^ -j/v^^ REPORTS OF DEPARTMENTS THE DEPARTMENT OF MATHEMATICS 61 By William Edward Stora'. THE DEPARTMENT OF PHYSICS 85 By Arthur Gordon Webster. THE DEPARTMENT OF BIOLOGY 99 By Clifton F. Hodoe. THE DEPARTMENT OF PSYCHOLOGY 119 General Psychology 122 By Edmund C. Sanford. Psycho-Pathology 144 By Adolf Meter. Anthropology 148 By Alexander F. Ciiamrerlain. Pedagogy 161 By William H. Buunham. Philosophy 177 By G. Stanley Hall. THE LIBRARY 187 By Loi:is N. Wilson. REPORT OF THE TREASURER 199 By Thomas H. Gage. V vi Table of Contents SCIENTIFIC LECTURES DELIVERED IN CONNECTION WITH THE DECENNIAL CELEBRATION PAGE ISmile Picard, Professor of Mathematics at the University of Paris. 1. Sur I'Extension de quelques Notions Matliematiques, et en parti- culier de I'Idee de Fonction depuis un Siecle .... 207 2. Quelques VuesGenerales sur la Theorie des Equations Differentielles 224 3. Sur la Theorie des Fonctions Analytiques et sur quelques Fonc- tions Speciales 241 LuDWiG BoLTZMANX, Professor of Tlieoretical Physics at the University of Vienna. Ueber die Grundprincipien und Grundgleiehuugen der Mechanik. (Four Lectures) 261 Santiago Ramon y Cajal, Professor of Histology and Rector of the University of Madrid. 1. Comparative Study of the Sensory Areas of the Human Cortex . 311 2. Layer of the Large Stellate Cells 336 3. The Sensori-Motor Cortex 360 Angelo Mosso, Professor of Physiology and Rector of the University of Turin. 1. Psychic Processes and Muscular Exercise 383 2. The Mechanism of the Emotions 396 August Forel, Late Professor of Psychiatry at the University of Zilrich and Director of the Burgholzli Asylum. 1. Hypnotism and Cerebral Activity 409 2. A Sketch of the Biology of Ants . . ' 433 DEGREES CONFERRED, 1889-1899 453 TITLES OF PUBLISHED PAPERS 459 SPECIAL STUDENTS 565 HISTORICAL SKETCH. HISTORICAL SKETCH. Claek Univeksity was founded by the munificence of Jonas G. Clark, a native of Worcester County, whose plans, conceived more than twenty years before, had gradually grown with his fortune. His affairs had been so arranged as to allow long intervals for travel and study. During eight years thus spent, the leading foreign institutions of learning, old and new, were visited, and their records gathered and read. These studies centred about the means by which the highest culture of one generation is best transmitted to the ablest youths of the next, and especially about the external conditions most favorable for increashig the sum of human knowledge. To the improvement of these means and the enlargement of these conditions, the new University was devoted. It was the strong and exjDress desire of the founder that the highest possible academic standards be here forever maintained ; that special opportunities and inducements be offered to research ; that to this end the instructors be not overburdened with teaching and examinations ; that all available experience, both of older countries and our own, be freely utilized ; and that new measures, and even innovations, if really helpful to the highest needs of modern science and culture, be no less freely adopted ; in fine, that the great opportunities of a new foundation in this laud and age be diligently explored and improved. He chose Worcester as the seat of the new foundation after mature deliberation, — first. Because its location is central among the best colleges of the East, and by supplementing rather than duplicating their work, he hoped to ad- vance all their interests and to secure their good will and active support, that together they might take further steps in the development of superior education in New England ; and secondly. Because he believed the culture of this city would insure tliat en- lightened public opinion indispensable in maintaining these educational 2 Historical Sketch. standards at their highest, and that its wealth would insure the perpetual increase of revenue required by the rapid progress of science. As the first positive step toward the realization of these long- formed plans, Mr. Clark invited the following gentlemen to constitute with himself a Board of Trustees : — Stephen Salisbury, A.B., Harvard, 1856; Universities of Paris and Berlin, 1856-58; LL.B., Harvard, 1861; President Antiquarian Society since 1887; State Senator, 1892-95. Charles Devens, A.B., Harvard, 1838 ; LL.B., Harvard, 1840 ; Major-General, 1863 ; Associate Justice of the Massachusetts Superior Court, 1867-73 ; Associate Justice of the Massachusetts Supreme Judicial Court, 1873-77, and again, 1881-91 ; Attorney-General of the United States, 1877-81 ; LL.D., Colmnbia and Harvard, 1877; Died January 7, 1891. George F. Hoar, A.B., Harvard, 1846 ; LL.B., Harvard, 1849 ; United States House of Eepresentatives, 1869-77; Member Electoral Commission, 1876; United States Senate since 1877 ; Chairman of Judiciary Committee, 1891 — ; LL.D., William and Mary, Amherst, Harvard, and Yale. "William W. Eice, A.B., Bowdoin, 1846; admitted to Bar, 1854; United States House of Representatives, 1876-86 ; LL.D., Bowdoin, 1886. Died March 1, 1896. Joseph Sargent, A.B., Harvard, 1834 ; M.D., Harvard, 1837 ; London and Paris Hospitals, 1838-40. Died October 13, 1888. John D. Washburn, A.B., Harvard, 1853; LL.B., Harvard, 1856; Representa- tive, 1876-79 ; State Senate, 1884 ; United States Minister to Switzerland, 1889-92. Frank P. Goulding, A.B., Dartmouth, 1863 ; Harvard Law School, 1866 ; City Solicitor, 1881-93. George Swan, A.B., Amherst, 1847 ; admitted to Bar, 1848 ; Member of Worcester School Board, 1879-90 ; Chairman of High School Committee, 1887-90. The following gentlemen have been added to the Board since to fill vacancies caused by death. In place of Dr. Sargent: — Edward Cowles, A.B., Dartmouth, 1859; M.D., Dartmouth, 1862, and Col- lege of Physicians and Surgeons, N. Y., 1863 ; Assistant Surgeon, U. S. A., 1863-72 ; Resident Physician and Superintendent Boston City Hospital, 1872-79; Medical Superintendent McLean Asylum since 1879; Professor of Mental Diseases, Dartmouth Medical School, since 1885 ; Clinical Instructor in Mental Diseases, Harvard Medical School, since 1888. In place of General Devens: Thomas H. Gage, M.D., Harvard, 1852; President Massachusetts Medical Society, 1886-88. Historical Sketch. 3 On petition of this Board, the LegisUiture passed the following Act of Incorpokaxion. Chapter 133. commonwealth of massachusetts, in the tear one thousand eight hun- dred and eighty-seven. an act to incorporate the trustees of clark university in worcester. Be it enacted by the Senate and House of Representatives in General Court assembled, and by authority of the same, as follows : — Section 1. Jonas G. Clark, Stephen Salisbury, Charles Devens, George F. Hoar, William W. Rice, Joseph Sargent, John D. Washburn, Frank P. Gould- ing and George Swan, all of the city of Worcester, in the Commonwealth of Massachusetts, and their successors, are hereby made a corporation by the name of the Trustees of Clark University, to be located in said Worcester, for the purpose of establishing and maintaining in said city of Worcester an institu- tion for the promotion of education and investigation in science, literature and art, to be called Clark University. Section 2. Said corporation may receive and hold real or personal estate by gift, grant, devise, bequest or otherwise, for the purpose aforesaid, and shall have all the rights, privileges, immunities, and powers, including the conferring of degrees, which similar incorporated institutions have in this Commonwealth. Section 3. Said corporation shall have the power to organize said Univer- sity in all its departments, to manage and control the same, to appoint its officers, who shall not be members of said corporation, and to fix their com- pensation and their tenure of oifice ; and said corporation may provide for the appointment of an advisory board and for the election by the Alumni of said University to fill any vacancies in said board. Section 4. The number of members of said corporation shall not be less than seven nor more than nine, and any vacancy therein may be filled by the remaining members at a meeting duly called and notified therefor ; and when any member thereof shall, by reason of infirmity or otherwise, become incapable, in the judgment of the remaining members, of discharging the duties of his office, or shall neglect or refuse to perform the same, he may be removed and another be elected to fill his place, by the remaining members, at a meeting duly called and notified for that purpose. Section 5. This Act shall take effect upon its passage. House of Representatives, March 30, 1887, Passed to be Enacted. Charles J. Notes, Speaker. Senate, March 31, 1887, Passed to be Enacted. Halset J. BoABDMAN, President. During the previous five years, Mr. Clark had gradually acquired a tract of land, comprising over eight acres, located on Main Street, about 4 Historical Sketch. a mile from the heart of the city, with additional tracts near by. This land has considerable elevation above that part of the city, is a watershed sloping to the southeast, insuring sanitary excellence and a wide and picturesque view. A park reservation of about 25 acres, directly oppo- site, has been set apart by the city, and named University Park. Plans for a main building were submitted to the Board by Mr. Clark, which were approved, and its erection was at dnce begvm. The corner- stone was laid with impressive ceremonies, October 22, 1887. This build- ing is plain, substantial, and well appointed, 20-4 x 114 feet, four stories high and five in the centre, with superior facilities for heating, lighting, and ventilation, and has been constructed of brick and granite, and finished throughout in oak. On the whole it is a model of stability and solid work- manship. It contains a total of 90 rooms, and in its tower is a clock with three six-foot illuminated dials, which was presented by the citizens of Worcester. The elevations and ground plan are published, and the heat- ing, lighting, ventilation, walls, floors, etc., etc., are described in detail in the Third Official Announcement. On April 3, 1888, G. Stanley Hall, then a professor at Johns Hopkins University, was invited to the presidency. The official letter conveying the invitation to the president contained the following well-con- sidered and significant expression of the spirit animating the trustees : — " They desire to impose on you no trammels ; they have no friends for whom they vsish to provide at the expense of the interests of the institution ; no pet theories to press upon you in derogation of your judgment ; no sectarian tests to apply ; no guarantees to require, save such as are implied by your acceptance of this trust. Their single desire is to fit men for the highest duties of life, and to that end, that this institution, in whatever branches of sound learning it may find itself engaged, may be a leader and a light." This invitation was accepted May 1, and the president was at once granted one year's leave of absence, with full salary, to visit universities in Eiirope. This year was diligently improved in gathering educational literature and collecting information and advice from leading authorities. Many reports based upon this work have already been made in the Peda- gogical Seminary and more are in course of preparation. During the absence of the president a Chemical Laboratory was begun. This building in its main body has three stories, in its eastern wing four, in its southwestern two. It contains 68 rooms. The outer walls are 2 feet in thickness and the partition walls from 12 to 16 inches. All par- Historical Sketch. 5 titions are of brick, so thtit the building is nearly fireproof. The two large laboratories are 2-4x58 feet and 22 feet high. This building is also described with plans in the T^ird Official Announcement. The opening exercises were held in a hall of the University, seating 1500 people, on Wednesday, October 2, 1889. The late General Charles Devens presided, and made an opening address. Addi-esses were made by Senator George F. Hoar and the president. The founder of the University stated his purpose as follows : — " When we first entered upon our work it was with a well-defined plan and pxu-pose, in which plan and purpose we have steadily persevered, turning neither to the right nor to the left. We have wrought upon no vague concep- tions nor suffered ourselves to be borne upon the fluctuating and imstable current of public opinion or public suggestions. We started upon our career with the determinate view of giving to the public all the benefits and advan- tages of a university, comprehending full well what that implies, and feeling the full force of the general understanding, that a university must, to a large degree, be a creation of time and experience. We have, however, boldly assumed as the foundation of our institution the principles, the tests, and the responsibilities of imiversities as they are everywhere recognized — but with- out making any claim for the prestige or flavor which age imparts to all things. It has therefore been our purpose to lay our foundation broad and strong and deep. In this we must necessarily lack the simple element of years. We have what we believe to be more valuable — the vast storehouse of the knowl- edge and learning which has been accumulating for the centuries that have gone before us, availing ourselves of the privilege of drawing from this source, open to all alike. We propose to go on to further and higher achievements. We propose to put into the hands of those who are members of the University, engaged in its several departments, every facility which money can command — to the extent of our ability — in the way of apparatus and appliances that can in any way promote our object in this direction. To our present depart- ments we propose to add others from time to time, as our means shall warrant and the exigencies of the University shall seem to demand, always taking those first whose domain lies nearest to those already established, until the fvdl scope and pvu-pose of the University shall have been accomplished. " These benefits and advantages thus briefly outlined, we propose placing at the service of those who from time to time seek, in good faith and honesty of purpose, to pursue the study of science in its purity, and to engage in scientific research and investigation — to such they are offered as far as possible free from all trammels and hindrances, without any religious, political, or social tests. All that will be required of any applicant will be evidence, disclosed by examinations or otherwise, that his attainments are such as to qualify him for the position that he seeks." 6 Historical Sketch. After careful consideration it was decided to begin with graduate work only and in the following five departments : I. Mathematics. II. Physics, Experimental and Theoretical. III. Chemistry, Organic, Inorganic, Physical, and Crystallography. IV. Biology, including Anatomy, Physiology, and Paleontology. V. Psychology, including Neurology, Anthropology, and Education. Mathematics is sometimes called the queen of all the sciences. As the latter become exact, they approximate it, and are fructified by its spirit and its methods. Its antiquity, its disciplinary value, its rapid and recent development, make it obviously indispensable. Physics is the field of the most immediate application of mathematics, and deals with the fundamental forces of the material universe, — heat, sound, light, electricity, — and the imderlying problems of form and motion generally, with their vast field of application in such sciences as astronomy and dynamic geology. Chemistry, with its great and sudden development, revealing marvellous order and harmony in the constitution of matter, is rapidly extending its dominion over industrial processes. Biology, which seeks to fathom the laws of life, death, reproduction, and disease, that underlies all the medical sciences, in its broader aspects lias taught man in recent decades far more concerning his origin and nature than all that was known before. Psychology, or the study of man's faculties and their education, is a new field into which all the sciences are bringing so many of their richest and best ideas, which is now so full of promise for the life of man. A sub-department of Education was established in 1892, and the department of Chemistry was temporarily discontinued in 1894. To express more explicitly the character and policy of the institution, the Trustees voted to approve and publish the following statement : — " As the work of the University increases, its settled policy shall be always to first strengthen departments already established, until they are as thorough, as advanced, as special, and as efficient as possible, before proceeding to the establishment of new ones. "When this is done and new departments are established, those shall always be chosen first which are scientifically most closely related to departments already established ; that the body of sciences here represented may be kept vigorous and compact, and that the strength of the University may always rest, not upon the number of subjects, nor the breadth or length of its cur- riculum, but upon its thoroughness and its unity. Historical Sketch. 7 " This shall in no wise hinder the establishment, by other donors than the founder, of other and more independent departments if approved by the Trustees. " While ability in teaching shall be held of great importance, the leading consideration in all engagements, reappointments, and promotions shall be the quality and quantity of successful investigation." By thus limiting the work of the University in the beginning to five departments, it appeals only to advanced men who desire to specialize in one or more of these fundamental sciences, leaving college students who require a larger range of studies, as well as those who desire to devote themselves to language and literature, historical, technical, or pro- fessional studies, to go elsewhere. Hence our work must be post- graduate. Tliis requires the best professors and apparatus, more books and journals, and necessitates a system of fellowships, scholarships, and provisions for original research. It thus becomes a training-school for professors. This is the most expensive of all educational work, seeks the fewest but the best men from the mdest area, and to succeed must be helpful in elevating the academic standards of the country to a higher plane. It requires the highest degree of wisdom and foresight on the part of the Founder and the Trustees, and possibly some sacrifices of local sympathy and support at first, till the nature of the work is well understood. It requires the greatest effort and devotion to work on the part of the Faculty and students. But the cause is itself an inspira- tion. It appeals to the future, the country, and to the world, and seeks quality more than numbers. It is in the current of all the best tenden- cies in the best lands, and is now the ideal of perhaps every eminent man of science everywhere. The inauguration and steady maintenance of this clear and simple policy gives the University a reason for being, and a distinct individuality it could not otherwise have, and also a real leadership in this epoch of awakening and transition, which is the golden time of opportunity for new institutions, and brings them to the front. Such a period as the present gives the latter even greater relative influ- ence and prominence than would be possible in periods of less public interest in education. New institutions can .and should lead, set new fashions, and be the first upon the higlier planes. Older institutions are retarded by conservatism and must advance more slowly, but when they move they carry great momentum. This condition makes the present a moment of perhaps unprecedented opportunity, which 8 Historical Sketch. has been long looked for and long delayed, and whicli renders both funds and labor in this field more precious than they have been, or will be when it is past. We may all be content if our Uni- versity can transmit to future generations by means of its organization, plan, and methods the best and liighest educational tendencies and move- ments now stirring the souls of the best men of the world, and uniting men of all lands, races, creeds, and stations in a larger if not also a deeper consensus of belief than history has ever known before. Our University does not draw its chief earnings from, or do most of its teaching for, undergraduates, and oiu- so-called graduate students do not take undergraduate courses. This makes the proportion of expendi- ture to income very high here, and indeed we can admit and do justice to but comparatively few students. Most of those who come here have spent one or more years after graduation in teaching, or in study in Europe or elsewhere. Most of those who have been members here have already obtained professorships or other academic positions elsewhere. The proportion of such is hardly excelled by the Ecole Normale of Paris, the special function of which is to train professors from other collegiate institutions. Every student who obtains original results is expected to present them in the form of lectures to his department, and thus to acquire experience in teaching under criticism. The work of the educa- tional department deals with problems and history of higher educational institutions, and is adapted to all the body of fellows and scholars, and seeks to increase the efficiency of every man both as a teacher of his own specialty and in general helpfulness to the institution with which he is to be connected. Since the opening of the University not less than five hundred books, memoirs, theses, or articles ^ have been published by members of the Uni- versity, which attempt to make additions to the sum of hmnan knowledge. These contributions are of very different orders of value, but together they constitute a body of knowledge in which the institution takes special pride. Every member of the University is expected to make at least one long and serious effort of this kind. Indeed, had its publications no value as contri- butions to knowledge, its educational value is the highest for mature men. Such effort gets minds into independent action, gives a sense of authority and of true mental freedofti, which no amount of acquisition can bring. It brings out new powers of mind and of will, and, while one of the chief 1 A list of these publications will be found at the end of this volume. Historical Sketch. 9 marks by wliich true University work is distinguished from that of lower grades, is in the line of all present tendencies to place doing above know- ing from the kindergarten up. Work that is published enlarges the sphere of interests of the author, subjects him to the higher test of being judged by his peers elsewhere, and brings in the potent and salutary stimulus of wider competition. This baptism of ink has often marked a new birth of ideals and ability in young men. INIodern as distinct from earlier culture culminates in the man-making training of will and judg- ment thus given. Such work, too, gives teaching a new power and zest. Instruction to a fit few by an investigator who stands on the frontier and has once felt the light and heat in which discovery is wrought out is inspiring, and is very different from information imparted at lower levels by teachers further removed from the work of discovery and creation. Clark University is exclusively what is called in Europe a Philo- sophical Facility, or a part of one so far as yet developed, devoted to a group of the pure sciences wliich underlie teclinology and medicine, but does not yet apjily its work to these professional fields. These or a college course could be added with relatively less expense. Our method has brought us face to face with many new problems. Our efforts at solving some of these are described in the department reports which follow. Like all new institutions, we have not entirely escaped trials, but we trust we have learned their lesson, and shall be the better and stronger for them. Instead of dispersing our energies in university extensions, we have followed the opposite course of university concentra- tion, like the Ecole Pratique of France. Accepting the plain lesson of history that the best educational influences work from above downward, that universities create the material of culture, while lower institutions are the canals for its distribution, we have sought aid for the latter work by an educational sub-department and summer school. We are not like the Smithsonian Institute, the Naples school, the Reichsanstalt, academies of science, etc., devoted solely to research, but have to make our lectures more condensed and fewer than usual, because addressed to advanced men, and to devise ways of making seminary and laboratory, two of the noblest words in the vocabulary of higher education, more effective. We liave tried to effect systematic exchanges with foreign institutions, — and our library has profited largely from this source, — and have sought by all the above means to aid in giving to universities and to professors the position due them in a time when sciences have come to underlie all the 10 Historical Sketch. arts of peace and war, and when the world, in all its activities, industry and trade, professions, legislation, is coming to be more and more con- trolled by experts, thus trained to the frontier of theii" specialties. The degree of Doctor of Philosophy has been conferred upon can- didates, whose names, together with the dates of their final examinations and the subjects of their dissertations, are given later in this volume. Other historical facts are given in the President's Address at the Decennial Celebration. REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY. At least two years, and in most cases three years, of graduate work will be necessary for this degree. Examinations for it, however, may be taken at any time during the academic year when, in the judgment of the University authorities, the candidate is prepared. A prearranged period of serious work at the University itself is indispensable. For this degree the first requirement is a dissertation upon an approved subject, to which it must be an original contribution of value. To this capital importance is attached. It must be reported on in writing by the chief instructor before the examination, printed at the expense of the candidate, and at least one hundred cojDies given to the University. In case, however, of dissertations of very unusual lengtli, or containing very expensive plates, the Faculty shall have power, at the request of the candidate, to reduce this number of presen- tation copies to fifty. Such formal or informal tests as the Faculty shall determine, shall mark the acceptance of each student or fellow as a candidate for this degree. One object of this preliminary test shall be to insure a good reading knowledge of French and German. Such formal candidature shall precede by at least one academic year the examination itself. (See special rules below.) The fee for the doctor's degree is $25, and in every case it must be paid and the presentation copies of the dissertation must be in the hands of the Librarian before the diploma is given. In exceptional cases, however, and by special action of the Faculty, the ceremony of promotion may precede the presentation of the printed copies of Historical Sketch. 11 the dissertation. The latter, however, must always precede the actual presentation of the diploma. An oral but not a written examination is required upon at least one minor subject in addition to the major, before an examination jury composed of at least four members, including the head of the department and the President of the University, who is authorized to invite any person from within or without the University to be present and to ask questions. The jury sliall report the results of the examination to the Faculty, which, if it is also satisfied, may recommend the candidate for the degree. For the bestowal of this degree, the approbation of the Board of Trustees must in each case be obtained. They desire that the standard of requirements for it be kept the highest practicable, that it be reserved for men of superior ability and attainment only, and that its value here be never suffered to depreciate. It is to the needs of these students that the lectures, seminaries, laboratories, collections of books, apparatus, etc., are specially shaped, and no pains will be spared to afford them every needed stimulus and opportunity. It is for them that the Fellowships and Scholarships are primarily intended, although any of these honors may be awarded to others. SPECIAL RULES. I. Residence. — No candidate shall receive the degree of Doctor of Philosophy ^sdthout at least one year's prc'V'ious residence. II. Candidature for the Doctors Degree. — Every applicant for the doctor's degree shall fill out before October 15th the regular appli- cation blank provided at the office. This schedule shall be submitted to the head of the department and the instructor in the major subject. Before affixing their signatures they shall satisfy themselves, in such manner as they may desire, as to the fitness of the applicant. III. When countersigned, this schedule shall be filed with the President, who wUl appoint an examiner to serve with a representative of the major subject as a committee to determine the proficiency of the applicant in French and German. IV. In case of a favorable report by this committee the applicant shall be a regular candidate for the degree. V. Candidates complying with all preliminary conditions, including 12 Historical Sketch. the examinations in French and German, before November 1st, will be allowed to proceed to the doctor's examination at any time between May 15th following and the end of the academic year. VI. Dissertation. — The dissertation must be presented to the in- structor under whose direction it was before written, and reported upon by him before the final examination. In every case the dissertation shall be laid before the jury of examination, at the time of examination, in form suitable for publication. This provision shall not, however, preclude the making of such minor changes later as the chief instructor may approve. VII. The dissertation shall be printed at the expense of the can- didate, and the required copies deposited with the Librarian within one calendar year from the 1st of October following the examination. The candidate alone will be held responsible for the fulfilment of these conditions. VIII. The favorable report of the chief instructor, filed in writing with the Clerk of the University, shall be a sufficient imprimatur or authorization for printing as a dissertation. The printed copies shall bear upon the cover the statement of approval in the following words, over the name of the chief instructor : — ■ A Dissertation submitted to the Faculty of Clark University, Wor- cester, Mass., in partial fulfilment of the requirements for the degree of Doctor of Philosophy, and accepted on the recommendation of (name of the chief instructor). IX. Examinations for the Doctor'' s Degree. — The examinations for the doctor's degree may be held at any time during the academic year, provided that at least one academic year has elapsed since the completion of the preliminaries of candidature, except in the case of fulfilment of these conditions between the beginning of any academic year and November 1st of that year, to which case Ri;le V. applies. The examinations shall be held at such hours and places as the President may appoint. X. Examinations may also be lield during the regular vacations of the University, but for these an additional fee of five dollars to each examiner, and the reasonable travelling expenses of any examiners who are out of town, all payable in advance, will be required. XL All these special rules shall go into force immediately as far as practicable, and shall govern all applicants for degrees in the academic year 1899-1900. THE DECEXNIAL CELEBRATION. The work of Clark University is so technical and special that it is necessarily more or less withdi-awn from popular interest. It has no commencements, and comes in very little contact with the public or the press in Worcester, or indeed with collegiate institutions in other parts of the country. This is a disadvantage so far as local or general public interest in its work is concerned, but the fact that it does not exercise many of the usual functions of a college is also a distinct advantage to its scientific work. The close of the tenth year of its existence presented an opportunity to bring before the public, in a simple way, befitting at once its size and its quality, a presentation of the work it has accomplished in the past and of its hopes and needs for the future. Early last winter the President began to consider plans of marking this anniversary, and, with the efficient aid of the Faculty, they gradually took definite shape. A personal appeal was then made to a number of public-spirited and wealth}' citizens of Worcester, and the scheme was rendered feasible by the gen- erosity of the following gentlemen, who donated the sums affixed to their names: — Mr. Stephen Salisbury, f 1000 Mr. Philip W. Moen, 500 Mr. Thomas H. Dodge, 200 Mr. Edward D. Thayer, Jr., 200 Mr. Charles S. Barton, 100 Mr. John H. Goes, 100 Mr. Andrew H. Green, 100 Mr. Arthur 'SL Stone, 100 John O. Marble, M.D., oO Mr. C. Henry Hutchins, $500 Mr. WilHam E. Rice, 500 Mr. Orlando W. Norcross, 200 Mr. Matthew J. Whittall, 150 Mr. A. Swan Brown, 100 Mr. Loring Goes, 100 Mr. James Logan, 100 Mr. Joseph H. Walker, 100 Mr. Frederick L. Goes, 25 Charles L. Nichols, M.D., §25. 13 14 Decennial It was decided that the close of the tenth academic year should be celebrated (1) by courses of lectures delivered by distinguished foreign scientific men, (2) by public exercises, and (3) by an evening reception. A conference was then held concerning the most prominent leaders in Europe in branches especially cultivated at the University, and after some correspondence the following persons were invited to give from two to four lectures each : — Emilb Picard, Professor of Mathematics at the University of Paris. LuDWiG BoLTZMANN, Ppofessor of Theoretical Physics at the University of Vienna. Angelo Mosso, Professor of Physiology and Rector of the University of Turin. Santiago Ram(5n y Cajai., Professor of Histology and Eector of the Univer- sity of Madrid. August Foeel, late Professor of Psychiatry at the University of Zurich and Director of the Burghblzli Asylum. Under the direction of a committee consisting of Assistant Professor A. G. Webster and Professor W. E. Story, the following forms of invitation to the various parts of the programme were prepared: — Celebration. 15 cfA ta^ i^e^^^/t^ed/C n-^yiyOyl u^C^>^ cj^ 'e^ {C/(^'n4/i^Md^^ te^f^i^ce i>yt^ S't&aa'& vtjdAf. Celebration. 17 ^^ J , /^ 18 Decennial The invitations to the lectures were sent to such persons as were con- sidered to be particularly interested in the subjects in question, of whom over one hundred accepted. Many declinations were inevitable and expected, owing to the unfavorable season of the year and, perhaps in part, to the somewhat too short notice given. The lecturers all arrived in due season, and were entertained as follows : — Professor Emile Picard, by Professor W. E. Story. Professor Ludwig Boltzmann, by Assistant Professor A. Gr. Webster. Professor Angelo Mosso, by President G. Stanley Hall. Professor S. Eamdn y Cajal, by Hon. Stephen Salisbiu:y. Professor August Porel, by Dr. Adolf Meyer. The lectures were held in the large lecture-room on the first floor, and were well attended. Professors Picard and Cajal lectured in French, and Professors Boltzmann, Mosso, and Forel in German. Their lectui-es are printed in full elsewhere in this volume. Many social functions occurred during the week ending July 8. On Wednesday evening, Professor Story received informally the attendants on the lectures of Professors Picard and Boltzmann ; on Thursday evening President Hall gave a reception to all the visitors ; and on Friday after- noon and evening the whole company was entertained by Hon. Stephen Salisbury at the Quinsigamond Boat Club house. The second part of the celebration occurred on Monday morning, July 10, beginning at 10.30, in the University. The professors had adopted academic costume, and many distinguished guests were seated upon the platform. The exercises opened with prayer by the Rev. Alex- ander H. Vinton, Rector of All Saints' Church. A few extracts from congratulatory letters were read by Professor Story, which are printed elsewhere in this volume. Brief congratulatory addresses Avere made by President Faunce, of Brown University, repre- senting the New England college presidents ; and Professor Bowditch of the Harvard Medical School, representing the higher scientific institutions of the state. President Faunce said : — " I count it a very happy fact that the first occasion on which I am to officially speak, representing Brown University, is at this anniversary at Clark University. I bring you to-day greetings from an institution of the higher learning founded in 1764 to a university founded in 1887. It is Celebration. 19 safe to say that Clark University has done more to widen the confines of human knowledge than any other American college in one hundred and fifty years. " When Professor J. P. Cooke, of Harvard, applied to the Faculty for chemicals and apparatus for experiment, he was told he must secure the materials at his own expense, and that he must be responsible for any explosions or damage in consequence of his experiments. From that day to this is a long step. Our method of applying nature has been trans- formed witliin a very few years. The distance between Acliilles' coach and the English stage-coach is not the same as that between the stage- coach and the Empire State express. The difference between the Phoeni- cian galleys and the Bon Homme Richard is not the difference between the Bon Homme Richard and the modern battleship. The little world of Shakespeare has become one vast universe of learning, and the field has broadened almost infinitely in all directions, and the goal is the far-off divine event toward which the whole creation moves. " In this movement of scholarship the enrichment of one institution is the enrichment of all, the enfeeblement of one is the enfeeblement of all. You have received at this celebration, almost Spartan-like in its sim- plicity, the congratulations not alone of America, but of Berlin and ^Munich and Vienna, because your advance and success is the advance of all. Only geographically and superficially are the leaders of modern scholarship divided, and so we congratulate you, not because you have duplicated existing plants, but that you have filled a place hitherto unfilled and have broken new ground. " Here among all the institutions of learning you have not detracted from the success of other institutions, you have placed fresh laurels on the heads of each. All of us feel a warm interest and admiration for this University because of the simple, quiet, and noble work done within these walls." Dr. Bowditch said that he was quite unprepared to say much, and he thought it just as well, for he belonged, in the words of Dr. Holmes, to the "silent profession." He paid a tribute to the felicitous speech of Dr. Faunce, which left him little to say. Dr. Bowditch spoke of the great work in scientific research being conducted by the institution, and, after some wishes for its prosperitj', congratulated the youngest college in the name of the oldest college in Massachusetts. 20 Decennial Then followed the address by President Hall, printed elsewhere in this volume. The honorary degree of Doctor of Laws, honoris causd, was then con- ferred, for the first time, upon the five foreign professors in the following terms : — " By virtue of the authority vested by the Commonwealth of Massachu- setts in the Board of Trustees of Clark University, and by them dele- gated to me, I now create you Doctor of Laws, honoi-is causd, and by this token [presenting diploma] invest you with all the dignities thereunto appertaining." Brief responses were made, of which translations foUow. Ltjdwig Boltzmajstn. The problem of science is a twofold one : first, to advance our knowledge of nature independently of any practical application ; and second, to make practical applications of the knowledge gained. Although to a superficial observer it may seem that the latter is of greater importance, the develop- ment of humanity has shown in the most convincing way that the first kind of activity is not only of paramount importance, but that the leading role belongs to it. In fact, it is only thanks to the pioneers of science who, laying aside all practical applications, penetrate deeper and deeper into the essence and arrangement of the forces of nature, that humanity has obtained that sway over the laws of nature which makes possible the present practical achievements. The German universities have devoted themselves at all times to the nurture of pure science apart from its practical applications, although but one of the four university faculties is consecrated to it, and that one not entirely. It must be considered as a good omen, therefore, that here in America, which is usually taken to be the land of practical men, the ideal of a place entirely consecrated to the service of pure science, unattainable in Germany, has found its realization, so that I, who am body and soul a German professor, deem it a great honor to have conferred on me in this place, the greatest distinction which the University can grant. While desiring Clark University to flourish and tlirive in the intimate conviction that the whole scientific world is interested in her prosperity, I express my thanks to the President and all its members for the high honor bestowed upon me to-day. Celebration. 21 Santiago Ramon y Cajal. I OFFER my most cordial thanks to Clark University for the honorable distinction she has bestowed upon me in spite of my small deserts by granting to me the degree of doctor of laws by this learned body, the remembrance of wliicli will never fade from my memory. This honor I deem to be the prize of the greatest value which my modest researches have procured for me, and the one which will encourage me most in my worship of the laboratory tasks and of the study of nature. This honorary distinction, as well as the invitation which Clark University condescended to make me to take part in the conferences for solemnizing the tenth anniversary of its foundation, shows once more that the men of science know of no frontiers, and that they form a universal familj-, whose solidarity and fellow-feeling place them high above the wrangle of mate- rial interests and selfish struggles of nationalities. It was truly a happy idea to create in America a university of higher studies, devoted not only to the labor of teaching, but also very especially to giving impulse to pure science. It has been said many times, but never enough, that there is no lasting industrial progress if it is not connected, as a brook with its soiirce, with the creation of original science. No matter how great the practical genius of a nation, it is impossible for it to preserve its political, commercial, and industrial hegemony, luiless it comes out intellectually superior to other nations, unless it attends with equal care to the laboratory and to the mUl, to the ideas as well as to the inventions, to the philosophy and to the science which guide as well as to the art which carries out. This happy alliance between theory and practice is what places Ger- many to-day at the head of civilization. It would be easy to adduce num- berless examples of the supremacy which industry, founded on science, holds over empirical industry created at hajDhazard according to the inventive character of each nation. I will quote only two — the chemical industry of the aniline dyes created chiefly in Germany, which assures to that nation an immense wealth ; and the optical industry representing all kinds of apparatus (microscopes, photograpliical and astronomical object-glasses) which sprung up under the inspiration of the great matliematicians. Abbe, Rudolph, Goertz, and others, and which by its manifest superiority over that of other nations jjrocures to Prussia a monopoly which makes the whole world her tributary. 22 Decennial That is the right way, the only one which leads to glory, wealth, and power. I trust that the creation of Clark University may give the sio'nal for founding in America other similar institutions embracing a still laro-er number of brandies of science, and having as their primary object the wresting of secrets from nature, supplying industry and arts Avith principles and facts capable of fruitful applications, forming the research spirit of the new generation, freeing it from the clogs of routine and imitation, and finally forming the foimdation of a splendid civilization superior in groundwork, as well as in form, to that of the European nations. Atjgust Fokel. I THANK you heartily for the great honor you have bestowed upon me by conferring upon me the degree of doctor of laws, honoris causd, of Clark University. But I accept this honor less in my own person than as a representative of Switzerland at your celebration — in the name of my little fatherland. Although nowadays the Swiss Federation disap- pears beside the powerful republic of the United States, yet she prides herself still on being the little old mother of democracy, which has fought for her free rights for centuries, and has maintained them up to the present day. I offer my heartiest congratulations for the brilliant success which Clark University has achieved during the short time of its exis- tence in the high domains of philosophy, pedagogy, and of many a scientific foundation of social questions. But we must also offer our heartiest thanks and congratulations for the generous and magnificent gifts of American citizens for the furtherance of scientific and social progress. Allow me to add a wish. Let Clark University continue to pursue — under the successful guidance of her excellent President, Professor G. Stanley Hall — her researches in the regions of psychology and pedagogy together with those on the brain and its life, and thus to further the investigation and the building up of truth in the teeth of all prejudices. Let her help to bury the old roads of barren metaphysical dogmas and speculations, and thus develop in its entirety the only fruitful ethically built-up progressive method of scientific investigation in these domains, as a blessing to our posterity and for the good of a better and happier humanity. Celebration. 23 Angelo Mosso. I OFFER ray thaitks to Clark University for the honor bestowed upon me. I shall carry with me to Italy a happy remembrance of the many proofs of sympathy and friendship which I have received in the Uni- versity and the city of Worcester. It is not only the expression of my gratitude that I offer you, but also my great admiration for all that I saw in your University, and especially the development in experimental psychology under the ha^jpy impulse which the President has given to this branch of science. It is not only on my own account that I offer you my thanks, it is also because, on my return to Italy, I hope to found in the University of Turin a school of experimental psychology. Emile Picaed. I offer my heartiest thanks to the President and Professors of Clark University for the degree just conferred upon me. I have been also greatly touched by the honor you bestowed upon me by inviting me to give a few lectures during this academic celebration. Your desire was thus to bear witness to your sjTnpathy with men of science in France. We follow on our side, in France, with great interest the American scientific movement, and we rejoice in seeing closer relations established between our universi- ties and those of this country. Science treads its ascending march on different roads, and research work requires to-day the most varied apti- tudes. The initiative and the energy which are prevalent in this country will not be wanting in occasions for displaying themselves, and, in all branches of studies, the American scientists will be able to erect some- thing equivalent to those large telescopes by means of which your astrono- mers have made such beautiful discoveries. It is in the universities which, like this one, are devoted to research, that the scientific movement is bound to have its origin. From everything I have seen and heard for the last few days, I am certain tluit the eminent professors of this University devote themselves with success to this noble task, and I beg to offer my most sincere wishes for the continuance of the brQliant development of Clark University. The exercises concluded with prayer by Dr. Vinton. The closing exercise of the decennial was a reception which was attended by between five hundred and six hundred ladies and gentlemen 24 Decennial Celebration. of Worcester. The arrangements had been made under the direction of Assistant Professor Henry Taber and Professor William E. Story. The large lecture-room and corridors were decorated with festoons of green and white, the flags of the United States and of the native countries of the foreign guests, and with potted plants. A collation was served in the library, and many pieces of apparatus were exhibited in operation in the physical and psychological laboratories. The following persons received : President G. Stanley Hall, Miss Florence E. Smith of Newton Centre, Mass., Mrs. A. W. Beals of Stam- ford, Conn., Hon. Stephen Salisbury, Dr. Edward Cowles, Miss Gage, Professor and Mrs. William E. Story, Assistant Professor and Mrs. Arthur G. Webster, Assistant Professor and Mrs. Clifton F. Hodge, Assistant Professor Edmund C. Sanford, Miss Sanford, Assistant Professor Henry Taber, Dr. and Mrs. A. F. Chamberlain, the foreign lecturers, Senora Ramon y Cajal, and Frau Boltzraann. The press of Worcester gave very full and detailed accounts of all that transpired during the week except the scientific lectures, all of which were in foreign languages and upon very technical subjects. The following original documents have been bound and filed in the University library : — (1) The congratulatory letters, telegrams, etc. (2) The correspondence with the foreign lecturers, and the letters of acceptance and declination from American professors. (3) The letters of acceptance and declination to the reception in the evening. The weather was somewhat warm during the first few days, but was clear and cool on Saturday, Sunday, and Monday. The hospitality of Worcester people was all that could be desired. CONGEATULATIONS. The following extracts are taken from many hundred congratulatory- letters, personal, official, and from institutions and educators of all grades and many lands. Congratulations on the conclusion of the University's first decade, and best wishes for the successful continuance of the work it has undertaken. William McKinlet, Washington, D.C., President of the United States. The attraction will be strong to all who are interested in the great subjects which these distinguished men will discuss, or in intellectual eminence for its own sake. Your occasion will be the most distinguished gathering that will occur in all New England this summer. . . . The high plane of the work done at Clark University, the only institution in our country exclusively devoted to original research and the instruction of advanced investigators, so far as I am aware, is well known to all who have followed the course of the University. Modestly, and without ostentation, it has pursued its noble ideals. If, under your able direction, its means were more extensive, the University would, doubtless, become the centre of a still larger circle of influence in the training of men for the prosecution of original research and the conduct of similar work in other institutions. I trust that your own large plans and those of the foimder of the University may enjoy a complete realization, and that its future may be crowned with the high success which so great an enterprise rightly deserves. Felicitating the honored founder, yourself, the trustees, and your colleagues in the faculty upon the great occasion you are soon to celebrate, David J. Hill, Washington, D.C., Assistant Secretary of State. It is one of the chief regrets of my life that I cannot attend the celebration of Clark University. Be assured that no reason personal to myself has pre- vented my attendance. I have seriously considered the question of crossing the Atlantic for the purpose, and coming back here immediately afterward. But that seems impracticable. 2^ 26 Extracts from We have to congratulate the University upon ten years of success. It was not to be expected that an institution vrhose aim is to lift the university educar tion, not only of this country, but of the world, to a higher plane, and to break out a new and untrodden path, should command popularity in the beginning, or that its success should at once be recognized by the general public. But we have no cause for regret or for discouragement. Teachers whom we have edu- cated are found in institutions of the first class in all parts of the country, and all parts of the world have sent representatives to receive our instruction. This is largely due to the wise and far-sighted intelligence of the founder, and, next, to your own constant and self-sacrificing labors. There have been times during these ten years when we have been tempted to think that the people of Worcester have been cold, and have been lacking in the liberality which we had hoped from them when we started. But in looking at the history of other institutions which are now useful and flourishing, it will be seen that they had in the beginning a like experience. I remember well a time when it almost seemed impossible to get the people of Worcester to endow a public library. But the hour came and the man came, and our public library is now munificently endowed and is a model of library administration. The Polytechnic Institute had its day of small things. But the liberality of two citizens of Worcester of the same name and race, whose two lives seem almost like the prolonged life of one individual, came to its aid, and it is now doing its work with large endowments, and its scheme has been copied by other institu- tions all over the country. I do not for a moment doubt that the time will come when our endowments will enable us to maintain in the entire circle of univer- sity education the position which we have taken and hold with regard to a few subjects. Already an eloquent orator, formerly head of the National Catholic University at Washington, has referred to Clark as " that little institution in Worcester which has added a new story to university education, and ' Which allures to brighter worlds and leads the way.' " An eminent professor of science from the English Cambridge declared at a meeting in the British Association, in the presence of famous scholars from all parts of the world, that there is one thing that England envies America, and that is Clark University. There is nothing except the country itself which ought to inspire a deeper devotion in its children than a university. As time goes on this feeling, made up of love and gratitude, will be found in fullest measure among the alumni of Clark. As they go out to reap the harvests of success in life, they will repay to their alma mater, in their own way, the great debt they owe her. When that time comes I have no fear that her endowments will not be ample to accomplish the work she has undertaken. In the meantime those of us to whom the con- fidence of the founder has committed a share iu her administration must renew our own vows of fidelity to her service. Among the many public honors which the undeserved kindness of my fellow- Congratulatory Letters. 27 citizens has bestowed upon me, I count none higher than my selection as one of the first board of trustees of this institution. I trust that your celebration will be full of delight for those who gather there, that they will look forward with bright hopes to the future, and that an immortality of fame and usefulness may await the institution which now celebrates its tenth birthday. George F. Hoak, United States Senator. I learn from your formal letters of invitation that you are to celebrate the close of the first decade of Clark University. It is one of the most wonderful careers to be chronicled in the history of American education. I congratulate you on your eminent success in conducting youi- University in so ef&cient a manner toward the improvement and elevation of pedagogy in the United States. Your movement is all the more valuable because it challenges the aims and purposes of the present existing education. It is an elementary force in making the American teachers circumspect and reflective, and causing them to seek deeper principles on which to ground their practice and on which to im- prove it. Hoping that there will be a long series of equally useful decades in the history of Clark University and in your own successful directorship of that institution, W. T. H.VERIS, Washington, B.C., Commissioner of Education. I cannot refrain from offering my congratulations to the President, Trustees, and Faculty for securing the services of such distinguished lecturers, as well as for the marked success that has attended Clark University during the first decade of its existence. Willis L. Mooke, Washington, D.C., Chief of Weather Bureau. I must add my profound appreciation of the great work for the highest science that is being accomplished by you. The solid knowledge that consti- tutes " Science" is a rather slow growth — it can only advance in proportion as man frees himself from ancient errors and evolves higher powers of observation and reason. The fine work done at Clark, the excellent memoirs published by its professors, and now these attractive lectures, give us all the assurance that your labors for the highest attainments in the study and teaching of science will be abundantly rewarded. Cleveland Abbe, Washington, D.C., Weather Bureau. Congratulating you on the successful rounding out of the first decade of the University, and with best wishes for the success of the institution in the future, W. J. McGee, Washington, D.C., Smithsonian Institution. 28 Extracts from I send you most cordial greetings on the interesting occasion, and hope the future of Clark will be as successful as the past, and that your plans for scien- tific research may be realized in the fullest degree. Carroll D. Wright, Washington, D.C., Commissioner of Labor. One may well be envious of the gratification that the generous founder of Clark University must feel at the world-wide recognition of its achievements during the very first decade of its existence. To have established a just claim upon the regard of foremost men associ- ated with educational establishments in this country and in Europe is of great significance. The work that the University has done and is doing will continue to attract to its halls those rare geniuses who, impressed with the transcendent importance of the science of Pedagogy, of Physiology and Psychology, seek with unfailing diligence to penetrate their inmost depth. This work can scarcely fail to exer- cise a beneficial influence upon the schools of the country, and become a distin- guished attraction to the city which is fortunately the home of the University, whose citizens will give it welcome and encouragement and markedly recognize the munificence of its founder, as well as the labors of those who have in so brief a time established it among the foremost seats of learning. Andrew H. Green, 214 Broadway, New York City. As I shall not be able to be present during the exercises on Monday, July 10, celebrating the completion of the tenth academic year of Clark University, I desire to express in writing my feelings of sympathy and my strong desire for the success of the University, and also to extend to you and your co-workers my sincere congratulations on this auspicious occasion. It is i^robably true that the initiative step of the institution was not fully understood or appreciated by the jjublic, but during the past ten years it has, under your able and judicious direction, steadily pursued a course well calcu- lated to win its way to public confidence and to an abiding position among the most eminent and distinguished institutions of learning in the civilized world. The entire exercises attending the celebration are calculated to draw aside the mystic veil, and when the occasion shall have been numbered among past events, the general public will be led to see and know Clark University in the future as it has been seen and known in the past by distinguished foreign scien- tists and educators. Yes, rest assured. President Hall, that before the last hour of the present century has been struck by the unerring and mighty hand of time, Clark Uni- versity, the far-seeing, noble, and generous founder, together with the Univer- sity's learned and distinguished first president, will have been crowned by Congratulatory Letters. 29 truth and justice with the laurel wreath of victory, exalted merit, and uni- versal appreciation. Thomas H. Dodge, Esq., Worcester. James Brice begs to be permitted to offer his congratulations upon that occasion. Will you please convey to them my best wishes for the continued prosperity of Clark University. It has a high mission ; for gathering in new knowledge is a much nobler task than distributing that which is known, useful as the latter may be. I feel confident that when your present age is lengthened tenfold and your successors celebrate the centenary, they will hold up a great record of influence for good in the States and in the world. Pkopbssor Michael Foster, University of Cambridge, England. Though thus tardily, it is none the less heartily, that I congratulate you and your colleagues and fellow-citizens in this celebration — and this not simply on reaching your first natural period of retrospect, but on the worthy manner of the celebration also. You are certainly setting forth a feast of rare and varied intellectual fare, and thereby also giving a great lesson to us in the Old World of that return to the international unity of universities, which it is fitting that you in America should lead. Again accept these my best wishes for the cele- bration, with hearty congratulations upon your vigorous and productive youth — with confident hope also of your yet more productive maturity. Professor T. W. Geudes, University of Edinburgh, Scotland. Arthur Bienayme (Toulon, France) addresses to the President his most sincere prayers for the prosperity of the University. I address my wishes for the brilliant future of your University. Professor Alfred Binet, Pan's, France. I find it unfortunately impossible to avail myself of your invitation, for I certainly would have desired to enter into personal relations with men who join to their high science a largeness of view seldom to be met with. Professor Jdles Tannery, Paris, France. 30 Extracts from My congratulations on the completion of tlie tenth academic year of the Uni- versity, mth my best wishes for its increase and prosperity. Pkofessok Adolf Baginsky, University of Berlin, Germany. I avail myself of this occasion to express my heartiest wishes for the further prosperity of your University. I rejoice at the admirable way in which you are to celebrate the foundation of your institution, thereby showing that it is to remain what it has hitherto been: the home of scientific investigation and culture. Phofbssor Max Dessoie, Berlin, Germany. I express my heartiest wishes for the prosperity of your University, whose scientific activity has so soon won for it a high place among the universities of your country. Professor Benno Erdmann, Bonn, Germany. In your effort to unite the nations under the banner of unselfish science, accept my most cordial congratulations and wishes for prosperity. Professor Paul Flechsig, University of Leipzig, Germany. I request you to receive my sincerest congratulations to this academical solemnity, and the expression of my hope, that your institution, highly ad- vanced through many difficulties and sacrifices, may enjoy the most splendid prosperity for many secula. Professor Ernst Haeckf:l, University of Jena, Germany. I send to you and Clark University best wishes for success. Professor Felix Klein, University of Gottingen, Germany. Permit me to express my warmest wishes for the future prosperity of your University, which, called to life ten years ago, has already won such deserved success. Professor Kdhne, University of Heidelberg, Germany. Accept my heartiest congratulations on your approaching celebration, and may it be the dawn of a stUl more momentous era than the preceding one has already been. Professor Oswald Kulpe, University of Wilrzburg, Germany. Congratulatory Letters. 31 May the following decennium of Clark University be prosperous in its development and rich in scientifie results. Professor Lindemaxn-, University of M'dnchen, Germany. I express my good wishes on the occasion of the celebration. Professor Max Noether, University of Erlangen, Germany. I do not want to let slip the opportunity of expressing my best wishes for the University which has done so much for science, and is spoken of, particu- larly in Germany, with the highest respect and esteem. Professor Ranke, University of Munchen, Germany. With the best wishes for the growth and success of youi- University, Professor W. Rein, University of Jena, Germany. I offer my best wishes for the welfare and progress of the University. Professor C. Runge, Hannover, Germany. Permit me to send my heartiest congratulations on this celebration. Under your guidance Clark University has, in the ten years of its existence, already won for itself a high reputation in the whole scientific world. May the second decennium continue like the first to advance and increase science, and may it be granted to you, Mr. President, for many years to come to be the standard- bearer of the scientific labors of Clark University. Professor Hermann Schiller, University of Giessen, Germany. Wishing the University further prosperity and progress. Professor F. Schub, Karlsruhe, Germany. I remember my sojourn in America and the kind reception which I met with in Worcester. I should rejoice to have the opportunity to renew the hospitality shown me by yourself and by your colleagues. Professor E. Stddt, University of Greifswald, Germany. May the young University, which has already developed so auspiciously, continue according to the old saying : Vivat, floreat, crescat ! Professor Waldeter, University of Berlin, Germany. 32 Extracts from I should have also been especially desirous of bringing to you my own recognition of what has hitherto been accomplished and my cordial wishes for the future. I follow with great interest particularly the psychological works which proceed from your University and are published in the American Journal of Psychology. I have always received from them the impression that the psychological and pedagogical departments of your University belonged to the most important institutions of their kind. May Clark University complete the second decennium of its existence with like, and where possible, increasing glory ! Pkofessor W. Wundt, University of Leipzig, Germany. Accept my warmest wishes for the development of the University. Professor Ed. Wetr, University of Prague, Austria. I feel a great pleasure in congratulating your Clark University on the cele- bration of the festival ; and allow me to express the hope that your University may extend its activity with every year to the honor of its President, its Trustees, and all its Members. Professor S. E. Henschen, University of Upsala, Sweden. I beg you to receive my cordial congratulations on the occasion of the beautiful decennium which your University has completed. I hope that this seat of learning shall have a future correspondingly to the excellent manner in which it has begun its life. Pkofessor H. Hoffding, University of Copenhagen, Denmark. I beg to present my sincere congratulations upon the erection of a scientific centre, the decennium of which you are to celebrate in so fitting a manner. Professor Zeuthen, University of Copenhagen, Denmark. I send you the best wishes for the success of your celebrated University, Professor Vito Volterra, University of Turin, Italy. Eternal prosperity to the vigorous propagator of light. Professor Stephanos, University of Athens, Greece. Dk. Wesley Mills (McGill University, Montreal, Canada) wishes the Uni- versity every success in the future. Congratulatory Letters. 33 With best wishes for the continued prosperity of Clark University, Professor J. Squair, University of Toronto, Canada. President Angell (University of Michigan) congratulates them on the useful work which the University has already accomplished. With hearty congratulations for what you have already achieved as President of Clark University, and in fidl assm-ance of a great future before you, Henry Barnard, Hartford, Conn., Ex-U. S. Commissioner of Education. William W. Bird.sall (President Swarthmore College) desires to extend congratulations upon the completion of the tenth year of Clark University. I congratulate you most heartily on the splendid work which Clark Uni- versity has accomplished during the ten years of its existence. Nothing in our educational work has reflected greater honor upon the American system than the high ideals so successfully maintained at Clark University. President John E. Bradley, Illinois College. Good wishes to the University in all its undertakings, and congratulations to President, Trustees, and Faculty upon the completion of ten years of distin- guished usefulness. Professor C. L. Bristol, New York University. My deepest wish is that Clark may do as much more for the advancement of science and the deepening of the true university spirit in the next decennium as it has in the one now closing. Professor Edward F. Buckner, Teachers' College, New York City. I beg leave to extend to you my most sincere congratulations on the work that Clark University has accomplished under your guidance, since the time of its founding, ten years ago. As a Fellow of the University, I enjoyed opportunities for work that other institutions could not afford, and I found your efforts to provide books, instru- ments, and material as effectual as they were untiring. As a Graduate I have found inspiration in your zeal for the furtherance of all that can advance education and science. I have followed the development of the University with pride. The first high ideals have not been lowered, and Clark remains, as it was at its founda- tion, a University for Universities. Professor H. C. Bl-mphs, Brown University. 34 Extracts from No undertaking nor movement has made so clear and definite impress upon the educational thought of America nor established guiding lines of control so distinctly in pedagogical and psychological progress as the suggestions and tendencies which have emanated from Clark University. Though the institu- tion is yet in its infancy, though the students in point of numbers have been limited, yet its influence has penetrated every state in the Union, has entered practically every educational institution of the land, from university to kinder- garten, and has quickened the spirit of educational conferences, from those of national repute to those of the little teachers' meetings of the village school. Granting the truth of the educational view for which Clark University stands, and allowing for the singularly forceful methods of instruction by the President and Faculty within the institution, and the energy with which its mission has been prosecuted, it is nevertheless stUl a marvel that its influence should have become, in this brief space, so widespread and vigorous. The facts which stand prove the wisdom of the plan of an institution which should be exclusively graduate, selecting as its students a limited number of mature thinkers who should be inspired by the power wliich ever comes from the con- tact with original investigation and a faculty of original investigators. Frederick Burk, President State Normal School, San Francisco, and President Clark University Alumni Association of California. I send my best wishes for the success of the anniversary exercises and for the continued and enlarged prosperity of the University. President Nathaniel Butler, Colby University. Kindly accept my congratulations upon the completion of your tenth aca- demic year. Professor R. H. Chittenden, Director Sheffield Scientific School. I desire to congratulate the Faculty of the University on the great work accomplished within a comparatively short period. Brother Chrtsostom, Manhattayi College. We rejoice with your many friends in the successful rounding out of Clark University's first decade. It is a consolation to the generous benefactor that the world recognizes the merit of the Institution, which his munificence estab- lished and maintains. Coming into existence the same year, holding similar views as to the place of graduate work, having the highest ideals of university endeavor, striving earnestly to realize them in spite of all difi&culty, our two Universities have always felt strong attachments for one another, and a more than ordinary interest in one another's success. The Catholic University ten- Congratulatory Letters. 35 ders you its most cordial greeting oa this the daj' of your rejoiciag. It bids me extend to you and through you to the University its most sincere wishes for still higher and greater success in its chosen fields. Thos. J. CoNATY, Washington, D. C, Rector Catholic University. With many congratulations on the past ten years' work of the University, Pbofessok Charles R. Cross, Massachusetts Institute of Technology. Professor C. B. Davejjport (Harvard University) desires to express his appreciation of the brilliant example of research as a primary university func- tion which Clark University has for a decade set. The University and all connected with it are to be congratulated. Professor Ellebt W. Davis, University of Nebraska. I desire to express my appreciation of the splendid work done by Clark University during these ten years. Professor Nathaniel F. Davis, Brown University. I must content myself with rejoicing over the unique intellectual enterprise you are carrj'ing out. I may not be informed regarding such matters, but it seems to me you have accomplished a sort of scientific coiq) cVetat in getting such a group of scholars to come to America upon the occasion of your anniver- sary. As a disciple of Clark University, and an admirer of everything it stands for, I take pride in the impression that must necessarily be made upon Ameri- can scholarship by the visit of such men. I congratulate all of you, and hope that everything you desire in connection with the series of lectures may be realized. Professor George E. Dawson, Bible Xormal College, Springfield, Mass. President Drown of Lehigh University begs for his colleagues and for himself, to offer his hearty congratulations to the President, Trustees, and Faculty of Clark University on the completion of a decade of usefulness in the higher education, marked by distinguished services in many lines of original research. Permit me to express my admiration of the work you have done and are doing. Professor William I'. Durfee, Bubart College. 36 Extracts from Let me assure you that we are all grateful for what Clark University is do- in"- for sound education in this country, and I can only hope that you may have many successful years in the development of the work which you are doing. S. T. Button, Brookline, Mass., Superintendent of Schools. In the opinion of many who have studied there, the peculiar advantage of Clark University is mainly attributable to the close and personal relations between professors and students under which the work is conducted. The formal lecture delivered to a body of men in the class-room has but little of the stimulative force imparted by a conversational discussion with the man alone in the lecturer's private study, and too great praise can hardly be given to the members of the faculty of Clark for their constant and generous sacrifice of time to this most helpful method of instruction. The frequent assignment through- out the course of problems involving research leads to the best of training for the later performance of original work, and the presentation in the lecture-room of the results thus obtained gives experience in the work of the lecturer. In perhaps no other institution are these methods of the personal conference and the " colloquium " so constantly applied ; no doubt such methods are impossible in most larger universities at present ; and one can hardly imagine such a course followed with more kindness and devotion at any time than is now the case at Clark. Professor Frederick C. Ferry, Williams College. 