ercises of the Department were opened by an essay from Prof. WILLIAM M. THORNTON, Adjunct Professor of Applied Mathematics, University of Virginia, upon the

POSITION OF MODERN MATHEMATICAL THEORIES IN OUR HIGHER COURSES OF PURE MATHEMATICS.

If the English-speaking mathematician could accept the dictum of Charles Dupin, that progress in Science is true only when it reacts on the elementary treatise, and should judge from this point of view the achievements of the grand geometers of our century, he would be logically brought to a conviction as dismaying as surprising. His examination would bring to light an innumerable host of text-books. But in the expounders of a science which boasts of peculiar lucidity and irrefragable logic, he would find for lucidity contradictions and confusion, for logic bare assumption and vicious argumentation. In all the crowd of geometries he would find that what was true was not new but old as Euclid and Archimedes, while what was new was not true. In the mob of Algebras he would find only vain attempts to disguise poverty of matter by an overplus of impossible examples, to make mathematical athletes of boys in their teens, to fit on youthful Davids the ponderous armor of Saul. If he should consider the résults of this teaching when the student approached the applications of Mathematics he would find his severe judgment only too fully confirmed. The natural philosopher would tell him of pupils whom five, six, or seven years of training in pure mathematics had left incompetent to handle a simple question of Functional Analysis or grasp the most elementary theorems of Physics when clad in mathematical symbols or penetrate beneath the worthless garment of x's and y's, of series and cosines, to the divine truth of nature which they contained. The Engineer would boldly choose as his standards the Elements of Euclid and the Universal Arithmetic of Newton, and fling overboard all the pitiful compilations of a newer date. It is the business of the present paper to offer a scheme for reconciling these diverse interests: for satisfying the claims of pure science, answering the necessities of the Natural Philosopher, and equipping for his work the Engineer.

The elementary Mathematics present to us, as it were, two currents flowing side by side yet clearly separated: appealing to distinct intellectual sympathies yet acting and reacting each on the other. On the one hand we have Geometry and a series of illustrious names from Euclid to Chasles; on the other Analysis and an equal series from Diophantos to Sylvester. The first finds the origines of its history in the beginnings of civilization, and traces its descent from Egyptian through Greek, Arab, and Italian, to our own day. The second of more recent growth shows the first movings of its current in Diophantos, rushes through some underground channel to reappear in the Hindu, is filtered through the Arab into Spain and Italy, gains new volume and strength in Vieta, and thence forward domineers the stream. The conception of the infinitesimal as embodied in the mechanism of the Infinitesimal Calculus added

the last weapon to the armory of the geometer: the successors of NEWTON and LEIBNIZ had only to complete the conquests of their intellectual ancestors.

When we endeavor to untangle the currents which form this broad stream and trace them back each to the fundamental notion which is its spring, three conceptions present themselves appealing each to its own class of intellects, numbering its own force of workers, elaborated into its special department.

The first of these departments deals with symbols, and the operations performed on them irrespective of their representation. It is the Pure Mechanism of Analysis. Guided by Peacock's principle of the permanence of equivalent forms [a principle which has met with universal acceptation explicit or implicit] it develops in orderly succession from the simplest definitions the theorems of Algebra proper, Goniometry, and the Infinitesimal Calculus. I know of no treatise except Peacock's, in which this point of view is adopted and steadily maintained. Sporadic examples may be found as in the treatment of Indices, the definition of imaginary exponentials; but the treatise has yet to be written which will show the wonderful articulation of the whole system, and explain the true grounds of its development. But this treatise will yet be written; for the principle itself has obtained universal acknowledgment simply in virtue of its own intrinsic value. Witness the lucid statement by DURÈGE in the Introduction to his Theorie der Functionen einer complexen veränderlichen Grässe. "The external consecution and internal harmony in all the parts of Mathematics is due to strict adherence to this axiom, that in subjecting a newly-introduced notion to established operations, the fundamental laws of these operations are assumed to be permanent in application to the new conceptions. This really arbitrary assumption may be carried out as long as no contradiction arises from it."

