22 Prof. Challis on Newton's 



sition of the two sides of an equation, not only indicates 

 advancement in numerical calculation, but shows also that the 

 Arabians put in practice the above-defined processes (I.), (II.), 

 (III.) for acquiring information by calculation. In fact, the 

 formation of the equation implies, first, the making of an hy- 

 pothesis (putting x for the unknown quantity), thence deri- 

 ving the equation by arithmetical rules from the data of the 

 question, and, thirdly, the possession of some means of solving 

 the equation. The rules of calculation are obtained by means 

 of numerical indications of quantity, it being necessary for 

 that purpose to know the relative magnitudes of the quan- 

 tities. Letters might thence be put in place of numbers in 

 such manner as to generalize all particular instances of the 

 application of the rules ; and the result would be general arith- 

 metic, but not algebra. The essential principle of algebra, as 

 now understood, is to reason according to the arithmetical 

 rules without knowing whether a represents a greater or less 

 quantity than b. The Arabians, in forming and solving equa- 

 tions, do not appear to have put letters for known or given 

 quantities. This step was first taken by Vieta (in the latter 

 half of the sixteenth century), necessitating the use of the 

 signs of operation + and — , and, in logical sequence of the 

 use of such symbols according to ascertained rules, forms of 

 expression which may be called representations of negative 

 quantity and impossible quantity. From this time the science 

 of general algebra made rapid progress, our countrymen 

 Oughtred, Harris, Wallis, and Newton being conspicuous 

 among its promoters. When by their labours the rules of 

 operating with indices and expanding in series were estab- 

 lished, the way was cleared for discovering a new process in the 

 application of calculation for answering questions. This ad- 

 vance was first made by Newton, whether as exhibited in his 

 method of prime and ultimate ratios or that of fluxions. 

 Leibnitz invented the differential calculus, which is the sym- 

 bolic form of the new calculation which is most suitable for 

 general application. By means of the differential calculus 

 equations are formed the answers to which are no longer 

 values of unknown quantities, but forms of unknown functions 

 of variable quantities. 



In the meantime discoveries were made by observation and 

 experiment which called for the application of the new cal- 

 culus. Kepler ascertained by observational astronomy his 

 three famous laws of the planetary motions without being able 

 to assign any reasons for them. Galileo determined experimen- 

 tally (in opposition to the Aristotelians) the laws of the acce- 

 leration of bodies falling towards the earth by the constant 



