VI.2 Euclid
b. Alexandria, Egypt?, ca. 325 B.C.E.; d. Alexandria?, ca. 265 B.C.E.
Deduction; postulate; reductio ad absurdum
Nothing is known about Euclid’s life. In fact, his major work, the Elements, is now seen as a rather loose collection, with no strong authorial voice and no clear way of determining what, if any, Euclid’s original contributions were. Born in the cultural climate of Ptolemaic Alexandria, the text probably aimed at systematizing the current knowledge in some mathematical areas.
The Elements covers plane geometry (including the squaring of any rectilinear figure, the bisection of an arc, the inscription and circumscription of polygons in circles, the finding of a mean proportional), solid geometry (e.g., the ratio of spheres to one another, the five regular solids), and arithmetic, from relatively simple (the properties of odd and even numbers, prime number theory) to more complex (commensurable and incommensurable lines, binomials and apotomes).
The title hints at the foundational character of the text, which starts with definitions of mathematical objects (e.g., point, straight line, scalene triangle), postulates (e.g., all right angles are equal to one another) and common notions (e.g., the whole is greater than the part). These initial premises are not demonstrated—whether some postulates are demonstrable spawned debate in antiquity, and later led to non-Euclidean geometries. In a style which has been termed axiomatic -deductive, proofs tend to be general rather than specific; they use a restricted set of formulaic expressions, refer to a lettered diagram, and each of their steps is justified by appeal to undemonstrated premises, to previous proofs, or to very simple notions, such as the principle of the excluded middle. Some proofs use reductio ad absurdum: instead of directly showing that something is the case, they proceed to show that any alternative is impossible.
There are parts of the book that reveal the presence of different, less abstract, demonstrative procedures. For instance, one of the theorems establishing criteria for two triangles to have the same area refers to one triangle being “superimposed” on the other, with the reader effectively invited to verify that their areas are indeed equal. The appeal is to a mental operation, which is quite different from the logical step-by- step method found elsewhere. Again, book IX contains propositions on odd and even numbers, which are often seen as vestiges of Pythagorean mathematics, to be demonstrated with the help of pebbles. The coexistence itself of arithmetic and geometry has been puzzling for some historians, who have proposed a notion of “geometric algebra,” so that book II, ostensibly about squares and rectangles built on segments of straight lines, would in fact foreshadow modern equations.
As well as works on astronomy, optics, and music, the Data, which is about solving geometrical problems on the basis of some elements that are already given, is also attributed to Euclid. His fame is, however, inextricably linked to the Elements. The very absence of a strong authorial voice has perhaps facilitated other mathematicians’ interaction with the text, which has been appropriated, added to, interfered with, and commented upon since antiquity. This very plasticity helped to make it possibly the most popular mathematical book of all time. (For more about its impact on the early development of mathematics, see GEOMETRY [II.2], THE DEVELOPMENT OF ABSTRACT ALGEBRA [II.3], and THE DEVELOPMENT OF THE IDEA OF PROOF [II.6].)
Further Reading
Euclid. 1990–2001. Les Éléments d’Euclide d’Alexandrie; Traduits du Texte de Heiberg, general introduction by M. Caveing, translation and commentary by B. Vitrac, four volumes. Paris: Presses Universitaires de France.
Netz, R. 1999. The Shaping of Deduction in Greek Mathematics. A Study in Cognitive History. Cambridge: Cambridge University Press.
Serafina Cuomo