426 
EUCLID 
division of a circle alluded to by Proclus, it can scarcely have 
contained more than a fragment of Euclid’s original work. 
But Woepcke found in a manuscript at Paris a treatise in 
Arabic on the division of figures, which he translated and 
published in 1851. It is expressly attributed to Euclid in the 
manuscript and corresponds to the indications of the content 
given by Proclus. Here we find divisions of different recti 
linear figures into figures of the same kind, e.g. of triangles 
into triangles or trapezia into trapezia, and also divisions into 
‘ unlike ’ figures, e. g. that of a triangle by a straight line parallel 
to the base. The missing propositions about the division of 
a circle are also here: ‘ to divide into two equal parts a given 
figure bounded by an arc of a circle and two straight lines 
including a given angle ’ (28), and ‘ to draw in a given circle 
two parallel straight lines cutting off a certain fraction from 
the circle’ (29). Unfortunately the' proofs are given of only 
four propositions out of 36, namely Propositions 19, 20, 28, 29, 
the Arabic translator having found the rest too easy and 
omitted them. But the genuineness of the treatise edited by 
Woepcke is attested by the facts that the four proofs which 
remain are elegant and depend on propositions in the 
Elements, and that there is a lemma with a true Greek ring, 
‘ to apply to a straight line a rectangle equal to the rectangle 
contained by AB, AC and deficient by a square’ (18). Moreover, 
the treatise is no fragment, but ends with the words, ‘ end of 
the treatise ’, and is (but for the missing proofs) a well-ordered 
and compact whole. Hence we may safely conclude that 
Woepcke’s tract represents not only Euclid’s work but the 
whole of it. The portion of the Practica geometriae of 
Leonardo of Pisa which deals with the division of figures 
seems to be a restoration and extension of Euclid’s work; 
Leonardo must presumably have come across a version of it 
from the Arabic. 
The type of problem which Euclid’s treatise was designed 
to solve may be stated in general terms as that of dividing a 
given figure by one or more straight lines into parts having 
prescribed ratios to one another or to other given areas. The 
figures divided are the triangle, the parallelogram, the trape 
zium, the quadrilateral, a figure bounded by an arc of a circle 
and two straight lines, and the circle. The figures are divided