^ GIFT or 
 
FROM THE TRANSACTIONS OF THE ROYAL SOCIETY OF CANADA. 
 
 THIRD SERIES 1910 
 
 VOLUME IV. SECTION III 
 
 Mathematical Instruction in France, 
 
 By 
 
 RAYMOND CLARE ARCHIBALD. M.A., Ph.D. 
 
 Po-^l 
 
 OTTAWA 
 
 PRINTED FOR THE ROYAL SOCIETY OP CANADA 
 
 1011 
 
Section III., 1910. [ 89 ] Trans. R. S. C<^^;Q<^ 
 
 XII. Mathematical Instruction in Frajice. 
 
 By Raymond Clare Archibald, M.A., Ph.D., 
 
 Brown University, Providence, Rhode Island. 
 
 (Presented by Dr. E. Deville and read in abstract, 28 September, 1910.) 
 
 CoNTEKTS. 
 
 Introductory 90 
 
 General Remarks Educational System and Primary Instruction . . 91 
 
 Elementary Mathematical Instruction 
 
 The Lyc^es 93 
 
 The Baccalaureat 98 
 
 The Classes de Mathematiques SpecicUes 103 
 
 Higher Mathematical Instruction 
 
 The Sorbonne 108 
 
 The Licence 112 
 
 The Dipldme d'JEtudes Supirieures de Mathematiques 113 
 
 The Agregation des Sciences Math&matiques 113 
 
 The Doctorat 115 
 
 The ficole Normale Sup^rieure 118 
 
 The ficole Polytechnique 120 
 
 The College de France 121 
 
 Concluding Remarks on Mathematical Instruction 122 
 
 Teaching of Mathematics as a Profession in France 124 
 
 The American Mathematical Student in Paris 127 
 
 Authorities 131 
 
 APt*ENDlCES. 
 
 A* The Agregation des Sciences Mathematiques 133 
 
 B. Mathematical Courses offered in Universities outside of Paris, 
 
 1909-1910 151 
 
 For Later Publication. 
 
 C. The Doctorat ^s Sciences Mathematiques in France, 1811-1910. 
 
 D. A List of Mathematical Text Books in French Secondary Education. 
 
 276f>2(> 
 
90 ROYAL SOCIETY OF CANADA 
 
 Introductory. 
 
 As the result of remarkable progress during the past fifteen years, 
 a vigorous American School of Mathematics, of which the German 
 School may well be considered the parent, has been developing. Many 
 years must elapse before the offspring may exercise the authority and 
 influence made felt by such masters as Gauss, Riemann, Steiner, Weier- 
 strass and Klein. But meanwhile the process of evolution is proceeding 
 in thorough fashion. In preparing for higher mathematical education, 
 America has recognized the fundamental importance of the secondary; 
 organization, discussion, and criticism of home methods as well as study 
 of those of foreign countries, have been widespread in recent years. 
 But here, again, the preponderance of discussion in book and periodical 
 is of German methods. The series of reports of the Carnegie Founda- 
 tion for the Advancement of Teaching is doing considerable to spread 
 accurate information of a more general character. 
 
 Yet in spite of the predominance of German influence in the dis- 
 cussion, I have become convinced, after several months of observation, 
 that just as much might be beneficially acquired by the study of mathe- 
 matics and methods of mathematical training in France, as in any other 
 country. Has not this country produced Chasles, Monge, Pon'celet, 
 Cauchy, LaPlace, Hermite? What city beside Paris has to-day such a 
 large number of mathematicians of the first order? There are Poincar^, 
 Darboux, Goursat, Picard, Painleve, Appell, Jordan, Humbert, Borel, 
 Tannery, to mention only a few. What other country gives such a 
 course of mathematical training as is provided in the Classes de Mathe- 
 matiques Speciales of the French Lyc^es? Where else is the extraordi- 
 narily high standard of the agregation demanded of higher teachers in 
 the secondary schools? 
 
 Nevertheless when leaving Harvard some ten years ago with a view 
 to further mathematical study in Europe, and although more or less 
 familiar with such classic treatises as those of Darboux, Picard, Tannery, 
 Appell and Goursat, I scarcely even considered France, as a possible place 
 of location. The professors at Harvard who had studied abroad had 
 been trained in Germany and were thoroughly imbued with German 
 methods and ideals. The same was doubtless true of at least ninety- 
 five per cent of the mathematical professors in the larger American 
 colleges, and the same may be said to-day Why this neglect of France ? 
 
 For one thing the American student usually looks forward to 
 getting a doctor's degree in one or two years, while the idea is certainly 
 prevalent among us that if the French degree of doctor were at all 
 available for the foreigner, he could only expect to get it at the age of 
 forty-five or fifty, and after writing some monumental or epoch-making 
 treatise. Such ignorance is without doubt due in part to the excessive 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 91 
 
 difficulty of getting any reliable information about French as compared 
 with German universities. Even when the inquirer is in Paris the 
 difficulty does not wholly disappear. 
 
 It would seem then, that a service may be rendered to students and 
 university professors who have in prospect a sojourn in Europe for 
 mathematical study, if I should present a general view of the situation 
 in France, along with fuller details on topics which must be of especial 
 interest to every mathematician. 
 
 The plan of the paper is indicated by the table of contents. 
 
 General Remarks Educational System and 
 Primary Instruction. 
 
 To more thoroughly understand the methods and ends of mathe- 
 matical instruction in France it will be well to introduce here some brief 
 general remarks. 
 
 For educational purposes France is divided geographically into 
 arrondissements. The assemblage of government schools (primary, 
 secondary and superior) in each arrondissement, forms an academie over 
 which a recteur presides. We thus have the 16 academies of Aix-Mar- 
 seilles, Besangon, Bordeaux, Caen, Chambery, Clermont, Dijon, Gren- 
 oble, Lille, Lyons, Montpellier, Nancy, Paris, Poitiers, Rennes, Toulouse, 
 as well as a seventeenth at Algiers. With the exception of Chambery 
 these names correspond to the seats of the French universities, which 
 have from two to four faculties (law, science, letters, medicine) each, 
 the faculties of science and letters together corresponding to the German 
 philosophische Facultdt. In the academie first named above, the 
 faculties of law and letters are at Aix and the faculties of science and 
 medicine at Marseilles. 
 
 The assemblage of academies forms the Universite de France, at 
 the head of which is the Minister of Public Instruction, who is ex officio 
 the " Recteur de 1' Academie de Paris et Grand Maitre de PUniversit^ 
 de Paris.'' For the Academie de Paris there is a vice-recteur, 
 whose duties are the same as those of the recteurs of other academies. 
 Although nominally lower in rank than the heads of academies in the 
 provinces, he is in reality, the most powerful official in the educational 
 system. The position of the Minister of Pubhc Instruction being so 
 insecure by reason of changing governments, continuity of scheme is 
 assured by three lieutenants who have charge respectively of the 
 primary, secondary and superior education. They in turn have an 
 army of inspectors who report on the work and capabilities of the recteurs 
 and their academies as far as primary and secondary instruction is 
 concerned. 
 
92 
 
 ROYAL SOCIETY OF CANADA 
 
 This suffices at present to indicate the remarkably centralized 
 and unique character of the French educational system. It is theo- 
 retically possible for the most radical changes in any part of public 
 instruction to be immediately brought about by a stroke of the pen 
 on the part of the Minister of Public Instruction. 
 
 In the recuperation of the French nation during the past 40 years, 
 gigantic strides have been made in all departments of education; scores 
 of handsome and spacious new buildings have been erected, new chairs 
 have been endowed, new laboratories established and equipped while 
 in connection with special schools all over the country, scholarships 
 and prizes call forth and reward the best effort of the nation's youth. 
 Forty years ago the state spent 32 million francs for education. The 
 1909 budget of the Minister of Public Instruction and Fine Arts called 
 for 293 millions nearly two-thirds of this amount being allotted to 
 Primary Instruction. 
 
 Primary and superior instruction are free in France and over five 
 million children are now annually in attendance at the public primary 
 schools. Broadly speaking there are three classes of these schools 
 which give strictly elementary education: 
 
 A. Ecoles Maternelles. A sort of kindergarten for children of 
 both sexes from 2 to 5 or 6 years old. 
 
 B. Ecoles Primaires EUmentaires for pupils 7 to 13 years of 
 age. The course is divided as follows: 
 
 Age 
 
 Section Fiifa^Dtine . ... 
 
 5 or 6-7 
 
 
 Cours El^mentaire 
 
 I^re Ann6e 
 
 7-8 
 
 
 He " 
 
 8-9 
 9-10 
 
 Cours Moyen 
 
 ISre " 
 
 
 He " 
 
 10-11 
 
 Cours Sup^rieur 
 
 I^re " 
 
 11-12 
 
 
 He " 
 
 12-13 
 
 On completion of the cours moyen the pupil receives a certificat 
 d' Etudes primaires elementaires. This certificate or its equivalent 
 is required of every child in France. A very small proportion of those 
 receiving it take up further work in the cours superieur, in the lyc^es or 
 in 
 
 C.--Ecoles Primaires Superieures. These are for children of the 
 labouring class who do not aspire to a classical education, but who 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 
 
 93 
 
 wish to prepare themselves for industrial, commercial or agricultural 
 careers. 
 
 In what follows it will be supposed that the discussion is limited 
 to the education of boys. 
 
 Elementary Mathematical Instruction. 
 The Lyc^es. 
 
 The present system of secondary education in France dates from 
 the great reform of 1902 and is carried on for the most part in Lycees 
 and Collhges communaux which are to be found in nearly all cities. 
 Because of their pre-eminence we shall consider the former only, which 
 are under control of the state. Here the boys, who come from families 
 in comfortable circumstances, may enter as elhves at the age of five or 
 six years and be led along in their studies till they receive the Bacca- 
 laureat at the age of 16 or 17. Many lycees have still more advanced 
 courses to prepare for entrance into such schools as the Ecole Normale 
 Sup^rieure, ficole Polytechnique, ficole Centrale, ficole Navale, ficole 
 de Saint Cyr, etc. The pupils at the lycees are of four kinds: 1st. 
 Externes, those who come to the lycees for classes but board and lodge 
 outside; 2nd. Internes or pensionnaires, eleves who live entirely in 
 the establishment; 3rd. Demi-pensionnaires who usually reside at 
 a distance but take their mid-day meal at the lycee; 4th. Externes 
 surveilles, that is externes who work out their lessons under the eye 
 of the preparateur in the salle d' etude of the lycee. In the whole of 
 France rather less than one third of the lycee pupils are internes. At 
 lycee Louis le Grand, Paris, in 1907, 275 of the 909 ^l^ves were 
 internes; on the other hand at the Saint Louis, which is quite near, 
 there were 504 internes out of a registration of 854. 
 
 The expenses of the pupil vary greatly with the class and the lycee 
 in which he happens to be. The following table exhibits the range of 
 cost (in francs per year) , for some of the principal cities of the provinces 
 and for the better lycees of Paris. 
 
 
 Bordeaux, Lyons, 
 Marseilles, Toulouse. 
 
 Paris. 
 
 Externes *'. 
 
 70-^50 
 
 90-700 
 
 
 Externes Surveilles. . . 
 
 110-540 
 
 130-790 
 
 
 Demi-pensionnaires 
 
 370-850 
 
 500-1200 
 
 
 Pensionnaires 
 
 700-1200 
 
 900-1700 
 
 
 The lower price in each case is for the classe enfantine, the higher 
 for the special classes open to bacheliers. 
 
94 
 
 ROYAL SOCIETY OF CANADA 
 
 Instruction in fully equipped lycees may be divided into four 
 sections: I, Primary; II, Premier Cycle; III, Second Cycle; IV, Classes 
 de Math^m,atiques Speciales. 
 
 I. Pnmary. The classes in this section are named as follows: 
 
 Age from 
 
 Classes Enfantine 
 
 Onzieme 
 
 5 
 
 
 Preparatory Division 
 
 Dixieme 
 
 6 
 
 
 Neuvieme 
 
 7 
 
 Elementary Division 
 
 Huitieme 
 
 8 
 
 
 Septieme 
 
 9 
 
 In a general way this course corresponds to that which leads to the 
 certificat d'etudes primaii'es elementaires , but while the latter was 
 designed as a more or less complete unit in itself, the former is laid out 
 on broader lines and has in view further studies which the boy will 
 follow up. According to the plans d'etudes, it would seem as if one 
 essential difference were introduced by instruction in a modern language 
 in the neuvieme, huitieme and septieme. In reality, however, the 
 modern language classes are so conducted in the sixieme of the Premier 
 Cycle that both kinds of students are taught together. 
 
 II. Premier Cycle {sixihme-troisihme) . This cycle of four years 
 constitutes an advanced course for students who have finished 
 their primary studies, and is the first part of secondary education 
 proper. It offers a choice between two lines of study, the one character- 
 ised by instruction in Latin with or without Greek, the other in which 
 no dead language is taught. The former is selected by the parent who 
 wishes to prepare his boy for the department of letters in the ficole 
 Normale Superieure or for the career of classical professor, lawyer or 
 doctor. The latter is likely to be chosen for the boy who is particularly 
 interested in science or who has a commercial career in view. 
 
 III. Second Cycle. This leads, normally, to the BaccalaurSat, 
 at the end of three years' study, in one of four different sections. The 
 student of Latin and Greek in the quatrieme and troisieme has passed 
 into the "Latin-Grec" section, the student of modern languages 
 into the " Science-Langues vivantes" section, while this section as 
 well as those of ''Latin-Langues vivantes" and "Latin-Science" have 
 been filled by students who have studied Latin (but not Greek) , during 
 the four years of the Premier Cycle. The scheme will be clearer in 
 tabular form. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 
 
 95 
 
 
 I\ipils who learn Latin, with or without Greek. 
 
 Pupils who learn no 
 dead language 
 
 Age 
 from 
 
 
 
 PREMIER 
 CYCLE. 
 
 4 years 
 
 Sixi^me A (Latin). 
 
 Sixidme B 
 
 10 
 
 Cinquieme A (Latin). 
 
 Cinquidme B 
 
 11 
 
 .j^ 
 
 Quatridme B 
 
 12 
 
 Quatrifeme A (Latin Greek) Quatri^me (Latin) 
 
 Troisifeme A (Latin Greek) Troisifeme (Latin) 
 
 Troisi^me B 
 
 13 
 
 
 
 1 
 
 Pupils who give up the 
 study of Latin. 
 
 i 
 
 Sciences-Langues 
 
 
 Latin -Grec. 
 
 Latin-Langues 
 
 Latin -Sciences 
 
 SECOND 
 CYCLE 
 
 3 years 
 
 Second A 
 
 Second B 
 
 Second C 
 
 Second D 
 
 14 
 
 Premiere A 
 
 Premiere B 
 
 Premiere C 
 
 Premiere D 
 
 15 
 
 PhilosophieA 
 
 Philosophie B 
 
 Math^matiquesA 
 
 Math^matiques D 
 
 16 
 
 Let us now observe a little more closely just what is involved in 
 this display, in the matter of studies and demands made upon the eleve. 
 As an important examination which we shall presently describe comes 
 at the end of the Premise, our present analysis will not pass beyond 
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 24 
 27 
 28 
 23 
 2l 
 26 
 28 
 
 * Those who select Greek are exempted from 2 hours 
 in modern languages and one in drawing. 
 
 ** 12 lectures of 1 hour, 
 t Optional. 
 
 There are several features of this scheme (we shall refer to four), 
 which are particularly interesting. 
 
96 ROYAL SOCIETY OF CANADA 
 
 1. The prominence given to the study ot French throughout, 
 
 2. That all Aleves at the age of 10 or 11 commence the study of 
 modern languages (English, German, Italian, Spanish, Russian, or, in 
 Algiers, Arabic), and continue it during six years at least, before matri- 
 culating into schools of university grade. Not only do they get glimpses 
 of the best things in the literature of the language, but also learn to speak 
 the language with considerable freedom and remarkable correctness 
 in pronunciation. How many university graduates with us get a 
 training which leads to this result? The direct method is employed 
 and no word of French is ever spoken in the advanced classes. The 
 majority of the Aleves choose German, as this is required of all candidates 
 for entrance into such military schools as the ficole Polytechnique and 
 ficole de St. Cyr. On the other hand there is an increasing number tak- 
 ing up the study of English which is required for the ficole Navale. 
 
 3. The small number of hours devoted to mathematics in the 
 Premier Cycle, in II, A, B, and in I, A, B, of the Second. Yet there can 
 be little doubt that the situation might be summed up in even stronger 
 terms than in the recent report of the trustees of the Carnegie Founda- 
 tion, when comparing the methods of American and German schools: 
 "... lack of efficient teaching is one of the most expensive national weak- 
 nesses, and that the inefficiency of our school system is in great measure 
 due to this lack is evident. For example, mathematics is a subject 
 which has been a standard study in our schools from the beginning. 
 Students who pass through our high schools and enter college spend 
 in the nine years corresponding to the period covered by the German 
 gymnasium, seventy-five per cent, more of time of instruction on 
 mathematics and yet receive a training vastly inferior to that of the 
 gymnasium." Practically all the professeurs titulaires in the French 
 lycees, even those in charge of the very elementary classes, are agreg^s 
 in the subjects which they instruct. Just what this means in mathe- 
 matics we shall explain later, but suffice it to remark here that even the 
 demands made upon those who wish to become gymnasien professors are 
 nothing like as severe. 
 
 Another feature of mathematical instruction which is particularly 
 interesting to us is, that from the troisieme on, that is, from the time 
 the boy is 13 or 14 years old, instruction is given entirely by lecture. 
 Indeed, even in classes before the troisieme when a text book is generally 
 in the hands of the el^ve, he is required to take notes " pour preciser " 
 the various topics. By such methods, searching questioning and fre- 
 quent "tests," on the part of the professor, and rigid inspection, kindly 
 expressed praise or cutting public reprimand on the part of the proviseur 
 (director of the lycee) , there is no possibility of learning parrot-fashion, 
 as is so prevalent in our schools no room for the shirker or the boy who 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 97 
 
 does not try his best; reasoning powers and independence of thought 
 must be constantly exercised. The eleves are encouraged to consult 
 the various text-books to be found in all the lycee libraries and for those 
 less bright this may be almost a necessity from time to time; but on 
 personal inspection in different lycees I found the note books of Aleves 
 of 14 or 15 alike remarkable for their neatness and completeness. The 
 habits thus gained in the lycee stand in good stead when the student 
 reaches the university. The rapidity of the lecturer and the complexity 
 of his theme seem to make little difference, for at the close of the hour 
 the whole is in the note books as neat as copper-plate. There is surely 
 a lesson to be learned here for improving our secondary and superior 
 education. 
 
 4. The large number of hours in class recitation which may not at 
 first appear very imposing. But we cannot fail to be astonished that 8 
 hours per day (in class and in preparation of lessons) may be demanded 
 from eleves in the premier cycle, and lOJ in summer, 10 in winter from 
 those in the second cycle. The law further explicitly states that 
 there is no limit to the number of hours which may be demanded of 
 the eleves in the Classe de Mathematiques Speciales. When we later 
 come to look more closely at their programme we shall not be surprised, 
 but nevertheless wonder, how these undoubtedly happy and healthy 
 young men of 17 or 18 have survived the treatment. In more advanced 
 lycee courses as well as at the universities I was also impressed 
 with the almost appalling intensity and seriousness of the auditors 
 the strife is too strenuous, the competition too keen, to admit of a 
 moment's levity or wandering thought. But when the lesson is over, 
 every care is instantly banished and the national gaiety is once more in 
 evidence. 
 
 To return to our table. We remark that the two groups of eleves 
 who elect sciences on entering the Second Cycle have the same number 
 of hours per week in mathematics indeed the courses are identical. 
 To give greater definiteness to our ideas as to their general attainments 
 let us consider the programme of studies for Premiere D, when the 
 boy is 15 or 16 years old. 
 
 French. Lectures and questions on the principal French writers 
 of the nineteenth century. Study of selections from prose writers and 
 poets, from moralists, orators, politicians, scientists and historians 
 of the sixteenth, seventeenth, eighteenth and nineteenth centuries. 
 
 History. Political history of Europe in the eighteenth century. 
 Detailed history of France at the close of the eighteenth century. 
 
 Geography. Detailed study of France, its geological constitution, 
 its climatology, physiography, topography, economic and military 
 organization; its colonies, etc. 
 
 Sec. III., 1910. 7. 
 
98 ROYAL SOCIETY OF CANADA 
 
 > 
 
 Physics. Optics, electricity. 
 
 Chemistry. Of the carbon compounds. 
 
 German. Selections from the dramatic poetry of Schiller, Goethe, 
 Kleist and Grillparzer. Extracts from the prose works of Wieland, 
 Goethe, Schiller, Auerbach, Freytag, Scheffel, etc. 
 
 English. Shakespeare's Julius Caesar and Macbeth, extracts from 
 Milton, Addison, Goldsmith, Wordsworth, Byron, Coleridge, Dickens, 
 Macaulay, Eliot, Tennyson and Thackeray. 
 
 Algebra. Equations and trinomials of the second degree. Calcu- 
 lation of the derivatives of simple functions; study of their variation 
 and graphic representation; study of rectilinear motion by means of 
 the theory of derivatives; velocity and acceleration; uniformly changing 
 motion. 
 
 Geometry. Solid. 
 
 Descriptive Geometry. Elements. 
 
 Trigonometry. Plane, including the use of four or five place 
 logarithim tables, the solution of triangles and trignometric equations. 
 
 In passing it may be worth noting that the Latin course for Premiere 
 A, B, C, includes the study of selections from Cicero's letters and ora- 
 tions, from Livy, Seneca, Tacitus, Lucretius, Virgil and from Horace's 
 Satires and Epodes. The Greek course for Premiere A, considers ex- 
 tracts from Xenophon's Memorabilia, from Plato, Demosthenes, Homer, 
 iEschylus, Sophocles, Euripides, Aristophanes, etc. 
 
 The Baccalauriat. 
 
 Having finished the Premiere, the eleve presents himself for exam- 
 ination under conditions which once more emphasise the unity of the 
 French educational system. This is the examination for the first part 
 of the state degree known as the Baccalaureat. 
 
 A peculiar feature of this examination is that it is not held in the 
 lyc^eg but at the university of the academie to which the particular 
 lyc^e belongs.^ As various civil and practically all government posi- 
 tions, except those in post and telegraph offices are only open to bachel- 
 iers, the state introduces into the body of examiners some who are wholly 
 independent of the lycees. These examiners are the professors in the 
 universities. 
 
 Since our future mathematicians are to come from Premiere C and 
 D we shall give a few particulars concerning their examination. All 
 examinations for the baccalaureat are held in July and October 
 at the ending of one school year and the beginning of the next. The 
 
 * As there is no university at Chamb6ry, the candidate presents himself before 
 a faculty of either Lyons or Grenoble. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 
 
 99 
 
 examiners of the candidates from Premiere C are six in number, three 
 of whom are university professors and three professors from the lyc4es 
 or colleges; for Premiere D there are but two university professors in 
 addition to three from the lycees. The examinations in all sections 
 are both written and oral. Here is the scheme of examination which 
 practically covers what the eleve has studied in earlier years. 
 
 Premiere C (Latin-Sciences). Written. 1st, a French composition 
 (3 hours) ; (the candidate has a choice of three subjects) ; 2nd, a Latin 
 translation (3 hours); 3rd, an examination in Mathematics and Physics 
 (4 hours). Oral, (about three quarters of an hour). 1st, explanation 
 of a Latin text; 2nd, explanation of a French text; 3rd, examination 
 in a modern language questions and answers being necessarily in this 
 language. Questions in 4th, History; 5th, Geography; 6th, Mathe- 
 matics; 7th, Physics; 8th, Chemistry. 
 
 Premiere D (Sciences-Langues Vivantes). Written. 1st, a French 
 composition (3 hours) ; 2nd, a composition in a modern language 
 (3 hours); 3rd, examination in Mathematics and Physics (4 hours). 
 Oral (about three-quarters of an hour). 1st, explanation of a French 
 text; 2nd, two tests in modern languages, one of which must be either 
 English or German. Questions in 3rd, History; 4th, Geography; 
 5th, Mathematics; 6th, Physics; 7th, Chemistry. 
 
 On registering at the secretary's office and paying the fifty francs 
 necessary for the above examinations the student has the option of 
 depositing his livret scolaire which contained a full record of his work 
 in the lycee for two or three years previously. If the pupil thus shows 
 a good record but fails to get the necessary fifty per cent on his written 
 paper he is nevertheless admitted to the oral. Otherwise no eleve 
 who has not passed the written examination can present himself for 
 the oral. On the other hand if he has passed the written examination, 
 but failed at the oral, he may try another oral examination (within a 
 year), without repeating the written part. 
 
 The searching character of the tests prepares us for a large number 
 of failures. Here is the record for 1909. 
 
 
 Number 
 
 of 
 
 Candidates 
 
 AdSf to i N-b 
 
 Oral 1 ^^^^^ 
 
 Percentage 
 Passed 
 
 
 July. 
 
 Oct. 
 
 July. 
 
 Oct. 
 
 July. 
 
 Oct. 
 
 July. 
 
 Oct. 
 
 
 Latin-Grec. . . . 
 
 2506 
 
 1262 
 
 1423 
 
 759 
 930 
 729 
 915 
 
 1097 
 1293 
 
 537 
 
 44 
 
 42 
 42 
 
 
 Latin-Langues Vivantes. . 
 
 3147 
 2717 
 
 4088 
 
 1683 
 
 1605 
 
 708 
 
 41 
 
 Latin-Science 
 
 Science-Langues Vivantes. 
 
 1247 
 
 1619 
 2110 
 
 1350 
 1741 
 
 570 
 731 
 
 49 
 
 46 
 
 1860 
 
 42 
 
 39 
 
 Philosophic 
 
 5824 
 
 2639 
 
 3764 
 1995 
 
 1911 
 
 3144 
 
 1572 
 ^642^ 
 
 54 
 
 59 
 
 Math^matiques 
 
 3163 
 
 1128 
 
 790 
 
 1762 
 
 56 1 57 1 
 
100 ROYAL SOCIETY OF CANADA 
 
 We observe that less than fifty per cent, of the pupils get through 
 on the first examination^ while a similar percentage of the remainder 
 fail and are required to return to the Premiere once more or to wait for 
 another year.^ Those who have been successful return to the lyc6e to 
 prepare for the second part of the baccalaureat. A choice of two courses 
 (which may be slightly varied), is open to them, the one Philosophic 
 A or B, the other, Mathematiques A or B. We shall only refer to the 
 latter which has been supplied with pupils from the Premiere C and D. 
 There they had 26 and 28 recitation hours per week. This has now 
 been increased to 27^ and 28^. There has been an increase in the num- 
 ber of hours devoted to mathematics, physics and chemistry, but a re- 
 duction in the amount of study of modern languages. Latin no longer 
 enters. The programme for Mathematiques A is in outline as follows: 
 
 Philosophy (3 hours). I. Elements of Scientific Philosophy: 
 introduction, science, method of mathematical sciences, method of 
 the sciences of nature, method of moral and social sciences. IL Ele- 
 ments of moral and social philosophy. 
 
