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prolonged. He always considers the whole circle as described. It is unfortunate that the circle is not the true fundamental line in geometry, so that his defective consideration of the straight line might have been of less consequence.

This general idea of a curve which at any

point of it exhibits some uniform property is expressed in geometry by the term "locus." A locus is the curve (or surface, if we do not confine ourselves to a plane) formed by points, all of which possess some given property. To every property in relation to each other which points can have, there corresponds some locus, which consists of all the points possessing the property. In investigating the properties of a locus considered as a whole, we consider any point or points on the locus. Thus in geometry we again meet with the fundamental idea of the variable. Furthermore, in classifying loci under such headings as straight lines, circles, ellipses, etc., we again find the idea of form.

Accordingly, as in algebra we are concerned with variable numbers, correlations between variable numbers, and the classification of correlations into types by the idea of algebraic form; so in geometry we are concerned with variable points, variable points satisfying some condition so as form to a locus, and the classification of loci into types by the idea of conditions of the same form.

Now, the essence of coordinate geometry is the identification of the algebraic correlation with the geometrical locus. The point on a plane is represented in algebra by its two coordinates, $x$ and $y$, and the condition satisfied by any point on the locus is represented

by the corresponding correlation between $x$ and $y$. Finally to correlations expressible in some general algebraic form, such as $ax+by=c$, there correspond loci of some general type, whose geometrical conditions are all of the same form. We have thus arrived at a position where we can effect a complete interchange in ideas and results between the two sciences. Each science throws light on the other, and itself gains immeasurably in power. It is impossible not to feel stirred at the thought of the emotions of men at certain historic moments of adventure and discovery–-Columbus

when he first saw the Western shore, Pizarro when he stared at the Pacific

Ocean, Franklin when the electric spark came

from the string of his kite, Galileo when he

first turned his telescope to the heavens. Such moments are also granted to students in the abstract regions of thought, and high among them must be placed the morning when Descartes lay in bed and invented the method

of coordinate geometry.

When one has once grasped the idea of coordinate geometry, the immediate question which starts to the mind is, What sort of loci correspond to the well-known algebraic forms? For example, the simplest among the general types of algebraic forms is $ax+by=c$. The sort of locus which corresponds

to this is a straight line, and conversely to every straight line there corresponds an equation of this form. It is fortunate that the simplest among the geometrical loci should correspond to the simplest among the