422 
EUCLID 
exactly agree, the differences, however, affecting the distribu 
tion and numbering of the propositions rather than their 
substance. The book begins with definitions of the senses 
in which things are said to be given. Things such as areas, 
straight lines, angles and ratios are said to be ‘ given in 
magnitude when we can make others equal to them’ (Defs. 
1-2). Rectilineal figures are ‘ given in species’ when their 
angles are severally given as well as the ratios of the sides to 
one another (Def. 3). Points, lines and angles are ‘given 
in position ’ ‘ when they always occupy the same place ’ : a not 
very illuminating definition (4), A circle is given in position 
and in magnitude when the centre is given in position and 
the radius in magnitude (6) ; and so on. The object of the 
proposition called a Datum is to prove that, if in a given figure 
certain parts or relations are given, other parts or relations are 
also given, in one or other of these senses. 
It is clear that a systematic collection of Data such as 
Euclid’s would very much facilitate and shorten the procedure 
in analysis ; this no doubt accounts for its inclusion in the 
Treasury of Analysis. It is to be observed that this form of 
proposition does not actually determine the thing or relation 
which is shown to be given, but merely proves that it can be 
determined when once the facts stated in the hypothesis 
are known ; if the proposition stated that a certain thing is 
so and so, e.g. that a certain straight line in the figure is of 
a certain length, it would be a theorem ; if it directed us to 
find the thing instead of proving that it is ‘given’, it would 
be a problem; hence many propositions of the form of the 
Data could alternatively be stated in the form of theorems or 
problems. 
We should naturally expect much of the subject-matter of 
the Elements to appear again in the Data under the different 
aspect proper to that book ; and this proves to be the case. 
We have already mentioned the connexion of Eucl. II. 5, 6 
with the solution of the mixed quadratic equations ax±x 2 = h 2 . 
The solution of these equations is equivalent to the solution of 
the simultaneous equations 
y±x = a l 
xy — b 2 ) ' 
and Euclid shows how to solve these equations in Propositions