 History and Geography (3 J hours). Modern Languages (2 hours). 
 Physics and Chemistry (5 hours) . Natural Science (2 hours) . Practical 
 Exercises in Science (2 hours). Draunng (2 hours). Hygiene (12 
 lectures of 1 hour). 
 
 Mathematics. (8 hours) : 
 Arithmetic. Properties of integers ; fractions ; decimals ; square 
 roots; greatest common divisors; theory of errors; etc. 
 
 Algebra. Positive and negative numbers, quadratic equations 
 (without the theory of imaginaries) , progressions, logarithms, interest 
 and annuities, graphs derivatives of a sum, product, quotient, square 
 root of a function, of sin x, cos x, tan x, cot x. Application to the 
 study of the variation and the maxima and minima, of some simple 
 functions, in particular functions of the form 
 
 ax- + bx + c 
 
 ; xHpx + q 
 
 a^x^ + b^x-hc^ 
 
 when the coefficients have numerical values Derivative of the area 
 of a curve regarded as function of the abcissa (the notion of area is 
 assumed).^ 
 
 Trigonometry. Circular functions, solution of triangles, applica- 
 tions of trigonometry to various questions relative to land surveying . 
 
 ^ For some it may have been the third or fourth trial. 
 
 ^ There are certain exceptional cases which I shall not consider. 
 
 3 The following note is attached to the resume in the official programme, " Le 
 professeur laissera de cote toutes les questions subtiles que souleve une exposition 
 rigoureuse de la th^orie des d^riv^es; il aura surtout en vue les applications et ne 
 craindra pas de faire appel a Tintuition." 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 101 
 
 Geometry. Translation, rotation, symmetry, homology and simili- 
 tude, solids, areas, volumes, poles and polars, inversion, stereographic 
 projection, vectors, central projections, etc. 
 
 Conies. Ellipse, hyperbola, parabola, plane sections of a cone or 
 cylinder of revolution, etc. 
 
 Descriptive Geometry, Rabatments application to distances and 
 angles projection of a circle sphere, cone, cylinder, planes, sections, 
 shadows. application to topographical maps, etc. 
 
 Kinematics. Units of length and time. Rectilinear and curvi- 
 linear motion. Translation and rotation of a solid body. Geometric 
 study of the helix, etc. 
 
 Dynamics and Statics. Dyn&iiiics of a particle, forces applied to 
 a solid body, simple machines in a state of repose and movement, etc. 
 
 Cosmography. Celestial sphere, earth, sun, moon, planets, comets, 
 stars Co-ordinate Systems, Kepler's and Newton's laws, etc. 
 
 One of the most striking things in this scheme, as compared with 
 American method, is to find arithmetic taught in the last year of the 
 lycee course. Note too, that from the Cinquieme on, it has been taken 
 up in connection with instruction in geometry and algebra. Indeed, 
 this method of constantly showing the interdependence or interrelation 
 of the various mathematical subjects was one of the interesting and 
 vaulable characteristics of French education as I observed it. For 
 example, I happened to be present in a classroom when the theory and 
 evaluation of repeating decimals was under discussion. After all the 
 processes had been explained, problems which led similarly to the con- 
 sideration of infinite series and limits were taken up. By suggestive 
 questioning a pupil found the area under an arc of a semi-cubical para- 
 bola and the position of the centre of gravity of a spherical cap. With 
 us it is not till the graduate school of the university that the boy is 
 taught the true inwardness of such processes as long division and 
 -extraction of roots; but in France, arithmetic is taught as a science and 
 the eleve leaving the lycee has a comprehending and comprehensive 
 grasp of all he has studied. 
 
 The increasing general interest in practical education is reflected 
 in the French method of teaching geometry with frequent illustration 
 involving discussion of the form or relation between the parts of objects 
 met with in every day life. Rather curiously, the method employed 
 in at least some German gymnasien, of demanding that a pupil demon- 
 strate even the more complicated propositions of geometry without any 
 reference to a figure on a blackboard, does not seem to obtain in France. 
 Curiously, because there can be no doubt of the fine exercise of mental 
 concentration required of the members of the class who first build up in 
 imagination the construction as indicated by one of their number and 
 
102 ROYAL SOCIETY OF CANADA 
 
 afterwards follow or criticize his proof; moreover the average French 
 boy could certainly soon become an expert in such mental gymnastics. 
 We remark that most of the mathematical subjects mentioned above 
 are more or less foreign to our secondary education. Instruction in 
 geometrical conies (courbes usuelles), is infrequently given by us, even 
 in universities. Again, the ordinary mathematical student who goes 
 up for his doctor's degree in America may have the vaguest idea of 
 what is even meant by Descriptive Geometery. True, it is a regular 
 course for our training of the engineer; but not, unfortunately, of the 
 mathematician. On the other hand the French mathematical student 
 has had at least four years of Descriptive Geometry, two of them before 
 receiving his baccalaur^at. The subject is required for admission into 
 many government schools. 
 
 We note that the idea of a derivative is familiar to the lyceen 
 during the last two years of his course. Why we so generally shut out 
 the introduction of such an idea into our first courses in analytical 
 geometry and theory of equations is, to me, a mystery. Finally, I 
 would remark that the classes in Math^matiques A last two hours, 
 with the exception of five minutes for recreation at the end of the first 
 hour. The professor thus has sufficient time to amplify and impress 
 his instruction. 
 
 At the close of the last year of the Second Cycle, the eleve takes the 
 examination for the second part of the baccalaur^at. The same general 
 conditions prevail as for the first part. Under no conditions whatever 
 can an eleve try the second part till he has passed the first. The jury 
 of four contains two university professors. The written examinations 
 in mathematics, physics and philosophy are each three hours long; 
 the oral covers what has been studied the year previously. If successful, 
 a dipolma now called the baccalaureat de V enseignement secondaire, is 
 granted to the eleve by the Minister of Public Instruction. The eleve 
 thus becomes a hachelier. The diplomas in all four sections are of the 
 same scholastic value. The charge made for diploma and examination 
 is 90 francs. By reference to the foregoing statistical table, it will be 
 noticed that more than forty per cent, of the candidates failed to pass 
 at each of the examinations in 1909. 
 
 Because of the similarity of title used in the different countries, 
 the Frenchman does not generally understand what the title Bachelor 
 of Arts implies nor is it easy to make any concise statement in explana- 
 tion. Little exaggeration can be made, however, in placing the bachelier 
 on a plane of scholastic equality with the Sophomore who has finished 
 his year at one of the best American universities. Certain it is that the 
 bachelier in Latin-Grec has done as much of the dead languages, philos- 
 ophy and history as is required in the whole pass course of the ordinary 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 103 
 
 Canadian university. The same may be remarked of the bachelier of 
 Latin-Sciences in modern languages and Latin. When it is further 
 remembered that it is possible for the average Canadian boy to get 
 his B.A. with small effort one inclines to place the baccalaur^at, with its 
 rigorous and impartial tests, even higher. No guessing of possible 
 questions and "cramming" for the same, so common in America, can 
 qualify a youth to pass an examination in France. 
 
 Another thought which the examinations for the baccalaur^at 
 suggest, is the superiority in one respect of Canadian education over 
 that in the United States where a great source of weakness would be 
 removed by the adoption of our plan, under which the examinations for 
 promotion from one grade to the next are conducted by the supervisor 
 of education, not by the teacher. The pressure brought to bear upon 
 teachers to promote ill-prepared pupils is thereby eliminated and this 
 pressure is a fruitful source of demoralization in American public schools. 
 
 Finally, does not the French system, as worked out by a great body 
 of educationists, suggest both the kind and method of a much needed 
 reform in our university requirements for the B.A. degree? A large 
 number of bacheliers, as we have seen, have studied no dead language. 
 They may proceed to the Universities and after a time be made doctors 
 in mathematics or natural science without being required to study 
 any dead language. Why may not the same obtain with us? What 
 advantages can be claimed for the study of dead languages, as taught 
 by us, which may not be equally claimed for modern languages? The 
 Harvard authorities apparently see none, as they have not required any 
 dead language after matriculation, for many years past. 
 
 The Classes de Mathematiques Speciales. 
 
 IV. If the bachelier who is proficient in mathematics be not 
 turned aside by circumstances or inclination, to immediately seek a 
 career in civil or government employment, he most probably proceeds 
 to prepare himself for the highly special and exacting examination 
 necessary for entrance into one of the great schools of the government. 
 The method of this preparation exhibits a very peculiar feature of the 
 French system. Whereas with us, or with the German, the boy who 
 has finished his regular course in the secondary school goes directly 
 to some department of a university for his next instruction, the bachelier, 
 who has a perfect right to follow the same course, returns to his old 
 lycee (or enrolls himself at one of the great Paris lycees, such as Saint 
 Louis, Louis le Grand or Henri IV) , to enter the Classe de Mathematiques 
 Speciales preparatoire which leads up to the Classe de Mathematiques 
 Speciales. The latter is exactly adapted to prepare students for the 
 ficole Normale Superieure, the ficole Polytechnique and the bourses de 
 
104 ROYAL SOCIETY OF CANADA 
 
 licencsK Only a small proportion of the lycees (34 out of the 115), 
 have this Clause; but with the exception of Aix they are to be found in 
 all university towns. On the other hand, yet other lycees have classes 
 which prepare specially for the less exacting mathematical entrance 
 examinations of the ficole Centrale, ficole de Saint Cyr, ficole Navale, 
 etc. But the number of eleves who on first starting out deliberately 
 try to pass examinations for these schools is small, in proportion to the 
 number who eventually reach them after repeated but vain effort to 
 get into the ficole Polytechnique or the ficole Normale Superieure. 
 Just what makes these two schools famous and peculiarly attractive 
 will appear in a later section. It has been noticed that when the eleve 
 has won his baccalaureat he may immediately matriculate into a uni- 
 versity, and although it might be possible for him to keep pace with the 
 courses, in mathematics, at least, it would be a matter of excessive 
 difficulty. There is then in reality, between the baccalaureat and the 
 first courses of the universities, a distinct break, bridged only by the 
 Classes de Mathematiques Speciales. 
 
 The eleves who enter the prSparatoire section of this class are, 
 generally,^ bacheliers leaving the- classes de Mathematiques; in very 
 rare instances, there are those who come from the classe de Philosophie. 
 Natural science, history and geography, philosophy indeed practically 
 every study except those necessary for the end in view, have been 
 dropped and from this time on to the agregation and doctorat all 
 energies are bent in the direction of intense specialization. This is 
 the most pronounced characteristic of French education to-day. In 
 mathematics, instruction now occupies 12 instead of 8 hours. New 
 points of view, new topics and broader general principles are developed 
 in algebra and analysis, trigonometry, analytical geometry and 
 mechanics. Physics and chemistry are taught during six hours instead 
 of five. Add to these, German, 2 hours; French literature, one hour; 
 descriptive geometry, 4 hours; drawing, 4 hours. After one year of 
 this preparatory training the eleve passes into the remarkable Classe 
 de Mathematiques Speciales. ^ 
 
 Eight years of strenuous training have made this class possible for 
 the young man of 17 or 18 years of age, who is confronted with no less 
 than 34 hours of class and laboratory work per week and no limit as 
 to the number of hours expected in preparing for the classes! 
 
 When first I looked over the programme it seemed a well nigh im- 
 possible performance for one year. Surely no other country can show 
 anything to compare with it. Although it would be of interest to 
 
 * Pupils who are not bacheliers, but who are preparing to enter the Ecole 
 Centrale, are also admitted into this class. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 106 
 
 reproduce the programme in full, to do so would take up a dispropor- 
 tionate space in a sketch of this kind. Moreover, many parts of it are 
 given in Appendix A in connection with the agregation examination 
 requirements. I shall, therefore, merely touch on a few of the points 
 of interest. The number of hours per week are distributed as follows: 
 Mathematics, 15; physics, 7 (2 in laboratory); chemistry, 2; descriptive 
 geometry, 4; drawing, 4; German, 2; French, 1. The scope of the 
 mathematical work may be judged from some books which were pre- 
 pared with the needs of such a class especially in view. 
 
 B. Niewenglowski, Cours d'algebre, I, 382 p.; II, 508 p.; Supple- 
 ment G. Papelier Precis de g^om^trie analytique, 696 p. Girod 
 Trigonometrie, 495 p. P. Appell Cours de mecanique, 650 p. 
 X. Antomari Cours de geometrie descriptive, 619 p. 
 
 If anything, this list underestimates the work actually covered 
 by those who finally go out from the class. Tannery's Legons d'algebre 
 et d'analyse (I, 423 p., II, 636 p.), might well replace Niewenglowski's 
 work while Niewenglowski's Cours de geometrie analytique (1, 483 p.; 
 II, 292 p.; Ill, 569 p.), represents the standard almost as nearly as 
 Papelier's volume. Another treatise on mechanics widely used is that 
 of' Humbert and Antomari.^ 
 
 When we further realize that the books in this list, which represents 
 the work for only one of a half dozen courses, are covered by the pro- 
 fessor in about six months the last three months of the year are given 
 over to drill in review and detail we begin to get some conception of 
 what the Classe de Mathematiques Speciales really stands for. In his 
 instruction the professor is officially "recommended" " de ne pas 
 charger les cours, de faire grand usage de livres, de ne pas abuser des 
 theories generales, de n'exposer aucune theorie sans en faire de nom- 
 breuses applications poussees jusqu'au bout, de commencer habituelle- 
 ment par les cas les plus simples, les plus faciles a comprendre, pour 
 s'elever ensuite aux theoremes generaux. Parmi les applications 
 d'une theorie mathematique, il conviendra de preferer celles qui se 
 presentent en physique, celles que les jeunes gens rencontreront plus 
 tard dans le cours de leurs etudes soit theoriques, soit pratiques; c'est 
 ainsi que, dans la construction des courbes, il conviendra de choisir 
 comme exemples des coiirbes qui se presentent en physique et en 
 mecanique, comme les courbes de Van der Waals, le cycloide, la chainette, 
 etc., que, dans la theorie des enveloppes, il conviendra de prendre 
 comme exemples les enveloppes qui se rencontrent dans la theorie des 
 engrenages cylindriques, et ainsi de suite. Les eleves devront etre 
 
 ^ Further details about these various volumes, as well as of many others, may 
 be found in Appendix D. 
 
106 ROYAL SOCIETY OF CANADA 
 
 interroges en classe, exerces aux calculs num^riques, habitues k 
 raisonner directement sur les cas particuliers et non a appliquer des 
 formules. En resume, on devra d^velopper leur jiigement et leur 
 initiative, non leur m^moire." 
 
 In France, as everywhere else, the success of the system depends 
 much on the personality of the professor. A Paris lycee instructor 
 who had a genius for getting hold of his boys has recently died. No 
 less than 35 of his pupils were admitted to the ficole Polytechnique in 
 a single year. The ordinary professor has to be content with a half 
 or a third of this number. But the success of a class is, by happy ar- 
 rangement, not left to depend wholly upon a single man. Take, for 
 example, lycee Saint Louis, which is the greatest preparatory school 
 in France for the ficole Normale Superieure and the ficole Polytech- 
 nique. There are four Classes de Math^matiques Sp^ciales and for all 
 the members of these classes, conferences, interrogations and individual 
 examination are organized. These exercises, which complete the daily 
 instruction, are conducted by one of the professors in the lycee itself, 
 or by one of those from the College de France, the Sorbonne, the ficole 
 Polytechnique, the ficole Normale, from other lycees or from the 
 colleges. Incapables are thus speedily weeded out. Of perhaps greater 
 value than the solidity of the training got in this way is the fact that 
 the interest of the 6leve is sustained. 
 
 Just a word about the calculus course. This is practically equiv- 
 alent to the first course in the best American universities. The integ- 
 ration of differential equations of the first order in the cases where 
 (1) the variables separate immediately, (2) the equation is linear, as 
 well as of linear differential equations of the second order, constant 
 coefficients, (a) without second member, (6) when the second member 
 is a polynomial or a sum of exponentials of the form Ae^^ , is taken up. 
 
 With the end of the year the 6leve has his first experience of a 
 concours. Previously he has found that it was necessary only to make 
 a certain percentage in order to mount to the next stage in his scholastic 
 career; but now it is quite different. In 1908, 1,078 pupils tried for 
 admission into the Ecole Polytechnique, but only 200, or 19.5 per cent., 
 were received; for the department of science in the Ecole Normale 
 Superieure, 22 out of 274, or 8 per cent., succeeded. In each case the 
 number was fixed in advance by the Government according to the 
 capacity of the school; the fortunate ones were those who stood 
 highest in the examinations, written and oral. In the case of the 
 Ecole Polytechnique, the written examinations were held in all the 
 lycees which had a Classe de Mathematiques Speciales. The 387 can- 
 didates declared admissible were then examined orally at Paris, and 
 from them the 200 were chosen. Similarly for the ficole Normale, the 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 107 
 
 written examinations are conducted at the seats of the various acad- 
 emies and the oral at Paris. Since 1904 the concours passed by the 
 ficole Normalians has been that for the bourses de licence, open to 
 candidates of at least 18 years of age and not more than 24. Certain 
 dispensations in the matter of age are sometimes granted. The value 
 of the bourse, for the section of science, is from 600-1,200 francs a year 
 and is intended to help the student to prepare for the licence and other 
 examinations required of prospective professors in the lycees and 
 universities. The candidates leading the list in the concours are sent 
 to the ficole Normale Sup^rieure for from three to four years. It is 
 necessary for the six or seven other boursiers to prepare for future ex- 
 aminations at the various universities of the provinces. Their bourses 
 last regularly for two, and exceptionally for three, years. 
 
 But to return to our eleves of the Classe de Mathematiques Sp6ciales. 
 At the end of the first year, when 18 years old, they usually present 
 themselves for the concours of both the bourse de licence and the ficole 
 Polytechnique, the examinations in the former being more strenuous 
 and searching. Only from 2 to 5 per cent, succeed on the first trial. 
 The others then go back to the lyc6e and take another year in the 
 Classe de Mathematiques Sp^ciales. Many points not fully understood 
 before are now clear, and at the end of the second year from 25 to 28 
 per cent, are successful. The persevering again return to their Classe 
 and try yet a third time (the last permitted for the bourse de licence) ; 
 but it is a matter of record that less than one-half of those who enter 
 the Classe de Mathematiques Speciales succeed even with this trial. 
 This is usually the last trial possible for entry into the ficole Polytech- 
 nique, as the young man who has passed the age of 21 on the first of 
 January preceding the concours may not present himself. The re- 
 mainder of the students either seek for entrance into government schools 
 with less severe admission requirements, and thus give up their as- 
 pirations to become mathematicians, or else continue their studies at 
 the Sorbonne. The candidate who heads the list in each of these 
 concours has his name widely published. In the case of the bourse de 
 licence he is called the cacique, and he very frequently tops also the 
 ficole Polytechnique list. 
 
 If the work of the Classe de Mathematiques Speciales is so enormously 
 difficult that only 2 to 5 per cent of its members can, at the end of one 
 year, meet the standard of requirements of the examinations for which 
 it prepares, why is not the instruction spread over two? Since nearly 
 all the mathematical savants who now shed lustre on France's fair 
 fame have passed from this remarkable class on the first trial, there 
 can be no doubt that the answer to this question may be found|in 
 the fact that the government ever seeks her servants among the elite 
 of the nation's intellectuals. 
 
108 ROYAL SOCIETY OF CANADA 
 
 Higher Mathematical Instruction. 
 
 V enseignement sup^rieur is carried on in universities, great 
 scientific establishments and special schools. We shall consider in 
 particular, the mathematical instruction as given at the Sorbonne, 
 the ficole Normale Superieure, the ficole Polytechnique and the College 
 de France. 
 
 The Sorbonne. 
 
 The University de Paris consists of the faculties de droit, de medecine, 
 des sciences, des lettres. (Faculties of Catholic and Protestant theology 
 were suppressed in the years 1885 and 1906 respectively.) The faculties 
 of science and letters have their offices, lecture rooms, laboratories and 
 special libraries in a building now called the Sorbonne. This building 
 contains also the headquarters of the officers of the Academie de Paris 
 and of the university administration, the museums, the main university 
 library, the ficole Pratique des Hautes fitudes, the ficole des Chartres 
 and the great amphitheatre capable of seating 3,500 persons and adorned 
 with a large allegorical painting, " the masterpiece of Puvis de Chavannes 
 and one of the finest decorative works of our time." 
 
 The present Sorbonne, completed less than a decade ago, is an 
 immense and magnificent edifice, erected to replace the old Sorbonne 
 (the outlines of which may be seen in the courtyard), dating from the 
 time of Richelieu. To make room for the newer building, the older 
 was (in 1885) torn down, with the exception of the chapel, which 
 picturesquely nestles in the midst of the new structure. The name 
 Sorbonne harks from the time of the confessor of St. Louis, Robert 
 de Sorbon, who in the thirteenth century founded a sort of hostel for 
 the reception of poor students of theology and their teachers. This 
 soon acquired a high reputation as the centre of scholastic theology, 
 and the name came to be applied to the faculty itself, which continued 
 to exercise great influence on French Catholicism tiown through the 
 centuries. It was suppressed, along with some twenty other univer- 
 sities, during the Revolution. But under Napoleon, in 1808, the Sor- 
 bonne was re-established as the seat of the monster Universite de 
 France, which embraced all the universities, secondary schools, etc., 
 in the country. The details of this organization did not prove accept- 
 able, and in 1896 the arrangement explained in the early part of this 
 paper came into effect. 
 
 Judged by the number of students, the Universite de Paris is the 
 largest university in the world. In January last, 17,512 students had 
 registered. Nearly half of these were law students, while of the remain- 
 der, 1,845 were pursuing work in one or more of the twenty-three 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 109 
 
 departments of the faculty of science.^ The instructors in French 
 universities are of six classes. The chair es magistrates are held by 
 professeurs titulaires; no less than eleven of these at the Sorbonne, are 
 in the departments of mathematics. Then there are professeurs 
 adjoints (of whom there may not be a number greater than one third 
 of the chaires magistrales) , charges de cours, mattres de conferences, 
 charges de conferences and maitres de conferences adjoints. Just what 
 scholastic status or state recognition is implied in these titles I shall 
 consider later; but it may be remarked here that all professeurs, although 
 theoretically appointed only till the age when the law requires them 
 to be pensioned off, are in reality appointed for life. Those at the 
 Sorbonne, at least, are known the world over, because of their eminence 
 in research and exposition. Here are the names of the chaires, of the 
 incumbents and of the courses offered during the past year : 
 
 1 Geometrie superieure Darboux. 
 
 I Semester. 30 lectures of 1 hour. " Theory of triply orthogonal systems." 
 
 Largely as in the new edition of Darboux's work on this subject and 
 as in selected parts of his Th^orie des Surfaces. 
 
 2 Analyse superieure et algebre superieure Picard. 
 
 II Semester. 30 lectures of 1 hour. "Determination of integrals of 
 
 partial differential equations of the second order with variou condi- 
 tions as to limits." 
 
 3 Calcul differentiel et calcul integral Goursat. 
 
 I and II Semesters. 60 lectures of 70 minutes. This course is practically 
 that given in Goursat's Cours d'Analyse Math^matiques, Tomes I-II, 
 new edition. 
 
 4 Applications de I'analyse a la geometrie Raffy. 
 
 I Semester. 30 lectures of 75 minutes. This course is a slight expansion of 
 Raffy's book on the subject. 
 
 5 Theorie des fonctions Borel 
 
 I Semester. 15 lectures of 1 hour. The announced subject of this course 
 was "Definite Integrals and some of their Applications," but the 
 treatment was more of series. 
 
 6 Astronomie mathematique et m^canique celeste Poincare. 
 
 I Semester. 30 lectures of 1 hour. "Movement of the Celestial Bodies 
 
 about their centre of Gravity." 
 
 7 Astronomic physique Andoyer. 
 
 II Semester. 30 lectures. 
 
 8 Physique mathematique et calcul de probabilites . . . Boussinesq. 
 
 I Semester. 30 lectures of 1 hour. "Mechanical Theory of Light." 
 
 II Semester. 30 lectures of 1 hour, "Reflection and refraction of a 
 
 pencil of light at the limit common to two homogeneous media." 
 
 ^ The total number of students at all the universities in France, in January 
 1910, was 41,044 as compared with 52,456 in Germany. 
 
110 ROYAL SOCIETY OF CANADA 
 
 9 Mecanique physique et experimentale Koenigs. 
 
 I and II Semesters. 60 lectures of 1 hour. "Moteurs thermiques." 
 I Semester. 12 lectures of 1 hour. Theoretical Kinematics. 
 
 10 Mecanique rationnelle^ ^Painleve. 
 
 I and II Semesters. 60 lectures of 1^ hours. The lectures in this course 
 practically cover Appell's Traits de Mecanique rationnelle, Tomes I-III, 
 
 11 Mathematiques gen^rales^ ^Ippell. 
 
 I and II Semesters. 45 lectures of 1 hour. This course is essentially that 
 given in Appell's Elements d'analyse mathematique. 
 
 Physique geiierale is demanded of all advanced mathematical 
 students. I add the subjects taught. 
 
 Physique Bouty. 
 
 I Semester. 30 lectures of 1 hour. Thermodynamics and 
 Electrolysis. Blondelet's text is an equivalent. 
 
 Physique -Pellat (Leduc). 
 
 12'^^ I Semester, 15 lectures of 1 hour, "Electrostatics, Ohm's law, 
 Electrodynamics, etc." This course covers Pellat's text. 
 Physique Lippmann. 
 
 II Semester. 30 conferences of 1 hour. " Gravity, Capillarity, 
 Acoustics, Optics." 
 
 Beside these courses, which are open to the public without even the 
 formality of registration, there are certain conferences et travaux pratiques 
 '' cours fermes " for those regularly matriculated. As far as we are 
 interested in them they are: 
 
 13 Geometrie superieure Cartan 
 
 I Semester. 15 conferences of 1 hour. 
 
