Famous Problems of Geometry and How to Solve Them

by Benjamin Bold

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 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.

Genres: Mathematics, Nonfiction

124 pages, Paperback

First published March 1, 1982

Reviews

[Valerie · Rating 5 out of 5]

More fun than suduko.

[Clintweathers]

Have I mentioned how much I love (and have for decades) Dover? Such great books of geekery, at really great prices.

This is another one, for sure. This is just what it says on the tin: Some of the great geometry problems, with solutions. You really don't need much more than highschool geometry to follow along, and it's fascinating to see how the solutions form.

Plus, there's something just damned cool about a math form where you can literally build anything from the ancient world using nothing more than a straight edge and a compass.

[Saman · Rating 5 out of 5]

The book, as the name suggests, is about the (brief) history of some famous problems in Euclidean Geometry, about the possibility of drawing geometric shapes using only compass and unmarked ruler.
The problems and solutions are explained well, attractive, brief and enlightening. I guess it is accessible to a high school student and still a nice read for people with stronger background.

[Maurizio Codogno · Rating 3 out of 5]

Lo dico subito, a scanso di equivoci: il titolo di questo libro è in effetti fuorviante. È infatti vero che si parla dei tre problemi classici irrisolti della geometria greca, vale a dire la duplicazione del cubo, la trisezione dell'angolo e la quadratura del cerchio, oltre che alla costruzione dei poligoni regolari di un numero qualunque di lati; ma non viene raccontato come li si risolve, o se si preferisce si racconta di come si possono risolvere "barando", cioè senza limitarsi a usare riga e compasso, la prima senza graduazioni e il secondo che non può riportare distanze.
Il testo, almeno per noi italiani, risulta interessante per alcuni punti che generalmente non vengono insegnati nei corsi della matematica, come la traccia della dimostrazione che ha portato Gauss a costruire l'eptadecagono regolare; nel complesso però non è che sia chissaché. D'altra parte, le dimostrazioni più complicate vengono omesse, quindi è comprensibile anche per uno studente all'ultimo anno del liceo o al primo anno di una facoltà scientifica che abbia una buona infarinatura di analisi e non si spaventi a vedere l'equazione ciclotomica.

[Roberto Zanasi · Rating 5 out of 5]

Le costruzioni con riga e compasso, l'impossibilità della risoluzione, mediante riga e compasso, del problema di Delo, della trisezione dell'angolo, della quadratura del cerchio. Infine, la costruibilità mediante riga e compasso dei poligoni regolari. Il tutto ben spiegato e dimostrato, con anche alcuni esercizi (risolti alla fine).