Read Chapter 4 of "A Calculating People" by Patricia Cohen, Princeton Univ. Press. You can view it here.

Solve the following problems from Pike's Arithmetic.

1. Given that the period of the earth is 1 year and Mars' mean distance to the sun is 1.524 times that of earth, determine the period of Mars. [Hint: Use Kepler's 3rd Law.]

Solve the following problems from Pike's Arithmetic.

2. Bought 9 Chests of Tea, each weighing 3 Cwts. 2 qrs. 21 lb. at 4£ 9s per Cwt what came they to? Here are some conversion factors: 4 qrs (quarters of a hundred weight) = 1 Cwt (hundred weight) ; 28 lb = 1 qr ; 20s (shillings) = 1£ ; 12d (pence) = 1s.]

3. Christ-Church, in Boston, has 8 bells; how many changes may be rung on them? [A change is a permutation.] Does Christ-Church still have 8 bells? What is the connection with Paul Revere?

4. Nine Gentlemen met at an Inn, and were so pleased with their host, and with each other, that, in a frolic they agreed to tarry so long as they, together with their host, could fit every day in a different position at dinner; pray how long, had they kept their agreement, would their frolic have lasted?

You can see the 5th edition (1832) of Pike's Arithmetic here. A copy of the first edition is available. If you wish to see it, ask me.

Assignment for Wednesday, August 20

Begin reading Boyer's A History of Mathematics. Start with the Forward by Isaac Asimov. Asimov was a very popular science fiction writer. He also wrote on various aspects of science. Read Chapters 1, 2 and 3.

Assignment for Monday, August 24

1. What do you regard as the three chief shortcomings of Egyptian mathematics? Explain.

2. What do you regard as the three chief contributions of Egypt to the development of mathematics. Explain.

3. Express 2/103 as a sum of two unequal unit fractions, and write these in Egyptian hieroglyphic notation.

4. Through duplication and mediation (that is, successive doubling and halving) find 101/16. Express the result in Egyptian hieroglyphic notation.

5. Show that if n is a multiple of 5, then 2/n can be broken into the sum of two unit fractions, one of which is half of 1/n.

Ishango Bone, a Mesolithic tally stick

Assignment for Monday, August 31

Begin reading Chapter 4.

Assignment for Wednesday, September 2

1. Compare, as to significance and possible influence on later civilizations, the geometry and trigonometry of the Babylonians with that of the Egyptians. (Your answer should be approximately 1 page.)

2. Write the number 0.0862 in Babylonian notation.

3. Verify that if (c/a)² is 1;33,45 and b=45 and c=1, 15, then a, b, c form a Pythagorian triad.

4. Solve the following Old Babylonian problem: Ten brothers receive 1;40 minas of silver, and brother over brother receive a constant difference. If the eighth brother received 6 shekels, find how much each earned. (There are 60 shekels in a mina.)

5. Plutarch (about 100 A.D.) claimed: if a triangular number is multiplied by 8, and then 1 is added to the product, then the result is a square number. Prove this. Then illustrate it geometrically for the second triangular number t₂.

Assignment for Monday, September 14

Begin reading Chapter 5. We will finish our discussion of Greek mathematics next week.

Assignment for Monday, September 21

Assignment for Monday, September 28

Remember that Paper 1 is due next Monday. You are encouraged to talk with me about your work before it is due.

Assignment for Wednesday, September 30

1. Describe at least three respects in which the mathematics of Apollonius differs from that of Euclid. Describe at least three aspects in which their works are similar.

2. Find three numbers such that the product of any two added to the square of the third gives a square. [Hint: Let the numbers be x, 4x + 4 and 1, so that two of the conditions are satisfied. This problem is from Book III of Diophantus's Arithmetica.]

3. Prove by mathematical induction that the sum of the first n Fibonacci numbers with odd indices is given by the formula

F(1) + F(3) + F(5) + ... + F(2n-1) = F(2n).

Assignment for Wednesday, October 14

Assignment for Wednesday, October 21

Assignment for Wednesday, November 4.

1. While searching the true shape of a planet's orbit, Kepler examined the curve ρ = 2 r cos³ θ (expressed in modern, polar coordinates, r is a constant). Sketch the curve carefully. Express it in Cartesian coordinates.

2. Historian J.W.L. Glaisher wrote: ``It is rather surprising that the method of quarter squares which depends upon the formula ab = (1/4)(a+b)² - (1/4)(a-b)² should have escaped the notice of Napier and his contemporaries." He remarked that a table of quarter-squares up to 200,000 enables 5-figure multiplication. Explore this idea. Construct a partial table of quarter squares to compute the product of 0.11372 and 0.52146. Work only with integers. Explain how to handle decimal points.

3. Recall that Nap.log(N) is the number n such that N = 10⁷(1-10 ⁻⁷)ⁿ. Derive a formula for Nap.log(M/N) in terms of Nap.log(M), Nap.log(N) and Nap.log(1).

4. Should Descartes be credited with the discovery of rectangular coordinates? Give evidence.

Assignment for Monday, November 16.

1. What were three of the chief sources of support of mathematicians in the seventeenth century? For each, give an example.

2. Verify Wallis's formula on page 381 (bottom) for the case n=3.

3. Compare the contributions of Newton and Leibniz to mathematical notations.

4. Leibniz argued that 1-1+1-1+... is 1/2. What was his argument? Would it be acceptable today? Justify your answer.

Study Guide for Final Exam

I will draw questions for the final exam from those below.

Egyptian Mathematics:

Describe the evidence on which we base our knowledge of Egyptian mathematics.

What mathematical problems were of interest to the ancient Egyptians? What is the origin of the word "geometry", and why is the word appropriate in view of the Egyptian contributions to the subject?

Describe the relative advantages and disadvantages of the number notations of the Egyptians.

Babylonian Mathematics:

Describe the evidence on which we base our knowledge of Babylonian mathematics.

Describe four significant contributions of the Mesopotamians to mathematics. Justify your answer, of course.

Describe the relative advantages and disadvantages of the number notations of the Babylonians.

Greek Mathematics:

Who was Thales? What discoveries are attributed to him?

Who was Pythagoras? What beliefs did the Pythagoreans share? What mathematics did they discover?

What is construction by compass and straightedge? What were the Delian problems?

Describe three of Zeno's paradoxes. How did they challenge Pythagorean ideas?

Describe some of the major contributions of Euclid's Elements.

Give Euclid's proof that there are infinitely many prime numbers.

Medieval Mathematics:

Discuss Muslim contributions to science and mathematics after the fall of Constantinople.

Who was Leonardo of Pisa? Describe the "Liber abaci." How did "Fibonacci numbers" arise there?

Renaissance Mathematics:

What philosophical changes came with the Renaissance, and how did they affect science and mathematics?

Who was Luca Pacioli? Describe the "Summa de arithmetica geometrica, proportioni and proportionalita."

Who was Albrecht Durer? What mathematics interested him? Why?

Cardano, Tartaglia and Ferrari are all associated with the solution of the cubic equation. Explain their roles.

How did "imaginary numbers" arise from cubic equations? Who is credited with their discovery?

Carefully state Kepler's three laws of planetary motion.

Discuss Galileo's ideas about infinity.

Who was John Napier? How did he define logarithms? What was the role of Henry Briggs?

The Development of Calculus:

Who was John Wallis? How did he compute the area under the curve y = xⁿ ?

Who was Isaac Barrow? What did he know about the relationship of derivatives and integrals?

How did Isaac Newton find a series expansion for the square root of 1+x ?

Describe the relationship between Newton and Leibniz. Why was there controversy?

Your research:

What are the main points of each of your research papers?