less than n·D, and that A:B > C:D if and only if for some m and n, m·A > n·B and m·C ≤ n·D.
Euclid’s theory deals only with proportionalities among magnitudes, not with ratios between
pairs of magnitudes, but it is a simple matter to reformulate the theory of Book 5, by
treating ratios A:B as “cuts” in the system of positive fractions m/n. In Book 6 Euclid applies
the theory of proportion to geometric entities and develops the notion of similarity. Books
7 – 9 introduce numbers as objects of study using a separately developed theory of propor-
tion. The major topic of the very difficult Book 10 is a classification of straight lines A which
are called irrational relative to a given straight line R if both A and R and SQ(A) and SQ(R)
are incommensurable.
Book 11 develops basic ideas of solid geometry. Book 12 uses a method, which
is called the “method of exhaustion,” to prove a series of sophisticated results, the
simplest of which is prop. 2: if C and C’ are circles with diameters d and d’, then
C:C’ :: SQ(d):SQ(d’).
In Book 13 Euclid constructs the five regular solids, triangular pyramid, octahedron,
cube, icosahedron, and dodecahedron, circumscribes spheres around them, and character-
izes their edges relative to the diameters of the circumscribing spheres using in the last three
cases the classification of Book 10.
It is clear from Proklos (In Eucl. pp. 65–68 Fr.) that Euclid’s Elements had more than
one predecessor, starting with a work of H K. It is also clear that
much of the contents of the Elements is based on the work of others, most clearly
E (Books 5 and 12) and T (Books 10 and 13). Nevertheless, Euclid’s
Elements is an outstanding achievement which replaced all of its predecessors and sources,
and became both an inspiration and a foil for much of the subsequent history of Western
mathematics.
Ed.: J.L. Heiberg, and H. Menge, Euclidis Opera Omnia, 9 vv. (1883–1916);
P. Ver Eecke, trans., Euclide, L’optique et al catoptrique (1938); DSB 4.414–437, I. Bulmer-Thomas;
B. Vitrac, trans., Euclide, Les Elements 4 vv. (1990–2001); J.L. Berggren and R.S.D. Thomas,
trans., Euclid’s Phaenomena (1996); DPA 3 (2000) 252–272, B. Vitrac; C.M. Taisbak, trans., Dedomena
(2003).
Ian Mueller
pseudo-E, E 15 ⇒ I M’
Euclidean Sectio Canonis (300 – 260 BCE?)
“Division of the Monochord” (= kanonos katatome ̄ = Sectio Canonis), a short text on mathemati-
cal harmonics ascribed in most MSS to E. Fragments are quoted by P
(title and authorship: In Ptolemaei Harmonica Commentarium 98.19 Düring; preface: 90.7–22;
props.1–16: 99.1–103.25) and B (De Institutione Musica iv). Its authorship, date and
unity of composition have been long debated: a logical error in prop. 11 has been used as
evidence against Euclid’s authorship, but arguments for dating it substantially later than
Euclid, and for excising the preface and two (or four) final propositions as late accretions,
have not met with consensus.
The text as we now have it is comprised of five types of material: (1) a discursive preface
attempting to derive mathematical harmonics from a physical acoustics which can account
for the behavior of strings (an essential connection in order for the monochord to be used
demonstratively in props.19–20); (2) nine purely mathematical propositions demonstrating
EUCLIDEAN SECTIO CANONIS