northern axis of the ecliptic; which is to the right hand, if the moon has north latitude ascending, or south latitude descending; otherwise to the left. 549. III. In case the axis of the moon's way, and the earth's axis, are both on the same side of the axis of the ecliptic, the complement of their difference; but when on contrary sides, the complement of their sum to 90o is an angle, which call E. 550. IV. The time that the moon's centre is at E is before the true conjunction, if the earth's axis is to the right hand of the northern axis of the ecliptic, and the moon's latitude at the conjunction north, otherwise after; but if the moon's latitude at conjunction is south, the contrary. 551. V. When the northern axis of the moon's way is to the right hand of the earth's northern axis, the motion in the direction of the earth's axis is from north to south for any time before E, but from south to north for any time after E; otherwise, the contrary. 552. VI. In order to reduce the latitude of the place, and the horizontal parallax of the moon, to the spheroidal figure of the earth, subtract the logarithm of the less from that of the greater axis of the earth (or of any numbers expressing their ratio), the remainder added to the co-tangent of the latitude in the sphere, gives the tangent of an arc; to which add again the remainder, the sum is the co-tangent of the latitude in the spheroid; or, to twice the remainder add the co-tangent of the latitude in the sphere, the sum is the co-tangent of the latitude in the spheroid. 553. From the sine of the said arc subtract the co-sine of the latitude in the spheroid, and to the remainder add the logarithm of the seconds contained in the moon's horizontal parallax in the sphere (being the equatoreal horizontal parallax), the sum will be the logarithm of the moon's horizontal parallax in the spheroid in seconds, or radius of the disc. 554. Take out from the tables the following sixteen logarithms, and place them one under another, as below: Call the co-sine complement arithmetical of the The logarithm of the moon's sidereal horary Add 6.44370 to e, the sum call The co-sine of the angle of the moon's way The arithmet. comp. sine of the angle E To the sine of the angle of the earth s axis add The arith. comp. of the sum of e and f, call The sine of the angle E The sine of the star's declination The co-sine of the latitude of the place in the The logarithm of the seconds in the radius of The sine of the latitude The co-sine of the declination Collect the following sums; the sum of i, k, l, m, is the logarithm of a number of seconds of time before or after the conjunction that the moon's centre is at E, by Rule IV. from which you have the apparent time E. The sum of s, t, u, is a constant logarithm F, the seconds answering thereto G; the sum of h, i, k, is a constant logarithm H, the seconds corresponding thereto I. The sum of n and o is a constant logarithm A B The sum of r and s C D The sum of q, r, and s Now, to find the apparent distance of the centres of the moon and star, reduced to the plane of projection, at any interval of time from E, proceed thus: 555. VII. To the constant logarithms A and B add the logarithm of the assumed interval of time in seconds, the number of seconds answering to the sums call a and b, and to the constant logarithm C add the sine, and to D the co-sine, of the hour from the star's culminating, the number of seconds answering to the sums call c and d; then, if the time is before E, and the hour before the star's culminating, or the time after E, and the hour after culminating, the difference of a and c, otherwise their sum, is a num◄ ber of seconds, which call K. 556. VIII. Take the sum of G and d, when the star's declination and the latitude of the place are of the same denominations; but if of different denominations, the difference; which sum or difference call L. 557. IX. Take the { sum difference } of I and b as the motion in the direction of the earth's axis is from north to south}, by Rule V. and the moon's latitude at the conjunction north; but if south, the contrary; which sum or difference call M: then take the difference of L and M, except when the difference of G and is to be taken for L*, and d is greater than g; in which case take the sum of L and M, unless it should happen at the same time that b is greater than I; and the difference of them is to be taken for *Which is, when the assumed time is nearer twelve hours from culminating than it is to the time of culminating for then, to C and D, in Rule VII. the sine and co-sine of the hour from the star's being on the meridian below the horizon, is added to obtain and d; and then, in Rule VIII. the difference of G and dis taken instead of the sum. M, then the difference of L and M must still be assumed; this sum or difference will be one leg, and K the other, of a right-angled triangle, by which may be found the hypothenuse, which is the apparent distance of the centres at the time required. N.B. For the distance of the centres at the time E, c is equal to K, and I is equal to M. 558. In case equal intervals of time are taken from the time E, a and b will be doubled, tripled, &c. and, therefore, need only be computed once for any number of instants: and the time between the instants converted into degrees, being added to, or subtracted from, the time in degrees of the first instant, reckoned from culminating, or twelve hours from it, as the times recede from, or approach to, the one or the other, will produce the hour angles at their respective instants. 559. When the apparent distance of the centres of the moon and star, approaching to, or receding from, each other, is equal to the semidiameter of the moon, the beginning or immersion, and end or emersion, of the occultation takes place.... In any of these operations, when the indices of the logarithms exceed ten, &c. reject the tens. 560. The chief advantage of this method is, that after the apparent distance of the centres is obtained for any instant of time, as many more as is thought proper may be derived from it, with the greatest facility, by four analogies only; whereas, in the common way, by parallaxes, three times the labour, at least, is requisite to arrive at the same accuracy*. This method is capable (with slight alterations) of being applied with great benefit to the computation of solar eclipses, as may be seen in Dr. Hutton's Miscellanea Mathematica, where the rules were first given by Mr. J. Keech, and accompanied by examples, the results of which agreed exceedingly nearly with the actual observations of Professor Hornsby and Mr. Reeves. CHAPTER XX. On the Transits of Mercury and Venus over the Sun's Disc. ART. 561. IF the orbits of the inferiour planets, mercury and venus, were in the same plane with the orbit of the earth, it is obvious that every time either of these planets was in conjunction with the sun, it would seem in actual contact with the body of the lu inary; or, since the unillumined or darkened side of the planet would be then towards the earth, it would appear like a small dark spot upon the sun's disc. But since the orbit of mercury makes an angle of 7°, and that of venus an angle of 3° 23', with the orbit of the earth (Art. 356.), neither of these planets can pass over, or transit, the sun's disc, unless at the time of conjunction the planet be so near its node, that its geocentric latitude is less than the apparent semidiameter of the sun. Now if we know the time in which a transit happened at either node, we may determine when there will be another transit at the same node, by considering that, before another transit can so happen, the earth must perform a certain number of complete revolutions through its orbit, in very nearly the same period that the planet performs a greater certain number of complete revolutions, which will bring both the earth and inferiour planet in the same situation relatively to the sun. The respective numbers of the revolutions of the earth and |