX. PROBLEM XI.

Any great circle cutting the primitive being given, to describe another great circle which shall cut the given one in a proposed angle, and have a given arch intercepted between the primitive and given circles.

If the given circle be a right one, let it be representCCCCXLV ed by APC (fig. 24.); and at right angles thereto draw fig. 24. the diameter BPM; make the angle BPF equal to the complement of the given angle, and PF equal to the tangent of the given arch; and from the centre of the primitive with the secant of the same arch describe the arch Gg. Through F draw FG parallel to AC, meeting Gg in G; then from the centre G, with the tangent PF, describe an arch no, cutting APC in I, and join GI. Through G, and the centre P, draw the diameter HK; draw PL perpendicular to HK, and IL perpendicular to GI, meeting PL in L; then L will be the centre of the circle HIK, which is that required.

Fig. 25.

Fig. 20.

But if the given great circle be inclined to the primitive, let it be ADB (fig. 25.), and E its centre: make the angle BDF equal to the complement of that given, and DF equal to the tangent of the given arch, as before. From P, the centre of the primitive, with the secant of the same arch, describe the arch Gg, and from E, the centre of the oblique circle, with the extent EF, describe an arch intersecting Gg in G. Now G being determined, the remaining part of the operation is performed as before.

When the given arch exceeds 90°, the tangent and secant of its supplement are to be applied on the line DF the contrary way, or towards the right; the former construction being reckoned to the left.

Any great circle in the plane of projection being given, to describe another great circle, which shall make given angles with the primitive and given circles.

Let ADC (fig. 26.) be the given circle, and Q its pole. About P the pole of the primitive, describe an -arch m n, at the distance of as many degrees as are in the angle which the required circle is to make with the pri mitive. About Q the pole of the circle ADC, and at a distance equal to the measure of the angle which the required circle is to make with the given circle ADC, describe an arch on, cutting mn in n. Then Then

SCHOLIUM.

It will hence be an easy matter to construct all the various spherical triangles. The reader is, however, referred to the article Spherical TRIGONOMETRY, for the method of constructing them agreeably to this projection; and also for the application to the resolution of problems of the sphere. For the method of projecting the sphere upon the plane of the meridian, and of the horizon, according to the stereographic projection, see the article GEOGRAPHY.

SECTION II.

Of the Orthographic Projection of the Sphere. THE orthographic projection of the sphere, is that in which the eye is placed in the axis of the plane of projection, at an infinite distance with respect to the dia

meter of the sphere; so that at the sphere all the visual rays are assumed parallel, and therefore perpendicular to the plane of projection.

Hence the orthographic projection of any point is where a perpendicular from that point meets the plane of projection; and the orthographic representation of any object is the figure formed by perpendiculars drawn from every point of the object to the plane of projection.

This method of projection is used in the geometrical delineation of eclipses, occultations, and transits. It is also particularly useful in various other projections, such as the analemma. See GEOGRAPHY, &c.

PROPOSITION I. THEOREM I.

Every straight line is projected into a straight line. If the given line be parallel to the plane of projection, it is projected into an equal straight line; but if it is inclined to the primitive, then the given straight line will be to its projection in the ratio of the radius to the cosine of inclination.

Let AB (fig. 27.) be the plane of projection, and Fig. 47. let CD be a straight line parallel thereto : from the extremities C, D of the straight line CD, draw the lines CE, DF perpendicular to AB; then by 3. of xi. of Eucl. the intersection EF, of the plane CF.FD, with the plane of projection, is a straight line: and because the straight lines CD, EF are parallel, and also CE, DF; therefore by 34. of i. of Eucl. the opposite sides are equal; hence the straight line CD, and its projec tion EF, are equal. Again, let GH be the proposed straight line, inclined to the primitive; then the lines GE, HF being drawn perpendicular to AB, the intercepted portion EF will be the projection of GH. Through G draw GI parallel to AB, and the angle IGH will be equal to the inclination of the given line to the plane of projection. Now GH being the radius, GI, or its equal EF, will be the cosine of IGH; hence the given line GH is to its projection EF as radius to the cosine or inclination. COROLLARIES.

Orthographic pro

COROLLARIES. jection of 1. A straight line perpendicular to the plane of prothe Sphere. jection is projected into a point.

Fig. 28.

