Course Overview: The course will meet in Salomon 202 in C hour (MWF
10:00--10:50) according
to the University Calendar from 22 January to 5 May 1997.
Assigned texts will include coursepacks and handouts containing excerpts
from primary
sources in English
translation; A Source-Book in Mathematics, 1200--1800, by D.
J. Struik;
The History of Mathematics: an Introduction, by Victor Katz; and
an introductory calculus textbook, Calculus Lite, by Frank Morgan
(recommended, not required). There will be readings for every class (with
few exceptions)
and written homework assignments every week (with few exceptions). There
will be
5 in-class or take-home quizzes over the course of the semester, and a final
10--15-page research paper due 12 May.
| Date | Topics |
| W 1/22 |
Introduction.
(What is calculus? How do we approach it?) Discussion of pre-calculus
mathematics, students' mathematical backgrounds. Explanation of course
requirements,
course objectives. Focus on original sources, historical development, and
mathematical understanding. |
|
Readings: Early mathematics in Katz, Ch. 1 (up to 1.3) |
|
Assignment 1 (for 24 January): Translation of Babylonian cuneiform
tablet |
|
Assignment 2 (for 29 January): "Translation" of Babylonian and Indian
quadratics
algorithms into algebraic terms. (Second Babylonian cuneiform tablet (AO 8862).
Baudhayanasulbasutra selections.) |
| F 1/24 |
Before Calculus: Practical mathematics in ancient civilizations.
(We explore the basic mathematical tools that ancient cultures developed for
problem-solving, and link them to some of our modern tools such as notation.)
Babylonian tablet and ancient number systems. Geometrical measurement:
demonstration
of Babylonian area calculation. Measurement of circles: Biblical, Chinese
rules for
pi. (Explicit discussion of algebraic "translation" of pre-algebraic
works.) |
|
Readings: Katz 2.1--2.3 |
| M 1/27 |
Philosophical limitations to mathematical knowledge.
(We discuss the evolving definition of mathematical knowledge and its shocking
consequence: the discovery that we can know exactly that there are things
we cannot know exactly.)
Discussion of homework: algorithmic procedures. What is number? The
Pythagoreans:
numerical cosmology, figurate numbers, means. Commensurability and
irrationality.
Inapplicability of mathematics to motion. |
|
Readings: Aristotle, Physics, Book VI |
| W 1/29 |
Developing mathematics within the limitations, I.
(The ancient Greeks ask: If we concede that there are limits to what
mathematical
knowledge can tell us, what can we know within those limits?)
Discussion: do the infinite and the infinitesimal exist? What are they? How
do we
separate what is mathematics from what isn't?
The axiomatic deductive method and the geometrical approach: strict
definitions of
number, separation of number from magnitude. |
|
Readings: Prepare for Friday's quiz |
| F 1/31 |
In-class quiz.
Identification and definition of concepts; problem-solving; short-answer
questions. |
|
Readings: Katz, pp. 54--56; Euclid's Elements (Coursepack 2):
Introduction ("Euclid and the Traditions About Him");
Book I, Definitions, Postulates, Common Notions;
Book II, Proposition 14 (construction of geometric mean);
Book XII, Propositions 1--2 |
|
Assignment 3 (for 5 February): Classical Greek demonstrations |
| M 2/3 |
Developing mathematics, II.
(What kind of problems can Greek mathematics solve and still be confident
that it
produces exact knowledge?)
Difficulties with area and volume. Rectification, quadratures, cubatures,
maxima and minima.
Famous problems and their constraints.
Chords and regular polygons. The method of exhaustion. |
|
Readings: Katz on Archimedes, pp. 95--96, 104--108; Archimedes (Coursepack 2),
Introduction ("Archimedes");
On the measurement of the circle, Propositions 1--3 |
| W 2/5 |
Archimedes.
(The greatest mathematician of antiquity comes up with some clever answers.)
Discussion of On the measurement of the circle. Pi.
"Infinitesimal" areas
and volumes. The volume of the cone. |
|
Readings: Euclid,
Book III, Definitions, Proposition 16; Katz on Apollonius, pp. 108--121;
Apollonius, (Coursepack 2) Introduction ("The Author and his own Account of
the Conics"); Conics, Propositions 1--4, 12--13 |
|
Assignment 4 (for 12 February): Archimedes and Apollonius |
| F 2/7 |
Curves other than the circle.
