13

though those parts which are retained exhibit no alteration in thenumbers which they contain. In the general Table for ParabolicMotions there are some corrections, and two new columns of dif-ferences are introduced, which required no great labour for theircomputation, hut which were very advantageous additions: theynot only facilitated the use of the table, but formed a check forsecuring its accuracy.

In the first edition Halley had said “hactenus nullus ex ob-servatis parabolae leges respuit,” but in the second this sentenceis qualified and altered to “nullus ex hactenus observatis motusparabolici leges quoad sensum reeusat.” This modification wasmade with reference to a great and essential improvement in thelatter part of the treatise. For in examining the comets whichappeared to return, the true elliptical theory is there applied totheir motions, instead of the approximations that had been atfirst obtained from the parabola.

The second edition of the Principia was published in 1713,and Newton notices in it d the strong similarity which had beenfound in the elements of the comets of 1682 and 1607. Hepoints out also that if they should he the same “ haud difficilefuerit orbent ellipticum cometac hujus determiuare.” This ex-pression evidently indicates that the determination had not thenbeen made, for if it had, Newton would certainly have been in-formed of it. There is reason indeed to conclude, as we shallsee hereafter, that Halley undertook the work at a subsequent,but not very distant, period; for he had entered upon the inves-tigation, though he had not completed it, in 1715, about two orthree years before he finally committed the results to the press.

Carrying ourselves back to the time when the discovery of somagnificent a fact was breaking on his mind, it is interesting toendeavour if possible to trace the course of the author’s feelings.Halley having summed up his arguments on the comet of 1682,says, in the first glow of exultation, From hence I can undertake

dus ascend., Inclin. orbit® and Perihelion logarithm of mean motion, and the 9thin orbe; the 5th and 6th of 1705, contain- giving the distance of the perihelion froming the longitude and latitude of the pe- the node (or the difference of the num-rihelia, are omitted; the 7th, 8th, and bers in the 2d and 4th). The direct or9th, Distantia perih. a sole, Log. dist. retrograde motion of each comet is like-perih. a sole, and Temp , aequat. perih. wise marked, for which it has been men-Londini, are the 5th, Gth, and 8th of the tioned that there was not room in thesecond edition; and two new columns are page of the Phil. Trans,introduced into it, the 7 th containing the d p. 400.