PHYSICAL AND MATHEMATICAL SCIENCES. 189

physical knowledge. In the age of which we are nowtreating, when great changes had begun to operate onthe characters of the different branches, and on the re-lation hetw’een them, we find the progress of each muchmore closely dependent on that of the other. In propor-tion as the combination of mathematics with physical re-search w r as more generally recognised, the advances ofthe latter created a necessity for some material im-provements in the former: and this necessity, once felt,was the most effectual means of calling forth the re-sources of mathematical genius, to supply those me-thods which the exigencies of physical investigationdemanded.

The discovery of Kepler’s laws had exhibited astriking instance of the connection of phenomena bynumerical relations ; but the discovery had involvedan amount of labour, in the mere work of calculation,almost beyond belief. If we look only at the successfultheories of the author, the mass of figures throughwhich he worked to establish them was incredible; andthese theories only presented some general laws, towhich the increasing discoveries of astronomy soonadded the necessity of numerous supplementary investi-gations, all alike requiring toilsome and extensive com-putations, not only to verify them in the first instance,but afterwards to turn them to practical account in thevarious applications which were to be founded uponthem.

With such quantities, for instance, as the sines andtangents of the tables, taken only to five or six placesof decimals, so simple a calculation as merely finding afourth proportional is extremely troublesome if it occuroften ; and, for any long series of calculations, occasionsa most intolerable sacrifice of labour and, what is moreserious, of time ; while the extraction of roots andmore complex operations, often repeated, would soonbecome appaEing to the most resolute calculator. Infact, with the methods in use about the beginning ofthe seventeenth century, the single circumstance of the