1 take this means of expressing my interest in the noble success of the University, and of wishing it continuance and increase. Rabbi Charles Fleischer, Boston, Mass. Congratulating you and the University upon these years of achievement, Alice C. Fletcher, Washington, D.C. I rejoice in the prosperity of your institution because it is one which sends forth its light, not only for the few, but for the many. President Wm. Goodell Frost, Berea College. Though my stay with you was short, yet it meant the inspiration that took me abroad and pushed me on to undertake important work. Professor John P. Fruit, William Jexnell College, Congratulatory Letters. 37 The Johns Hopkins University sends its cordial greetings to the President, the Trustees, and the Faculty of Clark University, on the completion of its first decennium, with congratulations upon its successful maintenance of high ideals, and with best wishes for its continued prosperity and power. President Daniel C. Oilman. I must add my congratulations on the success of your work, and my good wishes for its continuance on even a larger scale. Professor George L. Goodale, Harvard University. Every educator especially owes a debt of gratitude to Clark for the fearless work it has done in breaking down blind prejudice and advancing the truth. Professor John Y. Graham, University of Alabama. You have certainly arranged a most dignified and impressive series of lectures — wholly congruous with the work which you have been doing during the decade. Professor Edward H. Griffin, Johns Hopkins University. You will please accept the assurance that I am very glad indeed that your institution, which has already done so much for the cause of progressive educa- tion, has thus shown its vitality and power of endurance. No doubt these ten years have meant much struggle and anxiety on the part of those whose heart was in the work. Others may be able to express their appreciation of this work with greater eloquence, but none can be more sincere and thankful than I am. Truly, if there is such a thing as a science of education in this country now, Clark University . . . (has) contributed the largest share toward this accom- plishment. To me (its) work has meant an awakening and uplifting hardly equalled by any other influences that have worked upon my soul. May your anniversary week be a thorough success. Professor M. P. E. Grossmann, Milwaukee, Wis. Allow me to extend cordial congratulations on the auspicious event. Professor Charles W. Hargitt, Syracuse University. I extend for the University of Maine hearty congratulations, and wish con- tinued prosperity for the future. President A. \V. Harris, University of Maine. 38 Extracts from I send you my hearty congratulations on your decennial celebration. Phesident Walter L. Hervet, Teachers' College, New York City. I congratulate you and the Trustees and Faculty upon these successful years of your University work, and upon this most appropriate mode of celebrating the anniversary. It is a mode worthy of universal following, and will, without doubt, be more and more adopted by our institutions of higher learning. Professor 6. H. Howison, University of California. Allow me to offer my congratulations to you especially, and to your associ- ates, for the marked success which has attended the career of Clark University. We have felt that it not only increases the resources of high education for youth, but it stands for progress and enlightenment in the commonwealth and the country at large. There is a justifiable pride on the part of those who love earnestness and progress in educational matters, as they review the past of this institution, into whose good name and wide scope of influence you have thrown so much of your personal enthusiasm as well as your scholarly ability. Kev. Edward A. Horton, Boston, Mass. I have many pleasant memories of a year's profitable work at Clark, and rejoice in the continued success of Clark University. Professor L. S. Huleurt, Johns Hopkins University. The programme presented is most attractive and inspiring. I congratulate you upon the successful work of the past ten years. Dr. Henry M. Hurd, Baltimore, Md., Superintendent Johns Hopkins Hospital. I can't help expressing to you my feeling of satisfaction, and repeating the satisfaction I heard such men as Cattell, Royce, Van Gieson, !Munsterberg, and Putnam express, with the excellent good taste and refinement of your little celebration. All the refinements of the world seem now to take refuge in the smaller things ; the bigger ones are getting too big for any virtue to remain with them. You have done something original and succeeded perfectly, from the point of view of the passive " assistant." Professor William James, Harvard University. I beg you to accept my heartiest congratulations. Each year, I sincerely believe, finds me more grateful and appreciative of the privileges I enjoyed at Congratulatory Letters. 39 Clark, and especially do I realize more and more what you yourself did for me. I trust you may be spared health and vigor many years to come in your labor, for you are doing a great work. Geokge E. Johnson, Andover, Mass., Superintendent of Schools. I have been very deeply interested in the work of Clark University, and in the way it has held to its high purposes regardless of pressure of all sorts in other directions. . . . Stanford congratulates Clark on ten years' noble work for high thought and accurate investigation. President David S. Jordan, Leland Stanford Jr. University. For myself and all the staff of the University of California, I send you hearty congratiUations and good wishes. You have not attempted to do as many things as some other universities, but what you have attempted you have done exceedingly well. If excellent work is the standard of true success, you have been successful among the foremost. May your achievements and your reputation gain still greater lustre, and your educational work confer still larger benefits on succeeding generations. President Martin Kellogg, University of California. May I say that I think you have taken a most admirable course in the char- acter of this celebration, and that I wish you every success, not only on this occasion, but in all the future years of the University. Professor J. S. Kingslbt, Tvfts College. I regret more than I can express my inability to be present at the decennial celebration of your noble institution, and to hear the splendid series of lectures which you have provided. Professor Joseph LeConte, University of California. It is a pleasure to me to join in the celebration of the first decade of Clark University. The method of celebrating the event is, I think, exceedingly fit- ting. I enjoyed several of the most interesting years of my life in the lecture- rooms and laboratories of Clark, and always recall them with great satisfaction. Professor J. S. Lemon, Washington, D.C., Columbian University. Clark University stands unique among the universities of this country in the work which it is attempting to do. No other institution has done more in the 40 Extracts from line of original investigation, nor given during the same period greater inspira- tion to the educators of the country. Professor G. W. A. LncKET, University of Nebraska. Permit me to congratulate heartily the President, Trustees, and the Faculty of Clark University upon the completion of the tenth academic year of the University. President George E. MacLean, University of Nebraska. We appreciate the great work done by Clark University, and send every best wish for the future. President James G. K. McClhre, Lake Forest University. I have the highest feelings of regard for Clark University, for I feel that I owe much to it. Its conception is the broadest and best of all our institutions, and I hope the time will come when it can broaden out, aud, all obstacles being removed, go on to its full completeness. Professor William S. Miller, University of Wisco7isi7i. When one thinks of the amount of light that has spread from Clark Univer- sity and of the place that it fills in American education to-day, it is hard to realize that no such institution was in existence ten years ago. Please accept my most sincere congratulations to this auspicious event, with the hope that a long series of years of vigorous activity may be granted to you, so that you may lead the University to ever new achievements, and continue to benefit the cause of education in the future, as you have so splendidly done in the past. Vivat, floreat, crescat. Professor F. Monteser, Neil) York University. It is with very great regret that I find it impossible to attend the rich cele- bration you have prepared for the friends of Clark University and of all the forward movements in science for which you have made Clark University stand, and wish the University long-continued and increasing prosperity. Professor E. H. Moore, University of Chicago. My participation in the treasures you offered was thus limited to one day — but this one day, with the three lectures I listened to, and the very interesting men I met, was so agreeable and valuable that I feel the desire to thank you warmly for the distinguished and exquisite pleasure. I take special pleasure in combining with my personal thanks my congratulations to the high success Congratulatorij Letters. 41 of the celebration and my very best wishes for the next ten years in the life of Clark University. Peofessok Hugo Muxsterbekg, Harvard University. Permit me to express here my sincere admiration and respect for the aims, ideals, and plans of Clark University ; these are of such an ideal character that they are bound to interest profoundly every man who loves science for its own sake. Professor J. TJ. Nef, University of Chicago. Permit me to offer my hearty congratulations on the work done and the progress made in the ten years of Clark's existence, to express the hope that the future may be marked by even greater achievements. Pkesldest Ctrcs Northrop, University of Minnesota. Clark University does well to celebrate in such a becoming manner the noble service which she has rendered to higher education in this country. May the next ten years be no less fruitful and helpful to those who have become accustomed to look to Clark University for inspiration and guidance. Professor F. W. Osborx, Adelphi College, Brooklyn, y. Y. I write to congratulate you most cordially upon your celebration of the com- pletion of the tenth academic year of Clark University. Professor Henrt F. Osbobh, Columbia University. Allow me to congratulate you upon these lectures, and also upon the remark- able results which you have been able to accomplish in ten years in connection Avith Clark University. Professor G. T. W. Patrick, University of Iowa. The Provost, Trustees, and Faculty of the University of Pennsylvania present their cordial congratulations to the President, Trustees, and Faculty of Clark University on the happy completion of the tenth academic year of the University. The President, Trustees, and Faculty of Clark University certainly deserve the thanks of all those interested in the cause of education. Professor George II. Price, Vanderbilt University. 42 Extracts from Pray accept congratulations on tlie completion of a decade of grand work, and on the prospects of even better work for the future. John T. Prince, West Newton, Mass., Agent State Board of Education. I send my heartiest congratulations on the great achievements of Clark University during its first decade, and express my sincerest desire that its use- fulness may continue and expand for many centuries to come. Professor Ernst Richard, Neio York City. President H. W. Eogers (Northwestern University) desires to c-stend the congratulations of Northwestern University, as well as his own, upon the great success of Clark University and the distinction it has attained in the academic world. James E. Eussell (Dean, Teachers' College, New York) wishes to convey to the President of the University his best wishes for the continued success and prosperity of the institution. . President L. Clark Seelye (Smith College) offers his hearty congratula- tions on the important educational work it has already accomplished. With sincere thanks and hearty congratulations on the auspicious occasion, Professor James Seth, Columbia University. Meanwhile I wish to join in the many congratulations I am sure you will receive upon the quiet and dignified, but none the less eminent, manner in which Clark University has carried on the work of the past decade, and upon the manner in which it has reflected honor upon American scholarship in science and philosophy. Albert Shaw, New York, N. Y., Editor Review of Beviews. We shall always be grateful for the work that has already been accomplished by the University, and especially for the ideals which it has brought to the colleges and universities of the West. With high personal regard and warmest congratulations from our faculty. President William F. Slocum, Colorado College. With best wishes for the success of the celebration and for the continued prosperity of your institution, President F. H. Snow, University of Kansas. Congi'atulatory Letters. 43 Allow me to congratulate the University upon its happy completion of ten years' life and work, and to wish it a long and prosperous future. Professor Frederick Stark, University of Chicago. I wish to send my cordial congratulations and my wish that the next ten years may witness the coming to the University of such ample endowments as will enable it to accomplish its high ideals. President James M. Taylor, Vassar College. Please accept my best wishes for continued prosperity. President W. 0. Thompson, Ohio State University. I do not like to let the present occasion pass without intimating to you my appreciation and admiration of the methods and aims of university work for which Clark University has stood in the past, and will, I hope, stand in a long and prosperous future. My recent visit to Worcester merely confirmed a belief which I have long held, — that the type of man that Clark University calls to its professorial chairs, and the type of man that it sends into active life at the close of its three or four years of graduate study, are types that represent the highest ideal of scholarship, and are the very salt of American society. I hope, most sincerely, that the coming celebration will prove to be the door through which you and the University pass to greatly extended activity upon your own high level. Professor E. B. Titchener, Cornell University. I wish to express my sincere appreciation of the service to education and investigation which Clark University is thus undertaking, a service which is eminently in harmony with the work of Clark University from the beginning. Professor James H. Tufts, University of Chicago. I wish to extend my hearty congratulations on the successful work of the University during the last ten years, and my best and most hearty good wishes. Professor John M. Tyler, Amherst College. Professor Henry B. Ward (University of Nebraska) extends to the President, Trustees, and Faculty his congratulations upon the occasion, and best wishes for the continued success of the institution. 44 Extracts from Congratulatory Letters. I express my sincere congratulations Professor Sho Watase. University of Chicago. With best wishes for the success of the University, Pkofessor J. B. Weems, Iowa State College. Please accept congratulations upon the honorable record of these ten com- pleted years. The distinguished service of yourself and the University have made the whole world your debtor. President B. L. Whitman, Washington, D. C, Columbian University. The Clark University ideal as I understood it, when connected with its early work, is the ideal which I place above any others thus far proposed, and I hope that it may find strong friends to help it forward. Professor Charles 0. Whitman, University of Chicago. Professor A. W. "Wright (Yale University) sends congratulations and best wishes for the prosperity of the University. I DECENNIAL ADDRESS. By G. Stanley Hall, President of the University. It has been said that decades are the best periods for studying historic tendencies because they are long enough to contain a rich array of facts and events, and short enough to be grasped by a single mind in the stage of its prime. The ten years since Clark University was opened, the close of which, by the cooperation of a few beneficent public-spirited citizens of Worcester, we have sought to mark in a very quiet but dignified way that should befit at once its size and its quality, constitute distinctly the most important decade in the educational history of this country. The mere index of a few of the well-known and accomplished facts of these years has an eloquence beyond all words. They have witnessed the establish- ment of the Catholic University at Wasliington, witli its strong faculty and its handful of picked graduates from the seventy Catholic colleges of the country, the only university in the land besides Clark devoted solely to graduate work, an institution related to us, not only Ijy a strong tie of sympathy in the struggle and sacrifice for ideals and liigh standards, but by my own long and personal friendship with the first rector, and by the fact that its present head was oiu' Worcester neighbor and kindly friend. The Leland Stanford University, now one of the richest in the world, was planned and endowed by a long-time friend of our Founder, and the wife of that founder lately told me that she still counts ours among her wisest and most trusted advisers. The University of Chicago, with possibilities of increase brighter and larger than any other, from the very first the most rapid academic growth in history, has leaped into existence with a Minerva-like completeness, owing in no small part its first impulse to higher creative work in science to the sagacity of the chief trustee of its Ogden Fund, our fellow-townsman, Andrew H. Green, and which is still more closely affiliated to us by the fact that so many of the leading members of its faculty honored us by doing three years of their best work here, and for which we still cherish a little of 46 46 • Decennial Address. the feeling of a poor but proud and noble mother for her great son. The new Methodist University at Washington has begun the unfoldment of large and well-matured plans, for the fulfilment of which the vitality of the strong religious body behind it is perhaps the best of guarantees. The millions already provided and about to be expended at the State University of California which will involve transformation and enlarge- ment perhaps greater than all that has hitherto been done there, very comprehensive and valuable as that is ; the magnificent new architectural installation at Columbia and the federation of so many other affiliated institutions about it, with all the possibilities of our greatest metropolis open before it ; the steady development, whether for good or for ill, of the plan of a great national imiversity, to which at least all state, if not private, colleges and universities may be tributary as feeders, and which shall command all the vast resources of science in Washington, unify them, and add the new vitalizing function of research and perhaps teach- ing, a scheme that has enlisted most of the educational leaders of the land and is sure of eventual fulfilment, — such are some of the events which have seemed to many to thi'eaten the academic preeminence of New England, and even of the East, in the future ; that have stirred to their very foundations the older and more conservative institutions, and caused transformations not all apparent from the outside, but which involve hardly less than an ultimate revolution of academic sentiments, methods, and ideals. Fellowships, not for the indigent who need supjaort while preparing for the professions, but to give leisure, opportunity, and incentive for full development to talent, the choicest of all national prod- ucts ; research, with books, apparatus, above all, leaders competent to guide and inspire ; new post-graduate departments for non-professional specialization, with their own laboratories and libraries ; seminaries where experts discuss the latest literature, best methods, instruments, and results of investigation ; new journals devoted to the speedy publication of such studies ; new chairs and topics ; a growing and ever widening distinction between receptive learning and active creation, — these and the gradual completion of a system that is truly national, and has not its apex in Europe, where hundreds of our graduates still go yearly to get what it should be a matter of simple patriotism to siipply at home, must suffice to mark the direction and progress of these years in which institu- tions and work alike are becoming more and more plastic to the changing and ever more imperative needs of learning and science which have them- Decennial Address. 47 selves celebrated triumphs during the decade which could not even be enumerated within the hour. It is no wonder that many old academic problems have become obsolete and new ones have arisen, and that pres- ent demands in men, methods, and instruments have changed fi'om those of ten years ago. Again, within this time a wave of doubt and opposition to the public supi^ort of intermediate education passed over the country, but the reac- tion against that tendency has made the last few years j^reeminently the age of high schools. More and statelier buildings than ever before, longer courses and more of them, many modifications suggested by committees and others, great increase in the number of students, rich and well-planned departures like the Tome Institute, Mrs. Emmons Blaine's new normal foundation, and several others contemplated or assured but not yet established, the new associations of high schools and colleges covering now all sections of the country, the ever increas- ing collegiate character of the work done in such institutions, and the consequent development of a distinct, and in some places urgent, small college problem, — all this shows that even secondary education, the last stage to be reached by reforms, has here been stirred and quickened as never before. If we extend our view to lower grades, we find all plastic and chang- ing. This stronghold of conservatism is invaded by the spirit of reform, often revolution, and sometimes even of rather wild experimentation. New journals, pedagogical chairs, new methods, the new school hygiene, broader views that relate teaching to all the great problems of science, statescraft, and religion, have arisen, which have brought the university and kindergarten and all between them into an organic unity, yet fitting all features of the system to the vast variety of individual differences of character, temperament, and ability. In this field, I think, the closing decade has witnessed a change greater than the preceding quarter of a century. New and better minds are enlisted, educational topics are of increasingly central interest in the press and more dominant in the church and pulpit, education is becoming more professional and scientific, recognizing the necessity of expert leadership and mastery, and is at last assuming its rightful and larger power, and its normal basal all-condition- ing place as at bottom a biological science, revealing to us how state, church, home, literature, science, art, and all else have their ultimate justification only in so far as they are effective in bringing human beings 48 - Decennial Address. to the ever fuller maturity of mind and body on which civilization de- pends, and which it is the work of education to accomplish in the world. This is increasingly necessary as our country grows in population and in territorial expansion, and educational progress is coming to be recognized by history as the chief standard by which to test all other advancement. Europe has progressed during this decade, although with less rapidity, along nearly all these lines, and the next decennial promises not less, but more advance. In such a time it is, indeed, glorious to be alive, and to be young is heaven, for hope is even brighter than memory. No time in the history of the country could have been more favorable than the beginning of tiiis period for a great and new university founda- tion. The epoch-making work of the Johns Hopkins University, which for the preceding decade had made Baltimore the brightest spot on the educational map of the country, and was the pioneer in the upward move- ment, had leavened the colleges and roused them from the life of mo- notony and routine which then prevailed, and kindled a strong and widespread desire for better things. The significance of the work of that institution can hardly be overestimated. But financial clouds had already begun to threaten this great Southern luminary, and there were indications that, if the great work it had begun was to be carried on, parts of it, at least, must be transplanted to new fields. It was at this crisis that our munificent Founder entered the field with the largest single gift ever made to education in New England, and one of the largest in the world, and vnth the offer of more to come, if suffi- cient cooperation was forthcoming. He selected Worcester as the site of his great enterprise with a piety to the region of his nativity worthy of the greatest respect and emulation, and in addition to the fulfilment of his pledges gave it the benefit of his own previous wide studies of educa- tion in Europe, and contributed wisely matured plans and constant per- sonal oversight and labor for years. It is as strenuously engaged in this highest of all human endeavors that the world knows him, and that we shall remember him, and I am sure that we all unite to-day first of all in sending him in the retirement his health demands (although it cannot assuage his interest to see the work of his hands prosper) our most cordial greetings and our most hearty congratulations. With a dozen colleges within a radius of one hundred miles doing graduate work, the plainest logic of events suggested at once a policy of transplanting to this new field part of the spirit of the Johns Hopkins Decennial Address. 49 University, and taking here the obvious and almost inevitalile next step by eliminating college work, although the chief source of income by fees was thereby also sacrificed, and thus avoiding the hot and sometimes bitter competition for students, waiving the test of numbers, and being the first upon the higher plane of purely graduate work, selecting rigorously the best students, seeking to train leaders only, educating professors, and ad- vancing science by new discoveries. It was indeed a new field wide open and in\-iting, the cultivation of which was needed to complete our national life ; the preliminary stages of its occupancy all finished, yes, necessary almost as a work of rescue for the few elite graduates who wished to go beyond college but not into any of the three professions, and who had had hitherto a pathetically hard time. The call to the President gave assurance of the highest aims and of perfect academic freedom, a pledge that has been absolutely kept. He was sent to Europe a 3'ear on full pay to learn the best its institutions could teach, and the Faculty that first fore-gathered here has never been excelled in strength, if indeed it has ever been equalled anywhere for its size. Story, an instructor at Harvard, colleague and friend of Sylvester, formerly acting editor of the chief mathematical journal of the country and co-head of his department at Baltimore, founder of another journal here, who has enriched his department by contributions, the list of which printed else- where in this volume tells its own story ; Michelson, who while here accepted an invitation of the French Government to demonstrate in Paris his epoch-making discoveries in the field of light, which he did while on our pay-roll — lately especially honored by learned societies at home and abroad, now head of one of the best-equipped and largest laboratories in the world, and still continuing his brilliant contributions to the sum of human knowledge ; Whitman, now head of another great university laboratory, trainer of many young professors, fovuider and editor of the best and most expensive biological journal, head of Woods HoU marine laboratory and summer scliool, one of the best of its kind in the world, himself a contributor to his science ; Michael, than whom America had not produced a more promising or talented chemist, the list of whose published works would be far too long to read here ; Nef, perhaps our most brilliant young chemist, and now head of one of the largest and best-equipped laboratories in the world, and with a power of sus- tained original work rarely excelled ; Mall, now full professor at the Johns Hopkins University, and head of the great new anatomical labora- 50 Decennial Address. tory and museum there, whose published contributions are admirable illustrations of both the great caution and boldness needed by a student in his field ; Boas, the leading American in physical anthropology, now a professor at Columbia ; Loeb, almost the first expert that this country could boast in the new physical chemistry in the sense of Ostwald, now head of his department in the University of the city of New York; Bolza, an almost ideal teacher, suggesting the great Kirchoff in the per- fection of his demonstrations ; the brilliant and lamented Baur, leader of the expedition to the Galapagos Islands made possible by the gift of Worcester's patron saint of so many good enterprises, Mr. Salisbury; Donaldson, now dean of the graduate school of the University of Chicago, author of the best handbook in English on the brain, with a caution, poise, and diligence befitting the successful investigator in that dangerous but fascinating field ; MuUiken, suddenly placed in a position of great difficulty, discharged its duties with rare ability and discretion for one so young ; Lombard, now professor in Michigan, genial, assiduous, a gifted teacher and enthusiastic student ; White, scholarly, able, a born teacher and student ; McMurrich, an untiring investigator and a lucid inquirer after knowledge ; those now here, who have since become so well-known, Burnham, Chamberlain, Hodge, Perott, Sanford, Taber, and Webster ; these, not to mention many others, then only fellows, but who have achieved so much in their work and positions since, — these are the men and others whose presence on this spot, whose high intercourse and whose stimulating personal contact with each other, whose ardor and devo- tion in the pursuit of knowledge, whose healthful emulation in achieve- ment, made this almost classic groimd and the cynosure of the eyes of all those in this country who love science for its own sake. With the wealth, wisdom, and interest of our Founder, with the high character and culture of our Board of Trustees, with the intelligence of such a community of old New England, with an atmosphere of intellectual free- dom, with unique and precious exemption from the drudgery of excessive teaching and examinations, with the youth of the Faculty, none of whom had reached the zenith of their maturity, with substantial and ample buildings, abundant and forthcoming funds for equipment, few rules and almost no discipline or routine of faculty meetings, the motto on our seal, fiat lux, our university color white, — is it any wonder if some of our young men saw visions and dreamed dreams, or perhaps in some cases fell in love with the highest ideals, or that the very memory of the first stage Decennial Address. 51 of our history is to-daj-, as it has been in darker hours, a most precious memory and a basis of an all-sustaining hope ? To these days of our prime to which our former students and profess- ors recur with jo}% and in whose breasts the processes of. idealization of them have alread}' begun, days which were pervaded by sentiments of joy and hope very like those which animated the best years of the Johns Hopkins University, we have often reverted since in soberer hours with longing thoughts of what might have been had the University continued in all its pristine strength. Not one weak, dull, or bad man in our Faculty, all given not only leisiu-e, but every possible incentive to do the very best work of which they were capable, with a Founder and a board of control who realized that a new endowment should do new things, and that the best use of money is to help the best men, we entered a field very largely new and with as bright prospects as we could wish. But life has its contrasts and competitions. The reductions of our force, which occurred at the end of the third year, sad to us almost beyond precedent, although helpful elsewhere, may be ascribed to fate, disease, or to the very envy of the gods. Some incidents should remain unwritten, but it should be known that our Trustees foresaw from the beginning of the year one of the gravest of crises, and met it with an unanimity, a wisdom, and a firmness which even in the light of all that has transpired since, I think, could not be improved on. The pain of it all has faded, the glad hand has been extended and accepted by nearly if not quite all who left us ; the lessons of adversity have been learned and laid well to heart, and we hope and believe that these and all their attendant incidents may be considered closed. Although nearly half our Faculty and students left us in the hegira, and our income had dropped in almost the same proportion, and only the departments of psychology and mathematics remained nearly intact, we fortunately had left in every department young men as promising as any in the land. Tliey needed simply to grow, and never has there been such an environment for a faculty to develop as in tliis " paradise of young professors," as a leading college president has called this University. To Darwin the greatest joy of life was to see growth ; and to see the unfold- ment of tliese youthful, intellectual elite, and to feel the sense of growth with them as all near them must, is a satisfaction almost akin to the rapture of discovery itself. Now the years have done their work, and our Faculty, although smaller, was never stronger, never more prolific. 52 Decennial Address. stimulating, and attractive to students, in proportion to its size, than it is to-day. There has never been such loyalty to the institution and its ideals, such readiness to endure the petty and the great economies now necessary, such, prompt and frequent refusals of larger salaries elsewhere, and so strong a sentiment that, so long as a man has growth in him, our incentive, opportunity, and plan of work are of more value than a large increase of salary. These changes involved, however, but little reduction of the number of instructors or of students, but materially decreased for a time the effi- ciency of the University. Since the end of the third year the President, who was not required to teach, has done full professorial duty in addition to that of administration, has established a seminary at his house three hours each week through the entire academic year, and founded and con- ducted at his own expense a new educational journal. The income-bear- ing summer school has been organized and conducted during the past seven years with the active and efficient cooperation of a large local advisory board under the direction of Colonel E. B. Stoddard and Charles M. Thayer, Esq., by which its social character, that has contributed much to its success, has been established on a high plane. The summer school represents only the departments of biology, psychology, and jjeda- gogy, is open to every one of either sex on the payment of a small fee, is popular rather than technical in its scientific character, has been numer- ously attended, and in all ways is directly in contrast with the work of the year. Hardly a ripple has marred the harmony within the Univer- sity during these last seven years, and every man, student and instructor alike, has been hard at work and enthusiastic for our own unique and individual method and plan. This institution must be judged from within and by educational and scientific experts, and the commendations which we have lately received from leading specialists, some of which are printed elsewhere in this volume, have been so numerous, spontaneous, and hearty in response to our invitation to be present, as almost to rival in cordiality and loyalty to the now so definitely developed Clark idea and Clark spirit that of our three alumni associations of the Pacific Coast, Illinois, and Indiana organized during the present year. Scientific work must be weighed and not measured, so that numbers tell but little. Clark University has been instrumental in training well- nigh three hundred professors or special academic instructors, has numbered Decennial Address. 53 over twelve hundred different persons enrolled in its summer school, not counting the hundi-eds who have attended more than one session. These, especially the former, are in a sense our epistles known and read of all men. The other output of a luiiversity like ours is its scientific work, and here we have five hundred publications based upon work done here, of which twenty-five are books. The University now publishes three journals, with hope of a fourth as a more permanent way of marking the beginning of its second decade. Small as we are, if our departments and students are measured by the significant criterion of the number of the doctorates annually conferred here, we rank among the best and largest institutions of the land. Al- though our fellowship funds have declined, and that, too, in the midst of a competition, which never existed or was hitherto dreamed of, our numbers of late years have slightly but steadily increased, although at the same time we could go on forever and do invaluable work of research and publication like the French Ecole des Hautes Etudes, or a few other Old World institutions, even if we had no students ; and, indeed, America may need in the future, if, indeed, she does not already, at least, one such academic endowment for research only. One thing, at least, is true so far, hardship has no whit lowered our aims or diluted our quality, but if anything has had the reverse influence ; and I fervently trust (and think I can speak on this point with confidence for the entire Faculty) that this may be the case throughout all the infinite future that endowments like this in a country like ours have a right to expect. Although influences are too subtly psychological to be traced, I am writing our history, and find it a most inspiring theme, and I believe it adds already a very bright and hopeful page to the records of higher education in the country, and one which history will brighten to epochal significance. It has, on the whole, in it one clear note, not of discouragement, but of hope and confidence. Have we duly considered, even the best of us, what a real university is and means, how widely it differs from a college, and what a wealth of vast, new, and in themselves most educative problems it opens? A college is for general, the university for special, culture. The former develops a wide basis of training and information, while the latter brings to a definite apex. One makes broad men, the other sharpens them to a point. The college digests and impresses second-hand knowledge as higlily vitalized as good pedagogy can make it, while the university, 54 Decennial Address. as one of its choicest functions, creates new knowledge by research and discovery. The well-furnished bachelor of arts, on turning from the receptivity of knowing to creative research, is at first helpless as a new-born babe, and needs abundant and personal direction and encour- agement before he can walk alone ; but when the new powers are once acquired they are veritable regeneration. He scorns the mere luxury of knowing, and wishes to achieve, to become an authority and not an echo. His ambition is to know how it looks near and beyond the frontier of knowledge, and to wrest if possible a new inch of territory from the nescient realm of chaos and old night, and this becomes a new and consuming passion which makes him feel a certain kinship with the great creative minds of all ages, and having contributed ever so little, he realizes for the first time what true intellectual freedom is, and attains intellectual manhood and maturity. This thrill of discovery, once felt, is the royal accolade of science, which says to the novice, stand erect, look about you, that henceforth you may light your own way with independent knowledge. This higher educational realm is full of new " phenomena of altitude." Faculties, instead of discussing and elaborating plans for commencement ceremonies, hearing recitations, preparing and then reading the results of examination papers, and carefully marking each individual exercise, grinding in the old mills of parietal regulations, discipline, and the rules of conduct needful to civilize the adolescent homo sapiens ferus, revising requirements for admission, tacking and shaping the policy to gather in more students and keep ahead of others in the struggle to get the best connections with high and fitting schools, are occupied with far different problems wherever the university spirit has a true and real embodiment. Here first of all men must be discriminated, and great issues hang upon the success in differentiating superior from indifferent young men. To detect the early manifestations of talent and genius in the different fields of intellectual endeavor, which some presidents and professors can, and others so eminently lack the power to do, is the crucial doorkeeping problem, where great privileges are to be awarded to great promise. This is almost a life and career saving function for not only the young professors and students, but for the university. Men are not equal, and there must be a touchstone of mental aristocracy to discriminate $500 from $10,000 men. Second, having selected these, the university should bestow freely its Decennial Address. 55 needed aid and equipment, and the professor his choicest time and knowledge, to perfect the precious environment by which the later stages of growth, so liable to be lost, but on the full development of which civilization itself hangs, and perfected. How to select the best, ripest, and most fruitful topics for investigation requires an almost prophetic ken in which differences in individual professors are immense. To study individuals enough to adapt each theme to each personality is another problem as new as it is delicate and difficult. The i-ight solution of both these is the large half of the work. The professor should give his best sueeestion, with no reservation for himself, and the able student should not be an apprentice to serve his master, but should be distinctly educated toward leadership himself from the first. Havinsr thus sown fit seed in fit soil, it must be watched and watered with constant suggestion. The best and newest literature ; the most effective and original apparatus that can be devised and if possible made on the spot ; how to insure in the best form and place the speedy publication of work and to bring it under the eye. of all experts ; how to avoid conflict and duplication ; how general or how special thesis sub- jects and work should be to best combine the two sometimes more or less divergent ends of discovery and education ; the requirements for perhaps the choicest of all degrees, the doctorate of philosophy ; the best modes of individual examination for it ; the number and relation of subjects required ; the migration of students so as to insure not only the best environment for each, but to give to professors not only in the same department, but in different institutions, the same stimulus that was felt when the elective system aroused the dry-as-dust professors to unwonted effort lest their class-rooms be left vacant ; the kindred ques- tion of the relative value of graduate work at home and abroad for each student and for each department ; the fit federation of graduate clubs and their thirty-five hundred members in the twenty-three Ameri- can institutions now recognized in the yearbook ; the great problem of printing and special journals together with interchange of monographs ; the vast now library problems of purveying for highly specialized, but very voracious, appetites which make the true university librarian a man of far different order from others, and gives him a wealth of new prob- lems of exchange, foraging, etc. ; to maintain the true relations between lecture work and individual guidance while duly emancipating the pro- fessors from the drudgery of elementary teaching and mass treatment 56 Decennial Address. of great bodies of students ; the many and wide-reaching differences between pure and applied science, and the practical methods by which this distinction is maintained ; the danger of great aggregations of students and the advantages of few ; the wide differences between the new kind of professor needed in the university and those in the college, where no provision is made for the advancement of learning, and the tests are mainly pedagogic ; the even greater contrasts between scholar- ship funds for the aid of poverty to professional careers, which are a doubtful advantage even in colleges where they belong, and the true university fellowship as above described ; the growing dominance and need of expertness in all fields for which graduate departments must prepare as well as for professorship alone, — these and many great ques- tions like them, destined more and more to eclipse all others which are just looming up, and for the irrigation and ventilation of which we hope to establish here soon a new educational journal — such questions con- stitute this opening field of what may be called the higher educational statesmanship. The hastiest glance at the situation on an anniversary like this would be incomplete unless we turned toward the future. Our own needs here are many and our wants urgent, but our faith is firm that in a community like this the time will soon come when no wills will be drawn by wealthy people without carefully considering the conclusion of the largest parlia- mentary report ever made, which fills near a score of volumes, was many years in the making, and describes all the public bequests ever made in England. The substance of the conclusion of that most competent tribu- nal that has ever spoken upon this subject is that the best of all uses of public benefactions is, not for charity to the poor or even the sick and de- fective, noble and Christlike as those charities are, not for loiver education or religion, beneficient as these are, but rather for affording the very best opportunities for the highest possible training of the very best minds in universities, because in training these the tvhole work of church, state, school, and charity is not only made more efficient, but raised to a higher level, and in this service all other causes are at the same time best advanced. I beg respectfully, but with all my heart and mind, to urge this conclusion by the highest human authority upon all those contemplating the bestow- ment of funds where they will accomplish most for the good of man. Our very best department is the library, which is so well endowed that we do not at present need to expend the income of the fund. In this Decennial Address. 57 respect the sagacity and benevolence of our Founder has been more than sufficient for our needs up to the present time, and our most efficient and courteous librarian has found many means and devices, new to the most advanced library science, of bringing out its utmost efficiency for our work, and of making it in all the pregnant sense of that word attractive to all who once come within the sphere of its influence. His work amply merits all the growing recognition that it and his rare personnel are so steadily gaining. His special report contains new suggestions and exi^eriences. The large and new demands upon the Public Library caused by the presence of an university for research which involved a material addition to its work, which is likely to increase in proportion to our growth, should be distinctly recognized. The special privileges needed by investigators have often been a strain upon the capacity of both its officers, its methods of administration and service, and the resources of its alcoves. The Public Library has on the whole well met the test, and I desire here to express, not only for myself personally, but for the other members of the University our gratitude to tlie city, the Trustees, and particularly to the accomplished head of the library itself, whose cooperation, with his able corps of assistants, has been a factor in an important pai-t of our work. Our two strongest departments are mathematics and psj-chology. These two, as has been often said, are the root and heart of all other branches. Mathematics is the grammar of all the sciences that deal with inanimate nature, and the study of the human mind and soul opens the field where all animate nature celebrates her highest triumph and which underlies all the humanities. While we could expend with profit much more than at present, perhaps the entire resources of the University, uj^on these departments, or perhaps, even upon each of them, they are best equipped and least in immediate need. We have books, journals, pro- fessors, means of speedy publication, and well-developed traditions, and can claim, we think with modesty, to be doing creditable work. Our greatest and most pressing need, according to the policy first formulated of strengthening the departments already established before founding new ones, is to enlarge the biology to an independent position, with due provision for botany and the related subject of paleontology. The foundations of a building for this group of studies is already laid on the grounds, and its completion, with an endowment of $150,000 or i200,000 with what we now have, would give us a strong department able 58 Decennial Address. to compete successfully with the best ; perhaps we may sometime dedi- cate such a building and department to the name of some honored public- spirited citizen of Worcester. Physics, like biology, now represented by a single able and promising man, needs enlargement to the same degree, with an annex department of astronomy and astrophysics, and for the same sum coidd, in addition to what we now have, be put upon a creditable footing. The chemical building, admirably lalanned after careful studies of all the best in Europe, and well equipped, especially for organic work, has no endowment, and needs for its full development the income of at least a quarter of a million of dollars. Anthropology, so greatly needed in this land, but so lacking in academic installation and tradition here, is already a precious germ with one worthy representative, has been cherished from the first with us, and it, too, needs enlargement and independence. If we pass over into the humanities, there are, of course, the two great groups of philology and literature, ancient and modern, and a historical group culminating in political economy, sociology, and a grand department of international law, nowhere adequately represented in this country, and for the establishment of which somewhere Senator Hoar, acting president of the Board of Trustees, the first citizen of Massachu- setts, competent to-day to fill any one of four professional chairs in any university, in learning, experience, character, and position more nearly the American Gladstone than any other, has been so distinguished an advocate. Education, now coming to be the largest philosophy of life and the natural field of applied psychology, needs a more adequate representa- tion, and with a quarter of a million of dollars for an ideal university school for childi-en, we would almost guarantee in five years to make this place an educational Mecca, by short circuit methods now well demonstrated but nowhere embodied, which would greatly increase the efficacy and reduce the expense and ease the labor of the lower grades of education in this country. Oiu- summer school has become one of the largest and highest grade institutions of its kind in the country, and appeals especially to heads of fitting schools, with whom it would be important for us to be en rapport if we had a college ; to normal schools, whose faculties are a growing field for the employment of our pedagogical graduates ; to Decennial Address. 59 young instructors in colleges, superintendents, parents, etc. If our two weeks could become a summer quarter counting toward a degree, and if the su m mer school could be adequately endowed and furnished, with the interest which one department of our work has already enlisted among the teachers of our country, the best of whom could spend their summer here in work, this, too, could be made an institution of which auy city or university might well be proud. We urgently need without delay the means for establishing a univer- sity printing office, where we can publish our journals at less expense and do our own printing ; and if this should grow to larger dimensions and develop a life of its own, that, too, might be welcomed. These needs are all on the university plane, where the beginnings already made are precious beyond words, wrought out as they have been with so much pain and labor, and the highest effort of so many choice spirits. May the day never dawn when this in our country most sorely needed and prayerfully cherished academic tradition shall fade or be broken. The investments of wealth and effort already made are too great, and achievements already attained and future promise too bright, to permit this ever to be an open question here. Satisfied, yes proud, as we are to-day to submit to Worcester, to sister institutions, and the country, the records of our work when compared with our means, we have lived, and even now live and walk, let us confess it, to a great extent in faith and hope, looking confidently to a future larger than our past has been, with steadfast and immovable conviction that our cause is the very highest of all the causes of humanity, but ready even ourselves, if need be, to labor on yet longer in the captivity of straitened resources, being fully persuaded that our redeemer liveth and that in due time he shall appear. i THE DEPARTMENT OF MATHEMATICS. By William Edward Story. PAST AND PRESENT STAFF. William Edward Story, Ph.D., Professor of Mathematics since 1889. OsKAR BoLZA, Ph.D., Associate in Mathematics, 1889-92. Henry Taber, Ph.D., Docent in Mathematics, 1889-92; Assistant Professor of Mathematics since 1892. Joseph de Perott, Docent in Mathematics since 1890. Henry S. White, Ph.D., Assistant in Mathematics, 1890-92. FELLOWS AND SCHOLARS. Henry Benner, Fellow in Mathematics, 1889-90. L. P. Cravens, Scholar in Mathematics, 1889-90. RoLLiN A. Harris, Ph.D., Fellow in Mathematics, 1889-90. J. F. McCuLLOCH, Fellow in Mathematics, 1889-90. William H. Metzler, Fellow in Mathematics, 1889-92. J. W. A. Young, Fellow in Mathematics, 1889-92. Levi L. Conant, Scholar in Mathematics, 1890-91. Alfred T. De Lury, Fellow in Mathematics, 1890-91. James N. Hart, Scholar in Mathematics, 1890-91. Thomas F. Holgate, Fellow in Mathematics, 1890-93. John I. Hutchinson, Scholar in Mathematics, 1890-91; Fellow in Mathe- matics, 1891-92. Frank H. Loud, Scholar in Mathematics, 1890-91. N. B. Heller, Scholar in Mathematics, 1891-92. Lorrain S. Hulburt, Fellow in Mathematics, 1891-92. John McGowan, Scholar in Mathematics, 1891-92. Ernest B. Skinner, Scholar in Mathematics, 1891-92. L. Wayland Dowling, Scholar in Mathematics, 1892-93; Fellow in Mathe- matics, 1893-95. John E. Hill, Fellow in Mathematics, 1892-95. Herbert G. Keppel, Scholar in Mathematics, 1892-93; Fellow in Mathe- matics, 1893-95. 61 62 Department of Thomas F. Nichols, Scholar in Mathematics, 1892-93 ; Fellow in Mathematics, 1893-95. F. E. Stinson, Scholar in Mathematics, 1892-93; Fellow in Mathematics, 1893-95. "W. J. Waggener, Scholar in Mathematics and Physics, 1892-93. Waeken G. Bullaed, Scholar in ]\Iathematics, 1893-96. Schuyler C. Davisson, Fellow in Mathematics, 1895-96. Frederick C. Ferry, Fellow in Mathematics, 1895-98. John S. French, Scholar in Mathematics, 1895-96; Fellow in Mathematics, 1896-98. E. W. Rettger, Fellow in Mathematics, 1895-98. tS. Edward Ryeeson, Fellow in Mathematics, 1895-96. Died March 25, 1896. Hugh A. Snepp, Scholar in Mathematics, 1895-96. James W. Boyce, Fellow in Mathematics, 1896-99. Heebeet 0. Clough, Scholar in Mathematics, 1896-97. A. Harry Wheeler, Scholar in Mathematics, 1896-99. Lindsay Duncan, Scholar in Mathematics, 1897-99. Frederick H. Hodge, Scholar in Mathematics, 1897-98; Fellow in Mathe- matics, 1898-99. Halcott C. Moreno, Scholar in Mathematics, 1897-98; Fellow in Mathe- matics, 1898-. Stephen E. Slocum, Scholar in IMathematics, 1897-98 ; Fellow in Mathematics, 1898-. John N. Van der Vries, Scholar in Mathematics, 1897-98 ; Fellow in Mathe- matics, 1898-. Frank B. Williams, Scholar in Mathematics, 1897-98; Fellow in Mathe- matics, 1898-. Elwin N. Lovewell, Scholar in Mathematics, 1898-99. Louis SiFF, Scholar in Mathematics, 1898-99. Orlando S. Stetson, Scholar in Mathematics, 1898-99. SPECIAL STUDENTS. George F. Metzler, Ph.D., Honorary Fellow in Psychology, 1891-92. Calvin H. Andrews, Mathematics and Pedagogy, 1894-95. Walter E. Andrews, Mathematics and Pedagogy, 1894-95. Whole number of students in mathematics in 10 years 44 Aggregate attendance (including 4 who remain in 1899-1900) ... 83 years. Average number of students per year 8 Average attendance per student 2 years. Mathematics. 63 Mathematics occupies a peculiar position relatively to the arts and sciences. It is, par excellence, an art, inasmuch as its chief function is to solve problems, — not such examples as are given in the text-books, and which serve only as exercises in the application of methods, but any problems that may arise in human experience and for whose correct solu- tion sufficient data are at hand. When any line of investigation, to whatever subject it may refer, has been carried so far that exact reason- ing may be applied to it, mathematics is the authority to which the results of observation are submitted for the final determination of their consistency and the conclusions that may be drawn from them, and fur- nishes the means of applying these conclusions to the prediction of phe- nomena not yet observed. No science and no branch of teclmology is exact, that is, capable of predicting with certainty what will happen under given conditions, unless it rests upon a mathematical foimdation. Astronomy, physics, and applied mechanics already have this foundation to a considerable extent, while the other sciences are still in the inductive stage, in which material is being collected with which, it is to be hoped, such foimdation -will ultimately be laid. Mathematics is also a science, inasmuch as it has accumulated a large body of systematic knowledge involving and leading to the methods that it employs in its solutions. These methods are of such a peculiar nature, differing so widely from other methods, that a special course of training is requisite if any one would learn to use them, and their number and variety have become so great that a lifetime would not suffice to acquire familiarity with them all. But new problems are continually arising and demanding new methods, and we need, therefore, a body of men who shall devote them- selves especially to the task of supplying this demand. While the col- leges are engaged in general liberal education, teaching a variety of subjects that develop the mental faculties (and no subject is more effi- cient than mathematics for this purpose) and make the student acquainted with his own tastes and powers, thus enabling him to determine the life- work for which he is best fitted, it is tlie special function of the university to extend the limits of human knowledge, and to train those wlio have unusual intellectual talents to employ them to the best advantage. We believe this object is best accomplished by an institution devoted solely to it, and whose teachers' energies are not diverted by the lower, though no less important, aims of the college. When the policy that should characterize this University was under 64 Department of discussion, the first point decided was that its work should be strictly post-graduate, and that it should not compete with other institutions in the work that is generally recognized as undergraduate. In accordance with this principle, the mathematical department fixed its standard of admission so as to require such a knowledge of mathematics as can be obtained in the average American college, and laid out upon this foun- dation a curriculum of its own, as extensive and as thorough as circum- stances allowed. In elaborating the details of this curriculum, we have kept in mind the fact that those who pursue post-graduate studies in pure mathematics almost always look forward to careers as professors in colleges or other higher institutions of learning ; and we have taken the view that, other things being equal, the ideal teacher is a master of his subject, not only conversant with the general principles of all its more important branches, the problems that have arisen in each, the methods that have been devised for the solution of these problems, and the results that have been obtained, but also unbiassed, ready and sound in judgment, and actively engaged in scientific research. We believe that the training that is best adapted to produce efficient specialists is also the training that is best adapted to produce efficient teachers of specialties. While desirous of supplying all possible facilities to those who wish to pursue studies in special branches, and to those who, already occupying permanent positions, have but a limited leave of absence, we have made it our chief object to provide a thorough training for those who, having just completed a college course, have not yet entered upon their life-work. This provision consists of such courses of lectures, seminaries, and indi- vidual assistance as should enable a faithful student endowed with the proper natural ability to satisfy the requirements for the degree of Doctor of Philosophy at the end of his third year with us. The requirements for this degree have been determined by our conception of the ideal teacher, as already stated. To acquire the necessary breadth of knowl- edge of mathematics as a whole, the candidate is expected to attend, during his first two years, specified courses of lectures on the general principles, methods, and results of all the more important branches of pure mathematics, to supplement these lectures by private reading, and to take an active part in the seminary. In the seminary, a special topic, more or less directly connected with the subject of some lecture, is as- signed, from time to time, to each student, who is required to read it up Mathematics. 65 and make an oral report upon it before the class. Advanced courses of lectures on special subjects that vary from year to year are also given, and each candidate for the degree is expected to attend a number of such courses. The student spends the greater part of his third year in the original investigation, under the constant personal guidance of one of the instructors, of a topic of his own selection. In preparing for this inves- tigation, he is required to make a practically complete bibliography of the subject, and to read all the more important available articles that have been written on it. The results of the investigation, embodied in a dis- sertation suitable for printing, must be submitted to the instructor under whose direction the work was done, and must receive his approval before the candidate will be admitted to the final examination for the degree. This approval will not be given imless the dissertation is satisfactory in form and completeness and the results are sufficiently novel and impor- tant to constitute a real contribution to science. The dissertation is, in fact, the main criterion bj- which the candidate is judged, and no amount of other work will compensate for its defects. The ability of our grad- uates to carry on research and the excellence of the work actually done is assured by the regulation that each dissertation accepted by us as worthy of the degree shall be printed with the explicit approval of a member of our Faculty. It is evident tliat, whereas any one that has the necessary preparation and taste for mathematics may profit by the advan- tages here afforded, only those who have a certain amount of mathematical genius can secure the degree. In making appointments to fellowships and scholarships we have endeavored to maintain the same high standard. We are on the lookout for mathematical geniuses ; but it is difficult to determine from the e^^- dence of others whether candidates come up to our standard or not ; so that we have adopted the general policy of giving the best appointments to those only that have been with us for at least one year, and about whom we are in position to judge for ourselves. Of course, this policy could not be carried out during the earlier years of the University, and its effect is apparent in the fact that, whereas seventy-five per cent of the students that entered the mathematical department during the first three years remained with us but one year, only twenty per cent of those that have been admitted during the last seven years left at the end of their first year. I do not mean to imply that those who left before completing our course were inferior in ability to those who remained three years, but 66 Department of we desire particularly to encourage men who can and wUl go forward to the degree. Nearly all of those who have studied mathematics with us have adopted teaching as a profession, two-thirds are now members of college faculties, and one-third are engaged in higher school work. Those who have received the doctor's degree have generally seciu-ed at once desir- able positions in which to begin their life-work, and most of them have already acquired for themselves, by distinguished ability, very decided influence in the institutions with which they are connected. Of those who have left without the degree fully one-half ought to have continued, and would have done so but for want of pecuniary means ; and we liave been obliged to turn away many men of very great promise on account of our inability to assist them in providing the means of subsistence during the unproductive period of student life. We could employ for fellow- ships, with decided advantage, ten times the amount now at our disposal. Although, as I believe, students wiU find here a broader post-graduate curriculum in mathematics and greater personal attention from the in- structors than at any other university in the country, we need greater facilities to make our course what it ought to be. Four-fifths of the in- struction in the department is now given by two men, and we are com- pelled to give in alternate years lectures on fundamental subjects that ouglit to be given every year. As I have said, we lay great stress upon the ability of our students to investigate ; but this faculty can be fully developed only under the personal guidance of one who is himself in the habit of investigating and who has the facilities and opportimities neces- sary for such work. A teacher's usefulness is greatly increased by the inspiration that comes from a personal identification with his subject, from the fact that he has ideas of his own about it, and that he has ex- tended it by his individual exertions ; and the investigator can have no greater incentive to search for new results than the opportunity to pre- sent his thoughts and discoveries to an intelligent and appreciative class in the lecture-room. But the necessity of teaching many subjects simul- taneously distracts the mind and is fatal to research. The ideal condi- tions for an instructor in an institution like this would be those under which he could teach one subject at a time, and that a subject that he was himself developing, and follow this subject with his class to such a point as to bring into evidence the scope and importance of his own work. To apply this method to the courses that are actually given here Mathematics. 67 would require the services of three additional instructors in mathematics. We are actually laboring under the disadvantage that some of the im- portant branches now taught by us are not of such paramount interest to any one of our instructors as to be the subject of his personal investi- gation. We are compelled to restrict ourselves to elementary courses in many branches that ought to be carried to a much higher point, and to omit altogether from our consideration applications of mathematics to statistics, to the arts, and to other sciences. Applications to physics re- ceive the attention of the physical department, to be sure, but the mathe- matical department ought to do much more than it is at present able to do in preparing students for higher work in physics. The niunber of instructors necessary for such advanced work as we do is not to be deter- mined b}' the number of our students, but by the number of subjects taught. Again, every expert investigator finds himself continually obliged to spend much time in details that could just as well be worked out by a younger man, to whom such work would be of immense advantage, not only as an exercise in the practical application of methods, but also as furnishing the opportunity for a prolonged study of the workings of an investigator's mind ; and example is worth more than precept in the development of the faculty of investigation. We ought to have the means of retaining our best graduates for a year or two as personal assist- ants to the instructors, during which period they might also be gaining experience in the class-room by teaching a few hours a week under the supervision of one of the regular instructors. Such work is not drudgery, and would be, I think, sufficiently attractive to an ambitious young man to induce him to remain with us on a moderate stipend while he is wait- ing for such appointment as may seem to liim desirable. It is almost universally assumed that a mathematician needs no mate- rial equipment other than brains, with, possibly, a few books. However true this assumption may have been some decades ago, — and I fancy that its truth then rested solely upon the difficulty of procuring sucli eqni])- ment, — it is not true now, as must be a2)pai'ent to any one who studied carefully tlie (iennan educational exhibit at the World's Fair in Chicago. Ten years ago our department started out with a fair nucleus for a mathematical library and a moderate collection of models, to which we have not been able to make many additions. We have very few of tlie older mathematical works that illustrate the history of the subject, and 68 Department of we need particularly complete sets of many important mathematical jour- nals and the transactions of learned societies. In these journals and transactions have appeared most of the original investigations to which, as investigators ourselves, we have continual occasion to refer, both for suggestions and to avoid apparent plagiarism and the unnecessary dupli- cation of research. We should also be greatly assisted in our class- work by a more complete collection of models. In short, what I have in mind as a model mathematical department for post-graduate work would have, say, four professors and assistant professors, each having his personal assistant, and at least two instructors of lower grade for the more elementary work, and would be provided with a complete mathematical library and with all the apparatus that it is now possible to procure, with suitable provision for the pm-chase of new books and apparatus as they appear in the market. These schemes are not incapable of realization, although, perhaps, opposed to the traditions of education in this country. This University has never had any traditions excepting such as were based upon high ideals. Its mathematical department was not modelled after that of any other institution, but was determined by the conception of what would constitute perfection in such a department. We have always lived up to our ideals, in so far as we have done anything, without regard to consid- erations of material interest. We are not here to do what is done else- where, and we do not acknowledge that it would be best for us to do what other institutions, in their experience, have thought wisest. We propose to adopt no temporary policy that we shall sometime want to abandon, confident that the ideal university of the future will be ideal from the very root and not a graft upon inferior stock. When the doors of the Universty were first opened to students, in the fall of 1889, the mathematical staff consisted of William E. Story, Pro- fessor, Oskar Bolza, Associate, and Henry Taber, Docent ; a year later it was increased by the appointment of Joseph de Perott, Docent, and Henry S. White, Assistant ; and in 1892 Drs. Bolza and White resigned their positions to accept Associate Professorships in the University of Chicago and Northwestern University, respectively, and Dr. Taber was promoted to an Assistant Professorship, thus leaving the department with practically the same teaching force as it had during the first year. The instruction has been given by lectures, seminaries, and individual conferences. The number of lectures (of fifty minutes each) was sixteen Mathematics. 69 a week the first year, nineteen and twenty a week in the second and third years, respectively, and about fourteen a week, on the average, each year since. In some years courses of lectures on certain mathematical subjects ha\ang important physical applications have been given by Assistant Professor Webster of the Department of Physics. The subjects of the lecture courses given during the ten years include the following : — 1. The History of Arithmetic and Algebra among various peoples from the earliest times to 1650 a.d. 2. Theory of Numbers (introductory). 3. Theory of I^ umbers (advanced). 4. Numerical Computations. 5. Theory of Quadratic Forms. 6. Finite Differences. 7. Probabilities. 8. Theory of Errors and the Method of Least Squares. 9. Theory of Functions of a Real Variable. 10. Linear Transformations and Algebraic Invariants (introductory). 11. Theory of Substitutions, with applications to algebraic equations (intro- ductory). 12. Theory of Transformation Groups. 13. The Application of Transformation Groups to Differential Equations. 14. Finite Continuous Groups. 15. Klein's Icosahedron Theory. 16. Simultaneous Equations, including Restricted Systems. 17. Theory of Functions of a Complex Variable, according to Cauchy, Rie- mann, and Weierstrass (introductory). 18. Definite Integrals and Fourier's Series (introductory). 19. Ordinary Differential Equations (introductory). 20. Ordinary Differential Equations (advanced). 21. Partial Differential Equations (introductory). 22. Elliptic Fimctions, according to Legendre and Jacobi (introductory). 23. Weierstrass's Theorj' of Elliptic Functions. 24. Elliptic Modular Functions. 25. Abelian Functions and Integrals. 26. Theta-Functions of Three and Four Variables. 27. Riemann's Theory of Hyperelliptic Integrals. 28. Riemann's Surfaces and Abelian Integrals. 29. Conic Sections by modern analytic methods (introductory). 30. Quadric Surfaces by modern analytic methods (introductory). 31. General Theory of Higher Plane Curves (introductory). 32. Plane Curves of the Third and Fourth Orders. 70 Department of 33. General Theory of Surfaces and T\visted Curves (introductory). 34. Surfaces of the Third and Fourth Orders. 35. Twisted Curves and Developable Surfaces (advanced). 36. Applications of the Infinitesimal Calculus to the Theory of Surfaces. 37. Rational and Uniform Transformations of Curves and Surfaces. 38. Enumerative Geometry. 39. Analysis Situs. 40. Hyperspace and Non-Euclidean Geometry. 41. Modern Synthetic Geometry (introductory). 42. Quaternions, with applications to geometry and mechanics. 43. Multiple Algebra, including matrices, quaternions, " Ausdehnungslehre," and extensive algebra in general. 