In the second of the three departments the notion of the continuous variable is fundamental and the orthodox method of treatment is the method of limits. Quantitates infinitæ, says Gauss, in ratiociniis analyticis eatenus admittendæ sunt, quatenus ad theoriam limitum reduci possunt. And the names of Gauss and Cauchy have consecrated the formal employment of a method with the fundamental principle of which no geometer has ever been able to dispense. But it was not the variable of Euler with which they dealt. These two, GAUSS the German "princeps mathematicorum" and CAUCHY the "chief ornament of the French Academy," in creating the theory of Functions of a complex variable laid the corner-stone for the edifice of modern Analysis. It is in the province of the historian of pure mathematics and beyond the scope of this paper to trace the development of this notion. Its first dawn in the mind of JEROME CARDAN gave place to the fuller light of GIRARD's quantités envellopées. WALLIS called them imaginaries. DE MOIVRE, JOHN BERNOULLI, the two FAGNUNOS, D'ALEMBERT, and EULER used them as keen and ready tools. But it remained for GAUSS and CAUCHY to throw a clear light on their true nature and give validity to processes which had before appeared a sort of clever jugglery. The pages of CAUCHY'S Analyse Algébrique and of the famous Exercises prove the power of the new analysis in the

hands of such a master. Like GOLDSMITH in literature, there was no part of our science that he did not touch, and none that he touched did he fail to adorn. He reorganized the so-called Theory of Equations, and compressed into a single Theorem all its fundamental truths. He constructed on a sure basis the entire Theory of Series, and left it in such completeness that the genius of an ABEL could add nothing to what he had written. He created the calculus of Residus, that giant calculus which dwarfs all others into pygmies. He brought the penetrating gaze of his genius to bear on geometry, and in its most familiar theorems discovered rich mines of new truth. The impulse which he communicated is still felt in all the parts of the science, and the boldness of his methods, the lucidity of his style, the rigor of his logic have penetrated and permeated the most remote branches of Analysis.

The third of these departments deals with the notion of number. Although the most ancient it is at the same time the subtlest, and has taxed the intellects of analysts from DIOPHANTOS to DIRICHLET. It bears the delusive titles of the Science of Number, the Higher Arithmetic, or Arithmology: but that student need be master of all the weapons of analysis who attempts its arduous paths. Yet for disciplinary value, for intrinsic beauty, for unity of development, it ranks among the first, if it be not first, of all the branches of Analysis.

When we submit to a similar examination the constituent parts of Geometry with a view to eliminate the ruling conception in each from the mass of details we discover here also three clear-cut divisions.

The first is the older or Euclidean Geometry, the product of the genius of the Greek. In the hands of EUCLID, APOLLONIUS, and ARCHIMEDES it attained its full development, perfected its methods, and completed the chain of its theorems. The force of SIMPSON, the genius of Legendre and CAUCHY, the keenness of DE MORGAN have done but little to modify the one or increase the number of the other. As an historical product it is worthy of the profoundest study, for disciplinary value it stands unrivalled by language or literature, science or art; while its serene beauty, its marvellous symmetry, the strange articulation of all its parts into an organic whole have won for it the homage and admiration of every rank in every age.

When we pass to the second of the three departments, the modern synthic Geometry, we turn as it were from an antique statue to a modern painting full of the warm hues of life. The one stands before us serene and unchangeable, the product of genius working under stern limitations. Its outlines are clear and sharp as those of a Greek temple. It rejects the infinite and deals only with the finite; it disallows the imaginary and deals only with the real; it scorns vagueness and generality and becomes the great teacher of precision. The other is like the Gothic cathedral whose spires shoot into infinity; it embraces in its frame the imaginary; it aims at the utmost generality. The same characters which differentiate ancient literature and modern, ancient art and modern, strangely enough differentiate ancient Geometry and modern. And as no modern has been able to revive Greek drama or Greek sculpture, so no modern has been able to give to the lost treasures of Greek geometry aught but a gal

