The non-Newtonian calculi are alternatives to the classical calculus of Newton and Leibniz. They provide a wide variety of mathematical tools for use in science, engineering, and mathematics.
are infinitely many non-Newtonian calculi. Like the classical calculus,
each of them possesses (among other things): a derivative, an integral,
a natural average, a special class of functions having a constant
derivative, and two Fundamental Theorems which reveal that the
derivative and integral are 'inversely' related. Nevertheless, most non-Newtonian calculi are markedly different from the classical calculus.
Of course, in the
classical calculus the linear functions are the functions having a
constant derivative. However, in the geometric calculus, the exponential
functions are the functions having a constant derivative. And in the
bigeometric calculus, the power functions are the functions having a
constant derivative. (The geometric derivative and the bigeometric
derivative are closely related to the well-known logarithmic derivative
and elasticity, respectively.)
The geometric and bigeometric calculi have been the two most often used non-Newtonian calculi. In each of these two calculi, the use of multiplication/division to combine/compare numbers is crucial.
NOTE. The six books on non-Newtonian calculus and related matters
by Jane Grossman, Michael Grossman, and Robert Katz are indicated below,
and are available at some academic libraries, public libraries, and
booksellers such as Amazon.com. On the Internet, each of the books can
be read and/or downloaded (free of charge) at HathiTrust and at the Digital Public Library of America; and they can be read (free of charge) at Google Books.
Michael Grossman and Robert Katz. Non-Newtonian Calculus, ISBN 0912938013, 1972. (Please see item  in the References section.)
A Quotation from Gauss
The following Carl Friedrich Gauss quotation is from Gauss, Werke, Bd. 8, page 298; and from Memorabilia Mathematica (or The Philomath's Quotation Book) (1914) by Robert Edouard Moritz, quotation #1215.
"In general the position as regards all such new calculi is this - That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able - without the unconscious inspiration of genius which no one can command - to solve the respective problems, indeed to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius' calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."
Three Reviews of the Book Non-Newtonian Calculus
The following review was written by David Pearce MacAdam, and appeared in the Journal of the Optical Society of America (Volume 63, January of 1973), a publication of The Optical Society, which is a member of the American Institute of Physics.
"This [Non-Newtonian Calculus] is an exciting little book, for two reasons: first, its content, and second, its presentation. The content consists primarily of a brief nonaxiomatic description of the fundamentals of various calculi ...The following excerpt is from a review written by Ivor Grattan-Guinness that appeared in Middlesex Math Notes (Volume 3, pages 47 - 50, 1977), a publication of Middlesex University in London, England.
"For each calculus, a gradient, a derivative, and an average are defined; a basic theorem (essentially a mean-value theorem) is stated; an integral is defined; and a fundamental theorem of integral calculus is stated. In Chapter 1, the authors review the essentials of Newtonian calculus and establish the format of presentation that they follow in the later chapters on the various non-Newtonian calculi. Classical problems studied by such men as Galileo and Newton provide examples of, and motivation for, the authors' work. The authors' reference to these problems serves to emphasize the relevance of their results to the concerns of modern science. The greatest value of these non-Newtonian calculi may prove to be their ability to yield simpler physical laws than the Newtonian calculus. Throughout, this book exhibits a clarity of vision characteristic of important mathematical creations.
"The authors have written this book for engineers and scientists, as well as for mathematicians. They have made a dramatic break with tradition and omitted all proofs and many of the mathematical details that place so much of contemporary mathematics writing out of the reach of scientists and engineers. Instead, they have included details that help develop an intuitive conception of their calculi and relate their calculi to well-knows classical problems. The authors apparently feel that their results are of sufficient importance to scientists and engineers to justify these departures from the more traditional style of writing in mathematics. The writing is clear, concise, and very readable. No more than a working knowledge of [classical] calculus is assumed. Mathematicians who feel that their results are of importance to scientists and engineers, but who find little interest among those workers for their results, might consider presenting their work as Grossman and Katz have done. This would do much to reopen channels of communication between mathematicians and scientists and do much to advance both disciplines."
"There is enough here [in Non-Newtonian Calculus] to indicate that non-Newtonian calculi ... have considerable potential as alternative approaches to traditional problems. This very original piece of mathematics will surely expose a number of missed opportunities in the history of the subject."
The following excerpt is from a review written by H. Gollmann that appeared in Internationale Mathematische Nachrichten (Number 105, 1972), a publication of Österreichische Mathematische Gesellschaft in Vienna, Austria.
"The possibilities opened up by the [non-Newtonian] calculi seem to be immense."
Thanks to David Lukas and Kenneth Lukas for their expert advice about the Internet and website construction. In 2003, they suggested the idea of using the Internet to broadcast information about non-Newtonian calculus. Dave, who set up the first NNC website, is a computer engineer. Ken, a talented musician, uses the computer to compose music and to create videos and movies. They knew that posting information on the Internet would be an excellent way to inform people about non-Newtonian calculus. That information has turned out to be of considerable interest to many researchers who until they read about it on the Internet knew nothing about non-Newtonian calculus, but subsequently found applications in various fields.