PHYSICAL AND MATHEMATICAL SCIENCES. 303
on a former occasion, we briefly referred. The idea ofNewton was a singularly happy instance of the uni-versal and exact applicability of his metaphor. When afunction increases or decreases up to a certain point, andthence again decreases or increases, at that point it isa maximum or minimum. It is obvious that its velo-city of increase or decrease varies up to that pointwhere it is absolutely nothing; and then begins to in-crease again. Thus, by finding the value of the quan-tities involved which will make the fluxion becomenothing, we find the maximum or minimum.
Another kind of maximum or minimum, aboundingin problems of the highest interest, but far more diffi-cult than the last mentioned, is that in which thefunction itself is required to be found which will bethe greatest or least under certain conditions. Oneclass of such problems we have before referred to, asknown by the name of iso-perimetrical problems. Butthe investigation, in its more extended form, exercisedthe talents of the two Bernoullis, and has received fromthem, and from subsequent analysts, the most completeinvestigation.
In a sketch like the present, it is of course impos-sible to enter into any adequate account of the differentbranches into which the direct or inverse parts of thecalculus extend themselves. It must suffice to say, thatthe latter, as it is the most difficult and important, soit is by far the most extensive. One very large and im-portant class of problems which it has to consider, arethose which involve two or more variables mixed upwith their differentials in any manner of combinationin an equation. The difficulty here is that of separatingthem, if possible, so that each variable shall stand com-bined with only its own differential; but the process isonly capable of being effected in certain cases ; andthough extensive classes of such differential equations,as they are termed, have been solved, especially by theearly labours of the two Bernoullis, and have subse-quently occupied the researches of the most eminent