Fig. 29.

2. Every straight line in a plane parallel to the primitive is projected into an equal and parallel straight line.

3. A plane angle parallel to the primitive is projected into an equal angle.

4. Any plane rectilineal figure parallel to the primitive is projected into an equal and similar figure.

5. The area of any rectilineal figure is to the area of its projection as radius to the cosine of its inclination.

PROPOSITION II. THEOREM II.

Every great circle, perpendicular to the primitive, is projected into a diameter of the primitive; and every arch of it, reckoned from the pole of the primitive, is projected into its sine.

Let BFD (fig. 28.) be the primitive, and ABCD a great circle perpendicular to it, passing through its poles A, C; then the diameter BED, which is their line of common section, will be the projection of the circle ABCD. For if from any point, as G, in the circle ABC, a perpendicular GH fall upon BD, it will also be perpendicular to the plane of the primitive: therefore H is the projection of G. Hence the whole circle is projected into BD, and any arch AG into EH equal to GI its sine.

Let the less circle FIG (fig. 29.) be parallel to the plane of the primitive BND. The straight line HE, which joins their centres, is perpendicular to the primitive; therefore E is the projection of H. Let any radii HI and IN perpendicular to the primitive be drawn. Then IN, HE being parallel, are in the same plane; therefore IH, NE, the lines of common section of the plane IE, with two parallel planes, are parallel; and the figure IHEN is a parallelogram. Hence NE= IH, and consequently FIG is projected into an equal circle KNL, whose centre is E.

COROLLARY.

tance of the parallel circle from the primitive, or the Orthograsine of its distance from the pole of the primitive.

PROPOSITION IV. THEOREM IV.

An inclined circle is projected into an ellipse, whose transverse axis is the diameter of the circle.

phic projection of the Sphere.

1. Let ELF (fig. 30.) be a great circle inclined to Fig. 30. the primitive EBF, and EF their line of common section. From the centre C, and any other point K, in EF, let the perpendicular CB, KI be drawn in the plane of the primitive, and CL, KN, in the plane of the great circle, meeting the circumference in L, N. Let LG, ND be perpendicular to CB, KI; then G, D are the projections of L, N. And because the triangles LCG, NKD are equiangular, CL3: CG':: NK': DK; or EC: CG:: EKF: DK: therefore the points G, D are in the curve of an ellipse, of which EF is the transverse axis, and CG the semiconjugate axis.

3. The eccentricity of the projection is the sine of the inclination of the great circle to the primitive.

Case 2. Let AQB (fig. 31.) be a less circle, incli- Fig. 31. ned to the primitive, and let the great circle LBM, perpendicular to both, intersect them in the lines AB, LM. From the centre O, and any other point N in the diameter AB, let the perpendiculars TOP, NQ, be drawn in the plane of the less circle, to meet its circumference in T, P, Q. Also from the points A, N, O, B, let AG, NI, OC, BH, be drawn perpendicular to LM; and from P, Q, T, draw PE, QD, TF, perpendicular to the primitive; then G, I, C, H, E, D, F, are the projections of these points. Because OP is perpendicular to LMB, and OC, PE, being perpendicular to the primitive, are in the same plane, the plane COPE is perpendicular to LBM. But the primitive is perpendicular to LBM; therefore the common section EC is perpendicular to LBM, and to LM. Hence CP is a parallelogram, and EC-OP. In like manner, FC, DI, are proved perpendicular to LM, and equal to OT, NQ. Thus ECF is a straight line, and equal to the diameter PT. Let QR, DK be parallel to AB, LM; then RO=NQ=DI=KC, and PRXRT EK XKF. But AO : CG :: NO : CI; therefore AÒ3: CG' :: QR': DK'; and EC: CG:: EKF : DK'.

COROLLARIÉS.

4. If

Orthogra- 4. If from the extremities of the conjugate axis of phic Pro- any elliptical projection perpendiculars be drawn (in the jection of same direction if the circle do not intersect the primithe Sphere. tive, but if otherwise in opposite directions), they will intersect an arch of the primitive, whose chord is equal to the diameter of the circle.

Fig. 32.

Fig. 33.

Fig. 34.

PROPOSITION VI. Problem I.

To describe the projection of a circle perpendicular to the primitive, and whose distance from its pole is equal to a given quantity.