(In the days before coordinates and graphs, how do we talk about curved
lines? Can
we discuss any of them mathematically? Which ones? How?)
Incommensurability of curved and straight lines.
Definition of curves in general and tangents. Conic sections. Spirals. The
mathematical
definition of the parabola. |
|
Readings: Archimedes, Quadrature of the Parabola,
Introduction, Proposition 1,
Propositions 18--24 |
| M 2/10 |
Curves, II; the end of classical mathematics.
(We continue with Greek mathematics' approach to curves. What ever happened to
Greek mathematics?)
Discussion of Archimedes' quadrature of the parabola. Ptolemy's
chord tables. The decline in late antiquity; the revival in
medieval Europe. |
|
Readings: Aristotle, Physics (Coursepack 1): Book IV, Sections
10--14,
Book VII, Section 4; Struik III, 1 (pp. 134--138): Oresme on the latitude
of forms |
| W 2/12 |
Mathematics in motion, I.
(Drawing pictures of change: late medieval mathematicians try to quantify
change and motion.)
Discussion of the latitude of forms. Qualities and quantities; rates of change;
the mean speed theorem. Series. Graphic depiction of motion; coordinate
representation. |
|
Readings: Struik II, 1 (pp. 55--60, al-Khwarizmi); Katz, Ch. 7 up through
Section 7.1;
Katz, Ch. 8, Sections 8.3.1, 8.3.2 |
|
Assignment 5 (for 19 February): Medieval mathematical methods |
| F 2/14 |
New tools from other cultures.
(Computational mathematics from other civilizations comes to medieval and
Renaissance Europe.)
Hindu-Arabic numerals. Development of algebra in India and Islam. Trigonometric
quantities. Transmission of algebra to the West: reactions. |
|
Readings: Descartes, pro-algebra (Struik II, 7--8, pp. 87--93);
Kepler, anti.
Kepler on infinitesimals (Struik IV, 2, pp. 192--197) |
| M 2/17 |
NO CLASS
|
| W 2/19 |
Pushing the envelope: new solutions, I.
(Archimedes lives! How were ancient techniques found and changed by their
inheritors?)
Rediscovery of classical mathematics.
Discussion of infinitesimal methods for areas and volumes. Abandonment of
strict
method of exhaustion. |
|
Readings: Cavalieri's principle (Struik IV, 5--6, pp. 209--219);
Roberval handout on indivisibles; Katz, Section 12.2.2 (pp. 435--437) |
|
Assignment 6 (for 26 February): "Pushing the Envelope" |
| F 2/21 |
Pushing the envelope, II.
("A very large number of very small pieces:" new approaches to calculating
areas and volumes.)
Discussion of methods of indivisibles. Cavalieri's principle. Philosophical
objections. The integral power law. |
|
Readings: Napier on logarithms (Struik I, 4, pp. 11--21);
Katz, Section 10.4 (pp. 374--383), and Section 12.2.6 (pp. 449--450) |
| M 2/24 |
Pushing the envelope, III; another new tool for computation.
(Number-crunching: replacing multiplication by addition.)
Logarithms. Napier and Briggs. Gregory of St. Vincent and the area under
the hyperbola.
Complaints about newfangled methods. The method of exhaustion redux. |
|
Readings: Galileo on velocity and acceleration (Struik IV, 4, pp. 208--209);
Katz, Section 10.5 (pp. 383--388) |
| W 2/26 |
Mathematics in motion, II.
(Mathematicians return to the discussion of motion and change, and draw
more pictures.)
Discussion of velocity and acceleration. Projectile motion. Mathematical
and graphical
descriptions of motion. |
|
Readings: Fermat and Descartes on analytic geometry (Struik III, 3, 5, pp.
143--150,
and pp. 155--157) |
|
Quiz #2 (for 5 March): take-home quiz |
| F 2/29 |
The beginning of general solutions, I.
(The amazing discovery that a lot of problems that look different are
really the same.)
