dividing a given angle into any number of equal parts, or in any given ratio. The chief difficulty in construction is to find the point where the curve intersects radius. The catenary is formed by a perfectly flexible chain, suspended from two fixed points, and is found to have the lowest center of gravity of all curves of the same length joining those points and hence, of all solids generated by curves of such length it will afford the greatest surface. The honor of its determination was contended for by four great geometers, the two Bernouillis, Leibnitz and Huygens, and the dispute elicited a great deal of unbecoming asperity. The latter, however, has derived far greater fame from a treatise on the pendulum than any to be had from the catenary. This, whereby he added to geometry the brilliant theory of evolutes, is ranked by many with the Principia itself. Subsequently Tschirnhausen submitted a fine conception of the generation of curves by straight lines. His famous caustics were made by the intersection of reflected or refracted rays of light. He also indicated other curves to be developed. from tension, which was carried out in the invention of De la Hire, whereby a given curve was made the peculiar basis for the locus of another. But space forbids further annotation on the geometry of curved lines and surfaces.

Wonderful as has been the development within the last century or two of the analytic method, the sedulous culture of which at one time threatened to drive the study of pure geometry from the schools, yet it is to be observed that the last few years have witnessed an equally striking revival of interest in the methods of the ancients. This has been greatly stimulated by English writers on mathematical subjects, and quite recently by some most valuable French contributions. The descriptive geometry of Monge, by showing the relation of plain figures to those in space, permits the deduction of the properties of magnitudes of three dimensions from those of two dimensions. This has been supplemented by Carnot's Geometry of Position, and Poncel's treatise on Projections. Lastly, the introduction of additional propositions touching Transversals, and the writings of Chasles on Duality and Homography, establishing a new theory of conic sections, have revived expectation of attaining greater results from the principles of pure

geometry than have hitherto been deemed possible. Among those, too, who deserve to be ranked high in the catalogue of modern contributors are several American authors, whose works have made a very profound impression upon the advance of this science. The Philosophy of Mathematics, by Bledsoe, is unequalled as a discussion of the principles which lie at the base of modern analytical investigations. A brochure upon arithmetical geometry by Professor Hickman, is attracting attention among thinkers given to such studies, and contains the germ of a very novel exposition of ratios from the arithmetical standpoint. Rotary Motion by Gen. Barnard, new editions of Geometry and Trigonometry, prepared by Prof. Chauvenet, and Hallowell's Geometrical Analysis, are also works displaying a ripe scholarship and that patient thought so indicative of mathematical genius. The Smithsonian publications likewise contain valuable papers to which attention may be well directed.

Having now reached the table-lands of geometry, from whence the loftier heigths of mathematics can be approached with ease and certainty, it may not be amiss to pause here and consider more critically upon what ground of absolute truth this whole attainment rests. From the axioms about equality to the equations about curved lines, we have seen an unbroken chain of deduction set forth, in which the human mind can find no flaw. Granted the premises, and the conclusions become irresistible. This fact alone has given rise to infinite speculation in regard to the intellectual attitude which identifies itself with such primary conceptions. Why should the mind accept these axiomatic truths as not to be denied under any circumstances? Let us see how this is, remembering meanwhile that all the axioms of mathematical reasoning are assertions of different conditions of equality. Going back to the earliest thought, it is evident that intelligence depends on discerning the difference between things present to the sense. To distinguish, the mind must classify, and this classification is merely the recognition of likeness or unlikeness as to some special properties or qualities. Now, if all these special properties in two different objects be exactly alike, then the mind fails to distinguish between them and this condition we designate by equality. Thus, we see the idea of equality is a negative

but necessary element of any intelligence. It is virtually a condition precedent to any reasoning. Again, this idea of equality is one which obtains with the intellectual faculty in any exercise of its functions, not only as regards things, but also as regards relations, in both concrete and abstract forms of thinking. Among perceptions of the senses whilst the truth of equality as to properties may be more or less affected by imperfections of sensation, equality as to relations being an abstract idea, loses all taint of such untruth, and becomes a mental negation of discrepancy between other mental cognitions. Numbers, measures, magnitudes, are in themselves quantitative relations, and the mind in accepting any axiomatic affirmations in regard to them, simply recognizes the dead points, so to speak, in its classification of their relations, the points where it fails to find differences. Hence it is, when we say things equal to the same thing are equal to each other, we assert a state of consciousness which is a primary condition of any intellectual action. The same is true of all other axioms associate with or evolved from that condition of likeness or unlikeness, so that the truth of mathematics, as far as its axioms are concerned, is seen to rest, not on a concrete, but an abstract foundation. Equality in all its phases becomes, therefore, an incontrovertible basis for reasoning, or further comparing, deducing, and classifying new relations.

