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the first day of the former year, and was designated by A, is the seventh day of the second year, and is designated by G; that which was the second, and was designated by B, in the former year, is the first, and is designated by A in the second, and so on. It therefore follows, that whatever letter is the dominical letter in any common year, the letter next preceding it in the alphabet is the dominical letter in the following year; except the former was A, in which case the second is G.
In every common year, the first day of March is the 60th day of the year, and consequently corresponded to the letter D. In bissextile years, on account of the intercalation, the 1st of March is the 61st day of the year; but the letter D was still made to correspond to it, and the letters for the remaining part of the year were arranged accordingly. It therefore follows, that, after the 29th of February, any given day of the week was designated by the letter of the alphabet next preceding that by which it was designated in the first two months. Consequently, a bissextile had two dominical letters, one of which appertained to January and February, and the other, which was the next preceding letter in the alphabet, appertained to the other ten months.
From what has been said it follows, that the dominical letters succeed one another in a retrograde order, that is, in the order G, F, E, D, C, B, A, G, F, f.; and that each bissextile has two
in the same order.
It is now usual to retain only the dominical letter in the calendar, and to designate the other days of the week by numbers or by their names.
391. Determination of the dominical letter. The year 1800, which was a common year, commenced on the fourth day of the week, and, consequently, the dominical letter was the 5th of the alphabet, which is E. From thence, taking into consideration that every four years, in which a bissextile is included, requires five dominical letters in a retrograde order, it is easy to find the dominical letter for any year in the present century. To do this, multiply the number of years above 1800, by 5, and divide the product by 4, neglecting the remainder. Divide the quotient by 7, and subtract the remainder from 5; or from 12, when the remainder is equal to or greater than 5. The last remainder is the
number of the dominical letter. In bissextile years, the dominical letter thus obtained is that for the last ten months of the year. The dominical letter for the first two months is the next following letter in the alphabet.
Delambre, in the 38th chapter of his Astronomy, has given the investigation of a formula for finding the dominical letter in any century, according to the Gregorian calendar.
392. Solar Cycle. The Solar Cycle is a period of 28 years, in which, according to the Julian calendar, the days of the week return to the same days of the month, and in the same order. The first year of the Christian era was the 10th of this cycle. Consequently, if 9 be added to the number of any year, and the sum be divided by 28, the remainder will be the number of the year of the solar cycle. When there is no remainder, the year is the 28th of the cycle.
393. Lunar Cycle. The Lunar Cycle, or, as it is sometimes called, the Metonic Cycle, is a period of 19 years, in which the conjunctions, oppositions, and other aspects of the moon, return on the same days of the year. The synodic revolution of the moon being 29.5305885 days, 235 revolutions are 6939.688 days; which differs only an hour and a half from 19 Julian years. The number by which the year of the lunar cycle is designated, is frequently called the Golden Number.
The first year of the Christian era was the 2d of the lunar cycle. Hence, to find the year of the cycle, for any given year, add 1 to the number of the year, and divide by 19. The remainder expresses the year of the cycle. If nothing remains, the year is the 19th of the cycle.
394. Cycle of the Indiction. The Cycle of the Indiction is a period of 15 years. This period, which is not astronomical, was introduced at Rome, under the emperors, and had reference to certain judicial acts.
To find the cycle of the indiction for a given year, add 3, and divide by 15. The remainder expresses the year of the cycle.
395. Julian Period. The Julian Period is a period of 7980 years, obtained by taking the continued product of the numbers, 28, 19, and 15. After one Julian period, the different cycles of
the sun, moon, and indiction, return in the same order, so as to be just the same in a given year of the period, as in the same year of the preceding period. The first year of the Christian era was the 4714th of the Julian period. Hence, if 4713 be added to the number of a given year, the result will be the year of the Julian period.
396. Epact. The Epact, as an astronomical term, is the mean age of the moon at the commencement of a year, or, in other words, it is the interval between the commencement of the year and the time of the last mean new moon; and is expressed in days, hours, minutes and seconds.
