14

confidently to predict the return of the same in 1758: “ Undeausint ejusdem reditum fidenter praedicere anno soil. 1758.” Inthe first English translation this is softened into “ I dare ventureto foretell.” As time advanced, some anxiety seems to have comeon him, lest he should not have sufficiently established his greatconclusion, and in 1715 the expression is reduced to “I think Imay venture to foretell.” It has been mentioned that he hadoriginally noted the returns in 1682, 1607, 1531, and 1456; behad by this time likewise remarked that a comet had been seenin 1305, which gave him the addition of a double interval, andhe afterwards found that a similar phenomenon had been noticedin 1380: he had therefore then six e consecutive appearancesat similar distances of time; he therefore in his second editionagain recovers his confidence, and says, “ priorem sententiampaulo audentius tueri ccepi:” and, to verify his hypothesis, beundertook to calculate the motions in an elliptical orbit.

If Halley had had from given observations to find directly anellipse which would answer to the places of the comet, he wouldmost probably have then been unable to succeed. In very ec-centric orbits a small error in the assumed places near the apsiswill make a great variation in the curve of motion. Lexell foundthe periodic revolution of the comet of 1769 to be between 449and 519 years; but M. Pingre from some observations deducednearly the same quantity, while from others his time was ex-tended to 1231 f : a revolution of 2279 years came out for thecomet of 1773, and Mr. Lexell justly concluded that it was im-possible to determine its orbit: “la plus legere erreur dans lesobservations pouvant en occasioner une tres considerable dans laduree de la revolution?.” Halley had not, indeed, in the pre-sent instance, to contend with one of these extreme cases, but itmay on the other hand be conceived how greatly his difficultieswere increased, both by the roughness with which astronomicalobservations in former times were too often made, and by thewant of rules for conducting the calculations. His previous in-quiries had however most fortunately inverted the problem forhint. He had not from the places observed in a single appear-

e Pingrd(Cometographie,vol. II. p. 133) have conjectured that several additionsextends these returns to 1230, 1155,1080, might even he made to this list.

1006, and possibly to 930 (vol. I. p. 354) f Pingri?, vol. II. p. 85.

and 855 (vol. I. p. 347). Other writers g Ibid. p. 93.