of which 8 M N A is the projection. Therefore (art. 30. Schol.) the arcs on the sphere, which are represented by MN and mn, are equal to one another. If the lines P' M m', P'N n' had been drawn from the exterior pole P', the arc m'n' would have been that which is equal to M N on the sphere.

PROPOSITION XVII.

42. To measure the angle contained by the projections of two great circles of the sphere.

M

P

If the given angle be at the centre of the primitive circle, it may be measured on that circle by the chord of the arc by which it is subtended. If it be at the circumference of the primitive and be contained between that circle and the projection Pms of a great circle which is inclined to it; find the centre R, or the pole p of the circle Pms; then (art. 32. Schol.) CR measured on a scale of tangents, or (art. 34.) cp measured on a scale

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m

R

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E

of tangents and the number of degrees doubled, will give the number of degrees in the angle of which AP m is the projection : or again, cm measured on a scale of tangents, and the number of degrees doubled, will give the complement of that angle.

If the angle be contained between the projections, as P Q S, MQ N of two great circles, the angular point not being at the centre or circumference of the primitive; find the projected pole p of P Q S, and the projected pole q of MQN, and draw the lines Qp E, Q 9 F, meeting the primitive circle in E and F ; then E F measured by a scale of chords will give the value of the angle of which MQ P is the projection. For p and q being the projected poles of P Q S and M QN, an arc of a great circle joining them will measure the angle P Q M between the circles; and arcs of projected great circles joining Q and p, Q and 9, being the projections of quadrants, Q is the projected pole of a great circle passing through p and q. But, by the Scholium in art. 30., and as in the last proposition, E F measures the arc of which one joining p and q is the projection; therefore it measures the angle PQ M.

PROPOSITION XVIII.

43. A line traced on the surface of a sphere representing the earth so as to make constantly a right angle with the meridian circles is either the circumference of the equator or

of some parallel of latitude; but a line traced so as to make any constant acute angle with the meridian circles will be a spiral of double curvature: and if such a curve be represented on a stereographical projection of a hemisphere of the earth, the equator being the primitive circle, the projected curve will be that which is called a logarithmic spiral.

M

For let MN be the plane of the equator, and P its pole; and let Pa, Pb, Pc, &c. be the projections of terrestrial meridians making equal angles a Pb, bực, &c. with one another; those angles being, by supposition, infinitely small. Let ABC &c. represent the projection of the curve line which on the sphere makes equal angles with the meridians: then the arcs AB, BC, &c., being infinitely small, may be considered as portions of great circles of

a

N

the sphere; and consequently, by the principles of the stereographical projection, the angles PAB, PBC, &c. on the paper are also equal to one another. But all the angles at P are by construction equal to one another; therefore all the triangles, considered now as rectilineal, will be similar to one another. Hence

AP : PB:: PB PC, PB PC:: PC PD, &c. Thus all the radii PA, PB, PC, &c. are in geometrical progression, while the angles APB, APC, APD, &c., or the arcs ab, ac, ad, &c. are in arithmetical progression: hence, by the nature of logarithms, the arcs ab, ac, ad, &c. may be considered as logarithms of the radii P B, PC, PD, &c., and the curve line ABCD, &c. may be considered as a logarithmic spiral.

Scholium. The curve on the sphere, or in the projection, is usually called a loxodromic line, and on the earth it is that which would be traced by a ship if the latter continued to sail on the same course, provided that course were neither due east and west, nor due north and south.

44. If a hemisphere be projected orthographically on the plane of the equator, that circle will be the primitive, and its pole will be the centre of the projection; the terrestrial meridians, or the declination circles, the planes of which pass through the projecting point, will be straight lines diverging, as in the annexed diagram,

from the centre or pole P, and the parallels of terrestrial latitude, or of declination, the planes of which are parallel to the plane of projection, will be the circumferences of circles whose radii, or distances from the centre, will evidently be equal to the sines of their distances on the sphere, in latitude or declination, from the pole.

P

45. If a hemisphere be projected orthographically upon the plane of a terrestrial meridian, or of a circle of declination, such circle will be the primitive; the projecting point will be upon the produced plane of the equator in a line passing through the centre of the projection perpendicularly to its plane. The equator AB, and the declination circle PS, which pass through that centre, as well as the parallels of latitude or of declination, being all perpendicular to the plane of projection, will be straight lines; and if, as in the

A

E

Ir n

Q

m C

RNM с

B

figure, the primitive circle represent the solsticial colure, the ecliptic will also be a straight line, as EQ, crossing the equator in C, the centre of the projection, which will then represent one of the equinoctial points. The straight line, PCS, will be the equinoctial colure; and, except this and the primitive circle, the meridians or declination circles, being the projections of circles inclined to the primitive, will (2 Cor. art. 26.) be ellipses.

