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cover any parallax, he concluded that it was situated beyond the orbit of the moon, and in the region of the planetary spaces. Tycho observed it from Uraniburg, and Hagecins from Prague, in Bohemia. These two places differ about six degrees in latitude, but are nearly on the same meridian. Both observers compared the place of the comet with the same star; but notwithstanding these advantages, both observations gave the same result, and it was hence concluded that their instruments could determine no parallax of the

comet.

We may see from what has now been stated, that the greatest obstacle to the discovery of truth, is a proper system to regulate our labors. Kepler's result was similar to that of Tycho Brahe. He had an opportunity to observe the return of Halley's comet in 1607, and also the great comet of 1618, and from these observations he concluded that comets move freely through the planetary spaces. Newton, in the third book of the Principia, shows that comets are situated beyond the orbit of the moon, but below the sphere of the fixed stars, and consequently in the solar, or planetary region.

Comets were at first supposed to move in straight lines. Kepler endeavored to represent their observed places by supposing them to move in right lines with variable velocities, and several other astronomers subsequently adopted the same hypothesis; and calculations were made in accordance with it respecting the motions of some comets which had previously appeared. This hypothesis, though unfounded in nature, very nearly represents the observed places of many comets, except in those parts of their orbits, near the perihelion. M. Cassini was in this way able to predict, for some weeks ahead, the places of a comet with considerable precision. Hevelins, a few years later, seems to have been the first to conclude that comets move in curvilinear orbits, and that the orbit might be a parabola. These conclusions he drew from his observations on the comet of 1665. He did not conjecture, however, what pos tion the sun occupied with respect to the curve. This question was not definitely settled till the appearance of the great

comet of 1680, when Newton and Dörfel proved the orbit to be a parabola and the sun to occupy the focus of the

curve.

The observations made by the elder Cassini on a comet which appeared in 1672, and which described a course in the heavens similar to that of the great comet of 1577, observed by Tycho Brahe, first led to the idea that this class of celestial bodies may describe reëntering curves in their motion about the sun, and form a part of the solar system. It was Newton, however, who demonstrated that a comet can move in any one of the conic sections, the curves which it describes depending on its velocity at a given place in its orbit."*

Although angular instruments of some kind were generally used by the better class of astronomers during the Middle Ages, yet observations on comets were sometimes made by estimating their distance from one or more fixed stars by means of the eye alone. When angular instruments were used the observations were frequently very inaccurate, so that it is with considerable difficulty that astronomers of the present day satisfy themselves fully with respect to the identity of recent comets and some that appeared sev eral centuries back.

Upon the revival of scientific studies in Europe towards the close of the fifteenth century, for some time the cultivation of astronomy was almost exclusively confined to Germany. Waltherus, a native of Nuremburg, was the first to introduce the practice of determining the place of a celestial body, by observing its altitude and its distance from two stars whose positions had previously been ascertained. Tycho Brahe, it is well known, had numerous large and accurate instruments with which he made a great many good observations. Hevelius, a wealthy citizen of Dantzic, Prussia, employed instruments similar to Tycho's, but they were more accurately divided. The difficulty of measuring time in those days, induced astronomers, in most cases, to resort to that class of observations which does not employ time as one of the elements. Down to this date (1680) the

*Principia, bk. i., section iii,

best observations reduced the probable error of a position so determined to one minute of arc.

In

After Newton demonstrated the fact that comets may move in conic sections, and that they actually did move in parabolas or very elongated ellipses, he saw the necessity of having some method by which the elements of the orbits can be determined. In order to effect this, he proposed two methods differing in their fundamental positions. * the first he supposes that we may regard a small portion of the orbit as a straight line, described with a uniformn motion, and that its segments intercepted by right lines drawn from the comet to the earth, at the times of the respective observations, are proportional to the intervals of time be tween these observations. The objection to this method is, that it gives results which are altogether indefinite. The second method is unobjectionable in theory, but the computations are so very complicated and laborious that the method has not been generally adopted.