Discussion of analytic geometry and coordinate systems. Algebraization of
locus problems. Introduction of equations for curves. Descartes' principle of
nonhomogeneity. |
|
Readings: Pascal on integration of sines (Struik IV, 11, pp. 238--241) |
| M 3/3 |
Beginning of general solutions, II.
(Triangles and trigonometry; application of new methods to new problems.)
Development of trigonometric "functions": chords, sines, tangents.
Discussion of Pascal's "integration" of the sine. Introduction of new curves:
the cycloid. |
|
Readings: Struik IV, 8, pp. 222--227 (Fermat on the method of "adequality") |
| W 3/5 |
General solutions, III: optimization and tangents.
(Can we solve all similar problems in the same way? Seventeenth-century
mathematicians try.)
Descartes' method of normals. Fermat's On maxima and minima.
Representing
the curve and finding the tangent. Maximum and minimum values. |
|
Readings: Struik IV, 14 (pp. 253--263, Barrow on the fundamental theorem) |
|
Assignment 7 (for 12 March): The beginning of general solutions |
| F 3/7 |
(Mid-semester) The relationship between tangents and areas.
(The crucial central notion of the calculus begins to emerge.)
Rectification.
Discussion of Barrow's work on quadrature and tangents.
Generalization to various curves. |
|
Readings: Handouts from Pascal's Thoughts and Barrow's
Geometrical
Lectures |
| M 3/10 |
Historical context of the calculus, I.
(Why is all of this mathematics happening now? How are these concepts
justified?)
Historical and philosophical background of the calculus in the early seventeenth
century: education, technology, motivation. |
|
Readings: Struik V, 1, 8 (pp. 271--279 (up through par. 1), Leibniz on
differential calculus,
and pp. 312--315 (up through Prop. II), L'Hospital's version of it) |
| W 3/12 |
The differential calculus, I: Leibniz.
(At last, a truly general algorithm for all (?) curves.)
Leibniz' New Method and basic rules of differentiation.
Interpretation
of Bernoulli and L'Hospital. |
|
Readings: Katz, section 12.5.3 (pp. 466--468) and section 12.5.6 (pp. 471--472);
Struik V, 5 (pp. 291--295); V, 7 (pp. 303--308 and 311--312)
|
|
Assignment 8 (for 19 March): Newton and Leibniz, et al. |
| F 3/14 |
The differential calculus, II: Newton.
(Simultaneously (?), another truly general algorithm.)
Newton's theory and basic rules of differentiation; motion, fluxions, and
moments. |
|
Readings: Review of Leibniz and Newton readings for 3/12 and 3/14 |
| M 3/17 |
Historical context, II: the great dispute.
(Who invented the calculus? Why are we fighting about it?)
Reception of the two calculus techniques and the argument over priority
and plagiarism. National loyalties (the "scurvy English" vs. the "Leipzig
rogues"); the Royal Society's inquiry. |
|
Readings: Struik V, 4 (pp. 284--290) |
| W 3/19 |
The differential calculus, III: further testing.
(Does the new "calculus of differentials" work for all the curves we can
think of?)
Differentiation of trig functions, the logarithm, other
functions (power series). Substitution and the equivalent of the chain
rule. |
|
Readings: None |
|
Quiz #3 (for 2 April): take-home quiz |
| F 3/21 |
Class visit to Lownes Collection of John Hay Library.
View early editions of Fibonacci, Roberval, Descartes, Newton, etc.
Discussion of transmission of mathematics and survival of these texts. |
|
Readings: In Simpson/Agnesi handout, read Simpson, pp. 9-13 (skim examples),
pp. 33-39 (Scholium), pp. 49-53; and Agnesi, pp. 60-63 |
| M 3/24 |
NO CLASS |
| W 3/26 |
NO CLASS |
| F 3/28 |
NO CLASS |
| M 3/31 |
The differential calculus, IV: more than just tangents.
(How do we classify curves and their characteristics to make the new calculus
even more efficient?)
Extrema, concavity, higher derivatives.
|
|
Readings: Agnesi handout, pp. i--iv ("The Plan of the Lady's System of
Analyticks"); Katz, section 12.7 (pp. 482--486) |
| W 4/2 |
Historical context, III.