vanized vitality. The attempt of CHASLES to restore the lost Porisms of Euclid is the saddest requiem over them that man could have penned. But it is fortunate that the claim of this illustrious geometer to honor does not rest on such a vain impossible undertaking. The Géométrie Supérieure, and the Sections Coniques are rivalled only by the barycentrische Calcul and the Kreisverwandtschaft of MöBIUS and supplanted by no existing work. The unity of its method, the lucidity of its proofs, the wide-reaching generality of its views, the classic beauty of its style commend the Géométrie Supérieure to the attentive study of all ages. But the modern Geometry does not confine its growth to the pages of CHASLES, MÖBIUS, STEINER, OF STANDT and PONCELET. In them it developed its own grand organism, and by the theory of Involution included the Geometry of Measure in the Geometry of Position: then continued its impulse and is now busy with the task of reorganizing the descriptive geometry of MONGE, and introducing into his constructive system its own generality, simplicity, and uniformity. It has reformed the whole department of Graphics and made it the resource of the Physicist and the Engineer in the most complicated problems. It has impregnated with vitality the theorem of the Polygon of Forces, which falling unnoticed from the pen of LAMI, in 1687, after the lapse of nearly two centuries has given birth to the science of Graphical Statics. The Cartesian convention of signs [in fact the natural product of the principle of permanence] which had been usurped by Analysis from the time of DESCARTES, in the grasp of BELLAVITIS, has enriched the elements with the Theory of Equipollences and put within the reach of the tyro the subtlest problems of the Ancient Geometry. The directive power of the imaginary, a power dimly foreseen by KUIN a century before (1750), Sir W. R. HAMILTON has developed into the calculus of Quaternions.

The third division is based on the convention of DESCARTES embodied in the system of Coördinates which bears his name, is equipped with all the weapons of a highly-developed system of Analysis and armed with the methods of the Modern Geometry. As we likened the Euclidean Geometry to an antique statue and the modern Synthetic Geometry to a modern painting we may complete the parallel by likening the Analytical or Coördinate Geometry to a modern engine. Its beauty is the beauty of incomparable fitness for its work, its simplicity is the simplicity of uniform and unvarying action, its gigantic power makes possible for the beginner in science the solution of problems which foiled the giants of an earlier epoch. "The analytical method," says CHASLES, "on account of its universality should be prescribed by preference if not universally." "Let us not be dazzled," says LAMé, "by the simplicity, the lucidity, the elegance of certain purely geometric demonstrations into substituting them for those analytical methods which in reality detected the theorems enunciated and which if properly presented are equally simple, equally lucid, equally elegant."

Having completed this swift survey of the broad field of mathematical science and separated its characteristic elements we have next to define the extent of the course of mathematics. The twofold object of such a course is this:-to afford the student a clear, systematic view of all the

branches of pure mathematics and to enable him to acquire so much of familiarity and dexterity in the use of its algorithm as will enable him to read fluently and with intelligent appreciation the original writings of the great masters of our science. The one of these objects fixes the extent of the theory to be expounded; the other prescribes the necessary amount of illustration. The second requirement will vary with the pupil; sufficient provision having been made it must be left to the intelligence of the teacher to increase or diminish or alter this amount. The first requirement however is fixed by the nature of the science and a brief mental review enables us to eliminate from the general classification of the mathematical sciences those branches which are fundamental. They are as follows:

Such are the departments which must be elaborated in every complete course of pure mathematics. The omission of any one of them leaves a gap which nothing can fill: the insertion of others would simply repeat principles already developed. And finally the mastery of the fundamental principles of these branches will leave the student prepared to cope with any mathematical treatise which may be presented to him. He need shrink not even from the great mathematical poem of Laplace, the Mécanique Céleste.

The defect of the courses of mathematics now taught in our schools, colleges, and universities is twofold. On the one hand in the exposition of the theory they are strangely blind, fragmentary, and illogical. On the other in the illustration of the theory they vainly attempt to atone for the first defect by a great excrescence of exercises and examples whose fatuous absurdity fills the mathematician with amazement and the student with dismay. "They bind heavy burdens and grievous to be borne and lay them on men's shoulders; but they themselves will not move them with one of their fingers." It would be an invidious task to single out any special treatise and expose in it defects which are well-nigh universal. It may suffice to say that after examining all the American works on algebra which have met his eye the reader of this paper has been unable to find one in which the whole theory of series was not erroneously presented, and this though the Analyse Algébrique was published more than fifty years ago [1821]. The compilers of text-books on the Elements of Geometry escape SCYLLA only to fall into CHARYBDIS when they endeavor to graft on the Euclidean stock the generality of modern methods. It is in the popular manual of the Infinitesimal Calculus