531. M. de la LANDE observes, that in the passage of Mercury over the sun in 1782, the aberration retarded the phases by computation, 6'. 34", as will ap pear by augmenting its longitude by 18",8, the aberration at that time, and di minishing that of the sun 20", which is always its aberration. Compute the phases by supposing each body to be at its true place, and at its apparent place at the same time, and the difference shows how much the aberration affects the time. Moreover, when we calculate the true geocentric place of a planet, we must add 20" to the place of the sun in the Tables of its motion, the place of the sun being put down as affected by aberration.

532. By Article 526, the aberration =0,00564 dm, if the earth's distance from the sun be unity; if therefore that distance be represented by 10, the aberration =0,000564 dm, from which the following Table was constructed, to be entered with the distance of the planet from the earth, and the angle described by the planet about the earth in 24 hours, in latitude, longitude, right ascension or declination.

If the distance of the body from the earth be greater than 10, as 37 for instance, find the value for 10 and then multiply it by 3, and to it add the value for 7.

A TABLE

To find the Aberration of a Planet or Comet, in Latitude, Longitude, Right Ascension or Declination.

Ex. Suppose the distance of a comet from the earth to be 43, and its apparent motion in 24 hours to be 2°. 15' in longitude; to find the aberration in longitude.

Enter with the distance 10 and daily motion 2°. 15', and we get 45′′,68, which multiplied by 4 gives 182",7; and by entering with the distance 3 we get 13",7; hence the aberration is 196",4.

To reduce the place of the body computed from the Tables to the apparent place, add the aberration, if the latitude, longitude, right ascension or declination of the body decrease, but subtract, if it increase; and the contrary, to reduce the apparent to the computed place.

CHAP. XXIII.

ON THE PROJECTION FOR THE CONSTRUCTION OF SOLAR ECLIPSES.

Art. 533. AS the ecliptic is inclined to the equator and cuts it in two oppo-
site points, the sun keeps continually approaching to one pole and receding
from the other by turns, and therefore to a spectator at the sun, the poles must
appear and disappear by turns. When the sun is on the north side of the
equator, the north pole must appear; and when on the south side, the south
pole. When the sun is in the equator, the plane of illumination is perpendicu-
lar to the equator, and consequently the poles will lie in the circumference of
the circle of illumination; when the sun comes to the tropic, the pole will ap-
pear in the middle of its path over the circle of illumination; and when the
sun comes to the next equinox, the pole will appear on the other side of the
circle of illumination. When the sun gets on the other side of the equator,
this pole will disappear, and the other will appear in like manner. Hence, to
a spectator at the sun, the apparent motion of the pole P is the same as if the
axis Pp of the earth had an annual conical motion PrQs, pnqm about an axis
GOF perpendicular to the ecliptic EOC, the angle POG being equal to the
greatest declination of the sun. As these circles PrQs, pnqm are parallel to
the ecliptic, their planes will pass through the sun, and therefore to a spectator
at the sun the apparent motion of the poles will be in the straight lines PQ, pq ;
and as P moves as fast in the circle PrQs as the sun does in the ecliptic, if P
be the place of the pole at the equinox, and we take the arc Po equal to the
sun's distance from that equinox, and draw vo perpendicular to PQ, o will be
the apparent place of the pole at that time. It is manifest that Pv may be set
off upon any circle described on PQ. Hence also, the angle which the axis
Ow makes with the plane of illumination must be equal to the declination of
the sun.
As this apparent motion of the pole over the enlightened disc of the
earth is caused by the motion of the earth in its orbit, the motion of the pole
over the disc will be in a direction contrary to the diurnal motion of the disc;
if therefore P be the position of the pole at the vernal equinox, and PrQ be its
motion over the disc of the earth to the next equinox, the diurnal motion of
the disc will be made in the contrary direction.

534. When the sun, and consequently the spectator who is supposed to be at the sun, is in the equator, the spectator being in the plane of the equator, and, as to sense, in the plane of all the circles parallel to it, they will all appear to be

FIG.

projected upon the circle of illumination into right lines parallel to each other. But when the sun, and consequently the spectator, is out of the equator, the equator, and all the circles parallel to it, being seen obliquely, will appear to be projected into ellipses upon the plane of illumination, as the eye may be considered at an infinite distance; and as the eye has the same relative situation to all these circles, the ellipses must be all similar. When the sun is on the north side of the equator, that part of the ellipse which is the projection of that part of the circle which lies between the north pole and equator on the enlightened hemisphere will be concave to the pole; but when the sun is on the other side of the equator, that part will be convex. That is, let P be the 117. north pole on the enlightened hemisphere, the sun being on the north side of the equator, and vxyz, ambn, the ellipses into which the equator and any parallel to it are projected; then amb is that part of the ellipse which the place on this parallel describes in the day, and the other part bna is that which is described in the night; and the place is at m at 12 at noon, and at ʼn at 12 at midnight. In this case, the other pole p must be considered as being on the other, or dark side of the earth. But if P be supposed on the dark side, and consequently p on the light side, or if the sun be on the south side of the equator, n will be 12 at noon, and m will be 12 at midnight. For if Pp be 118. the axis, LN the plane upon which the circle ab is to be projected, E the sun on that side next to the north pole; then drawing Eam, Enb, the point a, answering to noon, the sun being on the meridian, is projected at m, and the point b, answering to midnight, is projected at n; but when the sun is on the other side of ab, as at e, a is projected to n' and b to m', therefore n' represents noon and m' midnight. On account of the great distance of the sun compared with the radius of the earth, the lines Ea, Eb, and ea, eb may be considered as parallel, and therefore the circle ab is orthographically projected upon the plane LN into an ellipse, whose minor axis is mn, or m'n'.

FIG.

FIG.

119.

535. The next thing to be done is to determine the magnitude of the ellipse into which the circle ab is projected, and its position upon the plane of illumination. Let Pp represent the axis of the earth, asbt a circle of latitude to any place, IPNp the meridian passing through the sun, and LON the plane upon which the projection is made; then (533) the angle LOP is equal to the sun's declination; draw am, bn, vr perpendicular to LO, and (534) mn is the minor axis of the ellipse; let us be that radius of the circle ab which is parallel to the plane of projection, and it will be projected into a line equal to itself, and consequently it will be the major axis; hence, 2vs, or 2va, or 2 cos. lat. is the major axis of the ellipse; but mm (the projection of ab upon LN) : ab:: sin. mab, or POL the dec.: radius; that is, the axis major axis minor:: rad. : sir. declination. And to find the distance Or from the center of projection to the center of the ellipse, we have, rad.1: cos. Or the dec.:: 0 Or

: