Read The Nature of Mathematics Online

by Geeky Nigeria

ANALYTICAL GEOMETRY 43 

lipse is always of the form got from the above equation by 
putting 6 = and c = a. 

It may be mentioned that, long after these curves were in- 
troduced as sections of a cone, Pappus discovered that they 
could all be defined in a plane as loci of a point P which 
moves so that the proportion that the distance of P from a 
fixed point (S) bears to the perpendicular distance of P(PN) 
to a fixed straight line is constant. As this proportion is less 
than equal to, or greater than 1, the curve is an ellipse, 
parabola, or hyperbola, respectively. 

It will not be expected that a detailed account should here 
be given of the curves which result from the development of 
equations of the second or higher degrees between x and y. I 
will merely again emphasize some points which are, in part, 
usually neglected or not clearly stated in text-books. The 
letters "a, 6, . . . x, y," here stand for "numbers" in the 
extended sense. We have seen in what sense we may, with 
the mathematicians, speak of fractionary, positive, and negative 
" numbers," and identify, say, the positive number + 2 and 
the fraction -| with the signless integer 2. Well, then, the 
above letters stand for numbers of that class which includes in 
this sense the fractionary, irrational, positive and negative 
numbers, but excludes the imaginary numbers. We call the 
numbers of this class " real " numbers. The question of 
irrational numbers will be discussed at greater length in the 
sixth chapter, but enough has been said to show how they were 
introduced. In mathematics it has, I think, always happened 
that conceptions have been used long before they were formally 
introduced, and used long before this use could be logically 
justified or whose nature clearly explained. The history of 
mathematics is the history of a faith whose justification has 
been long delayed, and perhaps is not accomplished even now. 

These numbers are the measurements of length, in terms of 
a definite unit, like the inch, of the abscissae and ordinates of 
certain points. We speak of such points simply by naming 
their co-ordinates, and say, for example, that " the distance of 
the point (x, y) from the point (a, 6) is the positive square root 
of (x-a) 2 + (x-b) 2 . 

Notice that or, for example, is the length of a line. It is 
natural to make, as algebraists before Descartes did,jc 2 8tandprww- 

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