THE ELEMENTS. BOOK X 
411 
>ve that 
K 
r. 
rationals 
:es equal 
sa, or the 
,ble with 
mce of* a 
sum or 
fith one 
ible, and 
rding to 
the sign 
Proposi- 
thirteen 
ve other 
another, 
i also be 
equal to 
y'. We 
s proce- 
L4 prove 
straight 
■ it is an 
Die with 
e versa; 
nd with 
sctangle. 
ninators 
ng the 
respec- 
?-k P % 
line an 
infinite number of other irrational straight lines arise each 
of which is different from the preceding. k*p being medial, 
we take another rational straight line a and find the mean 
proportional V(k* pa); this is a new irrational. Take the 
mean between this and a', and so on. 
I have described the contents of Book X at length because 
it is probably not well known to mathematicians, while it is 
geometrically very remarkable and very finished. As regards 
its object Zeuthen has a remark which, I think, must come 
very near the truth. 4 Since such roots of equations of the 
second degree as are incommensurable with the given magni 
tudes cannot be expressed by means of the latter and of num 
bers, it is conceivable that the Greeks, in exact investigations, 
introduced no approximate values, but worked on with the 
magnitudes they had found, which were represented by 
straight lines obtained by the construction corresponding to 
the solution of the equation. That is exactly the same thing 
which happens when we do not evaluate roots but content 
ourselves with expressing them by radical signs and other 
algebraical symbols. But, inasmuch as one straight line looks 
like another, the Greeks did not get the same clear view of 
what they denoted (i. e. by simple inspection) as our system 
of symbols assures to us. For this reason then it was neces 
sary to undertake a classification of the irrational magnitudes 
which had been arrived at by successive solutions of equations 
of the second degree.’ That is, Book X formed a repository 
of results to which could be referred problems depending on 
the solution of certain types of equations, quadratic and 
biquadratic but reducible to quadratics, namely the equations 
x 2 ±2 yx. p ±v . p 2 = 0, 
and x 4 ±2 yx 2 . p 2 ±v. p 4 = 0, 
where p is a rational straight line and ¡x, v are coefficients. 
According to the values of y, v in relation to one another and 
their character {y, but not v, may contain a surd such as 
V in or v/(m/n)) the two positive roots of the first equations are 
the binomial and apotome respectively of some one of the 
orders 4 first ’, 4 second ’, . . . 4 sixth ’, while the two positive 
roots of the latter equation are of some one of the other forms 
of irrationals {A 1} A 2 ), [B 1 , i» 2 ) ... {F 1 , F 2 ).