the

great distance of the sun, we may conceive it to revolve about zr in the same manner as about PO, and consequently the shadow will be projected upon the plane abcd, in the same manner as the shadow of PO is projected upon the plane HKRV, and therefore the hour angles are calculated by the same proportion. This is a

horizontal dial.

106. Now, let NLzK be a great circle perpendicular to PRpH, and consequently perpendicular to the horizon at z, and the side

L

H

K

R

N

=

next to H is full south. Then, for the same reason as before, if the angles Np1, Np2, &c. be 15°, 30°, &c. the shadow of po will be projected into the lines 01, O2, &c. at 1, 2, &c. o'clock, and the angles NO1, NO2, &c. will be measured by the arcs N1, N2, &c. Hence, in the right angled triangle pN1, pN: the complement of the latitude, and the angle Np1=15°; therefore (Trig. Art. 210), rad. : tan. 15° : : sin. PN: tan. N1; in the same manner we find N2, N3, &c. Hence, for the same reason as for the horizontal dial, if zabc be a plane, coinciding with NLzK, and st be parallel to Op, st will project its shadow in the same manner on the plane zabc as Op does on the plane NLzK, and therefore the hour angles from the 12 o'clock line are computed by the same proportion. This is a vertical south dial. In the same manner the shadow may be projected upon a plane in any position, and the hour angles calculated.

107. In order to fix a horizontal dial, we must be able to tell the exact time of the sun's coming to the meridian; for which purpose, find the time (92) by the sun's altitude when it is at the solstices, that being the best time of the year for the purpose, because then the declination does not vary, and set a well-regulated watch to that time; then, when the watch shows 12 o'clock, the sun is on the meridian; at that instant, therefore, set the dial, so that the shadow of the gnomon may coincide with the 12 o'clock line, and it stands right.

108. Hence, we may easily draw a meridian line upon a horizontal plane. Suspend a plumb line so that the shadow of it may fall upon the plane, and, when the watch shows 12, the shadow of the plumb line is the true meridian. The common way

is to describe several concentric circles upon a horizontal plane, and in the centre to erect a gnomon perpendicularly to it, with a small round well-defined head, like the head of a pin; make a point upon any one of the circles where the shadow of the head falls upon it in the morning, and again where it falls upon the same circle in the afternoon; draw two radii from these two points, and bisect the angle between them, and the bisecting line will be a meridian line. This should be done when the sun is at the tropic, when it does not sensibly change its declination in the interval of the observations; for, if it do, the sun will not be equidistant from the meridian at equal altitudes. But this method is not capable of very great accuracy; for, the shadow not being very accurately defined, it is not easy to say at what instant of time the shadow of the head of the gnomon is bisected by the circle. If, however, several circles be made use of, and the mean of the whole number of meridians so taken be drawn, the meridian may be found with sufficient accuracy for all common purposes*.

109. To find whether a wall be full south for a vertical south dial, erect a gnomon perpendicularly to it, and hang a plumb line from it; then when the watch, as above adjusted, shows 12, if the shadow of the gnomon coincide with the plumb line, the wall is full south.

* Several methods of drawing meridian lines with ease and accuracy may be seen in a small tract, with the above title, written by Andrew Ellicott, A. M., pub

lished in 1796.-AMER. EDITOR.

CHAPTER III.

TO DETERMINE THE RIGHT ASCENSION, DECLINA. TION, LATITUDE, AND LONGITUDE OF THE HEAVENLY BODIES.

Art. 110. THE foundation of all astronomy is to determine the situation of the fixed stars, in order to find, by a reference to such fixed objects, the places of the other bodies at any given time, and thence to deduce their proper motions. The positions of the fixed stars are found from observation, by knowing their right ascensions and declinations (41); and these are found by means of the transit telescope and astronomical quadrant, as explained in my Treatise on Practical Astronomy; and then, by computation, their latitudes and longitudes may be found.

111. As the earth revolves uniformly about its axis, the apparent motion of all the heavenly bodies, arising from this motion of the earth, must be uniform; and as this motion is parallel to the equator (76), the intervals of the times, in which any two stars pass over the meridian, must be in proportion to the arc of the equator intercepted between the two secondaries passing through them, because (13) this arc of the equator contains the same number of degrees as the arc of any small circle parallel to it, and comprehended between the same secondaries; and therefore, if one increase uniformly, the other must. Hence, the right ascension of stars passing the meridian at different times, will differ in proportion to the difference of the times of their passing, that is, if one star pass the meridian 1 hour before another, the difference of their right ascensions is 15°. Hence, if the clock be supposed to go uniformly, we have the following rule: As the interval of the times of the succeeding passages of any one fixed star over the meridian: the interval of the passages of any two stars :: 360°: their difference of right ascensions*. By the same method we may find the difference of right ascensions of the sun or moon, when they pass the meridian, and a star, and, therefore, if that of the star be known, that of the sun or moon will; which conclusion will be more exact, if we compare them with several stars, and take the mean.

