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same unhappy condition as heretofore; though, on thedeath of the emperor Rodolph, he was nominated tocontinue in the office of imperial mathematician underhis successor Matthias ; and, at the same time, was ap-pointed to a professorship in the university of Linz , inlf)12. But, in addition to his pecuniary embarrass-ments, he was at this time labouring under domesticaffliction, in the loss, first, of a son, and then of hiswife: he seems, however, with his usual elasticity ofspirits, to have soon recovered from the blow, and wefind him giving a minute, and highly characteristic andamusing account, of his choice of a second wife.

His new residence at Linz was not long undisturbed ;he became involved in some disputes with the Romancatholic priests, apparently connected with the tenet oftransubstantiation; but it does not exactly appear howfar the question had any reference to his philosophicalspeculations: it ended, however, in his being excom-municated. He says, in one of his letters, “ The priestand school inspector have combined to brand me withthe public stigma of heresy, because in every questionI take that side which seems to me to be consonantwith the word of God .” Whatever the particular pointin dispute may have been, the real and unpardonableoffence probably was, that he had dared to think forhimself.

It was at Linz that his thoughts were particularlydirected to geometrical investigations; and, from theaccidental circumstance of noticing the blunders of anignorant gauger in measuring his wine casks, he was ledto investigate the subject of the mensuration of solids :this gave rise to some speculations, which afterwardsassumed a more peculiar and prominent interest.

Kepler published, in 1615, a tract on Stereometry ,in which the measurement of the contents of manysolids was proposed, which had not as yet fallen underthe notice of mathematicians : such, for example, werethe solids generated by the revolution of a curve, notabout its axis, but about any line whatever. This in-