MA177--Change, Probability, and Numbers
Topics in the Mathematics of the Eighteenth Century

Fall, 1997
Fernando Q. Gouvêa

In his essay on The Study of the History of Mathematics, George Sarton describes the history of mathematics as a "secret history." What he means is that, despite the importance of mathematics in western culture and especially in western science, most people know nothing about it. From the point of view of this vast majority, mathematics is an activity done in secret. "Yet that secret activity is fundamental; it is all the time creating new theories, which sooner or later will set new wheels moving, new machines working, or, better still, will enable us to obtain a deeper understanding of the mechanism of the universe."

The eighteenth century was a time in which the importance of mathematics was not at all a secret, at least among the leading thinkers. Almost all of the forgers of the Enlightenment either were deeply involved in mathematics (e.g., D'Alembert) or were deeply aware of the importance of mathematics (e.g., Voltaire and Diderot, though their opinions about whether this was a good thing were very different). The enlightenment program of using reason to understand the world grew out, at least in part, from Newton's successful use of mathematical reasoning to understand the motion of the planets, and throughout the eighteenth century we find both people attempting to use mathematics to understand the world. Mathematics was applied to everything from classical mechanics (what the physics books call "Newtonian mechanics" was really created in the 1750s) to the advantages and disadvantages of smallpox inoculation, and even to an investigation of the validity of determining "the will of the people" through the process of voting.

The goals of this course are two. First, we want to develop an overall understanding of the role of mathematics (and, to a lesser degree, of science in general) in Enlightenment thought. We will do this through a general survey of the mathematics of the eighteenth century, paying special attention to the connections between mathematical thought and what was going on in society. Second, we want to focus on three of the many themes that were important at the time: the notion of a derivative, the calculus of probabilities, and the study of "properties of numbers". These topics are relatively elementary, but at the same time should give you some idea of the range of mathematics developed in the eighteenth century. They range from foundational (the derivative) to practical (probability) to highly theoretical (numbers). We will study these topics "from scratch" (the only prerequisites are high-school mathematics and a willingness to think), to some extent by reading the original works.

Now to the nitty-gritty of a syllabus:

Office hours:  I will be in my office and available for questions, discussion, and general conversation on Mondays, Wednesdays and Fridays, from 2:30 to 4:00. If you can't come during any of these times, please call and make an appointment. You may find me in my office at other times, but I only promise to be there and available at the times above.

Please do not hesitate to come see me--in fact, I strongly encourage you to come. It is part of your education, and one of your privileges as a Colby student.

Where to find me:  Here's the basic data

If you need to reach me when I'm not in my office, email is the best method. If you prefer, feel free to call and leave a message.

How the class will be organized: Reading and discussion of the readings will be the center of this class. I will give occasional lectures, especially when we're setting up the mathematics, but for the most part your reading will be the engine that drives the course. In the schedule below, I have attempted to sketch out what to read on a weekly basis, but I will be making explicit reading assignments every class. The overall schedule gives you an opportunity to read in advance if possible, and also, since some weeks look heavier than others, to organize your schedule.

The readings will be of two kinds: descriptive writing about mathematics and mathematicians, and mathematical writing. The first kind is very much like any other piece of historical writing, but mathematical writing may be new for most of you. It is important to learn how to read mathematics. We will discuss this in class, but for now let me note that reading mathematics is a slow process of unpacking the meaning of very tersely worded logical argument. (It was less terse in the eighteenth century than it tends to be nowadays, but I'm sure you'll find it very compressed nonetheless.) It needs to be read actively, with pencil and paper at hand to check the author's assertions, to consider examples, to do computations. A couple of pages of Newton may take as long to understand and master as a short novel. Plan accordingly!

Texts:  The texts for this course are the book Science and the Enlightenment, by T. Hankins, and a rather hefty coursepack. Let me try to describe each.

First, Hankins' book is a survey of the history of science in the eighteenth century. The most important chapters for us will be the first two (on the Enlightenment in general, and on the subjects that were classified as mathematics at the time: pure mathematics, mechanics, astronomy) and the last (on the "moral sciences," which, somewhat surprisingly includes probability theory and Condorcet's work on social mathematics). The other chapters, however, are also worth a look, especially because the emphasize the strong distinction between the mathematical sciences ("Natural Philosophy") and the experimental sciences ("Natural History") in the eighteenth century. Knowledge was organized differently at the time, and though Hankins imposes a twentieth century grid (Physics, Chemistry, Biology), his text does explain that this was not how science was organized at the time.