 14 Calcul differentiel et integral, et ses applications geometriques 
 Raffy. 
 
 I and II Semesters. 60 conferences of 70 minutes. 
 
 15 Astronomie physique : travaux pratiques Andoyer- 
 
 II Semester. 30 conferences. 
 
 16 Mecanique physique et experimentale : travaux pratiques. Koenigs- 
 I and II Semesters. 30 conferences. 
 
 17 Mecanique physique : principes de la statique graphigne et de la 
 
 resistance des materiaux Servant. 
 
 I and II Semesters. 30 conferences of 1 hour. 
 
 ^ Appell is professor of mecanique rationnelle, and Painleve of mathematiques 
 generale, but for this year at least they have exchanged the subjects demanded by 
 their chairs. A possible explanation may be found in the fact that Appell has a 
 remarkable gift of clear exposition of elementary subjects. 
 
 ^ Pellat died early in the year and Leduc (professeur adjoint) was temporarily 
 given charge of his course. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 111 
 
 18 Mecanique rationnelle Cartan. 
 
 I and II Semesters. 60 conferences of 1 hour. 
 
 Algebre Blutel. 
 
 I Semester. 30 lectures of 1 hour. 
 Exercices de mathematiques generales Gamier 
 
 I Semester. 20 conferences of 1 hour. 
 Travaux pratiques de mathematiques generales Cartan. 
 
 II Semester. 15 conferences of 1 hour. 
 
 19 
 
 20 Physique generate Leduc. 
 
 1 and II Semesters. 45 conferences of 1 hour. 
 
 Cartan is a maitre de conference, and Bhitel (professor of mathem- 
 atics at lycee Saint Louis), Garnier (collaborator on the French edition 
 of the Encyklopcidie der Math. Wissenschaften) , and Servant, are charges 
 de conferences. Unlike the organization at the College de France 
 and in German universities where no examinations enter to disturb the 
 serene atmosphere, one of the chief functions of French universities is 
 to provide means for preparing students for two state examinations, 
 the licence and the agregation. These examinations (but especially 
 the agregation), demand that exceedingly comprehensive instruction 
 shall each year be given in a large number of special subjects. To this 
 end most of the professors devote themselves. As a consequence there 
 is a great sameness in the courses offered at the different universities 
 from year to year and Lyons is about the only one outside of Paris 
 which attempts to do more than meet the state requirements.^ At 
 the Sorbonne there is little annual variation in two-thirds of the main 
 courses; these are 3, 4, 7, 8, 9^, 10, 11, 12. An outgrowth of 
 them is a remarkable series of elegant treatises. But Borel, Darboux, 
 Picard and Poincare make frequent changes in the subjects on which 
 they lecture. We remark, that no professor gives more than one course 
 (two lectures per week), except in the four cases of those who direct con- 
 ferences and travaux pratiques; also that Darboux, Picard, Poincare, 
 Andoyer, Bouty, Lippmann, only lecture during one semester.- The 
 maitres de conferences or charges de conferences do not give new 
 courses and treat of subjects in which they are especially informed as 
 do the Privat-docenten in German universities. But their instruction, 
 as well as that of all others who direct cours fermes, is supplementary 
 to the work of those holding the chaires magistrates. Thus 13 sup- 
 plements Darboux's course 14 supplements 3 and 4 15 and 7 go 
 together so also 1G, 17 and 9 18 and 10 19 and 11 20 and 
 
 * Compare Appendix B. 
 
 2 Most of Borel's work is at the l^cole Normale Sup6rieure, which is part of the 
 University de Paris. 
 
112 ROYAL SOCIETY OF CANADA 
 
 12. Although there is nothing in the mathematical departments 
 of the universities of France which exactly corresponds to the German 
 Seminar, ^ the method of conducting the conferences at travaux pratiques 
 is, I believe, a French specialty. We also find it used in the lyc^es at the 
 ficole Normale Superieure, at the ficole Polytechnique, etc. Each week 
 the instructor gives out exercises which the students solve and hand in ; 
 these are returned with written comment and correction. The hour is 
 employed by calling some student to the board and leading him by means 
 of suggestive questioning to work out, generally in great detail, a piece 
 of analysis or a problem or a theorem either arising from, or nearly 
 related to, the main course. The manner in which this is carried through, 
 with its exacting demands as to form in statement and black board 
 presentation, is in the highest degree instructive. 
 
 The above list does not display all the mathematical courses offered 
 at the Sorbonne this year. Cartan had a special problem course for 
 candidates for the agregation and Bachelier, gave a cours libre of 20 
 lectures on the calculus of probabilities and its application to financial 
 operations. The number of cours libres varies from year to year.^ 
 
 The Licence. 
 
 When a student has finished any one of the groups of studies 
 such as (3, 4, 14) or (10, 18), he may pass an examination and 
 receive a certificat d' etudes swperieures. With the third certificat is 
 given the diploma licence es science. The choice of subjects is not 
 necessarily limited to those given above but may be selected (at 
 the Sorbonne) from a list of 23^ which includes general chemistry, 
 zoology, geology, etc. If, however, the student expects to teach in the 
 secondary schools his choice is greatly limited. The mathematician 
 must have certificats in calcul differentiel et integral, in mecanique 
 rationnelle and in physique generale, or a third certificat in mathematics, 
 excepting (11, 19). The physicist must have certificats in physique 
 generale (12, 20), in chemie generale and in mineralogie or mathema- 
 tiques generales (11, 19), or another subject of mathematical or physi- 
 cal science. The natural scientist must have certificats in zoology or 
 general physiology, in botany and in geology. The examination for 
 certificats may be taken twice in a year, in July or in November. It 
 consists of three parts, epreuve ecrite, epreuve pratique, epreuve ovale. 
 
 ^ Other subjects are treated in Seminar style at the Ecole des Hautes Etudes 
 which is an off-shoot of the College de France. 
 
 2 In July, 1910, the University of Paris accepted the offer of M. Albert Kahn 
 to bear the expense, for a period of five years, of a course on "The Theory of 
 Numbers." 
 
 ' This number varies with the university; at Dijon it is 12. 
 
ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 113 
 
 The first two are written examinations of about four hours each. Theo- 
 retical considerations abound in the ecrite while numerical calculation 
 is characteristic of the 'pratique. The oral lasts for 15-20 minutes and 
 is held before the jury of those professors who have the whole examin- 
 ation in charge. It is necessary to get fifty per cent to pass. The first 
 certificat and examinations cost 35 francs, the second and third 30 francs 
 each and the licence diploma 40 francs. 
 
 The Diplome d' Etudes Superieures de Mathematiques. 
 
 This diploma which was instituted by a decree of 1904 has not yet 
 been awarded to any one, although its equivalent, 4 certificats (one 
 chosen at option), is required of all candidates for the agregation. It 
 may be considered as a little doctarate. The conditions leading to the 
 diploma are twofold: 
 
 (a) That a suitable travail be written on a subject agreed upon by 
 the faculty. 
 
 (6) That satisfactory answers be given to questions on the travail 
 and on topics given three months in advance and relating to the same 
 part of mathematics. The travail may consist either of original re- 
 searches, or of the partial or total exposition, of a memoir or of a higher 
 mathematical course. In the latter case by "exposition" is meant 
 either a simplified resume of the memoir or of the course, or the detailed 
 development, where the result or method that the author or professor 
 presents has only been outlined. 
 
 The Agregation des Sciences Mathematiques. 
 
 This examination, unlike that for the baccalaur^at and licence, is a 
 concours as in the case for entrance into the ficole Normale Superieure and 
 the ficole Polytechnique. The number who become agreges each year is 
 fixed in advance by the Minister of Public Instruction according to the 
 needs of the lycees in the country. This number in recent years has 
 generally been 14, but in 1897 as few as seven were chosen. The smallest 
 number of competitors since 1885 was 54, in 1907; in 1909 there were 81; 
 the largest number was in 1893, when there were 134 young men eager for 
 13 places.^ The candidate for this examination must have four cer- 
 tificats; those in calcul differentiel et integral, mecanique rationnelle, 
 physique generate and a fourth chosen at pleasure among the remain- 
 ing mathematical subjects. As an equivalent of the fourth certificate 
 a diplome d^ etudes superieures de mathematiques may be presented. The 
 subject of the fourth certificate at Paris is usually Picard's Analyse Supe- 
 rieure et Algebre Superieure (2^) or Darboux's Geometric Superieure 
 
 ^ Compare the analytical table of Appendix A. 
 Sec. III., 1910. 8. 
 
114 ROYAL SOCIETY OF CANADA 
 
 (1, 13 and an epure). Poincar^^s course is chosen less frequently and 
 at present Borel's course may not be selected independently of others. It 
 is usually four years after leaving the Classe de Mathematiques Speciales 
 that the young mathematician first presents himself for the agregation, 
 i.e., when he is about 21 years of age. In this interval he has probably 
 spent a 3^ear in military service, worked off the examinations for the first 
 and third of the above mentioned certificates during the second year, for 
 the second and fourth during the third, while the fourth year was spent 
 in general review, study of teaching methods or other special direct 
 preparation for the agregation. This examination, which is unique 
 in its difficulty and exactions, is organized for selecting the most ef- 
 ficient young men in the country, to take charge of the mathematical 
 classes in the lycees. It consists of epreuves preparatoires and epreuves 
 definitives. The former are four written examinations, each of seven 
 consecutive hours in length! The first two of these are on subjects 
 chosen from the programme of the lycees in mathematiques elemen- 
 taires and mathematiques speciales. The last two, based on the work 
 of the candidates in the universities, are a composition sur Vanalyse et 
 ses applications geometriques and a composition de m^canique rationnelle. 
 These epreuves are held at the seats of the various academies of France. 
 Those who have reached a sufficiently high standard are declared 
 admissible. Their number is usually a little less than twice the possible 
 number to be finally received. In 1909 it was 27, but in 1905, 20; 
 while in 1887 there were only 15, from which 13 were selected. They 
 must present themselves at the Sorbonne for the epreuves definitives. 
 These consist of two written examinations and two legons. The writ- 
 ten tests are an epreuve de geometric descriptive, and a calcul numerique. 
 Their duration is fixed by the jury, but it is usually four hours for 
 each. The legons, which are supposed to be such as a professor might 
 give (during |-1 hour) in a lycee, are on subjects from (a) mathema- 
 tiques speciales, (b) the programmes of the classes, Secondes, Premihre, 
 C, D, and Mathematiques A, B. The subjects are drawn by lot, and for 
 each lesson the candidate has four hours to think over what he is going 
 to say. No help from any book or other source is permitted. The 
 unfortunate who has little to say is speedily adjourned. As a salve 
 for disappointment and as encouragement to try again, he receives 
 300 francs a year for three years because he had won a place among 
 those admitted to the second examination. 
 
 The names in the list of agreges are published in order of merit, 
 and those who head the list are likely to get the better positions. Many 
 of the instructors at the Sorbonne were first agreges. Appell, Picard 
 and Goursat were successively first agreges, 1876-78; Cartan and Borel 
 1891-92; Andoyer in 1884. Painleve was, however, a ninth agrege; 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 115 
 
 Blutel a fourth, and Gamier a second. There have been very excep- 
 tional cases (in 1885, 1886, 1895), when an agr^g^ was still in his 
 twenty-first year, but the average age for the past twenty-five years is 
 a little less than 26. There are also those who do not reach the goal 
 of their ambition till after they are 40, and have tried perhaps ten or 
 a dozen times. The difference between the salary of those lycee profes- 
 sors who are agreges and of those who are not has been emphasized still 
 more by the law just passed, which gives the former an annual bonus 
 of 500 francs. 
 
 If the agrege wishes to become a professor in a university he must 
 pass the examination for the doctorate, but only a very small propor- 
 tion of the agreges take this step during the period 1885-1904, 20 
 per cent. If, however, he wishes to teach in a lycee he may demand 
 such a position as his right. Among the candidates who are admissible 
 but not received are generally selected professeurs charges de cours (or 
 others in positions of inferiority in the lycee), who, however, after 
 20 years of service may be named professeurs-titulaires and be the 
 academic equals of their luckier comrades of years before. 
 
 Other details concerning the agregation, such as the programme 
 for the concours of 1910, the examination papers for the concours of 
 1909, are given in Appendix A. 
 
 The Doctoral. 
 
 What is the relative value of the French and German mathematical 
 doctorate? What the relative difficulty of obtaining it? are questions 
 which the average American post-graduate studen^ who is seeking to de- 
 cide between France and Germany for further study is sure to ask. Small 
 as is the proportion of students in a German university who present them- 
 selves for this degree, the number in France is far smaller. In the two 
 years 1906-08 Germany made 87 doctors in mathematics, while France, 
 with but 20 per cent, fewer students, created only 13.^ This difference 
 in numbers is doubtless principally due to the fact that the end in view in 
 France is entirely different. The Frenchman usually goes up for his doc- 
 torate with the expectation of drawing wide attention to his these. The 
 step is also necessary for everyone who aspires to be appointed a professor 
 in a university unless, perchance, he has become a member of the Institut 
 without having the degree. All except three of the French universities 
 offer the degree of doctorate in mathematics, but only eleven of them 
 have ever conferred it. Again, of the 331 degrees which have been con- 
 ferred by the existing universities, 296 have been granted by the Uni- 
 versite de Paris. This is, of course, very different from the results in 
 
 ^ There is of course no degree of Doctor of Philosophy in France. The equiva- 
 lent is explained later. 
 
116 ROYAL SOCIETY OF CANADA 
 
 Germany, where Berlin university turns out a very small fraction of the 
 doctors in any one year during 1906-08, less than 4 per cent. There 
 is also one other great difference between French and German univer- 
 sities, although the examinations for licendes rather than those for 
 doctors must be chosen to emphasize the point sufficiently. The diffi- 
 culty of obtaining those degrees common to most French universities is 
 much the same, and although Paris is the principal degree-conferring 
 centre, it is well established that there have been years when it was 
 more difficult to obtain the licence, in some departments, at certain 
 universities of the provinces than at the University de Paris. That 
 the personality of the professor should play an important role in deter- 
 mining the standard of excellence demanded is only natural; but as 
 it is the ambition of every professor in the provinces to make his 
 department important and to ultimately arrive at Paris, one can be 
 very sure that no one university in France will ever sink to the level 
 of at least two German universities, where, on account of lax demands 
 in study and thesis, even train conductors call out "Twenty minutes 
 wait to get your doctor." 
 
 But if such representative universities as Berlin, Gottingen, Munich 
 be selected in Germany and compared with that at Paris, two questions 
 suggest themselves: (1) Does the average doctor's thesis (which in 
 both countries is the essential performance on the part of the candidate 
 for the degree) indicate a higher standard of excellence in one country 
 than the other? (2) Admitting this to be the case, are the minimum 
 requirements in this country as low as the general requirements in 
 the other? By actual study of the theses, I am convinced that the 
 answer to the first question is decidedly in favour of Paris. One could 
 easily cite a number of French theses which were notable and extensive 
 contributions to mathematical progress, but it is only necessary to 
 refer the reader to the complete list of the theses, which is given in 
 Appendix C. Before answering the second question, I shall explain 
 more fully the nature of the French doctorate, the general conditions 
 under which it is available for the Frenchman and the possible modi- 
 fications of those conditions in the case of foreign candidates. 
 
 There are two doctor's degrees open to the mathematical student 
 in France: the first, doctoral es sciences mathematiques, conferred by 
 the State (doctorat d'etat), the second conferred by the universities 
 for the Sorbonne, doctorat de VUniversite de Paris. Only one 
 American, a woman, has won the former degree, which was created 
 in 1810, and only one American has also obtained the latter, which 
 was organized as recently as 1898. In both cases the th^se is the prin- 
 cipal requirement, and judging by the eight for which the doctorat 
 de I'Universit^ de Paris has been granted, the standard in this respect 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 117 
 
 is about as high as for the doctorat de Tetat. It is in the matter of 
 further requirement that the doctorat d'etat is more difficult. For 
 this degree there is no possible way of avoiding the various examin- 
 ations which lead up to the licence es science with mention of the 
 certificats: 1st, calcul differentiel et integral; 2nd, mecanique ration- 
 nelle, and 3rd, at the choice of the candidate. For the doctorat de 
 I'Universite^ only two certificats are required, and, in the case of 
 foreigners, very great latitude is permitted the faculty in accepting 
 equivalents for these certificats, in view of work done elsewhere. 
 In both cases only one year of residence is required. We can, then, 
 now answer the second question, proposed above, in the affirmative 
 for the doctorat de TUniversite, and in the negative for the doctorat 
 d'etat. 
 
 The analytical table given in Appendix C clears up misconception 
 as to the age of the French mathematical doctor. During the past 25 
 years, the average age has been 30, but a large number "sustained" 
 their theses between the ages of 23 and 25. The youngest doctor was 
 Joseph Louis Frangois Bertrand, aged 17, created in 1839; the oldest, 
 in 1882, aged 5b. 
 
 Only a small proportion of the agreges ever become doctors, ^ 
 and in but one case (1894) has a doctor become an agrege. Which title 
 calls for the greater ability in the getting?^ The two things are so 
 entirely different, it is perhaps difficult to understand why some say 
 the doctorate ranks the higher. The musician with great technical 
 talent only, may be allowed to have equal ability with the performer 
 less gifted in this direction but endowed with strong temperament 
 power of perception and interpretation which draws aside the veil 
 for the ordinary observer and discloses formerly hidden heights, beyond. 
 Yet it is the latter who particularly appeals to us. So while the tech- 
 nical skill of the agrege is admired, and the state gives him certain rights 
 denied the doctor, it is only the latter who, on showing power of dis- 
 closing the truths waiting for discovery from the foundations of the 
 worlds, has the opportunity to direct the nation's youth in the great 
 universities of the country. So much the more sought after the man 
 who combines in himself to a high degree both talents, the gift of 
 brilliant exposition and the genius for discovery. 
 
 The general procedure toward the doctorate is the same for 
 
 ^ According to a decree of 1906 the insignia of the docteurs de T university de 
 Paris is, " Epitoge a trois rangs d'hermine, avec les couleurs de Paris (bleu et rouge) 
 dans le sens longitudinal." 
 
 ^ Cf. Appendix A. 
 
 ^ The agrege in Law and Medicine stands much higher than the doctor in these 
 Departments. 
 
118 ROYAL SOCIETY OF CANADA 
 
 both kinds. A these worked out under general supervision of a 
 professor is formally approved by a committee of three professors 
 named by the doyen. Birth certificate, diploma as licencie, 148 printed 
 copies of the these and 145 francs, are deposited with university officials, 
 and the day fixed for appearing before the committee to publicly answer 
 such general questions on the these or other topic which the benevolent 
 ingenuity of the examiners may propound. Compared with those in some 
 other departments of the university, the examinations of the mathema- 
 tician is a very informal affair. It rarely occupies more than three- 
 quarters of an hour. The candidate is immediately told whether he 
 has got the note "honorable" or "tres honorable," is congratulated 
 and dismissed. The amount of help which the candidate for the doctorate 
 receives from the professor is much less in France than in Germany. 
 In fact he rarely approaches the professor except when he gets his 
 subject or is reporting progress. It is expected of the Frenchman 
 that the these represent his own work and thought. The doctor who 
 has presented a remarkable these and passed a brilliant examination 
 may have the full cost of examination and diploma remitted. A similar 
 rule applies to bacheliers. 
 
 The ficoLE Norm ale. Superieure. 
 
 This great institution is a part of the University de Paris and its 
 object is to mould the future professors of the secondary schools and 
 universities of France by appropriately supplementing the instruction 
 they receive at the Sorbonne and the College de France. There are 
 about 165 pupils, in the departments of science and letters, and prac- 
 tically all are internes. We have already remarked that the eleves in 
 the department of science are the pick of the boursiers de licence. In 
 1909, 22 out of 270 candidates were thus selected and slightly more than 
 one half devoted themselves to mathematics. All decided to live as 
 internes although it was optional for anyone to attend the ficole as 
 externe when the amount of the bourse, 1,200 francs, would have been 
 paid to him. Most of the pupils were 20 years old, had obtained the 
 bourse on second trial and had passed one of the two years military 
 service obligatory for every Frenchman by the loi de deux ans of 1905. 
 
 The course for mathematicians is three years, and is arranged as 
 follows. During the first year the Aleves go to the Sorbonne to hear 
 Goursat's course in calculus and differential equations (3) and Raffy's 
 applications of analysis to geometry (4) . Instead, however, of follow- 
 ing Raffy's conference (14), which goes to complete the regular univer- 
 sity student's training for the certificat, calcul differ entiel et integral , 
 they are drilled by Borel and Tannery for three hours per week at the 
 ficole Normale. They also take physique g^n^rale with Bouty, Lipmann 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 119 
 
 and Leduc (12, 20), and at the end of the year pass the examinations 
 for the certificats in these two subjects. In the second year they take 
 up mecanique rationnelle with Painlev^ (10) at the Sorbonne, and with 
 Hadamard in conference at the Ecole. They also " assist " at such a 
 course as Darboux's and Cartan's in geom^trie sup^rieure (1, 13), 
 pass the examinations and receive two more certificats. During the 
 third year the eleves follow their own inclination in the selection of 
 courses at the Sorbonne or College de France, while they are well 
 grounded at the Ecole, in descriptive geometry by Roubaudi, and in 
 pedagogy, of algebra and analysis by Tannery, of geometry by Borel. 
 Nearly all those of the first and second year also follow Picard's course 
 (2), at the Sorbonne, and those of the third year that of Borel (15). 
 
 The drill in conferences (IJ hours each), at the ficole Normale is 
 unequalled. In addition to the good points of those at the Sorbonne, 
 we here find in much smaller classes a great degree of intimacy between 
 students and professor, and a freedom of question and discussion. 
 When, then, at the end of the third year these elite in intellect present 
 themselves in the terrific competition of the agregation, we are not sur- 
 prised that they give a good account of themselves. Some do not succeed 
 at the first trial or for ten, or twelve years afterwards, but 60 per cent, 
 of the 300 agreges named during the last 25 years were Ecole Normalians; 
 in 1890 there were only 4 out of 12, but of the 96 competitors in 1898, 
 the 8 chosen were from this famous school. ^ 
 
 But as the whole end and aim of the ficole Normale are not only to 
 prepare its eleves for the agregation and hence for professorships in 
 the lycees, but also to prepare them as university professors, we find 
 many who have been encouraged to take up certain fields of research 
 and who have made good progress toward a these for doctorate. Not 
 only this, but from those who have succeeded in the agregation, are 
 chosen agrege preparateurs who are taken back to the ficole for still 
 another two years (sometimes three) , while they prepare finally for their 
 doctorate with all the attendant advantages, of the counsels from their 
 former masters, and of the great library collections of the city. The 
 agrege-preparateur de mathematiques is officially charge de la biblio- 
 theque of the school. 
 
 The life in the ficole is singularly pleasant and inspiring. Here 
 alone of all the institutions we have considered do we find among the 
 61dves anything approaching the comradeship, so characteristic of the 
 student relations in American colleges. Nor in after years are friend- 
 ships and interests thus formed easily changed, as the Association 
 Amicale des Anciens EUves serves as a strong bond of sympathy and 
 
 ^ Cf. Appendix A. 
 
120 ROYAL SOCIETY OF CANADA 
 
 a constant means of intercourse. The fine old building, its studious 
 atmosphere, three to five years of friendly rivalry with almost equally 
 brilliant companions, daily intercourse with the professors, could hardly 
 fail to have developed a young man's latent talents or to have inspired 
 him to his best effort. Rarely does one find in France a professor such 
 as Tannery^ who is so generally beloved and respected by his eleves 
 past and present. 
 
 All the great privileges of the ficole are occasionally open to foreign- 
 ers either as internes or externes. It is now however, somewhat difficult 
 to make arrangements to enter as interne because of the increasing 
 demands made on limited space by the needs of the state. The charge 
 of 1,200 francs per year made for pension complete is exactly the value 
 of the bourse which France gives to students of her own nationality 
 and which she expects to be refunded if after leaving the Ecole, the eleve 
 decides not to take up the career of a teacher in her schools. All eleves 
 are assured positions in lyc^es those who have become agreges, as 
 professeurs titulaires, the others as professeurs charges de cours. 
 
 The ficoLE Polytechnique. 
 
 This Ecole founded by Monge at the close of the Revolution and the 
 most popular of the great schools of France, is under the direct control 
 of the minister of war and not of the minister of public instruction. 
 Its pupils are recruited from the most diverse orders of society solely 
 because of merit determined by concours on leaving the Classe de Mathe- 
 matiques Speciales. Its object is to prepare them as military and naval 
 engineers, artillery officers, civil engineers in government employ, tele- 
 graphists and officials of the government tobacco manufactories. 
 All eleves are internes. The cost of the pension is about 1,100 francs 
 per year, of the trousseau, 600-700 francs, but there are an unlimited 
 number of bursaries covering both pension and trousseau so that no 
 poor boy of brilliant attainment is shut out. 
 
 As to the past of the school, until the latter part of the nine- 
 teenth century it was famous not only by reason of the great engineers 
 it produced but also for its distinguished mathematicians. Poinsot, 
 Poisson, Cauchey, Poncelet, Chasles, Lame, Leverrier, Bertrand, 
 Duhamel, Liouville, A. Serrett, Laguerre, Halphen, Hermite, Poincare, 
 not to mention a host of others, were all trained here. . But now, the 
 demands made on the engineer are so great, the eleves are only given 
 the merest glimpses of the vistas which open up in modem mathematics. 
 
 ^ Tannery occupies one of the 12 chaires magistrales in mathematics at the 
 University de Paris, and as sous-directeur of the ecole normale superieure is directeur 
 des Mudes sci-eniifiques. [Later note added in the proof: Tannery died November 11, 
 1910, and Borel was appointed to his position.] 
 
[aechibald] mathematical INSTRUCTION IN FRANCE 121 
 
 From being perhaps the leading school of the time with regard to 
 its output of brilliant mathematicians it has, then, sunk to a position 
 of wholly inconsiderable importance in this respect. Yet each year 
 four times as many talented young mathematicians try to get in here 
 as into the ficole Normale Superieure. The number of those who enter 
 the Ecole Polytechnique because of failure to get into the ficole Normale 
 is not perhaps very large ; nevertheless there are certainly some among 
 them who would have made good mathematicians but who do not 
 make good engineers. They form, however, an insignificant fraction in 
 comparison with the hundreds of graduates who by original choice have 
 succeeded to the brilliant careers open to them. 
 
