MthEd/Math 300– Williams Fall 2011 Review Questions Part 2

Section 1: Match the following descriptions with the correct ancient culture by circling B for Babylon, E for Egypt, or G for Greece. There may be more than one correct answer for some questions. One point each.

1. Most original sources for studying their mathematical history are BEG lost. Could be true of B and E too, I guess. 2. Based on the evidence we have, tended to solve problems with B E G linear equations by the method of false position. (B and G, too?) 3. Seemed to be concerned with the problem of trading bread and B E G beer of various strengths. 4. In at least one source, provided a very accurate approximation to B EG 2 . 5. Were aware of and used the Pythagorean Theorem. B E G 6. Had a strong preference for geometry over arithmetic or numbers. BEG 7. In most sources, used an approximation of   3 to determine the B EG area of a circle. 256 B G 8. In a number of sources, used the approximation of   to E 81 determine the area of a circle. 9. Had a well-established procedure for solving quadratic equations. B E G 10. Multiplied by doubling and adding. B E G 11. Divided by multiplying by reciprocals. B EG 12. Seemed to rely on tables do to do some of their arithmetic. BEG 13. Had the luxury of pursuing mathematical studies for the sake of the BEG intellect, rather than just for practical purposes. (Some say E, too) 14. Were concerned about astronomy. BEG 15. Had trouble dealing with the fraction 1/7 in their number system. B EG Section 2: Fill in the name of the Ancient Greek mathematician who best fits the description.

16. Wrote a book or collection of books called the Elements, of which we have no copy. Hippocrates 17. Demonstrated how to create a triangle that had the same area as a given circle. Archimedes 18. Worked to solve equations with positive rational number solutions in his book Arithmetica. Diophantus 19. Believed that “all is number.” Pythagoras 20. Was the first known woman mathematician. Hypatia 21. Developed the Method of Exhaustion. Eudoxus 22. Asked to have a sphere inscribed in a cylinder on his tombstone. Archimedes 23. Was a loon. Squared a lune. Hippocrates

24. Famous not for original mathematical work, but for compiling and organizing results already known into an Euclid axiomatic system. 25. Was the author of The Almagest Ptolemy 26. Wrote a definitive treatise on Conics Apollonius 27. Was the first known Greek mathematician and philosopher. Thales Section 3: Short Answer

1. Briefly describe the sources we have for our knowledge of the mathematics done in each of the following early cultures. Be sure to address whether the sources are original, and if not, describe how close to the originals they are.

a. Egypt

Ahmes or Rhind Papyrus, Moscow Papyrus, a few others.

b. Babylon

Numerous clay tablets of two major kinds: table tablets, and problem tablets.

c. Greece

No work original to the time. Mainly copies of copies or worse. 2. Regarding Euclid’s Elements:

a. Briefly describe the mathematical content of the Elements (include at least 5 major topics).

Plane geometry, including circles Solid geometry Ratio and proportion Number theory Geometrical algebra Construction of regular polygons Platonic solids

b. Why was this work so important? What “footprint” did it leave in today’s mathematics?

Two words: Axiomatic system. Many mathematicians became interested in math reading it, or learned from it. 3. Describe two aspects of Greek mathematics that most distinguish it from the mathematics of Egypt or Babylon.

Careful proofs Mathematics for mathematics’ sake Not much calculation (Archimedes being the possible exception)

Maybe others?

4. Briefly explain the distinction made in ancient Greece between magnitude and number.

Numbers are discrete, cannot be broken down indefinitely because you eventually came to a “1.” In this sense, any two numbers were commensurable because they could both be measured with a 1, if nothing bigger worked.

Magnitudes are continuous, and can be broken down indefinitely. You can always bisect a line segment, for example. Thus two magnitudes didn’t have to be commensurable (although of course they could be.) 5. Describe what it meant by two magnitudes being incommensurable.