44. Symbolic Logic. Courses designated as " introcluctory " are given at least as often as every other year, and attendance on them is required of all candidates for the degree of Doctor of Philosophy that take mathematics as their principal subject. The other courses, intended primarily for the more advanced students, have been given less frequently and with particular reference to the suggestion of topics for original investigation. In connection with his lectures. Assistant Professor Taber has con- ducted a weekly seminary for students in their first or second year, for the purpose of cultivating in them an active attitude toward the subjects treated, instead of the passive attitude usually resulting from hearing lectures. Topics related to those of the lectures have been discussed by the students, and their work has been criticised both with reference to rigor of demonstration and manner of presentation. In this way some of the advantages of the laboratory and the practice school are brought into the field of mathematics. Professor Story, with the assistance of the other instructors, has directed the more advanced students individ- ually in the systematic investigation of special topics that promised to afford opportunity for the discovery of new results and methods, — a task that has sometimes required the professor to hold weekly three-hour conferences with each of four students during nearly the entire academic year ; but we believe the results have justified this unusual expenditure of energy. The average annual number of students taking mathematics as their chief study has been about eight, the average duration of their residence was about two years, and more than one-third of them have received (or will undoubtedly receive) the Doctor's degree, which is a decided improve- Mathematics. 71 ment in every respect over the record of the first three years. The pub- lished investigations of these students are enumerated in the Bibliography at the end of this volume. The researches of an instructor in an institution of this kind are not to be judged solely by the number and magnitude of his printed papers, as many of them are naturally turned over, in a more or less incom- plete form, to his pupils for further investigation and more adequate presentation ; at least it seems most natural and desirable that an in- structor should suggest to his pupils subjects for investigation on which he has himself worked, and for whose treatment he has found adequate methods. My chief subjects of investigation have been : — 1. Hyperspace and Non-Euclidean Geometry. 2. Algebraic Invariants. 3. Curves on Ruled Surfaces, and Restricted Equations. 4. The History of Mathematics prior to the invention of the infinitesimal calculus, and 5. A Mathematical Curriculum for Primary and Secondary Schools. I have developed systematically the general theory of space of any number of dimensions from assumptions that are precisely analogous to those on which the scientific treatment of threefold space is usually based, and which we recognize as the results of experience. In accordance with this general theory, I have thoroughly investigated the properties of loci of the first and second orders and some special loci of higher orders. The introduction of the most general kind of measurement has then led me to an equally thorough study of parallel and perpendicular loci, the curvature of loci, areas, and volumes in the most extended sense. Tlie first part of these results has already apfjeared in the Mathematical Review, and I hope to publish the remainder within a short time. Ever since the appearance of Clebscli's " Theorie der binaeren alge- braischen Formen," toward the end of the year 1871, when I was study- ing in Berlin, I have taken a lively interest in the theory of algebraic invariants, — an interest that was greatly augmented by my association with Sylvester at the Johns Hopkins University in 1876. I had tliought all along that there ought to be a direct process by which all such inva- 72 Department of riants could be obtained, but my efforts to find it had failed. A course of lectures on invariants that I have given every year or two since the opening of Clark University caused me to renew my attemjjts, and the classic paper of Hilbert in the 36th volume of the Mathematische Annalen, in which a process devised by Mertens (and which 1 regarded as indirect, inasmuch as it involved quantities extraneous to the matter in question) suggested a new line of research, which happily led at length to the long-sought direct process. I then applied this process, as Hilbert had applied Mertens's process, to the proof of Gordon's theorem that all the invariants of any finite system of quantics of finite orders can be expressed rationally in terms of a finite number of such invariants. These results were published in the Mathematische Annalen and in the Proceedings of the London Mathematical Society. I have spent much time in trying to find, by means of the process, an extension of Cayley's for- mula for the number of linearly independent ground-forms of a single binary quantic (extended by Sylvester to any system of binary quantics) to the case of quantics involving three or more variables, but so far with- out success. In my lectures on surfaces of higher orders and twisted curves I have paid particular attention to the algebraic curves that lie upon a given algebraic surface. If the given surface is ruled, the curves on it can be classified in such a way that certain problems relating to a curve can be solved when the class of the curve is known. My investigations in this direction have been communicated to my students, some of whom have already solved such problems. In connection with my investigations on twisted curves, I have also made a systematic study of restricted equa- tions, and liave carried the determination of the orders of such systems much farther than had been done before. I have lectru-ed at various times on the early history of mathematics, with special reference to the development of arithmetical and algebraic symbolism, and have collected a large number of systems of such symbols, which I hope sometime to utilize for a monograph on the subject. In connection with a course of lectures delivered for two years at the Summer School, I arranged a mathematical curriculum for primary and secondary schools, which will be published when I can find the leisure necessary to prepare the explanatory text. At my request, Assistant Professor Taber has furnished the material for an account of his pei-sonal researches, which involves such a complete Mathematics. 73 and excellent history of the theory of matrices that it seems to me inad- visable to abbreviate it ; I therefore append it to this report at length, for the benefit of those readers who may be interested in the subject. Dr. Taber's researches have been devoted to the development of the theory of matrices, and its application to bilinear forms, multiple algebra, and theory of finite continuous groups. The calculus or theory of matrices was invented by Professor Cayley (see his " Memoir on the Theory of Matrices," Phil. Trans., 1858), and has proved an instrument of great power in the theorj- of linear transformation, bilinear forms, and for the investigation, generally, of the projective group. i In order to explain the work done by Dr. Taber in this direction, a few words of explanation mil be necessary to describe the work done by Cayley and others. Associated with any linear substitution c/ =\^ /lyX, (i = 1, 2, ••• 11) is the bilinear form -4 = \ \ o-ii^iyj-, which may be regarded as repre- senting this substitution, or vice versa ; and, in the theory of matrices, we do not need to distinguish between this linear substitution and the asso- ciated bilinear form, or between either and the matrix ( . . " ) \i,.? = 1, 2, ...w/ common to both. If now B denotes the bilinear form y j ^ij^iifii or its associated linear substitution, A ± B will denote the bilinear form n n / / ('^'y + ^vO^.y^i 01" it^ associated linear transformation ; and AB will denote the bilinear form 2^ / ( / '^it^iv l^tVii or its associated linear substitution (obtained by the composition of the linear substitutions A and B). Equivalence between two bilinear forms or linear substitu- 1 By means of this calculus very important results have been obtained by Cayley himself, by Sylvester, Frobenius, Foss, Weyer, Study, and others ; and, by methods essentially simi- lar, Kronecker obtained important theorems on the orthogonal group to which reference is made below. 74 Department of tions, A and B, is denoted by writing A = B. Further, in what follows, J will denote the identical transformation, represented by ^a-,t/„ and A-^ the form, or substitution, satisfying the symbolic equation AA-^ n n = A-^A = /; A will denote the bilinear form > 7 dji^il/j, transverse or conjugate to A = ^ N ciy^j^j^ and | A \ will denote the determinant of the matrix A. A is said to be symmetric if A= A, and alternate, or skew symmetric, if JL = — -4.^ Cayley was, perhaps, led to the invention of this calculus by his researches upon orthogonal substitution, Crelle (1846), Vol. 32. For in Crelle, Vol. 50, three years before the publication of his memoir on matrices, he expressed the results of these researches in the notation of matrices. Thus Cayley showed that the general expression for the proper orthogonal substitution in n variables is (^I— B)(^I+ B^~^, where B denotes an arbitrary alternate, or skew symmetric, linear substitution ; and this expression gives Cayley's determination of the coefficients of a proper orthogonal substitution in n variables as rational functions of the essential parameters, ^ n(n — 1) in niunber. Again, in liis " Memoir on the Automorphic Linear Transformation of a Bipartite Quadrate Function " (PM. Trans., 1858), Cayley showed that the general automorphic linear transformation (linear transformation into ■n n itself) of a symmetric (alternate) bilinear form A = y y o-if^iyj with 1 1 cogredient variables and of non-zero determinant, may be represented by (J. + X')~^(^A — X), where X is an arbitrary alternate (symmetric) bi- linear form. This expression gives in the first case (when A is symmet- ric) Hermite's determination of the general proper automorphic linear transformation of a symmetric bilinear form, and, in the second case (when A is alternate), Cayley's determination of the transformation into itself of an alternate bilinear form. Further, in this same memoir Cay- ley showed how to reduce, to the solution of a system of r? linear equa- tions, the rational determination of the w^ coefficients of the automorphic linear transformation of a general bilinear form A (neither symmetric nor alternate) with cogredient variables and of non-zero determinant. Namely, he showed that the general formula for such a substitution is 1 In the first case aji = ay, in the second aji = — atj (i, j = 1, 2, ■■• n). Mathematics. 75 (4 + X)-^ {A- X), where X satisfies the condition (1)-^ X + A'^X = 0. This result includes the determination of the general automorphic trans- formation of A, when A is symmetric and when A is alternate. It also includes Ca3dey's determination of the coefficients of an orthogonal substi- tution to which it reduces when A = T. In what follows Gr will denote the group of proper automorphic linear transformations of A (the x's and y's being cogredient), and Gr' the proper orthogonal group. A transformation T oi G (or of Gr') is termed singular if —1 is a root of its characteristic equation (naraelj', | T— pl\ =0); otherwise, non-singular. Every non-singular transformation of group G (or (?') is given by Cayley's formula, and may be termed a Cayleyan transformation of the group. ^ No singular transformation of group G is given by Cayley's expression or determination. But for A alternate, also when A is neither symmetric nor alternate provided | ^ ± ^ | =^0, Dr. Taber showed in 1894 (JProc. Am. Acad. Arts and Sciences, Vol. 29) that group G is generated by the Caylej-an transformations of the group, — each transformation T of this group being obtained by the composition of a finite number of Cayleyan transformations. In the same paper Dr. Taber also showed that the sub-group of orthogonal transformations of G is, similarly, generated by the non-singular orthogonal transformations of this sub-group, when A is alternate, and when A =^ ± A provided I i± ^ I ^ 0. This theorem is similar to a theorem relating to the orthogonal group (group (?') established by Kronecker in 1890 (" Ueber orthogonale Sys- teme," Sitzunysberich. d. Preuss. Akad.), who showed that this group is generated by the Cayleyan transformations of the group, each trans- formation T of this group being obtained by the composition of two Cayleyan transformations, — the coefficients of each of the Cayleyan transformations being rational functions of the coefficients of T. In 1895 QMath.Ann., Vol. 46) Dr. Taber showed that, if A is real and alternate, every real transformation T oi Q can be obtained by the com- position of two real Cayleyan transformations of this group. This theorem was obtained independently and extended widely by Dr. Loewy, who in 1896 (^Math. Ann., Vol. 48) showed that, if A is irreducible ' For the case in which A is symmetric, the determination of the coefficients of T, given by Cayley's formula, is properly Hermite's •, but it is not convenient to distinguish here between this case and the other two cases, namely, when A is alternate, or is neither symmetric nor alternate, when the determination is Cayley's. 76 Depcirtment of (which case includes that in which A is alternate), every transformation of (r, real or imaginary, can be obtained by the composition of two Cayleyan transformations of the group, and that, therefore, when A is irreducible, there is no transformation of the kind termed by Foss essentially singular,^ that is to say, which cannot be obtained by the com- position of two non-singular, or Cayleyan, transformations. For a reducible form A not every singular transformation of (? can be obtained by the composition of two Cayleyan transformations of this group. Nevertheless, Dr. Taber showed in 1897 (^Math. Review, Vol. 1) that in every case the Caylej^an transformations of Gr form a group by themselves ; that the composition of any number of Cayleyan transforma- tions of Gr results in a ti'ansformation that can be obtained by the composition of two Cayleyan transformations of this group; and that thus the composition of Cayleyan transformations never gives rise to an essentially singular transformation. It is to be noted that from Cayley's formula for a transformation T of (?, namely, T= {A + X)-' (^ - X) = (/- ^-'X)(7+ A-'Xy\ we derive X=A(1 - r)(l + Ty^; and, therefore, the parameters, namely, the coefficients of X, which enter into the determination of T, can be expressed rationally in the coefficients of T and of A.^ Similarly, in the memoir by Kronecker mentioned above, he has shown that the coefficients of the two Cayleyan transformations, whose composition gives the general transformation T of group G', can be expressed rationally in the coefficients of T. For A real, alternate, and orthogonal, Dr. Taber gave, in the paper in the Mathematische Annalen mentioned above, the determination of the coefficients of the two Cayleyan transformations Cj and Cj, whose composition gives any real transforma- tion T of (?, as rational functions of the coefficients of T and of A. This determination of Cj and C^ he has since extended to the case in which T is imaginary, and A any alternate bilinear form.^ Dr. Taber has pointed out that the transformations of Cr, both when A is irreducible and when A is reducible, are in general of two essentially 1 Abhand. d. k. Bayer. Akad. d. Wiss., II. CI., XVII. Bd., II. Abth. 1890, p. 77. 2 Between these parameters when A is neither symmetric nor alternate n^ equations persist. ' See papers to appear in Proc. Am. Acad, of Arts and Sciences, Vol. 35. Mathematics. 77 different kinds. The difference between the two kinds of transformations of (x is given by the following theorem : — (I.) If we designate a transformation of group G as of the first or sec- ond kind according as it is or is not the second power of a transformation of the group, then every transformation of the first kind is the mth power of a transformation of the group, for any positive integer m, and can he generated by the repetition of an infinitesimal transformation of the group. A transformation of the second kind, by definition not an even power of any transformation of the group, is always the (2m + 1)"" power of a transforma- tion of the group for any odd exponent 2m + 1. But no transformation of the second kind can be generated by an infinitesimal transformation of the group. (II.) Every Cayleyan transformation of group G is a transformation of the first kind ; whereas, a non-Cayleyan transformation is, in general, of the second klnd.^ Dr. Taber has also given the conditions necessary and sufficient that a transformation T of group Gr may be of the first kind for the case in which A is symmetric (which includes the case when A = I, in which case G- becomes G"), and for the ca.se when A is alternate.^ Dr. Taber has sho^vn that, if A is neither symmetric nor alternate and iThis was proved for the orthogonal group in 1894, Bull. Am. Math. Sac, Vol. Z. At the conclusion of this paper it was stated tliat a precisely similar theorem held for what is here designated as group G. In the Math. Ann., 1895, Vol. 46, the theorem was proved for group G when A is alternate ; for the case in which A is symmetric, in the Proc. Land. Math. Soc, 1895, Vol. 26; and for the general case, in the Math. Beview, 1897, Vol. 1. ^ For the orthogonal group, to which G reduces when A = I, the conditions necessary and siifficient that a transformation shall be of the first kind were given by Dr. Taber in a com- munication to the American Academy of Arts and Sciences, March, 1895. (See Proceedings, Vol. 30, p. 651.) The necessity and sufBciency of these conditions was afterwards shown in Proc. Lond. Math. Soc, 1895, Vol. 26, and the theory for the orthogonal group extended to group G for A symmetric. It was not explicitly stated in this paper that the conditions given for the orthogonal group hold for G when A is symmetric, being so obvious a conse- quence of the considerations adduced. This does not seem to have been recognized by Dr. Loewy, who refers to this paper but gives the necessary and sufficient conditions. Math. Ann., Vol. 48, when A is symmetric as an extension of Dr. Taber's theorem for group G'. For A alternate the necessary and sufficient conditions were given by Dr. Taber in a communication to the American Academy of Arts and Sciences, January, 1896. (See Pro- ceedings, Vol. .31, p. 349.) The necessity of these conditions has previously been sliown by Dr. Taber in the Math. Ann., Vol. 46. In Vol. 49 (1897), Dr. Loewy gave the conditions as sufficient, undoubtedly without knowledge of Dr. Taber's priority in the statement of this theorem. 78 Department of \A± A\^0, group & contains no transformation of the second kind. TMa theorem leads, for the case mentioned, to the following rational represen- tation of any transformation of this group, namely, iiA + xy\A-x^-]\ where (^A)-^ X + A~^ X = 0. Moreover, Dr. Taber has shown that the sub-group of orthogonal transformations of 6r contains no transformation of the second kind when A is alternate.^ The determination of the congruent transformations between two bi- linear forms is the natural generalization of the problem to determine the automorpliic linear transformations of ^. A determination of the trans- formations between A and B depending on the solution of a single equa- tion of degree n has been given by Dr. Taber (^Mathematical Review, Vol. 1, 1897), which holds for any case whatever in which A and B are both symmetric or both alternate. The theory of matrices, or bilinear forms, is closely related to the theory of Hamilton's linear vector functions. In the American Journal of Mathematics, Vol. 12, Dr. Taber has given a development of the theory of matrices, proving many of the fundamental theorems, from the point of view of Hamilton's theory. One of Sylvester's most important contributions to the theory of matrices was a general formula, given in the Comptes Rendus, Vol. 94, 1882, expressing any power, integral or fractional, of the bilinear form or matrix .4 as a polynomial in A of degree w — 1. Thus, ii B = A>^, where jx is any fraction, and if p^, p^, . . . p„ are the roots of the characteristic equation of A, we have B= 20" (^ - PJ)(^ - Ps^ - (^ - P--0 * iPl-P2)iPl-p3)-(.Pl-pn) By means of this theorem the determination of a matrix or linear substi- tution whose jLith power is equivalent to A is reduced to the solution of a single algebraic equation of degree n. This formula was afterwards ex- tended by Sylvester to any function of the matrix A."^ Thus we have 1 See Bull. Am. Math. Soc, Series 2, Vol. 2, pp. 5 and 161. ^ Johns Hopkins Univ. Circulars, No. 28, Vol. 3, p. 34. Mathematics. 79 Neither of these formulae applies unless the roots of the characteristic equation of A are all distinct. For the general case, in which the roots of the characteristic equation have any given multiplicities, a formula for /(A) has been given by Dr. Taber.^ Thus, if the distinct roots of the characteristic equation are pj, p^, ••• p„ respectively of multiplicity m^, m^, — vir, and if ^<" = ^j'" ••• ^,_i"*^,+/" ••• ^/", where 4,'"' denotes [(A - p,r)"'i - (pj - p,r'r\'"j : [( - irKpj - p.)""'"^], then /(.)=£[/(,)..(. -..)^ . ... . ^-A0rp fvgf)]^,. For ?Wj = »Z2 = •■• nir = 1, this reduces to Sylvester's formula. The theory of matrices stands in a very special and important relation to the theory of higher complex quantity (multiple algebra). Namely, a class of systems of complex numbers with n"^ units arises from the theory of lineal transformation, — that is to say, a matrix of n^ elements gives rise to a system of w^ units e^ with the special multiplication table e„ ej^ = e^, e„ ejti = for/ ^ k. Multiple algebras (systems of complex numbers) of this class have been termed by Mr. Charles S. Peirce quadrate alge- bras, or quadrates ; and Peirce has shown that the p units of any system of complex numbers (the p units of any multiple algebra) can be expressed linearly in terms of the n^ units of a quadrate.^ Whence it follows that the tlieory of any system of complex numbers is identical with the theory of the combination by multiplication, addition, and subtraction, of a certain system of p matrices. The first quadrate algebra, namely, that with four units, is identical with the quaternions with the imaginary (Hamilton's bi-quaternions), as was first explicitly pointed out by Professor Benjamin Peirce. That is to say, by substituting for the original units e^ a certain system of four linearly independent linear functions of the four units we obtain a system of com- plex numbers, 1, i, j, k, which can be substituted for the original units, and whose midtiplication table is ^=j^ =k'^= — 1, li =tl, etc., ij = —ji = k, etc. Let now i', j', k' be a new system of quaternion unit vectors having the multiplication table i'^ = p = k'^ = - 1, i'f = - fi' = k', etc. And let a third system of units be formed by the combination of these two sys- I Math. Ann., Vol. 46, p. 668. See also Proc. Am. Acad, of Arts and Sciences, 1893, VoL 27, p. 46 et seq. " See Am. Jour. Math., Vol. 4, pp. 122 and 125. 80 Department of terns, it being assumed that each of the one system of quaternion unit vectors is commutative with each unit vector of the other system. That is to say, that iV = i'i, if =j'i, etc. We get thus sixteen units, 1, i, j, k, i', J', k', and the nine binary products ii', ij\ etc. Dr. Taber has shown that the system of units thus obtained is identical with the quadrate of six- teen units. The same is true if we had combined the four original units of the quadrate with four units, namely, e„ (r, s = 1, 2) with a similar system of another quadrate, viz., e'„ (r, 8 = 1, 2), — assuming that e„^',„ = e',„e„. The resultant system has sixteen units, and is the quadrate with sixteen units. ^ Dr. Taber has established a general theorem including the one just given. Namely, he has shown that, if rt = mp, the quadrate of n? units is a compound of two quadrates severally witli m units and p units, the units of one quadrate system being commutative with each unit of the other quadrate.'^ Whence it follows that if the prime factors of n are Sj, Sj, •••8„and n= Sj^i^j'^^-.-S^ v, the quadrate of n^ ujiits is a compound of /ij quadrates each with h-^ units, /lij quadrates each with ^2^ units, etc. The general projective group holds a jjosition of special importance in Lie's theory of finite continuous groups. For the adjoined group V of any finite continuous group G, by means of which the sub-groups of G are determined, will, if the equations of transformation of this group are properly chosen, appear as a sub-group of the general projective group. Thus the theory of matrices is of importance in the investigation of certain problems of Lie's theory, since this calculus furnishes a convenient instrument for the treatment of the general projective group. The chief theorem of Lie's theory states that if a system of infinitesi- mal transformations satisfies certain conditions, they generate a group with continuous parameters, each of whose finite transformations can be generated by an infinitesimal transformation of the group. ^ In 1892 Professor Study made the extremely important discovery that this theorem is subject to certain limitations, — showing that an exception to this theorem existed in the case of the special linear homogeneous group in ' Am. Jour. Math., Vol. 12, p. 391. 2 Ibid. This theorem was obtained independently, but subsequently, by Professor Study. See " Math. Papers of Internat. Math. Congress of 1893," p. 378. ' Transformationsf/ruppen, Vol. 1, pp. 75, 158 ; Continuierliche Gruppen, p. 390. Lie originally defined a finite continuous group, substantially ( Trans. Grp., p. 3), as a group with continuous parameters. Ultimately, he assumed that in a continuous group as thus defined each transformation can be generated by an infinitesimal transformation of the group ( Con- tin. Grp., p. 379). Mathematics. 81 two variables, namely, that not every transformation of this group can be generated by an infinitesimal transformation of the group. ^ Subsequently, in 1893 (J.m. Jour. Math., Vol. 16), Dr. Taber showed that the orthogonal group in n variables (for n S 4) also presents an exception to Lie's theorem; and in 1895 gave, in a communication to the American Academy of Arts and Sciences, the conditions necessary and sufficient that a proper orthogonal substitution may be generated by an infinitesimal orthogonal substitution.^ For ?i > 2 also, the special linear homogeneous group in n variables is continuous onh' in the neighborhood of the identical transformations. For two variables. Study gave the conditions necessary and sufficient that a transformation of this group may be generated by an infinitesimal trans- formation of this group. Dr. Taber gave, in 1896 (^Bull. Am. Math. Soc, Series 2, Vol. 2, p. 231), these conditions for n variables ; also the conditions necessary and sufficient that a transformation of the special linear homo- geneous group may be the mth power of a transformation of this group. From these conditions it appears that the wth power of any transforma- tion of this group can be generated by an infinitesimal transformation of this group; and that the transformations of this group can be divided into as many genera as there are prime factors of n. Thus, if S is a prime factor of w, there are transformations of this group whose n/Bth power, but no lower power, can be generated thus.* Dr. Taber has shown that the following groups are not continuous, except in the neighborhood of the identical transformations, namely, the group G, mentioned above, for A symmetric or alternate, and in general when A is neither sjTnmetric nor alternate, provided either | ^ -f- ^ | or \A — A\iii equal to zero.* For all these groups the infinitesimal trans- formations satisfy Lie's criterion. Dr. Taber has also shown that the following groups are continuous, namely, group Gr when | J. ± ^ | ^ 0, the sub-group of orthogonal trans- formations of (?, for A alternate, and the group of automorphic linear n n transformations of a bilinear form ^ = / / (^a^iyji of non-zero deter- minant, the x's and y's being contra-gredient.^ 1 Leipzige Berichte, 1892. ' See Proc, Vol. 30, p. 551. This result is referred to above on p. "7. » See Bull. Am. Math. Soc, Series 2, Vol. .3, p. *d Santoed: On Reaction-times when the Stimulus is Applied to the Reacting Hand. Ibid., Vol. 5, pp. 351-355 (1893). The experiments bring into question the statement of Exner that reactions are slower when the stimulus is applied to the reacting hand. Hancock: A Preliminary Study of Motor Ability. Pedagogical Seminary. Vol. 3, pp. 9-29 (1894). The Relation of Strength to Flexibility in the Hands of Men and Children. Ihid., Vol. 3, pp. 308-313 (1895). The first is a study of the spontaneous movements of school children from five to seven years old, — of the swayings and tremors displayed in efforts to stand still with eyes open or closed, or to hold the hand or forefinger still, — movements analogous to those of nervous disease. The second paper shows for the persons tested (20 men, 22 boys, and 11 girls), greater flexibility in the hands of the men as measured by the extent to which the joints could be flexed voluntarily. Both papers are of avowedly pedagogical interest. Lancaster : Warming Up. Colorado College Studies, Vol. 7, pp. 16-29 (1898). Based upon ergographic experiments. Sensation and Perception. Scripture : Einige Beobachtungen iiber Schwebungen und Differenz- tone. Philos. SUidien, Vol. 7, pp. 630-632 (1892). A brief experimental study of beats and difference tones produced by forks sounding separately on either side of the head. 128 Department of Dkesslar: On the Pressure Sense of the Drum of the Ear and "Fa- cial Vision." Am. Jour, of Psy., Vol. 5, pp. 3-44-350 (1893). The study shows that the faculty of the blind of recognizing the presence or absence of neighboring objects, which has been credited to some sort of obscure visual sensation in the skin of the face, or to sen- sations of pressure mediated by the drum of the ear, is probably a matter of hearing. Kkohn : An Experimental Study of Simultaneous Stimulation of the Sense of Touch. Journal of Nervous and Mental Disease, N. S., Vol. 18, pp. 169-184 (1893). Based chiefly ou experiments made in the Clark laboratory. Leuba: a New Instrument for Weber's Law, with Indications of a Law of Sense Memory. Am. Jour, of Psy., Vol. 5, pp. 370-384 (1893). Weber's law demonstrated in the classification of artificial stars. The law of sense memory suggested is that memories of intensities of sensation tend to shift toward the middle of the usual scale of intensities. Dresslar : A New Illusion for Touch and an Explanation for the Illusion of Certain Cross Lines in Vision. Ibid., Vol. 6, pp. 275-276 (1894). This illusion is similar to that of the Poggendorff illusion in vision, and the obvious explanation in the case of the touch illusion is extended to the visual one. Sanford: a New Visual Illusion. Science, Feb. 17, 1893. A visual illusion involving false judgments. Dresslar : Studies in the Psychology of Touch. Am. Jour, of Psy., Vol. 6, pp. 313-368 (1894). (Dissertation.) The study is in three sections : 1. On the Education of the Skin with the ^sthesiometer, particularly of its bilateral effects ; 2. Experiments on Filled and Open Space for Touch, showing that filled space seems larger when the finger moves over it, or when the extents compared are moved under the resting finger ; 3. On Apparent Weight as affected by Apparent Size and Shape — tests upon school children and adults. Psyclwlogy. 129 Circulation and Respiration. Dawson : Effects of Mental States upon Circulation. (Records in the instructor's hands but not worked up as yet.) Preliminary note in the Proc. of the Am. Psychological Ass'n, Psychological Review, Vol. 4, pp. 119-121 (1897). An extended study made with the plethysmograph applied simultane- ously to the hand and eye. Whipple : The Influence of Forced Respiration on Psychical and Physical Activity. Am. Jour, of Psy., Vol. 9, pp. 560-571 (1898). The effect of very rapid breathing on eight simple tasks involving sensory or motor activities, or both. Effects slight in most cases ; phj-sical strength and endurance seem to be increased, while discrimi- native powers seem to be depressed. Comparative Psychology. Klixe : jNIethods in Animal Psychology. Ibid., Vol. 10, pp. 256-279 (1899). Discussion of methods, and presentation of the results of experiments upon vorticellae, wasps, chicks, and white rats. Small, W. S.: Notes on the Psychic Development of the Young White Rat. Ihid., Vol. 11, pp. 80-100 (1899). The study consists of a careful record of the bodily and mental development of the white rat from birth onward for a number of weeks. Studies on Miscellaneous Topics. Calkln's, Mary Whiton : Statistics of Dreams. Ihid., Vol. 5, pp. 311-343 (1893). A careful analytical and statistical study of dreams, recorded immedi- ately after waking by two subjects during a period of six or eight weeks. An effort to get as full a picture as possible of normal dream- life. LrCKEY : Some Recent Studies of Pain. Ihid., Vol. 7, pp. 108-123 (1895). A review of recent literature on the physiology and psychology of pain. 130 Department of Miles, Caeolinb: A Study of Individual Psychology. Ihid.,\o\. 6, pp. 534-558 (1895). A questionnaire study of a number of special points, made on one hundred students in Wellesley College. Such topics are considered as : How do you know your right hand from your left ? How do you con- centrate attention ? Fears as children ? Things causing anger ? Favor- ite color ? Earliest memories ? Early ideals ? etc. (This study and the preceding, though not experimental, were made in connection with the work of the laboratory.) Drew: Attention: Experimental and Critical. Ibid., Vol. 7, pp. 533-576 (1896). (Dissertation.) The experimental portion of this paper consists of three sections : 1. Reaction and Association Times with Differing Degrees of Distraction ; 2. A Qualitative Study of Associations with Full and with Distracted Attention; 3. A Study of the Apparent Order of nearly Simultaneous Stimuli with variously Directed Attention. Hylan : The Fluctuation of Attention. Psychological Review, Mono- graph Supplement, No. 6, pp. 1-78 (1898). An experimental and expository paper, the experiments approaching the question in several different ways. HuEY : Preliminary Experiments in the Physiology and Psychology of Reading. Ayyi. Jour, of Psij., Vol. 9, pp. 575-586 (1898). Tests of rate of reading in vertical and horizontal directions, of the importance for recognition of the first and last parts of words, and of the actual movements of the eye in reading, determined by apj^aratus attached to the eye. This study was continued during the year 1898-99, with results that are nearly ready for publication. Technical Matters. The following papers have been chiefly concerned with technical matters and apparatus. SCKIPTURE : Psychological Notes. Ibid., Vol. 4, pp. 577-584 (1892). On the method of regular variation; The least perceptible variation in pitch ; The faintest perceptible sound ; Notation for intensity ; A con- stant blast for acoustical purposes ; Some psychological terms. Scripture : An Instrument for Mapping Hot and Cold Spots on the Skin. Science, Vol. 19, p. 258 (1892). Psychology. 131 Dbesslak: a New and Simple iMetliod for comparing the Perception of Rate of Movement in the Direct and Indirect Fields of Vision. Am. Jour, of Psy., Vol. 6, p. 312 (1894). SAJ4T0RD: A Simple and Inexpensive Chronoscope. Ibid., Vol. 3, pp. 174-181 (1890). A New Pendulum Chronograph. Ibid., Vol. 5, pp. 384-389 (1893). Some Practical Suggestions on the Equipment of a Psychologi- cal Laboratory. Ibid., Vol. 5, pp. 429-438 (1893). Notes on New Apparatus. Ibid., Vol. 6, pp. 575-584 (1895). The Vernier Chronoscope. Ibid., Vol. 9, pp. 191-197 (1898). While these studies have been going on in the laboratory, the work in philosophy and education, and in the non-laboratory sections of psychology, has been carried forward with perhaps even greater vigor. President Hall, Dr. Burnham, Dr. Boas, Dr. Chamberlain, Dr. Mej-er, Messrs. MacDonald, Strong, Gilmau, and others, have lectured on various aspects of the history of philosophy, pedagogy, psychiatry, aesthetics, criminology, and anthropology. Some account of the work in education, anthropology, and psychiatry will be found below in the special reports of Drs. Burnham, Chamberlain, and Meyer; the rest will be spoken of here. The work of instruction has been carried on by means of seminaries as well as lectures, and to a great extent also in the more informal but most effective way of personal conference with individual students. It is not possible from data now at hand to give a complete list of the courses given by President Hall, but at different times he has lectured upon the History of Philosopli}', Ancient, Mediaeval, and Modern (taking philosophy in a sense wide enough to include psychology, education, and medicine) ; on Cosmology, on General Psychology, on Morbid Psychology (with clinics at the Worcester Lunatic Hospital), on Genetic Psj-chology (both in tlie animal series and in the child), Educational Philosophy and Practice, Child Study, Adolescence, Curricula, Teacliing of Special Sub- jects, and upon other pedagogical topics. In addition to these lectures, 132 Department of President Hall has, almost from the first, conducted a weekly seminary, meeting in the evening at his own house. Here members of the depart- ment have reported on the progress of their investigations and received the benefit of mutual criticism, or have united in the study of some special author or topic. Notes of the discussions of the seminary during a period when chief attention was given to Plato have been published by Dr. H. Austin Aikins in the Atlantic Monthly (September and October, 1894), under the title, "From the Reports of the Plato Club." Presi- dent Hall has also directed the research of the greater part of the men in the department, recommending topics, methods, literature, and lines of thought, and in some cases has gone so far as to enter into joint author- ship Avith the students, taking their incomplete results and putting them into shape for publication. In the first years after the opening of the University, President Hall was assisted in the philosophical teaching by Dr. Alfred Cook, Dr. B. C. Burt, and JNIr. C. A. Strong as Docents. During the year 1889-90, Drs. Burt and Cook gave courses on Greek philosophy and on modern philos- ophy from Locke to Kant; and in 1890-91, Mr. Strong gave a brief course on the history of psychology among the Greeks from Thales to Aristotle, — an abstract of the lectures being later published in the American Jour- nal of Psychology, Vol. 4, pp. 177-197 (1891). During 1892-93, :Mr. Ben- jamin Ives Gilman, as Instructor in Psychology, lectured on Pleasure and Pain, and pursued independent investigations on the theory of musical consonance. Abstracts of liis lecturf^s are to be found in the American Journal of Psychology, Vol. 6, pp. 1-60 (1893). Mr. Arthur MacDonald, as Docent in Ethics, devoted himself to theoretical and practical studies in criminology, lecturing on that topic during the first year of the Univei'sity and conducting a seminary, with occasional lectures, during the second. Since 1891 all the philosophical teaching of the department has been done by President Hall himself. The research of this section of the psychological department has been devoted for the most part to questions that are too large and too unman- ageable for successful treatment in the laboratory, — questions of the origin and development of mental life in the race and in the child, of adolescence and sex, of emotion, of religion, and the like. Its scope and nature will be apparent from the following list of studies: — Psychology. 133 Child Study and Psychogenesis. Tbacy : The Language of Childhood. Am. Jour, of Psy., Vol. 6, pp. 107-138 (1893). The Psychology of Childhood. Boston, 1893. 94 pp. (Includes a reprint of the preceding.) (Dissertation.) The first paper is a careful study of extant data on the physiology, phonetics, and psychology of infant language, together with new mate- rial gathered by the author. The second is a similar treatment of sensation, emotion, intellection, and volition as they appear in very young children. Shaw : A Test of Memory in School Children. Pedagogical Semi- nary, Vol. 4, pp. 61-78 (1896). An account of tests made with a carefully prepared story, which was read to the children to test memory and lines of greatest interest. Statistics of about seven hundred papers from children ranging from the third year of school life to those in the higher classes of the high school. Hall and Ellis: A Study of Dolls. Ihid., Vol. 4, pp. 129-175 (1896). A study of the various aspects of the interest in dolls and of ways in which they are used in play, based upon numerous replies to two ques- tionnaires. Small, M. H. : The Suggestibility of Children. Ibid., Vol. 4, pp. 176-220 (1896). A record of experiments both on groups of children and on separate individuals, together with a large number of returns from a question- naire, with pedagogical inferences and applications. CuKTis : Inhibition. Ibid., Vol. 6, pp. 65-113 (1898). (Dissertation.) The four sections of the paper present: 1. A Summary of Facts and Theories, Psychological, Biological, and Neurological ; 2. An Account of the Influence of Different forms of Activity on one Another ; 3. A Study of Restlessness in Children ; and 4. Pedagogical Inferences from the Fore- going. The third section gives results of experiments and observations by the author together with questionnaire returns. The term " inhibition" is taken in a very wide sense. Paktridge: Reverie. Ibid., Vol. 5, pp. 445-474 (1898). A study of 337 questionnaire returns on day dreams and related phe- nomena. The physical signs, the subjective state, the causes and condi- 134 De^jartment of tions, the content, and the awakening are considered. An appendix contains records of the efforts of 330 children to describe an imaginary animal, and of an attempt to gather statistics as to hypnagogic images from upward of 800 children. Dawson : A Study of Youthful Degeneracy. Ihid., Vol. 4, pp. 221-258 (1896). A careful stiidy of about 60 degenerate youths (including 26 boys and 26 girls from the state reform schools of Massachusetts) as to Vitality, Head and face configuration, Anomalies of physical structure, Keenness of senses. Intellectual ability. Parentage, and Environment. Hall: Some Aspects of the Early Sense of Self. Am. Jour, of Psy., Vol. 9, pp. 351-395 (1898). A study of the growth and development of self-consciousness based on questionnaire returns. Making acquaintance with hands, feet, and other parts of the body, external and internal ; influence of dress and adornment ; experiences with mirrors ; various pet names ; childish conceptions of the soul ; questionings of children about their own identity, present reality, etc.; the effect of social environment, beginning espe- cially with the mother. Psychology of Religion. Daniels: The New Life: a Study of Regeneration. Ibid., Vol. 6, pp. 61-106 (1893). (Dissertation.) A study of adolescence in its anthropological and psychological as- pects, with special reference to conversion and other religious experi- ences occurring at that period, the whole being an effort to show the means by which the fundamental truths of religion and theology may be restated in accord with science and life. Letjba : A Study of the Psychology of Religious Phenomena. Ibid., Vol. 7, pp. 309-385 (1896). (Dissertation.) Based upon noted cases of conversion found in religious literature, on material gathered by questionnaire and in personal interviews. The headings of the first part are : The religious motive. Analysis of con- version, Sense of sin. Self-surrender, Faith, Justification, Joy, Appear- ance of newness. The second part treats of the current doctrines of justification, faith, will, determinism, and the doctrine of the grace of God as related to the experiences described. An appendix contains a number of the cases in full. Psychology. 135 Stakbuck: a Study of Conversion. Ihid., Vol. 8, pp. 268-308 (1897). Coutributions to the Psychology of Religion : Some Asi^ects of Religious Growth. Ibid., Vol. 9, pp. 70-124 Q 897). (Dissertation.) The first paper is a study of sudden conversions ; the second of more gradual changes of a similar character. Both are based almost exclu- sively on questionnaire returns ; the first on 137 cases, the second on 195. The topics in the first paper are : Age of conversion, j\Iotives and forces leading to conversions, Experiences preceding conversion, The change itself, Post-conversion phenomena, Other experiences similar to conver- sion, General view of conversion. Those of the second paper are: Statistics of material, Adolescent phenomena. The period of reconstruc- tion. External influences. Cases without marked stages of gro^vth, Adult religious consciousness, Ideals, Significance of the facts. Leuba : The Psycho-physiology of the Moral Imperative. Ibid., Vol. 8, pp. 528-559 (1897). An analysis of the phenomena of conscience, together with argument to show that the "moral imperative" is the psychical correlate of cer- tain activities of the cerebro-spinal system (taken as the neural basis of the Ufe of relation) as opposed to activities of the sympathetic system (taken as the neural basis of the vegetative and emotional life). Philosophy £ind Criticism. KiKKPATRiCK : Observations on College Seniors and Electives in Psychological Subjects. Ibid., Vol. 3, pp. 168-173 (1890). A study of questionnaire returns from college seniors as to their rea- sons for studying philosophical and psychological subjects, benefit gained, authors most impressive, and special topics foimd most interesting. Hall: Contemporary Psychologists. I., Prof. Eduard Zeller. Ibid., Vol. 4, pp. 156-175 (1891). An account of the life and writings of Zeller. Fkasek : Visualization as a Chief Source of the Psychology of Hobbes, Locke, Berkeley, and Hume. Ibid., Vol. 4, pp. 230-247 (1891). The Psychological Foundation of Natural Realism. Ibid. ,Yo\. 4, pp. 429-450 (1892). 136 Department of The Psychological Basis of Hegelism. Ibid., Vol. 5, pp. 472- 495 (1893). These papers are the result of an effort toward a " psychology of phi- losophy." The first two trace the influence of concepts derived from vision and from touch on the philosophic schools in question, and the third the influence of those derived from galvanism. Bailey : Ejective Philosophy. Ihid., Vol. 5, pp. 465-471 (1893). An attempt to describe briefly the philosophical "signs of the times." Leuba : National Destruction and Construction in France as seen in Modern Literature and in the Neo-Christian Movement. Ibid., Vol. 5, pp. 496-539 (1893). A review of these topics under the following heads : Artist sensual- ists, The quest for new sensations, Nihilism and pessimism. School of the decadents. Literary critics, Chronicles, The tormented, The Neo-Chris- tian movement. Allin : The " Recognition-theory " of Perception. Ibid., Vol. 7, pp. 237-248 (1896). Recognition. Ibid., Vol. 7, pp. 249-273 (1896). The first paper is a critique of a theory of perception widely held in the past and present ; the second is an analytical, critical, and expository account of the mental experience of recognition. Mental and Physical Peculiarities. SCKIPTUEE : Arithmetical Prodigies. Ibid., Vol. 4, pp. 1-59 (1891). Accounts of a large number of phenomenal calculators collected from widely scattered sources ; analysis and discussion of their mental pecu- liarities, and pedagogical inferences. Krohn : Pseudo-chromsesthesia, or the Association of Colors with Words, Letters, and Sounds. Ibid., Vol. 5, pp. 20-41 (1892). A summary of literature with presentation of several new cases, and a discussion of the theory of the phenomenon, followed by a bibliography. LiNDLEY : A Preliminary Study of some of the Motor Phenomena of Mental Effort. Ibid., Vol. 7, pp. 491-517 (1896). A study, on the basis of a questionnaire and special tests, of the tricks and peculiarities of movement and posture that accompany mental effort. Psychology. 137 LiNDLEY AND PARTRIDGE : Some Mental Automatisms. Pedagogical Seminary, Vol. 5, pp. 41-60 (1897). A questionnaire study of 495 cases of such mental automatisms as the avoidance of stepping on cracks, counting objects unnecessarily, group- ing objects like small patterns in wall paper into regular figures, and the picking out the middle one of rows of objects. Phillips : Genesis of Number Forms. Am. Jour, of Psy., Vol. 8, pp. 506-527 (1897). A study, based on over 2000 cases (974 school children, and nearly 700 normal school pupils and adults personally questioned), showing the almost miiversal presence of number forms, though often in very rudi- mentary condition. COLEGROVE : Individual Memories. Ibid., Vol. 10, pp. 228-255 (1899). (Dissertation. ) The paper is a study of some sixteen hundred replies to a question- naire on earliest memories, period of life best remembered, forgetful- ness and false memories, aids to memory, etc. This paper is an extract from a more extended work on memory in general. Emotion. Hall : A Study of Fears. Ibid., Vol. 8, pp. 147-249 (1897). Discussion of the chief fears of seventeen hundred peojile mostly under twenty-three years of age, together with description of methods used in reducing the original reports for general treatment. Fears of high places and falling, of losing orientation, of being shut in, of water, of wind, of celestial objects, of fire, of darkness ; dream fears ; shock ; fears of thunder, of animals, of eyes, of teeth, of fur, of feathers; special fears of persons, of solitade, of death, of diseases ; moral and religious fears ; fear of the end of the world, of ghosts ; morbid fears ; school fears ; and the repression of fears, — are all treated in separate sections. Hall and Allin: The Psychology of Tickling, Laughing, and the Comic. Ibid., Vol. 9, pp. 1-41 (1897). A study based upon about seven hundred qxtestionnaire returns. The following rubrics are treated : The Physical act of laughing. Tickling, Animals and their acts. Recovery from slight fear. Laughter at calamity, Practical jokes. Caricature, Wit, Laughter at what is forbidden or secret, at the naive and unconscious. Animal laughter. Miscellaneous items, and Notes on literature. 138 Department of Hall: A Study of Anger. Ihid., Vol. 10, pp. 516-591 (1899). A general summary of very widely gathered literary material, followed by a discussion of over two thousand questionnaire returns ; General descriptions of the state, Causes (with many sub-heads), Subjective variations, Physical manifestations (with many sub-heads), Anger at inanimate and insentient objects. Venting anger, Reaction, Control, Treatment, Miscellaneous aspects. Miscellaneous Topics. MacDonald : Ethics as Applied to Criminology. Journal of Mental Science, Vol. 37, pp. 10-16 (1891). Criminal Aristocracy, or the Maffia. Medico-Legal Journal, Vol. 9, pp. 21-26 (1891). Le Rossignol : The Training of Animals. Am. Jour, of Psy., Vol. 5, pp. 205-213 (1892). A review of literature on the subject. Kbohn : Facilities in Experimental Psychology at Various German Universities. Ibid., Vol. 4, pp. 585-594 (1892); Vol. 5, pp. 282- 284 (1892). Notes on Heidelberg, Strasburg, Zurich, Freiburg, Munich, Prag, Ber- lin, Leipzig, Halle, Jena, Bonn, and Gottingen. Lemon: Psychic Effects of the Weather. Ibid., Vol. 6, pp. 277-279 (1894). A preliminary note on the general question. Scott : Sex and Art. Ibid., Vol. 7, pp. 153-226 (1896). The study traces the higher enthusiasms of art and religion, as well as the passions of sex, to the " fundamental quality of erethism found in every animal cell." Beginning with erethism, the following topics are discussed: Specialization among cells. Separation of the sexes. Radia- tion, Selection, Combat, Courting, Fear and anger. Sex and care for young, The aesthetic capacity, Courting instinct in the lower races, Tattooing, Clothing, Shame, Jealousy and fear, Symbolism and fetich- ism, Phallicism, Modern phallicism. General features and laws of court- ing, Degeneration, Perversion, Ecstasy, .Esthetics, Conclusion. Scott : Old Age and Death. Ibid., Vol. 8, pp. 67-122 (1896). (Dis- sertation.) Old age and death treated from biological and physiological stand- points, together with discussion of 226 returns to a questionnaire designed Psycliolocjy. 139 to bring out the ideas of young people and others with regard to the aged, to death, and to a future life. Partridge : Blushing. Pedagogical Seminary, Vol. 4, pp. 387-394 (1897). A questionnaire study (120 cases, all normal school pupils) : Objective and subjective aspects, After-effects, Physiology, Psychology, Blushing and sex. Partridge: Second Breath. Ibid., Vol. 4, pp. 372-381 (1897). A study based upon about two hundred questionnaire returns. The following are the headings: Physical second breath, Mental second breath, Over-play and abandon in children. Reaction, Physiology of second breath. LxNDLEY : A Study of Puzzles with Special Reference to the Psy- chology of Mental Adaptation. Am. Jour, of Psy., Vol. 8, pp. 431-493 (1897). (Dissertation.) The subject is introduced by a consideration of the biology and psy- chology of play in general, followed by the classiiication of puzzles. The time and conditions of greatest interest in puzzles are treated on the basis of questionnaire returns. This is followed by a report of extended experiments made upon school children to discover their growth in abil- ity to deal with the difficulties presented by puzzles. Klixe: The ^Migratory Impulse vs. Love of Home. Ibid., Vol. 10, pp. 1-81 (1898). (Dissertation.) A biological and psychological study combining the results of experi- ments upon animals with those of a questionnaire. Such topics as the Influence of temperature, Spring fever. Migrations of wild and domestic animals and of man. Wandering tendency in men, women, and children, Love of home, and homesickness, are treated. QUANTZ : Dendro-psycboses. Ibid., Vol. 9, pp. 449-506 (1898). A study on material gathered from biology, anthropology, and ques- tionnaire returns of the psychic influence of experiences ^vith trees. Biological evidence of man's descent from arboreal ancestors, Psychical reverberations from ancestral experience, Tree worship, The life tree. The tree in folk-medicine. The tree in child life. The tree in poetry. BOLTOX, F. E. : Hydro-psychoses. Ibid., Vol. 10, pp. 171-227 (1899). (Dissertation.) A study, similar to the last, on the psychic effects of experiences with water : Evidences of man's pelagic ancestry, Origin of animal life, Ani- 140 DejKirtment of mal retrogression to aquatic life, Water in primitive conceptions of life, in philosophical speculation, Sacred waters. Water deities. Lustrations and ceremonial purifications by water, Water in literature, Feelings of people at present toward water. GoDDAKD : The Effects of Mind on Body as evidenced by Faith Cures. Ibid., Vol. 10, pp. 431-502 (1899). (Dissertation.) "Christian Science," "Divine Healing," hypnotism and other forms of mental treatment of disease are briefly considered ; and " Mental Sci- ence," taken as a type, is treated fidly from data gathered by extended correspondence and from hospital records. In the remainder of the paper the following topics appear : Positive testimony of the influence of mind on disease. Failures in the practice of mental therapeutics, Hypnotism as a therapeutic agent, Theory of mental therapeutics, Psy- chological problems suggested. Resume and conclusions. Street : A Genetic Study of Immortality. Pedagogical Seminary, Vol. 6, pp. 267-313 (1899). (Dissertation.) A study of the origin and characteristics of ideas of the soul, im- mortality, heaven, and a future life, made on the basis of the reports of the thoughts of deaf mutes before training, on about five hundred replies to a questionnaire, and on other material. Biological, psychologi- cal, and moral aspects of the belief in immortality are also considered. Besides the studies of these lists, which have been printed, a number more have been made and are in the hands of the instructors practically ready for publication. Others still have been made and the data sub- mitted without complete writing out ; a good part of these will ultimately be made use of either in themselves or as the basis for further research along the same lines. After this outline of work done in the past, a few words may be per- mitted with reference to the future of the department. This, like its past, must be closely connected with the general progress of psychological science, and the question naturally becomes that of the directions in which progress may be most reasonably expected. Let me begin, as before, with the laboratory. It seems to me that the two lines of greatest promise, conceding readily Psychology. 141 the importance of continuing research along lines already undertaken, are those of comparative and of individual psychology. Work lias already been begun in both fields. Especially in comparative psychology much has already been done by the biologists, but much remains yet to be done. There is surprisingly little accurate knowledge of the mental life of even the commonest animals ; there are many anecdotes, but not many reliable observations, and very few experiments. In this field lie the questions of instinct and heredity, belonging alike to psychology and biolog}', to which run back so many of the most fundamental and practical of even strictly psychological questions. Much may also be expected from the full intro- duction into psychology of tlie comparative method which has so broad- ened and enriched other sciences in which it has been a^jplied. The conception of mind, as of something not narrowly human or confined to a few higher animals, but as in some sort present in all animals, even the lowest, with a history as long as evolution, opens up vistas to which psy- chologists have been too little accustomed. Much surely is to be expected from this closer alliance of psychology with biology. While the theoretical interest of comparative psychology is thus hardly to be overestimated, the practical interest of the efforts toward an individual psychology is hardly less important. We know something about the mental differences of our fellow-men, but we know very little about them in a scientific way. What underlies temperament? What are the laws of the gro^vth of character ? Why do some pupils do well with some teachers and not with others ? What is the best treatment for reform school boys ? How shall one deal with exceptional and peculiar children in the family? Individual psycliology ought to answer such questions as these, and many others. It is clear, of course, that many of these questions extend far beyond the possibilities of the laboratory, but the methods and standpoint and training of the laboratory will play no small part in their final solution, and justify attacking them from that side. Closely connected with individual psychology, but lying a little fur- ther from the laboratory, is another field which might be called the " psychology of the permanent apperceptive groups " — the study of tlie mental attitudes, that is, that result from the fundamental experiences of life, a study of apperception which does not stop at demonstrating the fact of mental habit, but goes on to investigate the effect of one sort of mental habit upon the rest ; how, for example, the fact of fatherhood or 142 Department of a severe sickness may alter character distinctly and permanently. These topics have not been neglected, but many questions remain that would well repay the worker of proper equipment and insight. Coordinate with these are the study of the more complex emotions, of religion and of aesthetics, all of which promise much and should have an important place in a psychological department as a counterweight to the laboratory. It is on the data obtained from the study of these topics and those of the last group, with others like them, that true mental and moral hygiene must rest. Fortunately, here also we have beginnings. Beyond these again, there are topics of great popular interest, like those of Christian science and psychical research, upon which the lay- man has a right to ask an expert opinion from science, and on which psychology, after careful investigation, can and ought to speak. What any particular department of psychology can do in realizing these promises of the future, must depend upon the resources in men and materials that it can command. Work in comparative psychology can be begun at once wherever suitable accommodations can be provided for the animals, — proper housing, cages, aquaria, and such attendance as shall insure the health and happiness of the animals, which are essential factors in any reliable study of their behavior, — and a properly qualified observer can be secured. The first of these requirements is easier to fill at present than the second, for as yet too few persons have equipped themselves both as psychologists and naturalists, but this lack will not long exist if the subject is taken uj) in earnest. For the portion of individual psychology that comes within the scope of the laboratory, there is need of new instru- ments of at least a relative precision, many of which must yet be devised or slowly perfected by trial and failure, which involves a liberal subsidy. For any of the more general problems mentioned, the first requisite is men of proper natural equipment and training. Not every man of learn- ing is fitted to handle them, and those devoted to them must not be so much taken up with the routine and responsibility of elementary teaching, that they lack the time and spirit for ardent research. And these men, once secured, must be liberally supplied with such help in the way of books and other materials as they need. Of these three things, — quarters for comparative psychology, apparatus for individual psychology, and an enlargement of the staff, — the last is, in all ways, by far the most impor- tant. Competent and enthusiastic investigators can work with inadequate facilities, but no facilities can take the place of the men or of the freedom Psychology. 143 from routine teaching. The Clark department has already made such efforts in all these lines as its opportunities have permitted. Its ten years' history justifies the prophecy that, with enlarged opportunities, it would make more than commensurate return in an increase of the advanced teaching and research for which it was originally organized. PSYCHO-PATHOLOGY. By Adolf Meyer. It is hardly necessary to insist to-day on the remarkably suggestive influence which pathology has had on the biology of man, and especially on psychology. Many of the most fundamental changes in psychology are directly traceable to problems furnished by the study of abnormal life, clinical and post-mortem pathology, and experimental reproduction of diseases and of symptom-complexes. Under these conditions it is evident that the curriculum of a psychologist, and of biologists generally, is quite incomplete without, at least, some touch with results and problems of general pathology, and more especially of neuro- and psycho-pathology. Starting from the experience that certain types of psycho-pathology lead very promptly into paths which have nothing to do with biology, and put themselves directly on pre-biological traditions, it was considered best to develop a course which would begin with the principles of general pathology, the abnormalities of the most general biological factors, i.e. with a chapter properly belonging to any general biology. In this field, the experience in the domain of neurology and of psychiatry would have to be worked up more carefully, as far as possible in constant touch with the broader biological concepts. Medicine, barely deserving the attribute of an applied science, is not rich in literature breathing the biological spirit. To a great extent it stands on a pre-biological, materialistic standpoint, and the orthodox practitioner of medicine is usually anxious to keep to materialism and to profess ignorance of the psychological aspect ; and, again, many of those who look upon the psychological manifestations in their patients very rapidly acquire one of the traditional exclusive standpoints, danger- ously near certain mystical concepts. The psychology of hypnotism, of hysteria, even that of aphasia, give good instances of such tendencies. It is consequently desirable to build up a course from the elementary to the more difficult, and starting from the least contested foundations to proceed to the less comprehensible points. 144 Psycho- Pathology. 145 The plan outlined in the lectures and climes of the spring of 1897 gives an idea of the work. The course during the year covered the following ground : — 1. Introductory remarks on general biological conceptions. The gen- eral biological principles applied to the study of abnormal life. Relation between neurology and psychology, neuro-pathology and psychiatry, neuro- logical and psychical jjhenomena from the biological standpoint. Appli- cation of the point of view to alcoholic intoxication and to several forms of mental disease. Demonstration : Cases of Febrile Delirium, General Paralysis, Catatonia, and Idiocy. 2. Review of the general pathology on the ground of the aspect- hypothesis. The terms " disease," " residual," " defective formation," and "defective A7ila(/e." Clinical and post-mortem pathology and their share in general pathology. Only clinical pathology furnishes data on the psy- chological and physiological side. Plan of clinical study. Anatomical study. Our knowledge of the macroscopic and microscopic lesions of the nervous system and the underlying pathological processes, defective growth and nutrition, intoxication, abnormal function. Local disorders : Abnormal circulation, local intoxications, traumatic disorders, over activity, perverted function. Demonstration of abnormal brains and histological changes. 3. The general plan of the nervous system and illustrations of diseases of the various parts (levels). The neural tube ; the segmentary arrangement and the elements of the segments within the lowest level. The middle level apparatus — cerebellum, midbrain, and forebrain, and their afferent and efferent connections. Demonstrations : (1) Traumatic paralysis of the nervus peroneus. (2) Infantile paralj'sis. (3) Cases of hemiplegia. (4) Lead paralysis (Remak type). (5) Alcoholneuritis. (6) Locomotor ataxia. 4. The principles of localization. The meaning of the connections of neurones by numerous collaterals, of the " interruptions of the tracts by gray matter," of the term "centre." Description of the most important "centres," the lesions of the apparatus of mimic movements, the sensorimotor areas, the principal " sensory " projection fields. An outline of the principles of aphasia and its forms, of hemianopsia. Highest level symptoms. Demonstration : Hemiplegia with hemianop- sia ; two cases of hemiplegia with motor aphasia ; one case of sensory aphasia. Reference to a case of Brown-Sequard paralysis. 146 Psydw-Patliology, 5. General outline of mental diseases. Explanation of Kraepelin's classification. Illustration of a paradigm of mental disease : General Paralysis, its etiology, symptomatology, and principal types. Demonstra- tion of six cases. 6. Toxic psychoses and psychoses of distm-bed metabolism. Sum- mary of the data of psycho-physiological study of fatigue and intoxication furnished by the school of Kraepelin. Review of the methods and the results. Application to the clinical problems. Demonstration : Delirium Tremens, Subacute Alcoholic Insanity. Cretinism. Dementia Precox and Catatonia. 7. Periodic Insanity compared with the types of Verbloedungs- processe. Demonstration of further types of Catatonia and of Periodic Insanity; "Acute Mania," "Acute Melancholia." 8. Short sketch of Senile Dementia and demonstration of a case. Constitutional psychoses. Resume of the methods and aims of individual psychology (Cattell, Miinsterberg, Jastrow, Kraepelin, Gilbert, Binet et Henri, Guicciardi and Ferrari). Value of "types" of character or constitution. Their formation. Dominant ideas. Mysophobia as a type of Neurosis of Fear. Development of Paranoia ; cases of Paranoia. In the spring of 1896 a similar course of demonstrations had been given (see the outline, American Journal of Psychology, April, 1896, Vol. 7, pp. 449-450). In the spring of 1898 only one lecture was possible (on the methods of individual psj^chology, especially Kraepelin's work) and a short course of four clinics in the spring of 1899. The desire to extend the studies into research work has remained unfulfilled. Several attempts failed because the possibilities for such work were not mature, neither on the side of the hospital nor on the part of the University. The general principles of the work at Clark University tend toward the education of workers. So far the sub-department of psycho-pathology has been purely didactic, covered by the lectures of President Hall, on the topics which have specially attracted psychology, e.g., border-line phenomena, as seen in neurotic people, prodigies, and geniuses ; defec- tives, such as the blind, deaf, criminal, idiotic ; mental and nervous diseases, epilepsy, phobias, neurasthenia, hysteria ; morbid modifications of will, personality, emotion, etc., and by the above attempt at giving a course with clinics based on general pathology. The research work along these lines depends on two important condi- tions. For systematic work the organization of a clinic is necessary, Psycho- Pathology. 147 and on the part of the worker a fair knowledge of general and special jDathology (in its broadest sense — the knowledge of abnormal life, not merely pathological anatom}^ and baeteriolog}') is an absolute pre- requisite. A training in general and special pathology on the ground of a complete course of biology must be regarded as an absolutely necessary pre-requisite for research in psycho-pathology. Whether most courses of medicine offer what is needed, and whether a medical education should be required, is a matter of some doubt ; since much of the ordinary medical coui'se is business training rather than work in pathology in the true sense of the word, leaving out almost intentionally the broader asj^ects which we have to require more especially for research in our lines ; and most of the medical courses are so overburdened that the training in the history of human thought and philosophical criticism is completely crowded out, and this important safety-valve and balancing apparatus is almost missing in the medical curriculum. The other point, the creation of clinical possibilities, is not less difficult. Our attempt at the Worcester Insane Hospital has hardly matured sufficiently to allow of much research work. The work which forms the foundation of research must be done first, and the reorganiza- tion begun in 1896 is only just beginning to furnish the material for some studies suggested by Dr. Sanford, and some investigations on more closely psychiatric questions. The study of tlie most protracted disorders of human life requires such a patient spirit of work and an atmosphere of such tenacious adherence to solid working principles, that the predilection for fads and the haste for results are nowhere more lamentable. Should it be the good fortune of this department to get strengthened by the State, as well as by the University, a psychiatric clinic and research-station might grow up. Efforts of this character are being made in New York by an institute independent of the hospitals. Our plan is rather to develop the research-station on the basis of the clinical work. The constant con- tact with a field of experience such as a clinic offers furnishes the safest working basis and prevents one from running away with hasty specula- tion derived from too limited a number of facts. The best field for getting problems for work is that of actual observation, such as a clinic only can afford. To pick out curiosities merely will never lead to a psycho-pathology worth its name. ANTHROPOLOGY. By Alexander Francis Chamberlaik. The history of the Department of Anthropology at Clark University forms an important chapter in the history of the study of anthropology in America, since it was the first educational institution to distinctly recog- nize anthropology as a subject of graduate study leading to the degree of Doctor of Philosophy. The first official announcement of the University, published in May, 1889, included, under the work to be undertaken in the Department of Psychology, the following subjects : " The Psychology of Language ; Myth, Custom, and Belief anthropologically considered." With the opening of the academic year, anthropology was established as Section C of the Department of Psychology, and a laboratory and departmental library provided, with proper facilities for original investigation and research. The laboratory contained crania for practical study, necessary craniographic and craniometric instruments, together with the usual tools of the anthropologist working in the field. The library of the University, besides a special anthropological collec- tion, contains a very complete selection of the literature on applied ethics (criminology), embracing the chief works of the English, Italian, French, and German writers. In the psychological library will be found also many works relating to the subjects which anthropology and psychology treat of in common. In 1889 Dr. Franz Boas (now Professor of Anthropology at Columbia University, New York), a graduate of the University of Kiel, who was already well known through his researches among the Eskimo of Baffin Land and the Indians of Alaska and British Columbia, was appointed Docent in Anthropology, which position he held until the close of the academic year, 1891-92, when he assumed the duties of director of the sub-department of physical anthropology at the World's Columbian Exposition, taking with him his fine collection of crania. At the Uni- 148 Anthropology. 149 versity Dr. Boas continued his studies of the anthropology of the Northwest Coast, paying especial attention to a monograph on " The Mythologies of the North Pacific Coast," which he prepared for pub- lication, and to osteological studies of the material collected during his several journeys. In the summer of 1890 Dr. Boas was engaged in investigating the anthropology, ethnology, and linguistics of the Indian tribes of the coast of British Columbia, under the auspices of the British Association for the Advancement of Science. His report, presented to the Leeds meeting in 1891, treated of the customs and beliefs of the Bella Coola, who were shown to be of Salishan stock, besides containing a general review of the physical characteristics of the Indians of the North Pacific coast, with a discussion of the problem of mixed races. Studies of the Chemakum and Chinook languages were also continued and articles prepared for publica- tion. Early in 1890, the apj^roval of the school authorities having been obtained, an extensive series of anthropological measurements was begun in the schools of the city of Worcester, and carried to successful comple- tion. Preparations were also made for the inauguration of similar investigations in other parts of the Union and in Canada. These measurements were undertaken with the object of studying the growth of children as influenced by varying conditions. The investigations in Worcester were carried on by Dr. Boas, with the assistance of the follow- ing members of the Universitj^ : Dr. G. M. West, Mr. A. F. Chamberlain, Mr. T. L. Bolton, Mr. J. F. Reigart. In the spring of 1891 preparations were made for extensive anthropological measurements of the American Indians, under the auspices of the World's Columbian Exposition, Dr. Boas being placed in charge of the sub-department of physical anthro- pology. In prosecution of these investigations, the following students of the University, trained in the anthropological laboratory, were engaged during the summer: Dr. G. M. West, in Quebec and the maritime provinces of Canada ; Mr. T. F. Holgate, in eastern Ontario ; Dr. T. Proctor Hall, in western Ontario ; Mr. T. L. Bolton, in Idaho and Utah. Other observers were similarly employed in Alaska, British Columbia, the northwest territories of Canada, Labrador, Dakota, Wisconsin, Washington, Oregon, New Mexico, Yucatan, etc. The chief object of the extensive investigation thus begun is to show the distribution of types over the American continent, and to settle, if possible, disputed points regarding the physical anthropology of the Indians. In the 150 Anthropology. summer of 1891 Dr. Boas resumed his investigations of the Indians of British Columbia for the British Association, and also visited the last survivor of the Chinook tribe, from whom he obtained very valuable ethnologic and linguistic data. During the academic years, 1889-92, Dr. "Walter Channing, of Brook- line, Mass., Honorary Fellow of the University, carried on original investi- gations in the laboratory of the department. In November, 1890, Dr. G. M. West (afterward Instructor in Anthropology in the University of Chicago), a graduate of Columbia College, was appointed Fellow (and afterward Assistant) in Anthro- pology, and devoted himself to the consideration of its physical side, taking a large part in the anthropometric investigations begun in the Worcester schools. During the summer of 1891 Dr. West was engaged in anthropological measurements of the Indian tribes of Quebec and the maritime provinces of Canada. Appointed Assistant in Anthropology in 1891, he continued in that position until the close of the academic year 1891-92, when he became associated with Dr. Boas in the sub-depart- ment of anthropology in the World's Columbian Exposition, having charge of the anthropological investigations diu-ing Dr. Boas's absence in Europe. During the Docentship of Dr. Boas the lectures of the department were as follows : — 1. A course of lectures on : Physical Anthropology, Osteology, and particularly Craniology. The Physical Character of the living subject : Anatomy of Races. In connection with these lectures practical work was carried on in the laboratory. 