The coursepack contains a wealth of material, ranging from a survey of eighteenth century mathematics to some of the original material by Newton, Euler, and others. I will pass out a more detailed "guide to the coursepack materials" a few days into the semester.

Of course, you will also be making extensive use of the Colby library. The library has a number of books and reference works on the history of mathematics and the history of science, and it is important to become familiar with these tools and to learn to use them. As you will see below, there will be two "library assignments" which are designed to introduce you to the riches of our library.

Assignments:  There will be a variety of assignments during the course:

Help with writing:  since you will be doing so much writing for this course, you should be aware that Colby has a Writers' Center, located in Miller (check their web site). The Writers' Center tutors are trained to help you express your ideas in clear and effective prose; if you have any difficulties, consider using this Colby resource.

Exams:  we will have one midterm exam, on Monday, October 20, and also a final exam. These exams will cover both history and mathematics. I will provide more information as we get nearer to the exam date.

Grading:  Your grade will be computed as follows:

Cheating and Plagiarism:  You are encouraged to interact with others as you do the readings (in fact, it will be a lot more fun doing it that way). Your papers, however, must be your own. You may, of course, seek help from any and all sources, but in the end what you write must be a result of your own thought processes and your assessment of the source material. Do not quote without attribution, and do not state as fact the opinions of one of your sources. Footnotes and bibliographic references are required. Feel free to discuss any questions you have about this with me. Also, please read the Colby policy on academic honesty as stated in the College Catalogue.

A tentative schedule:  Here is the tentative schedule, with a list of readings for each week. This is very likely to change as we go along, but should give you some idea of what we will be covering and which texts will be read at each time.

Sept. 8-12
Introduction, what is mathematics, a sample of the mathematics of the eighteenth century. Readings: Bullock, "Literacy in the Language of Mathematics" (coursepack); Euler, "On a problem in Analysis Situs" (handout); Hankins, chapter I; Perl, "The Ladies Diary" (coursepack).
Sept. 15-19
Survey of eighteenth century mathematics, part 1: Newtonianism and classical mechanics. Readings: Boyer-Merzbach, chapter 19 (coursepack); Voltaire, selections from Letters Concerning the English Nation (coursepack); Rickey, "Isaac Newton" (coursepack); Hankins, chapter II.
Sept. 22-26
Survey of eighteenth century mathematics, part 2: applying mathematics to the real world. Readings: Boyer-Merzbach, chapters 20, 21 (coursepack); Hankins, chapter VI; from Fauvel-Gray, sections 14B, 14C (coursepack).
Sept. 29-Oct.3
Survey of eighteenth century mathematics, part 3: mathematicians and the French revolution, "social mathematics." Readings: Boyer-Merzbach, chapter 22 (coursepack); Calinger, survey of enlightenment mathematics, pp. 429-464 (coursepack).
Oct. 6-10
Measuring change: fluxions and differentials. Readings: Newton on fluxions, from Fauvel-Gray, section 12A (coursepack); Newton's "Lemma II" (coursepack); Berkeley, "The Analyst" (coursepack).
Oct. 13-17
What is a differential? Readings: D'Alembert on Differentials, from Calinger 482-485 (coursepack); Grabiner, "The Changing Concept of Change" (coursepack).
Oct. 20-24
Bringing it up to date: the modern notion of a derivative.
Oct. 27-31
Probability, part 1. Readings: Fermat and Pascal on probability, from Smith, pp. 546-565 (coursepack).
Nov. 3-7
Probability, part 2. Readings: Daston, "Putting Numbers to the World" and "Expectation and the Reasonable Man" (coursepack).
Nov. 10-14
Probability, part 3. Readings: Laplace, selections from A Philosophical Essay on Probabilities (coursepack).
Nov. 17-21
The properties of numbers. Readings: samples of Fermat's number theory, from Calinger pp. 341-346 (coursepack).
Nov. 24-28
Euler discovers Fermat. Readings: samples of Euler's number theory, from Calinger, pp. 506-509 (coursepack).
Dec. 1-5
One big theorem: Euler on sums of four squares. Readings: Euler, "Proof that Every Integer is a Sum of Four Squares", from Smith pp.91-94 (coursepack).
Dec. 8-12
Number theory today, and overall wrap-up.

Hints for Success

(Many thanks to Fred Rickey for a number of ideas in this syllabus, including a couple of assignments and the "Hints for Success".)

Fernando Q. Gouvêa ----
Last modified: Mon Dec 1 09:50:55 1997