Both could be measured by a common magnitude. By measured, we mean divided evenly by another magnitude, with nothing left over.

6. Describe briefly what the “method” was in Archimedes’ The Method.

It is a way of solving problems in area and volume by comparing line segments (i.e. slices of area) or circles (i.e. slices of volume) from two figures in terms of how they “balance” on a line segment acting as a lever. By clever use of ratios and proportion, Archimedes was able to use the Law of the Lever to these “slices” compared and therefore how the entire areas or volumes compared. Archimedes usually proved synthetically the results he found analytically using the “method.” 7. Describe the differences between Apollonius’ eccenter model for the sun’s motion and his epicycle/deferent circle model for the sun’s motion. What did Apollonius need in order to describe the motion of the planets?

The eccenter model had the sun following a circular path around the earth, but the center of that circular path was not the earth, but a point near the earth. This accounted for.....

The epicycle/deferent circle model had the earth at the center of the deferent circle, but the sun didn’t follow the deferent circle. Instead, it revolved around a point that followed the deferent circle, in a small orbit called an epicycle. This accounted for....

In order to account for the motions of planets, he needed to combine both models so that the deferent circles were not centered at the earth.

8. Describe some aspects of Ptolemy’s Almagest that were significant to the development of trigonometry.

Chord tables

The way the chord tables were constructed, using chord-versions of many of the rules we have in trigonometry, e.g. half-angle formulas, angle addition and subtraction formulas, the laws of sines and cosines, etc.

Calculations of planetary motions that used trigonometry, e.g. the solution of triangles using such things as the law of sines, etc. 9. Give three reasons Diophantus’ work in his Arithmetica might qualify him to be the Greek “Father of Algebra.”

1. Development of symbol system 2. Solutions of equations. 3. Knowing how to deal with negatives, combining like terms, etc. 4. General strategies for reducing and solving equations.

10. As Greek mathematics began its decline, its mathematicians did less original work. Instead, what did mathematicians like Pappus, Theon, and Hypatia mostly contribute to mathematics?

Commentaries on earlier works. (Examples?) 11. Describe the two meanings of analysis we have discussed, and contrast them with synthesis.

General “analysis” refers to methods for finding solutions to problems, or for finding the conclusions of theorems you wish to prove. Thus analysis means literally “taking things apart” to see how they work, or so you can understand them. In the context of theorems, it can also refer specifically to the method of beginning with what you wish to prove, and reasoning from that point in hopes you can arrive at the hypotheses, and then reverse the logical steps (this doesn’t always work, of course).

Synthesis refers to putting things together, as in, putting axioms and theorems together to create a proof.

12. What were the three classic problems of Greek antiquity? How were they eventually resolved?

1. Squaring the circle – constructing with Euclidean tools a square with the same area as a given circle.

2. Duplicating the cube – constructing with Euclidean tools the side of a cube that will have twice the volume of a cube with a given side.

3. Trisecting the angle – construction the Euclidean tools an angle that has exactly one- third the measure of a given angle.

All three of these were proved impossible with Euclidean tools. 13. Describe the method of exhaustion. Give the names of two mathematicians who used or developed this method.

Calculating an area (or volume) of a given figure by using figures of known area that fit inside (or outside) the given figure but can be modified or replaced by other figures that give a closer approximation. This is usually accompanied by a reductio ad absurdum argument that allows you to get an actual area, rather than a sequence of approximations. (You might want ot provide a nice example here).

Eudoxus developed it and Archimedes used it.

14. What is meant by geometrical algebra? Where do we read about it in Greek mathematical works? Give an example of a theorem of this kind.

Geometrical algebra is the subject covered in Book II of Euclid’s Elements. The topics include several theorems that give geometric results that are nevertheless useful in solving equations, or say things geometrically that we would now state algebraically.

Example: If there are two straight lines, and one of them is cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the sum of the rectangles contained by the uncut straight line and each of the segments.

(What algebraic rule does this correspond to?)