ancients, found to be 29.03 years. To find the mean diurnal motion of Saturn, in his epicycle, divide 20520 degrees (=57 circles) by 21551.5 days (=59.0048 years), and the quotient, 0°.95214, will be the movement required; 731.716 degrees divided by 21551.5 days give 0°.033486 for the planet's mean daily motion in longitude; and these two movements are equal to the mean daily motion of the sun about the earth: lastly, dividing 21551.5 days by 57, we have 378.09 days for the time of one mean periodical revolution of Saturn in his epicycle, or the interval between two oppositions. In like manner the periodical times and the mean movements of the other superior planets were found.

In determining the times and movements of the inferior planets Hipparchus considers that, as the centre of the epicycle described by each is always in a line drawn from the earth to the mean place of the sun, the mean movement of the planet in longitude, or the motion of the centre of the epicycle is equal to the mean motion of the sun; that is, to 0°.98563 daily: but, for the movement in the epicycle, he found, as is related by Ptolemy, that the anomalies of Venus were restored five times exactly, in 2919.75 days, and those of Mercury, 145 times in 16802.53 days; that is, the planets returned so many times to their greatest elongations from the sun, on the same side of that luminary, in those periods respectively; and, dividing the periods by the number of restitutions, he obtained 583.95 days and 115.875 days respectively for the periodical revolutions of Venus and Mercury in their epicycles in the interval between two such elongations; or, which was then considered the same thing, the interval between two consecutive inferior, or superior, conjunctions with the sun; then, dividing 360 degrees by the number of days in each of those periodical revolutions, we have, with respect to the sun, the mean daily motions of the planets in their epicycles; which are, for Venus, 0.61649 degrees and, for Mercury, 3.10667 degrees. But the return of an inferior planet two successive times to the point of maximum elongation on the same side, or to the point of like conjunction, is accomplished in a period equal to that of its revolution upon the circumference of the epicycle, together with

the time in which it would describe an arc of that circumference

subtending an angle equal to that of the sun's movement in the said period; for let E (Plate II. fig. 2.) be the earth; S"s", described about E as a centre, be part of the sun's orbit, and a M P be the orbit of the centre of the epicycle VT V'; also, let V' be the place of the planet at its inferior conjunction; then, while the planet has moved on the circumference of the epicycle in retrograde order from V' to v', the point of next inferior conjunction, the point S" has moved to s", having described the arc S" s" simply, if the planet be Mercury, and the arc S"s" together with a complete circumference, if the planet be Venus; in consequence of which, by drawing the line a M'b parallel to V м V', it is evident that V' must have described one circumference of the epicycle together with the arc b v', which is the measure of the angle b M' v', or of its equal S" ES". Now multiplying the number of days in the periodical revolutions above found, which are also the times of describing the arc S" s", by the daily movement of the sun, we have the value of the angle S" ES", or b м'v'; and to this adding 360 degrees, we obtain the angular movements of the planets, in their epicycles, in the times of those revolutions; consequently, by proportion, we get the mean times of the revolutions through the exact circumferences of the epicycles; which are, for Venus 224.71 days, and for Mercury 87.968 days; and, again, dividing 360 degrees by these times, the results will be 1.6021 degrees and 4.0923 degrees, the mean daily movements in the epicycles; these last are evidently equal to the sum of those above, and of the mean daily movement of the sun; and the periodical times just found agree almost exactly with those assigned to the sidereal revolutions of Venus and Mercury by the modern astronomers. The following table exhibits the periodical times and the mean movements of all the planets according to the theory of Hipparchus, but deduced, probably, by Ptolemy himself; for the latter observes that Hipparchus confined his investigations chiefly to the theories of the sun and moon; not having had so many good observations on the planets left to him by the ancients as he left to those who were to follow him.

a

Hipparchus was born at Nicæa, in Bithynia, but both Ptolemy and Theon relate that he made many celestial observations at Rhodes, and it is probable that his principal works were composed at that place. Besides the invention or improvement of trigonometry and the computation of tables for the purpose of facilitating its application in astronomical enquiries, he must have spent many years in making observations: his labours in deducing from them and from those of his predecessors the principal elements of the solar, lunar and, perhaps, of the planetary orbits must also have been immense, and we must add to these important objects the formation of his catalogue of 1080 stars; from all which we shall be justified in concluding that this distinguished philosopher was qualified to rank with the most celebrated of modern Europe. He died about the year 125 Before Christ.