 The course at the ficole Polytechnique is two years and mathematics 
 is taught each year. As at the Sorbonne, but in less effective manner, 
 the instruction is a combination of lecture and conference. Jordan and 
 G. Humbert are the professors but they are assisted by several inter- 
 rogateurs or rep etiteurs as at Lycee Saint Louis. Humbert's Cours d' Ana- 
 lyse (2 vols.), gives an idea of the course in analysis (2 years) ; then there 
 are also mechanics and machines ( 2 years) ; descriptive geometry (first 
 year); astronomy, geodesy (second year); physics, acoustics and optics 
 (first year) ; p%sics, thermo-dynamics, electricity and magnetism (second 
 year); etc. 
 
 The College de France. 
 
 This, the highest institution of learning in France, was founded by 
 Frances I, in the sixteenth century. It does not form part of the Aca- 
 demic de Paris, but is under the direct control of the minister of public 
 instruction. No fee or form of matriculation is necessary to attend the 
 lectures, no examinations are held and no degrees are conferred. It is 
 not necessary for its professors to hold any degree or to have passed any 
 specific examination. A man who holds only the degree of bachelier, 
 although not qualified to teach in secondary schools may, if otherwise 
 competent, be appointed professor here. Successive vacancies are 
 filled by the minister of public instruction who chooses between the 
 names of two candidates who have been recommended by the body of 
 professors occupying the 45 chairs. The professors have absolutely no 
 obligations apart from the delivery of lectures, and in some cases, those 
 with untroubled consciences have more or less evaded this requirement. 
 Such abuse of privilege led this year to a law requiring each professor 
 to give 40 lectures, distributed somewhat symmetrically over the two 
 semesters. The purpose of the College de France is to advance learning. 
 Within the limits of their chairs, the professors are absolutely free to 
 treat any part of their subjects, no matter how limited or how minute, 
 provided that they go to the bottom of it. 
 
122 ROYAL SOCIETY OF CANADA 
 
 The chairs and foundation which have interest for us are the follow- 
 ing: 
 
 Mecanique analytique et m^canique celeste "Theory of elastic 
 plates/' Hadamard. 
 
 Math^matiques "Transformation and multiplication of com- 
 plexes in elliptic functions." (Jordan) G. Humbert. 
 
 Physique generale et mathematique "Elasticity of solids and 
 fluids." Brillouin. 
 
 Physique generale at experimentale "General phenomena of 
 electricity and magnetism." Langevin. 
 
 Cours compl^mentaire " Hyperelliptic surfaces of the fourth 
 degree." Traynard. 
 
 This last course was established by a foundation of Mile. Peccot who 
 wished to be the means of encouraging young mathematicians. The 
 instructor must be a doctor of less than 30 years of age and he may not 
 lecture for more than five years. 
 
 Jordan has not lectured for several years and his duties have been 
 performed by the suppliant Humbert. We have here another peculiar- 
 ity of this institution. Jordan continues to draw two-thirds of his 
 salary while Humbert is remunerated with the remaining one-third. 
 
 The courses usually represent personal researches of the lecturers 
 and are well attended, particularly by eleves of the ficole Normale 
 Sup^rieure. 
 
 Concluding Remarks on Mathematical Instruction. 
 
 Unless I have greatly failed in my presentation, one thing which 
 may be readily inferred from what has gone before is, that no idea 
 could be more mistaken than the one so prevalent among us, that the 
 French are light-hearted, frivolous and at best superficial. Their 
 struggle for existence is severe and the competition is terribly keen. 
 As far as the mathematician is concerned and his training is by no 
 means exceptional, we have found that from the time the eleve leaves 
 the Premiere, that is when he was 15 years old, onward, he undergoes 
 most exacting examination at almost every turn. The successive 
 stages in his studies are very largely marked out for him and care is 
 constantly exercised to see that he make no false step and that he be 
 properly prepared to pass his examinations. The Universite de Paris 
 has appeared to be a great institution, "wonderfully organized, to 
 turn out a certain amount of a certain product, of a certain degree of 
 excellence, with the least possible loss of time and energy." The 
 strenuous directness of method and of achievement in this system 
 cannot help but impress us. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 123 
 
 Such are the standards set for those, who have prepared them- 
 selves as lycee professors, who direct the boy's education from the 
 time he is ten years old. How woefully low are our standards in com- 
 parison! Remark too, that after a boy is 6 years old he is taught 
 by men only. 
 
 We are also struck with the breadth of the future mathematician's 
 training. Although it is true that after the age of 16, the humanities 
 are set aside, yet physics, descriptive geometry, pure mathematics, 
 applied mathematics, all continue to occupy positions of importance. 
 I have indicated how, by peculiar method of instruction, all these 
 subjects are welded into a homogeneous whole, how that although 
 knowledge of wide range of fact is fundamental, it is the thorough 
 grasp of broad principles and the powers of ready application of those 
 principles to the most diverse kind of problems, that is made essential. 
 German influence has given us a great respect for fact, but the French, 
 if opportunity were given, would soon convince us that a fact, as a fact, 
 had little of interest, except in so far as it might be contributary to 
 the upbuilding of some system. To the Frenchman learning is not an 
 accomplishment, but "an honourable and arduous profession with all 
 its trials, all its heart-burning competitions, all its pitiless disdain of 
 weakness, all its stimulating rewards." This partly explains the se- 
 verity of the examinations. Every boy of remarkable intellect, be he 
 rich or poor, has the chance to have his talents developed to the ut- 
 most. From the time he is 11 or 12 years old till he is ready to step 
 into a position in a university, bursaries constantly reward his accom- 
 plishment. If in time he become professor in a provincial university, 
 his effort is in no whit relaxed; he looks forward to being promoted 
 to Paris. With this advancement accomplished, his intellectual ac- 
 tivity does not cease by any means, for he now hopes some day to 
 be numbered among the few members of the Academie des Sciences 
 of the Institut. Great as are the rewards and recognition of merit 
 here, there is still greater for him who, as Poincare, is pre-eminent, 
 namely, to be numbered among the ''immortals" of the Academie 
 Frangaise of the Institut. Men of such calibre and brilliance and un- 
 remitting intensity of application and purpose, are the professors at 
 the College de France and Universite de Paris. 
 
 Nothing in French universities takes the place of undergraduate 
 life in England and America; nor would we willingly attempt to adopt 
 their system, though it would certainly silence the frequent criticism 
 of our ordinary B.A. course, namely, that it is, to say the least, a poor 
 training for the future man of business; the student has few obliga- 
 tions to meet, no real obstacles to overcome; if the professors make 
 the courses difficult, he either rises in protest or seeks a college with 
 
124 ROYAL SOCIETY OF CANADA 
 
 easier requirements. We prefer to retain the genial, sympathetic re- 
 lations between the student and professor, to encourage the emotional 
 and sentimental life of the students with one another. 
 
 We may, however, still learn from France the advantages of in- 
 timate relation, in standards and scheme, of secondary and higher 
 education. How much of the first year in American universities is 
 wasted by getting freshmen into form, in teaching them how to work 
 and how to think for themselves! The French university professors 
 are in constant intercourse with the lycees, are the examiners of all 
 their graduates, are the authors of many of the text-books employed. 
 The mathematical training and equipment of the average writer of 
 secondary texts in France is of far higher order than that of the average 
 American author. ^ 
 
 Teaching of Mathematics as a Profession in France. 
 
 We have now seen how the mathematician is trained in France. 
 It remains to discuss the nature of the inducements which are offered 
 to young men to prepare themselves for giving mathematical instruc- 
 tion, and to see whether the inducements offered are sufficient to attract 
 the best talent of the nation. 
 
 The agreg^s are those specially prepared by the State for the 
 positions as professeurs titulaires in the lycees. Although this title is 
 not conferred regularly till the agrege has completed his twenty-fifth 
 year, those who are yoimger receive temporary appointment. The 
 salaries vary according to the classe of the professor. At Paris the lowest 
 salary is 6,000 francs per year, and the highest, 9,500. In this range 
 seven classes are represented; six, each differing from the one before 
 by 500 francs, and the hors classe, for which the salary is 9,500 francs. 
 Promotion from one class to another takes place by selection and by 
 seniority. From the sixth (the lowest classe) to the third, the number 
 of those who can be advanced each year by selection is equal to the 
 number which can be advanced by seniority. In the second and first 
 classes two advancements may be made by selection to one by seniority. 
 In choosing those for the hors classe, selection alone is taken into ac- 
 count. The promotions are made at the end of each calendar year, 
 and take place so that there are always 20 per cent, of them in the 
 sixth class, 18 in the fifth, 18 in the fourth, 16 in the third, 14 in the 
 second, and 14 in the first. This arrangement is obviously a happy 
 one, both by way of recognition of the merits of the unusually success- 
 ful teacher, as well as those of him whose service is rather characterized 
 by faithfulness. 
 
 * Compare Appendix D. 
 
[aechibald] mathematical INSTRUCTION IN FRANCE 125 
 
 In addition to the professeurs titulaires there are professeurs charge 
 de cours, who are usually selected from those ficole Normalians and 
 those admissible to the agregation, who fail to become agreg^s. After 
 20 years of service they may become professeurs titulaires and receive 
 the salaries we have indicated above. The government has, however, 
 this year passed a law which gives the higher reward to the agreg^. 
 It is to the effect that 500 francs per year shall be added to the regular 
 salary of every agreg^. The real range of salaries mentioned above 
 is then 6,500-10,000; in the provinces this reduces to 4,700-6,700. 
 For the professeurs charge de cours, the salaries at Paris vary from 
 4,500 to 6,000 francs; in the provinces, from 3,200 to 5,200. In the 
 Premier Cycle the professors have 12 hours of teaching per week, in 
 the second cycle and the Classe de Mathematiques Sp^ciales, 14-15 
 hours. Except for correcting exercises and filling out reports the 
 professors have absolutely no obligations outside of class hours. They 
 do not live in the lycees. The superintendence of the study of the 
 Aleves is carried on by repetiteurs, the more advanced of whom receive 
 at Paris 2,600-4,600 francs for 36 hours service per week. 
 
 Only a very small percentage of the agreges are also doctors,^ but 
 these few, as well as the more promising of those who are doctors 
 only, usually prefer to seek some of the minor positions in connection 
 with the universities. Maitres de conferences adjoints are selected from 
 among the doctors. Charges de conferences and maitres de conferences 
 are sought for (1) among agreges, (2) among doctors, but only the 
 latter may receive an appointment for more than one year. A charg6 
 de conferences at Paris receives 5,000-7,000 francs a year^ for 2-3 hours 
 per week of service. Even this amount is sufficient to enable a man 
 to live well; but when, in addition, the incumbent is professeur agreg^ 
 in a Paris lycee (as in two cases at the Universite de Paris at present), 
 his income may exceed the regular salary of a university professor. 
 For a good man there are also other sources of income from acting as 
 suppleant, examinateur or interrogateur. 
 
 From the charges de cours or maitres de conferences, who are at 
 least 30 years of age, who are doctors, who have seen at least two 
 years of service in a school of higher education, and who are distin- 
 guished for their services are appointed the professeurs adjoints of the 
 universities. They receive from 6,000-10,000 francs at Paris and 
 4,500-6,000 francs in the provinces. The salary of professeurs titu- 
 laires is 12,000-15,000 francs at Paris and 6,000 or 8,000 to 12,000 
 
 * Compare Appendices A, C. 
 
 2 According to a decree of June 25, 1910, charges de cours compl^mentaires and 
 maitres de conferences in the provinces, were thence forth to be of four classes, 
 and to receive 4,500-6,000 francs annually. 
 
126 ROYAL SOCIETY OF CANADA 
 
 in the provinces/ In recent appointment of professors, selection haS 
 been almost exclusively made from those who are both agreges 
 and doctors. That in exceptional cases the latter only is 
 necessary was illustrated by a recent appointment to Poitiers,^ 
 but it is quite unlikely that any professor will ever be promoted to 
 Paris who has not passed both examinations. As exceptional, note 
 that any member of the Institut may be appointed professor at a 
 university after six months of service in an establishment of higher 
 education. The professorship of highest honour in the gift of the 
 nation is at the College de France. Although the salary here is only 
 10,000 francs, the duties consist simply in delivering 40 lectures of 
 one hour each. In the universities the professor is expected, in general, 
 to give but one course of lectures, viz., that which is called for by his 
 chair. These lectures are delivered twice a week and last from an 
 hour to an hour and a half each. If the course continue through the 
 whole year, about 60 lectures are given ; but we have already remarked 
 that such men as Poincare, Picard, Darboux, give only half this number. 
 Remember, too, that many courses (practically all in the provinces) 
 are repeated year after year with little change, that the professors are 
 never called upon to arrange hours for conference with members of 
 the class or to correct students' exercises. 
 
 One decidedly disagreeable duty does, however, fall to their lot- 
 This is their obligation in the matter of various examinations. The 
 figures given in an earlier section (p. 99) show how formidable this may 
 be in the case of the baccalaureat alone, for the examiners as a whole. 
 At the present time, however, only about one half the work is done by the 
 university professor; and although his time is more or less broken into 
 from June 27 to August 10, and November 1-8, the whole number of 
 hours actually given up to the work by a single individual, in connection 
 with both the baccalaureat and certificats, does not exceed 55. The 
 whole number of hours which the professor gives to the State is, then, 
 85-145 per year. With such insignificant breaks in leisure for research 
 we can no longer wonder at the great productivity of many French 
 mathematical professors. The attractiveness of their positions is still 
 
 * Until the recent increase in the salaries of the university instructors in Germany, 
 70 per cent, of the full professors received less than 15,000 francs. On the other 
 hand there were three who received over 50,000 francs; and in any large German 
 faculty some full professors will generally be found who receive for teaching an 
 income from two to five times as large as some of his colleagues. These larger 
 incomes are due to special allowances from the government, to extra university 
 perquisites and to fees from the large body of students attracted by superior repu- 
 tation. As distinguished from the rest of the world, in this connection, Germany 
 pays an unusual amount for unusual merit. 
 
 ^ Compare Appendix B. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 127 
 
 further enhanced by other sources of income. Nearly all those at the 
 Sorbonne are members of the Academie des Sciences of the Institut de 
 France. As such they receive 1,500 francs annually. Since Poincare 
 is also member of the Academie Frangaise, this amount is presumably 
 doubled. Darboux, as secretaire perpetuel^ receives 6,000 francs. 
 Painleve, also a member of the Institut, has been elected a member 
 of the Chamber of Deputies, which will bring him in another 15,000 
 francs a year. 
 
 To such professorships the rising young mathematician may 
 aspire ; but as there are only fifty chairs in the whole country, the open- 
 ings are few and the progress toward them slow. 
 
 We cannot help but contrast the conditions of the American 
 professor, with at least 10-12 hours of lecturing per week, in several 
 departments of mathematics, not to speak of the demands made on 
 his time in correcting exercises and examination papers, and in ad- 
 ministrative work. Yet with all this burden, he is expected not only 
 to keep abreast of the times in his subject, but also to advance know- 
 ledge by his own researches. 
 
 The American Mathematical Student in Paris. 
 
 Many of the attractive features of mathematical study in Paris 
 have been already set forth in the foregoing pages, but I wish here 
 to briefly indicate a few others, as well as to give some special infor- 
 mation which may be helpful to the American student. 
 
 No one can be wholly insensible to the charm of Paris herself, 
 to the artisticity lavishly displayed by her people in sweeping shaded 
 boulevard, towering monument, imposing building, gorgeous decor- 
 ation, garden and embowered statuary, far-reaching park. From the 
 time of Caesar and Roman occupation of the Cite, historic associations 
 have multiplied, and now they cluster about every quarter. Galleries, 
 churches, palaces, Versailles, Saint Cloud, Chantilly, Fontainebleau, 
 illumine and vivify in thrilling fashion the printed accounts of happen- 
 ings of history. To the sympathetic student of the genius of the 
 people, their customs, their language, their habits of thought French 
 literature is vested with new dignity and charm and grace and subtle 
 meaning. Preeminent on the stage, strongly influential in the worlds 
 of art, among the foremost in all forms of scholarship the potential- 
 ities of fair France are great, both to educate and to refine. 
 
 But to benefit by such influences, as well as by the courses of 
 instruction, which run, for the most part, through the whole year, 
 the American student should plan to stay in Paris at least a year. 
 It would also be well for him to come as early in June as possible. At 
 this time there is no opportunity for attending university lectures, 
 
128 ROYAL SOCIETY OF CANADA 
 
 since the second semester closes about the middle of June; but he 
 can look over the ground, get acquainted and prepare generally for the 
 following autumn. The student who does not have friends in Paris 
 should go to the Bureau des renseignements at the Sorbonne, where he 
 will find some one who can speak English and give information as to 
 pensions, the various institutions of higher education, etc. Another 
 helpful bureau is the ComitS de patronage des etudiants etr angers. 
 There are a large number of students' associations, but probably the 
 only one which it will be found worth while to join is the Association 
 g^nirale des etudiants de Paris. This association has recently moved 
 into a handsome stone building used for over three centuries as lecture 
 hall for the faculty of medicine. Here may be found reading room, 
 library, fencing room, lounge rooms, etc. Members receive great 
 reductions on tickets for nearly all the theatres and in purchasing books 
 and other supplies. There are numerous social gatherings which pro- 
 fessors and alumni occasionally join. 
 
 Paris is apt to be uncomfortably warm during the summer months, 
 and unless the student is proficient in both speaking and writing the 
 French language, he will probably wish to seek out more enjoyable 
 quarters to carry on his studies. These may be found at such uni- 
 versity towns as Grenoble or Geneva, both beautifully situated, but 
 especially the latter, with endless possibilities in Alpine excursion. In 
 each, excellent summer courses, at small cost, are given specially for 
 foreigners. That at Grenoble lasts from July 1st to October 31st. 
 It would be well, however, to return to Paris a little before this latter 
 date, so as to settle the question of lodging somewhat before the be- 
 ginning of the scholastic year. 
 
 To get the most out of a sojourn in Paris, the American naturally 
 wishes, if possible, to get into a private family where he may enter, 
 to the full, into the spirit of the life and language of the people. This 
 is, however, a matter of much greater difficulty than in Germany, 
 where many are so ready to welcome the stranger to hearth and home. 
 In the case of the French the foyer is much more exclusive, and unless 
 mutual friends have intervened, a seat there is almost an impossibility. 
 The next best thing is to be in a small pension, where good French, 
 but no English, is spoken. It is a matter of ever-increasing difficulty 
 to find such a place. The charges near the Sorbonne vary from 150 
 to 250 francs per month; for 180 francs one may be excellently served. 
 A third method, not less expensive, is to rent a furnished room and dine 
 at a restaurant. A room may be procured for 30 to 65 francs per month. 
 But restaurant cooking and poor French frequently heard are un- 
 desirable features of this plan. It sometimes happens, however, that 
 meals alone (that is lunch and dinner) may be arranged for at a good 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 129 
 
 pension for 100-150 francs per month. Pensions on the other side of 
 the Luxembourg from the Sorbonne are usually the cheaper. As 
 some of the courses which the American will likely want to follow at 
 both the Ecole Normale Superieure and the Sorbonne commence at half- 
 past eight in the morning, it will be well not to live too far away. All 
 necessary expenses of the student who spends the year in Paris, with 
 summers elsewhere, ought not to exceed 750 dollars. 
 
 At the Universite de Paris the year commences about November 4th 
 and, in contrast to German methods, the lectures start immediately. 
 The student will find the opening addresses of the vice-recteur and 
 doyens, delivered in the great amphitheatre a day or two after the 
 semester has begun, of interest. Matriculation is a very informal, 
 though tiresome, affair. A college diploma and a certified French 
 translation of birth certificate should be taken to the Secretary's office. 
 It is with an air of considerable scepticism that the Frenchman receives 
 the statement that birth certificates are the exceptional possession of 
 the average Canadian or American. The difficulty, in the case of the 
 Canadian, is very easily solved on applying to the Canadian Commis- 
 sioner, rue de Rome, and doubtless like courtesy awaits the American 
 at the bureau of his Consul General. A passport is not necessary in 
 France, as in Germany. The fee of 30 francs gives the student all 
 rights of library, lecture and conference, for the whole year. To follow 
 the classes at the Ecole Normale it is only necessary to go through the 
 formality of applying for permission to the Vice-recteur of the Univer- 
 site de Paris. The letter received in reply should be presented to 
 M. Tannery, who will most cordially counsel and assist the newcomer. 
 The courses at the College de France commence about December 3. 
 The general holidays one week before and one week after New Year's 
 day, two weeks similarly arranged with reference to Easter are the 
 same as at the Universite de Paris. There is no break between the 
 first and the second semester, which begins March 1. All lectures 
 are freely open to the public. 
 
 Just what courses of those offered in these three institutions will 
 particularly appeal to the American student must naturally be both 
 a matter of taste and of previous study. I have given the details of 
 the courses in earlier pages. Appell, Goursat and Picard are especially 
 noted for the elegance and clarity of their presentation. But, except 
 from a pedagogical standpoint, Appell does not, at present, give any- 
 thing of interest to the American student ; while the first half of Goursat's 
 course will be found more or less of a review of earlier work. The 
 courses of Darboux, Poincare and Painleve are of a more advanced 
 nature and largely attended. Such conferences as those of Raffy^ and 
 
 ^ Raffy died since I wrote th^ above ; his death occurred June 9, 1910. 
 Sec. III., 1910. 9. 
 
130 ROYAL SOCIETY OF CANADA 
 
 Cartan at the Sorbonne, of Tannery, Borel and Hadamard at the ficole 
 Normale Superieure, should on no account be neglected. The training 
 and grounding they give is simply invaluable. 
 
 If the interests of the student wander into other fields, the oppor- 
 tunity for profit is just as great as in the department of mathematics. 
 There are the chairs in the Faculte des Sciences, such as physics, chem- 
 istry, biology, not to mention those held by other world renowned 
 savants in literature, history, philosophy, etc., of the Faculte des 
 Lettres. Any matriculated student in the Faculte des Sciences may 
 have his name inscribed in this Faculte without further charge. The 
 lectures of Reinach, Michel, etc., at the ficole du Louvre are open to all 
 and are of especial appeal to those interested in the various phases of 
 art. Indeed, as soon as one leaves special for general study, the riches 
 of intellectual treat on every hand lead to embarrassment of choice. 
 
 The book treasures and collections available for the student in 
 Paris are unequalled by any other city in the world. Chiefly by co- 
 operation of the Societe Mathematique de France, the Sorbonne pos- 
 sesses a remarkably complete collection of mathematical periodicals. 
 The officials of the library are exceedingly helpful and most generous 
 to the earnest student; not only do they grant admission to the peri- 
 odical section, but, occasionally, the privilege of exploring the general 
 stacks as well. Since the catalogue is poor, this facility for the searcher 
 is of inestimable value. The librarian is always ready to purchase 
 any standard work which is not in the library and which the student 
 specially needs. The Bibliotheque Nationale^ contains nearly all 
 mathematical periodicals lacking at the Sorbonne, a tolerably complete 
 set of French mathematical publications, as well as a representative 
 collection of those of other countries. When the need arises of con- 
 sulting older mathematical works, this library or the Bibliotheque 
 Mazarine is pretty sure to be able to supply the want. The Bibliotheque 
 Sainte Genevieve has, among others, a good collection of elementary 
 mathematical books. 
 
 Finally, under what conditions may an American mathematical 
 student in Paris proceed to the doctorate? The general question has 
 been fully considered in earlier pages, and only a few observations 
 remain to be made here. We have remarked that two degrees are 
 available, the doctoral d'etat and the doctorat de VUniversite de Paris. 
 In both cases the candidate must receive permission from the minister 
 of public instruction to present a university diploma as an Equivalence 
 de Scolarite of the baccalaureat. In both cases the These (which has 
 
 * A card of admission will be granted on presenting a letter from either the 
 Canadian Commissioner or the American Consul General, 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 131 
 
 always been written in French) is the principal thing required after 
 the student has received the necessary certificats. Although the French- 
 man works his out independently, the American student will not 
 derive less cordial help or suggestion from a French than, under similar 
 circumstances, from a German professor. He will find, however, that 
 this cordiality is very unlikely to expand in the former case, as in the 
 latter it is almost sure to do, to an invitation to be a guest in the home. 
 The question of certificats, with their exacting examinations, makes 
 the doctorat d'etat decidedly the more difficult. There is, however, 
 no reason why a student who has the ability to get his doctorate in 
 Germany in two years may not get the doctorat de T University de Paris 
 in the same time. Indeed, if he be well equipped, have a these well 
 under way before coming to France, and is prepared to " scorn delights 
 and live laborious days," this doctorate is a possibility in one year. 
 But since to 'those who know' no other doctor's degree in mathematics 
 has quite as high a standard as the doctorat d'etat, why does not some 
 Canadian aspire to be the first in the British Empire to win it? Or 
 why does not some young man from the United States feel spurred to 
 show his equality with one of his countrywomen? 
 
 Authorities. 
 
 For the study of primary and secondary education for boys, three 
 publications are essential: 
 
 Organisation pedagogique et plan d' etudes des ecoles primaires 
 ilementaires. 
 
 Plan d' etudes et programmes d' enseignement des ecoles primaires 
 superieures de gargons. 
 
 Plans d' etudes et programme d' enseignement dans les lycees et colleges 
 de gargons. 
 
 All departments of education are dealt with by the invaluable 
 Bulletin [hebdomadaire] administratif du ministre de Vinstruction pu- 
 blique, 1850- and Annuaire de Vinstruction publique et des beaux- 
 
 arts. The budgets may be found in the Journal Officiel, and statistics 
 of various kinds in the Annuaire Statistique of the Minister e du travail 
 et de la prevoyance socials direction du travail. 
 
 Of unofficial publications which I have sometimes found useful are : 
 
 Le Nouveau Baccalaureat de V enseignement secondaire; guide du 
 candidal. 
 
 Programmes des certificats d' etudes superieures. 
 
132 ROYAL SOCIETY OF CANADA 
 
 UUniversite de Paris et les etablissements Parisiens d' enseignement 
 swperieur. Livret de V itudiant, published by the Bureau des renseigne- 
 ments at the Sorbonne. 
 
 Annuaire de la jeunesse, of H. Vuibert. 
 
 Other authorities are indicated in Appendix C. 
 
 Interesting comment may be found in Klein's Vortnige uber den 
 mathematischen Unterricht an den hoheren Schulen (1907), and in Klein's 
 Elementar Mathematik vom hdheren Standpunkte aus Teil II: Geometrie 
 (1909. Pp. 456-77: "Unterricht in Frankreich ") . 
 
