HISTORY OF PHYSICAL ASTRONOMY.

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bounded by 263° 8' and 358° 41' was now wholly included between 297°and 333°.

It only now remained for him to compute the values of the mass, theeccentricity, and the longitude of the perihelion of the disturbing planet,for a number of particular values of the epoch contained between thoselimits which he had found to include all the admissible values, and thento determine the corresponding errors of the theory in 1690 and 1747.This calculation he executed for a great number of equidistant values ofthe epoch, but he was mortified to find that the errors were in all casesso considerable that they could not be accounted for by any probableerrors of observation, and the conclusion seemed to be inevitable, that itwas absolutely impossible to represent the irregularities of Uranus by thehypothesis of a disturbing planet. Thus it appeared to him that he wasall the while engaged in pursuing a phantom, or, to use the words of theillustrious Kepler on a similar occasion, “ all his labours vanished insmoke.”*

Notwithstanding this unexpected conclusion, Le Verrier was still reluctant to abandon the hypothesis of an exterior planet; for, in such abstruseand complicated inquiries, there may exist just reasons for supposing thatthe final results have been influenced in an inordinate degree by somehidden cause, depending either on the method of solution or on the na-ture of the problem itself. He did not therefore despair of rendering histheory consistent with observation by the detection of some peculiarity ofthis kind; nor did his genius fail to come eventually to his aid in thisperplexing emergency. He discovered, in fact, that the form of the ana-lytical expression for the mass was such, that a very small error committedin the observations of Uranus would affect, to an enormous extent, its nu-merical value corresponding to any given value of the epoch; whence itfollowed that the errors of the theory in 1690 and 1747, which had beencomputed by means of the values of the mass, and the other elements ofthe disturbing planet obtained by supposing the observations to be abso-lutely correct, had not been fairly represented by the results relative tothose quantities at which he had finally arrived. It appeared to him,therefore, that no legitimate conclusion could be deduced from the equa-tions of condition until the supposition of possible errors in the observa-tions was introduced into them. He now resumed the considerationof the three mean equations of 1758, 1793, and 1828, assuming that allthe observations which entered into their composition were affected withindeterminate errors. By a simple inspection of their forms he discoveredthat these equations could only be affected to a sensible extent by the errorsof the observations of 1715 and 1775 f. In addition, therefore, to the fourunknown quantities, relative to the disturbing planet, each equation con-tained these two errors represented by appropriate symbols. He now re-solved to assign a succession of values to the epoch in the two equations of

* Itaque causae physicse, cap. xlv. in furnos abeunt, De Motibus Stella Martis, cap. lv.Such are the terms in which Kepler announces the failure of his oval theory to accountfor the motions of Mars , a theory to which he long continued to cling with the most abso-lute conviction of its truth, and upon which he expended an almost incredible amount ofingenious reasoning and toilsome calculation. While pondering in great perplexity onthe cause of the failure, a happy inspiration of his genius revealed to him the grand truththat the orbit of the planet is an ellipse.

+ The errors of observation for 1810 and 1845 were also similarly calculated to"affect the equations; but, as they were in all probability very small, Le Verrier consideredthat he might safely dispense with taking them into account.