THE DATA 
423 
distribu 
ai! their 
te senses 
as areas, 
^iven in 
.1 ’ (Defs. 
en their 
sides to 
ì ‘ given 
’ : a not 
'position 
lion and 
;t of the 
en figure 
tions are 
84, 85 of the Data, which state that ‘ If two straight lines 
contain a given area in a given angle, and if the difference 
(sum) of them be given, then shall each of them be given.’ 
The proofs depend directly upon those of Propositions 58, 59, 
‘ If a given area be applied to a given straight line, falling 
short (exceeding) by a figure given in species, the breadths 
of the deficiency (excess) are given.’ All the ‘areas’ are 
parallelograms. 
We will give the proof of Proposition 59 (the case of 
‘ excess ’). Let the given area AB „ . 
be applied to AG, exceeding by the A J~~l 
figure CB given in species. I say / \ / 
that each of the sides HG, GE is / \ / / 
given. D FA 
Bisect DE in F, and construct / / /\ / 
such as 
rocedure 
n in the 
form of 
relation 
t can be 
pothesis 
thing is 
ire is of 
ed us to 
t would 
i of the 
»rems or 
on EF the figure FG similar and L L Z_V 
similarly situated to CB (VI. 18). A 
Therefore FG, GB are about the same diagonal (VI. 26). 
Complete the figure. 
Then FG, being similar to CB, is given in species, and, 
since FE is given, FG is.given in magnitude (Prop. 52). 
But AB is given; therefore AB + FG, that is to say, KXj, is 
given in magnitude. But it is also given in species, being 
similar to CB; therefore the sides of KL are given (Prop. 55). 
Therefore KH is given, and, since KG = EF is also given, 
the difference GH is given. And GH has a given ratio to HB; 
therefore HB is also given (Prop. 2). 
Eucl. III. 35, 36 about the ‘power’ of a point with reference 
to a circle have their equivalent in Data 91, 92 to the effect 
that, given a circle and a point in the same plane, the rectangle 
atter of 
different 
he case. 
II. 5, 6 
■ x 2 = h 2 . 
ution of 
contained by the intercepts between this point and the points 
in which respectively the circumference is cut by any straight 
line passing through the point and meeting the circle is 
also given. 
A few more enunciations may be quoted. Proposition 8 
(compound ratio): Magnitudes which have given ratios to the 
same magnitude have a given ratio to one another also. 
Propositions 45, 46 (similar triangles): If a triangle have one 
angle given, and the ratio of the sum of the sides containing 
>osi tions 
that angle, or another angle, to the third side (in each case) be