Euclid's Book on Divisions of Figures, with a Restoration based on Woepcke's Text and on the “Practices Geometriae” of Leonardo Pisano

Abstract

A TYPICAL problem of the Divisions is “to cut off a certain-fraction from a given triangle by a line drawn from a given point within the triangle.” Of the thirty-six propositions of the book, six are auxiliary, two deal with areas the boundaries of which are partly or wholly circular; the rest are concerned with the division of triangles and quadrilaterals. For several! reasons the treatise is very interesting; it is apparently cotaplete, the Arabic text¹ translated by Woepcke (Journ. As., 1851) seems to represent Euclid's text, and although the same cannot be said about the proofs supplied by Leonardo of Pisa (Fibonacci), they retain a great deal of the old Greek style. The peculiar fact that shows how, even early in the thirteenth century, geometry, as understood by the ancient Greeks, had become infected by arithmetic, is that Leonardo constantly gives numerical illustrations, and even refers (p. 41, note) to segments defining a given ratio as “numbers,” which we may be sure Euclid would not do in this context. Since the editor's translation of Leonardo is not absolutely literal, we must not lay stress on the passage (p. 61):—“Apply a rectangle equal to the rectangle zb bi to the line bi, but exceeding by a square; that is, to bi apply a line such that when multiplied by itself and by bi the sum will be equal to the product of zb and bi“,” the explanatory clause being possibly Dr. Archibald's; but however that may be, this sentence is a good illustration of the contrast between Greek methods and others.

Euclid's Book on Divisions of Figures, with a Restoration based on Woepcke's Text and on the œPractices Geometriae of Leonardo Pisano.

By Prof. R. C. Archibald. Pp. viii“88. (Cambridge: At the University Press, 1915.) Price 6s. net.

Enjoying our latest content?
Log in or create an account to continue

• Access the most recent journalism from Nature's award-winning team
• Explore the latest features & opinion covering groundbreaking research

Access through your institution

or

Sign in or create an account

Continue with Google

Continue with ORCiD