Differentials From a New Viewpoint

Extract

It is not my purpose to expound a new Theory of Differentials in the sense used by Professor G. Temple in his paper to the Mathematical Association in 1936 (G. Temple, “The Rehabilitation of Differentials,” M.G., Vol. XX, p. 120), but merely to embark on what he called the “comparatively trivial business” of giving an interpretation to the differential notation which pre-supposes the definition of derivatives, and to do so from the point of view of School rather than University teaching. A study of elementary textbooks on the Calculus shows two main schools of thought. The first, and older, of these regards differentials as infinitesimals, that is to say as variables which tend simultaneously to zero and whose ultimate ratio is their only interesting property. This presents great difficulties to beginners and savours too much of the early days when the theory of limits had not been developed. It has been largely ousted by the second school, which regards the differentials of independent variables as actual increments and defines the differential of a dependent variable (in terms of them) as what I like to call a “bogus increment”. It seems to me that this second interpretation is not as easy as it pretends to be, and that it suffers from serious drawbacks ; and the purpose of this article is to put forward a third interpretation which so far as I know is original and which is free from the objectionable features of the other two. Of what follows, the earlier part is a criticism of the second interpretation referred to, while the later part is an attempt to show how differentials can be more easily taught, and to demonstrate the adequacy of my interpretation.