 It is with great pleasure that I have also to acknowledge my in- 
 debtedness to a very large number of friends, acquaintances and officials 
 in the Sorbonne, the ficole Normale Superieure, the Ecole Polytechnique, 
 in the lyc^es an(J in the different departments of the ministere de 
 rinstruction publique. The invariable courtesy and obliging readiness 
 to place all possible material at my disposal, which my innumerable 
 inquiries called forth, constitute a very pleasant memory, among the 
 many, of a delightful year. 
 
 Paris, May 2, 1910. 
 
 [After my paper had been written, I saw, by chance, a reprint of an interesting 
 article, with the same title as this, published by Professor Pierpont about ten years ago, 
 in the Bulletin of the American Mathematical Society. It suggested several improve- 
 ments (in connection Avith questions of form and fuller development) which were then 
 made in my paper. Concerning recent literature reference may be given to : " France 
 as a Field for American Students " by S. Newcomb {Forum, xxiii., 320-326, May 1897. 
 French translation. Revue Internationale de V Enseignement, xxxiv,, 20-27, July 1897. 
 Cf. also Nation, \xiv., 400-01, Nov. 26, 1896) The chapter on " The Universities " in 
 B. Wendell's " The France of To-day " (second edition, 1908)" life at the Sorbonne " 
 by H. Jones {Nation, xci., 576-7, Dec. 15, 1910).] 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 133 
 
 APPENDIX A. 
 
 Agr^gation des Sciences Math^matiques. 
 
 As there are no mathematical examinations in any other country 
 to compare in difficulty with those to which the candidate for a French 
 agregation is required to submit, it has seemed to me that it would be 
 a matter of interest if fuller details of what is involved were set forth. 1 
 therefore subjoin: 
 
 I. The programme for the concours of 1910 (announced 9-11 
 months in advance). The examinations will be on topics 
 selected from this programme. 
 II. The examination papers for 1909. The four written exami- 
 nations, it will be observed, occurred on consecutive days. 
 The first paper may seem short for th.<i time allowed (seven 
 hours) ; but not when the enormously high standard in pre- 
 sentation and detail is taken into consideration. 
 III. A Table giving certain data respecting the agreges named during 
 the last twenty-five years, which show that the impressions 
 prevailing as to the age of the agrege, the number of Norma- 
 liens who become agreges, and the number of agreges who 
 became doctors are quite erroneous. 
 
 PART I. 
 PROGRAMME FOR THE CONCOURS OF 1910. 
 
 I. General Programme in Analysis and Mechanics. 
 
 Since the programmes for the certificats d' etudes superieures vary 
 among the different universities, the jury indicate in the programme 
 below the minimum of general knowledge which is supposed acquired 
 by the candidates in differential calculus, integral calculus and mechanics. 
 The subjects of the "compositions" in differential calculus, in- 
 tegral calculus and mechanics will be chosen from Nos., 1, 2, 3, 4, 
 5, 7, 8, 9, 14 and 15 of this programme; their scope will not 
 exceed the standard set by the subjects of problems proposed for the 
 corresponding certificats for the Licence. 
 
 Differential Calculus and Integral Calculus. 
 
 1. Fundamental Operations of Differential and Integral Calculus: 
 Derivatives and differentials; simple integrals, curvilinear integrals, 
 integrals of total differentials, double and triple integrals. 
 
 2. Applications of the Differential Calculus: Study of functions 
 of a real variable (formula of Taylor, maxima and minima, functional 
 determinants, implicit functions); Calculation of derivatives and 
 differentials; change of variables. Order of contact and genre of an area. 
 
/ 
 
 134 ROYAL SOCIETY OF CANADA 
 
 3. Applications of the Integral Calculus: Process of integration. 
 Length of an arc of a curve, plane and gauche areas, vohimes. Differ- 
 entiation and change of variables under the sign // Study of the 
 
 integral /^ f (x) dx when one of the limits or the function becomes infinite. 
 Formula of Green. Study of functions represented by certain series. 
 Properties of power series. 
 
 4 . Elements of infinitesimal Geometry : " Infinitesimal properties " 
 of plane and gauche curves (curve envelopes, curvature, torsion). In- 
 finitesimal properties of the surfaces; surface envelopes, summary of the 
 results on the transformations of contact; developable surfaces, ruled 
 surfaces, Meusnier's theorem; principal sections. Conjugate lines, 
 lines of curvature, asymptotic lines in any curvilinear co-ordinates. 
 
 5. Elementary Functions of a Complex Variable: Simple alge- 
 braic functions; circular and logarithmic functions. 
 
 6. Theory of Analytic Functions: Properties of the integral 
 
 f(z) dz. Series of Taylor and of Laurent. Poles, essentially 
 
 singular points, residues. Reduction of the hyperelliptic integrals. 
 
 7. Differential Equations of the first order: General solutions, 
 particular solutions, singular solutions. Simple types of integrable 
 equations. Integrating factor. Theorem of Briot and Bouquet on the 
 existence of the solutions in the cases where the known functions are 
 analytic. 
 
 8. Differential Equations and Systems of Equations of any order: 
 General solution, particular solutions, first solutions. Simple types of 
 integrable Equations. Linear Equations, 
 
 9. Integration of linear partial differential equations of the first 
 order. 
 
 10. Integration of differential equations {partial or total) of the 
 first order. 
 
 Mechanics. 
 
 11. Statics: Composition of forces applied at a point. Attrac- 
 tion of a spherical homogeneous shell at an exterior or interior point. 
 Elementary properties of the potential. Reduction of forces applied 
 to a solid body. Conditions of equilibrium of a solid body. Applica- 
 tions to simple machines. Funicular polygon. Suspension bridges. 
 Catenary. Principal of virtual work. 
 
 12. Kinematics: Velocity, acceleration. Movement of a plane 
 figure in its plane. Representation of the movement by the rolling of 
 a moving curve on a fixed curve. Movement of a solid body about a 
 fixed point. Representation of the movement by the rolling of a 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 135 
 
 moving cone on a fixed cone. Movement of a solid body in space. 
 Helicoidal movement. Relative movements. Theorem of Coriolis. 
 
 13. Dynamics of a Particle: Work. General Theorems. First 
 integrals of the equations of the motion. Application to the motions 
 of the planets. Movement of a point on a curve or on a surface. Pen- 
 dulum in a vacuum and in a resisting medium. Conical Pendulum. 
 Geodetic Lines. 
 
 14. Geometry of Masses: Centres of gravity. Moments of 
 Inertia. 
 
 15. Dynamics of Systems: General Theorems. First Integrals. 
 Energy, Stability of Equilibrium. Movement of a solid body about a fixed 
 axis. Pressure supported by the axis. Compound Pendulum. Move- 
 ment of a solid body about a fixed point. General movement of a solid 
 body. Law of friction and slipping. Application of the principle of 
 vis viva to machines. D'Alembert's Principle. Lagrange's Equations. 
 Relative motion. Percussions. 
 
 16. Canonical Equations: Theorem of Jacobi. 
 
 17. Hydrostatics: Equilibrium of a fluid mass. "Surfaces de 
 niveau." Pressure on a side plane. Archimedes' principle. Equi- 
 librium of floating bodies. 
 
 18. Hydrodynamics: General equations of the movement of a 
 fluid mass. Bernoulli's Theorem. Torricelli's Theorem. 
 
 Lessons. 
 
 Parts of the programmes from which are drawn the subjects of the 
 lessons. 
 
 1. Math^imatiques Speciales. 
 
 Series: Series of positive terms; character of convergence or 
 divergence drawn from the study of the expressions : 
 
 .^U. 
 
 ^' .1 Un. nPUr 
 
 Absolutely converging series. Convergence of series, with terms alter- 
 nately positive and negative, of which the general term decreases con- 
 stantly in absolute value and tends towards zero. Numerical examples. 
 General Properties of Algebraic Equations: Number of roots of 
 an Equation. Relations between the coefficients and the roots. Every 
 rational and symmetric function of the roots may be expressed rationally 
 as a function of the coefficients. Elimination of one unknown between 
 two equations by means of symmetric functions. Condition that an 
 equation has equal roots. Study of the commensurable roots. Des- 
 
136 ROYAL SOCIETY OF CANADA 
 
 cartes' Theorem. Complex numbers. De Moivre's Theorem. Trig- 
 nometric resolution of the binomial equation. 
 
 Functions: Function of a real variable, graphic representation, 
 continuity. Definition and continuity of the exponential function and 
 of the logarithmic function. Limit of (1 -f ^) when m increases 
 indefinitely in absolute value. Derivative of a function; slope of the 
 curve represented. Derivative of a sum, of a product, of a 
 quotient, of an integral power; of a function of a function. Derivative 
 of a* and of log x. Use of logarithm tables and of the slide rule. 
 Rollers Theorem, law of finite increments, graphic representation. 
 Functions of several independent variables, partial derivatives. Law of 
 finite increments. Derivative of a compound function. Derivative 
 of an implicit function (admitting the existence of this derivative). 
 Employment of the derivative for the study of the variation of a 
 function; maxima and minima. Primitive functions of a given function, 
 their representation by the area of a curve. 
 
 Functions defined by a power series with real coefficients. Interval of 
 Convergence: Addition and multiplication. In the interior of the 
 interval of convergence one obtains the derivative or the primitive 
 functions of the function, on taking the series of derivatives or of the 
 primitive functions (functions which pass to the extremities of the 
 interval are not considered). Examples: developments in series of 
 f^j i+xi> arctan x, log (1 x), log ^. Exponential series! Binomial 
 series. The equations y^ = y, and y^ (1 + x) = my serve to determine 
 the sum of two series. Development into series of a* , of arcsin x. 
 
 Curves whose equation is soluble or insoluble with regard to one of the 
 co-ordinates: Tracing. Equation of the tangent at a point; sub-tangent. 
 Normal, sub-normal. Concavity, convexity, points of inflexion. 
 As)anptotes. Application to simple examples and in particular to the 
 conies and to those curves of which the equation is of the second degree 
 with respect to one of its co-ordinates. 
 
 Curves defined by the expression of the co-ordinates of one of their 
 points as function of a prameter: Tracing. Numerical examples. 
 The curves of the second order and those of the third order with a double 
 point are unicursal. 
 
 Curves defined by an implicit equation: Equation of the tangent 
 and of the normal at a point. Tangents at the origin in the case where 
 the origin is a simple point or a double point. Discussion of the asym- 
 ptotes in the case of numerical examples of curves of the second and of 
 the third order. 
 
 Curvature. Envelopes. Developables. 
 
 Polar Co-ordinates: Their transformation into line co-ordinates. 
 Equation of a right line. Construction of curves; tangents, asymptotes 
 
ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 137 
 
 Applications (confined to the case when the equation is solved with 
 respect to a radius vector). Case of the conies. 
 
 Gauche Curves: Tangent. Osculating plane. Curvature. Appli- 
 cations to the circular helix. 
 
 Study of surfaces of the second degree with reduced equation: Con- 
 dition of the contact of a plane with the surface. Simple problems 
 relative to tangent planes. Normals. Properties of conjugate dia- 
 meters. Theorems of Apollonius for the ellipsoid and the hyper- 
 boloids. Circular sections. Rectilinear generatrices. The surfaces of 
 the second order are unicursal. 
 
 Dynamics. 
 
 1. Free Material Point: Principle of inertia. Definition of force 
 and mass. ^ Relation between the mass and the weight. Invariability 
 of the mass. Fundamental units. Derived units. Movement of a 
 point under the action of a force, constant in magnitude and direction or 
 under the action of a force issuing from a fixed centre: 1 proportional 
 to the distance; 2 in the ratio inversely as the square of the distance. 
 Composition of forces applied at a material point. ^ Work of a force, 
 work of the resultant of several forces, work of a force for a resulting 
 displacement. Theory of living force. Surfaces de niveau. Fields 
 and lines of force. Kinetic energy and potential energy of a particle 
 placed in a field of force. 
 
 2. Material Point, not free: Movement of a heavy particle on an 
 inclined plane, with and without friction, the initial veolcity acting along 
 the line of greatest inclination. Total pressure on the plane; reaction 
 of the plane. Small oscillations of a simple pendulum without friction; 
 isochronism. 
 
 Descriptive Geometry. 
 
 Intersection of Surfaces: Two cones or cylinders, cone or cylinder 
 and surface of revolution, two surfaces of revolution of which the axes 
 are in the same plane. 
 
 * It is admitted that a force applied at a material point is geometrically equal 
 to the product of the mass of the point by the acceleration that it impresses on the 
 point. 
 
 ^ It is admitted that, if several forces act at a point, the acceleration that they 
 impress on the point is the geometric sum of the accelerations that each of them 
 impresses on it, if acting alone. 
 
138 ROYAL SOCIETY OF CANADA 
 
 II. Lessons on the Subjects of the Programme of the Seconde 
 
 AND PrEMII^RE (C and D) AND MaTH^IMATIQUES A. 
 
 Seconde (C. and D.). 
 
 Algebra: Resolution of equation of the first degree in one unknown. 
 Inequalities of the first degree. Resolution and discussion of two 
 equations of the first degree in two unknowns. Problems; substitution 
 in equation. Discussion of the results. Variation of the expression 
 ax + b; graphic representation. -Equations of the second degree in 
 one unknown (theory of imaginaries not discussed) . Relations between 
 the coefficients and the roots. Existence and signs of the roots. Study 
 of the trinomial of the second degree. Inequalities of the second degree. 
 Problems of the second degree. Variation of the trinomial of the second 
 degree. Graphic representation. Variation of the Expression ^i^^^i i 
 graphic representation. Notion of derivative; geometrical significance 
 of the derivative. The sign of the derivative indicates the direction of 
 the variation; applications to very simple numerical examples and in 
 particular to the functions studied before. 
 
 Geometry : Simple notions of homothetic figures . Similar polygons . 
 Sine, cosine, tangent and cotangent of positive angles less than 2 right 
 angles. Metrical relations in a right triangle and in any triangle. Pro- 
 portional lines in the circle. Fourth proportional; mean proportional. 
 Regular polygons. Inscription in a circle of a square, of a hexagon; of 
 an equilateral triangle, of a decagon, of a quindecagon. Two regular 
 polygons of the same number of sides are similar. Ratio of their peri- 
 meters. Length of an arc of a circle. Ratio of the circumference to 
 the diameter. Calculation of t: (confined to the method of the peri- 
 meters). Area of polygons; area of a circle. Measure of the area of 
 a rectangle, of a parallelogram, of a triangle, of a trapezium, of any 
 polygon. Ratio of the areas of two similar polygons Area of a regular 
 convex polygon. Area of a circle, of a sector and of a segment of a 
 circle. Ratio of the areas of two circles. 
 
 Premiere (C. and D). 
 
 Geometry. 
 
 Translation: Rotation about an axis. Symmetry with respect 
 to a line. Symmetry with respect to a point. Symmetry with respect 
 to a plane. This second kind of symmetry is equivalent to the first. 
 
 Trihedral Angles: Disposition of the elements. Trihedral symme- 
 try. Each face of a trihedral is less than the sum of the other two. 
 Limits of the sum of the faces of a trihedral. Supplementary trihedrals. 
 Applications. Inequalities of the trihedrals. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 139 
 
 Homology: Parallel plane sections of polyhedral angles. Areas. 
 
 Polyhedra: Homothetic polyhedra, similar polyhedra. Prisms, 
 Pyramids. Summary of notions on the symmetry of the cube 
 and of the regular octahedron. Volumes of parallelopipeds and 
 of prisms. Volume of the Pyramid. Volume of a pyramid truncated 
 by parallel sections. Volume of a truncated triangular prism. Ratio 
 of the volumes of two similar polyhedra. ^Two symmetrical polyhedra 
 are equivalent. Sphere : plane section, poles, tangent plane. Circum- 
 scribed cone and cylinder. Area and volume. 
 
 Mathematiques A 
 
 Arithmetic: * Common fractions. Reduction of a fraction to its 
 simplest terms. Reduction of several fractions to a common denominator 
 Least common denominator. Operations with common fractions. 
 Decimal numbers. Operations (considering the decimal fractions as 
 particular cases of ordinary fractions). Calculation of a quotient to 
 a given decimal approximation. Reduction of an ordinary fraction 
 to a decimal fraction; condition of possibility. When the reduction is 
 impossible, the ordinary fraction can be regarded as the limit of an un- 
 limited periodic decimal fraction. Square of a whole number or of a 
 fractional number; nature of the square of the sum of two numbers. 
 The square of a fraction is never equal to a whole number. Definition 
 and extraction of the square root of a whole number or of a fraction to 
 a given decimal approximation. Definition of absolute error and of 
 relative error. Determination of the upper limit of an error made in 
 a sum, a difference, a product, a quotient, knowing the upper limits of 
 the errors by which the given quantities are affected. Metric System. 
 
 Algebra: Monomials, polynomials; addition, subtraction, multi- 
 plication and division of monomials and of polynomials. Equations of 
 the second degree in one unknown. Simple equations which are 
 equivalent. (The theory of imaginaries is not developed). Problems 
 of the first and second degree. Arithmetic Progressions. Geometric 
 Progressions. Common Logarithms. Compound Interest, annuities. 
 
 Trigonometry: Circular Functions. Addition and Subtraction of 
 arcs. Multiplication and division by 2. Resolution of triangles. 
 Applications of Trigonometry to various questions relative to the 
 elevation of planes. (The construction of the trigonometric tables is 
 not to be considered) . 
 
 Geometry: Inversion. Applications. Peaucellier's Cell. Polar of 
 a point with respect to a circle. Polar plane of a point with respect to 
 a sphere. Hyperbola: Trace, tangent; asymptotes; simple problems 
 on tangents. Equation of a hyperbola with respect to its axes. Plane 
 sections of a cone and of a cylinder of revolution. 
 
140 ROYAL SOCIETY OF CANADA 
 
 Vectors: Projection of a vector on an axis; linear moment with 
 respect to a point; moment with respect to an axis. Geometric sum of 
 a system of vectors; resultant moment with respect to a point. Sum 
 of the moments with prospect to an axis. Application to a couple of 
 vectors. 
 
 Descriptive Geometry: Rabatting. Change of plane of Projection; 
 rotation about an axis perpendicular to a plane of projection. Appli- 
 cation to distances and angles; distance between two points, between a 
 point and a line, between a point and a plane; the shortest distance be- 
 tween two lines of which one is vertical or at right angles to the plane, or of 
 two lines parallel to the same plane of projection; common perpendicular 
 to these lines. Angle between two lines; angle between a line and a 
 plane; angle between two planes. 
 
 Kinematics: Units of length and of time. Motion. Relative 
 motion. Trajectory of a point. Examples of motion. Rectilinear 
 motion; uniform motion; velocity, its representation by a vector. 
 Varied motion, mean velocity; velocity at a given instant, its repre- 
 sentation by a vector; mean acceleration; acceleration at a given 
 instant; its representation by a vector. Uniformly varied movement. 
 Curvilinear motion. Mean velocity, velocity at a given instant 
 defined as vectors. Algebraic value of velocity. Hodograph. Accele- 
 ration. Uniform circular motion, angular velocity; projection on a 
 diameter. Simple oscillation in a line. Change of the system of com- 
 parison. Resultant of velocities. Examples and applications. (Purely 
 geometrical applications are not to be insisted upon). Geometrical 
 study of the helix. Helicoidal motion of a body. Screw and nut. 
 
 Dynamics: Work of a force applied to a material point. Unit of 
 work. Work of a constant force, of a variable force. Elementary 
 work, total work. Graphical evaluation. Work of the resultant of 
 several forces. Theorem of forces acting on a material point. Simple 
 examples. 
 
 Cosmography: Moon. Apparent proper motion on the celestial 
 sphere. Phases. Rotation. Variation of the apparent diameter. 
 Eclipses of the moon and of the sun. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 141 
 
 PART II. 
 
 EXAMINATIONS IN THE CONCOURS FOR 
 
 1909. 
 
 (i) WRITTEN. 
 Mathematiques ^L^MENTAIRES."* 
 
 [Time, 7 hours; 7 a.m. -2 p.m]. 
 
 Given two circles, with centres O and 0\ radii R and R' ; these 
 circles are exterior to one another, and the common exterior tang^ents 
 are drawn, the points of contact being* A and A' y B and B\ whilst the 
 points of contact of the common interior tangents are C and C\ D and 
 D' y the points A and C being on either side of the line of centres where- 
 as the contrary takes place for the points A' and C if we have, as is 
 supposed, R<R'. The tangents A A' and CC cut in , the tangents 
 BB' and DD' cut in ^F, and the line EJ^ meets the line OO' in the point 
 G; the tangents A A' and >D' cut in /, the tangents BB' and CC cut 
 iny and the line 1/ meets the line OO' in the point K. Consider the 
 lines ACy BD and A'C ^ B'D'y which cross at the point K. 
 
 V . In order that the lines AC and B'D' become the coincident 
 line r, in which case the lines BD and A'C become the same line Sy it 
 is necessary and sufficient that the orthoptic circles of the two given 
 circles are orthogonal, which is equivalent to the metric relation 
 
 (The orthoptic circle of a circle is the circle which is the locus of points 
 from which one sees the given circle under a right angle). The point 
 G is then the middle of the segment OO' The preceding condition is 
 supposed fulfilled in all which follows. 
 
 2. If i? is a point of the line ;', the polars of this point with respect 
 to the two circles O and O' cut in a point S situated on the line ^ ; 
 
 3. The envelope of the line RS is a conic, which is to be de- 
 termined by metrical elements ; determine the principal tangents. The 
 locus of the orthocentre P of the triangle ORS is a conic, of which 
 it is required to find some remarkable points ; same question for the 
 triangle O'RS. 
 
 4. Suppose RMy RN and RM'y RN' the tangents drawn from a 
 point R of the line r to the two circles O and O' ; the plane being 
 oriented in the sense ABCD, let a, p and y, 8 be the angles, made with 
 an axis r by the half-lines of the tangents, situated on the same side 
 of the line r for each of the circles O and O' (these angles are found 
 again at O and O') ; setting 
 
 a-B Y + 8 B-a 8-y 
 
 ^- = ^' -^ =^', -^ = "' -^ = ^- 
 
 *See for solutions to questions in this paper Nouvelles Annales de Mathematiques 
 (4) IX, 455-67, 1909. 
 
142 ROYAL SOCIETY OF CANADA 
 
 Mechanics. 
 
 [Time, 7 hours ; 7 a.m. -2 p.m.] 
 
 A kite of weight P is subject to normal action by the wind, repre- 
 
 3 73" 
 sented by a force - P. In its position of equilibrium it is inclined 
 
 at an angle of 30" to the horizon ; it has an axis of symmetry on which 
 is its centre of gravity G and the centre O of the push of the wind ; O 
 is above G and OG equals 4 centimetres. 
 
 At a point A of the axis, below G, 40 centimetres, is attached a 
 string of length /; two other strings of length /' are attached in two 
 points B and C symmetrical with respect to the axis, the line BC^ equal 
 to Id is 29 centimetres above G. In the position of equilibrium these 
 three strings, flexible, inextensible and without mass are tight and 
 united in a point M at which is attached the string which holds the kite. 
 
 1. Find the relation which connects /, /' and d\ supposing these 
 lengths known calculate the tensions of the three strings. 
 
 2. The point M being 30 metres above the earth's surface, what 
 is the tension of the other extremity E of the string supposed fixed on 
 the earth, flexible, inextensible and of weight p per unit of length ; 
 determine p such that the tangent at E is horizontal (action of wind on 
 the string is to be neglected). 
 
 3. Under these conditions, suppose that the string, lengthened 
 from E^ unroll with friction of coefficient f along a helix traced on a 
 fixed cylinder of revolution, of which the axis is perpendicular to the 
 plane of the string, the radius of the cylinder being r and the pitch of 
 the helix h ; what will be the necessary force to maintain equilibrium, 
 this force being applied at the new free extremity of the string supposed 
 unrolled for a complete spiral ? (The weight of the part unrolled is to 
 be neglected). 
 
 4. The string holding the kite having the form found above (2) 
 and being supposed indeformable, place at the extremity situated on 
 the earth a runner \postillon\ subject to a force, the resultant of the 
 weight of the runner and of the action of the wind ; this force is constant 
 and is in the plane of the string; what condition must be fulfilled that 
 the runner move, supposing that there is a coefficient of friction 1 ? 
 
 Study the movement of the runner in the case where the force is 
 horizontal. (It is supposed that the runner is a material point moving 
 with coefficient of friction 1 on the material curve represented by the 
 string which is supposed indeformable. 
 
 6 July. 
 
[abchibald] mathematical INSTRUCTION IN FRANCE 143 
 
 Differential and Integral Calculus. 
 
 [Time, 7 hours ; 7 a.m. -2 p.m.] 
 
 Ox, OVf Oss being" three g-iven rectangular axes, consider a surface 
 S, of a single sheet. Suppose .y any portion of S, without any common 
 point with Oz and not having" a tangent plane parallel to Oz. 
 
 I. Suppose A the area of the projection of .y on the plane of xy ; 
 B the volume bounded by the area s, its projection A and the projecting 
 cylinder ; C, the volume bounded by the area 5" and the cone having 
 this area for base the origin for vertex ; D, the volume bounded by the 
 area .$ and by the conicoid which has the contour of s for directrix, Oz 
 for axis and xOy for director plane. 
 
 The quantities B^ C, D representing the volumes in question with 
 suitable signs, show that 
 
 (1) ZC=B-'2D 
 
 as long as the area .$ is not cut by certain lines situated on S. Show 
 also that the formula is still true without this last restriction, if the 
 elements of j5, in magnitude and sign, be always such that 
 
 B =ffz dx dy 
 (the double integral being" applied to the area A), and if at the same 
 time the elements of volumes C, Z>, are also affected by suitable signs 
 
 ("hz 'bz\ 
 
 p= --, ^= ^ I- Indicate as far as possible 
 
 the gfeometrical conventions of sign to which we are thus led. 
 
 II. The cone (supposed reduced to a single nappe) which bounds 
 the volume C, determines, on the cylinders of revolution of radius 1 
 which has Oz for axis, an algebraic area of which the elements will be 
 affected by the same signs as the corresponding" elements of C, in con- 
 formity to the preceding conventions : suppose E this area. 
 
 On the other hand turn s about Oz and designate by F the volume 
 of revolution thus generated ; by G, the area of the meridian section of 
 this volume, an element of /^ or of G being equally affected by a sign 
 (the same in the two cases) according to suitable convention. 
 
 Determine the surface S such that, for every portion s (without 
 point common with Oz or tangent plane parallel to Oz) taken on this 
 surface, we have the relation 
 
 (2) aA + bB+^cC+eE+ {- F-gG = 
 
 where a, b, c, e,/, g are constants. Show that S will verify a certain 
 partial differential equation of the first order of which the coefficients 
 are rational functions oi x^ y^ Zy p, where p= \lx^ +y^ (the radical being 
 taken as positive). Indicate (again geometrically) the determination 
 of the common sign to give to any element of F and to the corresponding 
 element of G such that this equation is the same for the whole surface 
 under consideration. 
 
