A Guide to the MA177 Coursepack

Since there are so many materials in the course pack, and since there's no overall numbering, I thought I should provide a guide. I've listed the coursepack materials in the order they appear.

General

The first set of materials give you a broad overview of what our course will be covering.

  1. J. Bullock, ``Literacy in the Language of Mathematics,'' American Mathematical Monthly, vol. 101 (1994), pp. 735-743.

    This article is about mathematics, and about what it means to be literate in the language of mathematics. We will use it at the beginning of the course.

  2. C. B. Boyer and U. C. Merzbach, A History of Mathematics, Second edition, Wiley, 1991. Chapters 19, 20, 21, 22.

    These chapters from the Boyer-Merzbach history cover the period we will be studying, and they will be useful to give a broad overview of the period. There is much more mathematical detail here than in Hankins' book; in a first reading, you should concentrate on the overall picture.

  3. T. Perl, ``The Ladies' Diary... Circa 1700,'' in Five Fingers to Infinity, ed. by Frank Swetz, Open Court, 1994, pp. 574-578. Originally published in Mathematics Teacher, 70 (April 1977), 354-358.

    The surveys often omit things we would probably like to know, since they concentrate on the ``central'' developments of a period. This article shows you a different side: how ordinary people interacted with mathematical ideas.

  4. R. Calinger, Classics of Mathematics, Prentice-Hall, 1995, pp. 429-464.

    This is another, shorter, survey of the Mathematics of our period. It is more tightly focused on the latter half of the eighteenth century; there's also less mathematical detail.

The impact of Newton and Leibniz

The work of Isaac Newton (and, to a lesser extent, of Gottfried Leibniz) had a fundamental impact on mathematical, scientific, and philosophical thought in the eighteenth century. It's important, then to know something about their work and how it was received in the eighteenth century.

  1. V. F. Rickey, ``Isaac Newton: man, myth, and mathematics,'' in Five Fingers to Infinity, ed. by Frank Swetz, Open Court, 1994, pp. 483-507. Original Publication: College Mathematics Journal, 60 (November 1987), pp. 362-389.

    This article is probably the best short survey of the life and work of Isaac Newton.

  2. Voltaire, Letters Concerning the English Nation, letters XIV, XV, XVI, XVII. Oxford University Press World's Classics edition, pp. 61-86.

    Voltaire was not a mathematician, but he was deeply interested in ``Newtonianism.'' This portion of his letters describes how he saw the impact of Newton's work in England.

  3. J. Fauvel and J. Gray, The History of Mathematics, The Open University/Macmillan (distributed in the US by the Mathematical Association of America), 1987, sections 14B and 14C.

    The real heirs of Newton and Leibniz were the mathematicians of the eighteenth century who turned Newton's ideas into a system that could be used for understanding Nature in a deeper and more precise way. This extract from the Fauvel-Gray sourcebook displays some of this work, including some of the discussion about the shape of the Earth.

Understanding change

These selections focus on what was then called ``the metaphysics of the calculus,'' that is, on trying to really understand the concept of a fluxion (or a differential, or a derivative).

  1. J. Fauvel and J. Gray, The History of Mathematics, The Open University/Macmillan (distributed in the US by the Mathematical Association of America), 1987, section 12A.

    Here are some of Newton's original words explaining his method of ``fluxions.'' If you find them hard to understand, you're not alone!

  2. Newton, Lemma II

    This is a selection from Newton's Principia which is referred to in Berkeley's critique. It is here both so you can see Newton's geometric account of his theory and so that you can understand Berkeley's comments.

  3. Berkeley, ``The Analyst''

    This is a portion of George Berkeley's famous critique of Newton's calculus. The basic thesis is that the conceptual foundations of Newton's methods is decidedly shaky. This vigorous and perceptive essay was enormously influential in the eighteenth century.

  4. (Dalembert, ``Différentiel'') R. Calinger, Classics of Mathematics, Prentice-Hall, 1995, pp. 482-485.

    In this article from the Encyclopédie, D'Alembert attempts to give a clear definition of a ``differential.'' Of all the eighteenth century attempts to clarify this notion, D'Alembert's comes closest to what eventually became the dominant view.

  5. J. V. Grabiner, ``The Changing Concept of Change,'' in Five Fingers to Infinity, ed. by Frank Swetz, Open Court, 1994, pp. 607-619. (Original publication: Mathematics Magazine, 56 (September, 1983), pp. 195-206.)

    This article is a survey of the history of the concept of a derivative (or fluxion, or differential), ranging from the seventeenth to the nineteenth century. We'll use it to survey the whole picture, and to bring us (somewhat) up to date.

How likely is it?

These selections focus on the theory of probability. The letters of Fermat and Pascal are usually cited as the first step towards probability theory. The essay of Laplace summarizes ``classical probability'' as it was at the end of the eighteenth century. Daston's essays are historical, focusing on important aspects of the theory and its relation to the real world.

  1. (Pascal and Fermat on the problem of points) D. E. Smith, A Source Book of Mathematics, Dover, 1959, pp. 546-565

    The beginnings of the theory of probability are found in these letters of Fermat and Pascal. The letters discuss how to divide the stakes in an interrupted game of chance.

  2. P. Laplace, ``A Philosophical Essay on Probabilities'', Dover, 1995, pp. 1-52.

    This selection from Laplace's non-technical book on probabilities gives you a feeling for how much happened during the eighteenth century (just compare with the Fermat-Pascal letters!). It is just a brief extract from a longer--and fascinating--book.

  3. L. Daston, ``Fitting Numbers to the World,'' in History and Philosophy of Modern Mathematics, ed. by W. Aspray and P. Kitcher, University of Minnesota Press, 1988, pp. 221-237.

    Daston, today's premier historian of probability in the eighteenth century, considers the relation between probability theory and the real world in the eighteenth century. She makes an interesting distinction between ``mixed mathematics'' and ``applied mathematics'' which we'll want to discuss.

  4. L. Daston, ``Expectation and the Reasonable Man'', chapter 2 of Classical Probability and the Enlightenment, Princeton University Press, 1988.

    This chapter from Daston's book on classical probability focuses on the role of the notion of ``expectation'' in probability theory. It also discusses the ``St. Petersburg paradox'' and the problem of smallpox inoculation, two important topics of debate in Enlightenment mathematics.

Playing with numbers

The idea of studying the ``properties of numbers'' goes back to Pierre de Fermat in the seventeenth century, but the theory seems not to have been fruitful until Euler picked it up. This section contains some samples of the work of Fermat and Euler on ``number theory.''

  1. (Fermat's Number Theory) R. Calinger, Classics of Mathematics, Prentice-Hall, 1995, pp. 341-346

    These extracts from Calinger's sourcebook give you a sample of Fermat's number theory.

  2. (Euler's Number Theory) R. Calinger, Classics of Mathematics, Prentice-Hall, 1995, pp. 506-509

    Same as the previous one, but now for Euler

  3. (Euler, sums of four squares) D. E. Smith, A Source Book of Mathematics, Dover, 1959, pp. 91-94

    Finally, one complete theorem: here is Euler's proof that every whole number can be written as the sum of four squares.



Fernando Q. Gouvêa ---- fqgouvea@colby.edu
Last modified: Tue Sep 16 09:21:37 1997