Famous Problems of Geometry and How to Solve Them

Benjamin Bold

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486137636
• Topic: Histoire
• Subtopic: Histoire des sciences

It took two millennia to prove the impossible; that is, to prove it is not possible to solve some famous Greek problems in the Greek way (using only straight edge and compasses). In the process of trying to square the circle, trisect the angle and duplicate the cube, other mathematical discoveries were made; for these seemingly trivial diversions occupied some of history's great mathematical minds. Why did Archimedes, Euclid, Newton, Fermat, Gauss, Descartes among so many devote themselves to these conundrums? This book brings readers actively into historical and modern procedures for working the problems, and into the new mathematics that had to be invented before they could be `solved.`
The quest for the circle in the square, the trisected angle, duplicated cube and other straight-edge-compass constructions may be conveniently divided into three periods: from the Greeks, to seventeenth-century calculus and analytic geometry, to nineteenth-century sophistication in irrational and transcendental numbers. Mathematics teacher Benjamin Bold devotes a chapter to each problem, with additional chapters on complex numbers and analytic criteria for constructability. The author guides amateur straight-edge puzzlists into these fascinating complexities with commentary and sets of problems after each chapter. Some knowledge of calculus will enable readers to follow the problems; full solutions are given at the end of the book.
Students of mathematics and geometry, anyone who would like to challenge the Greeks at their own game and simultaneously delve into the development of modern mathematics, will appreciate this book. Find out how Gauss decided to make mathematics his life work upon waking one morning with a vision of a 17-sided polygon in his head; discover the crucial significance of eπi = -1, `one of the most amazing formulas in all of mathematics.` These famous problems, clearly explicated and diagrammed, will amaze and edify curious students and math connoisseurs.

CHAPTER I

Achievement of the Ancient Greeks

USING GEOMETRIC THEOREMS, the Greek mathematicians were able to construct any desired geometric element that could be derived by a finite number of rational operations and extractions of real square roots from the given elements. To illustrate: Suppose we are given the elements a, b, and the unit element. The Greeks could construct a + b, a − b, a·b, a/b, a², and

.

PROBLEM SET I–A

Construct a + b and a − b, using the given line segments. The following diagram shows how to construct ab. If EG is constructed parallel to DF, then x = ab.

PROBLEM SET I–B

1.  Prove that x = ab.

2.  Using a similar procedure construct a²; a/b; a²/b The diagram below shows how to construct

.

A semicircle is constructed on LM as a diameter. NP is perpendicular to LM (P is the intersection of the perpendicular and the semicircle). Then

.

PROBLEM SET I–C

1.  Using the diagram above, prove

.

2.  Using a similar procedure construct

3.  Using the Pythagorean theorem construct line segments equal to

Using these constructions, the Greeks were able to construct the roots of a linear or quadratic equation if the numbers representing the coefficients were the lengths of given line segments.

PROBLEM SET I–D

Construct the root of ax + b = c, where a, b, c are given line segments.

To construct the roots of the quadratic equation x² − ax + b = 0 (a² > 4b), one can proceed as follows: Construct a circle whose diameter BD joins the points B(0, 1) and D(a, b). Then the abscissas of G and F (the points where the circle intersects the X-axis) will be the roots of the quadratic equation.

PROBLEM SET I–E

1.  Why do we use the restriction a² > 4b?

2.  Prove that the abscissas of G and F are the roots of x² − ax + b = 0. Hint: Show that the equation of the circle is

As pointed out in the Introduction, the Greeks (using the basic constructions outlined in this section) achieved considerable success in construction problems. Yet they left a number of unsolved problems for future generations of mathematicians to struggle with. The remainder of this book will be devoted to a brief history of attempts to solve these problems and their final solution in the nineteenth century.

PROBLEM SET I–F

Construct the positive root of x² + x − 1 = 0, given a unit length.

CHAPTER II

An Analytic Criterion for Constructibility

TO ANSWER THE QUESTION “Which constructions are possible with unmarked straight edge and compasses?” it is necessary to establish an analytic criterion for constructibility. Every construction problem presents certain given elements a, b, c . . . and requires us to find certain other elements x, y, z . . . . The conditions of the problem enable us to set up one or more equations whose coefficients will be numbers representing the given elements a, b, c . . . . The solutions of the equations will permit us to express the unknown elements in terms of the given element...