144 ROYAL SOCIETY OF CANADA 
 
 Volumes of parallelopipeds and prisms ; of the pyramid. Un- 
 necessary to consider truncated pyramid or prism. 
 
 Notion of the derivative. Geometrical interpretation. Applications. 
 
 Homothetic polyhedra. Similar polyhedra. (Programme of the 
 Premiere). 
 
 Problems of the second degree. 
 
 First lesson on regular polygons. 
 
 Work, kinetic energy for a particle. Simple examples. 
 
 Decimal numbers. Operations. Calculation of a quotient to a 
 given decimal approximation. 
 
 Resolution of triangles (omitting right angled triangles). 
 
 Multiplication and division of arcs by 2. 
 
 Summary of notions on the symmetry of the cube and of the regular 
 octahedron. 
 
 Rabatting. Applications ; angle between two lines, a line and a 
 plane, two planes. 
 
 Calculation of 11. 
 
 Upper limit of absolute error of a sum, of a difference, of a product 
 of two factors, of a quotient, of a square root. 
 
 Volume of a sphere. Spherical segment. 
 
 Tangent to a hyperbola. Asymptotes. Simple problems on 
 tangents. 
 
 Theory of moments with respect to, a point, an axis. 
 
 Math^matiques Sp^ciales. 
 
 Movement of a heavy particle on an inclined plane with and without 
 friction, the initial velocity being zero or directed along the line oi' 
 greatest inclination. 
 
 Power Series. Interval of convergence. Differentiation. In- 
 tegration. 
 
 Series of positive terms. Nature of the convergence or divergence 
 draw^n from the study of the expression 
 
 , V M , n^u, numerical examples. 
 
 Discussion of the commensurable roots of an equation with integral 
 coefficients. Examples. 
 
 Movement of a point attracted by a fixed centre of force in the ratio 
 of the inverse square of the distance. 
 
 Functions of several independent variables. Partial derivatives. 
 Formula of finite increments ; derivatives of a compound function. 
 
 Concavity, convexity. Points of inflexion (rectangular coordinates). 
 
 Elimination of one unknown between two algebraic equations by 
 means of symmetric functions. 
 
 Normal to an ellipsoid. 
 
ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 145 
 
 Establish the relations, 
 
 cos^ u= (l -\ 3- J cos* Xy cos* 7; = f 1 H ^ J cos'jj/, 
 
 1+tan^ 7?'' 
 
 1 + tan jj/ i?* 
 
 Verify by means of these relations, that the tangents J^M^ RN and 
 RM\ RN' form a harmonic pencil. The point R of the line r can also 
 be replaced by a point S of the line s, 
 
 2 July. 
 
 Math^matiques Sp^ciales.* 
 
 [Time, 7 hours; 7 a.m. -2 p.m.] 
 
 Given a parabola (/*) and a line {D) of which the equations with 
 respect to a system of rectangular coordinate axes, are : 
 
 and, suppose the surface (^) generated by a variable line (A) which 
 meets (P) in .(4 and (Z>) in a point ^, such that the distance AB is a 
 constant /. 
 
 r. Construct the projection on the plane XOY oi a section of the 
 surface by a plane parallel to the plane XOY \ construct the tangent in 
 a point of this projection and show that the curve obtained can be re- 
 garded as the locus of the middle points of the chords parallel to OX and 
 limited, on the one hand by a parabola of vertex O and axis OX^ on 
 the other hand by an ellipse of which the axes are in the direction OX 
 and OY. 
 
 2. Two kinds of lines A can be distinguished, according as the 
 abcissa of A is superior or inferior to that of /? ; in the preceding 
 sections separate the arcs which correspond to the generatrices of the 
 one system or the other and find the locus of the points which limit 
 these arcs. 
 
 3. Consider the solid limited by the surface (5) and by the planes 
 z-\-a = ^ z -1a = ^ \ find its volume and construct its apparent contour 
 on the plane ZOX. 
 
 4. Determine the orthogonal trajectories of the lines (A)- Through 
 a point Ay two lines (A) can be drawn to meet an orthogonal trajectory 
 in two points C and C"; show that this trajectory can be chosen such 
 that the sum ^C+^C" is proportional to the abcissa of A Can the 
 given constants be chosen such that only one orthogonal trajectory 
 meets all the lines (A) between their points situated on the parabola (P) 
 and on the line {D)? 
 
 3 July. 
 
 *The solutions of the questions in this paper are g-iven \n Revue de Math^matiques 
 Sp^ciales ]mv\, 1910; X, 532-540. 
 
 Sec. III., 1910. 10. 
 
146 ROYAL SOCIETY OF CANADA 
 
 Find the characteristic curves of the partial differential equation thus 
 obtained, by employing" the semipolar coordinates p, o> (polar coordinates 
 of the projection of the point on the plane xOy), z (coordinate of the 
 point). 
 
 Study the projections of these characteristic curves on the plane 
 xOy. Show that there exist characteristic curves which are situated on. 
 a cylinder of revolution with the axis Os, and discuss their form. 
 
 III. Suppose the constants 6, c, bound by the relation 
 
 (3) ^ + 3c = 
 
 and consider the curvilinear integral, 
 
 /= J [s ( c + ^,) - I] (xiy-ydx) p (1/+^) 8 
 
 taken from a point J/ to a point M' of the surface S, along the path L 
 situated entirely on this surface. Show that if S satisfies the condition 
 
 which has been imposed on it in the II. part, and if, under the sign , 
 
 the sign of the term in dz has been suitably chosen, the integral / does 
 not change its value, when M and M remaining fixed, we change in 
 a continuous manner on the surface, the line L traced between these 
 two points. 
 
 If, instead of the relation (3), the constants b, c are connected by 
 the relation 
 
 (4) ^ + 6c = 
 
 a property analogous to the preceding appertains to the integral 
 
 /= L ^^P ^^^ (xdy - ydx) - ^z^^-^ {^x'bx + yby) + P [e/p^a^r - (jc3v - y'bx)'\ 
 
 where P is a suitably chosen polynomial in p and c is one of the two 
 quantities + 1, - 1. 
 
 IV. Suppose further that the surface S contains a circumference of 
 which the plane passes through Oz and which has no point common 
 with Ojst, or with the cylinder of revolution considered above (end of 
 II. part). 
 
 On each of the characteristic curves for the different points of this 
 circumference take a finite arc, such that the portion S of 6" thus de- 
 termined does not contain any singularity. 
 
 Supposing given the value of the integral / (in the case of relation 
 (3)) or/ (in the case of the relation (4)), the length of a certain path L 
 joining M and M' and "situated on S, what are the other values that this 
 integral can acquire when L is scccessively replaced by all the other 
 paths which can be traced between the same points on ^ ? 
 
 Indicate the relation which exists between the radius of the circum- 
 ference the distance of its centre to Oz and the coefficients of equation (2)- 
 in order that the integral considered be unique under these conditions. 
 
 5 July.. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 147 
 
 FINAL EXAMINATIONS. 
 Numerical Calculation. 
 Consider the differential equation 
 
 Find the smallest value to give to m in order that the equation admits 
 a solution of which the representative curve, symmetric with respect to 
 Oy is tangent to Ox at the points A^ A' oi abscissae ^=+l,:v=-l, 
 the value y corresponding to, :v = being equal to 1 (point B) Determine 
 the points of inflection between A and A' of the representative curve of ^. 
 Find, as exactly as possible, the portion AB of this curve, the unit of 
 length being supposed equal to 40 divisions of the square employed. 
 
 Descriptive Geometry (Diagram).* 
 
 An equilateral hyperbolic paraboloid has for vertex the point 
 de cote 10 cm. and d' eloignement 10 cm. projected on the major axis of 
 the sheet; a principal parabola/* is horizontal and its focus de cote 
 10 cm. and d' eloignement 10 cm. is situated 1 cm. 5 m. to the right of 
 the vertex. 
 
 A second hyperbolic paraboloid has director plane, a plane of 
 profile; of rectilinear generativers there are: 1. the axis of the first 
 paraboloid; 2. a horizontal de cote 13 cm., of which the projection on 
 the plane of the parabola P meets the axis of this parabola 3 cm. to the 
 left of the vertex and the tangent at the vertex 3 cm in front of the 
 vertex. 
 
 Consider, on the one part, the region A of the space limited by the 
 first paraboloid and which corresponds to the interior of the parabola/*; 
 on the other part, the region B of the space limited by the second 
 paraboloid and which corresponds to the part of the horizontal plane 
 de cote 10 cm. situated in front of the axis of the first paraboloid. 
 
 Represent the solid bounded by the two paraboloids, by the hori- 
 zontal planes de cote 17 cm. and 2 cm. and by the plane of the profile 
 situated 10 cm. to the right of the vertex of the first paraboloid, the 
 solid part being always in the regions A^ B. 
 
 It is supposed that the planes of projection are transparent. 
 
 (2) ORAL. 
 Math^matiques 6l6mentaires. 
 Supplementary trihedral angles. Applications. 
 
 Symmetry with respect to a line, a point, a plane. (Programme of 
 the Premiere). 
 
 Relations between the coefficients and the roots of the equation of 
 the second degree. Applications. 
 
 * A solution of the problem in this paper is given in Revue de Math^ matiques 
 Sp^ciales Nov. 1910, XI, 4-2-45. 
 
148 ROYAL SOCIETY OF CANADA 
 
 Tangent at a point of a curve of which the coordinates are rational 
 functions of a parameter. Points of inflection. Singular points at a 
 finite distance. 
 
 Small oscillations of a simple pendulum without friction (iso- 
 chronism). 
 
 Intersection of a surface of revolution and of a cone. 
 
 Theory of envelopes in Plane Geometry. 
 
 Conjugate points in connection with a surface of the second order. 
 Conjugate planes. Pole and polar planes. Conjugate lines. 
 
 Symmetric and rational functions of the roots of an algebraic 
 equation. 
 
 Construction of a curve p =y(a)) in polar coords. (It is supposed 
 that lessons on tangents and asymptotes have been given). 
 
 Gauche curves. Tangents. Osculating plane. Curvature. Appli- 
 cation to the circular helix. 
 
 (1 \^ 
 1 H I . 
 m/ 
 
 Field line of force, function of force, surface de niveau. Theory 
 of kinetic energy at a point. 
 
 Theorem of Descartes. 
 
 Movement of a point under the action of a force issuing from** a 
 fixed centre and proportional to the distance. 
 
 Members of the Jury. 
 
 NiEWENGLOWSKi, Inspector General of Instruction President. 
 Hadamard, Professor^ University of Paris, 
 CoMBETTE, Inspector General of Public Instruction. 
 FoNTEN^, Inspector of the Academic. 
 Gr6vy, Professor^ Lycee Sai^it- Louis. 
 
[ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 149 
 
 PART III. 
 
 The Agreges des Sciences Mathematiques 1885-1909. 
 
 
 Number 
 
 Number 
 
 
 Number 
 
 
 Number 
 
 
 of 
 
 presenting 
 
 Number 
 
 of the 
 
 
 of 
 
 
 Agreges 
 
 themselves 
 
 admitted 
 
 Agr^g^s 
 
 
 Agr^g^s 
 
 Year 
 
 fixed by 
 
 at 
 
 to 
 
 who be- 
 
 Average Age 
 
 who had 
 
 
 Govt. 
 
 Concours 
 
 Oral 
 
 came 
 Doctors * 
 
 
 been 
 , El^ves 
 
 E.N.S. 
 
 1909 
 
 14 
 
 81 
 
 27 
 
 
 
 24+ [21-27] 
 
 7 
 
 1908 
 
 13 
 
 75 
 
 20 
 
 
 
 25 [22-35] 
 
 9 
 
 1907 
 
 14 
 
 54 
 
 25 
 
 
 
 27 [23-37] 
 
 7 
 
 1906 
 
 14 
 
 58 
 
 24 
 
 1 
 
 27+ [23-37] 
 
 7 
 
 1905 
 
 14 
 
 60 
 
 20 
 
 2 
 
 27 [21-33] 
 
 7 
 
 1904 
 
 14 
 
 72 
 
 26 
 
 4 
 
 28 [22-33] 
 
 10 
 
 1903 
 
 12 
 
 78 
 
 25 
 
 1 
 
 28+ [24-41] 
 
 8 
 
 1902 
 
 12 
 
 87 
 
 23 
 
 1 
 
 27 ]22-40] 
 
 9 
 
 1901 
 
 10 
 
 77 
 
 23 
 
 2 
 
 28 [23-37] 
 
 6 
 
 1900 
 
 8 
 
 76 
 
 20 
 
 
 
 .28 [23-34] 
 
 4 
 
 1899 
 
 8 
 
 86 
 
 16 
 
 1 
 
 25 [23-33] 
 
 6 
 
 1898 
 
 8 
 
 96 
 
 19 
 
 2 
 
 26 [21-33] 
 
 8 
 
 1897 
 
 7 
 
 93 
 
 18 
 
 3 
 
 25 [21-33] 
 
 3 
 
 1896 
 
 12 
 
 112 
 
 26 
 
 1 
 
 26+ [23-30] 
 
 6 
 
 1895 
 
 14 
 
 125 
 
 24 
 
 
 
 24 [20-35] 
 
 8 
 
 1894 
 
 11 
 
 126 
 
 20 
 
 2 
 
 26 [21-37] 
 
 7 
 
 1893 
 
 13 
 
 134 
 
 
 1 
 
 26 [22-29] 
 
 9 
 
 1892 
 
 12 
 
 125 
 
 19 
 
 3 
 
 24+ [21-27] 
 
 7 
 
 1891 
 
 13 
 
 
 22 
 
 5 
 
 26 [22-30] 
 
 7 
 
 1890 
 
 12 
 
 
 
 16 
 
 1 
 
 26 [23-30] 
 
 4 
 
 1889 
 
 13 
 
 
 
 
 5 
 
 23 [21-32] 
 
 7 
 
 1888 
 
 14 
 
 
 
 23 
 
 3 
 
 25+ [21-34] 
 
 9 
 
 1887 
 
 13 
 
 
 
 15 
 
 5 
 
 23 [21-36] 
 
 10 
 
 1886 
 
 13 
 
 
 
 15 
 
 6 
 
 23+ [20-31] 
 
 7 
 
 1885 
 
 12 
 
 
 
 22 
 
 2 
 
 27 [20-44] 
 
 6 
 
 Total.... 
 
 300 
 
 
 
 51 
 
 25i 
 
 178 
 
 ^ These figures were compiled in February, 1910. 
 
 The sixth column contains the average age; "24 + " means an age > 24 and 
 < 24J, "24 "is short for a number < 24 and > 23. The figures in square 
 brackets, [ ], show the range of ages for the year. 
 
150 
 
 ROYAL SOCIETY OF CANADA 
 
 APPENDIX B. 
 
 Mathematical Courses Offered in Universities Outside of Paris 
 
 1909-10. 
 
 There is nothing in France corresponding to the Universitdts- 
 Kalendar of Germany, and it is almost impossible to get any exact in- 
 formation about courses to be offered at even the Uuiversite de Paris, 
 until a couple of days before the Semester commences. The following 
 list is compiled from a variety of sources. It will be observed that 
 Lj^ons is the only university outside of Paris where any courses, over 
 and above those for the licence and agregation, are offered. 
 
 The letters in brackets after the names of the Academies, indicate 
 the Faculties of the Universities: La. = Law, S. = Science, Le. = 
 Letters, M. = Medicine. The numbers in brackets after the names of 
 Professors, are those in the list of doctors (Appendix C). An "A" 
 added in the brackets is an abbreviation for agrege. 
 
 No information is at hand regarding the Professors in the Uni- 
 versite d' Alger which was opened at the beginning of this year, with 
 the Faculties of Science and of Letters, and the mixed Faculty of 
 Medicine and Pharmacy. 
 
 Ajx Marseille (La. S. Le. M.) 
 Sauvage (142. A) 
 
 Charve (137) 
 Bourget (A) 
 Jamet 
 
 BESANgoN (S. Le.) 
 Lebeuf (231) 
 Carrus (272, A) 
 Andrade (181) 
 
 Franchebois 
 
 Bordeaux (La. S. Le. M.) 
 Cousin (209, A) 
 Delassus (217, A) 
 
 Picart (188, A) 
 Esdangon (262, A) 
 
 1. Calculus. 
 
 2. " Cours Compl^mentaire " 
 Mechanics. 
 
 Astronomy. 
 
 Cours Complement aire with Sauvage. 
 
 Astronomy. 
 Calculus. 
 
 1. Mechanics. 
 
 2. Cours Complementaire for En- 
 gineers. 
 
 Pr^parateur in Mechanics. 
 
 Calculus. 
 
 1. Mechanics. 
 
 2. Preparatory Mathematics. 
 Astronomy. 
 
 Prof. Adjoint and Maitre de Con- 
 ferences. 
 
{ARCHIBALD] MATHEMATICAL INSTRUCTION IN FRANCE 
 
 151 
 
 Caen (La. S. Le.) 
 Riquier (161, A) 
 Husson (268, A). 
 
 Villat 
 
 Calculus. 
 Mechanics. 
 
 Maitre de Conferences. 
 
 Clermont (Sc, Le.) 
 Pellet (122) 
 Guichard (151, A) 
 
 Dijon (La. Sc. Le.) 
 Baire (238, A) 
 Dwport (135, A) 
 
 Grenoble (La. S. Le.) 
 Collet (107) 
 
 Cotton (242, A) 
 Zoretti (264, A) 
 
 Lille (La. S. Le. M.) 
 Demartres (156, A) 
 Petot (171, A) 
 Clairin (253, A) 
 Boulanger (230, A) 
 
 Traynard (278, A) 
 
 Lyon (La. S. Le. M.) 
 Andre (117) 
 
 Flamme (170) 
 
 Vessiot (192, A) 
 
 Calculus. 
 
 1. Mechanics. 
 
 2. Astronomy. 
 
 Calculus. 
 
 1. Mechanics. 
 
 2. Astronomy. 
 
 1. Analysis. 
 
 2. Astronomy and Geodesy. 
 Mechanics. 
 
 Maitre de Conferences. Cours Com- 
 plementaires : 
 
 1. Analyse Superieure. 
 
 2. Math, centrales. 
 
 Calculus. 
 Mechanics. 
 
 Mathemetiques Centrales. 
 Prof. Adjoint and Maitre de Con- 
 ferences: Mechanics. 
 
 Maitre de Conferences. 
 
 1. Astronomy. 
 
 2. Elementary Mathematics (Con- 
 ference d'Agregation.) 
 
 1. Mechanics. 
 
 2. Math. Generales. 
 
 3. Mechanics (Conf. d' Agregation) . 
 
 1 . Differential Equations and Calcu- 
 lus of Variations. 
 
 2. Theory of Groups of Transforma- 
 tions. 
 
 3. Math. Generales: Algebra and 
 Calculus. 
 
 4. Higher Geometry (Conf. d'Agr^- 
 gation) . 
 
152 
 
 ROYAL SOCIETY OF CANADA 
 
 Le Vavasseur (198, A) 
 
 Wiernsherger (312) 
 
 Merlin 
 
 1 . Theory of Functions of a Complex 
 Variable and Geometrical Applica- 
 tions of Analysis. 
 
 2. Mathematiques Special (Conf. 
 d' Agregation) . 
 
 1. Mechanics. 
 
 2. Math. Generales: Analytical Geo- 
 metry. 
 
 Charg^ de Cours. Astronomy. 
 
 MoNTPELLiER (La. S. Le. M.) 
 Fabry (159, A) 
 Dautheville {157 , A) 
 
 Latt^s (274) 
 
 Calculus. 
 Mechanics. 
 
 Maitre de Conferences. 
 
 Nancy (La. S. Le. M.) 
 Floquet (126, A) 
 
 Vogt (177,A) 
 Hahn 
 
 1. Analysis. 
 
 2. Calculus. 
 Applied Mathematics. 
 
 Maitre de Conferences. Mechanics. 
 
 Poitiers (La. S. Le.) 
 Lesbesgue (256, A) 
 
 Boutroux (259) 
 
 1. Calculus. 
 
 2. Math. Generales. (Cours Com- 
 plementaires) 
 
 1. Mechanics. 
 
 2. Astronomy. 
 
 Rennes (La. S. Le.) 
 Lacour (215) 
 LeRoux (216, A) 
 
 Frechet (273, A) 
 
 Analysis. 
 Mechanics. 
 
 Maitre de Conferences. 
 
 Toulouse (La. S. Le. M.) 
 Drach (236) 
 Paraf (193) 
 Cosserat (176, A) 
 Buhl (248) 
 
 Blondel (A) 
 Saint-Blancat (276) 
 
 Calculus. 
 Mechanics 
 Astronomy. 
 Math. Generale. 
 
 Charge de Conferences. 
 Assistant Astronomer. 
 
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 ve/yne^ nzLxiif fiur -tcLpJuydi puc^'lMZdnZ' [T] et d'^on^'hn-ouA>4^^^^^J:^d't'nJk^sum^ 
 
 
 lo/yieAS^ 
 
 <djXyytala/nte^ 
 zf^efuide/r' 
 
 a icL dtiiU- 2cd coynjjo&iKond Icxitid , Onb etc- ^cdaJuJ a2hrdd&iX>u^ ouOL. atituu^e^ oxdUi [ otdve^ 
 aljiHcwUjM'Ut-) : 
 
 (SoLcLCcVcct^ 
 
 ^^TidAux 
 
 
 JixduenJz^ 
 %-ttA.dceke..^ 
 
 'yJiduJiri^a2'nxCrUM?axt^ G ^LoAt- ^C^iO . 
 
 diV. hnivu . Jle^didUfTT^dej Zita^riaiL) ai-idcif-na-ueJ d "^^ /^^ /ai/tc.'^ -liLcurhOiLed ^tcda^lis 
 
5- 
 
 dccf-cLtti . ( 07lcdjLlU^t^Jxm4M) -A. J ^ 
 'L^ Cjuxtdety . ^iu/me. de>J pxai^xIUZuz^j-ude^ ^ dej jdxiS'ynt^d ( JIc- jtcui jolucl- w vvtm^ydt- pwio/nUcU^ 
 ^ ie. iko-rtc dt^ pjUdrrvi^ J ( &Icl64<^ 3e toremiiw-j > 
 
 7L. UOUVlOrL- . JiyiatdjtM ivOTrw^Aitiau^ . Ooujcdui (k^rribiccuLti . ( J^ccMta^tuynC' 3*- Aixc/^niekz^ . 
 
 It 'Jdnaici'hne^ , dieloduyrL. t/nhu^ U.^ c^^eMCcitAxMl a6 itd icuU^yvtd de, 'C'^cj'UcdCon-^diA-' JcccmxL de^ti^ . ajtpucahiyni . 
 
 JV. CjcuiAcidc) . ole-ccrt^ dU'i' jUdi. juir , JJt. GcMu. 
 
 TL. O^Xl&iiJt/r , jhrnfyrude^ tec di^}^^,dia^yujicoUX^>^^ de- tcud&'UH^Jc- . - OUtpu iycda>rti ci "uc- -Oit^ 
 
 Ti . oodiWC^ . Ouv'TTicihiue^ e/n^ qv^nebca.--ypid/ya^ . 
 
 iL. CAorujo-Cvz , Uec^rt^ de^a^^UJZ futr jTL. (Sutccdet^ . 
 
 71. cOelmcU . diif^ni^ dAd^ jccdi- jtor 0Jt. 3zAx<yru . 
 
 71. AconeJC . c^tuyrt- dlAd^Zxitt -p^t/f dU, . Oyodie/r, 
 
 71 . Jvu/t'TLvvcti. decoTL- ddd^ MxdL Aiwi^ JiL, <J}t^de^ . 
 
 71. %'WLnAJ)(JL-, <:XeyCoru d^'d^jcLdL-HcfAr 3TL. B'oa^u^ , 
 
 71.. uixUj-yyurru) . CHMl/cdie^^^te^nM . d^J^v-rt- oleJ (t- mML -jt-ct/r dTt. OpUiVta-rt^ . 
 
 TL, (Sda- . dzuyrt- dUd-jdutt jtcu- Sit. Sefdu^ . 
 
 TL )lc|'ii^.cJcer'. Slfiou^eyrrte/nt de^ -Uu Euau.- . y^tcuc^ . 
 
 71-. UXcu/KCZdd-i^/fil-. JkidDU^nvU da. iieiuyyvd dx-cUU- [ dYLai^t^yyvaKotx^cd A j - 
 
 ^iL, Jl-tB^^E^ . JeldeJ -OiiUiudZorLS cL' u^^ HoytdUue^ 4a/rU J^-iZM'TUynt-: JovdiruyruMrxt' 
 
 '7V. XyctidMjt/r' . i^if/nuthdid t^n. 0(><ndcmyrLkc() j-vcdcL.Oi^ ; Hai^olixyn^ dz- id Co-tvd>y jvcvr -tcChj-toxt. U- ^>^ CLdtfmjs . 
 
 tern . 
 
 7J7. djdcvtt^ . 6n.i<ycu. deJ en^^dxipAve^ e/n- aioyndk-ce^ jdo/nz^ . 
 
 TL. (ho^wr. jlon/pncUe^ a, t'eliijv&Dldz^- 
 
 f(L- UfOrWriU^ . Lywudvu- dcd cocvdie^ jiMX/m^ . U^eveu'-HHtc^ . ca^cmj-MJ . ( CoindvrMiie^ i.uuu.a^nt^ ] . 
 vL. cfio/U^-vvcn^ > du^iu>e/yri,e'ntd'u/>i~/rotritj<'iijt'acho^t'd'-^^^ d HAy^ri^ cto-zbu^ p^Kt^ et p/U>^ 
 
 diori'ndjd- u- vx- didccnce.' . 
 
 -pdhi-ic i/rvdudt- edd/TTJc ncdHt. otu dixlaee^ ;)-LcU>cx.rd:, u/ru. 'Uun^ dz. hIlU ou^aijorucd fwruJi' . 
 yJt. (J)i2ma^ . dhyhnltc e 
 
 Jc^ruJe^ de^ ( 'f 4- dl. )***" . 
 
 y>L. jO/yitC . JetieJ ci tvt/yyud j^o^lbij^ . - dvicaJi^u^ cU C-O^^e^rcje/nu^ ejt cIl dcve/i^e/ncz. Uu^ de^ c'^i^da^ 
 deJ eaUTjzz6<s-uynJ : 
 
 IU^1 ^/u~" ^ ^^ 
 
JTLt^^^mc^Ka'^. <Sex^cm^ cUia^ faltl^ jtO/T 371. OyocL'^r~ , 
 jJl . u-ovjsX/ . ole^i^ cU^cL fctitt pxi4- Jll . (f^a^nutd . 
 
