Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/64

the greatest number of successive square roots in any term of $x$; this may be called the rank of $x$. We may then write

$x=a+b{\sqrt {B}}+b'{\sqrt {B'}}+\ldots$,

where $B,B',\ldots$ are all of rank not greater than $m-1$. We can form an equation which $x$ satisfies, and such that all its coefficients are rational functions of $a,b,b',B,B'\ldots$; for ${\sqrt {B}}$ may be eliminated by taking $(x-a-b'{\sqrt {B'}}-\ldots )^{2}=b^{2}B$, and this is of the form

$P_{2}-{\sqrt {B'}}P_{2}'=0$,

from which we form the biquadratic

$P_{2}^{2}-B'P_{2}'^{2}=0$,

in which ${\sqrt {B'}}$ does not occur. Proceeding in this way we obtain an equation in $x$ of degree some power of 2, and of which the coefficients are rational functions of $a,b,B,B',\ldots$, and are therefore of rank $\leq m-1$. This equation is of the form

$L_{1}x^{2^{s}}+L_{2}x^{2^{s}-1}+\ldots =0$,

where $L_{1},L_{2},\ldots$ are at most of rank $m-1$. If $L_{1},L_{2},\ldots$ involve a radical ${\sqrt {K}}$, the equation is of the form

${\sqrt {K}}(b_{1}x^{2^{s}}+\ldots )+(b_{1}'x^{2^{s}}+\ldots )=0$,

and we can as before reduce this to an equation of degree $2^{s+1}$ in which ${\sqrt {K}}$ does not occur; by repeating the process for each radical like ${\sqrt {K}}$, we may eliminate them all, and finally obtain an equation such that the rank of every coefficient is $\leq m-2$. By continual repetition of this procedure we ultimately reach an equation, such that the coefficients are all of rank zero, i.e. rational functions of $(a_{1},a_{2},\ldots a_{2r})$. We now see that the following result has been established:

In order that a point $P$ may be determinable by Euclidean procedure it is necessary that each of its coordinates be a root of an equation of some degree, a power of 2, of which the coefficients are rational functions of $(a_{1},a_{2},\ldots a_{2r})$, the coordinates of the points given in the data of the problem.

From our investigation it is clear that only those algebraic equations which are obtainable by elimination from a sequence of linear and quadratic equations correspond to possible Euclidean problems.

The quadratic equations must consist of sets, those in the first set having coefficients which are rational functions of the given numbers, those in the second set having coefficients of rank at most 1; in the next set the coefficients have rank at most 2, and so on.