1.54

HISTORICAL VIEW OF THE

eluded solids formed by the revolution of an arc of acurve about its chord, which are precisely of the formof casks and other vessels, so that the investigation hadan immediate practical object as supplying rules forgauging. Kepler proposed a vast number of varietiesof such solids for the consideration of geometers ; hewas himself unequal to the task of solving more than avery small number of the simplest of such cases : but,in doing this, though he had recourse to what was inreality the method of limits, yet he devised means ofavoiding much of the tediousness of the ancient processesby the introduction of the phraseology of infinitely smallquantities. He conceived a circle to be made up of aninfinite number of triangles, having their common vertexat the centre; and their infinitely small bases, portionsof the circumference : this was, in fact, the same thingas the method before mentioned of Archimedes , andassigned in the same way the area.

In 1619, Kepler reprinted his “ Mysterium Cosmo-graphicumand, nearly at the same time, publishedhis “ Harmonics.” We might, indeed, have been sur-prised at his thus deliberately re-editing his juvenilereveries, did not the last-named work of his matureyears fully equal them in extravagance. Of its verymiscellaneous contents, we shall not attempt even anenumeration, but shall only take occasion to notice howstrangely some bright flashes of real genius and philo-sophical truth break out here and there in the midst ofthe chaos. In the course of a speculation on the reasonwhy the heptagon is not employed in the actual con-struction of the universe, the author drops the hint, that“ the root of an equation which cannot be accurately-found, may yet be found within any degree of approx-imation by an expert calculator.” He expounds hisnotions of astrology; and, while he strongly condemnsthe absurdities of the vulgar belief, attempts to sub-stitute a system of celestial influences, in which he se-riously represents the earth as an enormous living animal,the tides being its act of respiration, and its vital sym-