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42 THE NATURE OF MATHEMATICS 

P, such that OP is the length of the circumference of a circle, 
say of unit diameter. For picturesqueness, we may imagine 
this point P got by rolling the circle along the re-axis from O 
through one revolution. The point P will fall a little to the 
left of the point 3^ and a little to the right of the point 3/^, 
and so on ; the point P is not one of the points to which 
names of fractions have been assigned by the process sketched 
above. This can be proved rigidly. If it were not true, it 
would be very easy to " square the circle." 

There are many other points like this. There is no fraction 
which, multiplied by itself, gives 2 ; but there is a length 
the diagonal of a square of unit side which is such that, if we 
were to assume that a number corresponded to every point on 
OX, it would be a number a such that a 2 = 2. We will return 
to this important question of the correspondence of points and 
lines to numbers, and will now briefly recall that " negative 
numbers'' are represented, in Descartes' analytical geometry, 
on the x-axis, by the points to the left of the origin, and, on 
the t/-axis, by the points below the origin. This was explained 
in the second chapter. 

Algebraical geometry gave us a means of classifying curves, 
All straight lines determine equations of the first degree be- 
tween x and y, and all such equations determine straight lines ; 
all equations of the second degree between x and y, that is to 
say, of the form 

ax 2 - + bxy + cy* + dx + ey +f = 0, 

determine curves which the ancient Greeks had studied and 
which result from cutting a solid circular cone, or two equal 
cones with the same axis, whose only point of contact is formed 
by the vertices. It is somewhat of a mystery why the Greek 
geometricians should have pitched upon these particular curves 
to study, and we can only say that it seems, from the present 
standpoint, an exceedingly lucky chance. For these conic 
sections of which, of course, the circle is a particular case 
are all the curves, and those only, which are represented by 
the above equation of the second degree. The three great 
types of curves the " parabola," the " ellipse," and the 
"hyperbola" all result from the above equation when the 
coefficients a, 6, c, d, e, f satisfy certain special conditions. 
Thus, the equation of a circle which is a particular kind of