PHYSICAL AND MATHEMATICAL SCIENCES. 219

ciples are involved ; and some of his expressions havebeen interpreted as a sort of anticipation of the ideaof prime and ultimate ratios and fluxions; but he didnot pursue the idea.

Cavalieri (born at Milan in 1598) was the friend anddisciple of Galileo, but much more profound in mathe-matics. In his hands the subject took a more regularand systematic form ; and he developed a method bywhich problems involving exhaustions might be solvedwith greater brevity and facility, in a work on “ Indi-visibles,” published in 1635.

The principle on which he sought to avoid the embar-rassments of the ancient methods, was by adopting thephraseology of areas made up of an infinite number ofparallel lines ; solids, of an infinite number of planes;and even lines themselves, of an infinite number ofpoints. Thus, when these series of planes or lines re-spectively were arranged according to a determinate law,and such that their sums could be assigned, the contentsof the solids and areas of the curves were then found.The phraseology was, of course, absolutely at variancewith the first definitions of geometry; but, as we observedbefore in a corresponding case, the question should ratherbe, Is there involved any misapprehension or contradic-tion in the things signified, ? It was, in fact, a way ofspeaking, undoubtedly incorrect, but introduced as aconvenient mode of avoiding circumlocution and pro-lixity. The lines and planes are the limits of theparallelograms and parallelopipeds, which, when offinite magnitude, put together in a diminishing series,form figures whose outline ascends as it were in steps.The limits of such figures are those whose boundariesare continuous lines, straight or curved according tocircumstances. Cavalieri conducted his reasonings in acompendious manner by the adoption of this incorrectmode of speaking: but it was in substance that oflimits; and might have all along been reduced to, andverified by, the method of exhaustions. Indeed, thereis no doubt that he himself took the exact and satisfac-