The distances of the parallel circles from the equator will be evidently equal to the sines of their latitudes or declinations; and the distances CM, CN, &c., cm, cn, &c., of the declination ellipses measured on the equator from its centre, or on each parallel of latitude or declination from its middle point, in the line PS, will be the sines of their longitudes or right ascensions; the radius of the equator and of each parallel being considered as unity.

46. When a portion of the surface of a sphere is projected gnomonically, and the plane of projection is a tangent to the sphere at one of the poles of the equator, the terrestrial meridians, or the circles of declination, and it may be added, every great circle of the sphere, since all pass through the projecting point, are represented by straight lines; the declination circles intersecting each

D

other, as at P, in the centre of the projection. The parallels of latitude or declination being the bases of upright cones of which the projecting point is the common vertex, are represented by circles having P for their common centre; and their radii are evidently equal to the tangents of their distances on the sphere from the pole of the equator.

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M P

7

47. When the plane of projection is in contact with the sphere at some point, as C, on the equator, the terrestrial meridians or circles of declination are represented by straight lines, as P S, M N, &c., since their planes pass through the projecting point; their distances from the centre, C, of the projection being equal to the tangents of the longitudes, or right ascensions, reckoned from that point on AB, which represents a portion of the equator. The parallels of declination, mn, pq, &c. are (art. 28.) hyperbolic curves; and

m

A

n

C

B

N S

their distances from the equator, measured on a declination circle, PCS, passing through the centre of the projection, are equal to the tangents of their latitudes, or declinations, on the sphere.

48. By placing the projecting point at a distance from the surface of the sphere equal to the sine of 45°, the radius of the primitive circle being unity, La Hire has diminished the distortion to which the surface of the sphere is subject in the stereographical projection, and on that account it is more convenient than the latter for merely geographical purposes. The most important circumstance in the globular projection, as it is called, is, that on a great circle whose plane passes through the projecting point, an arc equal in extent to 45 degrees, or half a quadrant, when measured from the pole opposite the projecting point, is represented by half the radius of the primitive circle.

p

M

In proof of this theorem, let the plane of projection pass through the diameter MN perpendicularly to the plane of the paper : let E be the projecting point in the direction of the diameter BA, perpendicular to MN, and at a distance from A, equal to the sine of 45° (= √ }), the semidiameter of the sphere being unity; and let BP be half the quad

B

E

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rantal arc MP B. Draw the line EP cutting MC in p; then cp will be equal to the half of CM, or Cp will be equal to Let fall PR perpendicularly on AB; then the angle PCB, being 45 degrees, PR and CR (the sine and cosine of 45°) are each equal to , and ER = 2 + 1; also EC = √ 1⁄2 + 1. Now the triangles ERP and ECp being similar to one another,

Multiplying both the numerator and denominator of the fraction by 2-1, the value of cp becomes, or cp is equal to half the radius of the primitive circle.

If lines were drawn from E to any other point in the arc M B so as to intersect CM, the distances of the intersections from c would be nearly, but not exactly, proportional to the corresponding arcs on MB; but, for ordinary purposes in geography, it is usual to consider them as such. Therefore, in representing a hemisphere of the earth on the plane of a meridian, the projections of the oblique meridians and of the parallels of latitude, which are respectively at equal distances from one another on the sphere, are usually made at equal distances from one another in the representation. The oblique meridians, and the parallels of latitude which, in the projection, should be portions of ellipses, are usually represented by portions of circles from which, in maps on a small scale, they do not sensibly differ.

49. The conical projection is used only in representing a zone of the sphere; the concave surface of the cone being supposed to be in contact with its surface on the circumference of a parallel of latitude, and the geographical points to be projected upon it by lines drawn from the centre.

V

Thus let the segment APS represent part of any meridian of the sphere, P the pole of the equator, the latter passing through the diameter AB perpendicularly to the paper; and let MN be the radius of the parallel of latitude which is at the middle of the zone to be projected. Let VM produced be a tangent to the circle APBS at the point м, and let it be the side of the cone P which touches the sphere on the circumference of the parallel circle MR. Then, the projecting point being at

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m

N

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G

E

n

с

S

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c, if the planes of the meridians be produced they will cut