Let PA p B (fig. 33.) be the primitive circle, and P, p the poles of the right circle to be projected. Then if the circle to be projected is a great circle, draw the diameter AB at right angles to the axis P p, and it will be that required. But if the required projection is that of a less circle, make PE, PF each equal to the chord of the distance of the less circle from its pole; join EF, and it will be the projection of the less circle required.

PROPOSITION VII. PROBLEM II. Through a given point in the plane of the primitive to describe the projection of a great circle, having a given inclination to the primitive.

1. When the given inclination is equal to a right angle, a straight line drawn throngh the centre of the primitive, and the given point, will be the projection required.

2. When the given inclination is less than a right angle, and the given point in the circumference of the primitive. Let R (fig. 34). be a point given in the circumference of the primitive, through which it is required to draw the projection of a great circle, inclined to the primitive in an angle measured by the arch QP of the primitive.

Through the given point R draw the diameter RCS, and draw GC g at right angles to it. Make the arch

GV of the primitive equal to QP, and draw VA at Orthogra right angles to GC; and in G g towards the opposite phic Preparts of C, take CB equal to AC; then, with the jection of the Sphere greater axis RS, and less axis AB, describe an ellipse, and it will be the projection of the oblique circle required.

3. When the distance of the given point from the primitive is equal to the cosine of the given inclination.

Every thing remaining as in the preceding case; let GV, equal to the given arch QP; then drawing the A be the given point, and AC the cosine of an arch diameter RCS at right angles to ACB, the ellipse described with the given axis RS, AB will be the projection of the inclined circle.

4. When the distance of the given point from the centre of the primitive is less than the semidiameter of primitive, but greater than the cosine of the given inclination.

Let D be the given point, through which draw the diameter ICi; and at the point D draw DL perpendicular to DC meeting the primitive in L; also draw LK, making with LD the angle DLK equal to the complement of the given inclination. Let LK meet DC in K; then will DK be less than DC. On DC as Walker on DK; through H draw a diameter of the primitive P• 159 a diameter describe a circle, and make DH equal to the Sphere, RCS, and describe an ellipse through the points R, D, S, and it will be the projection of the inclined circle.

PROPOSITION VIII. PROBLEM III. Through two given points in the plane of the primitive to describe the projection of a great circle.

1. If the two given points and the centre of the primitive be in the same straight line, then a diameter of the primitive being drawn through these points will be the projection of the great circle required.

2. When the two given points are not in the same straight line with the centre of the primitive; and one of them is in the circumference of the primitive.

Let DR (fig. 34.) be the two given points, of which R is in the circumference of the primitive. Draw the diameters RCS, and GC g, FDH perpendicular to it, meeting the primitive in Gg F. Divide GC, g C, in A, B, in the same proportion as FH is divided in D; and describe the ellipse whose axes are RS, AB, and centre C; and it will be the projection required.

3. When the given points are within the primitive, and not in the same straight line with its centre.

Let D, E (fig. 35.) be the two given points; Fig. 35. through C the centre of the primitive draw the straight lines ID, KE ¿; draw DL perpendicular to I i, and EO perpendicular to K k, meeting the primitive in L, O. Through E, and towards the same parts of C, draw EP parallel to DC, and in magnitude a fourth proportional to LD, DC, OE. Draw the diameter CP meeting the primitive in R, S, and describe an ellipse through the points D and R, or S, and it will also pass through E. This ellipse will be the projection of the proposed inclined circle.

PROPOSITION IX. PROBLEM IV. To describe the projection of a less circle parallel to the primitive, its distance from the pole of the primitive being given.

From

Fig. 36.

Fig. 37.

Fig. 38.

Plate

CCCCXLVI.

circle.

PROPOSITION X. PROBLEM V.

About a given point as a projected pole to describe the projection of an inclined circle, whose distance from its pole is given.

Let P (fig. 36.) be the given projected pole, through which draw the diameter G g, and draw the diameter Hb perpendicular thereto. From P draw PL perpendicular to GP meeting the circumference in L; through which draw the diameter L. Make LT, LK each equal to the chord of the distance of the less circle from its pole, and join TK, which intersects L 7, in Q. From the points T, Q, K, draw the liues FA, QS, KB, perpendicular to G g; and make OR, OS, each equal to QT, or QK. Then an ellipse described through the points A, S, B, R will be the projection of the proposed less circle.