Little seems to have been done in astronomy between the times of Hipparchus and Ptolemy; for we can hardly consider that the science gained any thing by the works of Hypsicles, Theodosius, Menelaus and the first Theon, who lived in that interval, since these are, apparently, only compilations intended for elementary instruction, and scarcely contain any subject which admits of application to practice. We proceed, therefore, to give some account of the works of Ptolemy, to whom the science is indebted for many considerable improvements.

245

XII.]

IMPROVEMENTS, ETC.

CHAPTER XII.

IMPROVEMENTS INTRODUCED BY PTOLEMY.

Instruments invented by Ptolemy.-Sun dials and clepsydra employed for measuring time.- Ptolemy's determination of the precession.-His catalogue of stars. His arguments for the rotundity and immobility of the earth.The obliquity of the ecliptic.—The length of the tropical year. The planetary orbits represented by eccentric deferents and epicycles.-The solar orbit according to Ptolemy the same as that of Hipparchus.-Formation of a table of the equation of time.-Elements of the lunar orbit at the epoch of Nabonassar.-Discovery of the second inequality of the moon's motion.Investigation of its value.-The diameters of the homocentric and eccentric circles, and of the epicycle.-The moon's parallax, and distance from the earth.-Ptolemy's lunar theory erroneous.

THE researches of Ptolemy, which are, chiefly, contained in the Megale Syntaxis or, as it was subsequently called, the Almagest, relate to the positions of the fixed stars, the length of the year, and the elements of the orbits of the sun, moon and planets; so that this work constitutes a general treatise on astronomy, and it is the more interesting to us as it remained the text-book of the schools both in the East and in Europe till the great revolution in the science took place by the promulgation of the system of Copernicus.

The instruments of observation used by Ptolemy or his contemporaries appear to have been of three different kinds: he describes one with which he measured, or proposed to measure, the zenith distance of the sun and the obliquity of the ecliptic, and which seems to have differed little from the meridional armillæ of Eratosthenes; it consisted of two rings, one moveable the interior circumference of the other, in the same upon plane, which was that of the meridian; the exterior circle was graduated in degrees with as many subdivisions as each degree would contain and the other carried two small gnomons, at the extremities of a diameter, which served to form the line of sight; the whole instrument was fixed on a pedestal and its verticity was ascertained by a plumb-line suspended from the top. Pto

lemy, also, describes a quadrant which he says may be made of wood or stone and with a moveable alidade carrying sights; like the meridional armilla it is said to have been intended to measure zenith distances but, probably, its radius was made much larger than that of the last mentioned instrument in order to afford more accurate or more minute subdivisions; and it is evident that these two instruments must have served precisely for the same purposes as our present mural circles and quadrants.

But it seems that neither of these instruments gave the zenith distances of celestial bodies with the degree of correctness which was then thought necessary, and Ptolemy invented another a which the second Theon afterwards designated the parallactic rods: this consisted of a pillar placed vertically on a foot, at the upper extremity was a joint on which turned a long alidade carrying the two sights, and these were circular apertures pierced in two plates, probably of metal; one of the apertures was recommended to be very small, and the other, rather larger than the visible diameter of the moon in perigeo, evidently, in order that, the whole disc of the sun or moon being visible within it, the position of the line of collimation might be nearly free from error: the alidade was kept in its position, when directed to the celestial body, by a third rod which, also, turned upon a joint in some part of the vertical rod and was graduated so as to shew the angle between the latter and the alidade; that is, the zenith distance required: as the graduated rod could not have been very extensive, we may consider that this instrument answered, in some measure, the purpose of one of our zenith sectors.

Another important instrument which was used by Ptolemy, and, probably also, by Hipparchus, was an astrolabe for taking the distances in longitude between the sun and moon, or between the moon and a star. It consisted of two rings fixed at right angles to each other, one in the plane of the ecliptic and the other in that of the solstitial colure; to these were added two other rings whose planes turned about the axis of the eclip