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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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