2. A course of lectures on : The Anthropology of Africa, embracing the consideration of the geographical distribution, physical characters, lantjuages, and culture of the native tribes. 3. A course of lectures on: The Application of Statistics to Anthro- pology. In the spring of 1892 Dr. West delivered a coiu'se of lectui-es on The Growth of School Children, based upon the results obtained in the Worcester schools. These lectures have been published in Science and the Archiv filr Anthropologie. In the spring of 1890 Mr. A. F. Chamberlain, a graduate of the University of Toronto, then Fellow in Modern Languages in University College, Toronto, who had been a student in ethnology under the late Sir Ajithropology. 151 Daniel Wilson, was appointed to the first fellowship created in anthro- pology in the University. Previous to entering upon the course of study for the doctorate, ^Ir. Chamberlain had made special investigations of the Algonkiau Indian languages, and these he continued, offering as his thesis an original monograph, "The Language of the Mississagas of Skugog," which was published in 1892. Other briefer essays in the same field have appeared in the ProceedmffS of the Canadiati Institute (Toronto), Canadian Indian, American Anthropologist, Journal of Ameri- can Folk-Lore, Proceedings of the American Association for the Advancement of Science, etc., during the years 1888-99. Time snatched from busy hours from 1891 to 1893 was devoted to original investigations in the language and folk-lore of the Canadian French, some results of which have been published in Modern Language Notes (Baltimore), Vols. 6-8. In 1892 was published the result of an extensive investigation of the use of " Diminutives in -ing," in the Platt- Deutsch (Low German) dialects, another study from which field, "Color Comparisons in the Low German Dialects," subsequently appeared. In the spring of 1891 Dr. Chamberlain delivered a brief course of lectures on " The Relationship of Linguistics to Psychology and Anthro- pology." In the fall of the same year he assisted in the anthropometric investigations carried on in the schools of the city of Worcester under the direction of Dr. Boas, and in April-ilay, 1892, superintended the measurements of some 15,000 school children in Toronto, Canada, the results of wliich work are being from time to time published (see Report of Commissioner of Education, 1896-97, Vol. 2) by Dr. Boas, under whose auspices it was carried out. From June to October, 1891, he was absent among the Kootenay Indians of southeastern British Columbia and Northern Idaho, having been selected by the committee of the British Association for the Ad- vancement of Science to carry on anthropological investigations among the Indian tribes of northwestern Canada. His report (discussing in detail the ethnography, physical anthropology, mythology, and language of this comparatively unknown aboriginal people) was presented at the Edinburgh (1892) meeting of the Association and printed, with an introduction by Horatio Hale, as the " Eiglith Report on the Northwestern Tribes of Canada" (London, 1892, 71 pp.). Other briefer studies, botanical, linguistic, mythological, psychological, based upon the material gathered during this expedition, have been published in the American 152 Anthropology. Anthropologist, American Antiquarian, Journal of American Folk-Lore, Verhandlungen der Berliner anthropologischen G-esellschaft, Archivio per V Antropologia, Am Ur- Quell, Science, etc. The great mass of material, however, is still in process of preparation for publication, and will, when complete, make some four good-sized treatises or volumes as follows : — 1. Kootenay Indian Art. An Interpretative and Comparative Study of some Three Himdred Drawings of Natural Objects, Human Beings, Animals, etc., made by various Indians of the Upper and Lower Kootenay. 2. Mythology of the Kootenay Indians. A Comparative and Interpretative Study of some Fifty Animal Tales and Legends of the Kootenay Indians. With original Indian Text, Translation, Explanatory Notes, etc. 3-4. Dictionary of the Kootenay Language, with Introduction on Grammar and Morphology. Part I., Kootenay-English ; Part II., English-Kootenay. As much time as could reasonably be spared from other duties has been devoted to the long and difficult task of compilation and revision of these original studies. During his tenure of the Lectureship in Anthropology, Dr. Chamber- lain has lectured twice a week throughout the academic year, the following courses having been delivered: — 1892-93. Mythology of the North American Indians. The syllabus and bibliography of this detailed interpretative study have been published in the " Third Annual Report of the President to the Trustees of Clark University," 1893, pp. 123-125, 141-161. Several of the lectures have appeared in full, or in abstract, in the Journal of American Folk-Lore. 1893-94. General Course : The Science of Anthropology in its Eelations to Psychology and Pedagogy. Special Coirrses : (a) Comparative Mythology of Ancient Greece and Italy ; (6) Child Life among Primitive Eaces, the American Indians especially. The introductory lecture of this course, under the title "Anthropology in Universities and Colleges," with brief historical bibliography, has been pub- lished in part in the Pedagogical Seminary, Vol. 3, pp. 48-60. An abstract of one of the lectures in course (6) has appeared as "Notes on Indian Child Language," in the American Anthropologist, Vol. 3, pp. 237-241 ; Vol. 6, pp. 321-322. 1894-95. Besides the course in General Anthropology, the following brief special courses were delivered : Anthropology and Ethnology of Sex ; The Child amongst Primitive Peoples ; Comparative Mythology of America and Anthropology. 153 the Old World; Psychology of Primitive Languages; The Beginnings of Art and Language; The JSsthetical Ideas of Primitive Peoples. The lectures on the " Psychology of Primitive Languages" were based upon original investigations among the Algoukian Indians of Canada, and the Koote- nay Indians of British Columbia, and abstracts of several of them have been published in the Aimrican Anthropologist, Vol. 7 (1894), pp. 6S-G9, 186-192 ; Verhandlungen der Berliner anthropologischen Gesellschaft, 1893, pp. 421-425, 1895, pp. 551-556 ; Archivio per I' Antropologia e la Etnologia (Firenze), Vol. 23 (1893), pp. 393-399. The lectures on "The Child among Primitive Peoples," delivered also in popular form at the Summer School in Julj^, 1894, have been elaborated and published as a book, with the title " The Child and Childhood in Folk-Thought " (New York, Macmillan, 1896). 1895-96. Besides the course in General Anthropology, the following special and briefer courses were delivered : Anthropometry of Children and Youth ; The Emotions and their Expression among Primitive Races ; The Idea of the Soul among Primitive Peoples ; Crime and Degeneracy among the Lower Eaces of Man; Origin and Develo^iment of the Family; Sociological History of Woman. Two of the lectures on " The Emotions and their Expression among Primi- tive Peoples " have appeared in part in the American Journal of Psychology, Vol. 10, pp. 301-305, "Fear," and Vol. 6, pp. 585-592, "Anger." 1896-97. Besides the course in General Anthropology, the following briefer special courses were delivered : The Philosophy of Primitive Mythologies ; Origin and Development of Social Institutions ; Race-Psychology ; The Anthropology of Childhood ; Civilization and Evolution. One of the lectures in the course on "The Philosophy of Primitive My- thology" appears, under the title "Folk-Lore and Mythology of Invention," in the Journal of American Folk-Lore, Vol. 10 (1897), pp. 89-100. 1897-98. Besides the course in General Anthropology, the following briefer special courses were delivered : The Anthropology of Sex ; Primitive Children and Children of Civilized Races ; Social Evolution ; Origin and Development of Primitive Religions ; Anthropometry. 1898-99. Besides the course in General Anthropology, the following special briefer courses have been delivered : Child Study in Italy, Variation and Degeneration, Heredity and Environment. Outside of the academic and summer .school courses the following lectures and addresses on topics of general interest have been delivered from time to time in Worcester and elsewhere : — 154 AnthropoloQTj. 1892. Aims and Methods of Anthropometry. Principals and Teachers of Grammar Schools, Toronto. 1892. Optimism. Canadian Club, Clark University, Worcester. 1893. Savage Views of Nature. Natui-al History Society, Worcester. 1893. The American Indian. Men's Association, Pilgrim Church, Worcester, Mass. 1894. Woman's Eole in the Development of Eeligion and Civilization. Fort- nightly Club, Woonsocket, E.I. 1895. The World's Debt to the Eed Man. Natural History Society, Sterling, Mass. 1895. The Mother and the Child in the Story of Eeligion and Civilization. South Unitarian Church, Worcester, Mass. 1896. Childhood. Conference of Lend-a-Hand Clubs, Lowell, Mass. 1896. The American Indian. Universalist Church, New Britain, Conn. 1896. The Making of Abraham Lincoln. South Unitarian Church, Worcester, Mass. 1897. Johanna Ambrosius. Leud-a-Hand Clubs, South Unitarian Church, Worcester, Mass. 1897. Youth. Lend-a-Hand Conference, Boston, Mass. 1897 Lincoln and Darwin. South Unitarian Church, Worcester, Mass. 1897. In Memoriam : Henry George. South Unitarian Church, Worcester, Mass. 1897. The Unitarian Church and Alcoholism. Conference of Unitarian Churches, Barre, Mass. 1898. Primitive Nature Study. Jacob Tome Institute, Port Deposit, Md. 1899. The Child and the Criminal. Monday Morning Club (Universalist Min. isters), Boston, Mass. At the meetings of various scientific societies, 1890-99, the following papers have been presented, those marked * having been published since their delivery: — 1. American Polk-Lore Society : — 1890. *Nanibozhu among the Otcipwe, etc. 1892. *Physiognomy and Physical Characteristics in Folk-Lore and Folk- Speech. 1892. Christ in Folk-Lore. 1893. Mythology of the Columbian Discovery of America. 1895. *Poetical Aspects of American Aboriginal Speech. 1896. *Folk-Lore and Mythology of Invention. 1898. *American Indian Names of White Men and Women. 2. Modern Language Association of America : — 1891. *The Use of Diminutives in -ing by some writers in Low German Dialects. Anthropology. 155 3. American Association for the Advancement of Science : — 1893. Primitive Woman as Poet. 1894. *Translation into Primitive Languages. (Abstract.) 1894. *Incorporation in the Kootenay Language. 1894. *Primitive Antliropometry and its Folk-Lore. (Abstract.) 1895. *Kootenay Indian Personal Names. *Word Formation in the Kootenay Language. 4. British Association for the Advancement of Science : — 1892. *Kootenay Indians. 1897. *Kootenay Indian Drawings. (Abstract.) 1897. *The Kootenays and their Salishan Neighbors. (Abstract.) 5. Berliner Anthropologische Gesellschaft: — 1893. *Wurzeln aus der Sprache der Kitonaqa-Indianer. 1895. *Beitrag zur Pflanzenkunde der Naturvolker America's. 6. International Congress of Anthropology (Chicago) : — 1893. *The Coyote and the Owl. (Tales of the Kootenay Indians.) Dr. Chamberlain has been a Councillor of the American Folk-Lore Society (1894), Secretary of the Anthropological Section of the Ameri- can Association for the Advancement of Science (1894), and one of the secretaries of the Anthropological Section of the British Association for the Advancement of Science (1897). Since 1894 anthropology has been represented on the programme of the Summer School of the University, and each year Dr. Chamberlain has delivered a course of twelve daily lectures upon anthropological questions and topics of more or less interest to the teacher and to the general public. These courses have been as follows : — 1894. Anthropology of Childhood. (The Child Among Primitive Peoples.) 1895. Pedagogical and Psychological Aspects of Anthropology. 1896. Anthropology of Childhood. (New Series.) 1897. Anthropological Aspects of Childhood. 1898. The Beginnings of Education and Educational Institutions. — Primitive Pedagogy. 1899. Educational Aspects of Human Evolution. At the various summer schools the following topics have also been popularly treated in evening lectures : — 1896. (a) The Philosophy of Childhood with the Poets. (6) The Genius of Childliood. 156 Anth'opologrj. 1897. (a) The Divinity of Childhood. (&) The Attitude of Primitive Peoples toward Nature. 1898. The Childhood of Genius. 1899. (a) The Prophecy of Childhood. (p) The Making of a Genius. (Abraham Lincoln.) Anthropology, while comparatively a new, is by no means an uncom- mon, subject of academic instruction, and the time has distinctly passed when it should be called upon to plead for its existence, or to make an oratio pro domo. Very many of the great European universities have specifically rec- ognized anthropology as worthy of the highest positions in their gift, and, in this country, institutions like Harvard, Columbia, Chicago, and the University of Pennsylvania have endued this department with the full dignity of a professorship. Moreover, nearly twenty other colleges and universities in America now offer instruction in anthropology, as such, while Sociology, one of the most important branches of the science, is to be found on the curriculum of all such institutions as ai"e making any efforts whatsoever to keep abreast of the times. Other branches of anthropology, such as Comparative Philology, Comparative Religion, Race Psychology, Anthropometry, Archaeology, Culture-History, etc., are finding more and more acceptance with the higher institutions of learning. Both with respect to original research and to academic lectures, the representatives of anthropology in American universities have no reason to fear comparison with the professors and instructors in any other branch of science, and their influence in broadening and humanizing some of the more belated and conservative of the kindred branches of human knowledge can hardly be overestimated. It is a significant fact that the latest and most complete academic recognition of anthropology, the promotion of Dr. Franz Boas to a profes- sorship in Columbia University, does just honor to one who began his academic career as a Docent in Clark University in 1890. How much of the interest in anthropology in other institutions of learning can be legiti- mately traced to this University, which, in 1892, conferred the first Ph.D. ever granted in America for research and investigation in anthropological science, cannot readily be ascertained, but its influence, both direct and indirect, has beeu, no doubt, as it still is, very great. Proofs of this are not wanting in the curricula of more than one of the higher institutions of learning, while the course in anthropology in the University of Illinois, Anthropology. 157 offered by Dr. Arthur H. Daniels, a graduate of Clark University, is directly due to the initiative and encouragement of the department of anthropology. Through the lectures delivered at the University and during the Summer School, the anthropological department has exercised an ever increasing influence, which has been added to by the appearance of one series of these lectures in book form. Another point of contact with the teaching profession throughout the country lies in the use of the depart- ment as a sort of bureau of information upon many and varied topics of educational science. During the last year, especially, very many requests for such information have been received and responded to, often in detail and as the result of personal research. To the students of the University the department of anthropology has always emphasized the great value of a bibliographical knowledge of the subject under investigation, and its services have always been at their disposal. In this University anthropology ranks as a branch of psychology, and to promote and advance it as such has been the constant aim and endeavor of its representative on the Faculty. The lectures have been such as to correlate with the instruction given in philosophy, psychology, and peda- gogy, and their object has been to furnish tlie students in those depart- ments with the most recent results of authi-opologieal investigations, and to imbue them with that wider and deeper thought that comes from the contemplation of the history of individual and of racial man, and to lay firm foundations upon which in }■ ears to come ma}^ rise a complete and per- fectly equipped department of anthropology. A glance at the theses and essays in the departments of philosophy and psychology will demonstrate the way in which the department has advantaged those who have pro- ceeded to their degrees in this University, such subjects as " Regeneration," "Dolls," "Migration," "Hydro-Psychoses," " Dendro-Psychoses," "Im- mortality," "Teaching Instinct," "Philosophy of Education," "Adoles- cence," "Degeneracy," etc., naturally calling upon anthroj^ologj* for its quota of fact and information, which has often been quite large and sig- nificant. Especially has this been the case since " Child-study" has loomed up so largely in the field of education, for questions of heredity and environ- ment, recapitulation, atavism and reversion, degeneration, variation, genius, and the like, must receive from anthropologj-, more or less, their true orientation and interpretation, — the science of the child would be help- less without the science of man, the story of the individual not half 158 Anthropology. understood without the story of the race. The greater prominence now being given to individual psychology, brings psychology also into closer and better touch with anthropology. That the first woman to hold a fellowship ui any department in Harvard University was an anthropolo- gist is a fact, which, taken in connection with the great amount of excel- lent original work done in anthi-opology by women, both in Europe and in America, augurs well for the future advancement of the science, when all institutions offering post-graduate instruction in anthropology and facilities for original investigation shall have been opened to women upon the same terms as to men. The composite character of the population of the United States, the existence within its borders of several entirely dis- tinct races, and the addition to these resulting from the recent acquisition of outlying and distant possessions, must inevitably tend to make anthro- pology more and more a real academic necessity, no less than a constant factor in the determination of national welfare and progress. Unless every sign fails, the history of anthropology in the next quarter century of American university life will compare in brilliancy with that of any other science similiarly stimulated and environed. At this University, anthropology has accomplished, as the record of the publications of the department shows, results out of proportion to its financial resoiu-ces and the facilities for investigation and research made possible thereby. With other departments in the University it has striven to overcome these serious handicaps as much as might be, and what has already been done must serve to indicate what can be done in the futui-e, if the department is generously and satisfactorily endowed. Nowhere else, perhaps, can the " sinews of science," rightly employed, give ampler or juster returns, if the past foreshadows the years to come. Clark University, the first institution in America to recognize anthro- pology as a fit and proper subject for post-graduate researches and inves- gation leading to the degree of Ph.D., and the first university to confer such a degree, can justly hope for that recognition which comes to the pioneers in all great educational movements. But before the department can labor at its best, it must have the best means of research and investigation, be equipped as well, at least, as any similar department in any other institution in the country. Given these, it can do as good work, or even better. The professorships at Harvard, Columbia, Chicago, and Pennsylvania, the Thaw Fellowship at Harvard, the library of 20,000 books and pam- AnthrojJology. 159 phlets in a single branch of anthropology at the University of Pennsyl- vania, and the laboratory and museum facilities of all these institutions which have come into such rich fruition dui'ing the last ten years, point the way for us, if the good work of the past is to increase and multiply. For comparison with the jiresent state of affairs at this University, the following data from the most recent official publications of the universi- ties concerned, institutions which offer post-graduate courses in anthro- pology and confer the degree of Ph.D. in that department, will suffice (sociology, etc., not included): — Harvard: Professor; Instructor; Thaw Fellow ($1050); Hemenway Fellow ($500) ; Winthrop Scholar ($200). Chicago : Associate Professor ; one Fellow. Columbia: Professor; two Instructors ; President's University Scholar ($150) ; one Fellow ; two Scholars. One cannot escape seeing the necessity of enlargement and further endowment at this University, if anthi'opology is to prosper fully. Before the great things of which it is capable can, in all their rounded completeness, be accomplished here, changes and improvements must take place, and the following are among those most needed or most desirable : — (1) The department must ultimately be dignified by the existence of a professorship, if it is to continue to hold its own among the similar departments in other great educational institutions. Anthropology can wait, as it has waited, but it scarcely deserves that refusal of academical advancement, which is, of necessity, bound up with straitened financial conditions. (2) A complete departmental library, which shall include all cur- rent periodicals and journals of anthropological interest and afford imme- diate access to the very latest American and foreign publications in all branches of anthropological science, is an absolute necessity. The ad- vantage of having aU these under one roof and procurable immediately after their issuance is inestimable. (3) A thoroughly equipped laboratory, for special researches and investigations, is also among the things first to be desired, and what investigators now, or formerly connected with the University, have done in tliis field is a full guarantee that such an addition to the facilities of the Universit}^ would be well utilized and appreciated. 160 Anthropology. (4) A museum, which shall contain materials and specimens illustrating the parallel development of the individual and the race, is also a desidera- tum, for this truly anthropological theory, so fecund for education and psychology, has yet to undergo that stern test which zoology, palaeon- tology, and geology have so successfully sustained. (5) Generous endowment of fellowships and scholarships (intra-mural and extra-mural) and other aids in investigation and field work is abso- lutely necessitated by any adequate instalment of anthropology. (6) More, perhaps, than is the case with most other departments, lib- eral allowances for clerical work and for ti'avelling expenses, the lack of which so often delays good studies and inconveniences good men, are nec- essar}', and the dei^artment must be congratulated on what has been achieved in the absence of all these. Often to be able means to accomplish. Judged both by the work accomplished here and the status of anthro- pology in other universities, the department has every reason to ask and every right to expect such increased endo-mnent as will enable it to make the next ten years of its existence as notable as the same period in the history of anthropology in any of the higher institutions of learning, European or American. PEDAGOGY. By William Henry Burnham. Soon after the opening of the University, President G. Stanley Hall entered upon the duties of Professor of Psychology and Education. During the first academic year no pedagogical courses were given, but toward the close of the year Dr. William H. Burnham, the writer of this report, was appointed Docent in Pedagogy and sent to Europe by the University to study educational institutions, methods, etc. During the year 1890-91, courses of lectures on pedagogy were given in the psycho- logical department by Drs. Hall and Burnham, and a seminary met weekly for the study and discussion of educational subjects. In 1893 the educa- tional courses were designated as a sub-department of psychology offering a minor for the doctor's degree. But the work has remained most inti- mately connected with that in psychology and antliropology. In any natural development of these three subjects, the subject-matter overlaps and is interrelated. In this University no attempt has been made to mark a line of division between them. Specially close has been the connection between psychology and pedagogy, most of the students in one subject taking courses in the other. Such vital connection of the two subjects has mutual advantages. Pedagogy is based upon psychology and owes to it the inspiration and stimulus to scientific work, and psy- chology owes to pedagogy the suggestion of some of its most fruitful fields of investigation. With a limited staff no attempt has been made to cover the whole field of pedagogy ; but by choosing specially important parts of the field, and by extending the courses over two or three years, an effort has been made to show how the subject should be studied. By this method lectures have been given on the history of the modern reform movement in educa- tion, begun on the one hand by the early Italian Humanists, and on M 161 162 Pedagogy. the other by Comenius, the present organization of schools in England, France, and Germany, the Evolution of the Teaching Profession, the Historical and Critical Study of Educational Principles, Mental and Physical Development, Educational Psychology, and School Hygiene, including the Hygiene of Instruction. Other courses have been given by Drs. Hall, Burnham, and Lukens on the following among other topics : History of Methods in Reading, Physical Education, Child-study, Adolescence, Ideal School, Herbartian Pedagogy, History of Cui-ricula, and leading present topics in education. A great variety of siibjects have been studied in connection with the seminaries, and the results of many of these studies have appeared in the Pedagogical Seminary, an educational journal edited by G. Stanley Hall and published at the University, beginning in 1891. The work of the department is best seen, however, by noting its aims, methods, and concrete results. The aim of the department is twofold : first, to give instruction and training to those who are preparing to be professors of pedagogy, super- intendents, or teachers in liigher institutions ; second, to make scientific contributions to education. These two ends are so closely related that the pursuit of one involves much of the work required for the other also. Suitable preparation for the course involves so much of general edu- cation as is usually indicated by the B.A. degree. A good reading knowledge of French and German is of vital importance, and an acquaint- ance with elementary psychology is desirable, it being taken for granted, of course, that those who intend to teach have adequate knowledge in their own special departments. Assuming that a student has adequate preparation, three things are essential for higher pedagogical training : first, a general knowledge of the organization of education in different countries and of literature in the field of education, including the history of education, psychology in its relation to education, and school hygiene; second, actual experience in teaching, together with observation of good teaching, and some direct study of educational institutions of different character and grade ; third, some experience in independent research, involving not only the thorough study of all authorities upon a subject, and of all work that has been done in the same field in different countries, but also original investigation leading to a scientific contribution. These three kinds of work may be done simultaneously or successively. Pedagogy. 163 In some of the best higher pedagogical seminaries in Germany they are done simultaneouslj-. Students study and report upon educational and psychological literatm-e. They visit classes of different grades, obser\'ing the work of regular teachers, and also teach in a practice school. At the same time they endeavor to investigate some special problem. In this University the study of educational literature, by lectures and inde- pendent reading, and the investigation of some problem, are usually car- ried on simultaneously ; but practical experience in teaching must be gained before or after the University course. There are some advan- tages in doing actual teaching simultaneously with the stud}- and investi- gation of educational problems. Direct experience in the school makes investigation more vital and practical, and is an important control in scientific research. But, while at present the University has no practice school, as a matter of fact, most of those who have been members of the educational department have had experience in teaching before comintr to the University ; and the lack of direct connection with the schools is in part supplied by visits to educational institutions. Moreover, there is no rigid line between instructors and students in the department. Both are teachers and learners in turn. Special emphasis is placed upon the importance of research ; and much of the time of the instructors is spent in consultation with individual students in regard to their investigations. President Hall especially has given a large amount of attention to direct- ing this work. The research undertaken has been largely in the field of genetic psychology and related subjects ; and the students have been assisted by the instructors in psychology, anthropology, and neu- rology. A great variety of topics, however, have been studied ; and a large part of the investigations have yielded results for publication. The papers ^ that have already appeared may be roughly classified as follows : — Contributions to the Physiology and Psychology of Development. BOHANNON, E. W. : A Study of Peculiar and Exceptional Children. Pedagogical Seminary, Oct., 1896, Vol. 4, pp. 3-60. Based upon answers to a questionnaire reporting over a thousand cases. ' Many of the papers mentioned In this list are quite as much products of the department of psychology as of that of pedagogy ; and, on the other hand, the pedagogical department has contributed to many of the psychological studies mentioned above. 164 Pedagogy. BOHANNON, E. W.: The Only Child in a Family. Pedagogical Seminary, April, 1898, Vol. 5, pp. 475-496. rrom a study of reports of 381 only children, it appears that only chil- dren are below the average in vitality and unusually subject to mental and physical defects of a grave character, and that, lacking the impor- tant education from the constant companionship of other children, they need special pedagogical care and training. BuRK, Fredekick: Growth of Children in Height and Weight. Am. Jour, of Psg., April, 1898, Vol. 9, pp. 253-326. A comprehensive resume of the numerous studies in this field, with a discussion of their pedagogical significance. BuRK, Frederick : From Fundamental to Accessory in the Devel- opment of the Nervous System and of Movements. Pedagogical Seminary, Oct., 1898, Vol. 6, pp. 5-64. A contribution to the physiology of development, especially a study of the evolution of hand movements in the development of the normal child. From a comprehensive review of the various neurological and psychological studies upon this subject, the author makes among others the following conclusion : that there is an early period in the develop- ment of each part or process when the purpose of education must be to follow the fixed innate heredity line of tendency (fundamental educar tion); that there follows a later period in an activity's development when it passes partially out of the control of racial habit and becomes more plastic to present environment (accessory education). Presented as a dissertation. BuRNHAM, Wm. H. : The Study of Adolescence. Ibid., June, 1891, Vol. 1, pp. 174-195. A brief introduction to the study of the adolescent problem. BURNHAM, Wm. H. : Individual Differences in the Imaginations of Children. Ibid., March, 1893, Vol. 2, pp. 204-225. Based upon literature and reports by students at the Worcester Normal School. Chrisman, Oscar : The Secret Language of Children. Science, Dec. 1, 1893, Vol. 22, p. 303. Pedarjogy. 165 Ckostvell, T. R. : Amusements of Worcester School Children. Pedagogical Seminary, Sept., 1899, Vol. 6, pp. 267-371. A study of the amusements of two thousand children based upon reports by the children. A contribution to the problem of variation in play as conditioned by age, sex, nationality, locality, and season. The results indicate as characteristic of the games of adolescence the coopera- tion of a number of individuals to secure a definite end, and the delight in contest in contrast with the individualistic amusements of earlier years. Hall, G. Stanley: Initiations into Adolescence. Pro£. Am. Antiq. Soc, Worcester, Mass., Oct. 21, 1898, Vol. 12, pp. 367-400. Includes a detailed account of certain rites of primitive peoples, and discusses the relation of adolescent instincts in religion. Lancaster, E. G. : The Psychology and Pedagogy of Adolescence. Pedagogical Seminary, July, 1897, Vol. 5, pp. 61-128. A comprehensive study by the questionnaire method. With a resiund of the work of others and practical suggestions. Presented as a disser- tation. YoDER, A. H. : The Study of the Boyhood of Great Men. Ibid., Oct., 1894, Vol. 3, pp. 1.S4-156. Based upon the study of a large number of biographies. Studies of Special Branches of Education from the Genetic Point of View. Ellis, A. Caswell: Sunday-school work and Bible Study in the Light of Modern Pedagogy. Ibid., June, 1896, Vol. 3, pp. 363- 412. An attempt to suggest the psychological method of religious instruc- tion, together with an historical sketch of the Sunday-school idea. Johnson, G. E. : Education by Plays and Games. Ibid., Oct., 1894, Vol. 3, pp. 97-133. Presents a classified list of about five hundred plays and games with a study of their educational value. HoYT, Wji. a. : The Love of Nature as the Root of teaching and learning the Sciences. Ibid., Oct., 1894, Vol. 8, pp. 61-86. Based chiefly upon literature, with pedagogical suggestions. 166 Pedagogy. LuKENS, Herman T. : • Preliminary Report on the Learning of Lan- guage. Ihid., June, 1896, Vol. 3, pp. 424-460. Traces the stages in a child's learning to talk, and presents much data in regard to pronunciation and the development of the sentence. LuKENS, Herman T. : A Study of Children's Drawings. Ihid., Oct., 1896, Vol. 4, pp. 79-110. A genetic study based upon original reports and observations. PHUiLiPS, D. E. : Number and its Application psychologically con- sidered. Ihid., Oct., 1897, Vol. 5, pp. 221-281. Includes a study of over two thousand arithmetic papers prepared by children in the schools, the results of a questionnaire research, a criti- cal estimate of many text-books, and a discussion of the general sub- ject from the genetic standpoint. Street, J. R. : A Study in Moral Education. Ibid., July, 1897, Vol. 5, pp. 5-40. Based upon the reminiscent answers of adolescents to a questionnaire. The results suggest the great role of imitation, instruction, the sentiments, and heredity in moral action, and emphasize the significance of habit. Street, J. R. : A Study in Language Teaching. Ibid., April, 1897, Vol. 4, pp. 269-293. Studies in School Hygiene. BuRNHAM, Wm. H. : Outlines of School Hygiene. Ibid., June, 1892, Vol. 2, pp. 9-71. Includes, besides a general survey of school sanitation, brief studies of such topics as fatigue, the period of study, school furniture, the hygiene of writing, etc. Btjrnham, Wm. H. : Bibliography of School Hygiene. Proc. N. E. A., 1898, pp. 505-523. A selected list of 436 titles. Cheisman, Oscar: The Hearing of Children. Pedagogical Semi- nary, Dec, 1893, Vol. 2, pp. 397-441. A resume of the investigations of the hearing of school children in different coimtries. Practically complete to the date of publication, with practical suggestions collected from different authorities. Pedayogy. 167 Deesslae, F. B. : Fatigue. Ibid., June, 1892, Vol. 2, pp. 102-106. An introduction to the general subject of mental fatigue. Dkesslar, F. B. : A Sketch of Old Sclioolhouses. Ihid., June, 1892, Vol. 2, pp. 115-125. A brief historical contribution to school hygiene. Principles, Methods, and Organization of Education. Ceoswell, T. R. : Courses of Study iu the Elementary Schools of the United States. Ibid., April, 1897, Vol. 4, pp. 29-4-335. Devoted especially to state and city courses and legal requirements. Ellis, A. Caswell : Suggestions for a Philosophy of Education. Ibid., Oct., 1897, Vol. 5, pp. 159-201. The closing chapter of an extended historical study of the philosophy of education presented as a dissertation. Hall, G. SiAifLEY : Child Study the Basis of Exact Education. Forum, Dec, 1893, Vol. 16, pp. 429-441. LtJKENS, Heeman T. : The Correlation of Studies. Educational Re- view, Nov., 1895, Vol. 10, pp. 364-383. Potter, J. R. : History of Methods of Instruction in Geography. Pedagogical Seminary, Dec, 1891, Vol. 1, pp. 415-424. Specially an account of German methods, based upon literature. ScEiPTUEE, E. W. : Education as a Science. Ibid., June, 1892, Vol. 2, pp. 111-114. A plea for experimental education with report of illustrative experi- ments. Seaes, Chaeles H. : Home and School Punishments. Ibid., March, 1899, Vol. 6, pp. 159-187. Based upon literature and the answers to a questionnaire. The Training of Teachers. Bcek, Feedeeick L.: The Training of Teachers. Atlantic Monthly, Oct., 1897, Vol. 80, pp. 547-561, and June, 1898, Vol. 81, pp. 769-779. 168 Pedagogy. BuENHAM, Wm. H. : Higher Pedagogical Seminaries in Germany. Pedagogical Seminary, Dec, 1891, Vol. 1, pp. 390-408. A sketch of the history and present character of the different kinds of pedagogical seminaries for training teachers for the higher schools in Germany, based on literature and personal observation. Btjenham, Wm. H. : Some Aspects of the Teaching Profession. Forum, June, 1898, Vol. 25, pp. 481^95. Hall, G. Stanley : American Universities and the Training of Teachers. Ih'id., April and May, 1894, Vol. 17, pp. 148-159, 297-309. Hall, G. Stanley: The Training of Teachers. IJic?., Sept., 1890, Vol. 10, pp. 11-22. Hall, G. Stanley : Research the Vital Spirit of Teaching. Ibid., July, 1894, Vol. 17, pp. 558-570. Phillips, D. E. : The Teaching Instinct. Pedagogical Seminary, March, 1899, Vol. 6, pp. 188-245. A study of the phenomena of leadership and teaching among animals and children, of the lives and motives of the great teachers, and of train- ing in relation to the teaching instinct, inckiding a contribution by the questionnaire method. Presented as a dissertation. Reigakt, J. F. : The Training of Teachers in England. Ibid., Dec, 1891, Vol. 1, pp. 409-415. A brief sketch based upon literature. Miscellaneous. BuEK, Frederick L. : Teasing and Bullying. Pedagogical Seminary, April, 1897, Vol. 4, pp. 336-371. Based on returns to a questionnaire. Hall, G. Stanley : Boy Life in a Massachusetts Country Town Thirty Years Ago. Proc. Am. Antiq. Soc, Worcester, Mass., Oct. 21, 1890, N. S., Vol. 7, pp. 107-128. An historical contribution showing the many-sidedness of the home education of the New England country boy. Pedagogy. 169 Hall, G. Stanley: The Case of the Public Schools. Atlantic Monthly, March, 1896, Vol. 77, pp. 402-413. Hall, G. Stanley: The Love and Study of Nature: a Part of Education. Agriculture of 3Iassachusetts, 1898, pp. 134-154. Lectures delivered before the ^Massachusetts State Board of Agri- culture at Amherst, Dec. 6, 1898. Treats of the child's attitude toward nature. Hancock, John A. : An Early Phase of the Manual Training Move- ment. Ibid., Oct., 1897, Vol. 5, pp. 287-292. A brief historical sketch of the old manual labor school. Johnson, G. E. : Contribution to the Psychology and Pedagogy of Feeble-minded Children. Ibid., Oct., 1895, Vol. 3, pp. 246-291. Reports result of tests of memory span, motor ability, and association, in feeble-minded children at the Massachusetts School for the Feeble- minded at Waltham, together -n^th an historical introduction and practi- cal suggestions for their education. KiSTLER, Milton S. : Jolm Knox's Services to Education. Education, Oct., 1898, Vol. 19, pp. 10.5-116. Kline, Linus W. : Truancy as Related to the ^ligratory Instinct. Pedagogical Seminary, Jan., 1898, Vol. 5, pp. 381-420. Includes a comparison of the physical condition of truants as shown by anthropometric tests with that of public school children. Sheldon, Henry D. : The Institutional Activities of American Chil- dren. Am. Jour, of Psy., July, 1898, Vol. 9, pp. 425-448. Based largely on returns to a questionnaire. Small, M. H. : Methods of manifesting the Instinct for Certainty. Pedagogical Seminary, Jan., 1898, Vol. 5, pp. 313-380. A comprehensive study of oaths based upon 2,263 answers to a ques- tionnaire and a vast amount of literature. Such have been the methods of the department, and such iji part the work done. The aim has been to treat a few subjects in a broad way, 170 Pedagogy. rather than to exhaust the field of conventional pedagogy. The necessity and the advantages of this method are obvious from a brief consideration of the subject of education, both theoretical and practical. Jean Paul Richter quotes the French artist who required from a good director of the ballet, besides the art of dancing, only geometry, music, poetry, painting, and anatomy. " But," he adds, " to write upon educa- tion means to write upon almost evei\ything at once ; for it has to care for and watch over the development of an entire . . . world in little, — a microcosm of the macrocosm. ... If we carried the subject still fur- ther, every century, every nation, and even every boy and every girl, would require a distinct sj^stem of education, a different primer and do- mestic French governess, etc."^ The subject of pedagogy is still more encyclopaedic to-day than when Jean Paul Richter wrote these words. Its foundation involves the whole physiology and psychology of develop- ment in the individual, and the history of culture in the race, and its suijerstructure includes, not only all the various forms and systems and methods of education, but the study of the influence of environment in the widest sense. The conventional views minimize both the difficulties and the impor- tance of the subject. It is said that pedagogy is applied psychology or applied child study, and again that pedagogy must get its norms from the history of education and from child study. This statement will do if one knows what it involves. The history of education means the liistory of civilization from its earliest traceable genesis among primitive peoples. It means a study of types of culture and the conditions of their develop- ment. In a word, it is a study of the evolution of education. Child study means, too, the study of the physiology and psychology of develop- ment in man. The science of development aims to give a complete description of all the stages of physical development from infancy to maturity, to show their sequence and their relation to the acquisition of organic, sensory, motor, and psychic processes. As far as psychology goes, it is genetic psychology, which means more than is frequently connoted by child study. Adidt psychology is one thing, relatively fixed, except for variations incident to environment or the individual. Child psychol- ogy, even for a single individual and a given environment, varies con- tinually because the individual is in the process of growth and rapid development of function. It is one thing for the infant, a very different 1 Richter, " Levana, or the Doctrine of Education," Author's Preface. Pedagogy. 171 thing for the child who can walk and talk, stiU another at that plateau in the curve of development that seems to come somewhere between nine and twelve for girls and ten and fourteen for bo3-s, still another for the adolescent. The variation is seen in the period of a single year, almost with the changing moons. This is true, not only on account of the grosser acquisitions, but is seen in the sequence of interests and acti%-ities. Child psychology is protean. It varies not only ^vith the individual and the environment, but especially with the stage of development. Further, the science of development includes comparative psychology. Not only must the child mind be compared with the adult mind, but the stages of devel- opment in the child should be compared with the stages of development in animals, the faculties of the child with those in animals, the motor ability and activities of the child with those in animals. And again, the stages of development in the child must be compared with those in the race; ontogenesis in relation to phylogenesis must be studied. All this is scientific study, not directly practical. Before deriving the norms for practical pedagogy, a propsedeutic study must be made. As Professor James has said: "Psychology is a science, and teaching is an art ; and sciences never generate arts directly out of themselves. An intermediary inventive mind must make the application by using its originality." This mediating function is represented by two somewhat vaguely defined branches of pedagogy — educational psychology, and the general principles and methods of education. Again, after the general principles of education have been derived from psychology and history, and the theoretical norms established, they must be verified by practical educational experiments. This brings us to the practical side of pedagogy represented by such subjects as the organizar tion of schools, the art of teaching, and special didactics. And parallel with the art of teaching in its derivation from the science of development is school h3'giene, which studies especially the conditions that favor the healthy development of the school child. Thus pedagogy is both theoreti- cal and practical, at once a science (at least potentially) and an art. Such is the subject which, as the Italian proverb runs, is always poor and naked, and, Ln the words of a German writer, has long sat as a drudge at the academic hearth, and whose highest recognition in the great univer- sities has usually been as the handmaid of philosophy. Everybody believes in education, yet few believe in pedagogy. The reasons for this are obvious. Apart from a few fundamentals that are almost common- 172 Pedagogy. place, pedagogy has lacked a solid body of scientific knowledge and universally accepted principles. Worse than tliis, it has lacked a definite method and a definite ignoi-ance. Most of the works on the history of education are padded with accounts of second-rate writers and second-rate books that happen to be labelled educational ; while the really great educators have often been neglected, and educational movements have been described as isolated currents in the progress of civilization, without regard to their vital connection with political, social, and industrial movements. The method has been the elementary method of studying and describing isolated facts without regard to historical jjerspective and causal relations; and even the works of the classic writers have been chiefly the repetition and recasting of a few old truths which had been forgotten or were ignored at the time in which the reformers lived. For example, Comenius two hundred and fifty years ago taught that we must study nature by the inductive method and adapt education to the sequence of the stages of natural development; but his writings were forgotten, and again and again the reformers have had to teach again to a new generation the simplest principles of the Comenian didactic. Most of the books on the educational systems of to-day, in like manner, consist of the barren details of organization and method, and the description often of inferior teachers and schools. The forces that have produced these teachers and schools, the significance of the educational movements, have not been seen; and the philosophical, social, and religious thought that has determined educational ideals has not been studied. These isolated facts are barren. Their real signifi- cance is in their relation to other facts. We cannot, for example, under- stand the educational events in England to-day unless we know something of the wider relations of the school movement. The wrangling over the question whether the parish of Eastbourne shall have a school board, or whether the school education of the parish shall continue to be supplied by voluntary schools, means a great deal more than a difference in taxes of a few pence in the jiound. This petty struggle is a part of the great movement for the disestablishment of the Church of England. The com- missioning of a new fellow for university extension work marks another step in the progress of the democratic ideal, which, no longer satisfied with provision for elementary education for every child, now demands also for each individual, according to his ability, a share in higher education. A new endowment for a technical school by the Worshipful Society of Gold- Pedagogy. 173 smiths, or the like, may indicate a new dread of democracy on the part of certain monopolists quite as much as any special interest in industrial education. Oxford itself, with aU its marvellous beauty and idealism, the stronghold of conservatism, cannot keep aloof from the great social, indus- trial, and educational movements outside. No better illustration could be chosen to show the progress of the democratic ideal in education. At the beginning of Queen Victoria's reign one could not even study at Oxford without subscribing to the Thirtj'-nine Articles. A few years ago Jowett advocated opening the university honors and emoluments to the world, admitting anybody to any university examination without restriction of sect, class, race, age, or residence. As was remarked at the time, if fifty or perhaps twenty years ago a radical undergraduate were to have made such suggestions, he would have stood a chance of being expelled from the university, as Shelley was, for blasphemy; now they are the last words of Jowett, quoted with apjiroval before the vice-chancellor. To miss these larger aspects is to miss everything of permanent value. Historical literature in education has relatively little importance for its direct practical teachings ; but the importance of the history of education as a culture subject can hardly be put too high. Education represents one of the deepest human interests, more vital than politics, and well- nigh as universal as religion. The history of education is the history of the development of civilization. It aims at nothing less than the study of the school as a factor in the development of culture in relation to the other factors in education, — the home, the church, the farm, the work- shop, the playground, and the rest. And it aims at the study of educa- tional movements in their genesis, and in relation to political, social, industrial, scientific, and literary movements. This involves not merely the study of educational writers so-called and school systems, but the study of types of culture and the causes that condition them. Likewise in the other parts of the field the failui-e to recognize the wider significance of the subjects studied, and the attempt to build sys- tems before the foundations were laid, have brought pedagogy into disre- pute. But in recent years the conviction has grown that educational problems must be studied inductively ; and, better still, important contri- butions by the inductive method have actually been made. This has put life into the subject and given hope for the future. Take child study as an illustration. The significance of the modern study of children is not merely the renewed emphasis on the old truth of adapting education to 174 Pedagogy. the stages of development, but the insight that the only way to make this principle vital is concrete inductive study to find out just what are the stages of natural development. Thus every fact in regard to general development or individual variation is deemed significant, and the student is willing to wait for a new science of development before attempting a permanent pedagogical system. During the past ten years the opportunities for truly scientific work in education have been shown as never before, methods of investigation have been demonstrated, and in part the foundations of a science have been laid. The things now needed are trained men and facilities. With them a solid content of scientific knowledge can be acquired that will place historical and social pedagogy on as firm a basis as general history and sociology, and genetic pedagogy on a scientific footing comparable to that of psychology. School hygiene has already its methods and a solid body of knowledge, but it needs special laboratories for instruction and research, either independent or in connection with psychology, physiology, and anthropology. The work in pedagogy in this University, although the practical aspects of the subjects studied have not been neglected, has been chiefly in the more scientific and theoretical parts of the field. It is not less important on this account. Pedagogical study, like research in any other field of history or science, is valuable for its own sake without regard primarily to practical results. It is its own justification and its own reward. With the nucleus of solid scientific contributions that now exists, no university can long afford to omit courses in education from its curriculum, whether they have any practical value or not. Such scien- tific studies, however, cannot be divorced from the practical art of educa- tion. The studies of children have emphasized the doctrine that the aim of childhood is its own development, and the best guarantee of normal maturity is normal childhood and immaturity ; in a word, they have emphasized the principles of normal development. But these principles are no longer pedagogical abstractions ; they are gi'eatly modifying the practical work of education, causing greater regard for individual chil- dren rather than uniform classes, for health rather than scholastic prod- ucts, for a psychological order of instruction adapted to the capacity and interests of children rather than logical sequence and articulation of grades. In a word, they are causing courses of study and methods to be reconstructed with regard to the one fundamental principle of fostering Pedagogy. 175 normal growth and development. To mention a few details, ten years ago school baths, adjustable seats and desks, and vertical script, were vagaries of university theorists ; to-daj^ they are deemed essentials in the best schools. Ten years ago suggestions of periodic disinfection of school apparatus and school text-books, of investigating pupils" individual capacity and power to resist fatigue, and of adapting education to indi- vidual capacity and interest, in elementary and secondary schools, were likely to be ridiculed ; now their soundness has been demonstrated by practical experiments. What part this University has had in this movement, it is not easy to say ; but it has always advocated such reforms in the regular courses of lectures ; many addresses on topics in school h3'giene and pedagogy have been given outside the University before schools and teachers' meetings ; students from this University have become school superintendents, teachers in secondary schools, professors of pedagogy or psychology in normal schools, professors of pedagogy in colleges and universities ; and teachers and educators from all parts of the country have attended lectures on pedagogy during the sessions of the Summer School. To make a department of pedagogy what it should be, it is necessary that the whole field of education be covered by lectures as far as possible, that the more elementary courses be given every year, that research should be extended to the multitude of topics that offer opportunity for study. Nowhere in the world is a complete course in pedagogy covering all the important parts of the field given. Here and there throughout this country and Europe are offered a few truly scientific courses, but the subject will hardly attain its due academic dignity until somewhere in one university are given courses which approximate an adequate treatment of the whole field. That this University might approximate a complete course in the subject are needed an addition to the staff, especially for the study of historical and social pedagogy, the establishment of special fellowships for educational research, a laboratory for school anthropometry and school hygiene, a great enlargement of the educational museum, a pedagogical library like that of the Munie pSda- gofjique in Paris, where educational literature of every kind, both good and bad, may be collected ; and, finally, a model school for the objectifi- cation of ideals, under the direction of competent teachers who should safeguard the interests of the pupils, but offer to university students op- portunities for observation, and in some cases for practice in school work. 176 Pedagogy. The aim of such a course in pedagogy, like that of the more limited course already given in this University, would be twofold : first, to con- tribute something to the body of knowledge in regard to education, the content of pedagogy ; and, second, to give practical training to students preparing to become teachers. These two aims are quite in harmony, for an essential in the training of a teacher is the development of those permanent professional interests and that professional apperception and prevision acquired by the study of the more scientific parts of pedagogy. \ PHILOSOPHY. By G. Stanley Hall. In addition to my own work in psychology and education, reported in the preceding articles by my colleagues, Drs. Hanford and Burnham, and in editing the American Journal of Psychology and the Pedagogical Seminary, I have lectured during the last eight years on the History of Philosophy. This course is felt to be of cardinal importance for those studying either psychologj' or education, to give them breadth of view, to teach what great problems have interested the race, and to give a repertory of general ideas that will ob\date some of the dangers of specialization. The course begins with a very brief survey of Oriental speculation, treats the pre-Socratic Greek thinkers with considerable detail and witli constant reference to their fragmentary texts. Great stress is laid upon Plato, and from a quarter to half of all his works are read aloud by the students in turn from Jowett's translation, and on these dialogues the examination for the doctorate is in some part based. Even for those who read some Greek, the use of the English translation is preferred, because more can be gained from Plato by men of this grade by extensive reading than by intensive and critical study of text. Discussions often arise in this work. Aristotle is treated in the same manner, and selec- tions and sometimes large portions of some of his works are read in English. From twelve to twenty lectures are given on the later schools, ending with Plotenus and Proclus. This usually concludes the work of the first year. Until two years ago the second year began with the rise of scholasti- cism and the third ended with Schopenhauer, Lotze, Hartmann, and con- temporary writers. Special effort has always been made to go considerably outside the stock text-book field and to deal to some extent with the history of science, with some reference to medicine and with very slight reference to literature, art, etc. The texts of Spinoza, Locke, Berkeley, N 177 178 Pliilosoiylxy. Hume, Kant, Ficlite, Schelling, Hegel, Schopenhauer, and Lotze have been used at different times and with very different results. Ethics, logic, metaphysics, and sesthetics are included in this course, and no special courses in any of these subjects have been given, although logical and ethical questions are treated in my psychological course. Considera- ble time is always given to epistemology. Two years ago, after considerable previous preparation, a course in Christology and Patristics was inserted between the ancient and the mod- ern course as above described. The life of Jesus was treated concisely and reverently from the standpoint of psychology, which is felt to be very different from that of the ciu-rent lives of Jesus. This course, although at present being repeated with amplification, is still too incomplete to warrant any final report upon its utility. On the whole this historic course, which occupies three years, is earnestly recommended to all stu- dents of psychology, religion, education, or any of the humanities, and has generally been taken by aU. During the past eight years I have opened my house one evening every week of the academic year to all students in the department of psychology and related themes from seven to ten o'clock. We began by discussing philosophical topics assigned beforehand to leaders in turn. One year most of the time of tliis seminary was devoted to reading and discussing Jowett's Plato. ^ Schopenhauer, Kant, and Hegel were tried for briefer periods, but gradually, as the numbers have increased and as the rule that each man should devote a portion of his time to some original investigation has prevailed, the evening has been occupied by each student in turn, who presents his thesis or subject, or a part of it, which is then freely discussed by the other members. The debates are often animated, as nearly every standpoint is represented. There are clergymen, young professors from other institutions, Hegelian idealists, Kantian epistemologists, and men of empirical science, and from these various directions nearly every subject is really illmuinated. Attendance is never enforced, and the light refreslunents served in the middle of the evening have never been an attraction, but only a welcome break from continued tension. The attendance for the last few years has rarely been ' See a somewhat disguised account of the first semester's work in two articles by H. Austin Aikins, entitled "From the Reports of the Plato Club." Atlantic Monthly, Sept. and Oct., 1894, Vol. 74, pp. 359-368, 470-480. Philosophy/. 179 under fifteen and rarely over forty, so that the entire freedom and infor- mality of conversation has been the rule. The themes assigned in a way described later have been presented here in so compact and forcible a way, that the seminary has been one of the most effective agents in my own education, and I think all its members share my sentiments in this respect. Here the new work on which each individual is spending so much of his year's time is pooled for the common benefit, the reader has the healthful stimulus of emulation in interesting his audience, acquires valuable practice in the methods of effective presentation, and always receives help in the way of new literature, references, the pointing out of defects in argument or method ; and conflicts are thus most surely avoided. Often other professors from the University attend, and the list of distinguished guests from abroad who have either participated in the discussions or introduced matter of their own is a long and dignified one. There is rarely any lack of interest or reluctance to discuss, and very infrequently is the animation too great for healthful mental circulation. Here nearly everything that has been done by the student members of this department of the University has been carefully wrought over, some of it more than once. Such stimulus I believe to be unsurpassed in educational value. The dialectic give and take of the conversational method, the mental alertness of debate, the charm of friendly intercourse upon high themes, which Lotze, like some of the ancients, thought the highest joy of life and the consummate fruition of friendship, are here combined in judicious propor- tions most favorable to growth. Some European seminaries are devoted to discussions of minute points ; in others the student is simpl)' a literary forager for the professors ; quite frequently some author is read ; but for our American needs, at least for Clark University, I think the method now settled upon is more educative than any other that I have seen. A word should be said concerning student lectures. At various periods during the decade each member of the department has been requested to take his turn in presenting some subject in due form before the class, taking my place at the lecturer's desk, and developing his theme with the aid of charts, blackboard, and specimens if need be ; and at the close of the lecture I have a personal interview, stating very frankly any faults of manner, automatism, voice, method of presentation, etc., liable to interfere with his usefulness as instructor or lecturer. More 180 PkilosopJiT/. often, in place of an original lecture, each man takes his turn in digesting with extracts some book or chapter of a standard work in the history of philosophy, with the same criticisms. This personal relation together with the many hours spent each week with individuals, elsewhere spoken of, has been, I believe, of great value. At the beginning of the year (or, for those who have already spent a year at the University, near the close of the spring term) careful lists of subjects which seem to the instructors in the department ripe for investi- gation are prepared. Each jots down all suggestions in this direction during the year, and all now meet to compare themes, consider whether they have already been treated, what new books and apparatus each wiU necessitate, by what paths each can best be approached and which are likely to yield the best and (what for thesis work is of great impor- tance) the most certain results of value. Conferences with each indi- vidual are then held and each is lU'ged to select some theme, either because it is congenial or because it represents a field he desires to enter, and to devote some considerable portion of the year to the effort to master it and to add something new, however small, to the sum of the world's knowledge. A really good subject has aspects or divisions that bring the student into contact with each professor in the department, and each gives everjrthing in the way of information, stimulus, and references that he possibly can. Our plan has always been to allow the student to print such work over his own name and to have full credit, although he usually makes acknowledgements at the close of his paper to his helpers. This plan we have found very congenial and stimulating to students, and it has avoided all questions of ownership rights in intellectual property. Again, a good subject must be midway between a very large and general and a very minutely special standpoint. The student must not waste his energy in vague generalities on the one hand, nor must he be shut up to some petty problem, perhaps fitting into and aiding the professor's special work, being thus unduly subordinated and apprenticed to him, as is so common in Germany. Fitting the problem to the man so that it will enlist all his interest and focus his knowledge and effort is half the work. In beginning more or less independent research like this, our best college graduates are often in a sense suddenly reduced back to infancy and need constant individual help to go alone. For the last ten years PMlosopliy. 181 most of several afternoons a week of my own time has been given in the laboratory, library, and conference room in trying to assist and direct j^oung men to launch out in some modest yet effective way, as becomes them, on the great life of discovery. Some, often the best scholars, are so tied to authority that it is hard for them to be brought to realize that the best things have neither been done nor said in the world, and that mastery of the text-book is not final. Others are strongly inclined to repeat experiments, multiply observations, and accumulate numbers, and find it hard to make a serious study of the real significance of their data. Some approach subjects with preconceived ideas and speculate in a deductive way, abhorring details which others get lost in. Every type of philosophical opinion and every shade of temperament, every de- gree of intellectual enterprise at almost every rate of progress, is repre- sented. Some are strong in the literary, historical, and antiquarian side of their topic ; others in its experimental technique or in statistical pres- entation and tabulation or in literary form ; some at once tend to lose themselves in aspects of the subject that are so large that, instead of com- ing to a conclusion in an academic year, they begin to anxiously plan a life work and anticipate remote difficulties; while others can see abso- lutely nothing in topics of great range and significance except some over elaborately fortified or proven fact. This form of modern university work is a new kind of high Socratic midwifery, but in my ojiinion it is the most beneficial of all the points of contact between professor and student. Some must be encouraged; others must be roundly scolded. Some would devote all their time to an interesting work of this kind, while others dawdle with it as a mere side issue of douljtful educational value. A few do not want it, but are con- tented with receptivity of what others liave done. Restless ones often seek change of theme, so that great discretion and great patience are needed in this work. Its rewards, however, are incomparably great. Having once discov- ered a fact or made ever so small an original contribution and had tlie baptism of printer's ink, the novitiate is henceforth a changed man. His ideals of culture, standards of attainment and excellence, and his methods of work are slowly revolutionized from this centre. Instead of being a passive recipient, his mind has tasted a free and creative activity which puts him on his mettle like the first taste of blood to a young tiger. He has learned that achievement and not possession is the end and aim ; his 182 Philosophy. mind has been brought to a focus in such a way that he now knows what real concentration means as never before. He realizes that almost every subject in the universe, if broadly seen, is connected with every other, and that the cosmos, like his own mind, is knit together into a unity of a higher order. In all his works and ways he is more independent and more inclined to seek, do, know, and experience for himself. By such personal conference with individuals at all stages of their preparation in such a work, which need not be a doctor's dissertation and often is not, I am con- vinced, after a decade of experience here and some years of the same work at the Johns Hopkins, that this is the highest criterion of an academic teacher's real efficiency in his vocation, and that it is as much above the mass teaching of the lecture-room as talent is above mere learning. The necessity of this work is one of the chief reasons why truly university work must always be done, if not at small institutions, at least in squads so small that they can be thus individualized. Ha\ang brought this work to some degree of completion, as should be done at the close of each academic year, even at some sacrifice of scientific quality (because educational values should take precedence even over this, where the two conflict), an indispensable requisite is publicity and that without delay. Any institution or department that confers a doc- torate upon the ground of a dissertation that is unpublished conceals that upon which the chief value of the degree rests. The older the student the more stress should be laid upon this part of the work as compared Mdth acquisition. In most departments, science is progressing so rapidly and work is so often duplicated that the necessity of announcing before- hand the theme of each research has often been urged, and any con- siderable interval between the completion of a work and its publication involves danger of anticipation by others, as well as general loss of value from the progress of science, which is always slowly leaving everything behind. Chiefly to avoid this danger the journals of this University were established, in which, without the cost to the students generally insisted upon elsewhere and with the advantage of a more or less extended international circulation among experts, everything can be speedily brought to the knowledge of those most interested and competent. To know that results will thus appear without delay is itself a real stimulus, and it is fortunate that evaluation of such work is coming to be a more and more prominent factor in determining appointments to univei'sity positions. The quality of mind which makes success here is infinitely Philosophy. 183 more inspiring to students, even of lower grades, than the rehearsal of second-hand knowledge perhaps many removes from its source. Very- much might be said upon the effect of research as a stimvdant to the teacher, while, from still another point of view, the fact that the instruc- tor has entered the great arena and submitted his productions to the critical estimate of experts, gives his pupils confidence in him as an authority and not a mere echo. The provision of a sufficient nxmiber of reprints for circulation among special journals that will notice each work, and for exchange with other productive workers or departments, is another one of the new university problems unknown to the college, to the fuller exploitation of which the new journal here contemplated and elsewhere spoken of will be devoted. Great importance has always been attached here to the methods of bringing students into immediate and personal contact with the latest literature, especially upon the topics of their theses or those related to the original researches upon which they may be engaged. The exchanges of the journals constitute a carefully selected list of nearly three score peri- odicals, all of which, besides those regularly subscribed for by the library, are immediately available. Besides these the journals receive a large number of the most important books and pamphlets within their field, especially from American, English, French, and German houses. These works together with the smaller resources of my own library, which mainly supplements that of the University, are at the disposal of students, who are often encouraged to write brief book notices for publication. The frequent personal conferences with each student in the department keeps the instructor's mind alert to find out and bring to the immediate notice of each anything bearing upon his theme. Meetings are occasion- ally held in my library, where I spend the evening going tlirough my shelves, taking out the books that I deem most important and that have helped me most, briefly characterizing each, and passing it around for actual inspection. If I had at my disposal an hour's time of a dozen of the most eminent men to utilize in such a way as would be of greatest benefit to me, I think I should ask them to do precisely this, for they would thus be giving me to some extent a key to their own intellectual activity and direction. Where this method is extended to monographs and pamphlet publications, the collection of which our s3'stem of exchanging theses promises to greatly enrich, its value is still greater for special students. 