The Greek Period --- Homework

Problems about Greek mathematics will illustrate that the Greeks achieve a full intellectual development comparable to any modern state. Their mathematics was more limited by what they would not accept that what they could not comprehend. You will see an assortment of problems ranging from essay type to actual geometric constructions, from number theory to modern Diophantine equations.

Early Greek mathematics

  1. If the very early Greeks has decided that the tangent line to a circle meets at a line segment, rather than a point, what simple contradiction about triangles could be proved? (See, §2.3, reference Democritus of Abdera in "The Origins of Greek Mathematics.")

  2. Find the analytic formula for the trisectrix. (You will need trigonometry.)

  3. Show that in a Pythagorean triple, if one of the terms is odd, then two of them must be odd and one even.

  4. Show that in a Pythagorean triple, if the largest term is divisible by 4, then so are the other two terms.

  5. Show that $\sqrt{5}$ is incommensurable. Use your argument to show that every non-square number is incommensurable.

  6. Given that an equilateral pentagon and triangle can be inscribed in a circle, show how to inscribe a 15-gon in a circle. (Note. This must be extablished strictly by Euclidean construction.)

  7. Show the other part of the double reductio ad absurdem argument for proving the Eudoxus theorem that the ratio of the areas of two circles are as the squares of their diameters. That is, you need to obtain the contradiction using circumscribed $6\cdot 2^{n}$- gons. (The proof is very similar.)

  8. Show the iterative nature of the subdivision of a line into extreme and mean ratio. That is, one the initial construction is complete, show how to subdivide the larger of the length into extreme and mean ratio, and so on.

  9. We know it is impossible to square the circle, but various figures can be "squared." Carry out the following constructions.

    1. Square any rectangle. (Hint. This is the same as solving the equation $x^{2}=ab$ geometrically, and in turn this amounts to determining a square root geometrically. Of course, there is a multiplication problem embedded here, namely finding the product $ab$, once again geometrically.)

    2. Square any triangle. (Hint. This is easy once you have "squared" the retangle. Why? Yet there is still some work to do.)

    3. From this show how to square any polygonal figure.

  10. Find a construction to solve the equation $6:3=4:x$.

  11. Find a construction of the division of a line into two portions with lengths in the ratio $\sqrt{2}:1.$

  12. Find a construction of the division of a line into two portions with lengths in the ratio $\sqrt{3}:1.$

  13. Generalize the problem above this result to find a construction of the division of a line into two portions with length in the ratio$\sqrt{p}:\sqrt{q}$

  14. There is every reason to believe that texts were used for several generations before they were replaced. Thus, unlike today, teachers would teach from the same text they learned from. Write a brief essay (1-2 pages) on the benefits that would accrue to both teacher and student from both teaching and/or learning from the same text. (Let us the assume that the text being used from is correct - also unlike today.)

  15. Give a classification of prime numbers. (e.g. Mercenne primes, Fermat primes). Include about ten classes. Cite when they were first used, and what if any applications are there, and whether it is known if there are an infinite or finite number of members in the class.

The classical and helenistic periods of Greek mathematics

  1. Find at least five axioms equivalent to the parallel lines axiom. Give the dates when they were proposed and the person that proposed them.

  2. Express the following propositions from Euclid's The Elements in modern terms. (You may need to consult with Euclid to get the meanings of terms.)

    1. Proposition VII-21. Numbers prime to one another are the least of those which have the same ratio with them.

    2. Proposition VII-28. If two numbers be prime to one another, the sum will also be prime to each of them.

    3. Proposition X-13. If two magnitudes be commensurable, and the one of them be incommensurab le with any magnitudek, the remaining one will also be incommensurable with the same.

    4. Proposition X-36. If two rational straight lines commensurable in square only be added together, the whole is irrational.

  3. Eudemus (c. 335 BCE) claimed to have added to the geometrical theory with a number of propositions. Here is one example: If a perpendicular is drawn to the hypotenuse from the vertex of the right angle of a right-angled triangle, each side is the mean proportional between the hypotenuse and its adjacent segment. Explain this propostion in modern terms.

  4. Concerning Ionian enumeration.

    1. What is the number MATH in hindu-arabic decimal (i.e. modern western) notation? Can there be any confusion of what it denotes?

    2. Determine the product MATHIs this different from MATH?

    3. What is a myriad? (Note the meaning of the term today.)

  5. In what sense can one consider Archimedes to be the inventor of calculus? Be sure to clarify what you mean by the term "inventor". Use the lecture notes and the myriad of Internet sources as your sources.

  6. Prove the Archimedes result that the area of a section of a parabola equals $\dfrac{4}{3}$ times the area of the inscribed triangle of the same height. (You may use calculus here.)

  7. With respect to a circle of radius $r$, Further, let $b_{1}$,MATH denote the regular inscribed MATH polygons, similarly, MATH for the circumscribed polygons. Prove the following formulae give the relations between the perimeters ($p_{n}$ and $P_{n})$ and areas ($a_{n}$ and $A_{n})$ of these $6\cdot 2^{n}$ polygons.

    1. MATH

    2. MATH

  8. Of the many innovations in Euclid, which one or two do you find are the most innovative?


  9. Prove Aristarchus' theorem that if MATH, then MATH (You may use any mathematical techniques you know. Calculus helps. Can you prove it using only geometrical methods?)

  10. A result known to and used by Aristarchus is that if MATH, then MATH decreases and MATH increases. Show these propositions.

  11. Using induction, prove that
    MATH
    and so on.


  12. Using the parabola, show how to find $\root{3}\of{2}.$ (Hint. You will need two parabolas with orthogonal axes.) $\ $ Can you generalize to find the cube root of any number?

  13. Prove a generalization of the Pythagorean theorem ála Pappus using triangles instead of paralellograms. [You need to get the correct statement first. This will come by mimicing the Pappus proof for triangles.]

  14. Prove that the Pappus generalization of the Pythagorean Theorem yields the Pythagorean Theorem in the case of a right triangle $\triangle ABC$. That is the "parallelogram" constructed in the Pappus proof is indeed a square upon the hypotenuse. (Hint. Here's an approach. Show $KM\perp BC$ $\Longrightarrow $ $BD\perp BC$ and then MATH. To do this, discover an importance congruence. )
    hmk_greek__38.png



  15. Give a proof of the Zenodorus theorem that if a circle and regular polygon have the same perimeter, the circle has the greater area. (Hint. One part of the above problem helps.)

  16. Find two numbers in mean proportion between $a$ and $b$.

  17. Find two numbers in mean proportion between $5$ and $9$.

  18. A sequence $\{a_{n}\}$ is in arithmetic progression if $a_{n+1}-a_{n}=d,$ is constant for all $n$. If $d>0$, the sequence is increasing, and if $d<0,$ the sequence is decreasing. Show than if a given arithmetic sequence is decreasing then
    MATH
    (Hypsicles, 2$^{\text{nd}}$ century BCE)

  19. Apply the Heron method to approximate the cube root of 45. How accurate is the approximation?

  20. Apply the Heron method to approximate the cube root of 1450. How accurate is the approximation?

  21. Show that the solution to any cubic equation can be determined by intersecting a hyperbola with a parabola.

  22. Prove that any two triangles inscribed in a circle having the same base have the same vertex. (This is Proposition III -21 of Euclid. See the diagram below.)
    MATH

  23. From the previous problem prove the conclusion that the maximum angle subtended from a point on a line to a segment of a line must occur at the point where a circle passing through the endpoints of the segment is tangent to the line.
    MATH


    MATH

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