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could be placed on cither the instruments or the observers ofthose days. The positions of the comet are likewise given byApian in azimuth and altitude, the former of which must havethen been very difficult to observe, and of the latter there isnone even so great as 11 j° from the horizon. It is not thereforewonderful that the elliptical results did not answer beyond a cer-tain similarity, which however was sufficient to confirm Halley ’sconviction of the identity of the comet. By reducing the periodictime in this instance again to 75 years, it was found that a con-sistent approximation might be produced; but he thought itvain to attempt more, when the data were of that description,which admitted of no certainty in the minuter quantities whichmight be deduced from them.

To have shewn in this manner that the phenomena of thethree comets might be produced by motion in an elliptic orbit,which was adapted to the intervals at which they had appeared,would have been no trifling addition to what Halley had pre-viously achieved; but he did not content himself with it. Therewere variations in each set of elements, which might have beenconsidered as objections to his theory ; and by examining thesemore minutely, he confirmed the truth of his conclusions, andadded precision to the details which they involved.

It is known that in consequence of perturbing forces theaphelia of the planets advance, and their nodes recede; whereasthe perihelion and nodes of this comet (if supposed to be thesame) had moved in the contrary direction; but Halley pointsout that the comet’s motion was in each case retrograde, andconsequently, that the same causes; which produced the effectsjust described on the planets, would make its aphelion recede andits nodes advance.

Again it might be objected, that the inclination was differentat each appearance, and that the revolution between 1531 and1607 was more than a twelvemonth longer than that between1607 and 1682. This inequality indeed occurred to him at firstas the greatest impediment to the admission of his theory ; but in1705 he had pointed out the general cause, which might haveaffected the duration of the periodic motion, and he now goesmore particularly into the question. He calculates the numberof days, by which the motion of Saturn may be altered by theattraction of Jupiter, and points out that the effect must be much

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