The Elements of Euclid

Abstract

Euclid’s Elements are considered by far the most famous mathematical oeuvre. Comprising about 500 pages organised in 13 books, they were written around 300 B.C. All the mathematical knowledge of the period is collected there and presented with a rigour which remained unequalled for the following two thousand years. Over the years, the Elements have been copied, recopied, modified, commented upon and interpreted unceasingly. Only the painstaking comparison of all available sources allowed Heiberg in 1888 to essentially reconstruct the original version. The most important source (M.S. 190; this manuscript dates from the 10th century) was discovered in the treasury1 of the Vatican, when Napoleon’s troops invaded Rome in 1809. Heiberg’s text has been translated into all scientific languages. The English translation by Sir Thomas L.Heath in 1908 (second enlarged edition 1926) is completed by copious comments.

“At age eleven, I began Euclid, with my brother as my tutor. This was one of the greatest events of my life, as dazzling as first love. I had not imagined that there was anything as delicious in the world.” (B.Russell, quoted from K.Hoechsmann, Editorial, π in the Sky, Issue 9, Dec. 2005. A few paragraphs later K.H. added: An innocent look at a page of contemporary the orems is no doubt less likely to evoke feelings of “first love”.)

“At the age of 16, Abel’s genius suddenly became apparent. Mr. Holmbo¨e, then professor in his school, gave him private lessons. Having quickly absorbed the Elements, he went through the Introductio and the Institutiones calculi differentialis and integralis of Euler. From here on, he progressed alone.” (Obituary for Abel by Crelle, J. Reine Angew.Math. 4 (1829) p. 402; transl. from the French)

“The year 1868 must be characterised as [Sophus Lie’s] break- through year. … as early as January, he borrowed [from the Uni- versity Library] Euclid’s major work, The Elements …” (The Mathematician Sophus Lie by A. Stubhaug, Springer 2002, p. 102)

“There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures … from the plan laid down by Euclid.” (A. De Morgan 1848; copied from the Preface of Heath, 1926)

“Die Lehrart, die man schon in dem ¨altesten auf unsere Zeit gekommenen Lehrbuche der Mathematik (den Elementen des Eu- klides) antrifft, hat einen so hohen Grad der Vollkommenheit, dass sie von jeher ein Gegenstand der Bewunderung [war] … [The style of teaching, which we already encounter in the oldest mathemati-cal textbook that has survived (the Elements of Euclid), has such a high degree of perfection that it has always been the object of great admiration …]” (B.Bolzano, Gr¨ossenlehre, p. 18r, 1848)