"The general truths concerning relations of space which depend upon the axioms and definitions contained in Euclid's Elements, and which involve only properties of straight lines and circles, are termed Elementary Geometry: all beyond this belongs to the Higher Geometry. To this latter province appertain... all propositions respecting the lengths of any portions of curve lines; for these cannot be obtained by means of the principles of the Elements alone. Here then we must ask to what other principles the geometer has recourse, and from what source these are drawn. Is there any origin of geometrical truth which we have not yet explored? The Idea of a Limit supplies a new mode of establishing mathematical truths. ...a curve is not made up of straight lines, and therefore we cannot by means of any of the doctrines of elementary geometry measure the length of any curve. But we may make up a figure nearly resembling any curve by putting together many short straight lines, just as a polygonal building of very many sides may nearly resemble a circular room. And in order to approach nearer and nearer to the curve we may make the sides more and more small more and more numerous. ...by multiplying the sides we may approach more and more closely to the curve till no appreciable difference remains. The curve line is the Limit of the polygon; and in this process we proceed on the Axiom, that "What is true up to the limit is true at the limit." ... thus the relations of the elementary figures enable us to advance to the properties of the most complex cases. A Limit is a peculiar and fundamental conception, the use of which in proving the propositions of the Higher Geometry cannot be superseded by any combination of other hypotheses and definitions. ...The ancients did not expressly introduce this conception of a Limit into their mathematical reasonings, although in the application of what is termed the Method of Exhaustions they were in fact proceeding upon an obscure apprehension of principles equivalent to those of the Method of Limits."