184 Philosophy. Genetic psychology, which one sub-department of this University so conspicuously represents, is far larger than the child-study of mothers' clubs or teachers' associations. It is simply the entrance of Darwinism into the field of mind. Underneath it lies the great transforming concep- tion that the soul is as complex, as old, and as gradually unfolded as the body, and like it must be studied comparatively in view of all that the psychic life of the lower or even the lowest organisms can teach us. The new methods cross-section the old classification methods which make memory, will, perception, imagination, etc., so many faculties, and seek to trace the origins of the higher mental powers to their faintest beginnings near the dawn of animal life. The most fundamental activities are those whose roots extend lowest down in the scale of existence, and these are also they that send their tops highest. The conception that mind, as we know it in consciousness, has been developed out of something very differ- ent that, like organic forms, tends to vary and change indefinitely is a new conception and is sure eventually to reconstruct out of new and old elements a far larger and more adequate city of Man-soul with reformed administrative, educational, religious, and other functions. This move- ment appears in biology in the tendency to study psychic phenomena in the most rudimentary and microcosmic organisms. It appears again in the new and careful studies of instinct in the higher animals, where con- ditions can be varied and educational experiments conducted with great precaution and detail. Another root of the genetic movement is in the anthropology of myth, custom and belief among primitive and savage peoples ; another in the studies of degenerative types among the defective classes, where decay has inverted the evolutionaiy order. It is on this foundation that the child-study movement rests, and its amazing development cannot be adequately explained without a due ap- preciation of this wider field. The minute observation and annotation, the measuring and weighing of a single child, or the collective study of one topic upon the basis of returns from very large numbers of children with the help of questionnaires, anthropometric work with its carefully wrought out averages, — all this appeals to the instinctive love of children; out of it has grown the new conception of childhood as the most generic period of life, wherein the limitations of individuality are not yet so pain- fully apparent as in adults, and it has given us new conceptions concern- ing the nature of genius, the laws of growth, the origin of fear, anger, love, the conditions of health, the nascent periods of maximal interest in Philosophy. 185 special lines and topics, until at last education seems likely to have under it a far more solid and scientific foundation than it has ever yet attained. While tliis subject has as yet occupied but a slight and recent portion of our curriculum, so much has already been accomplished as to warrant the very fairest hopes for the future. Among the first results likely to be witnessed are the gradual transformation of the methods of teaching and of investigating the problems of the special philosophical disciplines some- what analogous to the transformation of anatomy and morphology under the influence of embryology. How far this movement will extend among the other university studies, and whether with or without any new coor- dination of the successive stages of individual growth with the historic development of different philosophical systems as first presented by Hegel, it is impossible to foretell. THE LIBRARY. By Louis N. Wilson, Librarian. From the foundation of the University the library has been consid- ered an important factor and has received a great deal of attention from the Founder, President, and Faculty. Immediately upon his ap- pointment, the President requested eacli member of the University to draw up a list of books in his special field, laying j^articular stress upon important serials and sjjccial monographs. These lists were carefully collated, duplicates weeded out, and arranged in order for purchase. The total number of items amounted in June, 1889, to upward of fifteen thousand, a very large proportion being books and journals in foreign languages. In order to secure for the University the best possible rates, lists of standard works, both in sets and single volumes, were submitted for competition to a number of well-known booksellers both in this country and in Europe. This necessitated some delay, but it was fully warranted by the resultant saving in cost. To illustrate this point, the figures submitted by five firms for an identical list of 742 items are given here, viz.: $1806.30, $1810.90, $1971.86, $2038.89, $2106.41, showing a maximum difference of $360.11. After carefully comparing all the lists sent in, and taking into con- sideration the condition of the books offered, orders were placed with firms in New York, Boston, London, Paris, Berlin, Leipzig, and Vienna. During the past few years, owing to our very peculiar and constantly changing customs and postal regulations, it has become more and more desirable to import from Europe through some responsible bookseller in this country, in order to avoid the frequent and often vexatious annoy- ances consequent upon individual importations. Having decided upon a particular bookseller, orders were freely placed with the understanding that the library should receive the lowest possible rates consistent with good service, and from time to time lists were sent to other firms in order to be assui'ed that the agreement was faithfully carried out. A recent 187 188 The Library. test of this kind showed the following quotations for thirty-five volumes, fl05.26, $107.57, 1120.00. In general, the plan has worked exceedingly well. During the summer of 1889, while these orders were being executed, Mr. Clark placed the first books in the library by donating about thirty- two hundred volumes. A large proportion of these, on history, biogra- phy and travel, were given with the original bookcases as they had stood in his own private library. Another part of the collection con- sisted of the following sets of bound periodicals, almost all complete down to the end of 1883 : Atlantic Monthly, Blackwood's Magazine, Cen- tury Magazine,' Cornhill Magazine, Edinburgh Revieiv, Fortnightly Revieiv, Ge7itlemans Magazine, Harper's Magazine, LittelVs Living Age, Macmillan's Magazine, North American Revieiv, North British Revieiu, Notes and Queries, Popular Science Monthly, Putnam's Magazine, Quarterly Review, and Scribner's Monthly, also a set of the Report on the Scientific Results of the Voyage of H. M. S. Challenger, during the years 1872-76. Yet a third part consisted of a large number of rare old books, some of which are fine examples of early printing when there was no title-page, no pagination, date, or printer's name, and where the initial letters were omitted to be inserted later by hand. Of these fine old volumes, the following are mentioned as examples : — Paulus de Sancta Maria Scrutinium scripturarvun. Probably the oldest book in our library, with no title-page, colophon, pagination, or signatures. Eubricated throughout. Eationale divinorum ofliciorum. Supposed to have been printed at Basle in 1474-75. Astexanis Suma. Libri VIII., de preceptis, de virtutibus et viciis ; de sacra- meutis de sacro penitentie, de sacramento ordinis, de excommunicatione ; de matrimonio. Venetiis, 1478. Koberti Caraczoli de Licio de timore divinorum judiciorum ac de morte. Nuremberge, 1479. Alberti de Padua expositio Evangeliorum dominicalium totius anni et concor- daucia quatuor evangelistarum in passionem dominicam a Nicolao Viuckel- spickel. Ulm, 1480. Sancti Thome de Aquino ordinis predicatorum super quarto lihro sententiarum preclarum opus. Venetiis, 1481. Liber moralitatum elegantissimus' magnarum rerum naturalium liunen anime dictus. 1482. Sancti Hieronimi Vitas Patrum Sanctorum Egiptiorum. Ntirnberg (Koburger) 1483. The Library. 189 Blondi Flavii historiarum ab inclinatione Romanorum Imperii, libri XI. Venetiis per Octavianum Scotuin. 1483. Johannis de Turrecremata questionum dignissimarum cum solutionibus earum- den, etc. Davantrice, 1484. A work of the celebrated Spanish Dominican Juan de Torquemada. Legende de sancti composte per Jacobo de Voragine. Yenetia, 1484. An old German almanac beautifully printed in red and black and pasted on one of the covers of Hieronimi Vitce Patrum. It runs from 1486 to 1579, and was probably printed at the earlier date. Summa Rainerij de Fisis. Venetiis, 1486. Liber Cronicarum cum figuris et imaginibus ab initio mundi usque nunc tem- poris Impressum ac finitum in vigilia purificationis Marie in imperiali urbe Augusta a Johanne Schensperger. Anno ab incarnatione domini 1497. The so-called Nuremberg Clironide, with numerous woodcuts by Wolge- muth, the master of Albrecht Diirer. Sermones Pomerii de Tempore Hyemales et Estivales et sermones quadragesi- males per Helbartum de Themeswar. Hagenaw, 1502. With rubricated initials. Pauli Jovii elogia virorum bellica virtute illustriiun veris imaginibus supposita, quae apud Musaeum spectantur. Florentiae, 1551. Ramusio, Prime volume, & Terza edizione delle navigation! et viaggi. Vene- tia, Giunti, 1563. The first volume of Ramusio's well-known collection of voyages and travels, containing among other things Pigafetta's log during the first voyage around the world under Magalhaes. Missale Romanum, ex Decreto Sacrosancti Concilii Tridentini restitutum, Pii V. Pont. Max. jussu editum. Venetiis, apud Juntas, 1602. The Bible : that is the Holy Scriptures contained in the Old and New Testa- ment. London, 1610. A copy of the so-called Breeches Bible. Missale Romanum, ex Decreto Sacrosancti concilii Tridentini restitutum, Pii V. Pont. Max. jussu editum et dementis VIII. auctoritate recognitum. Ingol. stadii, 1610. Montanus (Arnoldus) De Nieuwe en Onbekende Weereld of Beschryving van America. Amsterdam, 1671. An old description of America in Dutch. Esquemeliug (John) and Ringrose (Basil), History of the Bucaniers of America. London, 1695. Esquemeling, who spent many years at Tortuga, gives here a very graphic account of the buccaneers. Armenian Bible. Venice, 1805. Fleeing from the persecution of their ortho- dox brethren, the Catholic Armenians of the mechitaristic order established themselves at the island of San Lazzaro, granted them by the Republic of Venice. Many a learned volume issued from their press, of which this is a specimen. New Testament in Lettish language. Mitau, 1816. Select Fables ; with cuts, designed and engraved by Thomas and John Bewick, previous to the year 1784 ; together with a Jlemoir and a descriptive cata- 190 The Library. logue of the works of Messrs. Bewick. Newcastle, 1820. Thomas Bewick is considered the restorer of wood engraving in England. Cookson (Mrs. James). Flowers drawn and painted after nature in India. 1834. In addition to a number of books presented to the library by Presi- dent Hall, we are indebted for gifts to the following citizens of Worces- ter : Hon. George F. Hoar, Mr. Henry J. Howland, Hon. Henry L. Parker, Mr. Samuel H. Putnam,^ the late Hon. W. W. Rice, Hon. Stephen Salisbury, Hon. John D. Washburn, and Hon. John E. Russell of Leicester. To receive the books temporary wooden stacks were erected in the main library room, and so substantially were they constructed that they are still serviceable. Solid oak shelving was put up on both sides of the reading-room, adjoining the main library room, with a three-foot pro- jecting shelf three and a half feet from the floor, upon which the current numbers of periodicals are displaj^ed. To the problem of cataloguing and classification, always a difficult one, both the President and the members of the Faculty gave a good deal of time and attention. It was felt that the scheme of classification must not be too rigid, and that nothing should be allowed to interfere with the free use of the books by all members of the University. The books were first carefully classified upon the shelves by departments, and marked as follows : — ■ A. Works or General Reference. I. Psychology. B. Journals. J. Philosophy. C. Mathematics. K. Ethics. D. Physics. L. Criminology. £. Chemistry. M. Anthropology. F. Zoology. N. Education. G. Physiology. 0. Botany. H. Pathology. The various subdivisions in each department may be inferred from that of the mathematical department. lA copy of "Jiistini historici claj-issimi in Trogi Pompeii historias Libri XLIIII." Venice, Jenson. 1470. Duke de Noailles' copy of the editio princeps. " Virorum Ulnstrium vitm ex Plntarcho Grceco in Latinum VerscE Solertiqce, cura emendatm fceliciter expliciut : ^' per Nicolaum Jensen Gallicum Venetiis Ipressce. 147S, die. II Jamiarii. 2 vols. "The Scientific Papers of James Clerk Maxwell." Edited by W. D. Niven, F. R. S. The Univer- sity Press, Cambridge, 1890. 2 vols. The Library. 191 C. — Mathejiatics. In Mathematics, C, the books are grouped in ten divisions, designated by the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, immediately following the let- ter C ; every division is subdivided into sections of which each is desig- nated by a second numeral following that indicating the division. The cipher, 0, always denotes a miscellaneous division or section. The math- ematical works are arranged on the shelves in accordance with the fol- lowing classification, the subdi^^sions of which, however, are not all used at present. The italicized part of each title is that printed on the sliding shelf label. C 1. History axd Philosophy. CI. CI. CI. CI. Bibliography. History. Biography. Philosophy. C 2. Collections. C 2. 1. Works, complete and select. C 2. 2. Compendia, Dictionaries. C2. 3. Tables. Formulce. C3. Symbolism and Operation. C 3. 1. Symholic Methods. C3. 2. Operations. C 3. 3. Multiple Algebra (ref. C 9). C 3. 4. Symbolic Logic. C 3. 0. Miscellaneous Symbols. C 4. Arithseetic. C 4. 1. Elementary Arithmetic. C 4. 2. Continued Fractions. C 4. 3. Numerical Series. C 4. 4. Finite Differences and Smu- mation. C 4. 5. Permutations and Combinor tions. C 4. 6. Probabilities. C 4. 7. Theory of Numbers. C5. Algebra. (For Multiple Alge- bra see C 3. 3.) C 5. 1. Elementary Algebra. C 5. 2. Determinants. C 5. 3. Tlieory of Equations. C 5. 4. Simultaneous Equations. C 5. 5. Transformation. C 5. 6. Invariants. C 6. Infinitesimal Calculus. C6. 1. Limits and Infinite Series. C6. o Functions of a Beal Varia- ble. C6. 3. Differential Calculus. C6. 4. Integral Calculus. C6. 5. Total Deferential Equations. C6. 6. Partial DZ/ferential Equa- tions. C6. 7. Functions Derived from Differential Equations. Spherical Harmonics. C6. 8. Calculus of Variations. C 7. Theory of Functions. C7. 1. General Tlieory. C7. 2. Algebraic Functions. C7. 3. E.rponential and Trigono- metric Functions. C7. 4. Elliptic Functions and In- tegrals. C7. 5. Hyperelliptic and Abelian Functions and Integrals. C 7. 6. Various Functions (fuch- siennes, etc.). C 7. 7. Functions of Several Varia- bles. 192 The Library. C 8. Geometry. C8. 1. Elementary Geometry and Trigonometry. C8. 2. Analysis Situs. C8. 3. Analytic Geometry in Gen- eral. C8. 4. Projective Geometry. Mod- ern Synthetic Geometry. C8. 6. Special Systems of Geomet- ric Analysis. C8. 6. Plane Loci in particular. C8. 7. Loci hi S Dimensions in par- ticular. C8. 8. Hyperspace and NonrEucli- dean Geometry. C8. 9. Applications of Geometry. As B is the general designation sively devoted to one department letter of the department to which it B C. Mathematical Periodicals B A. Miscellaneous Periodicals C 9. Extensive Algebra (ref. C 3. 3). C 9. 1. Geometric Representation of the Lnaginary. C 9. 2. Quaternions. C 9. 3. Geometric Algebras (Clif- ford). C 9. 4. Ausdelmungslehre (Grass- mann). C 9. 5. Equipollences (Bellavitis). C 0. Miscellaneous. C 0. 1. Apparatus. C 0. 2. BecrecUions, Games, Puzzles, etc. C 0. 9. Paradoxes and Paradoxers. Circle-squaring, etc. of periodicals, each periodical exclu- is designated by B, followed by the belongs, thus: . Transactions of learned societies, etc. So long as the number of books in any section is very small, they are grouped under the division to which that section belongs, and are desig- nated only by the number of that division. All books which refer to several divisions are placed in the division C 2 (collections), and all books referring to several sections of any one division are grouped under that division, unless they refer but slightly to more than one division or section. Volumes of a set are not separated, but the whole set is classed as if it were a single volume. Otherwise, every book is placed in the narrowest division or section to which it belongs. The library has two card catalogues : — I. An author's catalogue arranged alphabetically with miscellaneous and anonymous sections, so that nearly all books in the library are represented in it. II. A subject catalogue which is at the same time a shelf and an inventory catalogue. This is arranged as follows : Every volume and every pamphlet has its card, so that each card represents a volume. All the books are classified and arranged upon the shelves according to the departments, divisions, and subdivisions, but under each subdivision The Library. 193 books are placed alphabetically by authors. While each case, tier, and shelf is permanently labelled, the demarcation between the subdivisions is made by sliding shelf label holders bearing the subject, division, and sub- division. These label holders being movable, the subdivisions can easily be enlarged as new books are added. In mathematics, for instance, C 1, history and philosophy, comes first, with the first subdivision, CI, 1, bibliography. First on the top shelf, and therefore first in the catalogue drawer set apart for these tiers, comes bibliography, beginning with authors in A, and so on through the alphabet to the end of the subject. Then come history, biography, etc., on through mathematics and the other departments, the order of cards being identical with the order of the books upon the shelves, reading down the tiers as down a printed page. In the drawers the book cards are separated by red cards projecting on the right above the others, and on these projections the tier and shelf divisions are marked ; they are also separated by blue cards projecting above the others on the left-hand side, on which the subjects are marked. Whenever the position of any book is changed, it is only necessary to make a corresponding change in the position of its card. The shelf posi- tion of each book is marked in pencil, not upon these cards, but upon each card in the author's catalogue, and in the book itself, in order that it may be readily found and replaced. New books, after being entered in the author's catalogue, are kept in a case reserved for them for a few weeks before being permanently placed on the shelves and entered in the inventory catalogue. A full list of all serial publications taken by the library is kept in a special drawer of the catalogue case, so that a person unfamiliar with the library may ascertain, with very little trouble, what periodical publications are to be found here. Worcester is fortunate in possessing an excellent Public Library of more than 120,000 volumes, and well supplied with serial publications. In the early years of the University, it was the hope of the Founder that we might confine our purchases to such books and journals as were not to be found in the Public Library, and that the two might suf)plement each other ; this plan was largely carried out in the earlier years, but later the needs of our students demonstrated the necessity of the duplication of the more important scientific publications, though we still depend upon the Public Library for works of a less special character, and our students 194 The Library. have availed themselves of the library privileges thus extended to them to the fullest extent. Besides its indebtedness to the Worcester Public Library the Uni- versity is under great obligations to the following for frequent loans : Library of the Surgeon-General's Office, Washington, D.C. ; Library of Harvard University ; The City Library Association of Springfield, Mass. ; Boston Public Library ; Public Library, Cleveland, Ohio ; Trinity College Library and Case Memorial Library, of Hartford, Conn. ; Library of Yale University ; Forbes Library, Northampton, Mass. ; Library of Vassar College ; and many others. Several of these libraries have freely lent us books and volumes of serial publications, often of the greatest importance to those engaged in research work. No library, however large, can hope or expect to be prepared to meet all the calls upon it, and a glance at the diverse and advanced character of the publications issued from this University ^ shows how varied and numerous are the demands upon this department. To the Library of the American Antiquarian Society we are especially indebted for the kindly spirit of cooperation invariably shown. While strictly a reference library, its officers have ever been ready and willing to make reasonable exceptions in aid of the cause of historical and scientific research. The library is a veritable laboratory, and is looked upon as a work- room, and not as a museum with contents too sacred to be profaned by use. It is a favorite meeting-place for professors and students, where the heads of departments meet their men to direct their reading and demonstrate to them how to make the best use of a well-selected collection of scientific books. The books are readily accessible to every member of the University, and there is no limit to the number that may be taken out. Each one enters the volumes he takes out upon a printed form pro- vided for that purpose ; if not returned at the end of ten days, they are renewed by the librarian for another period of ten days, at which time they must be returned, but may be taken out again upon the following day. The library is open to all persons outside the University who are interested in any of its lines, and its books are freely lent to such persons, who are thus placed for the time being upon the same footing as mem- bers of the University ; and, while we borrow during term time an aver- age of fifty volumes a month, we lend as freely. The library is rich in 1 See Bibliography at tlie end of this volume. The Library. 195 certain special lines, and is often used by experts from other universities, state and national institutions. President Hall has an exceptionally fine private library, especially rich in pamphlets and special monographs in the various fields of philosophy, psychology, and education. During these ten years all students have been permitted to draw upon it as freely as upon the University libi'ary, and the efficiency of this department has been largely due to Dr. Hall's broad-minded and liberal conception of the function of the printed vol- ume. In his various courses he frequently gives demonstrations of books, pointing out the best books in each subject, the best to buy, the best to read, emphasizing and explaining the strong points in each, etc. In spite of the absolute freedom of the library, the loss of books has been surprisingly small. Once a year the books are carefully checked by means of the shelf cards, and in very few years have the losses amounted to more than two or three volumes. The missing volumes one j' ear fre- quently turn up later, so that a careful estimate recently made shows the actual money value of the books lost in ten years to be less than fifty dollars. Almost all who are interested in libraries have ideals as to the future development of their special fields, and the librarian has attempted, in the course of the past ten years, to formulate an ideal of an university library. He alone is responsible for his views, and is encouraged to state them here by the fact that the President and Faculty have given him the greatest freedom and their warmest support in all matters pertaining to his de- partment. The ideal library should be housed in its own building, and not rele- gated to rooms in a building constructed for other purposes. In con- structing such a building, the chief end in view should be to provide every facility for the xise of books, and this end should never be sacrificed for arcliitectural features or artistic purposes. Each department in tlie University should have a working library in its owai rooms, but whatever books are placed in these department libraries should be duplicated in the main library. Tlie building should be large enough to allow the book shelves to be arranged around the rooms, leaving the greatest amount of open space in the centre. Movable working desks, liberally supplied Avith 196 The Library. conveniences for writing, and containing ample drawer space for note-books and papers, are much to be preferred to the large fixed tables usually found in library buildings. The shelving should be of the most approved modern type, insuring economy of space and the proper care of the books, and the highest shelf within easy reach from the floor. The rooms should be provided with every possible convenience, including a sufficient num- ber of comfortable chairs, mth cozy nooks and corners inviting to a quiet half-hour ^vith a book, when one would other^vise be disinclined to read. That the light shoidd be good, the ventilation absolutely perfect, and the attendants have but one pui-pose — the service of the readers — are obvi- ous essentials. In these days of rapid multijjlication of new libraries and enlargement of many older ones, there is a great demand for complete sets of serial publications, and many of the important journals are growing rapidly scarce and difficult to obtain. It is, therefore, particularly desirable in an institution of this character to procure, as soon as possible, full sets of all the serial publications in its various departments and on all allied subjects, and every effort should be made, and no expense spared, to procure all the scientific contributions by specialists in the work represented here, or in departments likely to be of service in research work. The current numbers of all these publications shoiild be placed before the members of the University promptly, as it is imperative that those engaged in original investigation be advised of the latest literature on the subject, or of the work others are doing along similar lines. A most important part of a good library is its catalogue. The day has gone by when men can afford to spend hours in hunting among a mass of books to ascertain what the library possesses upon a given subject, or to rely upon the memory of the librarian and attendants, be the}^ ever so erudite. While, therefore, the aim should be to keep in printed and card form a list of all the books and articles that have been written upon a given subject, nothing should be allowed to interfere with the prompt cataloguing under subject headings of everything that the library pos- sesses. Two questions always arise here, first, " Where can I find a list of all printed matter upon my subject ? " and secondly, " How much of that printed matter is to be found in this library ? " A complete card catalogue can be so arranged as to answer perfectly these two questions. In this, as in every well-regulated library, printed forms should be provided to encourage readers to make suggestions and complaints to the The Library. 197 library committee ; the latter, in no case, to pass through the hands of the librarian. The subject of binding is always an important one, and we feel very keeulj' the need of united action on the part of all the libraries of the city in this respect. A careful inquiry has developed the fact that between 84000 and S5000 is expended yearly by the various institutions in this city for this purpose. There are unmistakable signs that the art of book- binding, which has for ages commanded the services of eminent crafts- men, as well as of men and women eminent in art, is receiving increased attention from book lovers here, and the time may not be far distant when this question will be taken up by a committee representing the different libraries. There would seem to be no reason also why the various institutions should not, in the near future, devise a system of cooperation, as is already proposed in Toronto, by means of which the resources of all the libraries in the city could be drawn upon by each. EEPORT OF THE TEEASUEER. At the first meeting of the Trustees of Clark University, May 4, 1887, Mr. Clark proposed to give : — (1) " The sum of $300,000 (payable as the same shall be needed) to the Gen- eral Working or Construction Fund to be applied in the erection of buildings and equipping them with such appliances and facilities as may be deemed necessary for putting the University in good working order." (2) " The sum of $100,000, the income of which shall be devoted to the support and maintenance of a University Library." (3) " The sum of $600,000, the iacome of which is to be devoted to the general uses of the University in its support and management, and which for the sake of convenience may be called the University Endowment Fund." "The Library and the Endowment Funds are never to be diminished, and no part of the principal is in any event ever to be applied to the objects to which the income of each is to be devoted. If by any accident or loss, either of said funds shall at any time become impaired, then the income of each of said funds shall be added to the principal until such impairment is made good and the funds restored to their original amounts." In addition to the foregoing gifts, Mr. Clark then and subsequently conveyed to the Trustees of the University, real estate, the valuation of which on the books of the assessors of the city of Worcester is .flSSjCOO. In the Treasurer's Annual Statement for the year ending August 31, 1899, which follows, is an account of the Library and University Endow- ment Funds. The amounts expended for construction and equipment of buildings under the terms of Mr. Clark's first proposal have been as follows : — 199 200 Report of Construction of the Main University Building . . $159,780.60 Construction of the Chemical Laboratory . . . 56,131.94 Equipment of the Main Building 18,480.28 Equipment of the Chemical Laboratory .... 14,801.47 Apparatus and Supplies 29,082.73 $278,277.02 Additional land was purchased by Mr. Clark for the University at an expense of $12,233.04 The balance to make up the proposed $300,000 . . 9,489.94 was subsequently expended in the additional equip- ment of the different departments. A statement of the expenses of the several departments for the years 1890-98, inclusive, including the amounts expended in the original equip- ment above mentioned, is appended. 1890. 1891. 1892. 1893. 1894. Mathematics . Physics . . . Chemistry . . .? 6,664.49 17,214.20 25,334.24 28,083.29 13,604.17 750.00 15,568.04 5,829.00 9,067.43 3,860.00 Z 7,235.00 7,320.98 7,491.00 15,429.70 11,400.00 1,550.00 5,733.41 2,900.00 5,162.92 4,560.00 $ 7,356..50 6,768.46 6,298.46 12,732.58 7,059.16 1,151.25 1,279.84 3,000.00 4.183.77 7,240.00 $ 6,926.40 3,567.78 2,693.26 3,676.47 7,666.03 1,586.13 1,334.45 3,800.00 8,983.01 5,280.00 $ 5,905.64 2,330.30 1,337.64 2,066.20 Psychology . Education . . Lihrarv . 6,584.00 1,826.87 2,596.33 Administration Expense . . Fellowships . 2,600.00 3,773.51 4,980.00 ,1125,974.86 868,783.01 157,070.02 $45,513.53 $34,000.49 Mathematics . Physics . . . Biology . . . Psychology . . Education . . Librai-y . . . Administration Expense . . . Fellowships 1895. f 5,900.00 2,329.07 2,072.74 6,015.46 1,312.29 1,628.72 2,600.00 3,434.13 4,740.00 (,032.41 1896. 5,900.00 2,393.03 2,200.00 7,010.00 1,250.00 1,740.16 2,600.00 4,319.80 4,620.00 $.32,032.99 1897. 5,900.00 2,948.73 2,300.00 7,010.00 1,250.00 2,456.00 2,600.00 4,237.82 3,420.00 J32,122..55 1898. $ 5,900.00 2,173.00 2,054.24 6,676.33 1,2.50.00 3,508.48 2,600.00 3,190.93 1,500.00 $28,852.98 the Treasurer. 201 In addition to the endowment and gifts, which have already been referred to, Mr. Clark has given to the University for its general purposes : — 1889-90 $12,000 1890-91 60,000 1891-92 26,000 1892-93 18,000 $106,000 The University has received from Mrs. Eliza W. Field "a fund of $500 to be called the John White Field Fund, the income of which is to provide for the minor needs of a Scholar or Fellow." There was also presented to the Trustees of the University by Hon. George S. Barton of Worcester $5000, the income of which is to be devoted to the aid of " some one or more worthy native born citizens of the city of Worcester, who may desire to avail themselves of the advan- tages of the institution." Hon. Henry L. Parker, in the summer of 1892, in behalf of many citizens of Worcester, presented the University with a tower clock and the sum of $781.30 to provide for its maintenance, which fund is known as the Clock Fund. REPORT OF THE TREASURER TO THE TRUSTEES FOR THE YEAR ENDING AUGUST 31, 1899. To THE Trustees of Claek University, Gentlemen, — I have the honor to submit herewith my annual report for the year ending August 31, 1899. The total receipts of the University from Sept. 1, 1898 to Aug. 31, 1899, inclusive, were .... $48,59.5.53 The total disbursements during the same period were . . . 37,130.27 Leaving a balance on hind Sept. 1, 1899, of $11,465.26 202 Report of (A.) The items of income are as follows : — Gross Income of the University Endowment Fund .... $28,407.33 Gross Income of the Library Fund 5,258.46 Gross Income of the University 1,586.00 Gross Income of the Summer School, 1899 1,388.50 Subscriptions to the Fund for the Decennial Celebration . . 4,150.00 From the Field Fund 20.00 Balance from previous year 7,785.24 Total $48,595.53 (B.) The expenditui-es have been as follows : — For the Department of Mathematics $ 6,300.00 For the Department of Physics 2,641.11 For the Department of Biology 2,012.25 For the Department of Psychology 7,966.82 For the Department of Education 1,250.00 Administration 2,700.00 Expense 4,729.87 Field Scholarship 20.00 Expenses of Summer School 889.85 Expenses of the Decennial Celebration 3,156.85 Library Expenses .......... 3,474.08 Sinking Fund 700.00 Jonas G. Clark on account of premiums 900.00 Accrued interest repaid 389.44 $37,130.27 (C.) The incidental earnings of the University from fees, etc., were . $ 1,586.00 (D.) Account of the Summer School for 1899 : — Keceipts $ 1,388.50 Expenses 889.85 Balance carried to University Account $ 498.65 (E.) Subscriptions to the Decennial Celebration : — Receipts $ 4,150.00 Expenses 3,156.85 Balance on hand appropriated to the publication of this volume . $ 993.15 the Treasurer. 203 (F.) The University Endowment Fund is invested as follows Oregon Railway and Navigation Co., 4s . West Shore R. R. Co., 1st Mtg., 4s, 2361 . City of Cambridge, Sewer Loan, 6s, 1905 . Norwich and Worcester R. R. Co., 4s, 1927 Rutland R. R., 1st Jltg., 6s, 1902 . Wilkesbarre and Eastern R. R., 1st Mtg., 5S; 1942 Hereford Ry. Co., 4s, 1930 Chicago and Eastern Illinois R. R., 1st Consol iSrtg., 6s, 1934 .... 1st Mtg. Sink. R, 6s, 1907 . Wayne Co., Michigan, 4s . Northern Ohio Ry. Co., 1st Mtg., 5s Lowell, Lawrence, and Haverhill St. Ry., 1st Mtg., 5s Worcester and Suburban St. Ry., 1st JItg., 5s Worcester and Marlboro St. Ry., 1st Mtg., 5s Atchison, Topeka and Santa Fe Ry. Co., . Gen. Mtg., 4s . . . $18,500.00 Adj., 4s ... . 10,000.00 Certif. Gen. Mtg., 4s . . 250.00 Second Ave. R. R. Co., New York, 1st Consol Mtg., 5s, 1948 .... 15 shares Worcester National Bank 71 shares Norwich and Worcester R. R. . Deposit in Worcester Co. Inst, for Savings Deposit in Five Cents Savings Bank 100 shares Fitchburg (preferred) 35 shares New York, New Haven, and Hart- ford R. R 100 shares Worcester Traction Co. (preferred) New England Yarn Co., 5s . . . Lake Shore Collaterals, 3^s Invested in premiums .... Cash in Worcester National Bank . Book value. Market value. Sept. 1, 1899. $110,000.00 $112,750.00 75,000.00 84,750.00 20,000.00 22,600.00 75,000.00 84,000.00 25,000.00 26,500.00 9,800.00 10,600.00 9,350.00 10,000.00 10.000.00 13,700.00 1,000.00 1,145.00 30,000.00 31,200.00 3,000.00 3,180.00 15,000.00 15,750.00 6,000.00 6,240.00 10,000.00 10,400.00 25,000.00 18,500.00 8,800.00 250.00 25,000.00 30,000.00 2,250.00 2,700.00 14,603.50 15,620.00 5,000.00 5,000.00 10,000.00 10,000.00 10,300.00 11,800.00 6,982.50 7,630.00 10,700.00 10,450.00 11,000.00 11,495.00 50,000.00 50,000.00 15,230.00 28,920.25 28,920.25 $614,136.25 $634,480.25 204 Report of The gross income of the University Endowment Fund was $28,407.33 There was paid from this : — To Sinking Fund to provide for premiums . To Jonas G. Clark on account of premiums Accrued interest repaid .... $700.00 900.00 389.44 Leaving net income carried to University Account (G.) The Library Fund is invested as follows : — 50 shares Washington National Bank, Boston 25 shares Tremont National Bank, Boston . 50 shares Merchants' National Bank, Boston 60 shares National Bank of Republic, Boston 50 shares Union National Bank, Boston 50 shares Second National Bank, Boston . 50 shares New England National Bank, Boston 50 shares Atlas National Bank, Boston. 61 shares State National Bank, Boston 15 shares Suffolk National Bank, Boston . 50 shares Eliot National Bank, Boston 50 shares National Bank of Commerce, Boston 60 shares Boylston National Bank, Boston 43 shares Old Boston National Bank, Boston 10 shares City National Bank, Worcester . 15 shares Norwich and Worcester R. R. stock Northern Ohio R. E. Bonds, 5s . 15 shares New York, New Haven, and Hartford R. R Invested in premiums .... Deposit in Worcester National Bank . The gross income of the Library Fund was : — From dividends and interest .... Rebate on bank tax, Balance carried to Library Expense Account $1,989.44 $26,417.89 Book value. Market value. Sept. 1, 1899. $ 5,527.00 $ 6,000.00 1,766.00 (in liquidation) 7,934.60 8,600.00 7,994.88 8,750.00 6,829.50 7,150.00 9,162.50 8,850.00 8,237.50 7,825.00 6,293.50 6,750.00 6,938.01 7,167.50 1,527.21 1,650.00 6,598.00 7,150.00 5,552.62 5,625.00 6,530.75 6,850.00 4,527.63 5,074.00 1,500.00 1,500.00 3,000.00 3,300.00 4,000.00 4,240.00 2,992.50 3,270.00 150.00 2,273.05 2,273.05 $99,335.25 $100,024.55 $4,085.77 1,172.67 $5,258.44 the Treasurer. 205 (H.) The Librarj' Expense Account : — Unexpended balance from previous years . . . $3,091.18 Credits for books sold 412.38 Income of the Library Fund for 1899 .... 5,268.46 $8,762.02 The expenses, including $900 for administration, heat and light, were 3,886.46 Leaving a balance Sept. 1, 1899, of . . . $4,875.56 The George S. Barton Fund, deposited in the Worces- ter Co. Inst, for Savings, amounts to . . $7,239.24 Income during the year 278.43 (J.) The John White Field Fund, deposited in the Worces- ter Co. Inst, for Savings, amounts to . . . $653.22 Income during the year 25.74 (K.) The Clock Fund, deposited in the Five Cents Savings Bank, amounts to $878.40 Income during the year 33.93 (L.) The Sinking Fund, to provide for premiums, is de- posited in the Worcester Five Cents Savings Bank, and amounts to $2,670.42 (M.) The salaries of the University Faculty were . . $19,990.00 (N.) Fellowships and Scholarships $1,310.00 (0.) Salaries of employees $2,135.00 Apparatus and supplies $870.18 Respectfully submitted, Thomas H. Gage, Treasurer. 206 Report of the Treasurer. We have examined the books and accounts and securities of Clark University, and find them to be correct and as stated in the foregoing treasurer's report for the year ending August 31, 1899. James P. Hamilton, T. H. Gage, Jr., Auditors. 2^-'>yKJi'<^aM^ THE HELIOTVPE PRINTING CO., BOSTON LECTURES ON MATHEMATICS. By Professor Emile Figaro. Premiere Conference. Sur TExtension de quelques Notions MathSmatiques, et en particulier de VIdSe de Fonction depuis un Steele. Mes premiers mots seront povir adresser mes remerciments au Conseil de cette Universite qui m'a fait I'honneur de m'inviter a ces fetes et m"a charge de prendre la parole devant quelques mathematiciens americains. C'est un honneur auquel je suis tres sensible, car nous savons en France que les etudes mathematiques se developpent rapidement en Amerique et nous suivons ce mouvement avec une tres vive sympatliie. Votre American Journal of Mathematics compte parmi les journaux periodiques les plus importauts et renferme de remarquables memoires, et je lis toujours pour ma part avec grand profit et interet le Bulletin de la Societe mathematique americaine, excellente revue historique et critique qui tient ses lecteurs au courant des travaux les plus recents. J'ai appris aussi que cette Societe allait fonder un nouveau recueil destine a des memoires plus etendus ; je ne doute pas qu'il ne soit appele a un brillant avenir. Dans les trois causeries que nous aliens avoir ensemble, je ne puis songer a aborder un sujet special qui demanderait une pre- paration particuliere. Nous allons rester dans les generalites et jeter un rapide coup d'oeil sur I'extension de quelques notions mathematiques et en particulier, de I'idee de fonction depuis un siecle. I. Toute la science mathematique repose sur I'idee de fonction c'est a dire de dependance entre deux ou plusieurs grandeurs, dont I'etude con- stitue le principal objet de I'analyse. II a fallu longtemps avant qu'on se rendit compte de I'etendue extraordinaire de cette notion ; c'est la d'ailleurs une circonstance qui a ete tres heureuse pour les jirogres de la Science. Si Newton et Leibnitz avaient pense que les fonctions continues n'ont pas necessairement une derivee, ce qui est le cas general, le calcul 207' 208 Emile Picard : differentiel n'aurait pas pris naissance ; de meme les idees inexactes de Lagrange sur la possibilite des developpements en series de Taylor ont rendu d'immenses services. Sans vouloir trop geueraliser, on pent dire que I'erreur est quelquefois utile, et que, dans les epoques vraiment creatrices, une verite incomplete ou approchee pent etre plus feconde que la meme verite accompagnee des restrictions necessaires ; I'histoire de la science confirme plus d'une fois cette remarque et, pour rappeler encore Newton, il est heureux qu'il ait eu au debut de ses recherches pleine confiance dans les lois de Kepler. Les geometres du siecle dernier, sans remonter plus haut, ne raffinaient pas sur I'idee de fonction ; pour eux, une fonction d'une variable est une fonction qu'on pent representer par une courbe formant un trait continu ; ce sont ces fonctions qu'Euler appelait functiones continuce. La question de la representation d'une fonction arbitraire sous une forme aualytique dans laquelle interviennent seulement les operations fondamentales de I'arithmetique effectuees un nombre fini ou infini de fois, se posa, semble-t-il pour la premiere fois a propos du probleme des cordes vibrantes. D'Alembert avait donne I'integrale de I'equation sous la forme /(a; + aJ) + 0), en designant par k une constante fixe, a un grand caractere de generalite, et il en est de meme du theoreme de M. Camille Jordan sur la legitimite du developpement pour les fonctions a variation bornee. Le memoire de Riemann sur les series trigonometriques est celebre dans I'histoire de ces series ; on pent dire en deux mots, pour le carac- teriser, qu'il abandonne le point de vue de Dirichlet, et qu'au lieu de chercher des conditions suffisantes, sa principale preoccupation est de trouver des conditions necessaires. A un autre point de vue encore, le memoire de Riemann marque une date parce qu'il continue cette revision des principes du calcul infinitesimal commencee par Abel et Cauchy ; la distinction entre les fonctions iutegrables et les fonctions non integrables y apparait pour la premiere fois, et on peut dire qu'il resulte des travaux de Riemann qu'il y a des fonctions continues n'ayant pas de derivees. On doit a M. G. Cantor la reponse a une question importante : une f onction peut-elle etre representee entre et 2 tt de plusieurs manieres par une serie trigonometrique ? En d'autres termes, zero peut-il etre represente par un developpement trigonometrique ou les coefficients ne soient pas tous nuls ? Independamment du resultat lui-meme, le memoire de M. Cantor est digne d'interet parce que, dans une question depuis longtemps posee, des notions concernant les ensembles de points viennent jouer un role utile. Etant donne un ensemble de points entre et 2 tt, M. Cantor appelle ensemble derive I'ensemble de ses points limites, et on peut definir ainsi de proche en proche les derivees successives d'un ensemble. Si la derivee W"" d'un ensemble se reduit a un nombre limite de points, I'ensemble sera dit de la n""" espece. M. Cantor etablit que si dans I'intervale (0, 2 7r) une serie trigonometrique est nulle pour toutes les valeurs de a; a I'exception de celles qui correspondent aux points d'un ensemble d'espece n, pour lequel on ne salt rien de la serie tous les coefficients seront nuls. II. J'ai insiste, peut-etre un pen longuement, sur les series trigone metriques. Independamment de leur importance dans les applications et particulierement en physique mathematique, elles ont joue un role considerable dans revolution de la notion de fonction ; c'est leur etude qui a appele I'attention sur des circonstances, qui ne nous etonnent plus aujourd'hui, mais qui paraissaient jadis invraisemblables, comme, par Premiere Conference. 211 exemple, ce fait que la limite vers laquelle tend una serie de fonctions continues pent n'etre pas egale a la valeur de la serie en ce point. Les precautions a prendre dans la derivation des series ont ete aussi sug- gerees par les series trigonometriques ; on peut faire remonter a cet exemple les nombreuses recherches effectuees depuis Cauchy sur la deri- vation et I'integration des series, auxquelles M. Osgood ajoutait il y a quelques annees un important complement dans son memoire sur la convergence non-uniforme. Le developpement d'une fonction en serie trigonometrique est aussi le type le plus simple de developpemeuts tres generaux qui se presen- tent dans les applications ; Fourier, ici encore, a ete un precurseur. L'etude du refroidissement d'une sphere, en supposant que la tempera- ture ne depende que du temps et de la distance au centre, I'a conduit a un developpement ou, au lieu des lignes trigonometriques des multiples a;, 2 a;, •••, nx de la variable, figurant les lignes trigonometriques de ajz, ajX, •■•, a„a;, les a designant les racines en nombre infini d'une certaine equation transcendante, et il a asquisse une theorie de ces sortes de developpements. Cette etude a et6 reprise par Cauchy dans plusieurs memoires qui forment une des applications les plus remarquables de ce que le grand analyste appelait la calcul des residus. Sous des conditions tres generales relatives a I'equation transcendante, Cauchy a demontre en toute rigueur la legitimite des develojjpements pour une fonction satisfaisant d'aillaurs aux conditions da Dirichlet, et ainsi se sent trouves cousiderablement generalises les resultats du memoire classique de I'il- lustre geometre allemand. D'autres develoj^pements d'un caractere encore plus general se ren- contrent en physique mathematique, et ont fait I'objet des travaux de Poisson, de Sturm et de Liouville et da bien d'autres, mais ici se pre- sentent, au point de vue de la rigueur complete, des difficultes que Ton a reussi a surmonter que dans un petit nombre de cas. Je citerai seule- ment I'axample tres simple du refroidissement d'un mur indefini dont les faces extremes sont maintenues a la temperature zero ; on suppose d'aiUeurs que la chaleur specifique soit une fonction de I'abscisse x cor- respondant a chaque tranche, de telle sorte que Ton a pour la tempera- ture V I'equation aux derivees partialles 212 Emile Picard: oii Aix) est ime fonction continue et positive de x dans I'intervalle (a, J) de I'epaisseur du mur. Envisageons I'equation lineaire ordinaire et les valeurs positives de k en nombre infini, A^j, ^j' "•i ^n "'i pour les- quelles il existe une integrale de I'equation precedente s'annulant en a et 6. A chaque valeur de A;j correspond une integrale yi(x') de cette equation (determinee a une constante pres), et le probleme qui se pre- sente est de developper uue fonction fix) s'annulant en a et 5 sous la forme La demonstration rigoureuse de ce developpement resulte des der- nieres recherches de M. Stekloff, s'aidant des travaux anterieurs de M. Poincare sur les equations de la physique mathematique. II semble bien qu'U soit indispensable pour I'entiere rigueur de supposer que/(a;) a des derivees des deux premiers ordres ; nous sommes loin d'atteindre ici a la generalite des conditions de Dirichlet pour le developpement en serie trigonometrique qui rentre d'ailleurs comme cas particulier (celui ou A(x) est une constante) dans le cas precedent. m. L'histoire des developpements en series que je viens de retracer rapide- ment nous donue un remarquable exemple de I'intime solidarite qui unit a certains moments I'analyse pure et les mathematiques appliquees. En plus d'une occasion, ce sont ceUes-ci qui out donne I'impulsion en posant les problemes, et c'est un fait assurement remarquable que des questions concernant les cordes vibrantes ou la propagation de la cbaleur aient conduit les geometres a approfondir la notion si complexe de fonction. L'histoire de la science mathematique offrirait d'ailleurs des le debut des exemples analogues ; nos facultes d'abstraction ne trouvent primi- tivement a s'exercer qu'en partant de certains faits concrets, et c'est sans doute en reflechissant aux procedes empiriques des praticiens egyptiens leurs predecesseurs que les premiers geometres grecs creerent la science geometrique. Mais ces vues risqueraient de m'en trainer trop loin. Je tiens seulement a ajouter qu'il ne faudrait pas professer une opinion trop systematique sur cette marche parallele de la theorie pure et des applica- tions, comme le faisait avec Laplace, Fourier, Poisson la brillante ecole Premiere Conference. 213 fran^aise de physique mathematique du commencement de ce siecle. Pour eux, I'analyse pure n'etait que Finstruiuent, et Fourier, en annon- gant a I'Academie des sciences, les travaux de Jacobi, disait que les ques- tions de la philosophic naturelle doivent etre le principal objet des meditations des geometres. " On doit desirer, ajoutait-il, que les personnes les plus propres a perfectionner la science du calcul dirigent leurs tra- vaux vers ces hautes applications si necessaires au progres de rintelli- gence humaine." Ce desir tres legitime ne doit pas etre exclusif ; ce serait meconnaitre d'abord la valeur philosophique et artistique des mathematiques ; de plus des speculations theoriques sont restees pendant longtemps eloignees de toute application, quand un moment est venu ou elles ont pu etre utilisees. Ou n"en peut pas citer d'exemple plus memorable que le concept des sections coniques elabore par les geometres grecs, qui resta inutilise pendant deux mille ans, jusqu'au jour oii Kepler s'en servit dans I'etude de la planete Mars. Les questions s'epuisent pour un temps, et il n'est pas bon que tous les chercheurs marchent dans la meme voie. Peu d'annees apres que Fourier ecrivait les lignes que je viens de rappeler, apparaissait Evariste Galois qui aurait, s'il avait vecu davantage, retabli I'equilibre en ramenant les recherches vers les regions les plus elevees de la theorie pure, et ce fut un malheur irre- parable pour la science frangaise que la mort de Galois, dont le genie allait exercer une action si profonde sur les parties les plus varices des mathematiques. Avec cette digression, nous semblons etre bien loin, messieurs, de notre promenade a travers I'idee de fonction depuis le commencement de ce siecle. Elle n'etait cependant pas inutile, pour montrer qu'un moment devait arriver ou les speculations sur la theorie des fonctions de variables reelles se poursuivraient sans souci immediat des applications et pren- draient de plus en plus un caractere philosophique. Nous avons dejii dit qu'il resultait indirectement des travaux de Riemann qu'une fonction con- tinue n'a pas necessairement une derivee. Weierstrass donna le premier exemple d'une fonction continue n'ayant de derivee pour aucune valeur de la variable, et il fit connaitre au sujet des fonctions continues une proposition qui nous ramene aux developpements en series, mais ici les termes sont des polynomes. D'apres Weierstrass, toute fonction con- tinue dans un intervalle peut etre developpee en une serie de polynomes qui est absolument et uniformement convergente dans cet intervalle. La demonstration de I'illustre geometre est tres compliquee ; elle prend 214 Emile Picard : comme point de depart une integrale consideree par Fourier dans la theorie de la chaleur, qui permet d'obtenir la fonction consideree comme la limite d'une fonction transcendante entiere dependant d'un parametre, quand celui-ci tend vers zero. C'est de la que Weierstrass deduit la possibilite de representer d'une nianiere approchee par un polynome toute fonction continue dans un intervalle fini, d'oii se tire alors de suite le resultat enonce. On pent arriver beaucoup plus rapidement au theoreme de Weierstrass en partant de I'integrale classique de Poisson dans la theorie des series trigonometriques ; elle montre facilemeut que la fonction, supposee definie dans un intervalle moindre que 2 tt, peut-etre representee avec telle approximation que Ton voudra par une serie limitee de Fourier, et on passe de suite a une representation approchee par un polynome ; celle demonstration s'etend a des fonctions continues d'un nombre quelconque de variables. M. Volterra est arrive aussi trea simplement au theoreme qui nous occupe en remarqiiant qu'une fonction continue est representable avec telle approximation qu'on voudra par une ligne polygonale convenable ; celle-ci conduit a une serie de Fourier uniformement convergente, et en la reduisant a un nombre suffisamment grand mais limite de termes on retombe sur le resultat indique plus haut. Le theoreme de Weierstrass presente un reel interet philosophique, en meme temps qu'il pent avoir quelque utilite au point de vue du cal- cul pratique ; on en a aussi quelquefois fait usage pour la demonstration de certaines propositions. Les developpements en series de polynomes speciaux sont d'un grand interet, mais ils ne peuvent s'appliquer qu'a des fonctions satisfaisant a des conditions particulieres. Ainsi, dans son memoire sur I'ap- proximation des fonctions de tres grands nombres, M. Darboux a etudie les developpements d'une fonction suivant les polynomes de Jacobi provenant de la serie hypergeometrique. Les conditions sont encore celles de Dirichlet ; pareillement aussi dans le cas ou la fonction devient infinie, elle doit rester integrable. H y a cependent une difference quand la fonction devient infinie pour les points extremes. Dans le cas des polynomes de Legendre, une fonction qui deviendrait infinie d'un ordre egal ou superieur a -I pour x = ±1 ne serait pas developpable, quoique les coefficients aient un sens. IV. Si nous revenons aux fonctions prises dans toute leur generalite, on reconnait vite la necessite d'etablir avec un soin extreme certaines pro- Premiere Conference. 215 positions que Ton accorde aisemeut pour les fonctions usuelles. C'est ce qu'avait deja reconnu Cauchy dans son Analyse algebrique ; les travaux de Hankel, le memoire de M. Darboux sur les fonctions discontinues, le beau livre de M. Dini et les etudes plus recentes des geometres italiens montrent bien les precautions necessaires dans ce genre de recherches. Ainsi, une fonction de deux variables reelles peut etre continue par rap- port a a; et par rapport a y sans etre continue par rapport a I'ensemble des deux variables, comme M. Dini en a indique des exemples. Parmi les travaux les plus recents sur ces questions delicates, je m'arreterai un instant sur un memoire de M. Baire qui renferme de curieux resultats. L'auteur a reussi a trouver la condition necessaire et suffisante pour qu'une fonction f(x') d'une variable reeUe puisse etre representee par une serie simple de polynomes ; I'enonce suppose certaiues notions sur la discon- tinuite d'une fonction par rapport a un ensemble de points : une fonction peut etre ponctuellement ou totalement discontinue par rapport a cet ensemble. La condition obtenne est que la fonction soit ponctuellement discontinue par rapport a tout ensemble parfait. M. Baire se pose aussi une question singuliere sur les equations lineaires aux derivees partielles. Envisageons I'equation Si je vous demandais quelles sont les fonctions satisfaisant a cette equation, vous me repondriez sans doute que les fonctions de x — y repondent seules a la question. M. Baire n'en est pas absolument sur ; il remarque que la theorie du changement de variables suppose la con- tinuite des derivees qu'on emploie ; si on suppose seulement I'existence des derivees -^ et -^ de la fonction cherchee /, on ne peut pas faire le dx dy ./' 1 f changement de variables classique. II faut une analyse delicate pour etablir que la fonction /, supposee continue par rapport a I'ensemble des variables x et y, et satisfaisant a (1) est une fonction Ae x — y \ la conclu- sion reste douteuse si / est seulement continue par rapport a x et par rapport a y. Au point de vue geometrique les recherches generales sur les fonctions ne sont pas non plus sans interet ; elles nous apprennent a nous defier de nos conceptions les plus simples. Quoi de plus simjjle semble-t-il qu'une courbe dont les coordonnees a; et^ sont des fonctions continues d'un para- metre t variant entre a et h. M. Peano a cependant montre qu'on peut 216 Emile Picard : choisir ces deux fonctious de telle sorte que, quand t varie entre a et 5, le point (x, y) puisse prendre une position quelconque dans un rectangle. A certains points (x, y) pourront correspondre d'ailleurs, dans I'exemple de M. Peano, deux ou quatre valeurs de t. Ce resultat est au premier abord deconcertant ; il derange nos idees sur les surfaces et sur les courbes. Voici encore un resultat singulier obtenu tout recemment par M. Lebegue ; il y a d'autres surfaces que les surfaces developpables qui sont applicables sur un jjlan. On peut a I'aide de fonctions continues obtenir des surfaces correspondant a un plan de telle sorte que toute ligne rectifiable du plan ait pour correspondante une ligne rectifiable de la surface, et la surface n'est cependant pas reglee. De tels exemples montrent la subtilite des recherches auxquelles doivent se livrer aujourd'hui ceux qui veulent approfondir la notion de fonction prise dans son extreme generalite. Ces etudes sont en bien des points intimement liees aux speculations sur la notion meme de nombre. Nous rejoignons ici une ecole de philosophic mathematique qui s'est briUamment developpee depuis quelque trente ans, ecole qui se livre a une minutieuse analyse sur la nature du nombre. On ne peut s'empecher d'etre frappe du nombre considerable de publications parues dans ces dernieres annees et se rapportant a cette mathematique philosophique ; elles sont bien en accord avec les tendances generales de I'epoque oii nous vivons, et oil I'esprit humaiu applique dans des directions variees une critique de plus en plus penetrante. Ces speculations raffinees ont meme penetre dans I'enseignement elementaire, ce qui est a mon avis tres regrettable. Mais il ne s'agit pas ici d'enseignement ; je ne recherche pas non plus I'interet que ces etudes presentent pour le philosophe ; il me parait tres reel, et on doit souhaiter que de jeunes philosophes s'engagent dans cette direction apres s'etre inities serieusement aux mathematiques. Je ne veux me placer qu'au point de vue de la mathematique. De bons esprits contestent que les speculations dont je parle aient quelque impor- tance pour les mathematiques positives et ils craignent de voir beaucoup de talent depense dans des recherches steriles. Je comprends tres bien leurs craintes mais je ne partage pas entierement leur avis. II y a lieu sans doute de faire des distinctions. Certaines questions sont d'un interet pureraent philosophique et n'auront jamais vraisemblablement la moindre utilite pour les mathematiques, corame, par exemple, de savoir si la priorite appartient au nombre cardinal ou au nombre ordinal, c'est a dire si I'idee de nombre proprement dit est anterieur a celle de rang ou si c'est Premiere Conference. 217 I'inverse. Mais dans d'autres cas, il n'en est plus de meme ; ainsi il est vraisemblable que la theorie des ensembles de M. Cantor, que nous avons deja rencontree deux fois sur notre cliemin, est a la veille de jouer un role utile dans des problemes qui n'ont pas ete poses expres pour etre une application de la theorie. Ne regrettons done pas cet effort hardi sur I'idee de nombre et sur celle de fonction, car la theorie des fonctions de variables reelles est la veritable base de I'analyse mathematique. II faut bien, il est vrai, reconnaitre que la notion generale de fonction est tres vague, et nous ne pouvons obtenir des resultats de quelque etendue qu'en faisant des hypotheses particulieres. Qu'est ce qui a guide plus ou moins consciemment dans le choix de ces hypotheses ? II resulte de ce que nous avons dit sur les rapports entre I'analyse et les applica- tions aux phenomenes naturels, que celles-ci ont plus d'une fois guide le mathematicien dans son choix. Une hypothese essentielle a ete celle de la continuite. Suivant le vieil adage "natura non facit saltus" nous avons le sentiment, on pourrait dire la croyance, que dans la nature il n'y a pas de place pour la discontinuite. II est utile quelquefois de conserver le discontinu dans nos calculs, par exemple quand nous regardons comme nuUe la duree du choc en mecanique rationuelle, ou quand nous reduisons a une surface les couches de passage dans plusieurs questions de physique; mais nous savons que, pour si petite qu'elle soit, les chocs ont une certaine duree et les physicians nous ont appris a mesurer I'epaisseur des couches oil se produisent dans plusieurs phenomenes des variations tres rapides. L'idee de derivee s'impose deja moins ; elle repond cependant au senti- ment confus de la rapidite plus ou moins grande avec laquelle s'accomplit tel ou tel phenomene. L'hypothese relative a la possibilite de la deriva- tion d'une fonction a done une origine analogue a celle de la continuite. Je ne veux pas dire qu'au point de vue du nombre l'idee de continuite soit aussi claire au fond qu'elle en a Fair, mais il ne s'agit ici que de la notion du continu physique tiree des donnees brutes des sens. Dans d'autres cas, on ne voit pas de cause du meme ordre dans la particularite imposee a la fonction ; il en est ainsi, ce me semble, pour la propriete des fonctions dites analytiques c'est a dire des fonctions qui dans le voisinage d'une valeur arbitraire de la variable peuvent etre developpees en series de Tajdor. Les fonctions etudiees les premieres, comme les fonctions rationnelles, I'exponentielle, les lignes trigonome- 218 Emile Picard: triques, jouissant de cette propriete, I'attention se sera sans doute trouvee appelee sur elle ; et ensuite la facilite avec laquelle cette hypothese a permis d'aborder certaines questions a fait acquerir aux fouctions analy- tiques une importance considerable. C'est done a leur commodite dans nos calculs qu'elles doivent le grand role qu'elles jouent. On ne salt pas d'ailleurs, pour une fonction definie seulement pour les valeurs reelles de la variable, quelles sont les conditions de legitimite du developpement en serie de Taylor. Une fonction de x pent avoir des derivees de tout ordre pour toute valeur de la variable, et n'etre cependant pas developpable. On doit a M. Borel un resultat remarquable con- cernant les fonctions d'une variable reelle definie dans un certain inter- valle et ayant dans cet intervalle des derivees de tout ordre. Si I'intervalle est (— 7r, + tt), la fonction peut etre representee par un developpement de la forme (-4„a;" + -B„ cos nx + On sin nx). Ces diverses remarques m'amenent a dire un mot d'une ecole de geo- metres qui ne veulent rien voir en dehors des fonctions analytiques, et d'une maniere plus generale de I'importance, peut-etre exageree, qu'a prise dans les travaux modernes la theorie des fonctions analytiques. C'est mutiler singulieremeut I'analyse que de vouloir se borner a des deve- loppements aussi particuliers que les series entieres, alors que Ton peut former tant de developpements d'une autre nature qui ne peuvent jamais etre representees par de telles series. Sans doute, les fonctions les plus usuelles sont analytiques, et on pourrait nous demander de citer des exemples dans la solution desquels interviennent des fonctions non analy- tiques, tandis que les donnees sont analytiques. lis ne sont pas courants ; ce sont les equations aux derivees partielles qui probablement les four- niront le plus facilement. Le suivant, du a M. Borel, me parait digne d'etre signale. Envisageons I'equation — 7. — a — - = f(x, V), dx^ dy^ -^ ^ ^-^ oh a est une irrationnelle convenablement choisie, et /(a;, ^) une certaine fonction analytique de a; et «/ de periode 2 ir pour x et «/. Pour I'equation de cette forme citee par M. Borel, il y a une seule solution periodique et cette solution n'est pas analytique. Soit a un nombre incommensurable Premiere Conference. 219 tel que — i etant Tune quelconque des reduites du developpement de a en fraction continue, on ait I rrii — n^a \ < 6-"'^-"" on forme 4) (x, «/) = 2 a"'}"* cos (m?x) cos in?y') (a < 1, 6 < 1). C'est une fonction non analytique. Posons d'autre part la fonction i/r sera analytique. Done si on prend I'equation (1) a priori et qu'on cherche une solution periodique, en x et y, il n'y en a qu'une ; c'est (^ qui n'est pas analytique. C'est encoi-e, en se plagant a un autre point de vue, qu'il parait mauvais de reduire la theorie des fonctions a la theorie des fonctions analytiques. 11 y a de nombreuses questions, ou le fait pour les donnees d'etre analytiques ne donne aucune facilite pour la solution, et ou on risque, en portant trop son attention sur cette nature des donnees, de cherclier la solution dans des voies sans issues. Pour le probleme du refroidissement de la barre dont je parlais plus haut, qu'importe que les fonctions donnees ACx) et f^x) soient ou non analytiques? Ce n'est pas tout ; il y a un dernier point sur lequel je tiens a insister. II peut arriver que la circonstance d'avoir a fairs a des fonctions analytiques conduise a une solution, mais il se peut que celle-ci ne se presente pas sous la forme la plus favorable, forme a laquelle ou arrive au contraire en faisant ab- straction de la nature analytique des donnees. La theorie des equations differentielles fournirait des exemples a I'appui de cette assertion ; bornons nous a citer le theoreme fondamental du Calcul Integral relatif a I'ex- istence de I'integrale de I'equation differentieUe -^=f{x,y^. Ce sont les demonstrations ne supposant pas que la fonction / soit analytique, qui donnent le plus grand intervalle comme region ou I'integrale est certainement determinee ; I'analyste, qui suppose analytique la fonction reelle f(x^ y) et veut n'envisager que des series entieres, est conduit par son mode de demonstration a un domaine plus restreint. J'ai simplement eu poui- but dans ce qui precede de montrer qu'il ne faut pas restreindre systematiquement la notion de fonction. D'une maniere generale, admirons des systemes tres bien ordonnes, mais mefions nous un peu de leur apparence scolastique, qui risque d'etouffer I'esprit 220 Emile Picard: d'invention. II ne s'agit pas, bien entendu, de nier la grande importance actuelle de la theorie des fonctions analytiques, mais il ne faut pas oublier qu'elles ne forment qu'une classe tres particuliere de fonctions, et on doit souhaiter qu'un jour vienne ou les mathematicians elaborent des theories de plus en plus comprehensives ; c'est ce qui arrivera peut-etre au siecle prochain, si I'idee de fonction, dont je vous ai bien incompletement esquisse Thistoire, continue son evolution. Mais, pour le moment nous sommes encore au dix-neuvieme siecle ; j'aurai I'occasion demain et apres demain de faire amende honorable aux fonctions anal3^iques, qui depuis trente ans out fait, comme vous savez, I'objet de travaux considerables. "VT. Nous venons de voir les vastes perspectives qu'ouvre I'extension de plus en plus grande de la notion de fonction. II faudra certainement montrer dans cette voie beaucoup de prudence, et ne pas entreprendre avant I'heure des recherches qui resteraient steriles ; mais il n'est pas douteux qu'un jour viendra ou Tanalyste sentira le besoin d'etendre le domaine de ses recherches. L'extension de I'idee de fonction n'est pas la seule qu'aient poursuivie en ce siecle les mathematiciens qui s'interessent aux principes de la science ; la question des quantites complexes a vive- ment excite I'interet, d'autant plus qu'une certaine obscurite planait sur elle, qu'entrainait le mot un peu mysterieux de quantites imaginaires. Le sujet ne presente plus rien aujourd'hui de mysterieux. Dans un memoire publie en 1884 Weierstrass a developpe une theorie des nombres complexes. II suppose que Ton considere des nombres de la forme ^A + ^2'^'i H Ha;„e„, ou les X sont des nombres reels on imaginaires ordinaires. Les e sont de purs symboles. On fait I'hypothese que la somme, la difference, le pro- duit et le quotient de deux nombres de Tensemble font eux-memes partie de cet ensemble. Les produits epe,j (j», q = 1, 2, ■••, «) sont done des expressions -2^p, g lineaires et homogenes en gj, e^, •••, e„ qui jouent le role essentiel dans la theorie. Weierstrass suj^pose de plus que les theoremes dits commutatif et assoeiatif subsistent taut pour I'addition que pour la multiplication. Pour I'addition, ils sont verifies d'eux-memes ; pour la multiplication, ils s'expriment par les egalites ah = ha, (,ah) ■ c ■■ (hc\ Premiere Conference. 221 a, 5, c etant trois nombres quelconques de I'ensemble. Ces conditions conduisent a certaines relations entre les coefficients des formes lineaires Ep^q. A tout systeme de formes Ej,^^ verifiant ces conditions corre- spondra un ensemble de nombres complexes. Les nombres complexes que nous venons de definir different seulement en un point des nombres complexes ordinaires. Quand n est superieur a deux, il peut exister des nombres differents de zero dont le produit par certains autres nombres est nul. Weierstrass appelle ces nombres des diviseurs de zero. M. Dedekind a montre qu'en general les calculs avec ces nombres complexes se ramenaient aux calculs de I'algebre ordinaire ; d'une maniere plus precise, si le carre d"un nombre ne peut etre nul sans que ce nombre soit nul, on peut aux n unites complexes primitives substituer n autres unites (le determinant de la substitution n'etant pas nul) de telle sorte que pour ces nouvelles vmites e\, e\, •••, e'„, on ait d'ou Ton conclut que les calculs relatifs aux nombres complexes prece- dents se ramenent a des calculs relatifs aux nombres reels ou complexes ordinaires. Nous avons admis que les lois commutative et associative subsis- taient dans I'algebre precedente. On s'est place a un point de VTie plus general en supposant que, seule, la loi associative subsistait [c'est a dire (a6)(? = a(hc)'\. On a alors une algebre beaucoup plus generale ; celle-ci est completement determinee par le systeme des expressions lineaires Ep^q. Un exemple celebre d'un systeme a quatre unites fij, e^, e^, e^ est fourni par les quaternions d'Hamilton gj = 1, avec les relations ^2 = h eg = j, e^ = k, t^=f = k^ = -l ij =- ji = k jk = - kj = i /fct=- ik=j. Une remarque tres interessante de M. Poincare ramene toute la theorie des quantites complexes a une question concernant la theorie des groupes. Elle consiste en ce qu'a chaque systeme d'unites complexes correspond un groupe continu (au sens de Lie) de substitutions lineaires 222 Emile Picard : a n variables, dont les coefficients sont des fonctions lineaires de n para- metres arbitraires, et inversement. Cette idee a ete approfondie par M. Scheffers qui a ete ainsi conduit a partager les nombres complexes en deux classes, suivant que le groupe qui leur correspond est integrable ou non integrable. A cette derniere classe appartient le groupe corres- pondant aux quaternions, et ceux-ci sont les representants les plus simples de cette categoric de nombres complexes. Le rapprochement entre la theorie des groupes de Lie et les nombres complexes fait disparaitre le mystere qui semblait planer sur ceux-ci, et la veritable origine des symboles est ainsi bien mise en evidence. On pent se demander si ce symbolisme est susceptible d'accroitre la puissance de I'Analyse. En France, les geometres qui s'intei'essent a ces calculs sont tres peu nom- breux ; je sais qu'au contraire en Angleterre et, je crois aussi, dans ce pays les quaternions sont tres apprecies. Je ne les ai pas assez manies moi-meme, pour me rendre compte si leur emploi en mecanique ou en physique mathematique simplifie les calculs d'une maniere tres appre- ciable ; il y a probablement la surtout une affaire d'habitude. Le point vraiment interessant serait de savoir si ces quantites complexes presen- teront un jour quelque interet pour 1' analyse generale, comme il arrive pour les imaginaires ordinaires. Les essais tentes jusqu'ici dans cette voie ne paraissent pas avoir ete heureux ; mais, maintenant que le lien avec la theorie des groupes est completement mis en evidence, il n'est pas impossible que de nouvelles tentatives n'aboutissent a quelque resultat interessant. Les idees de nombres reel ou complexe, la notion de fonction sont a la base meme de I'aualyse ; il y a encore une autre notion que le travail mathematique de ce siecle a conduit a elargir considerablement. L'idee d'espace forme la matiere meme de la geometric ; elle aussi a ete sou- mise a une critique penetrante qui a renouvele les bases de la geometric. Je n'en referai pas I'histoire depuis Gauss, Bolyai et Lobatschevski, his- toire tres souvent racontee, ni ne prendrai parti dans les querelles que se font encore a ce sujet les philosophes. Je veux dire seulement un mot de I'interet qu'ont cu pour les mathematiques les speculations sur la nature de respace. Dans le memoire celebre de Riemann, apparaissent pour la premiere fois les notions relatives a la courbure de I'espace dans les differentes directions, c'est-a-dire les — -^ — -^ — - fonctions invariantes caracteristiques d'une multiplicite a n dimensions ; une vive impulsion Premiere Conference. 223 a ete ainsi donnee a la theorie des formes quadratiques de differentielles. Pour ne citer qu'un exemple, j'indiquerai seulement la forme qui donne le carre de I'element d'arc dans la geometrie de Lobatchevski ; et il est interessant de rappeler le role qu'elle a joue dans las recherches de M. Poincare sur la formation des groupes fuchsiens. Apres Rie- mann, Helmholtz posa la question sur un autre terrain: son idee fonda- mentale consiste a porter I'attention sur I'ensemble des mouvements possibles dans I'espace dont on fait I'etude. Le grand physicien traitait ainsi par avance de problemes se rattachant a la theorie des groupes. Celle-ci n'etait pas encore creee a Fepoque oii Helmholtz ecrivait son memoire ; il a commis quelques erreurs apres tout secondaires, mais il n'en a pas moins la gloire d'avoir le premier regarde une geometrie comme I'etude d'un groupe. Lea recherches d'Helmholtz furent reprises completement par Lie ; elles lui offraient une magnifique occasion d'ap- pliquer son admirable theorie des groupes de transformations. Dans ces etudes, I'espace est a priori regarde comme une multiplicite, et, en prenant le cas de trois dimensions, un point est defini par trois quantites (x, y, z). Un mouvement dans I'espace n'est autre chose qu'ime trans- formation a;' =f(p!, y, z), y' = (.^, y^ 2)1 «' = V^(^. y. z) valable pour une portion de I'espace. On suppose que tous les mouve- ments possibles forment un groupe a six parametres, qu'ils laissent invariable une fonction des eoordonnes de deux points quelconques, qu'enfin le mouvement libre soit possible, comrae disait Helmholtz. Lie demontre alors que I'espace euclidien et les espaces non euclidiens sont les seuls qui satisfassent a ces conditions. Au point de vue ou s'est place Lie, I'etude des principes de la geometrie pent etre regardee comme epuisee, mais il se borne a considerer une petite portion de I'espace. Clifford et Klein ont appele I'attention sur la question de la connexite de I'espace qui est extremement interessante ; nous ne savons rien sur la connexite de I'espace oii nous vivons. On peut aussi chercher a appro- fondir le postulat de I'espace regarde comme une multiplicite, et sub- ordonner la conception metrique de I'espace a la conception projective avec von Staudt, Cayley et Klein ; mais je dois me contenter de rappeler ces directions diverses. 