 *-^7L. J^aiyii/- . -^DTvte^uie^.ti.ifn- de^ de^a:^ <>ujdfccuJ aU^ >iA)<H,i^ui<nt^ dc>rdi te^ txxeJ <iO'yub da/nS 
 
 ->nM:hcaj, iZ' -yuynt^btjey de4 cJtUMuJ Jia/jvijt'cJxhM) cle^X qtc 'eluJ cuin.^>rtt fte^c/ntiJ d^- dh^i'nUni/f^ 
 cwec^ CMdUlicU^ . 
 
 ^Ifvey dxaili^ de^-pcnJ:^ D eA^ vn'clnez. ci IiS" Jur u^fLUX/n^ hvoti^OTd^oU- et otu d'ele^^-e^ ifnA' 
 cuth dM>iu^, e^Z/ uv -ficuL-cvvULcvnyh cZz^ dju)d^ d- <^ccucHy } cu^ fuxJ^^^p-o/r uz^ jtoi/H^ du^ pxa/n- 
 ^7urUai>rvtciZ' de^ pA<}-i'edii>rL^ xtihctL- d, ^^D'^/tn^ d^ djivilt' du^ a^tcon^d oux^ ^ e^ di g^*^Jfn- om^^ 
 aS4<k>iAJ du^heu-t oc^xe^ de. -Ux^jLuUie^ . (?eUe^ dxodt- e^yt ^te^c^ondk^e^-nc^r la^ jia/uth-viL T e/rui 
 -Hcyvrvt ^dxio^ d dMjyiU^ du- -judri^ dt, jtXirhdL cj-uU- cardceyfvh -uz^ OMcurid oujce^ de, icu jeuCLle^ . 
 
 Ufu CATnJi^dd^U-- U' d^^c^xe^ jtic/uxl^tuHU^ L^^ -Hotr iCL^LC- dz- i.cL jtcoudoie^ I aui^ Co^ - 
 
cU'Mxi/ruie^' du X-fVtM^HJje4'' en^ JT/cOpctwyv- i^eyvuccuk^ (ie^Uent.e'nJC' ve^ ifvUoU^ e^ae^rwUe^ har Lo(^ 
 
 M.M . V 
 
 . Mciext 
 
 M.m^. C, . gcmzt 
 
 I 
 J 
 
 s 
 
 J 
 
 s 
 
 
 yfO . J!eicvu^ 
 A] QcdecuMCy 
 It. Mu^ctt 
 
 ^ro^amj^tt. fbow^ie Lo^^cMVc^ oli^'19'ii . 
 Cornj^A)<)iJXori6 iortttd 
 
 LMytcut AjJfvtcn^tUt el CoIcaaJL inlicyral . 
 
 auo/ruL U/Tix, ti/f-nde^ cni- M^ jfo-rtydA4>rU-44A}tem^ <^rifi^^ doi^trUiJaL. de^ Qtee^fL- . 
 CiXult^- dx<) j<nvc^tuyriA ^eAD-^jM-yde^ jta/ir dtd ^JjiyCcd . 
 
olio/n^ C3iymuai4ji&i , -Uans^ ae. couA^u^ut' f^uattid oAu^rtHbonAcj-uz^ e^t^ ci>v%olo7Vfvied cuAyOiuamA cfuHomcfuei . 
 'fe coLi ^Hc u^ acnpnu^ dont amalutZcnicd . 
 
 Mc, 
 
 UUhL. 
 
 CMJ^ 
 
 ^11 J toXurji^.^^. 
 
 cUd jo^ce^ applCaueAd d tuv c<yipj dotide^ , .^ UfncUuorLtf d-' eauiucit. d <^ C(ydii ^oUpU, . LXjTjJUccdipny ouccc^ 
 UoVuaAyU. fu/yvCc.ulciAAL- - \f<mh (^uMte^ndu^ . ^utoneMi.. OnjL>%e.yt^j^ d^ -vuilvcuZ- 'S>iAdluL . 
 
 ^iUfi-^-c- . LXcceleA.cdi<yrv- 
 t/lLou4>t,f'riA^^ d'u/TU- jiau/tJ- j-vUt^ru- dci/n4 ^^yn^ jUcori^ . Jl&jv^^e/nhiulo-n^ dec '>ru>uA>-eym^nA:^ jzd/r iL- 
 
 JTuHiVe^ynervt d Wi^ cctM <Mldc cuddxiA" d '-u/n' -p^n^ juce^ . Cn^ft/ied-e^f-ubdu^ri^ du mtmA?^^^rL-e'n4^ iiO^ 
 
 J7l<>uA>e^me/rvk d'um. cotjui :K>udt- da^ri^ -{'cMutc^ . JlLpuA>R^yne/nJ:. -HeUcoldaL . 
 Jltotu>e4ne^nh vucddfo . Ohiiyti^*<z^ de. Soxlou^ . 
 
 Xshi/yu'tyied aeyrUA-cuuc^. 
 
 dnhianauJ Hnz/rriUA^ du iq-twdccrui du- 'yncn-u^t^yte^nJ^ SiHpZLCcduyyi-. cue- frvo-tuOB^nuemM^ dfi^ jvOx^mJiJ . 
 
9 
 
 (pn^v-iJ^rm^ ac/ntfLOuuac~. D'rtIe.'aa.<}Ued H/ve/yrvie^itd . Sm^icda. . <Jicu)iLU^ at- 1 tcj-uiulyvc . 
 UTi/HA^^e^rnzriC cL'^nn. cO-^ijtd i^otZae^ ommha/t d '-iAm^ oux^ R<co . Jkz<^&iond duJvfunte^ jicvr / 
 
 <^7louA>u^ne/n^ c^mXou^ cL'^i^i^ru e^)^^^ ^kmAe- > 
 
 LAWnUccuXo^rL clu^ ifre<yui^-tl^ cleJ joxc&d ptA^a^ ocu^t- 'yyutdfu/nx^ . 
 
 uTlctWPe/yrte/nh A.^uxJkXi . (Jt/uuj^i^m^ . 
 
 1G CpaiAxxhu>rL6 (uuvorUcj-vied . 
 
 C^cjuiuox^ d^'-Lvnt. 'TyLci^t^ HuuiiL . u-ivuvued de^ nM^cocju-. tJteJ-dum- <^ur um^ hcuzvi/ JMa/niy ^ 
 J%4yricUu-^ cl' cLn-ciU/Kne^^ - CH^uiyU>U^ clti aoxjtd puMoinU . 
 
 ^^a-vuxtUm^ Ai'n.ixcatS du- ^yyvo-Wi^cmeynt a'^uynje-- 'yn^^^e^ jhtcitde^ . 
 
 J/t 
 
 tyCyOruS . 
 
 o^ cvriddio4v i'teeeJ^JiXi-i^^-^ ikAl'ti&oontiy yfvou/r oi-uz. de4A/oc^ 1rwVu'n2>'yriM> f i^} ^ ^ (^) '^ dtq/d^ 
 -^itAjttdifa -iv et-a cus/nk- u/ru jvloiJi an^a/yuL corruma/n-' diA^dieu/r' dt^ d^.u^ -n- t^ aiA^ '-u. e-cCAjte^ 
 dd^iA^oc- -iMftwHC-h*^ A. et 3 dey d^^ated ^tJzMieddf^ -ft. n- -eZ a--n^ -ieZd cht-u 'cryv ouc^ ide'n.tiaiie^me^-, 
 
 mut- et (Hi^ndihicj-vLz. de^ AxicVrud o ' e^^n^UT^e^ 'Uihx>-vtyrve^ie^nt'nt <i^n^ ji>TictLOrL-Jji^ coeji-cCyierds . _ 
 Ou^fnHyncdzo-ri^ d -o/ne^ 'Vytcotvrui^s^ e^rd^ dM/x> zchtuxkicyn^ oUu -nuHte^tudci hyrudio-rui ^-a^fnidbUci-u^ . 
 
 L,X)ndddarv jum/}^ a-u '-u/n^ ea^^udzorL- cu^t deJ^ui-clne^ -coicuej . 
 Jiedk^AyChe^ de^ A^i^ci/mJ c^-nt^rne/HJ-ufuWjdd 
 
 (Dnicfre^yytL^ dz- uJt^Cct/d^Jt . , Jlaiid^'uJ corriMdeojLd . Jvi/yyutXe^ de^ JTLcHAnc^. 
 
-10 
 
 
 -rrU/niyyTva^: 
 
 e^ du^ jra^-cuUllfuftZdc^ jtci^ -tcUvtvcnt- d. ztu^vd aa^ de^ du/ntitkie^ . 
 
 uLUzi ct ikn!u/yrLe^ oleJ :^<d4Mj de^ -ia^ aict-rLeJkie^ id^e^trve/fvtcuicL-' 
 
 cJoruiCoru }-efi/ni(^ -pa/f -iMxi^ &tfilt. e^nluw (L co-eM^cie^nh nieJU 'On^tc^^>culc- dt^ 
 
 ^iddlUcnv et >r^AAiiij^ucclJM>rv- . uLl'^iAttex^eAMT dt ^'i/nte.tVPillt' dz- c^Hu^ti^e^Uy , -on. CfDUe-nt 
 nd- deAM>4^ ou^ -vtd -fOnchvrtd -jt'U'trdhA>ej de- -let. -uyricuon- ^au jvu/nci^n^::^ lou iic^ue^ de^ de^u^v^e^ ot^ 
 de^ fcnvchcrrU ji/u^yrvitii>^ ( on- ^ha. ^ '<>i:,c4Ajte^tcL -HcU dt^ ce^ ei^tu- <te- -jicOiie- ou^cc^ a^xlkli^t^Jje^ dz- 
 
 J-tklC- eyCoHortt/ntceMe. . Je^iit. du- i>i/rt^m- . ole^ la-Lutdiot-iJ 'H' - H eh u' [i ^ Xj ^ r>x\\ 
 hti/ynzJit^nt, di^ OJltt/i/rrvOrUA- itd -^cynyyru^ dzd dtu/3C- ffeWe^ , 
 
 Ojii}JUpH^Krvt/njb^e/n. ii-i^de- d^^ <^^ , ^ ouic^ i:>iYL. cCy - 
 
 JrJjt^/vcdlan- dzi taixciLtunU diis^fLZ/ntieJUcd dAA^ hJUAnie/r' ^xd/u^ -^"^ daynj -u^ cctd -otc -teJ 
 
 -'Jrdi^a^uiiicy-yi^ ds^ i' -iortusdi^^n- diju^t^rdluiL -u/ytSpUn^ da- d-u^ond ord/tz- d- coeMC(Ue^nAi 
 
 -^ornymz- d ' (ia>h/U4^M-C'n4 dt^ -tay jot^u- A V *'*' - 
 
 ChyiLryiJid424z4 , uijthL^cadiotu ci de4 i^xu^njt^ d-i^rvkleJ e^ en^-j^oLvdiuJu-e/r a. dzJ c^^n4^44^ eh 
 
 a. de4 oowtLe^ dcmJi' < ' exttiodicyTU- z^ d,u^ tieJ^^yrud^ dt^Ad^futr "UtpAvcnA;^ cl iy-t-^^nt- dtS cootdmunU-S . 
 
 Jlcbuyn^ -du.cxi'rvcdiii Ju/)' ltd jvounti d d^'-i^rdhynC out, -m-CMi^^^ oLu coxyi^don/n-te^ tumtvcimi4, 
 
 .^tcuvMf^ ct/nJuxn/yyuyrutftu^ dt- cji-udke ^ytom-ttre^ . iMUvEicducri'- ctu- >ux-hJw-th ^nJicM//ru>rvuiu^ >lc- CHUuic 
 pdi/nh btC 'i4^[ne cUoite ^dActuahu, dt4>ite4 exJi.hxvid:krid^ cM^^yriJ^^ -M^dctouc- u^yvtcu^cc . 
 
 fJCctMun^ A/nhut yrr<>n4^u&^ dc^ eju-dtt^^ H<nrd^ d^ -tdyytCy c^rua-ue^ cm- dz- a-uoUke^ zh/ytae'ridi^ 
 
-7/ 
 
 mxc- au^ ca^ -^ -/ ' ea-uculcyn^ eA^ ^ti<k>un-^ par 'utHHc'vt oui^ -rayi^n- iie-ueur-") . __ (ut^ cU^ 
 Jm4. 
 
 
 Lo'>rvj^od'iJZo^yu aj -iKce^ied . . Lc^yyvh^^ituyru^ cio^ dcc^eAyCUumJ "i^iyi^e^ cut- ccxJ oic 'i^ 
 
 Jtcdca 
 
 tu^. 
 
 tcchJunii 
 
 cv M/rt^ OLOU^ 
 
 Ci)ruiiticrnJ -j-i-c-c^d^ excited cL 'ax^iyUiCDic- -i^ndc.h-t'nAayfAe^ cLe^i ji:?^ceJ -vrvtejUzdMuJ O^e^n-xortbi^^ 
 
 Ck:j-uix>cuen4X^ cIl diAxco <Hi^i-ryiM> cle^ jint^ cdiftlt ofuJ^^ a. uax. c-c\H4 JixU'cic^ , 
 
 orUM^ubxfru dt ^iVifcuc^ : uuo3> cSneJ ocl oulirLciteJ^ cSiiz- ou. cqu^iwa^^JiMLJcu^ cUx^>clu^^ 
 dtucc- i^ivJMxU cL^ 'rnHnuhunu cUynt u^ cux^ ikmt douU u/ru ^ytiM^vC^ -ham-- 
 
-ft 
 
 
 CLlaevriy , 
 
 pJUtnie/T 
 
 * yteJ-ouii^ion^ aid ^c^t.uxZu>rt^ cLcjvce^ntur de^%e^ a -t^m^^ Utc<m'ru4Jt^ 
 
 Uxjl^'te^rtcz^ eJt ifiQ^tz3 cU4 ^'ca^u/ne^ ci Wz^ exxticUiori^ du- JecotuL cUd.tc- . 
 
 ^ doLticdlo'rv, dl^ t' ecch/u^iloru ; ^te:p^u^e/nJuh>pn--cm^tHJ^ 
 
 JUflLorv dt- ICL d^xi\>-c^ ; ^CouniH-udiott- Pte^orf^e^ciH^it^ de- icC' deA4A>e-' . -Ze- ift'on^ dt^ txu dzuAf-e^ 
 i''ru}Ucj<.te^ le- Jen6 da,- uu Vcc^iauorL- / 0LhjuUc^3Uu>rL' a- dt^ e^XJi^ripJ^j ^yia^nve^u^t^uJ -^buj if-i/nvftieJ 
 
 Qflorrietrio . 
 
 -^hyiu>e^ru^%t, cU, 'c.cd:cLticrrt,- ctutour- d'-uyn. -hoirvt:: [ da<n^ -ce- -hioi/ruj . X>out aitvuxce^^e^TJc^ 
 
 d -oiyrtey ^tk^X^nMcJiorL- . 
 
 jZoiuni^ -^nMeJ iytA/f t ' n4?Tn<ythiti<^ . Jvvuootu-^ i)Le/yn^tevbUd , 
 
 exjuiJlcdtn/xL , dcu duuacne,,^, du p.e'tJkAii-caa^>rLZ^ . (j^eua> jiA^cxo-n^ re^LuIu.'U d'tvru ^-mZme^ 
 Sity dc ooU^ i>-onJb ifLyyL&tcUfleJ . iJlcULH<yt^ dc tCHAn^ ^iKinvi-bcc^ , '^xnta^ueM/T d ' u,rr^ cvu> dt 
 
 dxi^ 'yrd^^odJ^, deJ hitimMked j . 
 
 LJMue^ de^ -4vovucu>->ue^ cU>vc- dcu ac%cie,- . JVleJ-t-w^^e- d^e- tcukC' dM- r^^J^ci^rtau^ ^ du. 'fict^txtt. 
 dU^<>cin.oi'fn/*tU^ -. diM, -Svuii/Hrile, , dAA,- ijvcvhlxe,- , d'-H^n- fuym-o^crru^ aueXcAy-na-uz^ - 
 
 'JlCchJvo'U:. deJ ctUe^ dt- d^Ui^ac HcvuoyO-yve^ <f^^rn^ta/vteJ 
 
 LAMiCy d' "U/t^ .p^xUjapyi^ <^<Ay^ie^ ixynAJ^ice^ . (Divu, d'^u^rv cc^d^ , d'-u/K' d-cdku/i^ ^ d'iMt^ i^e^- 
 ^yyvem^ dt,- ae^uii,- . jx-cipj^uyth dej ou^u<i oU- deH^i^oo C-e^octe^ . 
 
 TJ^trruZrv C -etJ) . 
 
 
 
U\:><y\^YU>VnAAJi^ Je^Jkxrrvd 'pMX'ne^ jtcvuulue6 i:i'a/nciU^ j^v^1Jld>U^ cA^Jud 
 Jlcnunvd ^HnnyyrwCui^ ^yu/r ltd du^mibUe^ cUu awe- eh ^ I'oetcu'cUz^ tiauMjeA" . 
 
 JloryLvt<> de.c4/yrwoua:^ . U-i%c/uxbcorTJ [ e^v aortdi^cUyvoi/n^ ie^ fuicchjorU du-Uncue^ tcrrvyn^ 
 CaJ jt^oociXiudu.e/r' cLe^ J^u)i<>bu>'yLd -uytdc^nctM^ J . cwuui d ' tA^'n- at^A>-Ue/t'vt ci- -u^yu. OLMT^u>au/yKvodj>cn^ 
 at.Ci/h>^U- do-yvrue^ . 
 
 xTLe^d'Ucaon-' d'u/hte^ j%,cutu>TU- eytdincu/iC- e/n^-yutcluyn^ dlci^yn-cil^ c^uiiytioru di^ fvc^^buibi . 
 ''cc-t^a'Ut- -tcu 'liducbuyn^Cdt- -UrUva^^ote^ ^ leu ytcudunv- cnd-Uicu^ve^ p^^uh elht^ -^t^outdJey ccm^ryta^ 
 d!ou ZUrvobt- d'-voyi^ -vvcLctuyru dJ'C'iyyvcu^ -ni^tlod^cfu^ -iXtumiite-^ ^ 
 
 LtviAjiy^ cL 'iAyyu n-co^n^t^ erute/T' (?u^ jnctcJ44>-ri''rtcutc- ; ct?>njuy>ltu>yu d^^ ux^tc de^ id i^<yryi^yn^ 
 
 (^tj-Crubuyn^ ct- c-ocb^LOttduyn^ d^ -loi^^vcLciyne caftA.^.^^ d' Hyru -yunruyCyt- ^e^^die^ <>u^ jftct-cbA^yn.^yvixlte' Ji 
 ^une. aHjTA.ooc4ATrcdii>rL. de^dmcud^ dcynyyUe^ . 
 
 (^ncoxt^-de- tcL t.Cixdnz^ CwcAdo dy'dyn- e^nbce^^ d u^ru^ <^^nLU- 
 
 p^tZd 
 
Ik 
 
 
 3 
 
 JlXuxUfUi^c^iiuyri^ eh di^HM<>ru jutr Z . 
 ute^^vtuJion^ deJ ttuuructud 
 iAjtfvUC'cdioruf de^ -ta^ Akiouyn^^ynitu^ cxu^ dive^tA&d cf-ue^^Uond '^uaJivcd au^ ive- de^ ^hicun^ . 
 
 ije4>ryv<ibcic^ . Jn^e^iM^rru . Ctn^ddcoduynS . ClHhcuLeAl de^ Jtcuccelueir' . 
 
 ct uru- 
 
 CaucJk'Oi^ de^ TCL' jicinaJ^d^ . 
 
 J&cbionA pZonz^ d 'lAAu cSn^ Pit d 'um- C44l4Atd^t^ de- ^tu'-imitUnv < 
 
 cJw-itctvc>Tl^ CC^nvicULC^ ^ia/fv dA^ -ujjoi^oui- . U^^t^-p^^dlM^' d'-U'^p4>i/ntf d'u/ne- dn^ouV^ 
 
 d' 444ve^ --tccm^ . tJoint dt^ jiodi^ d'^u^ne^ dn4>-lti^. Jl^t^adiA^-e- de deu^ dn4yiJx4 H<t'Lcul<Ue6 . olcafne^ 
 de. Uiih. d Wz^ -hMSLrv- , 
 
 L^cyyuC^oiicrL. dc- UV dnvitc dy t''i/yiJkmA^ d 'ti^n^-Hloun' . 
 
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 -fuH/nt . ^yyv(yyruo^ -HO/T -^cUUKycA:^ d^ -tAm- cuc. 
 
 CU 'U/ri^ 
 
 jvoi/nt . Jomi-m. de4 '^^^^yma/ncd jux/r -tcUihAytt/ a- -M/rv- cxax^ < CtpJ^^ccdicru- a. -u/n. cckip^ dfi^ i>ui/L6. 
 
 yuybtti(nt. cvudcu/r d'-iA/n^ a^u^ p^t^tHe/ndACAi.ioUre^ d- U/f^ pzo/rv- dt, yhn/o^ecCUyfu- , 
 
 CX'hpMC'oJxori, CXAA^ dA^%ci4^yC^ eh (xcuiL ctncue^ ; oLCytot/fvct' de^ Uu^^oty ruH/tuJ , d '^uru -fun/rt^ cL 
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 cvnate, dty Uau/X^ juccrU . 
 
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 <s>nu>teymje^ dc4 jo'u^ iM.^>^ jtimr u4%-p4>ln^ ^ncdi/Uel^ . 
 Cc^iMc^ de^ -luyn^ ^ de^ ^ined^ . 
 
 
 d-uU^ie^ du- Jvixtanci/mymC' jtcuyi' -2^ c{>ruou-^ d^^ T^'cxyrT/yue. ifu4Ji)a.'niS^ 
 
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Mathmatical Instruction and the Professors 
 
 of Mathematics in the French 
 
 Lycees for Boys. 
 
 By R. C ARCHIBALD. 
 
 Professor of Mathematics at Br own University. 
 
Mathematical Instruction 
 and the Professors of Mathematics 
 in the French Lycees . 
 for Boys 
 
 By 
 
 R. C. Archibald 
 
 Brown University 
 
 Providence 
 
 R. I. 
 
Reprint from School Science and Mathematics. Vol. 13. Jan- 
 uary and February, 1913. 
 
 MATHEMATICAL INSTRUCTION AND THE PROFESSORS 
 
 OF MATHEMATICS IN THE FRENCH LYCEES FOR 
 
 BOYS.^ 
 
 By R. C. Archibald, 
 Professor of Mathematics at Brown University. 
 
 The general scheme of the French educational system and the 
 position of the lycee in this system are topics, the consideration 
 of which the title of my paper, strictly speaking, excludes. And 
 yet, to give appropriate setting to the main themes and bases 
 of comparison with our own schools, brief reference to these 
 topics seems not wholly uncalled for in this connection. 
 
 For educational purposes France is divided geographically into 
 arrondissements. The assemblage of government schools (pri- 
 mary, secondary and superior) in each arrondissement forms an 
 academie over which a recteur presides. There are thus the 16 
 academies of Aix-Marseilles, Besangon, Bordeaux, Caen, Chani- 
 bery, Clermont, Dijon, Grenoble, Lille, Lyons, Montpellier, 
 Nancy, Paris, Poitiers, Rennes, Toulouse, as well as a seven- 
 teenth at Algiers. With the exception of Chambery these names 
 correspond to the seats of the French universities. 
 
 The assemblage of academies forms the Universite de France, 
 at the head of which is the Minister of Public Instruction, who is 
 ex officio the ''Recteur de I'Academie de Paris et Grand Maitre 
 de rUniversite de Paris." For the Academie de Paris there is 
 a vice-recteur, whose duties are the same as those of the recteurs 
 of other academies. Although nominally .lower in rank than the 
 heads of academies in the provinces, he is in reality, the most 
 powerful official in the educational system. The position of the 
 Minister of Public Instruction being so insecure by reason of 
 changing governments, continuity of scheme is assured by three 
 lieutenants who have charge respectively of the primary, second- 
 ary, and superior education. They in turn have an army of in- 
 spectors who report on the work and capabilities of the recteurs 
 and their academies as far as primary and secondary instruction 
 is concerned. 
 
 This suffices at present to indicate the remarkably centralized 
 and unique character of the French educational system. It is 
 theoretically possible for the most radical changes in any part of 
 
 lAbridgment of a paper presented at the mid-winter meeting of the Association 
 of Mathematical Teachers in New England, held at Brown University, Providence, 
 R. I., February 3, 1912. 
 
44 SCHOOL SCIENCE AND MATHEMATICS 
 
 public instruction to be immediately brought about by a stroke 
 of the pen on the part of the Minister of Public Instruction. 
 
 The present system of secondary education in France dates 
 from the great reform of 1902 (important modifications were in- 
 troducted in 1905 and 1909) and is carried on for the most part in 
 Lycees and Colleges communaux which are to be found in nearly 
 all cities. Because of their pre-eminence we shall consider the 
 former only, which are under control of the state. Here the 
 boys, who come from families in comfortable circumstances, may 
 enter as Sieves at the age of five or six years and be led along 
 in their studies till they receive the Baccalaureat at the age of 16 
 or 17.2 Many lycees have still more advanced courses to prepare 
 for entrance into such schools as the cole Normale Superieure, 
 ficole Polytechnique, cole Centrale, cole Navale, cole de 
 Saint Cyr, etc. 
 
 Instruction in fully equipped lycees may be divided into four 
 sections : I, Primary; II, Premier Cycle; III, Second Cycle; IV, 
 Classes de Mathematiques Speciales. 
 