Historical and philosophical background of the calculus in the late
seventeenth and
early eighteenth century. |
|
Readings: Struik V, 2--3 (pp. 281--284, Leibniz); V, 7 (only pp.
308--311, Newton);
V, 10 (only pp. 324--326, Bernoulli) |
|
Assignment 9 (for 9 April): Techniques of the calculus. |
| F 4/4 |
The integral calculus, I: Leibniz and Newton again.
(The two inventors present their versions of the other half of the calculus.)
Leibniz and Newton on integration; using the fundamental theorem
to find areas. |
|
Readings: Handout from Newton's Method of series and fluxions |
| M 4/7 |
The integral calculus, II.
(Systematizing the techniques of integration.)
Finding the area under various curves; tables of integrals.
Rectification.
|
|
Readings: Katz, section 13.1 (pp. 494--506) |
| W 4/9 |
Historical context, IV.
(Making an algorithm into a discipline.)
The standardization and expansion of the procedures of calculus.
Notation; textbooks; inclusion in the curriculum; new problems. |
|
Readings: Katz, section 12.5.5 (pp. 470--471); section 12.6.2 (pp. 475--477);
(re-read) Struik V, 10 (all, esp. "absolute quantities") |
|
Assignment 10 (for 16 April): Calculus issues in the 18th century |
| F 4/11 |
The integral calculus, III.
(Further techniques of integration.)
Substitution and integration by parts. |
|
Readings: Handout from Simpson; (re-read) Katz, section 13.1 (introduction),
section 13.1.1 |
| M 4/14 |
The integral calculus, IV.
(Theme for final paper: study a renowned problem solved by calculus and write a
10--15-page paper explaining the mathematics and discussing its historical
development.
A list of suggested topics and a description of desired goals and methods for
the paper will be provided.)
Other applications of the integral.
|
|
Readings: Handout from Swift; Struik V, 12 (pp. 333--338, Berkeley) |
| W 4/16 |
Historical context, V: philosophical objections to the calculus.
("The emperor has no clothes, and he's dividing by zero:" complaints that
the "new analysis" has abandoned mathematical certainty.)
Newton and Berkeley. Fluxions and the "ghosts of departed quantities."
|
|
Readings: Katz, section 13.2 (pp. 506--519) |
|
Assignment 11 (for 23 April): Researching sources for final paper |
| F 4/18 |
The calculus explosion, I.
(Is there anything calculus can't solve? Eighteenth-century mathematics
takes off.)
The differential equation; power series and their development. |
|
Readings: Struik V, 15 (pp. 345--351); (re-read) Katz, section 13.2 (pp.
506--519) |
| M 4/21 |
The calculus explosion, II.
The exponential and the logarithm. |
|
Readings: Katz, section 14.4 (pp. 572--576) |
| W 4/23 |
Historical context, VI: differing philosophical approaches.
(Guest lecture by Prof. Joan Richards, on development of Continental vs.
English 18th-c.
mathematics.) |
|
Readings: Katz, section 13.5 (pp. 525--533); Struik V, 14 (pp. 341--345,
d'Alembert);
V, 19 (3) (pp. 388--391, Lagrange) |
|
Assignment 12 (for 30 April): TBD |
| F 4/25 |
Rigorization of the calculus, I.
(Can we restore mathematical certainty to the calculus? Sure; just change the
definitions.)
Cauchy and the limit; definition of the derivative. |
|
Readings: TBD |
| M 4/28 |
Rigorization, II.
(More changing of definitions.)
Definite integrals: Cauchy and Riemann. |
|
Readings: TBD |
|
Quiz #4 (for 5 May): take-home quiz |
| W 4/30 |
Rigorization, III.
(The dreaded epsilon-delta limit finally appears.)
Foundations of the limit: Heine and Weierstrass, epsilon-delta definition. |
|
Readings: TBD |
| F 5/2 |
Conceptual changes after the 19th century.
(How did all this become the calculus we know today? Is it going to stay
that way?)
Standard approach to the rigorous calculus. Nonstandard analysis;
Robinson and infinitesimals. |
|
| M 5/5 |
Conclusion. |