Robert Woodhouse

"Taylors method... has no recommendation from its neatness and perspicuity, but is justly censured by John Bernoulli for its obscure conciseness."

"Although I am not aware of having omitted any thing that is requisite to the full explanation of the subject, yet I cannot flatter myself that it will be thoroughly understood from this Work alone. For, in general it may be laid down as true, that no doctrine, of novelty and intricacy, can be completely taught by a single Treatise. It seems to be indispensably necessary for the student, that the subject should be put under several points of view: that if not apprehended under one, it may be under another."

"There is another point... and that is the method of demonstration by geometrical figures. In the first solution of Isoperimetrical problems, the Bernoullis use diagrams and their properties. Euler, in his early essays, does the same; then, as he improves the calculus he gets rid of constructions. In his Treatise [footnote: Methodus inveniendi, &c.], he introduces geometrical figures, but almost entirely, for the purpose of illustration: and finally, in the tenth volume of the Novi Comm. Petrop. as Lagrange had done in the Miscellanea Taurinensea, he expounds the calculus, in its most refined state, entirely without the aid of diagrams and their properties. A similar history will belong to every other method of calculation, that has been advanced to any degree of perfection."

"To history we shall adhere no farther, than is sufficient to preserve an unbroken series of methods gradually becoming more exact and extensive; the series beginning with the first rude, though perfectly just, method of James Bernoulli, and ending with Lagranges exquisite and refined Calculus of Variations."

"The methods of the Bernoullis and of Taylor, were held, at the time of their invention, to be most complete and exact. Several imperfections, however, belong to them. They do not apply to problems involving three or more properties; nor do they extend to cases involving differentials of a higher order than the first: for instance, they will not solve the problem, in which a curve is required, that with its radius of curvature and evolute shall contain the least area. Secondly, they do not extend to cases, in which the analytical expression contains, besides x, y, and their differentials, integral expressions; for instance, they will not solve the second case proposed in James Bernoullis Programma if the Isoperimetrical condition be excluded; for then the arc s, an integral, since it =\int \!dx \sqrt(1+\frac{dy^2}{dx^2}), is not given. Thirdly, they do not extend to cases, in which the differential function, expressing the maximum should depend on a quantity, not given except under the form of a differential equation, and that not integrable; for instance, they will not solve the case of the curve of the quickest descent, in a resisting medium, the descending body being solicited by any forces whatever."