But besides axioms, it will be said, there are also definitions which contribute as essential elements of geometrical reasoning. These definitions are certain distinct qualities which are predicated of their subject matter to the exclusion of all other qualities. Now, it has already been shown that reasoning itself is the selecting, comparing, and classifying for future reference of objects according to certain distinct qualities, and that it is the possession of this power of attending to a part of the perception instead of the whole which constitutes the basis of all generalization. We can consider length, therefore, as the element of a line with which we are to deal, or straightness, or perpendicularity just as much as we could any or all other properties, and by regarding it always as so conditioned, use it as a counter in reasoning just as well as if we were surrounded on all sides by lines that had only length and no other property. Of course,

if at any time in the chain of argument we were to restore to the line its other qualities, the reasoning would become vicious and erratic, but so long as we dealt with it in the one aspect alone, no mistake can transpire. To say we cannot so conceive of a line because no such thing as a line without breadth ever actually occurs in nature, and our ideas can only be images of sensations before experienced, is as absurd as to say we cannot conceive of infinite space or time, because we have had no sensible experience of such a concept before. We know that in conceiving of space at all there is no limit we can fix but of itself gives out the idea of a beyond, and that this beyond without a possible limit, this negation of limit to space, this stripping it of the accidental trappings of sensation, is in fact the idea of infinity with which we deal and operate and reason. Why, then, may not the line be reduced in like manner to the simple function of length, and so also be dealt with and reasoned about? It would seem as if a denial of this would be tantamount to a denial of all power of abstract reasoning, which, a fortiori, we do know, through our own consciousness, is possessed. So also the assertion that two straight lines cannot inclose a space, which has often been said not to be demonstrable, nor axiomatic, becomes, under this view, a mere recognition of the definition of a right line as correlated with the definition of space, that is the idea of the line being simply extension in one direction, two straight lines would be but extension in two different directions, and the different directions, whilst they may intersect, cannot thereafter meet again without violating the condition of directness. Their prolongation in different directions after intersecting precludes the idea of meeting, for then the directions would not be different, as they would both aim at the same point. But to enclose space they must meet again, because enclosure means connectedness of outline, which is thus shown to be inconsistent with the quality of straightness or directness. And the same may be said of every other definition of geometry, which is as absolutely true, in so far as it relates to a single property, whether of point, line, or surface, as it could be if it embraced them all. To say there is no point with only position and without magnitude in all nature, and therefore the basis of geometric truth is illusory, is merely to trifle with the primary concepts

of the human mind, as much so as to say that all reasoning which deduces evolution in social or political forms of thought is an illusion because we can know only individual men; yet we do not embrace each individual man of that society in our conception.

But it has been said if this be the truth of mathematics as found in geometry, if so great a body of human reason can be thus erected, which shall defy the ages with its indestructibility and prove a perpetual source of delight and instruction to mankind, out of a few manifest relations of equality and a few conventional definitions, why is it that the same processes of ratiocination cannot be applied to other departments of human thought: to law, to logic, to theology? In the first place it may be replied that very many branches of knowledge have already partaken largely of the accuracy and absoluteness of mathematical demonstration, by virtue of a rebuilding from the bottom upon a few simple relations. It is true of chemistry, crystallography, optics, light, heat, motion, and many other of the exact sciences, and to which exact sciences additions are being constantly made. In the second place, it may be said that a complete answer to this question is neither altogether easy in the giving nor entirely satisfactory when given. One cause undoubtedly is, that the fundamental ideas of many lines of thought require much more elaborate conditioning and defining, and by so much restrict the exact reasoning to be applied to them within narrower limits. Another may, perhaps, be found in the fact that the primary concepts and origins of the moral and social sciences are opinions about which different persons are apt to disagree, no matter how simple are the first impressions. But after all, the question still thrusts itself forward from age to age, fascinating to the great thinker and hopeful to the great teacher: can there not be built up out of the reasonings of men on all these subjects masses of truth which shall be unassailable as mathematics. Who shall say? Perhaps the day is already dawning in which this shall be done, and perhaps the worker is even now laying broad the foundations upon which shall arise this greater glory of a near future.