* A small correction must here be applied for the aberration of the star, in or. der to get the true difference of right ascensions, as will be explained; because there is a small difference between the true and apparent places.

112. Now, to determine the right ascension of a fixed star, M. Flamstead proposed a method, by comparing the right ascension of the star with that of the sun when near the equinoxes, the sun having the same declination each time; and as this method has not been noticed by any writers, we shall give an explanation. Let AGCKE be the equator, ABCWE the ecliptic, S the place of the star, Sm a secondary to the equator, and let the sun be at P, near to A, when it is on the meridian, and take CTPA, and draw PL, TZ perpendicular to AGC, and ZL is parallel to AC, and the sun's declination is the same at Tas at P. Observe the meridian altitude of the sun when at P, and also the time of the passage of its centre over the meridian;

observe also at what time the star passes over the meridian, and then (111) find the apparent difference Lm of their right ascensions. When the sun approaches near to T, observe its meridian altitude for several days, so that on one of them, at t, it may be greater, and on the next day, at e, it may be less than the meridian altitude at P, so that in the intermediate time it must have passed through T; and, drawing tb, es perpendicular to AGCE, observe, on these two days, the differences bm, sm of the sun's right ascension and that of the star; draw also so parallel to Zo. Then, to find Zb, we may consider the variation both of the right ascension and declination to be uniform for a small time, and consequently to be proportional to each other; hence, vb (the change of meridian altitudes in one day): ob (the difference of the meridian altitudes at t and T, or the difference of declinations) :: sb (the difference of sm, bm found by observation) : Zb, which, added to bm, or subtracted from it, according to the situation of m, gives Zm, to which add Lm, or take their difference, according to circumstances, and we get ZL, which, subtracted from AĞC, or 180°, half the remainder will be AL, the sun's right ascension at the first observation, to which add Lm, and we get the star's right ascension at the same time. Instead of finding iZ, we might have found sZ, by taking TZ-es for the second term, and thence we should have got Zm. Thus we should get the right ascension of a star, upon supposition that the position of the equator had remained the same, and the apparent place of the

-E

star had not varied in the interval of the observations. But the intersection of the equator with the ecliptic has a retrograde motion, called the Precession of the Equinoxes; also, the inclination of the equator to the ecliptic is subject to a variation, called the Nutation; and, from the aberration of the star, its apparent place is continually changing; these must therefore be allowed for, by considering how much they have varied in the interval of the observations; but these are not subjects to be treated of in an elementary treatise. Having thus determined the right ascension of one star, that of the rest may be found from it (111).

113. The practical method of finding the right ascension of a body from that of a fixed star, by a clock adjusted to sidereal time* is this: let the clock begin its motion from Oh. O′ 0′′, at the instant the first point of aries is on the meridian; then, when any star comes to the meridian, the clock will show the apparent right ascension of the star, the right ascension being estimated in the time, at the rate of 15° for an hour, provided the clock is subject to no error, because it will then show, at any time, how far the first point of aries is from the meridian. But as the clock is necessarily liable to err, we must be able, at any time, to ascertain what its error is, that is, what is the difference between the right ascension shown by the clock, and the right ascension of that point of the equator which is at that time on the meridian. To do this, we must, when a star, whose apparent right ascension is known, passes the meridian, compare its apparent right ascension with the right ascension shown by the clock, and the difference will show the error of the clock. For instance, let the apparent right ascension of Aldebaran be 4h. 23' 50" at the time when its transit over the meridian is observed by the clock, and suppose the time shown by the clock to be 4h. 23′ 52", then there is an error of 2" in the clock, it giving the right ascension of the star 2" more than it ought. If the clock be compared with several stars, and the mean error taken, we shall have, more accurately, the error at the mean time of all the observations. These observations being repeated every day, we shall get the rate of the clock's going, that is, how fast it gains or loses. The error of the clock, and the rate of its going, being thus ascertained, if the time of the true transit of any body be observed, and the error of the clock at the time be applied, we shall have the right ascension of the body. This is the method by which the right ascension of the sun, moon, and planets are regularly found in observatories.

* A clock is said to be adjusted to sidereal time when it is adjusted to go 24 hours from the time a fixed star leaves the meridian till it returns to it, or it is the time of a revolution of the earth about its axis.