PROPOSITION XI. PROBLEM VI.

To find the poles of a given projected circle. 1. If the projected circle be parallel to the primitive, the centre of the primitive will be its pole.

2. If the circle be perpendicular to the primitive, then the extremities of a diameter of the primitive drawn at right angles to the straight line representing the projected circle, will be the poles of that circle.

3. When the projected circle is inclined to the pri

mitive.

Let ARBS (fig. 36, 37.) be the elliptical projection of any oblique circle; through the centre of which, and C the centre of the primitive, draw the line of measures CBA, meeting the ellipse in B, A, and the primitive in G, g. Draw CH, BK, AT perpendicular to G g, meeting the primitive in H, K, T. Bisect the arch KT in L, and draw LP perpendicular to G g; then P will be the projected pole of the circle, of which ARBS is the projection.

PROPOSITION XII. PROBLEM VII.

To measure any portion of a projected circle, and conversely.

1. When the given projection is that of a great circle.

Let ADBE (fig. 38.) be the given great circle, either perpendicular or inclined to the primitive, of which the portion DE is to be measured, and let Mm be the line of measures of the given circle. Through the points D, E, draw the lines EG, DF parallel to Mm; and the arch FG of the primitive will be the measure of the arch DE of the great circle, and conversely.

2. When the projection is that of a less circle parallel to the primitive.

Let DE (fig. 39.) be the portion to be measured, of the less circle DEH parallel to the primitive. From fig. 39. the centre C draw the lines CD, CE, and produce them to meet the primitive in the points B, F. Then the

intercepted portion BF of the primitive will be the Orthogrameasure of the given arch DE of the less circle DEH. phic Pro3. If the given less circle, of which an arch is to be jection of the Sphere. measured, is perpendicular to the primitive.

Let ADEB (fig. 40.) be the less circle, of which Fig. 40. the measure of the arch DE is required. Through C, the centre of the primitive, draw the line of measures M m, and from the intersection O of the given right circle, and the line of measures, with the radius OA, or OB, describe the semicircle AFGB; through the points D, E, draw the lines DF, EG parallel to the line of measures, and the arch FG will be the measure of DE, to the radius AO. In order to find a similar arch in the circumference of the primitive, join OF, OG, and at the centre C of the primitive, make the angle m CH equal to FOG, and the arch m H to the radius C m will be the measure of the arch DE.

4. When the great projection is of a less circle inclined to the primitive.

Let RDS (fig. 41.) be the projection of a less circle Fig. 41. inclined to the primitive, and DE a portion of that circle to be measured. Through O the centre of the projected circle, and C the centre of the primitive, draw the line of measures Mm; and from the centre O, with the radius OR, or OS, describe the semicircle RGFS; through the points D, E, draw the lines DF, EG parallel to the line of measures, and FG will be the measure of the arch DE to the radius OR, or OS. Join OF, OG, and make the angle m CH equal to FOG, and the arch m H of the primitive will be the measure of the arch DE of the inclined circle RDS.

The converse of this proposition, namely, to cut off an arch from a given projected circle equal to a given arch of the primitive, is obvious.

The above operation would be greatly shortened by using the line of signs in the sector.

It seems unnecessary to insist farther on this projection, especially as the reader will see the application of it to the projection of the sphere on the planes of the Meridian, Equator, and Horizon, in the article GEOGRAPHY; and to the delineation of Eclipses in the article ASTRONOMY. The Analemma, Plate CCXXXV. in the article GEOGRAPHY, is also according to this projection; and the method of applying it to the solution of astronomical problems is there exemplified.

SECTION III.

Of the Gnomonic Projection of the Sphere.

In this projection the eye is in the centre of the sphere, and the plane of projection touches the sphere in a given point parallel to a given circle. It is named gnomonic, on account of its being the foundation of dialling: the plane of projection may also represent the plane of a dial, whose centre being the projected pole, the semiaxis of the sphere will be the stile or gnomon

of the dial.

As the projection of great circles is represented by straight lines, and less circles parallel to the plane of projection are projected into concentric circles: therefore many problems of the sphere are very easily resolved. Other problems, however, become more intricate on account of some of the circles being projected into ellipses, parabolas, and hyperbolas. PROPOSITION