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HISTORICAL VIEW OF THE

terms of the other; each two of these values determinea point in the curve; and, by consequence, we succes-sively trace out the whole locus, as successive corre-sponding values are taken.

Thus, in general, any indeterminate equation withconstant co-efficients would, when solved for one of thevariables, always give a peculiar locus, under certaininvariable conditions both as to extent and position,determined by the constant quantities. An excellentillustration of the idea is afforded by the familiar con-sideration of a map, in which the latitudes and longi-tudes, always measured from a given origin, beingassigned, for successive points, along a coast, or a river ;for example, we can lay down on the map the formand outline of that shore, or the course of that river.And if the latitude and longitude of each point had somefixed and invariable relation which connected them to-gether, and, however the actual values might alter, yetthat relation should remain the same for all points inthe course of the line traced out, the announcement ofsuch relation would constitute the equation of the locus.We have before observed, that the business of the geo-meter is first by his definition to frame some constructionof a geometrical figure, and then by demonstrative de-duction from the conditions of such construction toestablish its various properties. The regular curvedlines of which geometry takes cognisance, are such asresult from such particular mode of construction, in-stances of which we have already noticed. From theseconstructions their properties are deduced. The ancientgeometers investigated many such curves; but theywere all defined upon perfectly independent construc-tions, and had no principle of any common relation.Thus, the discovery of each particular curve, and theinvestigation of its properties, cost the geometer a dis-tinct effort of invention, and demanded a separateexpenditure of intellectual energy; and even whensuccessful, he was as often indebted to chance as to hisown sagacity.