224 Emile Picard : J'ai seulement, messieurs, voulu montrer dans cette conference quelles perspectives ouvre aux chercheurs I'extension de nos idees sur les fonc- tions, sur le nombre et sur I'espace. Si I'elaboration mathematique est aussi feconde au siecle prochain qu'elle I'a ete en ce siecle, I'analyse differera beaucoup dans cent ans de ce qu'elle est aujourd'hui ; on maniera peut-etre couramment les fonctions les plus extraordinaires, et on verra tres clair dans des espaces ayant beaucoup de dimensions et des connexites elevees. Pour se representer I'etat de la mathematique en I'an 2000, il faudrait I'imagination de I'auteur de "Looking Backward"; il est malheureux que M. Bellamy dans son roman ne nous ait pas parle des mathematiques a cette epoque. Comme I'humanite, s'il faut Ten croire, aura alors beaucoup de loisirs, les mathematiques seront sans doute extremement florissantes et les problemes qui nous arretent aujour- d'hui ne seront plus que des jeux d'enfauts pour nos successeurs. Seconde Cokfebekce. Quelques Vues GenSrales sur la Theorie des Equations DiffSrentielles. Jb voudrais aujourd'hui jeter un coup d'oeil sur la theorie des equa- tions differentielles, qui joue en analyse un role considerable et dont les progres importent vivement a ses applications ; c'est un domaine tres vaste et j'eprouve quelque embarras a faire un choix entre les directions si diverses oii s'est developpee cette theorie. Les geometres du siecle dernier ne paraissent pas s'etre preoccupes d'etablir rigoureusement I'existence des integrales des equations differentielles; ils integraient, quand ils le pouvaient, les equations qui se presentaient dans leurs recherches, sans se soucier de ces theoremes d'existence, comme on dit aujourd'hui, auxquels nous attachons beaucoup d'importance. C'est a Cauchy que Ton doit les premieres recherches precises sur ces questions ; le champ en est tres vaste, et il ne I'a pas parcouru en entier, mais, au moins dans le cas ou les fonctions et les donnees sont analytiques, il a indique les principes qu'ont suivis tous ses continuateurs. Dans les theoremes relatifs a I'existence des integrales, on emploie des methodes differentes suivant que les equations et les donnees sont supposees on non analytiques. I. Pla5ons nous d'abord dans le premier cas, de beaucoup le mieux elabore. L'idee essentielle de Cauchy consiste dans la consideration des Seconde Conference. 225 fonctions majorantes. On salt que les difficultes resident surtout dans la demonstration de la convergence de certaines series entieres que les equations differentielles permettent de former. Cauchy y parvieut par des comparaisons avec d'autres equations facilement integrables. Pour les equations differentielles ordinaires, il n'y avait a faire apres Cauchy que des simplifications de forme, et, pour le cas d'une seule equation aux derivees partielles, quel que soit le nombre des variables, le grand geo- metre avait indique aussi les points essentiels de la demonstration, que Mme. Kovalevski, dans un memoire reste classique, a presentee sous une forme tres simple. Le theoreme fondamental est alors le suivant : Si on a une equation aux derivees partielles d'ordre n relative a une fonction z de p + 1 variables independantes x, x^, •••, Xp et que I'equation con- tienne la derives d'ordre n, • — , une integrale sera en general determinee si on se donne pour x = a les valeurs de z et de ses derivees par rapport a X jusqu'a I'ordre n — 1; ces donnees sont des fonctions holomorphes de OTj, a-j, •••, Xp dans le voisinage de aj, a^, •••, «,• On peut done dire, en s'ap23uyant sur cet enonce que I'integrale generale de Tequation consideree depend de n fonctions de p variables independantes. C'etait un point auquel on tenait beaucoup autrefois de savoir de combien de fonctions arbitraires dependait I'integrale generale d'une equation aux derivees partielles ; certains resultats paradoxaux avaient cependant deja appele I'attention comma les formes diverses de I'integrale generale de I'equation de la chaleur — = -, qui se presentait tantot avec une, tantot avec dx^ dy deux fonctions arbitraires. De tels resultats ne nous etonnent plus aujourd'hui, quand il s'agit comme ici de fonctions analytiques. Nous n'avons qu'a nous rappeler qu'un nombre fini quelconque de fonctions a un nombre quelconque de variables independantes ne presente pas, au point de vue arithmetique, une plus grande generalite qu'une seule fonc- tion d'une seule variable, puisque dans I'un et I'autre cas I'ensemble des coefficients des developpements forme simplement une suite enumerable. Aussi s'explique-t-on que M. Borel ait pu etablir que toute integrale analytique d'une equation aux derivees partielles a coefficients analy- tiques peut etre exprimee a I'aide d'une formule ne renfermant qu'une seule fonction arbitraire d'une variable reelle. Nous venons de considerer une seule equation aux derivees partielles. L'etude des systemes d'equations differentielles presentait de plus grandes 226 Emile Picard : diificultes. Une premiere question est tout d'abord restee longtemps sans reponse ; il etait possible de se demander s'il pouvait exister des systemes qui comprennent un nombre illimite d'equations distinctes c'est a dire ne pouvant pas se deduire par differentiation d'un certain nombre d'entre elles. M. Tresse a etabli qu'un systeme d' equations aux derivees partielles etant defini d'une maniere quelconque, ce systeme est uecessaire- ment limite, c'est a dire qu'il existe un nombre fini s, tel que toutes les equations d'ordre superieur a s, que contient le systeme, se deduisent par de simples differentiations des equations d'ordre egal ou inferieur a s. II importait ensuite de se rendre compte de la nature des elements arbitraires figurant dans I'integrale generale. Mme. Kovalevski n'avait examine que certains systemes composes d'equations en nombre egal a celui des fonctions inconnues et resolubles par rapport aux derivees d'ordre le plus eleve de chacune des fonctions, ces derivees etant relatives a une meme variable x. M. Riquier d'abord, puis M. Delassus ont donne sous des formes differentes la solution du probleme dans le cas general ; M. Delassus arrive par des changements de variables a obtenir une forme canonique completement integrable, et montre que I'integration d'un tel systeme a m variables se rameue a I'integration successive de m systemes de Mme. Kovalevski contenant successivement 1, 2, •••, m variables; c'est en partant de cette propriete qu'on pent demontrer facilement I'existence des integrales analytiques, et determiner les fonctions et constantes initiales en nombre fini dont dependent ces integrales. II semble y avoir eu longtemps chez les mathematiciens quelques hesi- tations sur ce qu'ou devait entendre par integrale generale d'une equation aux derivees partielles. Si Ton se borne aux cas ou il ne figure dans les equations que des elements analytiques, et si Ton n'envisage que les inte- grales analj'tiques, on considere aujourd'hui, conformement a I'opinion de M. Darboux, qu'une integrale est generale, si on pent disposer des arbitraires qui y figurent, fonctions et constantes, de maniere a retrouver les solutions dont les theoremes de Cauchy et de ses successeurs nous ont demontre I'existence. Anterieurement, Ampere s'etait place a un autre point de vue ; dans son grand memoire sur les equations aux differences partielles, il s'exprime ainsi : " Pour qu'une integrale soit generale, il faut qu'il n'en resulte eutre les variables que Ton considere et leurs derivees a I'infini que les relations exprimees par I'equation donnee et par les equa- tions que Ton en deduit en la differentiant." II est bien clair qu'il s'agit de relations ne renfermant aucune des quantites arbitraires qui figurent Seconde Gonf hence. Til dans rintegrale consideree. Les avis etaient partages entre les geometres, et on se demandait s'il y a identite entre la definition d' Ampere et celle de Cauchy. M. Goursat a montre bien nettement, sur differents exemples, qu'une integrale peut etre generale au sens d' Ampere sans etre generale au sens de Cauchy. II ne faudi'ait pas conclure des divers travaux qui precedent, que, tout en envisageant seulement des integrales et des equations analytiques, I'etude des conditions determinant les integrales d'un systeme d'equations aux derivees partielles soit actueUement achevee. Les theoremes generaux indiques font connaitre certaines donnees qui determinent une integrale, mais celle-ci peut etre determinee par une infinite d'autres conditions. II n'est pas douteux que les types a trouver de ces theoremes d'existence sont en nombre infini. Prenons I'exemple tres simple de Tequation ■+ a \- b -—+ cz = \). dx dy dx dy Une integrale est determinee par la condition de se reduire pour x=0 a une fonction donnee de y, et pour y = a une fonction donnee de x : voila un genre de determinations d'une integrale qui ne rentre pas dans les conditions du theoreme general de Cauchy. Les conditions tres variees, qui peuvent determiner les integrales des equations aux dif- ferences partielles appeUent encore de nombreuses recherches. n. Nous venons de nous placer au point de vue de la theorie des fonctions analytiques. Comme je le disais liier, il y a souvent grand interet, non seulement a un point de vue pliilosophique, mais meme en quelque sorte au point de vue pratique, a adopter des hypotheses plus generales. C'est encore a Cauchy que I'on doit pour les equations differentielles ordinaires la demonstration de I'existence des integrales sans supposer les equations analytiques. Sa methode, bien naturelle et bien simple, consiste a regarder les equations differentielles comme limites d'equations aux differences. On peut fairs sur cette methode de Cauchy une remarque tres interessante ; elle est susceptible de fournir des developpemcnts en series des integrales qui convergent tant que les integrales reste7it continues, et laissent continues les coefficients diffSrentiels. En ce sens, elle est 228 Emile Picard: superieure aux autres methodes qui ont ete proposees. Ainsi, pour prendre un exemple, soit le systeme d'equations dxi dt = XiC^x^, x^, — , a;„) (i = 1, 2, • • • n) ou les -X" sont des polynomes. On pent representer les integrales de ce systeme prenant pour ^ = les valeurs x^°, x^, •••, x^ par des deve- loppements de la forme P^(x^, x^, ..., x,\ t) + - + P„ixO, x^\ -, x,\ + - les P etant des polynomes en x.^,x^, ••■,x„° et t, et ces developpements sont convergents tant que les integrales restent des fonctions continues de t. D'autres methodes ont ete proposees pour demontrer I'existence des integrales, comme la methode des approximations successives qui donne pour les series une convergence tres rapide, mais ces series ne convergent pas necessairement dans tout le champ ou les integrales sont continues. Pour une equation differentielle ordinaire d'ordre «, on suppose generalement, quand on veut etablir I'existence des integrales, qu'on se donne pour une valeur de x les valeurs de la fonction et de ses derivees jusqu'a I'ordre w — 1, mais on pourrait prendre beaucoup d'autres don- nees ; et c'est ce qui arrive notamment dans les applications du calcul des variations. Ainsi pour une equation du second ordre, il arrive qu'une integrale soit determinee par les conditions de prendre pour a-^ la valeur 1/q et pour Xj la valeur y^. On a peu travaille jusqu'ici dans cet ordre d'idees, et cependant maintes conditions initiales sont aussi interessantes que celles adoptees dans le theoreme general classique. Les recherches entreprises dans cette voie ont conduit a quelques resultats par I'emploi de methodes d'approximations successives, et on a pu ainsi reeonnaitre des cas singuliers de divergence dans I'emploi de ces methodes d' approximation. Si nous passons maintenant aux equations aux differences partielles, les equations et les donnees n'etant pas necessairement analytiques, nous nous trouvons dans un domaine tres etendu ou on n'a fait que les pre- miers pas. II faut deja quelque soin pour etablir I'existence des inte- grales de I'equation lineaire Seconde Conference. 229 sans supposer que X(^x, y') soit analytique. Pour les equations d'ordre superieur, il n'y a qu'un petit nombre de types pour lesquels on puisse detinir avec precision ce que Ton entend par integrale generale. lis ont generalement pour origine des problemes de geometrie infinitesimale ou de physique mathematique ; les variables et les fonctions restent ici reelles. Prenons, comme exemple, I'equation + a— + 0— + cz = dxdy dx dy ou a, b, sent des fonctions continues de x et y, sur laquelle Riemann a ecrit quelques pages extremement remarquables. Soit un arc de courbe MP tel que toute parallele ii Ox et a Oy le rencontre au plus en dz un point ; nous nous donnons les valeurs de z et — sur cette courbe. II y aura une integrale et une seule, continue ainsi que ses derivees partielles du premier ordre, satisfaisant aux conditions donnees, et elle sera definie dans le rectangle de cotes paralleles aux axes et ayant Met F pour sommets opposes. On voit combien cet enonce est d'une nature plus precise que ceux qui ont ete donnes anterieurement en nous pla^ant au point de vue de la theorie des fonctions analytiques, oii pour une equation comme celle-ci on etablit seulement I'existence d'une solution dans le voisinage d'une courbe, voisinage determine avec tres peu de precision. L'exemple si simple que nous avons choisi montre encore qu'il n'existe pas toujours d'integrale continue ainsi que ses derivees premieres satisfaisant aux conditions donnees sur un arc de courbe ; il en sera ainsi quand sur cet arc il y aura une tangente parallele a I'un des axes. Voici un second exemple dans le meme ordre d'idees ; on pent relativement a I'equation dx'^ dy'^ dz^ se donner les valeurs de m et de 3- pour les points d'un cercle C situe dans le plan z = Zo ; I'integrale ainsi definie est determinee a I'interieur des deux cones de revolution passant par la circonference et de gene- ratrices paralleles a celles du cone 3^ + y^ — z^ = 0. Les conditions determinant une integrale peuvent prendre des formes tres diverses. Ainsi des conditions de continuite sont susceptibles de remplacer certaines donnees : c'est un fait auquel nous sommes tres 230 Emile Picard: habitues, mais qui n'en est pas moins tres remarquable. L'equation du potential a provoque dans cette voie de nombreuses recherches, et le theoreme fondamental auquel Riemann a donne le nom de Dirichlet, apres avoir ete approfondi par Schwarz et Neumann, a encore fait recem- ment I'objet des recherches de M. Poincare. Des problemes analogues ont ete poses et resolus pour un grand nombre d'equations, par example pour l'equation 3% fl^M Su du _ ^ ds? dy^ dx by pour laquelle une integrale continue est determinea par ses valaurs sur un contour ferme dans toute region ou le coefficient c est negatif ; de telles questions ne sont d'ailleurs pas limitees aux equations lineaires. Ces divers exemples caracterisent bien la nature des theoremes d'exis- tence des integrales, quand on ne se place pas au point de vua de la theorie des fonctions analytiques. II y a la un ordre immense de re- cherches egalement interessantes pour la theorie pure et pour les appli- cations de I'analyse. Sans meme aborder de questions entierement nouvelles, que de points seraient a reprendre dans les travaux celebres des physiciens geometres de la premiere moitie du siecle, de Fourier, de Poisson, de Cauchy meme, si on voulait y apporter la rigueur que Ton exiga aujourd'hui en mathematiques. Je dois ajouter d'ailleurs, comme transition entre las deux directions relatives aux generalites sur les equations aux derivees partielles, qu'il existe des classes tres etendues d'equations dont toutes les integrales sont analytiques. Citons les equations lineaires d'ordre n a deux variables independantes : dans una region du plan ou toutes les caracteristiques sont imaginaires, toute integrale bien determinee et continue ainsi que ses derivees partielles jusqu'a I'ordre n est necessairament analytiqua. II y a aussi de nombreuses equations non lineaires ayant toutes leurs integrales analytiques. Je viens de parler des caracteristiques d'une equation ; c'est la un sujet en connexion etroite avec les theoremes generaux d'existence qui viennent de nous occuper. Les caracteristiques sont certaines multipli- cites jouissant de proprietes particulieres relativement a une equation donnee, multiplicites singulieres en ce qu'elles ne definissent pas ime integrale contrairament a ce qui arrive en general pour les multiplicites contenant les memes elements. Tandis qua la notion de caracteristiques Seconde Conference. 231 est aujourd'hui tres nette pour les equations ou systemes d'equations a deux variables independantes, elle a encore besoin d'etre approfondie dans le cas de plus de deux variables. rn. Si, quittant les generalites relatives a I'existence des integrales, nous voulons parler de la recherche effective des integrales et de I'etude d'equations particulieres, Tembarras est grand de tenter des classifica- tions dans un ensemble considerable de travaux, et nous sentons combien nos classements sont toujours defectueux par quelque endroit. Peut-etre pourrait-on tout d'abord distinguer Tancienne ecole mathematique, et le mot "a/iej'ewne" ne veut pas dire qu'elle ne continue pas a prosperer. C'est I'Ecole d'Euler, de Lagrange, de Monge dans son immortel ouvrage sur les applications de I'analyse a la geometric, d' Ampere dans son celebre memoire de 1817 sur les equations aux differences partielles. En France, cette ecole des analystes geometres pour qui les problemes de geometric infinitesimale sont I'occasion de belles recherches anal}i;iques, a pour chef M. Darboux. Ses LcQons sur la Th^orie des surfaces sont aujourd'hui un livre classique qui a rappele I'attention sur des questions quelque temps negligees. Relativement a I'integration effective des equations du second ordre, pendant de longues annees apres la publication du memoire d'Ampere, il n'avait ete rien ajoute d'essentiel a la theorie developpee par le grand geometre. En 1870, M. Darboux publia un memoire renfermant des vues profondes et originales qui est fondamental dans I'liistoire de cette theorie. Depuis cette epoque, divers geometres ont developpe des methodes plus ou moins analogues. M. Goursat vient de rassembler dans un ouvrage considerable les methodes proposees, en y ajoutant ses decou- vertes personnelles sur ces questions difficiles. On peut caracteriser toutes ces recherches, en disant qu'on s'y propose de trouver explicite- ment des integrales avec le plus grand degre possible d'indetermination. Quelquefois, les methodes sont des indications de marche a suivre quand telle circonstance heureuse se presente, et on cherche des classes d'equa- tions pour lesquelles il en soit ainsi ; dans d'autres cas, on renonce au moins temporairement a I'integration complete, et on recherche des solu- tions de plus en plus etendues au moyen de transformations convenables comme, par exemple, celles de M. Bianchi pour I'equation des surfaces a courbure constante. Les idees du grand geometre norvegien, Sophus Lie, dont la science 232 Emile Picard. deplore la perte recente, ont exerce aussi depuis vingt ans une grande influence dans I'etude des equations differentielles sous le point de vue qui nous occupe en ce moment. La theorie des groupes de transforma- tions, une des plus belles creations mathematiques de ce siecle, est venue apporter un element incomparable de classification ; elle a permis de faire une vaste synthese en donnant une origine commune a des notions eparses qui paraissaient sans liens. Je disais tout a I'heure que nos classifications se plient difficilement a la complexite des choses. Certains problemes se trouvent a un confluent, ou se rencontrent I'ancienne Ecole de Monge et d' Ampere et FEcole plus recente qui se rattache a la theorie moderne des fonctions. Monge avait integre I'equation des surfaces minima, et c'est la un de ses titres de gloire. Ses formules ont ete transformees par Weierstrass, et alors a apparu le lien entre la theorie des fonctions d'une variable complexe et la theorie des surfaces minima. Un probleme appelle vivement I'attention dans cette theorie : c'est le pi'obleme de Plateau relatif aux surfaces minima passant par un contour donne. II a ete resolu seulement dans des cas tres speciaux ; je crois qu'en exercant la sagacite des analystes il sera quelque jour I'ocGasion de progres importants dans I'analyse generale. IV. J'ai surtout parle jusqu'ici des equations aux derivees partielles. La theorie des equations differentielles ordinaires est plus speciale, d'autant que quelques uns ont une tendance a la regarder comme un chapitre de la theorie des fonctions analytiques. Apres les remarques que j'ai faites hier, je n'ai pas besoiu d'ajouter que ce n'est pas la mon opinion ; je vous ai indique plusieurs problemes qui ne relevent en rien de la theorie des fonctions analytiques, et il me suffira de citer encore I'extension des idees de Galois aux equations differentielles. Ceci dit, il n'est pas douteux que les progres de la theorie des fonctions analytiques ont exerce la plus heureuse influence sur certains points de la theorie des equations diffe- rentielles ordinaires. Je ne fei'ai que rappeler le memoire celebre de Puiseux sur les fonctions algebriques, dans lequel etudiant a un point de nouveau les plus simples des equations differentielles a savoir les quadratures, il revele I'origine de la periodicite des integrales de differen- tielles algebriques. Les recherches de Briot et Bouquet ne sont pas moins classiques ; les auteurs y etudient les circonstances singulieres qui peuvent se presenter dans une equation du premier ordre quand le coeffi- Seconde Conference. 233 cient differentiel devient infini ou indetermine. II faut se reporter a pres de cinquante ans en arriere pour bien juger ce memoire, oil pour la premiere fois est mis en evidence le role des points singuliers dans I'etude des fonctions ; ces notions nous sont bien familieres aujourd'hui, mais nous ne devons pas oublier que ce sont les memoires de Puiseux et de Briot et Bouquet qui en ont montre la haute importance. II semble que le memoire de Briot et Bouquet aurait du etre immediatement I'origine de travaux dans la meme voie, mais bien des annees se passerent avant qu'il ne fut repris et complete. C'est en AUemagne, sous I'influence de I'enseignement de Weierstrass que nous vo3"ons d'abord reparaitre Tetude des singularites des equations differentielles, et cela pour les equations differentielles lineaires. II est vraiment curieux que Briot et Bouquet, apres avoir traite le cas plus diiBcile des singularites d'une equation non lineaire, fut-elle du premier ordre, n'aient pas songe a s'occuper des equa- tions lineaires, laissant a M. Fuchs I'honneur de fonder une theorie, dont Tillustre geometre allemand a fait lui-meme des applications du plus haut interet, et qui a provoque un nombre immense de recherches. On remplirait des bibliotheques avec les memoires composes depuis trente ans sur la theorie des equations lineaires. Je ne puis songer a vous parler des nombreuses classes d'equations dont I'etude a ete faite. En restant dans les generalites, je rappelle seulement que I'etude des points singuliers presente une graude difference suivant que ce point singulier est regulier, comme dit M. Fuchs, ou presente les caracteres d'un point singulier essentiel. Ce dernier cas est de beaucoup plus difficile ; M. Thome a forme des series satisfaisant formellement a I'equation, mais qui en general ne sont pas convergentes. Remarquons a ce propos que Briot et Bouquet ont les premiers montre qu'une equation differentielle pouvait conduire a une serie en general divergente ; lenr exemple bien simple est I'equation a^-^ = ax + by ax verifiee par une serie entiere dont le rayon de convergence est nul. Cette petite constatation a appele I'attention sur un fait d'une importance capitale, et qui ne se rencontre que trop frequemment dans les applica- tions; les developpements purement forraels sont nombreux en meca- nique analytique et mecanique celeste, ou lis font le desespoir des geometres. Pour les equations lineaires, ces developpement ont un certain interet, comme I'a montre M. Poincare, au point de vue de la 234 Emile Picard. representation asymptotique des integrales. On peut d'ailleurs obtenir et de bien des manieres, une representation analytique des integrales autour du point singulier. Je dois eufin mentionner, relativement aux points singuliers irreguliers, les recherclies de M. H. von Koch qui a tire tres heureusement parti dans cette question des resultats obtenus sur les determinants d'ordre infini. Revenons aux equations du premier ordre. Briot et Bouquet ont surtout etudie les singularites en faisant les reductions au type ou / est holomorphe et s'annule pour a; = 0, y = 0, et leurs recherches ont ete depuis completees par la connaissance de la forme analytique des integrales au voisinage du point singulier. Le cas plus complique de Tequation x-^ = f(x,y^ (m>2) (1) n'avait fait jusqu'a ces derniers temps I'objet d'aucune recherche depuis les quelques lignes que lui avaient consacrees Briot et Bouquet. Cette etude vient d'etre reprise simultanement par M. Horn et par M. Bendix- son. Ces auteurs se servent d'une methode convenable d'approximations successives dont j'indiquerai le principe. Nous supposons expressement que X reste reel et se rapproche de zero par valeurs positives, et posons fix, y') = hy + F(x, y} F ne contenant pas de terme du premier degre en y independant de x. Si la partie reelle de b est positive, I'equation precedente a une infinite d'integrales tendant vers zero en meme temps que x, et elle n'en a qu'une quand la partie reelle de h est negative. Les deux cas peuvent etre traites en faisant les approximation successives x-^=ly^ + F/+... ^=a'x + b'y+... (2) ou les seconds membres sont des developpements suivant les puissances de X et y, et convergents pour a; et y assez petits. Le point 2;= 0, y = (i correspond-il a une position d'equilibre stable ? II est impossible ac- tuellement de repondre a cette question. II y a peut-etre quelques mecaniciens qui croient que la nature de I'equilibre depend seulement 238 Emile Picard : des termes du premier degre dans le second niembre. Nous nous garde- rons bien de leur en vouloir, car c'etait au fond I'erreur de Lagrange, mais il est clair qu'en reduisant les equations a la partie lineaire, on peut avoir une solution stable qui cesse de I'etre quand on retablit les termes d'ordre superieur. Les equations (2) presentent une particularite curi- euse qui merite d'etre signalee. On peut se proposer de trouver une integrale premiere F etaut en liolomorphe en x, y, x', y', et commengant par des termes du second degre. Or on trouve une telle fonction F au point de vue formal, mais la serie ainsi obtenue ne converge pas en general. J'ajoute que, si la force dependait non seulement de la position du point mais de la vitesse, c'est-a-dire si dans (2) les seconds membres dependaient aussi de x' et y', la recherche de la fonction F ne pourrait plus generalement etre effectuee, mais il serait plus facile de repondre a la question relative a la stabilite. Quand on a aucune notion de la grandeur de I'intervalle pour lequel les fonctions definies par les equations differentielles sont continues, on peut cependant trouver des developpements valables pour tout le temps pendant lequel les fonctions resteront continues. J'ai dit tout a I'heure que Ton pouvait deduire de tels developpements de la methode classique de Cauchy ; c'est lii un resultat interessant, mais malheureusement il n'a guere qu'un interSt theorique, car il semble bien difficile de deduire de ces developpements quelques renseignements sur le champ ou les integrales restent continues. II y aura cependant des cas ou certaines proprietes auxiliaires des equations permettent d'avoir des renseignements sur le champ ovi les integrales restent continues. Que Ton prenne, par exemple, les six equations classiques en p, q, r, y, y', y" relatives au mouvement d'un solide pesant suspendu par un point ; I'integrale des forces vives et I'integrale y^^. y'\ y"^ = const, permettent de reconnaitre que les six fonctions precedentes resteront finies pour toute valeur du temps, et nous sommes alors assure que pour ce probleme la methode de Cauchy donne des developpements valables pour toute valeur du temps. Seconde Conference. 239 VI. A I'ordre d'idees qui nous occupe, se rattachent les travaux de M. Poincare sur les solutions periodiques, et sur les solutions asymptotiques. L'etude des solutions periodiques d'une equation differentielle presente un interet particulier. Je connais peu d'exeraples ou on puisse trouver directement une solution periodique. Dans ses travaux sur ce sujet, M. Poincare procede par voie indirecte ; il profite de la presence d'une con- stante tres petite dans les equations, et il raisonne par continuite en par- tant d'une solution periodique pour la valeur zero de cette constante. II serait a desirer que Ton piit penetrer par une autre voie dans l'etude des solutions periodiques. Quant aux solutions asymptotiques a une seule solution, leur etude resulte de developpements analytiques simples ; mais I'existence dans certains cas particuliers de solutions doublement asymp- totiques, c'est a dire de solutions asymptotiques pour t = — oo k une solution periodique et de nouveau asymptotiques pour t = + cc k cette nieme solution etait extremement cachee, et leur decouverte a demande un effort considerable. L'etude des courbes definies par les equations differentielles est sur- tout une etude qualitative. Si Ton considere d'abord une equation du premier ordi'e et du premier degre. ^ = ^ (X et F polynomes en x et «/) (2) l'etude des points singuliers generaux se deduit des resultats de Briot et Bouquet. Ces points se partagent en trois t3rpes, que M. Poincare ap- pelle des cols, des noeuds et des foyers. Un point singulier d'une nature deja plus compliquee est fourni par ce que M. Poincare appelle un centre, qui en general presente de I'analogie avee les foyers mais autour duquel dans certains cas I'integrale constitue une courbe fermee. On a alors un exemple de solutions periodiques dont la periode depend des conditions initiales. Les travaux les plus recents sur les points singuliers de courbes integrales de I'equation (2) sont dus a M. Bendixson ; le savant geometre suedois a etabli en particulier que s'il existe pour I'equation (2) une courbe iutegrale allant a I'origine avec une tangente determinee, toutes les courbes integrales allant a I'origine y parviendront avec des tangentes determinees. L'etude des courbes integrales ne doit pas etre bornee au voisinage des points singuliers ; ou doit chercher a se rendre compte de leur forme 240 Emile Picard: sur le plan tout entier ou sur la sphere en faisant una perspective. Si Ton chemine, pour I'equation (2), sur une courbe integrale, qu'arrivera- t-il ? Cette courbe peut etre fermee de telle sorte qu'on reviendra au point de depart ; elle peut aussi avoir un des foyers comme point asymp- tote. Elle peut avoir encore pour courbe asymptote une courbe fermee satisfaisant d'ailleurs a I'equation diff erentielle. Ces courbes fermees, que M. Poincare appelle cycles limites jouent un role capital, et c'est dans les cas oil il est possible de se rendre compte de leur position que la dis- cussion de I'equation peut etre faite d'une maniere complete. Pour les equations du premier ordre mais de degre superieur les dif- ficultes sont beaucoup plus grandes. L'etude des points singuliers ge- neraux a ete faite ; elle trouve en particulier son application dans des problemes comme celui des lignes de courbure d'une surface passant par un ombilic. L'etude des courbes dans tout le plan est singulierement compliquee par un fait qui ne pouvait se rencontrer pour les equations du premier degre. II peut arriver qu'une courbe integrale couvre une aire, a'est a dire puisse se rapprocher autant qu'on voudra d'un point arbitraire dans une aire. D'apres les difficultes que presentent encore les equations du premier ordre, il est clair que pour les equations d'ordre superieur au premier l'etude qualitative des integrales soUicitera longtemps encore I'effort des chercheurs. Au point de vue analytique, une circonstance importante est e noter. Tandis que pour le premier ordre, on peut tirer parti dans quel- ques cas comme celui des centres de certains developpements en serie, il arrive au contraire ici dans les cas correspondants que les developpements analogues sont purement formels ; nous en avons vu un exeraple en par- lant tout a I'heure de la stabilite de I'equilibre. Remarquons a ce propos que les questions d'iustabilite sont beaucoup plus faciles a traiter que les questions de stabilite comme il resulte des interessantes recherches de M. Liapounoff. Quand il y a une fonction des forces I'equilibre est stable si, pour cette position, la fonction des forces est maxima. Quant aux positions d'equilibre pour lesquelles cette derniere condition n'est pas remplie, on les a toujours regardees comme instables, mais leur instabilite n'avait pas ete demontree. M. Liapounoff I'a etablie en particulier pour le cas que I'on peut appeler general ou la non existence du maximum de la fonction des forces se reconnait par les termes du second ordre. Je citerai seulement un exemple relatif aux courbes integrales d'une equation d'ordre superieur au premier. Dans un memoire recent, M. Troisihne Conference. 241 Hadamard N-ient d'etudier les lignes geodesiques des surfaces a courbures opposees et a connexion multiple ayant un nombre limite de nappes in- finies. II etablit que les tangentes aux lignes geodesiques passant par un point de la surface, et restant a distance finie, forment un ensemble par- fait non continu. Ce resultat est interessant au point de vue de la dispo- sition des lignes geodesiques de la surface ; il montre qu'il existe des lignes geodesiques se rapprochant d'une geodesique fermee determinee, puis abandonnant celle-ci pour se rapprocher d'une autre, puis passant a une troisieme, et ainsi de suite indefiniment. II montre de plus que Tallure des courbes integrales pent dependre dans certains cas, des pro- prietes discontinues je veux dire arithmitiques des constantes d'integra- tion. C'est sur cette idee que je veux m'arreter ; dans la theorie des equations differentielles conime en maintes parties des mathematiques, les recherclies sont obligees de prendre un caractere arithmetique. C'est VarithmStisation des mathematiques dont parlait M. Klein dans un article recent. J'ai essaye, messieurs, en restant dans les generalites et sans prendre aucune classe particuliere d'equations, de faire une sorte de carte geogra- phique sommaire de la theorie des equations differentielles. Beaucoup de voies sont ouvertes et dans des directions tres varices ; sur plus d'un point, les questions sont seulement posees, mais elles paraissent bien posees ; et nous nous rendons compte, ce qui a son prix, de la nature des difficultes qu'il faudra vaincre. C'est une etroite alliance entre les dis- ciplines les plus diverses qui amenera maintenant de nouveaux progres. II n'est plus i^ermis aujourd'hui au geometre inventeur d'etre I'liomme d'un seul point de voie, et il faut nous resigner a de grandes complica- tions. C'est un privilege que les sciences mathematiques partageront probablement dans I'avenir avec d'autres sciences. Esperons seulement que des hommes de genie viendront, de loin en loin, donuer au moins pour un temps I'illusion de la simplicite. Tkoisieme Conference. Sur la ThSorie des Fonctions Analitiques et sur quelques Fonctions SpSciales. La theorie des fonctions de variables complexes est devenue aujourd'hui une branche considerable de I'analyse mathematique. Elle doit son bril- lant essor a la decouverte de quelques propositions generales parmi les- quelles se trouvent au premier rang les theoremes de Cauchy sur les 242 Emile Picard: integrales prises le long d'un contour. Ces lois generales des fonctions analytiques apj^liquees a des fonctions speciales donnent souvent avec facilite leurs principales proprietes. L'application de ces lois constitue une methods synthetique, et des resultats auxquels avaient conduit une longue serie de transformations de calculs apparaissent quelquefois avec une evidence intuitive. La theorie des fonctions elliptiques en oifre un memorable exemple, et n'y a-t-il pas quelque chose de merveUleux a integrer avec M. Hermite le long d'un parallelogramme de periodes et a obtenir ainsi d'un trait de plume les principales proprietes des fonctions doublement periodiques ? La fagon dont Riemann pose et resout dans sa dissertation inaugurale le probleme des integrales abe- liennes n'est pas moins digne d'etre meditee comme exemple d'une methode synthetique dans la theorie des fonctions. I. II n'est plus douteux aujourd'hui que les principes essentiels qui sont a la base de la theorie n'aient ete connus de Gauss. On sait que celui-ci ne publia pas ses recherches sur ce sujet. On ne pent guere admettre qu'il n'en ait pas saisi la haute importance ; fidele a sa devise " pauca sed matura " il attendait sans doute de s'etre livre a une plus longue elabora- tion, quand Cauchy fit connaitre ses decouvertes. On doit done regarder Cauchy comme le veritable fondateur de la theorie appelee a un si grand avenir ; non pas certes qu'il I'ait presentee sous une forme didactique. Ouvrant des voies nouveUes, son esprit toujours en travail se souciait peu de donner a ses conceptions une forme parfaite. On suit le travail d'invention dans maintes publications de Cauchy, notamment quand on parcourt dans ses ffiuvres Completes les notes innombrables extraites des Comptes-Rendus. Dans la theorie qui nous occupe, une place a part doit etre faite a I'idee fondamentale d'etendre la notion de I'integrale definie en faisant passer la variable par une succession de valeui-s imaginaires ; cette conception a ete la source des plus belles decouvertes, et la represen- tation d'une fonction par une integrale le long d'un contour ferme gardera a jamais le nom d'integrale de Cauchy. Le point de depart de Riemann se rapproche beaucoup de celui de Cauchy ; il est tres philosophique de prendre comme base les deux equa- tions simultanees 9m _dv du _ dv dx dy dy dx Troisihne Conference. 243 et de reduire ainsi la theorie des fonctions d'une variable complexe a I'etude de ces deux equatious simultanees aux derivees pailielles. En meme temps apparaissent les liens entre cette etude et plusieurs questions de physique mathematique comme le mouvement permanent des fluides sur un plan et celui de I'electricite sur une plaque conductrice ; et tons ces problemes sont susceptibles d'etre generalises si au plan simple dans lequel se meut la variable (x, y) on substitue le plan multiple de Riemann. Les deux relations ecrites plus haut amenent a considerer I'equation Am = 0, equation qui contient toute la theorie des fonctions d'une variable com- plexe, et parmi les problemes qu'on peut se poser sur cette equation le plus celebre est celui de la determination d'une integrale par ses valeurs sur un contour ferme. Une application d'une autre nature concerne la geometrie ; je veux parler du probleme des cartes geographiques qui amene a la question de la representation conforme d'une aire sur une autre. "Weierstrass a edifie la theorie des fonctions de variables complexes sur une autre base que Cauchy et Riemann, en partant des developpements en series entieres ; en France, ces developpements avaient ete aussi envisages par M. ^leray qui n'avait pas connaissance des legons de Weierstrass. Le memoire public en 1876 par I'illustre analyste de Berlin, qui a fait connaitre a un public plus etendu les resultats developpes depuis long- temps dans I'enseignement du maitre, a ete le point de depart d'un grand nombre de travaux sur la theorie des fonctions. Cauchy avait deja obtenu d'importants resultats sur le developpement en sommes ou en produits infinis de certaines categories de fonctions. II etait reserve a Weierstrass et a ses disciples de traiter ces questions dans toute leur generalite. La decomposition des fonctions entieres, c'est a dire des fonctions uuiformes et continues dans tout le plan, en facteurs primaires est un des plus admii-ables theoremes de I'analyse moderne ; chacun de facteurs primaires est le produit d'un facteur lineaii-e par une exponentielle. Les developpe- ments des fonctions uniformes en sommes et en produits infinis ont fait ensuite I'objet d'un grand nombre de travaux parmi les quels il faut citer tout particulieremeut le memoire de Mittag-Leffier qui a aborde ces prob- lemes avec la plus grande generalite possible. Je rappellerai aussi un memoire de M. Runge auquel des recherches toutes recentes viennent de redonner de I'actualite, ou se trouve en particulier etabli que toute fonction holomorphe dans un domaine connexe peut dans ce domaine etre developpee en une serie de polynomes. Cauchy et ses disciples fran^ais, en etudiant la theorie des fonctions 244: Emile Picard: uniformes, n'avaient pas penetre dans I'etude de ces points singuliers appeles aujourd'hui points singuliers essentiels, dont le point 2=0 pour 1 la fonction e" donne I'exemple le plus simple. La consideration des facte ors primaires permit a Weierstrass de montrer que dans le voisinage d"un point essential isole une fonction uniforme peut se mettre sous la forme d'un quotient de deux fonctions uniformes n'ayant pas de poles dans le voisinage de a ; Weierstrass montra aussi que dans le voisinage d'un tel point la fonction s'approche autant que Ton veut de toute valeux donnee. On a plus tard complete ce resultat, en etablissant que dans le voisinage d'un point singulier essentiel isole la fonction prend rigoureuse- ment une infinite de fois toute valeur donnee, une exception seulement etant possible pour deux valeurs particulieres au plus. La demonstra- tion de ce theoreme se deduit de la consideration d'une fonction pre- sentant precisement la propriete qu'on veut demontrer etre impossible ; cette fonction est la fonction modulaire de la theorie des fonctions elliptiques, mais ses points singuliers ne sont pas isoles. Un coroUaii-e du theoreme indique conduit a la proposition suivante relative aux fonc- tions entieres : si, pour une fonction entiere G-{z) il existe deux valeurs a etb telles que les deux equations Gr(z) = a et G-(z) = h aient seulement un nombre limite de racines, la fonction G-(z') est un polynome. De nombreuses tentatives ont ete faites pour demontrer directement les theoremes precedents sans recourir a la theorie des fonctions ellip- tiques. Pour le theoreme sur les fonctions entieres, M. Hadamard avait reussi a I'etablir quand, la fonction entiere etant representee par > o^m^^i -. m=0 on a (a„)<-r- — —•, a etant positif. Plus recemment M. Borel est [1 • 2 •-• my arrive a le demontrer pour toutes les fonctions entieres et meme a le generaliser considerablement. Les travaux de M. Hadamard et de M. Borel publics dans ces dernieres anuees sont extremement remarquables. Dans ces recherches, une notion importante introduite par Laguerre, celle du genre d'une fonction entiere, joue un role capital ; ce qui fait I'iuteret de cette notion, c'est qu'elle est intimement liee a la distribution des racines de la fonction. M. Poincare avait fait le premier la remarque que le genre d'une fonction entiere est en relation etroite avec I'ordre de grandeur de la fonction pour les grandes valeurs de la variable. M. Hadamard a cherche une limite du genre a I'aide des coefficients du Troisihne Conference. 245 developpement, et il a ^tabli que si le coefficient de a;" est moindre que J-, la fonction est de genre E en designant par ^ + 1 I'entier (1 • 2 ••■ my immediatement superieur a X. 11 a reussi aussi a demontrer que, en designant par ^(jn) une fonction croissant indefiniment avec w, si le coefficient a„ decroit plus vite que -— la »*"'* racine a un module ^ ^ [(^(m)]"' ^ superieur a (1 — e)(^(p) ou e est infiniment petit pour p = ca . De ses resultats, M. Hadamard a fait une belle application a I'etude de la distri- bution des racines d'uue fonction celebre consideree par Riemann dans son memoire sur les nombres premiers. Dans son travail sur les zeros des fonctions entieres, M. Borel a eu surtout pour objet la demonstration de I'impossibilite de certaines iden- tites. Soit /*(?■) une fonction positive croissant indefiniment avec r. Designons par 6r,(3) une fonction entiere dont le module maximum pour (a) = r est inferieur a e""'', et IIi(z) une fonction entiere dont le module maximum est superieur a [mC*") ]'"*"% '^ etant positif ; Tidentite ne peut avoir lieu que si tous les G- sont identiquement nuls. En particu- lier pour n = 2, une pareille identite ne peut exister, G-^ etant une constante, (?i et G-^ des polynomes : c'est le tbeoreme enonce plus haut sur les fonctions entieres. Apres ces resultats sur les fonctions holomorphes dans tout le plan, revenons aux series entieres dont le rayon de convergence est fini. Une telle serie donne, pour employer le langage de Weierstrass, un element de fonction, en supposant bien entendu que le rayon de convergence n'est pas nul. L'extension analytique d'un tel element joue un role capital dans la theorie de Weierstrass ; il est dans cette etude du plus haut interet d'avoir des renseignements sur les singularites de la fonction sur le cercle de convergence. Le memoire de M. Darboux sur I'approxima- tion des fonctions de tres grand nombres, les recherches plus recentes de M. Hadamard, de M. Borel et de M. Fabry out conduit a des resultats d'un haut interet. Je ne veux signaler qu'une consequence curieuse, entrevue deja par M. Pringsheim : c'est qu'une serie entiere a en general son cercle de convergence comme coupure. On salt que Weierstrass a le premier indique un exemple d'un serie entiere ne pouvant etre prolongee analytiquement au dela de son cercle de convergence, et cet exemple 246 Emile Picard: detourne provenait de la theorie des fonctions elliptiques. 11 est vrai- ment singulier que Ton ait eu autrefois quelques diiiicultes pour trouver des exemples de ce que Ton doit considerer maintenant comme la circon- stance la plus frequente. Parmi les methodes proposees pour I'etude de la serie prolongee au dela de son cercle de convergence, il en est deux qui sont particuliere- ment simples. La premiere, employee par M. E. Lindeloff repose sur la theorie de la representation conforme ; la seconde utilise la notion de serie divergente sommable resultant des travaux de M. Borel. Cette notion semble devoir jouer dans plusieurs questions d'analyse un role important. J'en indiquerai en deux mots le principe. Soit une serie, Mq + M^ + ••• M„ + ••• ; on lui associe la fonction de a: u„a^ M(a) = Mq + «!« + r^ H h u^a" L'expression 1-2 ' ' 1.2-w CO = Cu(a')e'''da + pent avoir un sens quand la serie initials est divergente ; on la regarde alors comme la limite de la serie. En appliquant cette notion a la pro- gression geometrique qui represente , et en se servant de I'integrale de Cauchy, on est alors conduit a une expression analytique qui dans bien des cas represente la fonction dans une aire exterieure au cercle de con- vergence. Je ne puis songer a rappeler, ne fi\t-ce que d'un mot, les etudes les plus importantes faites tout recemment sur le prolongement analytique. Arre- tons nous seulement sur un resultat que vieut de publier M. Mittag-LeiBer. Considerons, avec I'eminent geometre suedois, un element de fonction dans son cercle de convergence, et sur chaque rayon suivons la fonction jusqu'a ce que nous rencontrions un point singulier, celui-ci pouvant d'ailleurs etre a I'infini. On ne garde sur chaque rayon que la portion comprise entre le centre et le premier point singulier, et on obtient ainsi une aire que M. Mittag-Leffler appelle Vetoile correspondant a I'element de fonction. II montre qu'on peut obtenir une representation de la fonc- tion dans toute I'etoile, sous la forme d'une serie ayant pour termes des polynomes en x dont les coefficients sont lineaires par rapport aux coeffi- cients du developpement initial ; de cette fagon, quand on a en un point la valeur d'une fonction analytique et de toutes ses derivees, on peut Troisieme Conference. 247 obtenir a I'aide de ces seules donnees une representation de la fonction valable dans toute une etoile. Ce resiiltat ponrra peut-etre avoir un cer- tain interet pour la theorie des equations differentielles ; il faut toutefois observer que dans ce cas la methode de Cauchy, comme nous I'avons dit bier, conduit au meme resultat. Ainsi les series considerees bier (page 18), constituent des developpements valables dans une etoile. Nous avons, dans ce qui precede, considere un element de fonction, c'est a dire que la serie ,,,,«, ^i x avait un rayon de convergence different de zero. Si la serie precedente ne converge que pour a; = 0, elle ne represente rien et il semble qu'il n'y ait aucun probleme a se poser a son sujet. Cependant nous avons donne bier des exemples d'equations differentielles conduisant a de tels developpe- ments ; la derivee d'un ordre quelconque m de certaines integrales dans un certain angle ayant I'origine pour sommet tend vers 1 • 2 •••m • «;„ quand X tend vers zero a I'interieur de Tangle convenable A. Ces conditions re- latives aux valeurs des derivees ne f)euvent manifestement determiner une seule fonction dans Tangle A pres de Torigine, car on pent a une pre- miere fonction ajouter une exponentielle de la forme e"^" (« etant convena- blement cboisie) dont toutes les derivees sont nulles a Torigine ; mais, en appliquant sa metbode de sommation des series divergentes, M. Borel est conduit a imposer une condition supplementaire et a obtenir alors, dans des cas etendus, une fonction unique determinee par la serie divergente (1). n. Les divers travaux que je viens de rappeler montrent avec quelle activite les analystes se sont occupes dans ces derniers temps des gene- ralites concernant les fonctions analytiques d'une variable. La tbeorie generale des fonctions de plusieurs variables avance beaucoup moins rapidement ; les questions qui se posent ici sont beaucoup plus difficiles, tant en elles-memes que par le defaut d'une representation qui fasse image. Nous suivons une variable complexe sur son plan, mais avec deux variables complexes nous nous trouvons dans un espace a quatre dimensions, oil de plus les diverses coordonnees ne se presentent pas symetriquement. Au lieu de deux equations, nous avons quatre equa- tions aux derives partielles auxquelles doivent satisfaire deux fonctions de quatre variables. L'elimination d'une des fonctions conduit pour Tautre a un systeme de quatre equations aux derivees partielles qui 248 Emile Picard : remplace I'equation cle Laplace, mais qui n'a pas ete etudie directement comme cette derniere equation. II semble qu'on ne puisse pour ce sys- teme se poser aucun problems analogue a celui de Dirichlet et de Rie- mann ; nous ne trouvons ici aucune analogie entre le cas d'une variable et celui de deux variables. A un autre point de vue, le developpement de Taylor a deux variables peut bien servir a definir un element de fonction, mais nous n'avons rien d'analogue au cercle de convergence. Que sont les regions de conver- gence pour un tel developpement? II faudrait considerer des surfaces dans I'h^perespace a quatre dimensions ; aucune regie n'etant connue a cet egard, on se borne a considerer deux cercles assez petits dans les plans respectifs des deux variables, cercles a I'interieur desquels la serie est convergente. Les theoremes generaux sur les fonctions analytiques de deux variables complexes sont peu nombreux. Une remarque souvent utile a ete faite il y a longtemps par Weierstrass ; elle a en quelque sorte pour objet de mettre en evidence, dans une fonction de n variables holomorphes autour de x^ = 0, •••, a;„= 0, et s'annulant pour ces valeurs des variables, la partie de la fonction qui s'annule. Weierstrass montre que autour && x^= ••• = x^ = (i la fonction peut se mettre sous la forme d'un produit de deux facteurs holomorphes, dont I'un ne s'annule pas a I'origine et dont I'autre est un polynome par rapport a I'une des variables. Une autre proposition d'une demonstration delicate est due a M. Poincare et a pour objet de generaliser le theoreme de Weierstrass relatif aux fonctions uniformes d'une variable n'ayant a distance finie que des poles, fonctions qui peuvent se mettre sous la forme d'un quotient de deux fonctions entieres. Pareillement une fonction de deux variables qui, pour toutes les valeurs finies des variables presente le caractere d'line fonction rationnelle peut etre mise sous la forme d'un quotient de deux fonc- tions entieres. Ce beau theoreme a ete etendu par M. Cousin, qui a suivi une toute autre voie, aux fonctions d'un nombre quelconque de variables. On doit encore a M. Poincare un resultat bien saillant : je veux parler de I'extension aux integrales doubles du theoreme fondamental de Cauchy relatif aux integrales simples prises le long d'un contour. II n'y a pas de difficulte a definir une integrale double d'une fonction F(x, y) de deux variables comj^lexes a; et y ff^(^^ y')<^^dy Troisieme Conference. 249 sur un continuixm a deux dimensions situe dans I'hyperespace a quatre dimensions qui correspond aux deux variables complexes. Si le con- tinuum est ferme, et qu'on puisse le reduire a una ligne ou a un point sans que F cesse d'etre continue, I'integrale sera uulle. Ce resultat conduit a poser vm grand nombre de questions. Si F est une fonction rationelle, il y a lieu de considerer les rSsidus de I'integrale double ; ces residus s'expriment par des periodes d'integrales abeliennes ordinaires. Si F est une fonction algebrique de x et y, on aura a envisager les pSriodes de I'integrale double, et on voit s'ouvrir un vaste champ de recherches. On s'apergoit d'ailleurs bien vite que si certaines analogies subsistent avec le cas d'une variable, il en est beaucoup .d'autres qui disparaissent entierement. Des integrales le long d'un contour out donne a Cauchy le nombre des racines d'une equation contenues dans ce contour, mais dans la question correspondante du nombre des racines communes a deux equations simultanees, les integrales doubles n'ont aucun role a jouer ; ce sont des integrales triples etendues a un certain continuum a trois dimensions qui interviennent dans cette recherche. Je parlais tout a I'heure de la dissymetrie qui se presente au point de vue reel dans la theorie des fonctions de deux variables complexes. II etait interessant de rechercher si il n'est pas possible de generaliser les deux equations aux derivees partielles de la theorie d'une fonction d'une variable. Le probleme est evidemment indetermine ; tout depend de la propriete de ces equations sur laquelle on porte specialement son attention. On peut se placer au point de vue suivant : rechercher tous les systemes d'equations aux derivees partielles relatifs a n fonctions de n variables independantes et telles que, si Wj, u^, •••«„ et Hj, v^, •••v„ de- signent deux solutions quelconques, les v considerees comme fonctions des u satisfassent au meme systeme. Cette propriete appartient evidem- j , ,. du dv 5m dv j 111 ment aux deux equations -r— = -r-> -r~ — ~'^- La recherche de ces svs- ^ dx oy ay dx '' temes peut se faire d'une maniere reguliere, et peut se deduire de la connaissance des certains groupes d'ordre fini ; ainsi tous les systemes du type precedent d'equations aux derivees partielles du premier ordre pourront etre obtenus a I'aide des groupes lineaires et homogenes a n variables. II est possible que, parmi tous ces systemes, il en est qui presentent quelque interet particulier, et avec lesquels on puisse edifier une theorie plus ou moins analogue a la theorie d'une fonction d'une variable complexe. Le cas de n = 3 ne donne rien d'interessant ; pour 250 Emile Picard: w = 4, on pourrait prendre d'abord le groupe lineaire qui donne naissance aux quaternions, il lui correspond un systeme d'equations differentielles qui presente peut-etre quelque interet. Cette extension de la theorie des fonctions d'une variable complexe n'est pas la seule qui ait ete proposee. M. Volterra a cherche dans une autre voie en considerant des fonctions de ligne, ce qui I'a conduit a d'interessantes relations differentielles et a quelques problemes analogues a ceux de Dirichlet. L'avenir dira si ces extensions sont simplement des curiosites ou si elles presentent quelque interet general. III. Quittons maintenant les generalites et jetons un coup d'oeil sur quelques fonctions speciales. II n"en est pas qui aient ete plus etudiees que les fonctions algebriques d'une variable ; c'est en faisant leur etude que Puiseux, dans un niemoire reste celebre, a appele I'attention sur I'interet que presentait la consideration de la variable complexe. On a quelque peine a se representer qu'il a paru merveilleux que Vi et — Vi puissent etre considerees comme deux determinations d'une meme fonc- tion; c'est dans ce memoire aussi qu'apparait pour la premiere fois I'origine de la periodicite. La theorie des fonctions algebriques est devenue un confluent ou se rencontrent les notions les plus diverses ; chacun, suivant ses gouts, pent y trouver les points de vue qu'il prefere. Avec les methodes de Weier- strass, nous trouvons la precision extreme qui caracterise son ecole, et le souci constant de n'introduire aucune consideration etrangere a la theorie des fonctions fiit ce au prix de detours longs et penibles. Celui qui aime le langage et les formes de raisonnement de la geometric analytique suivra BrUl et Noether dans leur theorie si feconde des groupes de points. Ceux enfin qui recherchent les grands horizons auront plaisir a lire Riemann qui, avec la merveilleuse conception de la surface qui porte son nom, rend, pour ainsi dire, intuitifs les points les plus delicats de la theorie. Ce serait d'ailleurs une vue etroite que de regarder seulement la belle conception de Riemann comme une methode simplicative. Pour Riemann, le point essentiel est dans la conception a priori de la surface conuexe, formee d'un nombre limite de feuillets, et dans le fait qua une telle surface congue dans toute sa generalite correspond une classe de courbes algebriques. De plus, on pent envisager des surfaces de Riemann a un nombre infini de feuillets, et les travaux de Poincare ont montre le Troisie'me Conference. 251 role utile qu'elles peuvent jouer dans I'etude des fonctions non uniformes. On sait aussi I'importance qu'avait pour Riemann le probleme de la repre- sentation conforme ; le cas de la representation conforme des aires a connexions multiples a ete traite par M. Schottky dans un tres beau memoire oil I'auteur se montre disciple de Weierstrass, mais qui se rat- tache naturellement a I'ordre d'idees de Riemann. A una aire plane percee de p trous, envisagee comme ayant une face superieure et une face inferieure correspond une courbe algebrique de genre p ; la question de la representation conforme de deux aires revient alors a la correspondance entre les points de deux courbes algebriques. Aux courbes algebriques se rattaclient des fonctions extremement re- marquables d'une variable ; ce sont les fonctions que M. Poincare appelle fuchsiennea et que M. Klein designe sous le nom de fonctions automorphes. Pour les courbes des genres zero et un, on pent exprimer les coordonnees par des fonctions uniformes d'un parametre, meromorphes dans tout le plan (fonctions rationnelles et fonctions doublement periodiques). II etait naturel de chercher, pour les courbes de genre superieur a un, une representation parametrique par des fonctions uniformes. Des tentatives varices ont probablement ete faites pour resoudre cette question, en clier- chant a realiser cette expression par des transcendantes n'ayant que des poles a distance finie. De telles tentatives, on le sait aujourd'hui, ne pouvaient reussir, car on peut etablir que, entre deux fonctions uniformes dans le voisinage d'un point qui est pour chacune d'elles un point singulier essentiel isole, ne peut exister une relation algebrique de genre superieur ^ I'unite. Les transcendantes a employer sont d'une nature beaucoup plus compliquee ; les unes ont un cercle comme coupure au dela duquel elles ne peuvent etre prolongees analytiquement, les autres sont definies dans tout le plan, mais elles ont sur un cercle une infinite de points singu- liers essentiels formant, d'apres la denomination de M. Cantor, un ensem- ble parfait qui n'est pas continu. Les celebres memoires de M. Poincare sur les fonctions fuchsiennes et les belles recherches de M. Klein sur le meme sujet forment un des plus beaux chapitres ecrits dans ces vingt dernieres annees sur la theorie des fonctions. Les fonctions automorphes forment une generalisation extremement etendue et remarquable des fonc- tions modulaires etudiees par M. Hermite dans la tlieorie des fonctions elliptiques, et des fonctions considerees par M. Schwarz en faisaut dans certains cas I'inversion du rapport de deux solutions de I'equation hyper- geometrique. Toute cette theorie est d'ailleurs etroitement liee a la 252 . Emile Picard : theorie cles equations lineaires, et c'est un des resultats les plus saillants obtenus par M. Poincare qu'avec des transcendantes analogues aux fonc- tions fuchsiennes on puisse integrer les equations differentielles lineaires a coefficients algebriques n'ayant que des points singuliers reguliers (au sens de M. Fuchs). Parmi les transcendantes se rattacliant aux fonctions algebriques citons encore les integrales de fonctions a multiplicateurs etudiees tout particulierement par M. AppeU. Ce sont des fonctions n'ayant sur la surface de Riemann que des poles ou des points singuliers logaritlimiques, et dont toutes les determinations se deduisent de I'une d'entre elles par des substitutions de la forme (u, au + b')\ elles generalisent par suite les integrales abeliennes pour lesquelles les a sont egaux a I'unite. Un beau resultat obtenu par M. Appell est que ces fonctions se presentent dans la recherche des coefficients des fonctions abeliennes de deux variables quand on les developpe en series trigonometriques. On a aussi recherche les cas ou I'inversion d'une integrale de fonction a multiplicateurs conduit a une fonction uniforme, mais la conclusion a ete negative, c'est a dire que dans ce cas la courbe algebrique est necessairement du genre zero ou du genre un, et la fonction uniforme obtenue se ramene ou des transcendantes connues. rv. Les equations differentielles forment une mine inepuisable pour ob- tenir des fonctions speciales. Les equations lineaires ont ainsi conduit a des fonctions jouissant de proprietes bien definies. Pour les equations non lineaires, M. Fuchs appela le premier I'attention sur les equations algebriques du premier ordre a points critiques fixes et montra comment on pent reconnaitre qu'on se trouve dans ce cas. M. Poincare fit voir ensuite qu'on pouvait ramener ce cas a des quadratures ou aux equations de Riccati. M. Painleve a etendu ces resultats en considerant les equa- tions du premier ordre dont les integrales n'ont qu'un nombre limite de valeurs autour de I'ensemble des points critiques mobiles. Une des con- clusions de ses recherches est que I'integrale, supposee transcendante, de toute equation algebrique du premier ordre qui satisfait a la condition precedente, est une fonction algebrique de I'integrale d'une equation de Riccati dont les coefficients dependent algebriquement de ceux de I'equa- tion donnee. On pent se proposer des problemes analogues pour les equations differentielles algebriques d'ordre superieur au premier. II se presente ici des difficultes considerables; I'une d'elles tient au fait suivant : Troisieme Conference. 253 tandis que toute transformation biuniforme d'une courbe algebrique en elle-meme (avec siugularites isolees) est necessairement biratiounelle, il pent arriver au contraire qu'une transformation biuniforriTe d'une surface algebrique en elle-meme ne soit pas biratiounelle. Une seconde difficulte, non moins grave, consiste dans rexistence possible de singularites essen- tielles mobiles. J'ai indique hier la distinction faite a cet egard par M. Painleve entre la classe generale d'equations ne possedant pas de tels points et la classe singuliere. En cherchant a etendre aux equations du second ordre a points critiques fixes la methode qui avait reussi a M. Poincare pour les equa- tions du premier oi'dre jouissaut de la meme propriete, on est an-ete imme- diatement par la premiere difficulte signalee plus haut, et c'est seulement dans le cas ou Tintegrale generale de I'equation est supposee dependre algebriquement des deux constantes d'integration que Ton jDeut pour- suivre I'etude sans de serieuses difficultes ; on retombe d'ailleurs sur des transcendantes deja connues. M. Painleve a fait une etude complete des autres cas qui peuvent se presenter ; I'integrale generale peut encore etre une fonction algebrique d'une seule des constantes, ou enfin dependre d'une maniere transcendante des deux constantes (de quelque fagon qu'on les choisisse). Ce dernier cas seul est irreductible aux transcendantes classiques, c'est a dire ne peut etre ramene aux quadratures et aux equa- tions lineaires. Ce cas se presente d'ailleurs effectivement, et M. Pain- leve a forme explicitement toutes les equations du second ordre de la forme ou B, est rationnel en «/', algebrique en y et analytique en x ; elles se laissent ramener a dome types canoniques tres simples. J'indiquerai seulement deux de ces equations pour lesquelles I'integrale generale est uniforme, y" = 2i^ + xy -\- a (a = constante numerique) L'integrale generale de I'une et I'autre equation est une fonction uni- forme et meromorphe de z dans tout le plan, et cette integrale est une transcendante vraiment nouvelle. Ces exemples precis montrent com- bien M. Painleve a pousse jusqu'au bout ses profondes recherches. Je me bornerai a dire, relativement aux equations du troisieme ordre, que I'integrale generale peut avoir des lignes de points singuliers essen- 254 Emile Picard: tiels. On en a facilement des exemples en considerant I'equation diffe- rentielle algebrique du troisieme ordre a laquelle satisfait una fonction automorphe d'une variable. V. Le champ des fonctions speciales de plusieurs variables complexes, dont I'etude a ete quelque peu approfondie, est assez limite. La theorie des fonctions abeliennes a fait I'objet d'un nombre considerable de tra- vaux qui sont trop classiques pour que je m'y arrete ici ; les memoires de Riemann et de Weierstrass, les etudes de M. Hermite sur la transfor- mation des fonctions abeliennes sont dans toutes les memoires. Apres les etudes faites sur les fonctions fuchsiennes d'une variable, U etait naturel de chercher des transcendantes analogues pour le cas de deux variables ; on devait d'abord se demander s'E existe des groupes discon- tinus contenus dans le groupe lineaire a deux variables / a'u + h'v + c a"u + h"v + c"\ .-, . [u,v; — - — — , -— ■ 1. c-i; \ au + ov + c ail + ov + c y Un seul exemple d'un tel groupe, mais bien peu utUe, s'offrait a I'esprit, celui du groupe a quatre periodes. Aucun exemple analogue au groupe modulaire ne se presentait, et il n'y avait rien a demander sur ce point a la theorie des fonctions abeliennes, au moins sous sa forme classique. Par quoi d'ailleurs se trouverait remplacee ici la condition imposee aux substitutions d'un groupe fuchsien, de conserver un certain cercle ? L'etude des formes quadratiques ternaires a indeterminees con- juguees vint permettre de former en grand nombre les exemples cher- chees. M. Hermite avait, U y a longtemps, montre I'interet au point de vue arithmetique des formes quadratiques binaires a indeterminees conjuguees ; les formes ternaires indefinies conduisirent a de nombreux groupes du type (1), discontinus a I'interieur d'une certaine hypersurface de I'espace a quatre dimensions. Cette surface remplace la circonference de la theorie des groupes fuchsiens. Les groupes du type precedent furent appelees groupes hyperfuchsiens ; on se rend aisement compte que leur recherche generale constitue, comme pour les groupes fuchsiens, un probleme uniquement d'ordre algebrique ; mais, toute representation geometrique faisant defaut, cette recherche directe serait tellement penible qu'elle est reellement impraticable. Aussi les exemples fournis par des considerations ao-ithmetiques sont-ils extremement precieux. Troisieme Con/Silence. 255 Aux groupes liyperfuchsiens correspondent des fonctions uniformes restant invariables par les substitutions du groups. Des exemples de fonctions hyperfuchsiennes d'une nature differente peuvent etre fournis par les series hypergeometriques de deux variables. Une telle serie, fouction de x et 1/ dependant de quatre parametres arbi- traires X, /i, b^, et h^ satisfait a un systeme de trois equations lineaires aux derivees partielles du second ordre, ayant trois solutions communea lineairement independantes. Designant celles-ci par Wj, co^, Wg, on pent chercher dans quels cas les quotients '"a "3 — = w, — = v donnent pour x et y des fonctions uniformes de u et v. Les conditions sont tres simples ; si on prend deux quelconques des quatre quantites \, ft, Sj et ^21 soit, par exemple \ et 6j, la difference \ + Sj — 1 doit etre I'inverse d'un nombre entier positif, et pareillement si on prend trois quelconques de ces quantites, soit \, /a et b^, la difference 2 — \ — /x — Jj est encore egal a I'inverse d'un entier positif. Je citerai I'exemple X = /li = Jj = ^2 = f pour lequel le polyedre fondamental du groupe est tout entier a VintSrieur de I'hypersurface limite. On pent generaliser les fonctions fucbsiennes en considerant d'autres groupes discontinus que les groupes hyperfuchsiens. Une substitution birationnelle entre deux variables m et d n'est pas necessairement lineaire, et ce serait un probleme interessant mais difficile de former tons les groupes discontinus au moins dans une certaine region de I'hyperespace (u, v) de substitutions birationnelles. En dehors des groupes lineaires (hyperfuchsiens) on a seulement considere jusqu'ici les groupes formes de substitutions de la forme / au + h \ ( a!v-\-b' \ et des substitutions ou u est remplace par une fonction lineaire de v et inversement. Ce sont les groupes hyverah4liens qui rentrent evidcmmeut dans les types des substitutions quadratiques ; il y a dans ce cas deux domaines frontieres. II y aura sans doute des decouvertes interessantes a faire un jour dans le champ tres vaste des groupes discontinus de sub- stitutions birationnelles, et des fonctions correspondantes (dans le cas * ou il en existera, comme il arrive pour les fonctions hyperfuchsiennes et hyperabeliennes). 256 Emile Picard: VI. Nous avons rappele tout a I'heure le brillant d^veloppement de la theorie des fonctions algebriques d'une variable ; les progres out ete beaucoup plus lents dans le champ de deux variables. C'est un sujet en pleine elaboration, et que Ton attaque de plusieurs cotes. Clebsch, se plagant au point de vue de la geometric analytique, signala le premier que, pour une surface algebrique de degre m, certaines surfaces d'ordre wj — 4 devaient jouer le role que jouaient les adjoiutes d'ordre m — 3 par rapport a une courbe de degre m. L'etude de ces surfaces d'ordre m — 4 a ete reprise par M. Noetlier dans un memoire de grande importance. En se plagant au point de vue de la theorie des fonctions, voici I'origine de ces surfaces. Si on cherche les integrales doubles J J -B(2^, «/, z^dx dy (/(x, y, z) = 0) restant toujours finies, integrales qu'on appelle les integrales doubles de premiere espece, on trouve qu'elles sont de la forme // Q(xy y, z)dxdy Q etant un polynome d'ordre m — 4. Le nombre Pg de ces polynomes lineairement independants est ce que Ton appelle le genre gSomStrique de la surface ; un pareil nombre est manifestement un invariant. Jus- qu'ici les analogies sont completes avec les courbes ; il y a des integrales doubles de premiere espece, comme il y a des integrales abeliennes de premiere espece. Mais une premiere difference va de suite se manifester. II faut calculer le nombre des arbitraires qui figurent dans les poly- nomes Q d'ordre m — 4 se comportant aux points multiples de la sur- face de telle maniere que I'integrale reste finie. Or on pent trouver par une formule precise le nombre des conditions ainsi entrainees, mais seule- ment pour un polynome d'un ordre suffisamment grand N; si done on fait dans cette formule iV= wi — 4, il est possible que Ton trouve un nombre different de p^ ; on designe le nombre que donne la formule a laquelle je fais allusion par p„, et on I'appelle le genre numerique de la surface. Le cas le plus general est celui ou p„=Pg; quand il n'y a pas egalite, on a p„(^t), y = x(J-)i 2 = •<|^(0 und nennen i das Argument oder die independente Variabele, x, y, z aber die dependenten Variabeln. Wir konnen es zu- nachst als hinlanglich sicher gestellte Erfahrungsthatsache betrachten, dass ein Korper nie aus einer Lage plotzlich verschwindet und im nachsten Zeitmomente in einer andern um Endliehes davon Ver- schiedenen wieder zum Vorschein koramt und dass dies auch von jedem Theile eines Korpers gilt, dass also <^, jj;, ^ continuirliche Functionen der Zeit sind, d. h. ihre Zuwiichse verscUwinden um so mehr je kleiner der entsprechende Zuwachs der Zeit ist. Die von den verschiedenen Lagen des Punktes A zu den verschiedenen Zeiten gebildete Curve nen- nen wir die Bahn dieses Punktes, denjenigen Theil derselben, welcher alien Lagen, die wahrend einer gegebenen Zeit durchlaufen werden ent- spricht den wahrend dieser Zeit zuriick gelegten Weg. Nicht ganz so sicher als die Continuitat der Functionen <^, ;jj;, i|r ist es, ob sie auch differenzirbar sind. Man driickte sich in der alien Mechanik folgendermassen aus. Es lege ein Punkt eines Korpers, wiihr- end einer sehr kleinen Zeit Si einen sehr kleinen Weg hs zuriick. Es sei nun a priori evident, dass sich wahrend dieser kleinen Zeit, die Umstiinde, unter denen sich der Korper befindet nur sehrwenig geiindert haben kon- nen, dass es daher, wahrend der niichst folgenden Zeit ht wieder einen sehr- nahe gleichen und gleich gerichteten Weg hs zuriicklegen muss, so dass also fiir kleine Zeiten sowohl der Weg als auch die Coordinatenzu- wiichse der verstrichenen Zeit proportional sein miissen. Man glaubte damals iiberhaupt, dass jede iiberall endliche continuirliche Function einen Differenzialquotienten haben muss. Weierstrass hat bekanntlich gezeigt, dass dies ein Irrthum ist. Bezeichnen wir z. B. mit y die Weierstrass- ische Keihe so nahert sich der Zuwachs des j/, der irgend einem Zuwachse des X entspricht an alien Stellen immer mehr der NuUe, wenn sich der betreffende Zuwachs das x der Nulle nahert und trotzdem nahert sich der Quotient beider Grossen niemals einer bestimmbaren Grenze. Bei der deductiven Darstellung ergibt sich hieraus wieder nicht die mindeste Schwierigkeit. Wir konnen ja dann unser Bild formen, wie wir wollen und einfach die Differenzirbarkeit von vornherein in dasselbe aufneh- men, es damit rechtfertigend, dass das Bild hinterher mit der Erfahrung stimmt. Aber jetzt ist es unsere Absicht von der Erfahrung auszugehn. Nun lehrt uns zwar diese, dass sehr hiiufig, wiihrend kleiner noch beob- achtbarer Zeiten der Weg eines Punktes eines Korpers um so genauer 288 Ludwig Boltzmann: der verflossenen Zeit proportional ist, je kiirzer diese ist, woraus wir wolil auf die Diiferenzirbarkeit der Functionen 0, %, A|r schliessen konnen. AUein wir keunen auch Beispiele sehr rascher Oscillationen uud konnen nicht exact beweisen, ob nicbt in gewissen Fallen Bewegungen vorhanden siud, wie z. B. die Wiirraebewegungen der Molekiile, welche durch eine der Weierstrass'schen Function iihnliche besser als durch eine Differen- zirbare dargestellt werden. Doch sind dies allerdings Diuge von gering- erer Wichtigkeit und wir woUen daher die Differenzirbarkeit der Coor- dinaten nach der Zeit unsern weitern Uberlegungen zu Grunde legen. Unter dieser Voraussetzung existiren die Ableitungen der Functionen <^, x^ "^ nach der Zeit. Wir nennen sie die Componenten der Geschwin- digkeit des Punktes A des Korpers. Die Geschwindigkeit selbst konnen wir in folgender Weise construiren : Es befinde sich der markirte Punkt des Korpers zur Zeit ^ in ^ zur Zeit t + htin B, so dass also OA, und OB die dazu gehorigen Lagenvectoren sind. Die Gerade AB ist daun das, was man die Differenz der beiden Vectoren nemit. Wir construiren nun einen Vector, welcher die Richtung AB hat und dessen Lange der Quotient AB dividirt durch St ist. Ferner suchen wir die Grenze, wel- cher sich dieser Vector in Grosse und Richtung nahert, wenn Bt immer mehr abnimmt. Die so bestimmte Lange ist die Geschwindigkeit, die Richtung aber, der sich der Vector nahert, die Geschwindigkeitsrichtung. Wir wollen hier noch eine Bemerkung anfiigen. Damit wir den Weg durch die verflossene Zeit dividiren konnen, mtissen beide durch reine Zahlen ausgedriickt sein und wir haben gesehen wie dies gesehieht. Wiihlen wir die Langeneinheit a mal so gross, so wird die Zahl, welche nun eine gewisse Liinge ausdriickt a mal kleiner. Es ist moglich, dass auch andere Grossen dieselbe Eigenschaft haben, dass sie durch a mal kleinere Zahlen ausgedriickt erscheinen, sobald wir die Langeneinheit a mal vergrossern. Von alien so beschaffenen Grossen sagen wir dann, dass sie die Dimension einer Liinge haben. Jede Liinge, (der Weg, die Coordinaten etc.) hat daher selbstverstiindlich die Dimension einer Liinge. Die Zahl, welche uns die Zeit t ausdriickt, ist natiirlich unabhiingig von der gewahlten Liingeneinheit, wird aber a mal kleiner, wenn wir die Zeiteinheit a mal grosser wiihlen und wir sagen von jeder Grosse, welche durch eine Zahl von dieser Eigenschaft ausgedriickt wird, sie habe die Dimension einer Zeit. Die Geschwindigkeit wird durch den Quotienten zweier Zahlen gemessen, woven der Ziihler die Dimension einer Liinge, der Nenner die einer Zeit hat. Sie ist also sowohl von der Wahl der Ziveite Vorlesung. 