Math History: Review for final

Mathematicians

These are some of the ones we discussed in class at one time or another. Know something about each of these people and their contribution to mathematics.

1. Medieval mathematics : Bramagupta, Al-Khwarizmi, Fibonacci, Oresme.

2. Renaissance mathematics : Cardan, Copernicus.

3. Pre modern : Fermat, Descartes, Napier, Cavalieri.

4. Calculus : Newton, Leibnitz, Bernoullis, L'Hospital, Euler, Cauchy.

5. Modern: Lobachevsky, Bolyai, Gauss, Bolzano, Poincare, Cantor, Hilbert

History Threads

Here are the various developement stories we became familiar with since midterm. Be able to discuss these.

1. The development of trigonometry

2. Solution of cubic equations

3. The developement of calculus

4. Platonic versus Aristotelian outllook and the axiomatic method.

4. Modern Geometry

5. Curve theory.

Problems

Here are some review questions.

How were conics used to solve cubic equations by Apollonius and later the the Arabs?

A bird sitting on the top of a 40 foot tower starts flying to a fountain located between the 40 foot tower and a 30 foot tower. At the same time, a bird on the 30 foot tower starts flying at the same speed toward the fountain also. If the towers are 50 feet apart and the birds reach the fountain at the same time, how far is the fountain from the 40 foot tower?

Suppose the sum of two consecutive integers is a square, then the square of the larger will equal the sum of two squares. Verify this assertion and illustrate with some examples.

Show that any two consecutive Fibonnaci numbers are relatively prime.

Why was the sine function at first called the half-chord?

Make a sketch of the celestial sphere, with Polaris at the top and the equator the horizontal great circle. Sketch in a small, but not too small, version of earth at the center of the universe. Now sketch in the ecliptic inclined at the appropriate angle intersecting the equator at a point V in front and A in back: these are the vernal and autumnal equinoxes . Mark the high and low points on the ecliptic S and W: these are the s ummer and winter solstices . Finally, draw arrows on the ecliptic and equator showing the direction the sun moves in this sketch.

Find the length of the noon shadow of a 20' pole in Lexington, Kentucky (38 degrees N), at the vernal equinox.

Theorem: Given a plane triangle ABC and 3 points X, Y and Z on the sides AB, BC, and CA of the triangle, then XC, YA, and ZB are concurrent if and only if

Draw a diagram for this theorem and prove the theorem.

State and solve the 2 + 1 line locus problem.

Find the linear substitution to make into which will eliminate the second degree term.

How would Newton begin to invert the series ?

Show that the curve , 0 <= x <= 1 is not rectifiable.