PHYSICAL AND MATHEMATICAL SCIENCES. 301

these ratios of the infinitely small elements the differ-ential coefficients.

There was, however, in any point of view, less dif-ficulty in the first conception of the idea than in showingits generality, and in proving that it applied to allsorts of functions. Both its inventors gave a completesystem of such rules. This constituted the direct, thefluxional, or the differential calculus: the process neces-sary for expressing the limiting ratio would be more orless long and difficult, according to the nature of theparticular function ; but in all cases it could be appliedwithout any inherent obstacle. There was, however,the reverse process to be considered and provided for;that is, what is practically the most important, whenexpressions are given involving fluxions or differentialsmixed up with other terms which are combinations ofthe variable, to find the quantity from which such ex-pression would he derivable, as its fluxion or differential,according to the rules before laid down. This reverseprocess was in most cases, except a few which wereobvious inversions of the preceding process, very diffi-cult, and in a large number no method appeared of ob-taining the result at all in any exact or finite terms.These, however, were the results actually wanted for allthe most important applications; here then the valueof serieses was apparent; here those forms before inves-tigated found their proper place in the system of ana-lytical mathematics : wherever the expressions in ques-tion could be reduced into a series of simple terms, eachof those terms separately could be easily made to givethe quantity which by differentiation would have pro-duced it, and the sum of all the terms, if a finite series,would give the exact value of the whole quantity gen-erated, or an approximation to it if the series were in-finite but yet convergent. This process was called theinverse calculus, or that affluents or integrals: accord-ing to Leibnitz ’s view it gave the sum of all the smallelementary parts ; hence the term integration.

The calculus then took its origin in the attempts to