Whosoever will go forth into forest or field where nature most luxuriates, will doubtless be lost in admiration at the many delicate shapes and shades which mingle before his vision; and whoever will reflect upon that scene as it trembles into change at each instant of time, will, perhaps, equally wonder how any pleasure can arise from witnessing such utter confusion of outline and color. Yet, poet and painter will tell you that beneath all such apparent wantonness there obtains a divine principle of order, that not the minutest tendril, nor climbing vine, nor rarest foliage, but has its definite form; and that not a luster anywhere weds the blossom, or shimmers on the leaf, but takes its tint from some wave of light of determinate curvature. And so, go where you will in the wide world of sensation, you find yourself environed by ever-shifting expression that lies behind the immediate impressions of touch and sight, seemingly infinite in variety and disorder yet all, when analyzed, subject to certain and simple classification. They may combine with each other on terms of such subtlety that you shall not be able to tell where direction begins or deviation ends, just as broken lines merge into roundness, or the red and the blue dissolve into purple twilight; but still it is given us to know that the elements of each specific form are there, and to define and calculate the relation of those elements is the province of geometry. With great propriety, therefore, it may be designated the science of all beauty, whether we regard the wealth of linear expression lucent in the shifting scenery of the skies, or the rigid formalism stored up in the treasure-house of crystallization; whether the flowing forms of motion, or the traceries pendent throughout all vegetation.

It is the talisman which works a miracle of reconciliation between seeming antagonisms of shape, and gives even

"To airy nothingness

A local habitation and a name."

From determining the bases of the most exact sciences it reaches up into the highest passages of art. Its analysis finds itself embodied everywhere in the most prosaic uses, contributes supreme direction to the most intricate mechanism, and imposes upon genius itself arbitrary laws of expression. Equally true is it, though it would perhaps offend the fair maiden to be told so, that the charm of her pouting lip or the grace of her rounded limbs, fascinating as they may be to the passionate soul, are yet resolvable into mere geometrical equation. Indeed, an outward manifestation of symmetry does establish a relation to the inner emotional nature which is one of the mysteries of our being. It is the combination of idea and sensation, which dawns gradually upon the mind, so that straightness, or evenness, or curvature come forth apparently as the reproduction of some anterior consciousness. Or as Taylor Lewis has finely said, "In the sight of a straight line even, there is beauty, interest, emotion, something of the soul's own, and this is because, like all beauty, it is in some way soul seeing soul." It is like the feeling of the philosopher, Aristippus, when, after his shipwreck he discovered a circle marked upon the sands: "Let us be of good cheer," said he, "I see mind." Experience here teaches a priori truths, strange as that may seem, and perhaps such is the best elucidation of the axiomatic basis on which this branch of mathematics reposes.

Geometry, however, lays its chief claim to the attention it has received amongst men, not so much on account of its exposition of the beautiful, or even the useful, as it does because of certain methods of reasoning which it has appropriated, enforcing thereby a rigorous logic throughout all its demonstration. So clear and convincing does it thus become to the intellect, that its conclusions, when once reached, are thereafter incontrovertible. This has led many great thinkers in past ages, and some even in the present time, to attempt to assimilate the principles of reasoning upon moral, ethical and political subjects to the exactitude of geometrical proofs.

The attempt, however, has been vain so far, and the din of disputation over controverted questions in every other field of inquiry has gone on century after century, whilst here alone continuity and extension from the earliest historic times to the present hour, evidence an unbroken progress.

Why this should be so, and how this has been so, are certainly problems worthy to enlist the profoundest interest of all who have any reverence for the achievements of the human mind.

The subject matter of the present paper has been desig nated "Geometry-Old and New." This designation is intended to embrace as well the principles which have been handed down from antiquity, as the remarkable additions which have been made by the moderns. Of course it will not be possible to enter upon any elaborate discussion of its many and varied demonstrations. That would require a vastly extended criticism, for there is no important proposition which might not itself invite lengthy review. Thus it is related of Sir Henry Saville, who in the seventeenth century founded a chair of geometry at Oxford, that he opened the exercises by thirteen lectures on the first book of Euclid. Extravagant as this may seem, those most familiar with the subject will readily understand how such lectures might have been both interesting and instructive. Here, however, it is only designed to trace out the general principles which underlie the science, giving occasional illustration to make them intelligible, and incorporating therewith something of a historic resumé, from its origins to the present day.

Sir William Hamilton, the most acute mathematical mind of this century has defined algebra to be the science of pure time, and geometry that of pure space. These are rather abstract than practical definitions, but they serve to indicate the general separation between the two branches of mathematics. The former concerns itself with numbers, as indicia of duration, and seeks to generalize symbolically its results. The latter deals with volumes, surfaces, lines, points and angles, and develops their relations from given properties. Geometry is divided into two parts. The first is called elementary geometry, and treats of those magnitudes whose elements are the right line and circle. This takes in all proportions relating to figures bounded by straight lines, circles,