No improvement on Newton's method was made for many years, and mathematicians failing in every attempt to discover a method by which an orbit might be accurately calculated without resorting to so extended numerical computations, were obliged to resort to a system of trial and error, and in this way to arrive at the elements of an orbit that satisfied the observations with sufficient accuracy. "Boscovich, however, undertook to solve this difficult problem directly; and the solution which he gave is remarkable as being the first in which the velocity of the comet in its orbit was regarded as one of the essential conditions. His method, as might be expected, is excessively complicated; and from the multiplicity of both algebraical formulæ and graphical operations which it presents, it has never been employed in the actual computation of an orbit."

Lambert next attempted to solve this problem, and among the formula which he found pertaining to the motion of a body in a parabola, is one that is very ele gant, and which gives the time of describing an arc in

*See Principia, bk. iii., prop. xli.

VOL. XVII.-NO. XXXIV. 23

terms of the chord and the extreme radii-vectores.* Astronomers still employ this theorem in every method in use for the solution of this problem, since it affords the means of correcting the hypothetical values which it is necessary to substitute in the equations of the problem. Lambert further showed how to determine at once whether the distance of the comet from the sun was greater or less than that of the earth, which abridges very much the preliminary calculations. Lambert's method was subsequently modified by the researches of Lagrange and Laplace, and the results to which they finally arrived are now in the main adopted as the basis of all analytical solutions. Dr. Olbers proposed another method for finding the elements of a parabolic orbit also (like the above) from three geocentric observations. He supposed the chord joining the comet's position at the extreme observations to be divided into segments by the radiusvector at the second observation, proportional to the intervals of time between the observations. This method, though simple, gives accurate results only when the intervals of time between the dates of the observations are very nearly equal. When the intervals differ considerably from each other, and when the comet is further from the sun than the earth, the results are very unsatisfactory; and for this reason Legendre devised a method suitable to all cases. Instead of assuming what Olbers does, he finds an expression for each segment of the chord in terms of the time. This method can be used when Olbers' fails, but the numerical calculations are far more complicated and tedious.

Having found all the elements of the orbit of a comet, whether they be elliptical, parabolic, or hyperbolic, the position of the comet in its orbit at any time is easily found, which gives its heliocentric place. An easy calculation transforms the position from heliocentric to geocentric longitude and latitude, or to geocentric right ascension and declination. In practice, however, it is found more convenient to investigate the position of the comet with respect to three rectangular coördinate plans, the equator

* See Tait and Steele "On the Dynamics of a Particle," p. 128.

being one, which will give us the rectangular coördinates of the comet referred to the centre of the sun, and subtracting from these the rectangular coördinates of the earth referred to the same point, we have remaining the geocentric rectangular coördinates, which may at once be transformed into polar coördinates giving us the distance of the comet from the earth, and its geocentric right ascension and declination. The several positions thus computed, being arranged in a table, form an ephemeris of the comet.

Having completed an ephemeris of a comet we can compare it with the observed places of it, and if these agree within the probable limits of error, it is concluded at once that the orbit is sufficiently well represented by a parabola. Should the computed and observed places differ considerably, elliptic elements are tried; and when these fail, hyperbolic elements are substituted. The motions of the great mass of comets are sufficiently well represented by a parabolic orbit, but a few have been found to move in ellipses and a still less number in hyperbolas.

The whole number of recorded appearances of comets, amounts to 787. The earliest is one given as 1770 B. C., but there is some doubt about it. The account is as follows: "St. Augustine has preserved the following extract from Varro:-There was seen a wonderful prodigy in the heavens worthy to be compared with the brilliant star Venus, which Plautus and Homer, each in his own language, call the "Evening Star." Castor avers that this fine star changed color, size, figure, and place; that it was never seen before, and has never been seen since. Adrastus of Cyzicus and Dion the Neapolitan refer the appearance of this great prodigy to the reign of Ogyges.'* This description, such as it is, may be presumed to be that of a comet, but no further particulars have been preserved."+

We must remember that only such comets as were visible to the naked eye, could be seen before the invention of telescopes. Watson gives the following numbers as a

* De Civitate, XXI. 8.

+Chambers' "Descriptive Astronomy," p. 354. Comets, p. 147.

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