CHAP. II. June-therefore, on the first of July 1852, or at the time that the moon fulls on or about the first of July, there must be a large eclipse of the moon, visible to all places from where the moon will then be above the horizon; and furthermore, 18 years and 11 days after this, that is, in the year 1870, on the 12th day of July, the moon will be again eclipsed; and, in this way, we might go on for several hundred years, but in time the small variations, which occur at each period, will gradually wear the eclipse away, and another eclipse will as gradually come on and take its place. Elements In the same manner we may look at the calendar for any year, take any eclipse, that is anywhere near either node, and run it on, forward or backward. Let us now return to the eclipse of July 12th, 1851. To decide all the particulars concerning a lunar eclipse we for the com- must have the following data, commonly called elements of the eclipse: putation of lunar eclipses. General directions to obtain the elements of eclipses. 1. The time of full moon. 2. The semidiameter of the earth's shadow. 3. The angle of the moon's visible path with the ecliptic. 4. Moon's latitude. 5. Moon's hourly motion. 6. Moon's semidiameter. 7. The semidiameter of the moon and earth's shadow. To find these elements, the approximate time of full moon is found from Table XI, and the tables immediately connected. For the time thus found, compute the longitude of the sun from Table IV, and the tables immediately connected, as illustrated by examples on page 254. Compute, also, the latitude, longitude, horizontal parallax semidiameter, and hourly motion in latitude and longitude, from the lunar tables, commencing with Table XVI, and following out the computation by a strict inspection of the examples we have given (rules, aside from the examples, would be of no avail); and, if the longitude of the moon is exactly 180° in advance of the sun,. it is then just the time of full if not 180°, it is not full moon; if more than 180°, it is past full moon. moon; It will rarely, if ever, happen that the longitude of the CHAP. II. moon will be exactly 180° in advance of the longitude of the sun; but the difference will always be very small, and, by means of the hourly motions of the sun and moon, the time of full moon can be determined by the problem of the couriers.* The moon's latitude must be corrected for its variation, corresponding to the variation in time between the approximate and true time of full moon. To find the semidiameter of the earth's shadow, where the Rule to find moon runs through it, we have the following rule: the semidia. meter of the To the moon's horizontal parallax, add the sun's, and, from earth's shathe sum, subtract the sun's semidiameter. This rule requires demonstration. Let S (Fig. 53) be Fig. 53. dow. the center of the sun, E the center of the earth, and Pm a small portion of the moon's orbit. Draw p P, a tangent to both the earth and sun; from p and P, draw PE and p E, forming the triangle p E P. By inspecting the figure, we perceive that the three Demonstra angles: SEp+pEP+m E P=180°. Also, the three angles of the triangle, PEp, are, together, equal to 180°; Therefore, SEp+pE P+m EP=P+p+pEP; Drop the angle, pEP, from both members of the equation, and transpose the angle SEp, we then have m EP-P+p-SEp. * Robinson's Algebra-problem of the couriers. tion of the rule. CHAP. II. But the angle, m Ep, is the semidiameter of the earth's What is meant by the shadow at the distance of the moon; SEp is the semidiameter of the sun; P, that is, the angle EPp, is the moon's horizontal parallax; and p is the horizontal parallax of the sun; therefore, the equation is the rule just given.* The angle of the moon's visible path with the ecliptic is alangle of the ways greater than its real path with the ecliptic, and depends, moon's visi- in some measure, on the relative motions of the sun and ble path with the ecliptic. moon. To explain why the real and visible paths of the moon are different, let AB (Fig. 54) be a portion of the ecliptic, and Am a portion of the moon's orbit; then the angle, m AB, is the angle of the moon's real path with the ecliptic. Conceive the sun and moon to depart from the node, A, at the same time, the moon to move from A to m in one hour, and the sun to move from A to 6 in the same time; join b and m, and the angle mbB is the angle of the moon's visible path with the ecliptic, which is greater than the angle mAB; which is the angle of the moon's real path with the ecliptic. On this principle we determine the angle in question. All the other elements are given directly from the tables. * Some writers have directed us to increase this value of the shadow by its one-sixtieth part, but we emphatically deny the propriety of the direction. CHAPTER III. PREPARATION FOR THE COMPUTATION OF ECLIPSES. WE shall now go through the computation in full, that it may serve for an example to guide the student in computing other eclipses. 1851, Six Luna. CHAP. III. tion of a lu- computed. Half Luna. The approx. imate time of fall moon We now compute the sun's longitude, hourly motion, and Sun's lon semidiameter for 1851, July 12, 19 h. 15 m. mean Greenwich time, as follows: 1851 OM. Lon. Lon. Peri. I. II. III. N. 9 8 32 39 9 8 22 24958250 025 648 31 129 454 310 27 gitude com puted, corresponding to the approximate time of full moon. 12 d 19 h 46 49 27 0 0 15m 0 37 2 371 28 19 2 0 485 732 151 677 CHAP. III. We now compute the moon's longitude, latitude, semidiDirection ameter, horizontal parallax, and hourly motions for the same, for comput- mean Greenwich time, as follows: the ing moon's true longitude. Equation FOR THE LONGITUDE. 1. Write out the arguments for the first twenty equations, and find their separate sums. With these arguments enter the proper tables (as shown by the numbers), and take out the corresponding equations, and find their sum. 2. Write out the evection, anomaly, variation, longitude, supplement to node, and the several arguments for latitude, in separate columns, corresponding to the given time, and write the sum of the twenty preceding equations in the column of evection. 3. Add up the column of evection first; its sum will be the corrected argument of evection, with which, take out the equation of evection (Table XXIV), and write it under the sum of the first twenty equations; their sum will be the correction to put in the column of anomaly. 4. Add up the column of anomaly, and the sum will be the moon's corrected anomaly, which is the argument for the equation of the center. With this argument take out the equation of the center from Table XXV, and write it under the sum of the preceding equations, and find the sum of all, thus far. Write this last sum in the column of variation, and then add up the column of variation; which sum is the correct argument of variation, and with it take out the equation for variation from Table XXVI. 5. Add the equation for variation to the sum of all the preceding equations, and the sum will be the correction for longitude, which, put in the column of longitude, and the whole added up, will give the moon's longitude in her orbit, reckoned from the mean equinox. 6. Add the orbit longitude to the supplement of the node, of the equi- and the sum is the argument of reduction to the ecliptic; it nox is some. times called is also the first argument for polar distance. With the argument of reduction take out the reduction from Table XXVII, and add it to the longitude. |