289 Langen als audi von der Zeiteinheit abhiingig, und wird a mal kleiner, weun die erstere a mal grosser, dagegen a mal grosser, wenn die letztere a mal grosser gewiihlt wird. Wir sagen dalier ihre Dimensionen sind : Liinge dividirt durch Zeit, was aber hiemit jeder geheimnisvollen oder metaphysischen Bedeutung eutkleidet ist. Man redet vielfach statt von dem Quotienten der Zahl welche die Zeit ausdriickt in die, welclie die Liinge ausdriickt, einfach von dem Quotienten einer Zeit, in eine Liinge. Man hat da den Begriff der Division erweitert und muss den Quotienten einer Zeit in eine Liinge ganz neu deliniren, geradeso wie man den Begriff einer negativen oder gebrochenen Potenz neu definirt und darunter einen Bruch respektive eine Wurzel versteht. Der Vortheil dieser neuen I^efi- nitiou besteht darin, dass man vielfach Rechnungsregeln, welche fiir die friihere Definition bewiesen wurden auf die neue Definition iibertragen kann. Man darf aber nicht a priori schliessen, dass dies von alien Rech- nungsregeln gilt ; es muss vielmehr die Ubertragbarkeit von jeder Rech- nungsregel besonders bewiesen werden. Ebenso ist es eine vollstilndig neue Definition, wenn wir unter der zweiten oder dritten Potenz eines Cen- timeters die geometrische Figur eines Quadrats oder Wiirfels von 1 cm. Seitenlange verstehen und es muss gerechtfertigt werden, in wie wait diese neue Definition zweckmiissig ist. Die Fixirung des Begriffs der Beschleunigung und ihrer Componenten nach den drei Coordinaten- richtungen hat nun nicht mehr die mindeste Schwierigkeit. Sei A C der Vector, welcher in Grosse und Richtung die Geschwindigkeit zur Zeit t, OD der, welcher sie zur Zeit t -\- U darstellt. Wir ziehen die Gerade CD, also die Differenz der beiden Vectoren. Dieselbe wird sehr klein sein, wenn ^t sehr klein ist. Wir erhalten aber eine endlich bleibende Gerade, wenn wir sie im Verhiiltnis der Zeiteinheit zur Zeit ht vergross- ern, wobei ihre Richtung unveriindert bleiben soil. Die Grenze, welcher sich der so vergrosserte Vector CD mit abnehmendem U niihert, heisst der Beschleunigungsvector, seine Liinge stellt die Grosse, seine Richtung die Richtung der Beschleunigung dar. Seine Componenten in den drei Coordinatenrichtungen heissen die Componenten der Beschleunigung. Man iiberzeugt sich in bekannter Weise, dass es die zweiten Ableitungen der friiher mit ;^, <^, i^ bezeichneten Functionen sind. Wir miissen daher die Voraussetzung machen, dass diese Functionen auch zweite Ableitungen haben. Man iiberzeugt sich auch leicht, dass die Zahl, welche die Grosse der Beschleunigung ausdriickt wieder sowohl von den gewiihlten Liingen als von der gewiihlten Zeiteinheit abhiingt und a mal 290 Ludivig Boltzmann : kleiner wird, wenn erstere a mal so gross, dagegen cfi mal grosser, wenn die Zeiteinheit a mal so gross gewiihlt wird. Wir werden dalier sagen, die Beschleunigung hat die Dimensionen : Lange dividirt durch das Quadrat der Zeit. Wir konnen wieder die Beschleunigung als solche definiren als den Quotienten einer Zeit in eine Geschwindigkeit oder des Quadrats einer Zeit in eine Lange ; diirfen aber die letzteren Defini- tiouen nur mit einer gewisseu Vorsicht anwenden, da sie Erweiterungen des Begriffs der allgebraischen Division darstellen, fiir welche die An- wendbarkeit der verschiedenen in der Algebra bewiesenen Rechnungs- regeln erst neu erprobt werden muss. Nachdem wir diese Begriffe moglichst an die Erfahrung ankniipfend entwickelt haben, miissen wir zur Aufstellung der Gesetze iibergehn, nach welchen die Bewegung der Korper geschieht. Wir werden da natiirlich wieder nicht mit Aufstellung der Gesetze fiir die Bewegung eines materiellen Punktes beginneu, da dieser eine reine Abstraction ist. Wir werden uns natiirlich auch nicht der Illusion hingeben, dass wir ohne alle Abstractionen auskommen. Wir konnen nach meiner Ansicht nicht einen einzigen Satz aussagen, welcher wirklich nur eine reine Erfahrungs- thatsache wiire. Die einfachsten Worte wie gelb, siiss, sauer etc., welche blosse Empiindungen anzugeben scheinen, driicken schon Begriffe aus, die bereits aus vielen Erfahrungsthatsachen durch Abstraction gewonnen worden sind. Wenn Gothe sagt, die Erfahrung ist nur zur Halfte Erfahrung so will er mit diesem scheinbar paradoxen Satze sicher aus- driicken, das wir bei jeder begrifflichen Auffassung der Erfahrung oder Darstellung derselben durch Worte schon iiber die Erfahrung hinaus- gehen miissen. Die oft aufgestellte Forderung, dass die Naturwissen- schaft nie iiber die Erfahrung hinausgehen diirfe, soUte daher nach meiner Ansicht dahin ausgesprochen werden, dass man nie zu weit iiber die Erfahrung hinaus gehen diirfe und nur solche Abstractionen ein- fiihren solle, die sich bald wieder an der Erfahrung priifen lassen. Wir werden auch nicht das Triigheitsgesetz an die Spitze stellen. Dieses mag theoretisch das einfachste Gesetz der Mechanik sein, physikalisch ist es keineswegs das einfachste, da es eine ganze Reihe von Abstractionen zur Voraussetzung hat, worauf ich schon friiher hingewiesen habe. Als die beiden physikalisch einfachsten Fiille erscheinen uns vielmehr erstens der der relativen Ruhe zweitens, der freie Fall eines schweren Korpers. Wie wir sahen, konnen wir einen Korper niemals gauz den iiussern Eiu- fliissen entziehen. Wenn nun solche Einfliisse vorhauden sind, von denen Ziveite Voi'lesiing. 291 jeder fiir sich allein eine Bewegung erzeugen wiirde, wenn aber unter dem vereinten Einflusse aller relative Ruhe gegen das Bezugssystem Platz greift, so sagen wir alle Ursachen der Relativbewegung compensiren sich. Ich konnte niich audi des gebrauchlicbsten Ausdruckes bedieuen, die Kraf te balten sich das Gleichgewicht, allein ich will absichtlich die gewohn- ten Ausdriicke vermeiden, well wir mit denselben unwUlkiirlich eine Jlenge von Vorstellungen verbinden, die sich dann, ohne dass wir es woUen, unkontrolirt in unsere Schlussweise einschmuggeln und so den Schein erwecken, als batten wir etwas bewiesen, was wir nur gemass unserer alten Denkgewohnheit und Ideenassociation ohne Begriindung beigefiigt haben. Ich will ausserdem das Wort Kraft vermeiden, ehe ich gleich- zeitig auch von der Masse sprechen kann. Endlich betrachten wir liier nur die relative Bewegung. Es kann aber ein Korper relativ gegen seine Umgebung ruhen, ohne dass sich die auf ihn wirkendeu Kriifte das Gleichgewicht zu halten brauchen wie ein Korper, der relativ gegen einen mit Beschleunigung sich bewegenden Lift ruht. Wir betrachten nun einen bestimmten Fall, wo die Ursachen der relativen Bewegung compensirt sind. Ein schwerer Korper sei an einen diinnen Faden aufgehiingt. Wir konnten da meinen, dass gar keine Bewegungsursachen vorhanden sind. Doch finden wir, dass sofort Bewegung eintritt, wenn ^vir den Faden entfernen. Es miissen also mindestens zwei Bewegungsursachen vorhanden gewesen sein, welche sich gegenseitig compensirten. Wenn wir die nach Entfernung des Fadens eintretende Bewegung analysiren, so finden wir, dass sie, wenn gewisse allgemeine Bedingungen erfiillt sind, sehr angenahert immer in derselben Weise vor sich geht. Diese allgemeinen Bedingungen sind folgende. Die Oberfliiche des Kcirpers darf nicht zu gross gegen dessen Gewicht sein, es darf keine heftige Luftbewegung um den Korper herum stattfinden, der Faden muss ohne Erschutterung durchgeschnitten oder ruhig durch Verbrennung oder sonst wie veruichtet worden sein. Dieselbe Bewegung tritt auch ein, wenn wir den Korper anfangs mit der Hand oder einer Zange oder einer sonstigen Vorrichtung halten und plotzlich ohne Erschiitterung sich selbst iiberlassen. Das Charakteristische aller dieser Anfangs- bedingungen besteht darin, dass siimmtliche Punkte des Korpers in den ersten Momenten der Bewegung sehr kleine Geschwindigkeiten haben. Wir konnen daher anniihernd voraussetzen, dass siimmtliche Punkte des Korpers im ersten Momente der Bewegung keinerlei Anfangsgeschwindig- 292 Ludivig BoUzmann: keit batten. Wenn diese Bedingungen erflillt sind, so lehrt die Erfalir- ung, dass der Korper stets fast genau nach denselben Gesetzen sich bewegt, wo immer er in der Niihe der Erdoberflache sicb selbst iiberlassen worden sei. Die Bewegung bestimmen wir dabei uatiirlich einstweilen relative gegen die Erde. Wenn wir uns noch auf einen nicbt zu grossen Theil der Erdoberflache bescbriinken, so ist aucb die Richtung der Bewegung iiberall dieselbe ; es ist die des Fadens, der friiher den Korper trug. Die Erfahrung lehrt nun fiir diese Bewegung die folgen- den Gesetze. Erstens der Korper bewegt sich parallel zu sich selbst, d. h. alle Punkte desselben legen in gleichen Zeiten, gleiche und gleichgerich- tete Wege zuriick. Da also die Bahn fiir jeden Punkt dieselbe ist, so kann man sie als die Bahn des gauzen Korpers bezeichnen. Zweitens, alle diese Wege sind geradlinig. Drittens, die Geschwindigkeit wachst fort- wilhrend, die Beschleunigung ist jedoch iiberall, zu alien Zeiten und sogar fiir alle Korper dieselbe. Dass diese Gesetze in der Natur nur mit grosserer oder geringerer Anniiherung realisirt sind, wurde bereits besprochen. Wir konnen nun dasselbe Experiment wiederholen, nur dass wir dem Korper im Momente, wo wir ihn sich selbst iiberlassen einen Stoss geben, oder sonst wie bewirken, dass er schon anfangs eine Geschwindigkeit hat. Da wir die Siitze vom Schwerpunkt und der Drehung der Korper noch nicht kennen gelernt haben, so miissen wir uns dabei auf die Fiille beschriinken, wo sich der Korper wieder parallel zu sich selbst bewegt. Es wird dies zwar nicht immer eintreten und wir konnen die Bedingungen dafiir, dass es eintritt noch nicht angeben, aber in vielen Fallen wird dies stattfinden und diese Fiille wollen wir vorliiufig allein betrachten. In alien diesen Fiillen legen wieder alle Punkte des Korpers gleiche Bahnen zuriick, welche wir also als die Bahn des Korpers bezeichnen konnen. Die ganze Bewegung kann wieder dahin beschrieben werden, dass die Beschleunigung immer vertikal nach abwiirts gerichtet und iiberall zu alien Zeiten und fiir alle Korper dieselbe ist. Da wir nun gesehen haben, dass die Bewegung, wenn wir sie an verschiedenen Stellen im Zimmer oder in dessen Umgebung beginnen lassen, immer in ganz gleicher Weise vor sich geht, so miissen wir schliessen, dass die Beweg- ungsursache, welche wir Kraft nennen, daselbst iiberall unveriinderlich dieselbe ist. Anderseits ist auch die Beschleunigung unveranderlich dieselbe, wir konnen daher schliessen, dass wenigstens in diesem speziellen Falle die Beschleunigung das fiir die Kraft Massgebende ist und well Dritte Vorlesung. 293 erstere iiberall vertikal nach abwarts gerichtet ist, so sagen wir auf den Korper ^v^rkt eine constante vertikal nach abwarts gerichtete Kraft die Schwere. Dritte Vorlesttng. Um tiefer in die Gesetze der Bewegungen einzudringen, miissten wir jetzt die niichst einfachsten Fiille betrachten. Ein naives Gemiit konnte da wohl meinen, dass wir nun die Gesetze nach denen ein Grashalm wiichst, untersuchen soUten. Leider aber wissen wir iiber diese noch heute fast gar nichts. Besser ware es schon die Gesetze der Wirkung gespannter Schniire, Federn etc. zu betrachten. Allein auch da treten die Beweg- nngsgesetze nicht in grosster Einfachheit hervor. Der historische Gang war vielmehr der folgende. Nachdem Galilei die Bewegungsgesetze soweit wir sie bisher betrachtet haben, gefunden hatte, suchte Newton sie vor allem auf die Bewegung der Gestirne anzuwenden und auch von ihm gilt, was Schiller von WaUenstein sagte: ^^Fiirwahr ihn hat kein Wahn betrogen als er auf warts zu den Sternen sah." Dem Laufe der Sterne hat er die Bewegungsgesetze abgelauscht, auf denen alle heute in der Technik und Machinenlelire benutzten Formeln ja iiberhaupt unsere ganze moderne Naturkenntnis basirt. Freilich bringt der Ubergang zur Sternenwelt manche Unbequemlichkeit mit sich. Erstens miissen wir um einfache Gesetze zu erhalten, unser altes Bewegungssystem, als welches der Erdkorper diente, verlassen und ein relativ gegen den Fixsternhimmel sich nicht drehendes Coordinatensystem wiihlen. Zweitens ist auch die Bedingung, das die Planeten sich parallel zu sich selbst bewegen nicht erfiillt. An ihre Stelle tritt der Umstand, dass ihre Entfernungen vom Beobachter so gross sind, dass ihre einzelnen Theile iiberhaupt nur schwer unterschieden werden konnen, so dass wir also in der ersten Anniiherung mit welcher wir uns wieder begniigen, iiberhaupt die Bahnen der ver- schiedenen Punkte eines und desselben Planeten gar nicht unterscheiden konnen. Wir konnen also wohl auch annehmen, dass die Gesetze dieselben wilren, wenn die Himmelskurper sich parallel zu sich selbst bewegten. Wir kommen also hier einestheils dem Begriffe des materiellen Punktes sehr nahe, da die Ausdehnung der bewegten Korper so klein gegen die Liinge ihrer Bahn ist, dass letztere fiir alle Punkte der Korpers merklich gleich wird. Anderseits aber sind wir von dieser Idee so weit entfernt als moglich, da wir es mit Korpern zu thun haben, die nichts weniger als materielle Punkte, vielmehr oft grosser als unser ganzer Erdkorper sind. 294 Ludivig Boltzmann: Die Beobachtung und Messung lehrt, dass sich im Weltraume haufig um einen Centralkorper ein System von Himmelskorpern bewegt, welche wir die Trabanten nennen. Wir erhalten die einfachsten Gesetze, wenn wir die Bewegung der Trabanten auf ein Coordinatensystem bezielien, dessen Anfangspunkt im Mittelpunkte des betreffenden Centralkorpers liegt nnd dessen Axen dreien fest mit dem Fixsternhimmel verbundenen Geraden stets parallel bleiben. Fiir die Bewegung der Trabanten gelten erfahrungsgemiiss die drei Keppler'schen Gesetze. Da beim freien Falle die Beschleunigung eine so wichtige Rolle spielte, so wollen wir auch in diesem Falle die Beschleunigung berechnen, welche irgend ein Tra- bant in seiner Bewegung erfiihrt. Diese Rechung ist sehr bekannt und ganz leicht. Es hat sie Kirchhoff in seinen Vorlesungen iiber Mechanik in sehr eleganter Form durchgefiihrt. Man findet aus dem ersten und zweiten Keppler'schen Gesetze, dass sie fiir jeden Trabanten zu jeder Zeit gegen den Centralkorper gerichtet und dem Quadrate des Abstandes r k von demselben verkehrt proportional, also in der Form — darstellbar ist. Aus dem dritten Keppler'schen Gesetze ergibt sich ausserdem, dass die Constante k von Centralkorper zu Centralkorper verschieden ist, aber fiir alle Trabanten eines und desselben Centralkorpers denselben Wert hat. Da wir schon bei der Schwere die Beschleunigung als das mass- gebende fiir die Bewegvingsursache oder Kraft erkannt haben, so wollen wir auch hier sagen, der Centralkorper iibt auf jeden Trabanten eine Kraft aus, welche die Richtung der vom Mittelpimkte des Trabanten gegen den des Centralkorpers gezogenen Geraden hat und der Lange dieser Geraden verkehrt proportional ist. Diese ist einstweilen sonst nichts als ein anderer Ausdruck fiir die Thatsache des Vorhandenseins dieser Beschleunigung. Newton hat diesen Satz sofort enorm verallge- meinert indem er annahm, dass iiberhaupt jeder Himmelskorper auf jeden audern ja jedes materielle Theilchen auf jedes andere eine solche Kraft ausiibt. Wenn daher ein Himmelskorper mehreren andern so nahe ist, dass er von ihnen eine merkliche Einwirkung erfiihrt, so haben wir den Fall, dass er gleichzeitig aus verschiedenen Ursachen verschiedene Be- schleunigungen nach verschiedenen Richtungen erfiihrt. Da wir die Beschleunigung durch einen Vector dargestellt haben, so ist es nicht die einzig notwendige, aber doch bei weitem die nahe liegendste, einfachste Annahme, dass sich diese Beschleunigungen wie Vectoren addiren. In der That zeigt sich, dass man unter dieser Annahme immer Ubereinstim- Dritte Vorlesimg. 295 mung mit der Erfahrimg erhiilt. Es ergeben sich die Storungen der Plane- ten untereinander, der Monde durch die Sonne und durch die Planeten in genauer Ubereinstimmung mit der Erfahiung. Man kann jetzt auch den Horizont erweitern und alle Himmelskorper auf ein und dasselbe mit dem Fixsternliimmel fest verbundene Coordinatensystem beziehen und erhiilt audi die Bewegung der Centralkorper gegen dieses Coordinaten- system in Ubereinstimmung mit der Erfahrung. Die Schwere erweist sicb als identisch mit der Anziehung des Erdkorpers auf den schweren Korper. Schliesslich zeigen die Erscheinungen der Ebbe und Flut, die Versuche von Cavendish, Maskelyne, Airj- etc. die Riehtigkeit der Aus- dehnung des Newton'schen Gesetzes auf die irdischen Korper. Da die wirkliche Beschleunigung immer die Vectorsurame der verschiedenen von den wirkenden Korpern erzeugten Beschleunigung ist, so folgt jetzt als spezieller Fall des Newton'schen Gesetzes, dass ein Korper, welcher von alien iibrigen so weit entfernt ware, dass keiner derselben eine Wirk- ung auf ihn ausiiben wiii-de, zu alien Zeiten die Beschleunigung Null erfiihre. Wir erhalten also erst jetzt das Triigheitsgesetz. Selbstver- stiindlich ist hiemit liber die Ursache der Newton'schen Kraft, ob die- selbe eine direkte Fernwirkung ist oder durch ein Medium vermittelt wird, nicht das mindeste prajudicirt. Wir konnten auch jetzt schon den Begriff der Masse ableiten. Die Massen zweier Centralkorper wiirden sich ja wie die ihnen entsprechenden Werte der Constanten k des Gravi- tationsgesetzes verhalten und durch den Cavendish 'schen Versuch konnte diese Definition auch auf irdische Korper ausgedehnt werden. Allein wir wiirden da die Proportionalitat der Constante h mit der als Triigheits- widerstand definirten Masse vorwegnehmen, was offenbar ein logischer Fehler wiire. Wir miissen daher zum Begriffe der Masse auf ganz an- derem Wege zu gelangen suchen. Wir haben bisher als das Massge- bende fiir die Kraft die Beschleunigung betrachtet. Es konnte nun als das einfachste erscheinen, die Grosse der Beschleunigung, welche ein Korper durch einen andern erfiihrt, einfach als die Grosse der Kraft zu bezeichnen, welche der letztere auf den ersteren ausiibt. Es geschieht dies auch manches Mai und man bezeichnet die so definirte Kraft als die beschleunigende Kraft. Allein im AUgemeinen ist es besser einen andern Begriff einzufiihren. Wir deukeu niimlich beim Worte Kraft in erster Linie an die Muskelanstrengungen, welche wir ausiiben konnen. Nun liegt freilich kein Grund vor, ja es ware ganz verkehrt anzunehmen, dass jedes Mai, wenn unbelebte Korper Kriifte aufeinander ausiiben etwas 296 Ludiuig Boltzmann: vorhanden sein miisse, was diesen Muskelanstrengungen irgendwie ent- spricht. Allein es wird sich doch empfehlen, wenn wir die Bezeich- nungen so wahlen, dass sie sich den durch diese Muskelanstrengungen erworbenen BegriflFen moglichst gut anschliessen. Wir sahen, dass alle Korper durch die Schwere die gleiche Beschleunigung erfahren. Wiirden Avir nun diese ohne weiteren Factor als Mass der Kraft wahlen, so ware die Kraft, welche die Schwere auf sie ausiibt, (das Gewicht) fiir alle Korper dasselbe. Nun lehrt aber die tiigliche Erfahrung, dass die Mus- kelanstrengung welche wir brauchen, um den Fall aufzuheben, fiir ver- schiedene Korper sehr verschieden ist. WoUen wir daher mit unseren Vor- stellungen im Einklang bleiben, so miissen wir sagen, dass audi die Schwere auf die verschiedenen Korper sehr verschiedene Kriifte ausiibt, dass aber die Korper von gi-osserem Gewichte dieser beschleunigenden Wirkung der Schwere einen grosseren Widerstand, den Triigheitswiderstand, die Masse, entgegensetzen, so dass erst in folge beider Umstande zusammen alle Korper die gleiche Beschleunigung erfahren. Ura die Masse in dieser Weise als Triigheitswiderstand zu defiuiren, miissen wir an verschiedene Korper die gleiche Kraft anbringen. Das Verbal tnis ihrer Massen kon- nen wir dann als das verkehrte Verhiiltnis der Beschleunigungen defi- niren, die sie durch gleiche Kriifte erhalten. Aber darin liegt eben die grosste Schwierigkeit wie man die Gleichheit der Kriifte, wenn diese auf verschiedene Korper wirken, ohne logischen Fehler feststellen soil. Man konnte zwei Korper dem Zuge gleich beschaffener gleich gespannter Schniire oder elastischer Federn unterwerfen. Allein da miisste man erst durch complicirte der Erfahrung entnommene Argumente als wahr- scheinlich hinzustellen suchen, dass gleich beschafifene Schniire auf zwei ganz verschiedene Korper dieselben Kriifte ausiiben, was gewiss nicht a priori evident ist. Wir konnten auch nach Mach einfach den Satz der Gleichheit der Wirkung und Gegenwirkung postuliren. Wenn dann bloss zwei Korper in Wechselwirkung begrilfen sind, so wiire die Gleich- heit der Kriifte, welche auf beide Korper wirken evident. Wenn sie sich zudem nur Parallelverschiebungen ertheilen, so wiire das Verhiiltnis ihrer Massen einfach zu definiren, als das verkehrte Verhiiltnis der Be- schleunigungen, welche an ihnen zu beobachten sind. Allein bei der Wirkung dazwischen gebrachter Schniire, Fiiden etc. haben wir eigent- lich schon iramer mehr als zwei in Wechselwirkung begriffene Korper und es wiirde auch die Deformation dieser Zwischenkorper in Betracht zu Ziehen sein. Der von Mach angenommene Fall konnte also in reiner Dritte Vorlesung. 297 Weise eigentlich nur bei director Femwirkung vorkommen und es wiire sehr misslich, wenn man vom rein empirischen Standpunkte aus die directe Fernwirkung a priori anuelimen miisste. Streintz sucht eine einwui-fsfreie Delinition in folgender Weise zu gewinnen. Er denkt sich irgend ein System beliebiger Korper. In demselben kommen zwei Korper Ky und K^ vor. Diese rulien im ersten Augenblicke und beginnen sich dann mit Beschleunigung aber jeder parallel zu sich selbst zu be- wegen. Es soil nun die Bewegung beider Korper dadurch aufgehoben werden konnen, dass man sie starr mit einander verbindet. Dies ver- wendet er als Kriterium, dass friiher auf jeden genau die gleiche Kraft wirkte, weil sich beide Kriifte durch blosse starre Verbindung jetzt auf- heben. Er uennt diese Begriffsbestimmung der Gleichheit der Kraft die statische. Sie hat das fiir sich, dass sie das Princip der Gleichheit der Wirkung und Gegenwirkung involvirt, wie man sofort sieht, wenn man den speziellen Fall betraclitet, dass das ganze System bloss aus den zwei auf einander wirkenden Korpern K^ und K^ besteht. Sie hat aber doch auch manches Willkiirliche. Dass durch die starre Verbindung die Wirk- ung der iibrigen Krafte nicht gestort wird, kann -wieder hochstens erfalirungsmiissig wahrscheinlich gemacht werden. Dass die Verbin- dungskriifte sich zu den iibrigen addiren, setzt schon gewisse Siitze der Statik voraus. Noch grosser wiirden die Schwierigkeiten, wenn, die Korper K^ und K^ anfangs in Bewegung begriffen wiiren. WoUte man da nicht von vornherein annehmen, dass die Krafte bloss von der rela- tiven Lage abhitngen, durch den aus der plotzlichen starren Verbindung resultirenden Stoss nicht gestort werden und Ahnliches, so miisste ihre Beschleunigung durch eine die Bewegung gestattende und auf beide Korper bloss beschleunigend wirkende plcitzlich eingeschaltete Feder aufgehoben werden. Hiilt man einmal an der Streintz'schen Vorstellungr fest, so hat die Definition der Massenverhaltnisses welter keine Schw^erig- keit. Die Massen der beiden Korper K^ und K^ verhalten sich dann umgekehrt, wie die Beschleunigungen, die sie ira ei-sten Falle, wo keiue staiTe Verbindung vorhanden war, erhielten, da ja damals auf beide gleiche Krafte wirkten. Natiirlich ist sowohl bei der Mach'schen als bei der Streintz'schen Definition noch immer erfordei'lich, sich auf besondere Erfalirungssiitze zu berufen, vermoge welcher das Massenverhiiltnis zweier Korper immer gleich ausfiillt, unter was immer fiir Umstanden man den hiezu dienenden Versuch angestellt haben mag und vermoge welcher das Verhaltnis der Massen der Korper K^ and K^ stets gleich 298 Ludwiy Boltzmann: dem Producte der beiden Massenverhaltnisse der Korper K^, K^ und K^^, K^ ist. Zu bemerken ist noch, dass wir nur das Verhaltnis zweier Massen bisher definirt baben. Um die Masse durch eine Zahl auszudriicken, miissen wir irgend eine Masse willkiirlich als eine neue Einheit wahlen. Von alien Grossen, welcbe daber durch Zablen ausgedriickt werden, deren Grosse von der Wabl der Masseneinbeit abbiingig ist, werden wir sagen dass sie gewisse Dimensionen beziiglich der Masse baben. Haben wir den Begriff der Masse in der einen oder andern Weise festgestellt, so bat die Definition der Kraft im gewobnlicben Sinne oder wie man aucb sagt, der bewegenden Kraft keine Scbwierigkeit mebr. Dieselbe ist das Product der Masse in die Bescbleunigungen und bat daber beziiglicb der Masse die Dimension eins. Da sicb die Bescbleunigungen wie Vectoren addi- ren, so gilt dies aucb von den Kraften, wenigstens insoweit wir diese bisber betracbtet baben. Dieser Satz vom Kraftenparallelogramm sowie die iibrigen bisber entwickelten Siitze, konnen nun aucb auf die Statik und Dynamik der durcb gespannte Fiiden oder durcb Federn erzeugten Druck und Zugkriifte iibertragen werden. Natiirlicb zunacbst bloss in dem idealen Falle, dass die Bewegung der einzelnen Tbeile der Faden und Federn nicht betracbtet wird und dass die bewegten Korper sicb stets parallel zu sicb selbst bewegen. Es konnte so z. B. die Mecbanik der Atwood'scben FaUmascbine mit Hilfe des bisber Entwickelten obne weiteres discutirt werden. Aus dem Umstande, dass sicb das Newton'sche Gravitationsgesetz in symetriscber Weise beziiglicb beider wirkender Korper aussprecben muss und dass die Anziebungsconstante K fiir alle Trabanten desselben Cen- tralkorpers gleicb ist, leitet man leicbt ab, dass diese gleich dem Producte der Massen der beiden wirkenden Korper in eine fiir das ganze Universum constante Grosse sein muss, wiibrend die Thatsacbe, dass alle Korper durcb die Scbwere die gleiche Bescbleunigung erbalten, scbon lebrt, dass das Gewicbt der Masse proportional sein muss. Wir sind aber noch sebr weit davon entfernt aus den bisber entwickel- ten Grundlagen siimmtliche Satze der Mecbanik ableiten zu konnen. Wir haben ja bisher bloss die Bewegung eines festen Korpers parallel zu sicb selbst betracbtet und haben den wichtigen Begriff des Angriffspunk- tes einer Kraft noch gar nicht gewonnen. Um diesen zu erbalten, um die Drebung der starren Korper, die Deformationen der elastiscben und die Bewegungen der fiiissigen behandeln zu konnen, miissen wir von neuen Dritte Vorlesung. 299 Thatsachen ausgehen. Wenn ein Faden an einem Korper befestigt ist oder eine Feder auf eine einzige Stelle desselben driickend wirkt, so gibt es stets eine ganz kleine Partie des Kdrpers, -welclie zuniichst von der Kraft afficirt wird. Losen wir diese los und stellen einen kleinen Zwischenraum zwischen ihr und den iibrigen Tbeilen des Korpers her, so wird derselbe erst wieder afficirt, wenn dieser Zwischenraum durch die Bewegung des kleinen abgetrennten Theiles sich ausgefiillt hat. Wir nennen daher diesen Theil die Angriffstelle und konnen sie wieder zu einem Angriffspunkte idealisiren. Wir miissen nun noch die bekann- ten Satze iiber die Versetzbarkeit von Kriiften an starren Korpern als idealisirte Erfahrungsthatsachen beifiigen. Mittelst derselben konnen wir dann in ebenfalls hinlanglich bekannter Weise die Sixtze iiber das Gleichgewicht von beliebigen Krjiften, welche auf einen starren Korper wirken, die Satze von den statischen Momenten ableiten. Wir schlagen hier insoferne einen analogen Vfeg ein, wie Streintz bei der Definition der Masse, als wir von der Statik ausgehen und erst von dieser zur Dynamik gelangen. Die Satze von den statischen Momenten haben wir da freilich zunachst bloss fiir eine begrenzte Zahl von Kraften bewiesen, von denen jede nur auf einen einzelnen Punkt des Korpers wirkt. Wir miissen dazu noch die Annahme hinzufiigen, dass man im Falle, wo die Krjifte den Korper oder einen ausgedehnten Theil desselben als Gauzes anfassen die Sache immer so ansehen kann, als ob sie auf sehr viele respective unendlich viele Punkte seiner Oberfliiche oder seines Innern gerade so wirken wiirden, als ob an jedem dieser Punkte eine ein wenig gespannte Schnur oder eine ein wenig driickende Feder befestigt ware. So muss mann z. B. von der Schwere annehmen, das sie gleichmassig auf alle Punkte des schweren Korpers wirkt. Einen andere Weg, auf welchem man den Ubergang von der Bewegung parallel zu sich selbst zur Drehbewegung versuchen konnte, will ich hier nur ganz kurz andeuten. Wir konnen aus dem Principe der Erhaltung der lebendigen Kraft folgenden Satz ableiten. Wenn auf einen festen Korper eine Kraft wirkt, die ihn nur parallel zu sich selbst zu bewegen sucht, so muss immer eine ihrer Richtung paraUele Gerade, welche wir die Angriffslinie nennen wollen, von solcher Beschaffenheit existiren, dass wenn man einen beliebigen Punkt des festen Korpers, welcher auf derselben liegt, festhalt, der Korper ins Gleichge^vicht kommen muss. In gleicher Weise kann man beweisen, dass, wenn zwei feste Korper K^ und K^ so in Wechsel- wirkung begriffen sind, dass jeder dem andern nur eine Bewegung parallel 300 Ludwig Boltzmann : zu sich selbst ertheilt, Wirkung unci Gegenwirkung gleich sein muss und die Angriffslinien zusammenfallen miissen. Denkt man sich dann einen Punkt A del" gemeinsamen Angriffslinien festgehalten, so muss das ganze System ins Gleichgewicht kommen. Jeden solchen Punkt konnen wir als Angriilspunkt der Kraft betrachten. An diesen Begriff des Angriffs- punktes, konnen dann ebenfalls die Satze von den statischen Momenten gekniipft werden. Hat man einmal diese Satze so oder so gewonnen, so muss man zur Zerlegbarkeit der Korper in Volumelemente iibergehen. Man flihrt wieder als Erfahrungssatz an, dass sehr viele Korper, wenigstens mit geniigender Annaherung in zwei Korper von je der halben Masse zerlegt werden, wenn man sie in zwei Tlieile von gleichem Volumen zerschneidet. Analog, wenn man sie in drei gleiche Theile theilt u. s. f . Denkt man sich dies ins Unendliche fortgesetzt, so gelangt man zu einem Satze, den man daliin aussprechen kann dass diese Korper aus unendlich vielen Volumelementen dv bestehen und die in jedem Volumelemente enthaltene Masse dm = pdv ist. Bei andern inhomogenen Korpern gilt dies wenig- stens nahezu fiir jeden kleinen Volumtheil des Korpers, so dass wir die- selbe Formel anwenden konnen, wenn wir p als von Punkt zu Punkt veranderlich betrachten. Was nun die Krafte anbelangt, welche die Volumelemente fester Korper aufeinander ausiiben, so muss man aimehmen, dass jedes Volum- element nur auf die unmittelbar benaehbarten wirkt und dass es auf alle der Trennungsfliiche anliegenden Punkte Krafte ausiibt, welche gerade so wirken, als ob daran ziehende gespannte Fiiden oder driickende, auf- gestiitzte Sttibe befestigt wiiren. Wenn die Trennungsfliiche eben und geniigend klein ist, so muss man zudem annehmen, dass diese Kriifte gleichmiissig auf alle der Trennungsflache anliegenden Punkte wirken. Diese Satze konnen wol kaum direct erfahrungsmassig bestiitigt werden und linden ihre Rechtfertigung nur in der nachherigen Ubereinstimmung der aus ihnen entwickelten Satze mit der Erfahrung. Wendet man den Satz von den statischen Momenten auf ein Volumelement an, so findet man, dass im Falle des Gleichgewichtes die auf ein zur a;-axe senkrechtes Fliichenelement in der ?/-Richtung wirkende Kraft gleich sein muss der auf ein gleiches zur ^-Richtung senkrechtes Fliichenelement in der a;-Richtung wirkenden Kraft, was wir den Satz X nennen woUen. Zu den bisher aufgestellten Annahmen welche wir uns als durch die Erfahr- ung geniigend motivirt dachten, sind noeh die folgenden hinzuzunehmen. Dritte Vorhsung. 301 Erstens, die elastische Kraft ist bloss von der augenblicklichen Gestaltver- anderung des betreffenden Korpers, nicht von den friilieren Zustanden desselben, noch auch von der Geschwindigkeit seiner Theilchen abhangig. Zweitens, jedes Volumelement bewegt sich nach den Gesetzen, welche wir bisher bloss fiir die Bewegung parallel zu sich selbst abgeleitet haben. Unter diesen Annahmen erhiilt man dann sofort die Gleichungen der gewohnlichen Elasticitiitslehre. Dieselben gelten natiirlich wieder nur fiir einen idealen festen Korper, alle festen Korper zeigen innere Reibung und elastische Nachwirkung, welche wir bisher ausgeschlossen haben. Auch der Satz, welchen wir den Satz X nannten, ist keineswegs a priori evident. Lord Kelvin hat sich einraal den Lichtiither, sonst ganz niit den Eigenschaften begabt gedacht, welche wir an festen Korpern wahrnehmen, nur dass er die Richtigkeit dieses Satzes X fallen liess. Wir wollen uns hier nicht in eine Discussion einlassen ob durch die Annahme Lord Kelvins das Verhalten des Lichtiithers erkliirt werden kann. Es geniigt uns, dass derselbe ohne alle inneren Widerspriiche Bewegungsgleichungen fiir die Volumelemente eines festen Korpers ausarbeiten konnte, fiir welchen der Satz X nicht gilt. Wir wollen jedoch vorliiufig bei Korpern stehen bleiben, welche den idealen Gleichungen der Elasticitiitslehre geniigen. Wenn solche Korper so wenig deformirbar sind, dass man sie als Starr betrachten kann und wenn durch beliebige Systeme derselben beliebige Bedingungsgleichungen realisirt sind, so kann man jetzt leicht nachweisen, dass fiir dieselben das vereinigte Princip der virtuellen Geschwindigkeiten und d'Alembert's gelten muss. Uenn wenn man alle Kriifte auch die elastischen ins Auge fasst, so verschwindet jedenfalls die Summe II [(„g_x)8..(»g-r)s,+(.»g-.).. da jedes Glied dieser Summe einzeln verschwindet. Da aber die Wirkuug immer gleich der Gegeuwirkung ist, so miissen die Glieder dieser Summe, welche sich auf die Wechselwirkuug der Volumelemente beziehen separat verschwinden, wenn diese starren Korpern angehoren also keiner relativen Lageniinderung fiihig sind, wahrend bei bloss einseitigen Verbindungen die bekannten Ungleichungen abgeleitet werden konnen. Dies kann auch auf Verbindungen iibertragen werden, die nur theihveise starr sind z. B. unausdehnsame Fliichen, Faden etc. ; denn diese konnen immer als Grenzfall sehr diinner elastischer Korper betrachtet werden. Man erhiilt 302 Ludwig Boltzmann : 80 das vereinigte Princip der virtuellen Verschiebiingen und d'Alembert's in der gewohnlicben Form. Erst aus diesem Principe konnen wir jetzt die Siitze von der Bewegung des Schwerpunkts, vom Triigheitsmomente etc. ableiten. Diese Siitze erscbeinen daber in unserem Systeme erst an dieser Stelle. Es kann dies nicbt anders sein ; derin darin bestebt ja das Wesen der inductiven Metbode, dass wir nicht den Begriff des materiellen Punktes als eines unausgedebnten mit Masse begabten Korpers postuliren, sondern die Scbliisse, welcbe man sonst mit HUfe dieses Begriffes macbt, erst ausfiibren, wenn wir zur Vorstellung des Volumelementes gekommen sind, welcbe wir eber der Erfabrung entnebmen zu konnen glauben, als die des materiellen Punktes. Wir konnen dann diese Siitze erst erbalten, wenn wir die Wecbselwirkung der Volumelemente bebandelt baben. Wir mussten freilicb scbon friiber an zwei Stellen vom Begriffe des matbematiscben Punktes Gebraucb macben, niimlicb als wir die Beweg- ung eines einzigen bervorgebobenen Punktes eines Korpers betracbteten und als wir Kriifte iingirten, welcbe an einem einzigen Punkte eines Korpers angreifen. Allein da war die Abstraction docb viel einfacber und klarer, als wenn wir das Ideal eines unausgedebnten mit Masse begabten Korpers bilden und dessen Drebung einfacb vemacbKissigen, obne dass wir die Gesetze der Drebung vorber kennen gelerut baben. Mancbe Satze konnten mr allerdings audi auf einem andern als dem eingescblagenen Wege gewinnen. Ein Analogon des Scbwerpunktsatzes konnten wir z. B. ableiten, indem wir ein System von ausgedebnten Korpern betracbten wiirden, zwiscben denen innere Krafte tbiitig sind und auf welcbe aucb iiussere Kriifte wirken, welcbe ibnen aber alle nur Bewegungen parallel zu sicb selbst ertbeilen. Nimmt man dazu nocb die Annabme, dass fiir die innere Kriifte Wirkung und Gegenwirkung immer gleicb ist, so wlirde ein dem Scbwerpunktsatze iibnlicber Satz fiir ein solcbes System in Wecbselwirkung begriffener ausgedebnter Korper folgen. Die Krafte, welcbe in Fliissigkeiten wirken, konnen als ein spezieller Fall, der in elastiscben Korpern wirkenden betracbtet werden und sie konnen daber ebenfalls nacb der im bisberigen auseinandergesetzten Metbode gewounen werden. Die Gestaltiinderungen der Fliissigkeiten konnen dann durcb die Bewegung der Volumtbeile derselben dargestellt werden, welcbe die entwickelten Gesetze befolgt ; nur dass die Deforma- tion des Korpers als Ganzes jetzt eine beliebig grosse sein kann. Wir baben biemit das Gebiet der eigentlichen mecbaniscben Erscbein- Dritte VorJesung. 303 ung«n erschopft. Bei den dissipativen Erscheinungen (elastische Nach- wirktmg. Reibung etc.) spielt bereits die entwickelte Warme eine Rolle. Wir konnen natiirlich die Form der friiliern Gleichungen wahren, indem wir zu den bisher abgeleiteten Kriiften noch Glieder von solcher Be- schaffenheit hinzu addiren, dass deren Summe genau gleich dem Werte der mit der Masse multiplicirten Beschleunigung wird. Diese Zusatz- glieder konnen wir dann inimer als Reibungskraft, Mittelswiderstands- kraft etc. bezeichnen, doeh hat diese Darstellung einen rein formalen Wert, wenn die Zusatzglieder in ganz complicirter Weise von der Bewegungsgeschwindigkeit, den friihem Zustanden etc. abhangen. Es bietet die Molekulartheorie da entscbieden mebr Anschaulichkeit, da sie die Zusatzglieder docb dutch langsame Drehung der IVIolekiile in neue Ruhelagen, Umsetzung der sichtbaren Bewegung in Molekularbewegung etc. einigermassen versinnlichen kann. Das Princip der virtuellen Ver- schiebung behiilt dann natiirlich, so lange es auf das Gleichgewicht ruhender Korper angewendet wird, seinen Sinn, da bei der Ruhe dissipa- tive Vorgiinge fehlen. Aber das d'Alembert'sche Princip ist auch zu einer leeren Formel herabgesunken, so bald sich in den Ausdriicken fiir die Krafte Glieder finden, welche selbst wieder von der Bewegung, von den vorhergegangenen Zustanden der Korper etc. abhangen. Uber die DarsteUung der elektrischen und magnetischen Erscheinungen will ich hier nur bemerken, dass dieselbe ebenfalls in die Form der mechanischen Gleichungen gebracht werden kann und muss, sobald diese Erscheinungen von Bewegungen ponderabler Korper begleitet sind. Des Naheren hier- auf einzugehn, ist jedoch nicht meine Absicht. Ich woUte in dem Bisherigen keineswegs eine consequente in sich abgeschlossene Darstellung der Mechanik vom inductiven Standpunkt geben. Ich wollte vielmehr bloss die Wege andeuten, auf denen eine solche vielleicht gewonnen werden konnte und namentlich die Schwierig- keiten aufdecken, mit denen ihre Durchfiihrung verkniipft ist, wenn man sich bestrebt. das innere Bild ebenso klar hervortreten zu lassen und consequent durchzufiihren, wie dieses bei der deductiven Behandlung moglich ist. Ich komme daher zu dem Resultate, dass unter den bis- herigen Darstellungsversuchen der Mechanik die deductiven, wie die von Hertz und die von mir in meinem citirten Buche gemachte vorzuziehen seien. Da aber diese deductive Darstellung wie schon zu Anfang gezeigt wurde, den Mangel hat, dass sie so lange Zeit hindurch gar nicht an die Erfahrung ankniipft und vielfach den Schein des Willkiir- 304 Ludwig BoUzmann : lichen erweckt, so wiirde es mich sehr freuen, wenn es jemanden geliinge, der deductiven Darstellung eine inductive an die Seite zu stellen, welche gleicli einfach and naturgemass vorginge und doch das innere geistige Bild in gleicher Deutlichkeit und Consequenz hervortreten liesse. Es Vierte Vorlesung. 305 ware dies wohl in einer kurzen Abhandlung kaum moglich, sondern nur in einem grosseren Buche, wo man den Grundprincipien sogleich die An- wendung auf alle speziellen FiiUe folgen lassen konute. Denn erst an der Moglichkeit der exacten Darstellung aller moglichen speziellen FtiUe erprobt sich die Klarheit mid Consequenz der Bilder, wie sich das am besten an der Hertz'schen Darstellimg zeigt, wo diese Anwendung auf spezielle Fiille fehlt. Sollten sich aber die Liicken, die sich in ineiner gegenwiirtigen Darstellung finden, nicht ausfiillen lassen, so wiirde mich auch dies freuen, denn es wiirde den definitiven Sieg der deductiven iiber die inductive Behandlungsweise bedeuten. Icli mochte gewissermassen die Vertreter der inductiven Riclitung einladen, alle Fehler, die sich in meiner gegenwiirtigen Darstellung finden aufzudecken, die Moglich- keit der genauen Durchfiihrung aller Schlussweisen, die ich hier nur kurz angedeutet habe, zu zeigen und ihre besten Krafte einzusetzen in dem Wettkampfe mit der deductiven Darstellung, damit beide mit ein- ander verglichen werden konnen und sich ini Wettstreite stets ausbilden und vervoUkommnen. Da der Energiebegriff nicht nur in der Mechanik, sondern in der ganzen Naturwissenschaft eine so wichtige Rolle spielt, so wiiren auch con- sequente Darstellungen der Grundprincipe der Mechanik vom Standpunkte der Energetik hochst erwiinscht, welche also nicht von den Begriffen der Beschleunigung und Kraft sondern von denen der lebendigen Kraft und des Potentiales auszugehen hatten. Doch miissten die betreffenden Bilder auch nach der deductiven oder inductiven Methode durchaus klar consequent und einwurfsfrei entwickelt werden und es miissten vollkom- men pracise Regeln gegeben werden, wie dieselben eindeutig auf alle speziellen Falle anzuwenden sind, ohne dass die Kenntnis der alten Mechanik dabei vorausgesetzt wird. Vierte Vorlesung. Die vierte Vorlesung begann der Vortragende mit der Vorzeigung des Modells fiir die Maxwell'sche Theorie der Elektricitiit und des Magnetis- mus, welches in dessen Buch ^^Vorlesungen iiber Maxwells Theorie der Elektricitiit und des Lichtes erster Theil " beschrieben ist. Es wurden alle dort erwiihnten Experimente mit gutem Erfolge durchgefiihrt. Hierauf gab er noch folgende Ubersicht iiber die das Princip der kleins- ten Wirkung und das Hamilton'sche Princip umfassenden Gleichungen. 306 Ludwig Boltzmann: Wenn wir die Falle einseitiger Verbindungen ausschliessen, so wird das vereinigte Princip der virtuellen Verschiebungen und d'Alemberts, wie wir sahen durch eiiie Gleichung ausgedriickt, welche wir erhalten, wenn wir den Ausdriick auf Seite 36 gleich Null setzen. Fiilirt man darin generalisirte Coordinaten ein und setzt Einfachheit halber voraus, dass sine Kraftfunctiou V besteht, welche aber die Zeit enthalten kann, so transformirt sich dieselbe in folgende Gleichung ^_^ + ^ = 0, dt dp dp wobei p irgend eine generalisirte Coordinate, q das dazu gehorige Mo- ment, T die gesammte kinetische Energie ist. Wenn jede beliebige Coordinate p zu jeder beliebigen Zeit t eine beliebige Variation hp erfiihrt, so kann man die letzte Gleichung mit Sp multipliciren und beziiglich aller p summiren. Im speciellen Falle, dass alle Zp integrable Functionen der Zeit sind, kann man noch mit dt multipliciren und iiber eine beliebige Zeit (von t^ bis t) integriren ; nach partieller Integration der dq/dt ent- haltenden Glieder folgt in dieser Weise : h(\T-V)dt = ^(,q^p-qM~y (1) wobei sich rechts die erstern Grossen auf die obere die letztern auf die untere Integrationsgrenze beziehea. 1. Hamiltons Princip der stationdren Wirkung. Aus der Fundamentalgleichung 1) folgt das Princip der stationaren Wirkung, wenn man die Grenzen des Integrals und die Coordinaten- werte filr dieselben als unveranderlich voraus setzt. Dann ergibt sich, wenn man setzt T-v=w, ("wdt = a, J}—=w folgende Gleichung : Sn = Oder BW= 0. O oder W haben also f iir die Bewegung dieselbe Bedeutung, wie V f iir das Gleichgewicht in der Ruhe. Die Bedingungen, welche den Grenz- wert von D, oder W unter den geschilderten Umstiinden angeben, sind mit den Bewegungsgleichungen identisch, wesshalb Helmholtz diese Gros- Vierte Vorlesung. 307 sen als kinetisches Potential bezeichnet. Fiir das Gleichgewicht in der Ruhe, bestimmen diese Bedingungen einen Grenzwert von V, da dann T=0 und F'von der Zeit unabliiiugig ist. Der Satz, dass fiir das Gleich- gewicht, F'ein Grenzwert ist, ist also ein ganz specieller Fall des Satzes Tom kinetische Potentiale oder des Hamilton'schen Princip der station- aren Wirkung, wie dieser auch genannt wird. 2. Ilamiltons Princip der variirenden Wirkung. Wir setzen in Gleichung 1) einmal nur die untere dann nur die obere, dann nur den Wert einer Coordinate fiir die untere, endlich diesen Wert fiir die obere Grenze des Integrales als veriinderlich voraus ; es folgen sofort die Hamilton'schen partiellen Differentialgleichungen : Es soil nun F'die Zeit nicht enthalten, also die Energie T + V sich mit der Zeit nicht andern. Wenn man dann in Gleichung 1) die Grenzen als variabel betraclitet, so transformirt man sie nach einigen Zwischen- rechnungen leicht in die folgende : 2 S r Tdt = f's ( r + F) dt + ^ (q5p - q^Sp^-) (2) wobei aber die Sp jetzt unter gleichzeitiger Variation der Grenzen fiir die Zeit und der Bewegung zu bilden sind. 3. Das alte Princip der kleinsten Wirkung. Setzt man in Gleichung 2) die Coordinatenvariationen fiir die Gren- zen von t gleich Null und nimmt ausserdem an, dass die Variation der Bewegung ohne Energiezufuhr geschieht also h{^T -\- V^= ist, so folgt h('Tdt = Q, also die alte Form des Princips der kleinsten Wirkung, welches in mancher Beziehung specieller, in so fern aber wieder allgemeiner ist, als das Princip der stationaren Wirkung, als es die Bewegungszeit als veriinderlich betraclitet. 308 Ludwig BoUmiann: 4. Analogien mit dem zweiten Hauptsatze. Wir wollen annehmen, class das letzte Glied der Gleichung 2) verschwindet. Es gilt dies nicht bloss, wenn an den Grenzen fiir die Zeit hp = Bpf^ = ist, sonderu auch wenn die Bewegung periodisch ist und t — tg die Dauer dieser Periode ist. Es gilt auch wenn die Verschiebungen siimmtlicher materielleu Punkte des Systemes in folge der Variation der Bewegung senkrecht auf der augenblicklichen Gescliwindigkeitsriclitung derselben steht. Bisher waren die Bp ganz -willkurliche Variationen. Wir wollen sie nun in folgender Weise erzeugt denken. 1. Mit dem Systeme, auf welches sich die Gleichung 2) bezieht, soil ein zweites Sys- tem verbunden sein, welches mit dem ersten in Wechsehvirkung steht und letzteres soil eine unendlich kleine Bewegung machen. 2. Ausserdem soil dem ersten Systeme eine unendlich kleine lebendige Kraft BQ zugefiihrt werden. Die in der Gleichung vorkommende Grosse B F'ist bloss die Veriin- derung von F'in folge der Lageniinderung der Punkte des ersten Systems. Sei S'F'die in Folge der Lagenanderung des zweiten Systems, so ist B'V die Arbeit der vom ersten auf das zweite System wirkenden Kriifte. Sie muss mit der zugefiihrten Energie BQ zusammen die gesaiumte Anderung BJE der Energie des ersten Systems geben. Es ist also BI]= BQ + B'V. Anderseits ist BI1= BT + BV + B' V, da S^die Anderung der kinetischen, BV+ B'V die Gesammtiinderung der potentiellen Energie ist. Aus bei- den Gleichungen folgt BQ = B (^T + V}. Setzen wir _ f'BQdt _ f'Tdt BQ = '^ und T = ^ , t-t, t-t, so folgt aus Gleichung 2) unter den gemachten Annahmen sofort ^=8 T lognat.(J>cf^J] wo die Analogic mit dem zweiten Hauptsatze deutlich zu Tage tritt. Thermodynamisches Beispiel : Unter dem ersten Systeme verstehen wir die Molekiile eines Gases, unter dem zweiten einen das Gas begrenzenden beweglichen Stempel, BQ ist die dem Gase zugefiihrte Wiirme. Mechan- isches Beispiel : Das erste System ist ein mit einer punktformigen Masse verbundener Magnetpol der gezwungen ist, sich in einer Ebene zu bewe- gen, das zweite System ein kurzer Magnet, um welchen der Magnetpol Vierte Vorlesung. 309 eiue Centralbewegung macht. Nun erfiihrt der Magnet eine kleine Drehung wodurch sich das Wirkungsgesetz der Centralbewegung andert und ausserdem der Magnetpol einen kleinen Stoss. Das Gesagte soil gewissermassen ein Schema sein, in welchem die versehiedenen dem Principe der kleinsten Wirkung verwandten Principe zusammengestellt sind. Es zeigt sich, dass die Analogien mit dem zweiten Hauptsatze weder einfach mit dem Principe der kleinsten Wirkung, noch auch mit dem Hamilton'schen identisch sind, aber sowolil zum einen, wie auch zum andern in sehr naher Beziehung stehen. Ich habe zu Anfang betont, dass die Entwicklung der Wissenschaft nicht immer in stetiger Verfolgung der alten Wege vor sich geht, son- dern sehr haufig durch plotzliche Einfiihrung ganz neuer Methoden und Ideen gefordert wird. Wo konnte fiir letztere Art der Entwicklung ein fruchtbarer Boden sein als in Amerika, wo alles neu ist, wo die Geschick- lichkeit des Geistes, Ungewohnliches zu unternehmen, die grossten unvor- hergesehenen Schwierigkeiten zu besiegen stete U bung findet, wiihrend wir in Europa wolgedrillt in den Bahnen der alt€n wissenschaftlichen Me- thode uns zwar mit grosserer Leichtigkeit und Sicherheit bewegen, als die Bewohner der neuen Welt, aber dem Ungewohnten und Neuen gegen- iiber verbliifft und unbehiilflich sind. Sicher werden daher nicht bloss die Amerikaner aus ihren rastlosen Bestrebungen die Pflege der reinen Wissenschaft zu fordern den grossten Nutzen ziehen, sondern auch die Wissenschaft wird durch die Mitwirkung der Amerikaner stets mehr und grossartiger gefordert werden. Auch ich fiilile den hohen bildenden Wert, den es fiir mich hatte meinen engbegrenzten heimatlichen Hori- zont durch die Bekanntschaft mit der grossartigen Natur und Cultur Amerikas zu erweitern, wol das fruchtbringendste Experiment, das ich je angestellt habe. Ich sage Ihnen daher meinen besten Dank fiir die hohe Ehre, welche Sie mir durch die Berufung zu diesen Vortragen erwiesen, und wiinsche nur, dass das von mir gebotene nicht ganz hinter der Grosse dieser Auszeichnung zuriickstehen moge. COxMPARATIYE STUDY OF THE SENSORY AREAS OF THE HUMAN CORTEX. By Santiago Ram6n y Cajal. In order to respond worthily to the gracious invitation with which Clark University has honored me, I ought to offer you, as was my original intention, a work of synthesis, a general summary of the present state of our knowledge of the minute anatomy of the nervous system. Unfortu- nately, the duties of my professorship, every day more pressing, have deprived me of the time necessary for the accomplishment of such a task, and have compelled me to moderate my ambition, and to limit it to presenting to j^ou a modest analytical contribution to our knowledge of the microscopical structure of the sensory centres of the human cere- bral cortex, a subject to which I have devoted the leisure of the past months. This subject is so vast and so difficult that, in spite of my efforts and the time devoted to it, I have been able to clear up, only a few points. Consequently, my contribution will be, to my utmost regret, a very incomplete one, treating, as it does, only the visual cortex as I have made it out in man and some of the higher mammals. I shall add, however, a few observations on the structure of other sensory regions. This anatomical study of the sensory areas of the cortex, at the present state of our knowledge, presents points of special interest, since, as you well know, neurologists who have interested themselves in the histology of the brain are divided at present into two camps, the unicists and the pluralists. The unicist doctrine, proclaimed by Mejiiert and reaffirmed quite recently by Golgi and Kolliker, supposes that all regions of the cortex possess essentially the same structure, functional diversity being due to diversity of origin of afferent or sensory nerves. This amounts to saying that cerebral specific energy of nerves is the necessary effect of the partic- 311 312 Santiago Ramon y Cajal : ular organization of each sense as well as of the special character of the stimuli received by the peripheral sensory surfaces, skin, retina, organ of Corti, etc. The pluralist doctrine, upheld recently by Flechsig, without rejecting the particular influence of connections with diiferent nerves, maintains that diversities of function result also from the particular structure of each cortical area. It is this latter opinion, as we shall presently see, that presents a closer agreement with the observed facts. In fact, my researches tend to prove that the topographical specialization of the brain depends not only on the quality of the stimuli analyzed and gathered up by the sensory mechanisms, but also on the structural adaptations which the cor- responding cerebral areas undergo; since it is very natural to suppose, even if one were to form an a priori judgment, that the cortical areas con- nected with the spacial senses sight and touch, which form exact images of the exterior world with fixed relations of space and intensity, have by accommodation to the stimuli received an organization different from that existing in cortical areas attached to the chemical senses of taste or smell, and from that which is appropriate to the chronological sense hearing, which gives only relations of succession, free from every spacial quality. We may add that if there exist in the human cerebral cortex, as Flechsig supposes, besides the sensori-motor centres, other regions, asso- ciation centres, characterized by absence of direct sensory or motor con- nections, it seems very natural also to associate to these important regions of the brain, with which are connected the highest activities of psychic life, a special organization corresponding to their supremacy in the hierarchy of functions. But we must not carry to an extreme the structural plurality of the brain. In fact, our researches show that while there are very remarkable differences of organization in certain cortical areas, these points of differ- ence do not go so far as to make impossible the reduction of the cortical structure to a general plan^ In reality, every convolution consists of two structural factors: one, which we may call a factor of a general order, since it is found over the whole cortex, is represented by the molecular layer and that of the small and large pyramids; the other, which we may call the special factor, particularly characteristic of the sensory areas, is represented by fibre plexuses formed by afferent nerve fibres and by the Visual Cortex. 313 presence at the level of the so-called granular layer of certain cell types of peculiar form. But, before proceeding to outline the general conclusions of an ana- tomico-physiological order, that result from all our researches taken together, permit me to present very briefly the facts of observation. Visual Cortex. The minute anatomy of the visual cortex (region of the calcarine fissure, sulcus cornu lobulus lingualis) has been already explored by sev- eral investigators, among whom we may make particular mention of Mey- nert, Vicq d'Azyr, Gennari, Krause, Hammarberg, Schlapp, Kolliker, et al. But their very incomplete researches have been performed by such insuffi- cient methods as staining with carmine, the Weigert-Pall method, or that of Nissl with basic anilines — methods which, as is well known, do not suffice at all to demonstrate the total morphology of the elements and the organization of the most delicate nerve plexuses. They led, however, in spite of the difficulties which stood in the way of these first analytical attemjats, toward a precise differentiation of the visual cortex from other regions of the brain. At the outset two characteristic differences attracted the attention of the first investigators into the structure of the visual cortex : first, the existence of a very thick stratum of granules, sub- divided into accessory strata by laminae of molecular appearance; and, second, the presence in the intermediate layers of the cortex of a white lamina formed of medullated fibres — which lamina may be seen with the unaided eye. This lamina, appearing in cross-section as a white line, has been named, in honor of the writers who first described it, the line of Gennari or Vicq d'Azyr. For the sake of brevity, we shall omit a detailed description and dis- cussion of the various layers admitted by the authorities on this region ; suffice it to mention in order the eight layers described by Meynert for tlie human cortex : First, molecular ; the second, layer of small pyram- idal cells ; third, layer of nuclei or granules ; fourth, layer of solitary cells ; fifth, layer of intermediate granules ; sixth, layer similar to the fourth, containing nuclei and scattered cells ; seventh, deep nuclear layer ; eighth, layer of fusiform cells. We may also mention the ar- rangement of layers recently described by Schlapp for the occipital cortex of the monke}' : (1) layer of tangential fibres ; (2) layer of exter- 314 Santiago Ramon y Cajal: mm mm nal polymorpliic cells ; (3) layer of pyram- idal cells ; (4) layer of granules ; (5) layer of small solitary cells ; (6) second layer of granules ; (7) layer poor in cells ; (8) layer of internal polymorphic cells. The investigations which I have made on the human cortex as well as on that of the dog and cat, by both the Nissl and Golgi methods, have led me to distinguish the fol- lowing layers : — 1. Plexiform layer (called molecular layer by authors generally and cell-poor layer by Meynert). 2. Layer of small pyramids. 3. Layer of medium-sized pyramids. 4. Layer of lai'ge stellate cells. 5. Layer of small stellate cells (called layer of granules by the authors). 6. Second plexiform layer, or layer of small pyramidal cells with arched axon. 7. Layer of giant pyramidal cells (soli- tary cells of Meynert). 8. Layer of medium sized pyramidal cells with arched ascending axon. 9. Layer of fusiform and triangular cells (fusiform cell layer of Meynert). You see that we have modified current nom-enclature by introducing terms which call to mind cellular morphology. For we believe that such trite expressions as "mo- lecular layer," "granular layer," must be Fig. 1. — Vertical section of tlie visual cortex ot man, calcarine sulcus, stained by Nissl's method — semischematic. 1. Plexiform layer. 2. Layer ot small pyramids. 3. Layer of medium-sized pyramids. 4. Layer of large stellate cells. 5. Layer of small stellate cells. 6. Second plexiform layer, or layer of small pyramids with arched axon. 7. Layer of giant pyramids. 8. Layer of medium-sized pyramidal cells with ascending axon. 9. Layer of fusiform and triangular cells. Visual Cortex. 315 banished once for all from scientific language, and they must be replaced by terms which point out dominant morphological characters in the nerve structures of each layer or some interesting peculiarity relative to the course and connections of the axis cylinder processes. The number of layers could be easily increased or diminished, because they are not sepa- rated by well-marked boundaries, particularly in Nissl's preparations. Thus the number of layers which I adopt is somewhat arbitrary. By distinguishing, however, nine layers, I have followed a criterion of indi- vidualization which seems to me the most convenient and suitable for my exposition of the cortex as a mechanism composed of elements at a cer- tain level which differ in special morphological features from those of neighboring levels. Besides, the number, extent, and size of cells in these laj^ers vary a little in the different median occipital convolutions, as does also the degree of definite nidification, according as we study the convex or concave aspect of the gyri. Our description relates generally to the cortex of the margin of the calcarine fissure, the region where structural differentiation of the visual cortex is most pronounced. Plexiform Layer. The plexiform or molecular layer is one of the oldest cerebral forma- tions in the phylogenetic series. It presents characters similar to those of the human cortex in all vertebrates except the fishes. This has been fully demonstrated by the researches of comparative histology under- taken by Oyarzun (batrachia), by myself (batrachia, reptilia, and mam- malia), by my brother (batrachia, reptilia), by Eddinger (batrachia, reptilia, aves), by CI. Sala (aves). In the visual cortex of man, the structure of this layer coincides perfectly with that which my own re- searches, as well as those of G. Retzius, have revealed in the motor region. The only modification which may be noted, visible even by Nissl's method, is its notable thinness in the margins of the calcarine fissure (except in the sulci, and here it appears somewhat thinned). This diminution in thickness, noted by authors generally, depends probably on the small number of medium-sized and giant pyramidal cells in the underlying layers, because it is well known that each i)yramidal cell is represented in the plexiform layer by a spray of dendrites. A similar opinion has been expressed by Bevan Lewis in order to explain irregularities in thickness of this layer in different regions of the cortex 316 Santiago Ramon y Cajal: of the rabbit and guinea-pig. The structure of the plexiform layer is very complex. From my own researches, confirmed largely by those of Retzius, Schafer, KoUiker, and Bevan Lewis, it follows that it consists of an interweaving of the following elements: (a) the radial branches of the small, medium-sized, and giant pyramidal cells, with which we must include in addition those of the so-called polymorphic cells ; (6) layer of terminal ramifications of the ascending axons of Martinotti ; (c) layer formed by the arborizations of the nerve fibres, terminal or collateral, which come from the white matter ; (d) layer of special or horizontal cells of the first layer (Cajal's cells, of Retzius) ; (e) layer of small and medium-sized stellate cells with short axons ; (/) layer of neu- roglia cells, well described by Martinotti, Retzius, and Andriesen. a. Terminal Arborizations of the Pyramidal Cells (Fig. 4). — As my observations have shown in case of the mammalian cortex, and tliose of Retzius for the human fcBtus, the radial trunk of the pyramidal cells does not end, as Golgi and Martinotti supposed, in a point entwined by neuroglia elements in connection with the blood-vessels, but in a spray of varicose dendrites covered with contact granules, spreading out some- times over a considerable area of the plexiform layer. In my first work on the cerebral cortex, I thought that the only cells whose terminal dendrites reached up to the first layer were ' the medium-sized, small, and giant pyramidal cells ; but my latest researches have enabled me to discover that all cells possessing a radial stem, without exception, including even those of the deeper layers, are represented in the plexi- form layer by a terminal dendritic arborization. It is without doubt an important structural law whose physiological import must be very considerable. We may observe that large trunks which ai-ise from the giant pyramids divide into a spray with very long and thick branches having their distribution in the deeper level, while the slender stems emanating from the medium and small sized pyramids form an arboriza- tion of numerous slender branches of limited extension and distributed particularly through the superficial laminte of the plexiform layer. This distribution, which is not absolutely constant, leads us to surmise that the terminal arborizations of each kind of pyramidal cell come into contact with special neuritic terminal arborizations in traversing this first layer. The trunk and end brush intended for the first layer appear not only in preparations made by the chromate of silver method ; for I have stained them perfectly with methylene blue (method of Ehrlich-Bethe) Visual Cortex. 317 in case of young animals, and also in adult gyrenceplialous mammals, such as the dog and cat. Besides, in good preparations by Ehrlich's method, particularly when fixation has been made a short time after the impregnation, one may see very distinctly the contact granules of the dendrites, processes which I was first to describe and whose existence has been confirmed by many investigators since. With methylene blue they present the same appearance as in Golgi preparations, i.e. they are slender and short, stand out at a right angle, are sometimes divided, and end freely in a rounded knob. This proves, accordingly, how groundless are all the gratuitous objections which have been brought against the preexistence of these appendages, as well as against their mode of termi- nation. Among the entirely arbitrary conjectures which have been made as to the disposition of these appendages we include also W. Hill's opinion, who considers them the fibres of a reticulum that is incompletely stained by means of the chromate of silver. We must proclaim emphatically that at present there is no method of staining cellular processes that is capable of disproving the agreeing results of the methods of Golgi, Ehrlich, and Cox. Whoever, having as a foundation the revelations of any one of these methods, has considered it possible to demonstrate the existence of such a reticulum has only exposed to view his own lack of experience in handling these important means of analysis. b. Special or Horizontal Cells of the Plexiform Layer. — These interest- ing elements, which I discovered in the cortices of the small mammals (rat, rabbit, guinea-pig), have been successfully investigated by Retzius in case of man, as well as by my brother in batrachians and reptiles, and by Veratti in the rabbit's embryo. They present in the visual cortex, where I have stained them very often, the same characters as in other regions of the brain. As I have already described these elements elsewhere, I shall give here only an outline, to which I may add a few remarks derived from my recent observations upon man (Fig. 2). Following the example of Retzius, when we study the horizontal cells by Golgi's method in a human fcetus from the seventh to the ninth month, or in case of a newborn babe, we notice that they are distributed throughout the entire thickness of the plexiform layer, but are especially numerous in close proximity to the pia. Their form is very variable, sometimes fusiform or triangular, and again stellate, with the angles extending out into the long processes. But the characteristic feature of these elements is due to the fact that their processes, which are variable 318 Santiago Ramon y Cajal. Visual Cortex. 319 in number and very large at their origin, give rise, after a few divisions, to an extraordinary number of varicose horizontal fibres, extremely long, from which spring at right angles numberless ascending secondary branches terminating in rounded knobs near the cerebral surface. Very often the superior surface of the cell body also gives rise to some of these ascending branches, which sometimes have a considerable thickness. In what way do these tangential fibres terminate ? Is it possible to discern among them certain processes possessing the characters of axons ? Upon careful examination of the best preparations obtained from cortices of human embrj-os, we discover easily that these processes, when they become very fine, have all the appearances peculiar to axons. There is no morphological distinction which would enable us to distin- guish the two classes or species of cellular processes. That which most strikes one is the enormous length of their horizontal fibres (tangential fibres of Retzius). One can follow them for two or three tenths of a millimeter without being able to discover their true termination. How- ever, in certain cases it is possible to demonstrate that the tangential fibres, after having given rise to a great number of vertical twigs, become thinner and finer, and finally subdivide into terminal branchlets, which spread out under the pia or in the superficial laminte of the first layer. On comparing these cells of the human brain with their homologues in the higher mammals (rabbit, cat, etc.), we discover that among the latter they give rise to a relatively small number of tangential branches, and that these extend a much shorter distance. This is the reason we consider the remarkable jirofusion and the extreme length of the hori- zontal fibres as one of the most characteristic features of the human cortex. Retzius did not succeed in staining the horizontal cells in man except in the foetal period. Accordingly, it was impossible to know what be- comes of these elements in the adult, and whether, as Retzius is inclined to think, all the processes that we find in the embryonic period persist. My recent researches on the cortex of infants fifteen months and even fifteen and twenty days old, in which I have been successful in staining the horizontal cells, suffice to furnish a few data which, if they do not solve the problem once for all, at any rate place the question in a some- what more favorable light. When we examine the plexiform layer of a babe fifteen days old, we find considerable changes in the horizontal cells. First of all. we 320 Santiago Ramon y Cajal : notice that they have become smaller, and that the tangential processes have diminished in diameter while they have become notably lengthened. But the peculiarity which most strikes the attention is the almost total disappearance of the ascending collateral branches. This atrophy begins in a progressive thinning of the processes and in the reabsorption of their terminal varicosities ; then the whole branch disappears, so that the only structures left are the horizontal fibres, whose ensemble forms throughout the thickness of the plexiform layer a system of parallel fibres of enor- mous length. There are places, however, where the ascending branches persist, but very much changed as to their direction, having become oblique instead of vertical, becoming branched several times, and termi- nating in the plexiform layer without reaching so far up toward the pia as before. In a word, most of the vertical branches seem to me to represent an embryonic arrangement corresponding to the interstices, for the most part vertical, between the epithelial cells of the cerebral cortex of the foetus, which proves once more, as I have demonstrated in other nerve centres, that during the period of evolution the neuron is the locus of a double series of functions: on the one side a vegetative building up of the dendrites ; on the other, reabsorptions and transformations of the cells which persist. Have the horizontal cells with which we are now concerned a true functional process ? In case this is so, what is the part played by these elements in the vast system of nervous relations established in the plexi- form layer ? In preparations of the human brain stained with chromate of silver, it must be confessed, it is not easy to solve this important question, since the purely morphological criterion, which is sufficient to distinguish the axon in other neurons, cannot be applied to horizontal cells, all the pro- cesses of which, on becoming finer, have the form of true axons. Thus, in spite of Veratti's affirmation, we believe that this method will shed no light upon the subject, even when applied to embryos. In order to ap- proximate to any solution of the problem, we must use a method capable of staining nerve prolongations in a manner to differentiate them from dendrites. It was only after using Ehrlich's methylene-blue method upon the motor and visual cortex of the cat that I became convinced that the horizontal cells have in reality a very long axon, which is provided with a medullary sheath. The other processes, which we have called horizontal fibres, represent true dendrites, as is shown b}' two peculiarities : the great Visual Cortex. 321 facility with which they take metliylene blue, and their pronounced vari- cosity after fixation with ammonium molybdate. We must repeat that this varicose alteration, which is a striking modification in the form of cellular prolongations, presents itself only in dendrites. The neurites maintain perfectly, with methylene blue, their normal contours, unless exposure to the air, necessary to obtain the selective staining, has been too long. As to the axon, it may be suQiciently well demonstrated in horizontal sections of the plexiform layer in the form of a pale blue fibre, except the initial portion and the nodes, which present a dark blue staining. This is a property of all parts of a fibre not surrounded with a medullary sheath. At the point of certain constrictions we may succeed in dis- covering a few collaterals springing out at right angles, provided also with myeline sheaths. Finally, one is sometimes so fortunate as to dis- cover in an axon of this kind true bifurcations situated at a great dis- tance from the cell of origin, but always in the plane of the plexiform layer. Unfortunately, the methylene blue does not stain the terminal nerve arborizations. This has prevented me from learning in just what way these axons terminate and with what axons they are dynamically associated. It is possible that certain heavy horizontal fibres come into contact with the horizontal cells, since they never bend downward toward the underlying layers, as do the medium-sized and finest medul- lated fibres. They belong probably to the terminal arborizations of Martinotti's ascending axons and, perhaps, also to the collaterals and terming,ls coming in from the white matter. c. Cells with a Short Axon (Fig. 3, G, I!,F'). — A few years ago, while studying the cerebral cortex of the small mammals, I discovered, besides the gigantic horizontal cells, other elements which I called polygonal cells. These are characterized by their stellate form and by their short axon, which ramifies and ends within the limits of the plexiform layer. These cells, whose existence neither Schiifer nor Lewis seem to have been able to confirm, — no doubt on account of the insufficiency of their attempts to obtain an impregnation of them, — are much more abundant than might have been supposed from my first observations. However, I must acknowl- edge that, they are not at all easily impregnated with chromate of silver and that, in order to find a sufficient number for study, we must make a great many attempts at staining them. On the other hand, Ehrlich's method stains them very readily in the dog and rabbit. In these animals 322 Santiago Ramon y Cajal . — and I think that it holds true also in man — the plexiform layer of the cerebrum is as richly supplied with elements with a short axon as the molecular layer of the cerebellar cortex. They occur in all levels of the layer and differ remarkably in size and shape. The majority of them are stellate and are comparable in size to other cells with short axons Fio. 3. — Cells and neuritic terminal arborizations in the 1st and 2d layers ; Tisual cortex of infant 20 days old. A and B, neuritic plexus, extremely fine and dense, situated in the layer of small pyramids; C, an analogous arborization, but not so dense; B, a small cell whose ascending axon forms a similar arborization ; E, spider-shaped stellate cell of the 1st layer ; F, G, cells with short axon branching loosely in the plexiform layer; a, axon. that occur in the deeper layers of the cortex. Others are smaller, resembling in their minuteness the granules of the cerebellum. But whether large or small, the morphological characters of these elements are very similar. Their dendrites are divergent, extremely branched, and distributed exclusively to the plexiform layer. Their neurites are Visual Cortex. 323 usually very short, subdivide in a most complicated manner in the neigh- borhood of the cell, but never cross the deep boundary of the first layer. From the point of view of the direction and length of their neurites all these elements may be classified into three varieties : (1) Stellar cells with horizontal neurite which becomes resolved after a varying distance, generally very long, into a terminal arborization which has the appear- ance of being connected with the terminal branches of the remote pyra- mids. (2) Cells of generally smaller size whose neurite branches either laterally or vertically from the cell body, but always at a moderate dis- tance (Fig. 3, (?, F^. (3) Very small cells (which I discovered recently in the human cerebral cortex) provided with numerous fine, divergent, and slightly branched dendrites, whose neurite, extremely slender, breaks up near its origin into a dense arborization, exceedingly fine and compli- cated. We shall designate these elements dwarf or spider-shaped cells. They may be found, as we shall see, in all the layers of the cortex (Fig 3, Ey To sum up : bearing in mind the form of cell bodies and formation and connection of axons, all the stellate cells of the plexiform layer, including the horizontal or special cells, seem to me similar to the stellate cells of the molecular layer of the cerebellum and to those which occur in the layers of the same name in the cornu ammonis and fascia dentata. Their function is probably to establish connections between terminal arborizations as yet imperfectly made out, possibly those formed by the ascending axons of Martiuotti, or the association fibres coming up from the white matter with the terminal branches of the pjTamidal cells. The function of the great horizontal cells would seem to be to establish connections between elements, that is to say between terminal neuritic arborizations and radial dendrites, separated by very considerable dis- tances ; while the medium-sized and small elements, with their short axons, would perform the same associative function at short or moderate distances. d. Martinotti's Ascending Fibres. — There is no lack of these in the visual cortex, although it has seemed to me that they are not so numerous as in other regions of the brain. Their terminal ramifications, well known from the researches of Martinotti as well as my own, occupy really the whole plexiform layer, where they extend over wide areas, distributing themselves preferably into its deeper levels and coining in contact with cells with short axons and, possibly, also with the large horizontal cells. 