The Epact, as given in the calendar, is nearly the age of the moon at the commencement of the year, expressed in whole days, and was introduced for the purpose of finding the days of mean new and full moon throughout the year, and thence the times of certain festivals. Without entering into any explanation of the reason of the rule, it must suffice here to observe, that the Epact for any year during the present century may be found by multiplying the golden number of the year by 11, adding 19 to the product and dividing the sum by 30. The remainder is the Epact for the
TABLES OF THE ELEMENTS OF THE ORBITS OF THE PLANETS AND OF THEIR MASSES AND DENSITIES.
397. Physical astronomy, in which the principle of universal gravitation is applied to the investigation of the motions of the heavenly bodies, and the various effects of their actions on one another, is a very extensive and, in many of its parts, very difficult department of science. A few propositions of an elementary character, and some general remarks and results, are all that will be here introduced.*
* The celebrated Principia of Newton was the first work on physical astronomy. At the present time, the prominent works on this subject generally, or on the moon S
398. The moon is retained in her orbit by the force of gravity diminished in proportion to the square of the distance from the earth's centre.
Let E, Fig. 56, be the centre of the earth, A a point on its surface, and GH a part of the moon's orbit, assumed to be circular. When the moon is at any point M in her orbit, she would, by the first law of motion, move on in the direction of the line MF, a tangent to the orbit at M, if she was not acted on by some force to turn her aside. Let L be her place in her orbit one second of time after she has been at M, and let LC and LD be drawn parallel to EM and MF respectively; and joining LM, let EI be drawn perpendicular to it, and, therefore, bisecting it in I. The line CL, or its equal MD, is the distance the moon has been drawn, during one second, from the tangent towards the earth at E. Now, as the distance a body moves in a given time is proportional to the force by which it is moved, MD may be taken as a measure of the force by which the moon is drawn towards the earth.
Put g = MD, G = the force of gravity at the earth's surface, or the distance a heavy body falls there, by this force, in a second, EA, the earth's radius, dEM, the moon's distance, p = moon's sidereal revolution in seconds, moon's horizontal parallax, and = 3.14159, &c. Then, assuming that the force MD, or g, is that of gravity diminished in proportion to the square of the distance from the earth's centre, we have,
Now, by similar triangles, we have EM: IM :: LM : MD, or 2EM: 2IM, or LM LM: MD, that is,
2d: LM :: LM : g................
But the chord LM does not sensibly differ from the arc LM, which is the distance described by the moon in one second. Hence, as 2d is the circumference of the moon's orbit, we have,
Substituting the value of LM in (B), it becomes,
in particular, are Laplace's Mécanique Céleste, Pontécoulant's Système du Monde, and Plana's Théorie de la Lune.
Taking the mean values of r, p, and 7,* we easily find the value of G to be 16.22 ft.; which is very nearly equal to its known value as determined by experiment. This conformity of the computed result with that obtained by experiment, may be regarded as establishing the truth of the proposition.
399. The planets are retained in their orbits about the sun, and the satellites in theirs, about their respective primaries, by forces directed in each case to the central body and varying inversely as the square of the distance from that body.
Assuming the planets and satellites to be retained in their orbits by forces directed and varying as stated in the proposition, it is proved, by a series of investigations that we shall omit, that their motions and periods must be in conformity with Kepler's Laws. Hence, as these laws were deduced from observations, and have been fully confirmed by subsequent observations, it follows that the proposition must be true.
400. Determination of the relative masses or quantities of matter in the sun and planets.
For a planet that has a satellite, let D be the mean distance of the planet from the sun, d the mean distance of the satellite from the planet, P and p the periodical revolutions of the planet and satellite respectively, and m the mass of the planet, that of the sun being regarded as a unit or 1.
Also, let ƒ force of gravity of a unit of mass at a unit of
* These are, 2r
41776044 ft., p, in seconds, 2360585, and