''The Ancient Tradition of Geometric Problems'' is a book on ancient

Greek mathematics Ancient Greek mathematics refers to the history of mathematical ideas and texts in Ancient Greece during Classical antiquity, classical and late antiquity, mostly from the 5th century BC to the 6th century AD. Greek mathematicians lived in cities ...

, focusing on three problems now known to be impossible if one uses only the

straightedge and compass construction In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

s favored by the Greek mathematicians:

squaring the circle Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square (geometry), square with the area of a circle, area of a given circle by using only a finite number of steps with a ...

doubling the cube Doubling the cube, also known as the Delian problem, is an ancient geometry, geometric problem. Given the Edge (geometry), edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first ...

, and

trisecting the angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...

. It was written by Wilbur Knorr (1945–1997), a

historian of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...

, and published in 1986 by

Birkhäuser Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (parti ...

Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, book ...

reprinted it in 1993.

Topics

''The Ancient Tradition of Geometric Problems'' studies the three classical problems of circle-squaring, cube-doubling, and angle trisection throughout the history of Greek mathematics, also considering several other problems studied by the Greeks in which a geometric object with certain properties is to be constructed, in many cases through transformations to other construction problems. The study runs from

Plato Plato ( ; Greek language, Greek: , ; born BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...

and the story of the Delian oracle to the second century BC, when

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

and

Apollonius of Perga Apollonius of Perga ( ; ) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention o ...

flourished; Knorr suggests that the decline in Greek geometry after that time represented a shift in interest to other topics in mathematics rather than a decline in mathematics as a whole. Unlike the earlier work on this material by Thomas Heath, Knorr sticks to the source material as it is, reconstructing the motivation and lines of reasoning followed by the Greek mathematicians and their connections to each other, rather than adding justifications for the correctness of the constructions based on modern mathematical techniques. In modern times, the impossibility of solving the three classical problems by straightedge and compass, finally proven in the 19th century, has often been viewed as analogous to the

foundational crisis of mathematics Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particul ...

of the early 20th century, in which

David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...

's program of reducing mathematics to a system of axioms and calculational rules struggled against logical inconsistencies in its axiom systems,

intuitionist In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of ...

rejection of formalism and dualism, and

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...

showing that no such axiom system could formalize all mathematical truths and remain consistent. However, Knorr argues in ''The Ancient Tradition of Geometric Problems'' that this point of view is anachronistic, and that the Greek mathematicians themselves were more interested in finding and classifying the mathematical tools that could solve these problems than they were in imposing artificial limitations on themselves and in the philosophical consequences of these limitations. When a geometric construction problem does not admit a compass-and-straightedge solution, then either the constraints on the problem or on the solution techniques can be relaxed, and Knorr argues that the Greeks did both. Constructions described by the book include the solution by

Menaechmus Menaechmus (, c. 380 – c. 320 BC) was an ancient Greek mathematician, list of geometers, geometer and philosopher born in Alopeconnesus or Prokonnesos in the Thracian Chersonese, who was known for his friendship with the renowned philosopher P ...

of doubling the cube by finding the intersection points of two

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

s, several

neusis construction In geometry, the neusis (; ; plural: ) is a geometric construction method that was used in antiquity by Greek mathematicians. Geometric construction The neusis construction consists of fitting a line element of given length () in between tw ...

s involving fitting a segment of a given length between two points or curves, and the use of the

Quadratrix of Hippias The quadratrix or trisectrix of Hippias (also called the quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is traced out by the crossing point of two Line (geometry), lines, one moving by translation (geometry), tran ...

for trisecting angles and squaring circles. Some specific theories on the authorship of Greek mathematics, put forward by the book, include the legitimacy of a letter on square-doubling from

Eratosthenes Eratosthenes of Cyrene (; ; – ) was an Ancient Greek polymath: a Greek mathematics, mathematician, geographer, poet, astronomer, and music theory, music theorist. He was a man of learning, becoming the chief librarian at the Library of A ...

Ptolemy III Euergetes Ptolemy III Euergetes (, "Ptolemy the Euergetes, Benefactor"; c. 280 – November/December 222 BC) was the third pharaoh of the Ptolemaic dynasty in Egypt from 246 to 222 BC. The Ptolemaic Kingdom reached the height of its military and economic ...

, a distinction between Socratic-era sophist

Hippias Hippias of Elis (; ; late 5th century BC) was a Greek sophist, and a contemporary of Socrates. With an assurance characteristic of the later sophists, he claimed to be regarded as an authority on all subjects, and lectured on poetry, grammar, his ...

and the Hippias who invented the quadratrix, and a similar distinction between

Aristaeus the Elder Aristaeus the Elder (; 370 – 300 BC) was a Greek mathematician who worked on conic sections. He was a contemporary of Euclid. Life Only little is known of his life. The mathematician Pappus of Alexandria refers to him as Aristaeus the Elder. Pa ...

, a mathematician of the time of Euclid, and the Aristaeus who authored a book on solids (mentioned by

Pappus of Alexandria Pappus of Alexandria (; ; AD) was a Greek mathematics, Greek mathematician of late antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem in projective geometry. Almost nothing is known a ...

), and whom Knorr places at the time of Apollonius. The book is heavily illustrated, and many endnotes provide sources for quotations, additional discussion, and references to related research.

Audience and reception

The book is written for a general audience, unlike a follow-up work published by Knorr, ''Textual Studies in Ancient and Medieval Geometry'' (1989), which is aimed at other experts in the

close reading In literary criticism, close reading is the careful, sustained interpretation of a brief passage of a text. A close reading emphasizes the single and the particular over the general, via close attention to individual words, the syntax, the order ...

of Greek mathematical texts. Nevertheless, reviewer Alan Stenger calls ''The Ancient Tradition of Geometric Problems'' "very specialized and scholarly". Reviewer Colin R. Fletcher calls it "essential reading" for understanding the background and content of the Greek mathematical problem-solving tradition. In its historical scholarship, historian of mathematics

Tom Whiteside Derek Thomas "Tom" Whiteside FBA (23 July 1932 – 22 April 2008) was a British historian of mathematics best known for his studies of the mathematics of Isaac Newton. He was a professor at the University of Cambridge and a fellow of the Britis ...

writes that the book's occasionally speculative nature is justified by its fresh interpretations, well-founded conjectures, and deep knowledge of the subject.

References

External links

*
The Ancient Tradition of Geometric Problems
' at the

Internet Archive The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...

{{DEFAULTSORT:Ancient Tradition of Geometric Problems, The Greek mathematics Books about the history of mathematics 1986 non-fiction books History books about ancient Greece