ON DIVISIONS OF FIGURES 427 
into two equal parts, or two parts in a given ratio ; or again, 
a given fraction of the figure is to be cut off, or the figure is 
to be divided into several parts in given ratios. The dividing 
straight lines may be transversals drawn through a point 
situated at a vertex of the figure, or a point on any side, on one 
of two parallel sides, in the interior of the figure, outside the 
figure, and so on ; or again, they may be merely parallel lines, 
or lines parallel to a base. The treatise also includes auxiliary 
propositions, (1) ‘to apply to a given straight line a rectangle 
equal to a given area and deficient by a square ’, the proposi 
tion already mentioned, which is equivalent to the algebraical 
solution of the equation ax — x* — 6 2 and depends on Eucl. II. 5 
(cf. p. 152 above) ; (2) propositions in proportion involving 
unequal instead of equal ratios : 
If a, d > or < b. c, then a : b > or < c : d respectively. 
If a : b > c :d, then (a + b) :b > (c + d) :d. 
If a :h < c :d, then (a — h) : b < (c — d):d. 
By way of illustration I will set out shortly three proposi 
tions from the Woepcke text. 
(1) Propositions 19, 20 (slightly generalized): To cut off’ 
a certain fraction (m/n) from a given triangle by a straight 
line drawn through a given point within the triangle (Euclid 
gives two cases corresponding to m/n = \ and m/n = §). 
The construction will be best understood if we work out 
the analysis of the problem (not given by Euclid). 
Suppose that ABC is the given triangle, D the given