324 Santiago Ramon y Cajal: Granting that the cells of origin for these fibres lie in layers of the cortex that contain sensory fibres, we might suppose that Martinotti's ascending axons represent intermediate links placed vertically between these sen- sory fibres and cells with short axon in the plexiform layer. And as these are connected, perhaps, with the dendrites of the pyramidal cells, the result would be that the sensory stimuli, entering the cortex in this indirect way, would be compelled to traverse two intercalated nerve cells before reaching the pyramids. e. Neuroglia Cells. — These conform in the visual cortex to the well- known types of other cerebral regions. We find accordingly : (1) Cells with long radii, the marginal cells well described by Martinotti, which lie just under the pia. They emit long, smooth, descending processes radiating across the plexiform layer, ending at different levels both of this and of the layer of small pyramids ; (2) Cells with short radii. These elements, long since described by Golgi, and described in de- tail by Retzius, by myself, Andriesen, Kolliker, and others, are charac- terized by their form, very often stellate or fusiform, by their location in all levels of the plexiform layer, and by the great number of their pro- cesses, short, spongy, branching, and bristling with innumerable contact granules, which penetrate into the spaces lying between the neuro-proto- plasmic plexus and are well spread over the interstices of the elements which must not come into contact. It is in virtue of this intricate rela- tion between these appendages and the cell bodies and dendrites, as well as for other reasons which we have not time to dilate upon here, that we attribute to the neuroglia elements with short processes an insulating role. According to my view, they prevent inoiDportune contacts, while their processes exercise due regard to all points of cells or fibres where contacts exist and nerve currents pass. Layer of Small Pyramids. This layer is well separated from the 1st, but blends by insensible gradations with the 3rd, or layer of medium-sized pyramidal cells (Fig. 4, 5). Examined in Nissl preparations this layer presents a great number of small pyramids, very poor in chromatic granules and separated by a plexus of fibrils much more dense than in the case of cells of the deeper layers. We find also, scattered irregularly, stellate or triangular cells Visual Cortex. 325 larger than the pyramids. These are the giant cells with short axon, as is shown in good chromate of silver preparations (Fig. 5, 2), C). We shall now discuss the cells of this layer, beginning with the pyramids. Pyramids. — The morphology and relations of these cells being well Fio. 4. — Small and medium-sized cells of the visual cortex of an infant 20 days old (calcarine salens). A, Plexiform layer ; B, layer of small pyramids ; C, layer of medium-sized pyramids ; a,. descending axon ; (, recurrent collateral ; c, dendritic trunk of giant pyramid. known since the researches of Golgi, Retzius, and m)'self, I shall limit my remarks to a bare mention of a few peculiarities of their disposition in the visual cortex. It will be noticed that these cells are generally smaller and more 326 Santiago Ramon y Cajal: numerous in the visual centres than in other cortical areas. Sometimes the more superficial cells are arranged in one or two regular files and separated from those beneath by a fine dense plexus of fibres. The small pyramids give rise to the following processes : an axial dendi'ite, often bifurcated near its origin, which runs to the plexiform layer and terminates in a spray of fine branches, which often ascend to the neighborhood of the pia ; basilar divergent dendrites, rather long and repeatedly branched ; and, finally, a fine descending axon, which, in most favorable specimens, can be followed down to the neigh- borhood of the white matter. From the initial portion of its course spring three, four, or a larger number of collateral processes, which traverse, with many subdivisions, in a horizontal or oblique direction, a very con- siderable extent of the second layer. From the small pyramids lying close to the plexiform layer, and even from some cells more deeply situ- ated, the first two collaterals recurve, ascending sometimes, as Schiifer has discovered, up to their termination in the first layer. However, this termination in the first layer is much less frequent than might be inferred from this authority's descriptions and dramngs. In our prepa- rations of the visual and motor cortex of a child a few days old and of a cat twenty-five days old, the great majority of the recurrent collaterals do not cross the boundary of the second layer. Here, in conjunction with many neurites belonging to cells with short axons, they assist in forming a very dense plexus, which contains in its meshes the primary dendrites of the small pyramids. Generally, — and this maybe considered as an answer to the authorities who strive to convert the recurrent course of the collaterals into an argument for the doctrine of the cellulipetal con- duction of these fibres (v. Lenhossek, Schiifer), — I may affirm that the vast majority of the initial neuritic collaterals — and I consider such all those that arise within the gray matter — always come into contact with some of the dendrites belonging to homologous nerve cells situated at dif- ferent levels of the same cortical formation. When the cells to which they correspond lie in the same or a deeper plane, the collaterals intended for them take a horizontal, descending, or oblique course ; but if the cells of the same category are situated in a more superficial plane than the point of origin of the collateral, they must describe a recurrent arc in order to reach their destination. Visual Cortex. 327 Later of Medium-sized Pyramids. Being a continuation by insensible gradations of the small pyramidal layer, it contains cells of precisely similar form, differing from the cells of the second layer only in their somewhat greater size, their longer radial dendrite, and, ordinarily, by a larger number of neuritic collaterals (Fig. 4, (7). In the deeper level of this layer may be observed — very seldom, however — large pyramidal cells, but not so large as those situ- ated in the seventh layer. Cells with Short Axon of the Second and Third Layers. — These elements, almost as numerous as the pyramidal cells themselves, may be seen scattered irregularly throughout the entire thickness of the two lajxrs. They are generally more numerous near the limits of these layers, that is to say, in the superficial portion of the second and in the deeper level of the third layer. Although in form and size these elements are very variable, and although there are transitional forms which make it often difficult to dis- tinguish between them and to subdivide them into well-pronounced types, still, by considering the size of the cell body and the character of the axon, they may be divided into the following five classes : (a) cells with short ascending axon ; (5) cells with short descending axon ; (c) cells with horizontal or oblique axon ; (c£) dwarf or spider-shaped elements ; (e) fusiform or bipanicled cells, whose axon breaks up into a fibrillar ai'borization. a. Cells with Ascending Axon (Fig. 5, a, B'). — As may be seen in Fig. 5, these cells belong to two principal varieties : (a) Gigantic cells, with long dendrites (Fig. 5, A, C). These are quite numerous in the visual cortex, where they occupy preferably the deep portion of the third layer. Their form is stellate, sometimes fusiform or triangular. From their angles arise several varicose, thick, and very long dendrites, often disposed as two brushes, the one ascending, the other descending. The axon takes its origin either from the cell body or from a dendrite. Sometimes it describes an arc, whose concavity is toward the surface, on its way outward to become resolved into an arborization of very few branches. The characteristic feature of this arborization is the enormous length and the horizontal or oblique direction of its terminal twigs. These traverse a very considerable portion of the second and third layers, where they make contact with numberless pyramidal cells. It Flo. 5. —Large stellate cells having short ascending axons, 2d and 3d layers, visual cortex, infant 15 days old. A, elements of the 3d layer with axons divided into long horizontal branches ; B, small cell with arched axon from the layer of small pyramids ; C, large cell with arched axon ; D, large cell from the boundary of the 1st layer ; F, cell with arched ascending axon branching ir^a most complicated manner; a, a, a, axons. Visual Cortex. 329 may be added that these gigantic cells may be recognized even in Nissl preparations by their stellate form and considerable size. They corre- spond, probably, to the globular cells of Bevan Lewis and other writers. (5) Medium-sized type : This is a fusiform or stellate cell, M-hose size does not exceed that of the small or medium-sized pyramids. It is characterized above all by its axon, which is slender and ascending, and which terminates in a complicated arborization with manj' varicose branches and with relatively small spread at varying levels of the second and third layers. As to the dendrites, they appear varicose and diverge in all directions, but usually do not extend to the first layer (Fig. 5, F, and Fig. 3, Z>). b. Cells with Descending Axons. — These are stellate, triangular, or fusi- form, of medium size, and provided with many ascending and descending dendrites. They occur chiefly, as has been pointed out by Schafer for other regions of the cortex, along the superficial boundary of the layer of small pyramids (Fig. 5, B, and 6, C). Their axons descend through the second and sometimes through the third layer, giving off to them a few collaterals, and terminate in a diffuse arborization throughout the different levels of these layers. Very frequently this axon, after descend- ing a certain tlistance, emitting a few collaterals to the layer of small or medium-sized pyramids, traces an arc with concavity toward the surface and ascends to terminate in an arborization, very complicated and with exceedingly varicose branches, in the layer of small pyramids close to the plexiform layer (Fig. 5, B~). As seen in Fig. 6, which reproduces certain cells of short axons from the visual cortex of the cat, these elements with descentling axons are very numerous in other g3Tencephalous mammals. We also find a variety of cell, recognized in man, p3'riform, uni-polar, whose single descending process gives rise to a bouquet of varicose dendrites and an axon (Fig. 6, a, J). The collaterals and terminal arborizations of these axons form in the cat a dense plexus throughout the superficial plane of the layer of small pyramids. The great number of cells with short axons which occur in the most superficial lamina of the layer of small pyramids has induced certain ^\■rit- ers, such as Schafer and Schlapp, to consider this transitional region as a special la3'er, which they call the layer of superficial polymorphic cells. AVe cannot subscribe to this innovation because, in spite of the great number of these cells, this transitional lamina contains also a large num- ber of small pyramids, that is to say, cells which, in addition to their 330 Santiago Ramon y Cajal. morphological varieties, have the same connections as ordinary pyramidal cells. Of course, if for the subdivision of the cortex into layers we take FiQ. 6. — Cells with short axons from the layer of small pyrataids, visual cortex of cat aged 28 days, a, 6, small pyriform cells with short descending axons ; c, cell with arched axon ; e, /, cells with descending axons distributed to the medium-sized pyramids of 3d layer. as our basis of classification the form of cell bodies, independently of other characters, we might be entitled to differentiate between the first and second layer consisting chiefly of stellate cells ; because in this region, as Visual Cortex. 331 is well known, the small pyi-amids have a stellate or triangular form. But, in assigning to an element a place in his classification, one must not decide from the form alone, which in case of superficially placed pyramids is a function of their position. In fact, we find that the form of these cells varies according to their proximity to the plexiform layer. The true characteristic of a pyramidal cell consists in the presence of a long axon extending down to the white matter and of a spray of dendrites (supported or not by an intermediate trunk) spreading up into the plexi- form layer. Now, in the light of such a criterion, it is easy to see that sufficient reason does not exist for making out of the most superficial pyramids a distinct category of cells to be used as a basis for the creation of a new cortical la3'er. c. Cells with Horizontal or Oblique Axoi (Fig. 7). — These elements, which are angular or fusiform, with their long axes more or less hori- zontal, possess few, but rather long, dendrites. Their axon arises gen- erally from the lateral aspect of the cell body or from a thick polar dendrite, takes from the first a horizontal or oblique direction and, after giving off a few collaterals, terminates, sometimes after extending to a considerable distance, in an arborization widely spread but with few branches. In certain cells of this category, it is shorter and subdivides in the immediate neighborhood of the ceU body (Fig. 7, ^, C). d. Dtvarf or Spider-shaped Cells. — Brought to our attention by CI. Sala in the corpus striatum of birds, mentioned also by my brother in the cerebral cortex of batrachians and reptiles, these strange elements are notably abundant and of very pronounced character in the cerebral cortex of man and gyrencephalous mammals. They are found irregularly scat- tered in all layers of the visual area. Their soma is very small, not ex- ceeding the diameter of the nucleus by more than five or six fi. About the nucleus is a thin lamina of protoplasm which is drawn out into a great number of dendrites, delicately varicose, radiating, slightly branched and short. The appearance of these dendrites is such that one might mistake the cell, at first sight, for a neuroglia corpuscle with short processes. But, examining them with a high power, we recognize at once that their slender dendrites do not possess collateral appendages (contact granules), so characteristic of processes of neuroglia cells. Finally, attentive examination reveals the axon, a delicate fibre, which becomes resolved immediately into a ver\' dense varicose arborization of incompar- able fineness. Often this terminal plexus is so extremely fine that it 332 Santiago Ramon y Cajal: appears through an ordinary objective as a yellowish or brownish spot in the neighborhood of the cell and resembling somewhat a granular precipi- Fio. 7. — Cells with short horizontal or oblique axons situated in the 2d and 3d layers, visual cortex of infant a few days old. A, B, cells with axons almost horizontal from 2d layer; C, D, E, cells with short axon diffusely branched ; F, U, I, pyriforra cells of the 1st layer, whose sig- nificance is still uncertain; G, small cell with very short axon diffusely branching within the 1st layer. tate. In some cases this arborization is coarser and can be seen with a Zeiss objective D or E. At the level of the superior boundary of the layer of small pyramids, in the visual cortex of the child and even of Visual Cortex. 333 other mammals, may often be seen a dense plexus of exceedingly slender branching fibrils. Their original fibre appears to come from the deeper levels of the 2d layer (Fig. 3, A, B, C). These terminal plexuses often take the impregnation irregularly, which gives the appearance of brownish or coffee-colored spots scattered and sometimes arranged in a row just underneath the plexiform layer. At first I was not successful in tracking satisfactorily the fibres of origin and, therefore, hesitated as to stating the significance of these interesting arborizations. Very recently, how- ever, in two or three fortunate specimens I have been able to demonstrate the connection between this plexus and the fine ascending axons of certain small cells situated in the deeper level of the 2d or outer level of the 3d layer. I am, therefore, now inclined to consider this intermediate, or sub- plexiform, nerve plexus as consisting of terminal arborizations intended for the small pyramids. The fibres of origin spring from more deeply situ- ated spider-shaped cells very hard to impregnate. I may add that these plexuses are not lacking in the cat and dog, although in these animals the fibrilhe are not so numerous nor so extremely fine as in the human brain. Permit me also to add that they occur in all regions of the cor- tex, although up to the present we have obtained the best impregnation of them in the visual area. e. Small Bipanieled Cells. — In the visual region, as well as in other areas, of the human cortex we find in profusion certain small cells vertically elongated. Their axon presents the very singular feature of breaking up into long slender brushes of terminal fibrillffi. At first, I thought that these singular cells were forms characteristic of the acoustic area, for here they are remarkably developed and very numerous. Further investiga- tion, however, has convinced me that they occur in all parts of the cortex, disposed in greatest numbers along the lower level of the 2d and 3d layers (Fig. 8 and Fig. 11, E, F}. As stated above, we are discussing the small spindle-shaped cells with poles radially disposed, which give rise to groups of dendrites, slender, unprovided with contact granules, very finely varicose, and often arranged in long ascending and descending brushes. In some cases these are so fine that on superficial examination they might be mistaken for delicate neuritic arborizations. But the most striking peculiarity of these cells concerns the subdivisions and course of their axons. This pro- cess is very delicate. It ascends or descends a certain distance, then gen- erally gives off a few collaterals at right angles which soon subdivide into 334 Santiago Ramon y Cajal. Fig. 8. — Small fusiform, bipanicled cells from auditory cortex of infant (1st temporal convolution). A, cell giving origin to a de- scending axon moderately branched ; B, cell whose axon breaks up into a number of pen- cils of very long ascending and descending fibrils; a, axon. (Examined with Zeiss apo- chromatic obj. 1.30.) ascending or descending fibrillse, and finally it breaks up into brushes of very slender filaments which run radi- ally, extending throughout almost the entire thickness of the cortex. As a whole this arborization with its initial collaterals forms one or several parallel brushes, the fibrils of which skirt the trunks of the pyramids and adapt them- selves to the cell bodies, over which they appear to creep, like the creeping fibres of the cerebellum on the branches and bodies of the Purkinje cells. In the brain of the human infant at birth these arborizations have not attained complete development and present but few vertical branchlets. It is not until twenty or thirty days after birth that we can observe the long and complicated terminal brushes. In certain areas, the acoustic, for ex- ample, each neurite may form as many as five ascending or descending brushes. The fibrils of which they consist are so delicate that in order to see them well we must use the highest apochromatic objectives. If now we consider all the differ- ent kinds of cells having short axons, of which we have given a somewhat fastidious description, from the point of view of their connections and their probable functions, we may character- ize them as special cells of association. The form of their cell body and the dis- position of the axon vary according to the number, form, and position of the cells to which they must convey nerve Visual Cortex. 335 stimuli. Thus cells with a horizontal axon must be intended to transmit impulses to elements, probably pyramidal cells, which occur at the same level in the cortex. Those whose axon is vertical, ascending or descend- ing, would naturally transmit impulses to elements of different layers. Those which are bipanicled would serve to associate dynamically a great number of pyramids in vertical series. Finally, the small, spider-shaped cells may have for their function association of groups of pyramids very close together. Unfortunately for this theory, we do not know from which nerve fibres all these elements of association receive their initial stimuli. Accordingly, we must be resigned to remain in ignorance as to the path of the afferent impulses and, as well, in regard to the special influence which these elements must exercise. It seems very probable, however, that their function consists not only in facilitating the spread of incoming stimuli, but also in adding to it something new, some specific modifica- tion which cannot now be determined. We shall return to this point in our general conclusions upon this work. But we may see from the above how many paths nature has opened up to render association of nerve impulses possible in every direction and through any distance. That which proves the importance of these association cells and leads us to surmise that they play an important psychic role is the fact that they are extremely numerous in the human brain. They are found in the animal brain as well, but are not numerous and are usually confined to the boundary of the 1st layer. I conclude here my exposition of the prosy topics that I chose as the theme of this lecture. And nothing remains except to thank you for the attention and good vnR which you have shown me in spite of the extreme dryness of the subject-matter. LECTURE II. Layer of the Large Stellate Cells. My recent researches in the visual cortex of man have led to the unex- pected discovery of certain large cells of stellate form possessing an axon which descends to the white matter. Figs. 9 and 10 represent very clearly the most common forms of these strange elements. They are differentiated immediately from pyramidal cells b}' their lack of a radial trunk. Generally speaking, the cell body is stellate, but there is no lack of semilunar, triangular, and even mitral forms. Their dendrites are thick and much branched, and extend in all directions, especially horizon- tally, without ever leaving the territory of the 4th layer. In man these processes are sparsely provided with contact granules, while they are very numerous in the homologous cells of the mammalia (cat and dog). As to the axon, it is rather large, arises from the inferior surface of the cell body, descends through the 4th layer, sometimes tracing here accom- modation curves, and after crossing the 5th, 6th, 7th and 8th layer, passes into the white matter and is there continued as a meduUated nerve fibre. In passing through the 4th and 5th layers it gives off three, four, or a larger number of, often, very large collaterals which end in arborizations extending over a considerable area in these layers. It is not uncommon to see these collaterals taking a recurrent course to become distributed in planes above the point of origin ; but in this they never trespass on the boundaries of the 4th and 5th layers. Finally, and this is a very frequent disposition in the adult cortex, this axon, after having given off its col- laterals, becomes notably finer. Taking into consideration its diameter, sometimes less than that of its first collateral, we might be led to mistake it for the latter rather than a true continuation of the axon. We shall return to this peculiarity, which is presented by many cells in the visual cortex. The stellate cells present a similar character in the adult human cortex, and I reproduce in Fig. 10 their principal types impregnated (long method of Golgi) in the case of a man thirty years old. The only 336 Visual Cortex. 337 difference that we remark between these cells iu the adult and infant brain is the greater development of the dendrites, which extend long distances in horizontal planes in the adult. Tlie volume of the soma also Fio. 9. — Layers 4 and 5, with portion of 6; stellate cells of the visual cortex, infant 20 days old, calcarine sulcus. A, layer of large stellate cells ; o, semilunar corpuscle ; '), fusiform horizontal cell ; c, cell with radial trunk ; e, cell with arched axon ; B, layer of small stellate cells ; /, horizon- tal fusiform cells ; , giant pyramid of 7th layer ; c,h, axons of small pyramids of 6th layer. a good part of the 7th layer. In the brain at birth their terminals present no special peculiarities ; but in one twenty days old I have found that a number of these arborizations surround the giant pyramids, form- ing terminal nests. Only their arrangement is not so definite here as in the motor region, where we find it wonderfully developed. (Compare with description below.) Visiial Cortex. 355 Eighth Layer. Examined in Nissl preparations this layer presents a mass of medium- sized pyramids and a remarkably dense formation of granules. Tliis is the reason Meynert and other writers have called this the layer of deep granules or inferior granular layer. Golgi's method reveals in this formation elongated cells of pyramidal form. They have the radial trunk continued, up to the plexiform layer and also descending basilar dendrites which become subdivided and end within the 8th layer. Among these there is no lack of fusiform or tri- angular cells, but they always present the long radial trunk which we find over the whole cortex (Fig. 22, C). In general form, it will be observed that these cells resemble true pyramids. However, the pecidiar behavior of their axons establishes a very clear distinction between them. As may be seen in the figure (22, i), this axon at first descends, then describes an arc, ascends into the 7th, 6th, and 5th layers, and finally ends in a horizontal arborization chiefly distributed to the layer of stellate cells, but a few of its branches go to the 5tli layer. From the loop of the axon, and in the course of its ascent, sirring several collaterals, which ramify in different planes of the 8th layer. In a few of these cells we may observe that, at the bend of the axon, a slender branch, similar to a collateral, is given off, which crosses the 8th and 9th layers and enters the white matter as a meduUated fibre (F'ig. 22, g). The great majority of these collaterals, however, terminate com- pletely within the 8th and 9th layers. At any rate, we must distinguish, considering the morphology of their axons, two kinds of cells : (a) cells with arched axon none of whose collaterals extend to the white matter ; (6) cells whose neurite divides, at the arch, into a fine descending branch, which becomes a meduUated fibre of the white matter, and into a larger ascending branch with its terminal arborization in the 4th or 5th layers. This arched arrangement of the axon in cells of the 8th layer appears very strange. It occurs not only in the infant brain, but in the visual cor- tex of the adult as well. It seems, at first sight, to violate all laws that govern the length and direction of the axons in other sections of the nervous system. And, what seems still more remarkable, all these whim- sical windings seem to subserve solely the purpose of sliortening the stretch between the cell body and the first collaterals given off by the arch. This same phenomenon occurs in many other nerve cells. Were FiQ. 22. — Seventh and 8th layers, visual cortex of cat, aged 20 days. A, deeper portion of layer of stellate cells ; B, layer of giant pyramids ; C, layer of medium-sized pyramids with arched axon; a, b, pyramids; c, d, small pyramids with axons distributed to 7th layer; g, tri- angular cell, whose axon gives rise to a large ascending collateral ; i, another whose axon forms an arch and ascends ; 1, pyramid with axon descending to white matter ; j, element from the deep- est levels of the layer of medium-sized pyramids (corresponding to layer of fusiform cells in man) which gives origin to a large axon that ascends possibly to the 1st layer. Visual Cortex. 357 it not for a deviation from our present theme, I might adduce very con- vincing instances of this tendency of the axoq, to take the direction most favorable for the nerve impulses which arise in the cell to very quickly reach the elements connected with their initial collaterals. Permit me also to add that the 8th layer contains giant stellate cells with ascending axon (Martinotti's cells), which runs to the plexi- form layer (Fig. 22, j), and also a similar but smaller cell, whose axon gives rise to an arborization between the neighboring cells. Ninth Layer. Coinciding closely with the so-called polymorphic layer of other authors, this layer contains elongated elements, fusiform, triangular, or ovoid, possessing a radial dendrite, extending up to the plexiform layer, and also one or several basal dendrites, which take a descending or oblique direction. Finally, these cells have an axon which descends in a straight line to the white matter; where, after giving off several col- laterals, it continues as a medullated fibre. There are also in the 9th layer a few fusiform cells with short radial dendrites and ascending axon and a number of stellate cells with short axon of the so-called Golgi type. In addition, the arrangement of the cells of the 9th layer varies greatly in different parts of a convolution. In the convex portion they are very numerous, fusiform, and slender, elongated and perfectly radial ; while opposite the sulcus they have a quite different form, are stouter, more variable, and frequently lie with long axis parallel to the white matter, i.e. perpendicular to their ordinary direction. Their peripheral processes perform the most whimsical contortions in order to become radial and reach the plexiform layer. Their axon appears frequently horizontal, describing a very open curve on its way to the white matter. All these forms and many others represent adaptations of the cells to the foldings of the cortex and to its varying thickness in dift'erent parts of a convolution. I will not impose further upon your indulgent attention with these tiresome enumerations of layers and forms of cells, in the mazes of which nature herself seems to have intended to lose the investigator and put his patience to the test. And I will close this tedious lecture with a 358 Santiago Ramwi y Cajal: succinct exposition of the anatomico-physiological inductions that seem to follow from my observations on the minute structure of the visual cortex of man and the mammalia. 1. The visual cortex of man and gyrencephalous mammals possesses a special structure very different from that of any other cortical area. 2. The visual region is characterized, above all, by fewness of giant pyramids and by presenting, at the level of the granular layer of other cortical areas, three distinct layers of cells of special form, to wit : the layer of large stellate cells, the layer of small stellate cells, and the layer of pyramids with arched ascending axon. 3. Gennari's or Vicq d'Azyr's stripe contains principally terminal arborizations of certain very large fibres, originating probably in the primary optic centres (external geniculate body, pulvinar, anterior cor- pora quadrigemina). 4. Since these optic fibres are distributed chiefly to the stellate cells of the -ith and 6th layers, it seems natural to consider these elements the substratum of visual sensation. 5. The innumerable cells with short axons in the 4th and 5th layers represent, probably, the intermediate links between the optic fibres on the one side and the stellate cells of the 4th and 5th layers and the pyram- idal cells on the other. 6. As these intermediate cells are often very small and have short axons, it may be that, besides their function of diffusing the incoming impulses through the cortex, they play also the special role of augment- ing the visual impulses by fresh discharges of nerve force, in order that they may reach, in sufficient strength, the cortical regions in which the function of commemorative recording of optical images occurs. The pathways for conveyance of visual residues from the median occipital region to the association centres in the parietal cortex are possibly repre- sented by axons of the stellate cells of the 4th and 5th layers. 7. Granting that the giant p3Tamids of other cortical regions give rise to motor fibres, it would follow that in the 7th layer they possess the same function. These cells, whose dendritic trunks come into con- tact with the optical plexus, 4th and 5th layers, serve pi-obably to mediate the reflexes of the eyeball and head (conjugate movements of the eyes) occasioned by elective stimulation of the visual cortex, a theory which would seem to be supported by the physiological experiments of Schiifer, Danillo, Munk, and others. Visual Cortex. 359 8. Granting that each giant pyramid comes into contact in the 4th and 5th layers, as well as in the first layer, with fibres that are proba- bly associative, we may suppose that motor discharges of these cells can be effected by two kinds of impulses : by ordinary optical stimulation and by stimuli of a volitional order, possibly coming from the association centres and reaching, finally, the plexiform layer. My own researches do not furnish grounds for further conclusions. Many points still remain to be cleared up ; but their complete eluci- dation will be the fruit of researches more detailed and exact than those I have been able to undertake. LECTURE III. The Sensoei-Motor Cortex. After the study that we have just made of the visual cortex, we can be more concise in our examination of the motor area. In all cortical regions we notice general structural characters and special features which constitute the physiognomy proper of each cerebral area. Naturally, the latter will be of more interest to us, and they will form the subject of the present lecture. I shall not stop here to give any history of researches undertaken upon the minute anatomy of the psycho-motor areas. A bibliography of the subject would be very long, tedious, and altogether superfluous, since it has already been provided in the recent studies of Retzius, Hammar- berg, and KoUiker. It will suffice to name, among those to whom we are most indebted for a knowledge of the structure of the motor cortex, Mey- nert, Baillarger, Kolliker, Krause, Betz, Lewis, Golgi, Martinotti, Retzius, Flechsig, Kaes, Hammarberg, Nissl, etc. All these writers have selected the psycho-motor cortex for special study ; and it is safe to assert that all our knowledge of the minute structure of the entire cortex has taken its character from this region, which some writers have denominated " typical." They have done this because it was thought at the time when the fundamental works of Meynert and Golgi appeared that in histologi- cal structure the whole cortex corresponded to a uniform design, present- ing only unimportant variations in different regions. Neither have I time to enumerate the layers which have been described for this cerebral region. Their number has varied under the pen of each writer with the animal and the method he has happened to employ. Thus Meynert, who made his observations on man, distinguished five layers ; Stieda, Henle, Boll, and Schwalbe limited their number to four; while writers like Krause admitted as many as seven. I myself, at the time of my investigations upon the small mammals, recognized four, naming them : (1) molecular layer ; (2) layer of small and medium-sized pyramids ; 360 Sensorl-Motor Cortex. 361 (3) laj-er of large pyramids ; (4) layer of polymorphic cells. This number, derived particularly from study of the small mam- mals, is not valid in the more complicated human cortex. To the four classical layers of smooth-brained mammals we must add one at least, the so-called granular layer of Meynert and other writers. This layer, situated in its very midst, divides the layer of giant pyramids into two, which we may call respectively the external, or superficial, and the internal, or deep, layers of giant pyramids. To sum up, the following are the layers which it is possible to recognize by Nissl's method in the human motor cortex (ascend- ing frontal and ascending parietal convolu- tions). To conform to our scheme in the visual cortex, we have altered the terminol- ogy for this region also. 1. Plexiform layer (layer poor in cells of Meynert, molecular layer of some writers). 2. Layer of small and medium-sized pyramids. 3. External layer of giant pyramids. 4. Layer of small stellate cells (gran- ular layer of the authors). 5. Internal, or deep, layer of giant pyramids. 6. Layer of polymorphic cells (fusiform and medium-sized pyramids of certain writers). Fio. 23. — Section of adult human motor cortex, stained by Nissl's method (semischematic). 1, pleziform layer ; 2, layer of small pyramids ; ?•, layer of medium- sized pyramids ; 4, external layer of giant pyramids ; 5, layer of small stellate cells ; G, internal layer of giant pyramids ; 7, layer of polymorphic cells or dcei) pyramidal layer of medium-sized cells; 8, layer of fusiform cells. Wmmm ■hmmM li ^ 4' ■j'j. ' 'if 362 Santiago Ramon y Cajal : These layers correspond particularly to the concave portions of the motor convolutions. Over the convexities the gray matter is thickened especially at the level of the polymorphic layer, which here appears divided into two sub-layers : an external, very rich in pyramidal and triangular cells (Fig. 23, 7) ; the other, internal, presenting, besides the heavy bundles of white fibres, fusiform cells disposed in parallel series (Fig. 23, 8). 1. Plexiform Layer. — This is similar in structure in the motor and visual areas. It contains, therefore : (1) dendritic arborizations of the pyramidal and polymorphic cells, that is to say, of all the cells of deeper layers (2, 3, 4, 5, 6) except stellate cells of the 4th layer and the cells with short axons scattered through the entire cortex ; (2) terminal arbori- zations of the ascending axons of Martinotti; (3) the ramifications of the recurrent collaterals which come up from the axons of certain small and medium-sized pyramids ; (4) the fibres, terminal and collateral, which arise from the white matter ; (5) stellate cells of variable size with short axon which ramifies within the 1st layer ; (6) the special, or hori- zontal, cells with long tangential dendrites ; (7) finally, neuroglia cells of the two well-known types, with long radiating processes close undei"neath the pia (Martinotti, Retzius, Andriesen, Bevan Lewis, et al.}, and type with short processes, located at all levels of the plexiform layer (Golgi, Cajal, Retzius, Martinotti). We shall not enter upon their descriptive details, since all the struc- tures present the same characters here as in the visual cortex. We shall merely add that in the motor cortex the plexiform layer is notably thick. It also contains a greater number of horizontal cells and terminations of the trunks of pyramidal cells (Fig. 25, A, B, C). Its greater thickness arises probably, as Lewis remarks, from the extraordinary number of pyramidal cells in the underljdng layers. 2. Layer of Small and Medium-sized Pyramids (Fig. 24, 2 and 3). — We shall not stop upon these, because they are so well known. Permit me merely to call to mind the fact that their radial trunk, often forked near its origm, makes its arborization in the plexiform layer ; while from the base springs a fine neurite which, in case of the small mammals, we can trace into the white matter. In the child's cortex this is made diffi- cult by the distance, but I have been fortunate on two occasions in fol- lowing this axon into the medullary substance, where it was continued as a meduUated fibre. The neuritic collaterals are also very numerous Sensori-Motor Cortex. 363 and a number of them may be seen to recur and make their arbo- rizations in the superficial lamina of the plexiform laj'er. Cells with Short Axons. — These are numerous, although it does not seem to me that they are so extremely abundant as in the visual region. In Fig. 25 I have reproduced some of these ele- ments which habitually occur in my preparations. We remark especially : a, a large stellate type, whose ascending axon subdivides into horizontal or oblique branches covering a great extent of the layer of small and medium-sized pyramids (Fig. 25, K}; b, a second type of similar form but whose axon forms its terminal arboriza- tion very close to the cell (Fig. 25, E} ; c, still another form with horizontal axon the superficial branches of which penetrate into the plexiform layer (Fig. 25, -Z>) ; d, arachniform cells with axons subdivided into dense plexuses (Fig. 25, F, G); e, fusiform, bi- panicled cells, which have been sufficiently described. Fio. 24. — Ensemble of layers of motor cortex of infaut aged one and a half mouths; Golgi's method (semischematio) . Layers are numbered as follows: 1, plexiform; 2 and 3, small and medium-sized pyramids; 4, super- ficial giant pyramids ; 5, granular or small stellate cells ; 6, deep giant pyramids ; 7, poly- morphic cells, or deep medium-sized pyramids. (In this figure I have not represented the deepest portion of the 7th layer.) 364 Santiago Ramon y Cajal: Having studied all these types and many others in the visual cortex, it is unnecessary here to enter upon a more detailed description. One Fig. 25. — Cells with short axons of the plexiform and small and medium-sized pyramidal cell layers from motor cortex of infant aged one month and a few days. A, B, C, horizontal cells of the plexiform layer ; D, cell with horizontal axon ; E, large cell with very short diffusely subdivided axon ; F, G, spider-shaped cells whose delicate axons form a dense plexus (G) up to the plexiform layer ; H, J, bipanicled cells. thing concerning the bipanicled cells I may add, viz., that in the motor cortex there appear to be two kinds : one, small cells provided with slender axon disposed in very delicate vertical pencils (Fig. 25, B^ ; the Sensori-Motor Cortex. 365 other consisting of relatively large cells having very long and thick den- drites and with an ascending or descending axon giving rise to terminal arborizations of extreme complexity, producing nests or terminal bas- kets about the bodies of the small and medium-sized pyramidal cells (Fig. 25, J"). Possibly this tj-pe, which I take to be a variety of the common bipanicled cell, is present over the whole cortex ; but as yet I have succeeded in finding it only iu the motor convolutions of the infant at over one month of age. 3. Superficial Layer of Giant Pyramids. — Being a continuation by imperceptible gradations of the above, this layer contains the well-known large pyramids of the writers. In addition to the observations of Betz, Le^vis, Golgi, and myself, however, I must add a single detail to their classical description. The radial process varies greatly as to the extent of surface it covers in the plexiform layer. When its dendrites must cover a large surface, the trunk forks near the cell, and the two trunks, deviating at an acute angle, ascend to give rise to two or more terminal sprays, in some cases at considerable distances apart. This amounts to saying that certain medium-sized and large pyramids stand related to a large number of nerve fibres in the 1st layer, while other cells of the same size have more limited connections (Fig. 24). In g}'rencephalous mammals, dog and cat, the superficial large pyra- mids are smaller than in the infant. They might be considered as a sub- ordinate element in the layer of medium-sized pyramids. Most frequently the only giant pyramids in the cat occur below the granular layer, — a layer which, I may add, is very slightly developed in this animal, being often blended with the layer of medium-sized pyramids. The number of superficial, medium-sized, and giant pyramids is very large in the motor area both in animals and man ; and this is one of its characteristic features. However, the regions designated by Flechsig as association centres possess also a notable number of large pyramids. From this feature alone it would be quite difficult to distinguish the frontal and parietal from the motor convolutions. The axon of the large and medium-sized pyramids descends, as is well known, to the white matter and is continued as one or two nerve fibres. I must call special attention to the fact that, as shown by my own researches, this fibre may fork usually into a fine branch which goes, probably, to the corpus callosum and a larger branch to the corpus stria- tum. This may be easily observed in the brain of a newborn mouse or 366 Santiago Ramon y Cajal : in one a few days old. It may also be seen that the fibre entering the corona radiata passes beyond the corpus striatum, giving off to it a few collaterals. It is thus well established that the axon of the large pyra- mids is true projection fibre which takes part in forming the pyramidal tract. But we must be on our guard about accepting the view of certain writers, — v. Monakow, for example, — who ascribe this role, participa- tion in the motor tract, exclusively to the giant pyramids, because I have demonstrated beyond all doubt, in the motor region of the mouse and rabbit, that a number of the axons of medimn-sized pyramids and many from polymorphic cells also penetrate the corpus striatum. I therefore consider as wholly arbitrary all the opinions which tend to attribute an exclusive function to elements in each distinct cortical layer. In the cor- tical layers, as well as in the ventral horn of the spinal cord, tliere occur together elements with axons of very diverse character and connections. The motor cell takes its place beside the associational cell along with the element whose axon or collateral goes to the corpus caUosum. There are, accordingly, in the cortex no " sensory layers " nor " motor layers " ; because, as we shall see in a moment, the great majority of the cells are related, either by their cell bodies or by their radial trunks, to the plexus of sensory fibres. We find thus reproduced the arrangement of the spinal cord, where all the cells, or almost all, come into contact with sensory fibres of the first or second order, and all represent links in the •chain of reflex connections. 4. Layer of Small Stellate Cells {Granular Layer of the Authors). — Stained by Nissl's method the layer of small stellate cells appears as a great number of nuclei surrounded with little protoplasm which contains a few fine granules of chromatin (Figs. 23, 5, and 24, 5). Most of these elements, the granules proper, are very small and globular or stellate in form. Others, I have observed, are comparable to small pyramids, being of triangular form and having a fine radial trunk. Nor is there any lack of stellate or fusiform cells of considerable size, which call to mind those of the visual cortex. All these elements appear to be mingled. However, in certain places I thought I could discover that the small globular ceUs are situated chiefly in the external plane of the layer, while the minute pyramids were more numerous in the deeper levels, but there are exceptions to this. But Nissl's method does not enable us to study the fine processes of •these elements. To this end we must have recourse to the chromate Sensor I- Motor Cortex. 367 of silver method, and by its application — especially in case of an infant fifteen to thirty days old, a time at which the reaction is easily obtained — I have been able to demonstrate the extreme complexity of the granular layer. Good preparations show that it consists of elements with very diverse characters, which in spite of their minor differences may be classed into two groups : (1) cells with long axons which extend down to the white matter; (2) cells with short axons which end within the granular la3'er or in layers above it. Cells with Long Axons. — These may be classed into two varieties, small pyramidal cells and medium-sized stellate cells. (a) The small pyramid is specially numerous in the deep level of the 4th layer (Fig. 26, A, B}. It has been figured by various writers, notably by Kolliker, although even he does not give us an}' information on the character of its axon. The cells are ovoid-pyramidal in form. They possess a radial trunk wliich extends up to the plexiform layer, where it ends in a few very slender varicose twigs without contact granules. It also has a few tiny descending or oblique dendrites which divide repeatedly. Finally, I have very often traced its axon to the white matter, in which it is continued as a slender medullated fibre. From its initial portion arise two, three, or four coUatei'als, some of which curve upward to distribute themselves through the 4th layer. In some cases the diameter of these collaterals is so large, compared with that of the axon, that they might be considered the real axons. (6) Stellate Cells. Very hard to stain, and possibly quite scarce. Their dendrites arise from the angles of the cell body and run in all directions, but are distributed exclusively to the 4th layer (Fig. 26, i)). Their axons spring from the inferior surface, descend almost in a straight line, and, after giving off a few large collaterals, very frequently arched and re- current, are lost in the white matter. These interesting cells, exactly similar to the stellate cells of the visual cortex, are also found in the motor cortex of gyrencephalous mammals, although, to judge from my own preparations, only in small numbers. Their presence would seem to indicate distinctively sensory regions of the brain. Elements with Short Axons. — These are also very numerous in the infant brain, representing, perhaps, the chief morphological factor of the 4th layer. Several varieties have been distinguished, of which the most common are the following : — (a) Stellate or Fusiform Cells of Medium Size. Their dendrites 568 Santiago Ramon y Cajal : diverge in all directions, but chiefly above and below, and end in the midst of the 4th layer. Their axon springs from the superior surface, ascends for a variable distance, and at varying levels of the layer of stel- Fia. 26. — Cells with long axons from 4th layer of motor cortex of infant aged one month. A, B, C, smaU pyramidal cells; D, large stellate cell; E, medium-sized pyramid; a, axon; b, c, large descending collaterals. late cells forms an arborization of horizontal or oblique branches of con- siderable length and distributed exclusively to the 4th layer. Very often the axon branches in the form of a T before proceeding to its ter- minal arborization, and from its initial part arise collaterals whose course SensorirMotor Cortex. 369 and terminations resemble those of the terminal branches. These cells, we may add, correspond in all points to the cells with ascending axons described for the 4th and 5th layers of the visual cortex (Fig. 27, A, C, D). (6) Fusiform, Triangular, or Stellate Cells. These are somewhat Fio. 27. — Cells with short axons from 4th layer of motor cortex of infant. A, B,C, cells, stel- late or fusiform, with ascending axon divided into lonj; horizontal branches ; £, arachuiform cell ; F, cell with axon distributed to layer of medium-sized pyramids. larger than the preceding. Their axon ascends to the plexiforra layer, in which it subdivides and terminates. In its ascent it supplies a few collaterals to the 4th and 3d layers. These elements, as we see, corre- spond to the so-called cells of Martinotti. In a few cells of this class the axon possibly does not reach the first layer, becoming lost in the layers below (Fig. 27, A"). 2b 370 Santiago Ramon y Cajal: (c) Small Stellate or Spider -shaped Cells. These possess fine and richly subdivided dendrites and also a neurite, which forms a very rich arborization close to the cell (Fig. 27, E). (ci) Bipanicled Cells. These have the characteristics already de- scribed in our study of the visual cortex. (e) Finally, in the cat and dog I have found a few stellate cells with very numerous dendrites, whose descending neurite forms a very dense and complicated arborization, for the most part in the 4th layer, but in some cases extending down to the deep layer of giant pyramids. Possibly these cells are homologous to the spider-shaped cells in man, which they resemble in the extraordinary richness of the plexus formed by the axon. It would then be necessary to suppose, however, that in the cat and dog these cells are much larger than in man. In order to complete my description, permit me to add that there is no lack of ordinarj^ pyramidal cells, in some cases large, scattered irregu- larly in the 4tli layer (Fig. 26, E^. In mammals like the cat and dog, and to a much greater degree in the rabbit, the profusion of pyramidal cells obscures our picture of the granular layer. Sensory Nerve Plexus of the 4th Layer. — One of the most significant facts which I have discovered in the motor cortex is a plexus of very large fibres whose numerous subdivisions occupy the 4th layer and extend even into the 2d and 3d. They probably enter the cortex from the corona radiata. As early as in my first work I called attention to these fibres as being different in diameter, direction, and origin from axons of pyramidal cells, but at that time I had not succeeded in deter- mining the region to which they are peculiar or the precise place of their termination in the cortex. My recent researches upon the brain of man and also small mammals enable me to add a few details to my description of some years ago (Fig. 28). First of all, I have been able to determine exactly their origin and position in the brain. These are both easy to observe in the brain of a rabbit at birth and still better in that of a mouse a few days old. In the mouse it may be seen especially well that certain large fibres (called by KoUiker, who has confirmed their existence, fibres of Cajal) proceed from the corpus striatum, enter the white matter, and often extend horizon- tally in it for great distances. In their course they throw off long col- laterals, which penetrate into the overlying gray matter. All these collaterals, as well as finally the original axon itself, ascend through the Kio. 28. — Plexus of heary sensory fibres from motor cortex of cat 25 days old. A, plexiform layer; B, layer of small and medium-sized pyramids; ('and D, layers of Rrannles and superficial layer of giant pjTamids ; i?, deep layer of fiiant pyramids ; >', layer of polymorphic cells; a, fibre from white matter; 6, asceudinj; collateral ; c, varicose tcrmiual arborization ; d, fibre directed to the plexiform layer, which appears to be distinct from the large fibres. 372 Santiago Ramon y Cajal: polymorphic layer, dividing once or twice, then, passing obliquely through intervening layers, form an arborization of heavy fibres within the layers of small, medium-sized, and large pyramidal cells. However, in the rat and rabbit these branches are most numerous in a relatively superficial plane, which corresponds probably with the granular layer of the human brain, — a layer that is not differentiated in the smaU mammals. We also find a relatively small number of branches that ascend to the plexiform layer. As to the cortical distribution of this plexus, we may also place on record a fact of interest. It never covers the whole cortex. It begins to appear some distance from the median fissure and disappears below long before reaching the olfactory area or limbic lobe. I have never observed it in the cortex of this sulcus, nor in the anterior portion of the frontal lobe, nor even in the region of the auditory or visual centres. I shall return to this matter in a future investigation, for I think it merits most thorough study ; because, if it can be confirmed in a positive manner and by other methods, we shall possess a criterion by which to distinguish between areas of association and projection in the cortex. The projection areas will probably be found to be not, as Flechsig thinks, those possessing fibres that go to the corpus striatum (since Dejerine and others have discovered these fibres in the so-called association centres) but those receiving sensory fibres. At the same time, the association centres will be characterized by the absence of these direct sensory connections. At any rate, I believe that even in the brain of the smallest mammal there are areas, of small extent it may be, specialized to store up the images or residues of the sensory projection centres. It would be most astounding if the brains of the small mammals possessed a different architecture from that of man, taking into consideration the fact that all the senses have the same essential structure in all mammals and that memory — visual, tactile, muscular, etc. — is just as necessary to their lives as to our own. The sensory plexus is highly developed in gyrencephalous mammals and in man. I have found it well impregnated in the brains of infants at birth and a few days old. Here it appears made up of large fibres having an oblique direction and a flexuous or even staircased course. After dividing several times in the 6th and 5th layers they give rise to a singularly extended arborization of horizontal fibres distributed chiefly to the layer of granules or small stellate cells. We thus see in the motor SensorirMotor Cortex. 373 cortex, as was the case in the visual, that the layer of granules is the principal focus of sensory impressions. From this terminus they are propagated by the numberless cells with short ascending axons to the layers above and especially to the medium-sized and giant pyramids. However, it must be acknowledged that the sensory plexus is not so narrow and well defined as the optic. For, although its greatest density occurs in the 4th layer, its terminal branches divide in their ascent to the superficial layer of medium-sized and giant pyramids. The fibres which extend up to the small pyramids in man are not numerous. It is for this reason that I cannot agree with Bevan Lewis in ascribing to them sensory functions. I do not wish to be understood to deny the sensory function of the small and medium-sized pryamids. According to my view, all the cells of the motor cortex are sensory because they all, possibly, come into contact either directly (cells of the 3d, 4th, and 5th layers) or indirectly, through the intervention of cells with short axons, with sensory terminal arborizations. But, since some cells send their axons to the pyramidal tracts, we are able to distinguish them as sensori-motor cells of the first order. The others, which send their neurites to other motor areas of the brain, possibly effect contact with sensori-motor cells of the first order located elsewhere. These cells of indirect sensori-motor communication we may be warranted in calling sensori-motor cells of tJie second order. It goes without saying that this distinction is purely hypothetical ; for no method enables us to determine the precise point within the brain where the axons of the pyramidal tracts of the corpus callosum or of bands of association fibres form their terminal arborizations. 5. Layer of the Giant and Medium-sized Pyramids. — In the adult human brain stained by Nissl's method, a section of the motor cortex reveals, below the granular layer, a layer of plexiform or granular aspect filled very thickly, but in no particular order, with a few giant and a great number of medium-sized pyramids (Fig. 29). Usually the giant pyramids are located near the 4th layer, forming there a few irregular ranks. Impregnated by Golgi's method, they appear similar to the same cells in other regions of the cortex, but differ in a few particulars. The body is generally conical, very much elon- gated, giving rise at the apex to a large trunk, often dividing near the cell, which terminates in the 1st layer in the usual manner. A group of long complicated dendrites diverges from its base, and from the sides 374 Santiago Ramon y Cajal . spring several very long horizontal processes which subdivide into ter- minal brushes, and these, intertwining with similar structures from neighboring cells of the same level, form a dense and very characteristic Fig. 29. — Deep layer of giaut pyramidal cells from motor cortex of infant aged 20 days. A, B, pyramidal cells; D, C, elements with short axons. protoplasmic plexus. It is the same arrangement we already know so well in the visual cortex, except that, instead of one plexus, there are many. The axon is large and, after giving oif very long collaterals to the 5th and 6th layers, it passes on to become a meduUated fibre of the white matter. The medium-sized pyramids are very numerous, much scattered, and Sensori-Motor Cortex. 375 occur in greatest profusion in the lower levels of the layer. They do not differ in character from the giant pyramids, except as to the lateral somatic dendrites, which are few and not characteristic. Besides the pyramidal cells the 5th layer contains a few other kinds of elements. From the point of view of their morphology the following are the more striking types. (rt) Cells which form Terminal N'ests. — These cells, very similar to those which give rise to the basket fibres of the cerebellum, are most numerous in the 5th layer between or below the giant pyramids. I have found them also in the layer of granules or small stellate cells. Their volume is small, similar to that of a small pyramid, and in form they appear stellate or triangular with very long and much-branched varicose dendrites. The neurite, however, presents the most distinctive feature. It ascends, forking close to its origin, and breaks up into a ramification of very many branches, ascending, oblique, or horizontal. After a few subdivisions, all these branches make their way to the giant and medium-sized pyramids to form very complicated varicose arboriza- tions close around their cell bodies and principal processes, after the manner of the terminal baskets of the cerebellum or the nests found in Deiter's nucleus. Each nest contains arborizations from several cells, and each basket cell helps to form a large number of nests (Fig. 30, c?). (6) Cells with a Diffusely/ Branched Ascending Axon. — This is a fusiform or stellate cell located at different levels of the 5th layer, to which it sends its dendrites. The axon ascends to the superior limits of the layer where it foi-ks, and its terminal branches form a loose horizon- tal arborization of an enormous extent and connected probably with the deep giant pyramids (Fig. 29, C. D). (c) Small Pi/ramids with Arched Axons. — Tliis cell, wliich I have studied particularly in the motor cortex of the cat, is entirely similar to the element which we found in the 6th and 7th layers of the visual cortex. The cells possess a fine dendrite which ascends to the first layer, where it ends in a very modest and delicate arborization. Their axon descends and, after giving off a few relatively long recurrent collaterals, appears to fork and end in the midst of the 5th layer. The branches which spring from the bend of the arch descend in some cases, but I have not been able to trace them down to tlio white matter. (d) Cells with Long Ascending Axon. — These are fusiform or tri- angular cells with long polar dendrites which never reach the first layer. 376 Santiago Ramon y Cajal : Their axon arises from the superior surface of the cell, and, after giving off a few branches to the 5th and 4th layers, it continues its ascent to the plexiforra layer and there makes its terminal arborization. 6. Layer of Polymorphic Cells. — This layer contains the same elements as the layer of the same name (9th) in the visual cortex (Fig. 31), that Fig. 30. — Pericellular terminal arborizations from the deep layer of giant pyramids, motor cortex (ascending frontal convolution) of infant aged 25 days, a, axons giving rise to oblique and horizontal branches ; 6, c, d, terminal nests. is to say, fusiform cells with two long polar dendrites, triangular cells, and true pyramids. Their axons all go to the white matter. Their ascend- ing trunks, which are never lacking, become very attenuated on account of the branches given off while passing through the 4th layer and reach the 1st layer as an exceedingly delicate fibril, which ends in a fine, slightly extended, notably varicose dendritic spray. In Fig. 31, I have reproduced the principal types of cells found in the polymorphic layer. Besides the medium-sized pyramidal and triangular types having long descending axons (Fig. 31, A, B'), there occur other Sensori- Motor Cortex. 377 forms in great numbers. These are fusiform or triangular cells whose axons penetrate into the superposed layers, furnishing to them a great Fio. 31. — Principal types of polymorphic cells from motor cortex of infaut aged20days. -1, B, cells with long axous extending to white matter ; C, D, E, fusiform cells with ascending axon ; H, giant stellate cell. number of branches. Some of these axons seem to end in the deep layer of giant pyramids, but others appear to pass beyond this. Finally, there 378 Santiago Ramon y Cajal: is no lack of arachniform cells (Fig. 31, J), cells with short axon of the sensory type of Golgi, whose axons form terminal arborizations in the layer under consideration. I may add that I have found in two cases giant stellate cells with heavy horizontal axon which gives off collaterals (Fig. 31, H^. I do not know the ultimate fate of this process and am unable to say whether these scattering cells form a constant feature of the motor cortex. Cortex op Acoustic, Olfactory, and Associational Areas. Unfortunately, my own researches are not as yet in a very advanced state in regard to these important cortical centres. So that any in- formation that I can give must necessarily be fragmentary and of little value. The acoustic resembles exactly the motor cortex as to general arrangement of cells and layers, but diliers from it in a few pecu- liarities : (1) by the fineness of the fibres forming the plexus at the level of the layer of granules or small stellate cells ; (2) in the profusion of bipanicled cells with their very delicate and complicated neuritic brushes ; (3) above all, by the presence of certain special cells scattered irregularly thi-ough the entire thickness of the cortex. The very large axon of these special cells extends in a horizontal or oblique direction, but I have not yet been able to determine exactly its manner of termination. These large cells are fusiform and lie horizontally. From their polar dendrites spring a number of fine ascending branches, which subdivide repeatedly but do not extend up as far as the plexiform layer. The olfactory cortex, that of the limbic lobe, is characterized by the following peculiarities : (1) the enormous development of the plexiform layer and the presence in it, in addition to its usual structures, of the antero-posterior fibres that come from the external root of the olfactory tract ; (2) the absence of the layers of small pyramids and granules ; (3) the presence of certain large horizontal cells below the plexiform layer ; (4) the peculiar form of the medium- and large-sized pyramids which emit from the deep end of the cell body a brush consisting of numerous much subdivided dendrites ; (5) above all, the fact that the sensory plexus, i.e. the fibres which come from the olfactory bulb, makes its terminal arborization exclusively in the plexiform layer and in the most superficial portion of that layer, corresponding to that of the small Sensor ir Motor Cortex. 379 pyramids. This significant fact, brought to light by the studies of Calleja, shows us that the sensory fibres do not end in the same level of the cortex in all regions. Hence, the layer specialized to serve as substratum for the phenomena of sensation may change its position in different sensory areas. Our task is now drawing to its close. My work upon the topo- graphical structure of the cortex has been fragmentary and leaves much to be desired. Many things, in fact, are still undiscovered. But, despite the very incomplete state of my researches and the narrow limits of the field they cover, I may di-aw a few anatomico-physiological conclusions, of which the chief are the following : — And first, as to the hierarchy of centres in the cortex of the human brain, comparing it with the mammalian brain, we may call to mind that, while it does not contain wholly new elements, it presents very distinc- tive characteristics, to wit : — 1. The enormous development of the horizontal cells of the plexi- form layer and the considerable length of their so-called tangential fibres. 2. The great abundance of cells with short axons scattered through- out the whole cortex, cells which form special varieties by reason of differences in their forms and the directions of their axons. 3. The presence of cells with short axons, very slender (bipanicled spider cells), with terminal arborizations whose delicacy is not apjjroached by anytliing found in any animal. 4. The considerable development of basilar dendrites of the pyram- idal cells. 5. The presence among the mid-layers of the cortex of a formation of so-called granular cells, a kind of focus occupied by enormous num- bers of pyramids with short axons descending, arched, and ascending. This granular formation is present in gyrenceplialous mammals, but in them it is very poor in cells with short axons and in small pyramids. In the smooth-brained animals it is almost wholly lacking. The human cortex has evolved, accordingly, along three different lines : by multiplying cells with long axons and, above all, those with short axons ; by decreasing the volume of cells and the diameter of cer- tain fibres in order to make possible within the limits of space a deli- cate and greatly improved organization ; finally, by varying and infinitely 380 Santiago Ramon y Cajal : complicating the external morphology of the nerve elements, undoubt- edly with the purpose of multiplying, in correspondence with their complexity, functional associations of all kinds. As to differences and analogies in regional structure, the following propositions may be regarded as established : — 1. The sensory as well as the so-called associational areas are made up by a combination of two orders of structural factors. The first order consists of common factors, which show very little modification. They are represented by the plexiform layers and the layers of pyram- idal and polymorphic cells. The second order comprises special fac- tors, structures peculiar to each cortical area. Their chief anatomical feature resides especially in the granvdar layer and is related mainly to the presence of particular centripetal fibres and of special types of cells with long axons (stellate cells of different kinds). 2. It seems probable that the common factors perform functions of a general order concerned, possibly, with ideas of representations of all kinds of movements related to the special sensations of which the cortical region is the seat. It seems also probable that the special anatomical factors of the sensory areas perform the function of elab- orating specific sensations (sensation of seeing, hearing, etc.) and also of conveying sensory residues to the so-called association centres, where they may be transformed into latent images. 3. Each sensory cortical centre receives a special category of nerve fibres (fibres of central sensory tracts). Their cells of origin, as has been shown by the researches of v. Monakow, Flechsig, v. Bechterew, and many others, reside in the particular nuclei of the medulla, corpora quad- rigemina, and optic thalami. It is precisely the presence of these sen- sory fibres of the second order that constitutes the prime anatomical characteristic of the centres of sensation or projection. 4. The absence of these sensory fibres, which come from the corona radiata, may be used in all mammals to distinguish the so-called associ- ation centres. These centres, which exist even in the mouse, also have a nerve fibre plexus distributed among their median layers (layer of gran- ules in the association areas in man). The fibres, however, which consti- tute them are very fine and appear to come from sensory centres of the brain. Possibly the cells about which these sensorio-ideational fibres terminate represent the substratum or, at any rate, the first link in the chain of nerve elements whose function is the representation of ideas. SensorirMotor Cortex. 381 5. Since we have seen that each afferent fibre in the sensory cortex comes into contact with an extraordinary number of nerve cells appar- ently scattered without any order, we must suspect that these relations conform to the preconceived design of a well-determined and constant organization. As, at present, it seems to be impossible to discover these relations, we may surmise that each sensory fibre comes into contact, directly or through other cells, solely with those pyramids whose stimulation is nec- essary in order to effect, after the manner of the reflex arc, movements coordinated and intentional. We may also surmise (supposing that the stellate cells of the tactile and visual cortex form the link between the sensory and ideational centres) that each sensory afferent fibre, bringing a unit of sensation (the impression received by a cone of the retina or by the terminal arborization of any peripheral nerve fibre), enters into rela- tion exclusively with the group of nerve cells entrusted with the func- tion of conveying this impression to a particular point in the associational cortex. Many other hypotheses are possible, but I must conclude for fear of tiring your kind and sympathetic attention and exhausting your patience. I fear that I have already made too free use of hypotheses and have pre- tended to fill the gaps of possible observations with arbitrary supposi- tions. It is a rule of wisdom, and of nice scientific prudence as well, not to theorize before completing the observation of facts. But who is so master of himself as to be able to wait calmly in the midst of darkness until the break of dawn ? Who can tarry prudently until the epoch of the perfection of truth (unhappily as yet very far off) shall come ? Such impatience may find its justification in the shortness of human life and also in the supreme necessity of dominating, as soon as possible, the jihe- nomena of the external and internal worlds. But reality is infinite and our intelligence finite. Nature and especially the phenomena of life show us everywhere complications, which we pretend to remove by the false mirage of our simple formulse, heedless of the fact that the simplicity is not in nature but in ourselves. It is this limitation of our faculties that impels us continually to forge simple hypotheses made to fit, by mutilating it, the infinite uni- verse into the narrow mould of the human skull, — and this, despite the warnings of experience, which daily calls to our minds the weakness, the 382 Santiago Ramon y Cajal : SensorirMotor Cortex. childishness, and the extreme mutability of our theories. But this is a matter of fate, unavoidable because the brain is only a savings-bank machine for picking and choosing among external realities. It cannot preserve impressions of the external world except by continually simpli- fying them, by interrupting their serial and continuous flow, and by ignoring all those whose intensities are too great or too small. I cannot conclude this, my third and last lecture, without a word of tribute to this great people of North America, — the home of freedom and tolerance, — this daring race whose positive and practical intelligence, entirely freed from the heavy burdens of tradition and the prejudices of the schools, which weigh still so heavily on the minds of Europe, seems to be wonderfully endowed to triumph in the arena of scientific research, as it has many times triumphed in the great struggles of industrial and commercial competition. 1 ^f^. V '^^ ^..^..^^^^^^-^ ^^Z;^ ;^^<%^^C_ r^^^ ^<^