 I. Primary.^ The classes in this section are named as fol- 
 lows : A jr 
 
 Age from 
 
 Classes enfantines Onzieme 5 
 
 Classes preparatoires Dixieme 6 
 
 Neuvieme 7 
 
 Classes elementaires Huitieme 8 
 
 Septieme 9 
 
 From the Dixieme to the Septieme 20 hours are devoted to 
 class recitation each week. In the Classes preparatories 3 hours 
 a week are taken up with Calcul, that is, principles of numera- 
 tion, elementary operations with integers, notions concerning the 
 metric system; intuitive geometry; simple exercises to enable 
 the pupil to draw the more elementary regular figures (square, 
 
 2The pupils at the lycees are of four kinds: 1st. Externes, those \vho come to the 
 lycees for classes but board and lodge outside: 2nd. Internes or pensionnaires, eleves 
 who live entirely in the establishment; 3rd. Demi-pensionnaires who usually reside 
 at a distance but take their mid-day meal at the lycee; 4th. Externes surveilles, that 
 is externes who work out their lessons under the eye of the preparateur in the 
 salle d'etude of the lycee. The expenses of the pupil vary greatly with the class and 
 the lycee in which he happens to be. The range of cost (in francs per year) (1) for 
 some of the principal cities (Bordeaux, Lyons, Marseilles, Toiilouse) of the provinces 
 and (2) for the better lycees of Paris is as follows: Externes (1) 70-450, (2) 90-700; 
 externes surveillis (1) 110-540, (2) 130-790; demi-pensionnaires (1) 370-850, (2) 
 500-1200; pensionnaires (1) 700-1200, (2) 900-1700. The lower price in each case 
 is for the classe enfantine, the higher for the special classes open to hacheliers. 
 Primary education (outside of the lycees and superior education) in France, is free. 
 
 ^Free primary instruction is given in Ecoles Primaries Elementaires for pupils 
 from 6 or 7 to 13 years of age. The course is divided as follows: Cours elementaire 
 (2 years), cours moyen (2 years), cours superieur (2 years). On completion of the 
 cours moyen the pu^il receives a certiHcatS d'etudes primaries elementaires. ^ This 
 certificate or its equivalent is required of every child in France. Many children 
 require considerably more than four years to get the certiUcat. 
 
MATHEMATICS IN THE FRENCH LYCRES 
 
 45 
 
 rectangle, triangle, circle) and different sorts of angles. In the 
 Classes elementaires, 4 hours a week are assigned to revision of 
 the preceding programme; decimal numbers; rules of three; 
 intuitive geometry by the aid of models. One hour a week is 
 given up to drawing. 
 
 II. Premier Cycle (sixieme-troisieme) . This cycle of four 
 years constitutes an advanced course for students who have 
 finished their primary studies, and is the first part of secondary 
 education proper. It offers a choice between two lines of study, 
 the one characterised by instruction in Latin with or without 
 Greek, the other in which no dead language is taught. The 
 former is selected by the parent who wishes to prepare- his boy 
 for the department of letters in the ficole Normale Superieure 
 or for the career of classical professor, lawyer or doctor. The 
 latter is likely to be chosen for the boy who is particularly interest- 
 ed in science or who has a commercial career in view. 
 
 III. Second Cycle. This leads, normally, to the Baccalaureat, 
 at the end of three years' study, in one of four different sections. 
 The scheme will be clearer in tabular form. 
 
 
 Pupils who 
 
 learn Latin, 
 
 with or without 
 
 Pupils who learn no 
 dead language 
 
 Age 
 from 
 
 
 VJi v^^iV. 
 
 PREMIER 
 CYCLE. 
 
 4 years 
 
 Sixieme A (Latin). 
 
 Sixieme B 
 
 10 
 
 Cinquieme A (Latin). 
 
 Cinquieme B 
 
 11 
 
 
 Quatrieme B 
 
 12 
 
 Quatrieme A (Latin Greek) 
 
 Quatrieme (Latin) 
 
 Troisieme A (Latin Greek) 
 
 Troisieme (Latin) 
 
 Troisieme B 
 
 13 
 
 
 ' 
 
 
 Pupils who give up the 
 study of Latin. 
 
 i 
 Sciences-Langues 
 
 
 Latin-Grec. 
 
 Latin-Langu. 
 
 -A- 
 
 L 
 
 ATIN-SCIENCES 
 
 SECOND 
 CYCLE 
 
 3 years 
 
 Second A 
 
 Second B 
 
 Second C 
 
 Second D 
 
 14 
 
 Premiere A 
 
 Premiere B 
 
 Premiere C 
 
 Premiere D 
 
 15 
 
 PhilosophieA 
 
 Philosophic B 
 
 MathematiquesA 
 
 Mathematiques B 
 
 16 
 
 Let us now observe a little more closely just what is involved in 
 this display, in the matter of studies and demands made upon the 
 eleve. As an important examination which we shall presently 
 describe comes at the end of the Premiere, our present analysis 
 will not pass beyond this grade. Here is the programme for a 
 week. 
 
 There are several features of this scheme (we shall refer to 
 
46 
 
 SCHOOL SCIENCE AND MATHEMATICS 
 
 four), which are par- 
 ticularly interesting. 
 
 1. The prominence 
 given to the study of 
 French throughout. 
 
 2. That all eleves at 
 the age of 10 or 11 com- 
 mence the study of 
 modern languages 
 (English, German, Ital- 
 ian, Spanish, Russian, 
 or, in Algiers, Arabic), 
 and continue it during 
 six years at least, be- 
 fore matriculating into 
 schools of university 
 grade. Not only do they 
 get glimpses of the best 
 things in the literature 
 of the language, but al- 
 so learn to speak the 
 language with consid- 
 erable freedomi and re- 
 markable correctness in 
 pronunciation. The di- 
 rect method is employed 
 and no word of French 
 is ever spoken in the 
 advanced classes. The 
 majority of the eleves 
 choQse German, as this 
 is required of all candi- 
 dates for entrance into such military schools as the ficole Poly- 
 technique and ficoie de St. Cyr. On the other hand there is an 
 increasing number taking up the study of English which is re- 
 quired for the ificole Navale. 
 
 3. The proportion of recitation periods devoted to mathemat- 
 ics. As the drawing courses are about equally of a free hand 
 and of a geometrical character, half the time given to drawing 
 may be counted as mathematical and the percentages of recitation 
 periods "required for mathematics during the four years of the 
 Premier Cycle are : In the Latin course 14 per cent, 14 per cent, 
 
 3 
 
 3 O 
 
 r^ 
 td>Co>bo>W> 
 
 
 
 1>>.CJiC0OiWOiW 
 
 sr'* 
 
 French. 
 
 11 
 
 go 
 
 . OS- Oi- -3- -Q 
 
 Latin. 
 
 ) 
 
 Greek. 
 
 OiWOiCnCnOiWOi 
 
 Modern Languages. 
 
 wtocowwcowco 
 
 History and Geography. 
 
 ?s 
 
 l_l. _lOI-J M M 
 
 Natural Science. 
 
 n !r 
 
 ^^ ^^ 
 
 Physics and Chemistry. 
 
 3 ^ 
 n 
 
 i_. M (-1 -> i ' '. 
 
 Moral Philosophy. 
 
 sa 
 
 m' !- i . . 
 
 Book-keeping. 
 
 -4 
 
 ^^ to ts M ^^ o M ts 
 
 Drawing. 
 
 
 
 Geometrical Drawing. 
 
 ^^M^s 
 
 Arithmetic. 
 
 o 
 3 
 
 wMhseo- . 
 
 Arithmetic and Geometry. 
 
 
 
 !>. '' 
 
 Algebra and Geometry. 
 
 o 
 
 : : : : ^: ^: 
 
 Writing. 
 
 -1 
 
 ^o^^^^^s^^^o^^^s 
 
 c^*.coco^seo^sw 
 
 Total No. Hours. 
 
 5" 
 
 pnca>ppK;> 
 
 
 
 
 French. 
 
 
 '. coco CO. Kk 1*^ * 
 
 Latin. 
 
 
 ! I I c;t. I I en 
 
 Greek. 
 
 
 <J^o<HSl-ao-Q^^ 
 
 Modern Languages. 
 
 
 : '. ^^^^! I n) ts 
 
 Ancient History. 
 
 
 ^^^o^s^^^o^s^^^^ 
 
 Modern History. 
 
 
 
 Geography. 
 
 I- * 
 
 coco- coco- 
 
 Physics and Chemistry. 
 
 
 O0 MtS- 
 
 Practical Exs. in Science. 
 
 : : : : j j j j 
 
 Geology. 
 
 H.2. 
 
 '.'.'. to'. '.'. ' 
 
 Supplementary Exs. in Latin. 
 
 ?r 
 
 ^s^o 
 
 Drawing. 
 
 o 
 
 M ^^ ^^ ^^ 
 
 Geometrical Drawing. 
 
 t-> 
 
 . . ^^ ts oi oi ts ts 
 
 Algebra and Geometry. 
 
 o 
 
 n 
 
 OlOl" '.''.'. 
 
 Geometry, Descriptive Geom. 
 Trig., Algebra. 
 
 
 ^^^^^o^^^^^^^^5 
 
 OOOll-'COCO-^*-*' 
 
 Total No. Hrs. 
 
MATHEMATICS IN THE FRENCH LYCES 47 
 
 22 per cent, 22 per cent; in the Modern Language course, 22.7 
 per cent, 22 per cent, 22 per cent and 22 per cent. In the Second 
 and Premiere of the Second Cycle the percentages run : in the 
 Latin-Greek course, 12.5 per cent, 4.5 per cent; in the Latin- 
 Modern-Language course, 12.5 per cent, 6 per cent ; in the Latin- 
 Science course, 23 per cent, 24 per cent; and in the Modern- 
 Language-Science course, 30 per cent, 30 per cent." To sum up 
 from the Dixieme to the Premiere the boy has spent 10.5 per 
 cent, 11 per cent, 19.4 per cent, 22.8 per cent of his class hours 
 in mathematical recitation according as he has pursued the 
 courses leading to Premiere A, B, C or D. This emphasis which 
 the French lay on mathematics is interesting and although the 
 percentages may be somewhat higher than in America the train- 
 ing received is vastly superior in France. The fact that prac- 
 tically all the professeurs titulaires in the French lycees, even 
 those in charge of the very elementary classes, are agreges in 
 the subjects which they teach means much. Just how much we 
 shall explain later, but suffice it to remark here that no other 
 country imposes as high scholastic standards for its professors of 
 secondary education. 
 
 Another feature of mathematical instruction which is particu- 
 larly interesting to us, is, that from the troisieme on, that is, 
 from the time the boy is 13 or 14 years old, instruction is usually 
 given entirely by lecture. Indeed, even in classes before the 
 troisieme when a text-book is generally in the hands of the eleve, 
 he is required to take notes "pour preciser" the various topics. 
 By such methods, searching questioning and frequent "tests," on 
 the part of the professor, and rigid inspection, kindly expressed 
 praise or cutting public reprimand on the part of the proviseur 
 (director of the lycee), there is no possibility of learning parrot- 
 fashion no room for the shirker or the boy who does not try his 
 best; reasoning powers and independence of thought must be 
 constantly exercised. The eleves are encouraged to consult the 
 various text-books to be found in all the lycee libraries and for 
 those less bright this may be almost a necessity from time to 
 time; but on personal inspection in different lycees I found the 
 note books of eleves of 14 or 15 alike remarkable for their neat- 
 ness and completeness. Th,e habits thus gained in the lycee 
 stand in good stead when the student reaches the university. The 
 rapidity of the lecturer and the complexity of his theme seem to 
 make little difference, for at the close of the hour the whole is 
 in the note books as neat as copper-plate. 
 
48 SCHOOL SCIENCE AND MATHEMATICS 
 
 4. The large number of hours in class recitation may not at 
 first appear very imposing; but we cannot fail to be astonished 
 that 8 hours per day (in class and in preparation of lessons) 
 may be demanded from eleves in the premier cycle, and lOJ in 
 summer, 10 in winter from those in the second cycle. The law 
 further explicitly states that there is no limit to the number of 
 hours which may be demanded of the eleves in the Classes de 
 Mathematiqiies Speciales. When we later come to look more 
 closely at their programme we shall not be surprised, but never- 
 theless wonder, how these undoubtedly happy and healthy young 
 men of 17 or 18 have survived the treatment. In more advanced 
 lycee courses as well as at the universities I was also impressed 
 with the almost appalling intensity and seriousness of the auditors 
 the strife is too strenuous, the competition too keen, to admit 
 of a moment's levity or wandering thought. But when the les- 
 son is over, every care is instantly banished and the national 
 gaiety is once more in evidence. 
 
 To return to our table. We remark that the two groups of 
 eleves who elect sciences on entering the second cycle have the 
 same number of hours per week in mathematics indeed the 
 courses are identical. To give greater definiteness to our ideas 
 as to their general attainments let us consider the programme of 
 studies from Premiere D, when the boy is 15 or 16 years old. 
 
 French. Lectures and questions on the principal French writ- 
 ers of the nineteenth century. Study of selections from prose 
 writers and poets, from moralists, orators, politicians, scientists 
 and historians of the sixteenth, seventeenth, eighteenth and nine- 
 teenth centuries. 
 
 History. Political history of Europe in the eighteenth century. 
 Detailed history of France at the close of the eighteenth century. 
 
 Geography. Detailed study of France, its geological con- 
 stitution, its climatology, physiography, topography, economic 
 and military organization ; its colonies, etc. 
 
 Physics^ Optics, electricity. 
 
 Chemistry. ^Of the carbon compounds. 
 
 German. Selections from the dramatic poetry of Schiller, 
 Goethe, Kleist and Grillparzer. Extracts from the prose ^vorks 
 of Wieland, Goethe, Schiller, Auerbach, Freytag, Scheffel, etc. 
 
 English. Shakespeare's Julius Caesar and Macbeth, extracts 
 from Milton, Addison, Goldsmith, Wordsworth, Byron, Cole- 
 ridge, Dickens, Macauley, Eliot, Tennyson, and Thackeray. 
 
 Algebra. Equations and trinomials of the second degree. 
 
MATHEMATICS IN THE FRENCH LYCES 49 
 
 Calculation of the derivatives of simple functions ; study of their 
 variation and graphic representation; study of rectilinear motion 
 by means of the theory of derivatives ; velocity and acceleration ; 
 uniformly changing motion. 
 
 Geometry. Solid. 
 
 Descriptive Geometry. Elements. 
 
 Trigonometry. Plane, including the use of four or five place 
 logarithm tables, the solution of triangles and trigonometric equa- 
 tions. 
 
 Having finished the Premiere, the eleve presents himself for 
 examination under conditions which once more emphasize the 
 unity of the French educational system. This is the examination 
 for the first part of the state degree known as the Baccalaurcat. 
 
 A peculiar feature of this examination is that it is not held in 
 the lycees but at the university of the academic to which the 
 particular lycee belongs.* As various civil and practically all 
 government positions, except those in post and telegraph offices 
 are only open to hacheliers, the state introduces into the body of 
 examiners some who are wholly independent of the lycees. 
 These examiners are the professors in the universities. 
 
 Since our future mathematicians are to come from Premiere 
 C and D we shall give a few particulars concerning their ex- 
 amination. All examinations for the baccalaureat are held in 
 July and October at the ending of one school year and the be- 
 ginning of the n,ext. The examiners of the candidates from 
 Premiere C are six in number, three of whom are university 
 professors and three professors from the lycees or colleges; for 
 Premiere D there are but two university professors in addition 
 to three from the lycees. The examinations in all sections are 
 both written and oral. Here \s the scheme of examination which 
 practically covers what the eleve has studied in earlier years. 
 
 Premiere C (Latin-Sciences). Written. 1st, a French com- 
 position (3 hours) ; (the candidate has a choice of three sub- 
 jects) ; 2nd, a Latin translation (3 hours) ; 3rd, an examination 
 in Mathematics and Physics (4 hours). Oral (about three- 
 quarters of an hour). 1st, explanation of a Latin text; 2nd, 
 explanation of a French text; 3rd, examination in a modern 
 language questions and answers being necessarily in this lan- 
 guage. Questions in 4th, History; 5th, Geography; 6th, Mathe- 
 matics ; 7th, Physics ; 8th, Chemistry. 
 
 *As there Is no university at Chambery, the candidate presents himself before a 
 faculty of either Lyons or Grenoble. 
 
50 SCHOOL SCIENCE AND MATHEMATICS 
 
 And similarly for Premiere D. 
 
 The searching character of the tests prepares us for a large 
 number of failures. Here is the record of the percentage of 
 candidates passed in (1) July, (2) October, 1909: Latin-Grec 
 (1) 44, (2) 42; Latin-Langues Vivantes (1) 41, (2) 42; 
 Latin-Sciences (1) 49, (2) 46; Sciences-Langues Vivantes (1) 
 42, (2) 39. We observe that less than fifty per cent of the pupils 
 get through on the first .examination'* while a similar percentage 
 of the remainder fail and are required to return to the Premiere 
 at once or wait for another year. Those who have been 
 successful return to the lycee to prepare for the second part of 
 the baccalaureat. A choice of two courses (which may be slightly 
 varied), is open to them, the one Philosophie A or B, the other, 
 Mathematiques A or B. We shall only refer to the latter which 
 has been supplied with pupils from the Premiere C and D. There 
 they had 26 and 28 recitation hours per week. This has now 
 been increased to 27% and 28%. There has been an increase in 
 the number of hours devoted to mathematics, physics and chem- 
 istry, but a reduction in the amount of study of modern lan- 
 guages. Latin no longer enters. The programme for Mathe- 
 matiques A is in outline as follows : 
 
 Philosophy (3 hours). History and Geography (3% hours). 
 Modern Languages (2 hours). Physics and Chemistry (5 
 hours). Natural Science (2 hours). Practical Exercises in 
 Science (2 hours). Drawing (2 hours). Hygiene (12 lectures 
 of 1 hour). 
 
 Mathematics. (8 hours) : Arithmetic. Properties of integers; 
 fractions; decimals; square roots; greatest common divisors; 
 theory of errors; etc. 
 
 Algebra. Positive and negative numbers, quadratic equations 
 (without the theory of imaginaries), progressions, logarithms, in- 
 terest and annuities, graphs derivatives of a sum, product, 
 quotient, square root of a function, of sin x, cos x, tan x, cot x. 
 Application to the study of the variation and the maxima and 
 minima, of some simple functions, etc. 
 
 Trigonometry. Circular functions, solution of triangles, appli- 
 cations of trigonometry to various questions relative to land sur- 
 veying. 
 
 Geometry. ^Translation, rotation, symmetry, homology and 
 
 5For some it may have been the third or fourth trial- 
 
 There are certain exceptional cases which I shall not consider. 
 
MATHEMATICS IN THE FRENCH LYCES 51 
 
 similitude, solids, areas, volumes, poles and polars,' inversion, 
 stereographic projection, central projections, etc. 
 
 Conies. Ellipse, hyperbola, parabola, plane sections of a cone 
 or cylinder of revolution, etc. 
 
 Descriptive Geometry. Rabatments application to distances 
 and angles projection of a circle sphere, cone, cylinder, planes, 
 sections, shadows application to topographical maps, etc. 
 
 Kinematics. Units of length and time. Rectilinear and curvi- 
 linear motion. Translation and rotation of a soHd body. Geomet- 
 ric study of the helix, etc. 
 
 Dynamics and Statics. Dynamics of a particle, forces applied 
 to a solid body, simple machines in a state of repose and move- 
 ment, etc. 
 
 Cosmography. ^Celestial sphere, earth, sun, moon, planets, 
 comets, stars Co-ordinate Systems, Kepler's and Newton's laws, 
 etc. 
 
 One of the most striking things in this scheme, as compared 
 with the American method, is to find arithmetic taught in the last 
 year of the lycee course. Note, too, that from the Cinquieme on, 
 it has been taken up in connection with instruction in geometry 
 and algebra. Indeed, this method of constantly showing the in- 
 terdependence or interrelation of the various mathematical sub- 
 jects was one of the interesting and valuable characteristics of 
 French education as I observed it. For example, I happened to 
 be present in a classroom when the theory and evaluation of re- 
 peating decimals was under discussion. After all the processes 
 had been explained, problems which led similarly to the consider- 
 ation of infinite series and limits were taken up. By suggestive 
 questioning a pupil found the area under an arc of a semi-cubical 
 parabola and the position of the centre of gravity of a spherical 
 cap. With us it is not till the graduate school of the university 
 that the boy is taught the true inwardness of such processes as 
 long division and extraction of roots; but in France, arithmetic 
 is taught as a science, not as an art, and the eleve leaving the lycee 
 has a comprehending and comprehensive grasp of all he has 
 studied. 
 
 We remark that most of the mathematical subjects mentioned 
 above are more or less foreign to our secondary education. In- 
 struction in geometrical conies {courbes usuelles), is infrequently 
 given by us, even in universities. Again, the ordinary mathe- 
 mathical student who goes up for his doctor's degree in America 
 may have the vaguest idea of what is even meant by Descriptive 
 
52 SCHOOL SCIENCE AND MATHEMATICS 
 
 Geometry. True, it is a regular course for our training of the 
 engineer; but not, unfortunately, of the mathematician. On the 
 other hand the French mathematical student has had at least four 
 years of Descriptive Geometry, two of them before receiving his 
 baccalaureat. The subject is required for admission into many 
 government schools. 
 
 We note that the idea of a derivative is familiar to the lyceen 
 during the last two years of his course. Why we so generally 
 shut out the introduction of such an idea into our first courses in 
 analytical geometry and theory of equations is, to me, a mystery. 
 Finally, I would remark that the classes in Mathematiques A last 
 two hours, with the exception of five minutes for recreation at 
 the end of the first hour. The professor thus has sufficient time 
 to amplify and impress his instruction. 
 
 At the close of the last year of the Second Cycle, the eleve 
 takes the examination for the second part of the baccalaureat. 
 The same general conditions prevail as for the first part. The 
 jury of four contains two university professors. The written 
 examinations in mathematics, physics and philosophy are each 
 three hours long; the oral covers what has been studied the year 
 previously. If successful, a diploma now called the baccalaureat 
 de Venseignement secondaire, is granted to the eleve by the Min- 
 ister of Public Instruction. The eleve thus becomes a hachelier. 
 Diplomas in all four sections are of the same scholastic value. 
 The charge made for diploma and examination is 90 francs. 
 More than forty per cent of the candidates failed to pass at each 
 of the examinations in 1909. 
 
 Because of the similarity of title used in the different countries, 
 the Frenchman does not generally understand what the title 
 Bachelor of Arts implies nor is it easy to make any concise state- 
 ment in explanation. Little exaggeration can be made, however, 
 in placing the bachelier on a plane of scholastic equality with the 
 Sophomore who has finished his year at one of the best Ameri- 
 can universities. 
 
 Furthermore, his training has been undoubtedly much more 
 thorough. After the age of 6 or 7 French boys are taught by 
 men.'^ These men have all studied at the University and have 
 passed the examen de licence. With very few exceptions now, 
 the instructors have also passed the extremely difficult Examen d' 
 agregation in the subjects they propose to teach. By comparison, 
 how woefully deficient our teachers of like grades ! The recently 
 
 'Girls are taught by women. Coeducation does not exist in the lycees. 
 
MATHEMATICS IN THE FRENCH LYCBES 53 
 
 published reports of the United States sub-committees of the 
 International Commission on the Teaching of Mathematics state 
 the case frankly. That precious years are often lost to our youth 
 by their inferior instruction is obvious to every one. As to Ex- 
 aminations no guessing of possible questions and ''cramming" 
 for the same, so common in America, can qualify a student to 
 pass an examination in France. The rigorous and impartial 
 tests for promotion are conducted, at least in part, by those out- 
 side the lycee and pressure brought to bear upon teachers to 
 promote ill-prepared pupils is unknown. According to a recent 
 report of the Carnegie Foundation for the Advancement of 
 Teaching, this is a "great source of weakness" and "a fruitful 
 source of demoralization in American public schools." 
 
 I should now like to tell you something of the fourth section 
 of lycee instruction, namely, the Classes de Mathematiques 
 Speciales. 
 
 If the bachelier who is proficient in mathematics be not turned 
 aside by circumstances or inclination, to seek immediately a 
 career in civil or government employment, he most probably pro- 
 ceeds to prepare himself for the highly special and exacting 
 examination necessary for entrance into one of the great schools 
 of the government. The method of this preparation exhibits a 
 very peculiar feature of the French system. Whereas with us, 
 or with the German, the boy who has finished his regular course 
 in the secondary school goes " directly to some department of a 
 university for his next instruction, the bachelier, who has a 
 perfect right to follow the same course, returns to his old lycee 
 (or enrolls himself at one of the great Paris lycees, such as Saint 
 Louis, Louis le Grand or Henri IV), to enter the Classe de 
 Mathematiques Speciales preparatoire which leads up to the Classe 
 de Mathematiques Speciales. The latter is exactly adapted to 
 prepare students for the !ficole Normale Superieure, the cole 
 Polytechnique and the bourses de licence. Only a small propor- 
 tion of the lycees (36 out of the 115), have this Classe; but with 
 the exception of Aix they are to be found in all university towns. 
 On the other hand, yet other lycees have classes which prepare 
 specially for the less exacting mathematical entrance examina- 
 tions of the ficole Centrale, ficole de Saint Cyr, ]&cole Navale, etc. 
 But the number of eleves who on first starting out deliberately 
 try to pass examinations for these schools is small, in proportion 
 to the number who eventually reach them after repeated but vain 
 effort to get into the ficole Polytechnique or the ficole Normale 
 
54 SCHOOL SCIENCE AND MATHEMATICS 
 
 Superieure. Just what makes these two schools famous and 
 pecuharly attractive will appear in a later section. It has been 
 noticed that when the eleve has won his baccalaureat he may 
 immediately matriculate into a university, and although it might 
 be possible for him to keep pace with the courses, in mathematics, 
 at least, it would be a matter of excessive difficulty. There is 
 then in reality, betw,een the baccalaureat and the first courses of 
 the universities, a distinct break, bridged only by the Classes de 
 Mathematiques Speciales} 
 
 The eleves who enter the preparatoire section of this class are, 
 generally, bacheliers leaving the classes de Mathematiques ; in 
 very rare instances, there are those who come from the classe 
 de Philosophic. Natural science, history and geography, philoso- 
 phy indeed practically every study except those necessary for 
 the end in view, have been dropped and from this time on to 
 the agregation and doctorat all energies are bent in the direction 
 of intense specialization. This is the most pronounced char- 
 acteristic of French education to-day. In mathematics, instruc- 
 tion now occupies 12 instead of 8 hours. New points of view, 
 new topics and broader general principles are developed in 
 algebra and analysis, trigonometry, analytical geometry and me- 
 chanics. Physics and chemistry are taught during six hours in- 
 stead of five. Add to these, German, 2 hours ; French literature, 
 one hour; descriptive geometry, 4 hours; drawing, 4 hours. 
 After one year of this preparatory training the eleve passes into 
 the remarkable Classe de Mathematiques Speciales. 
 
 Eight years of strenuous training have made this class possible 
 for the young man of 17 or 18 years of age, who is confronted 
 with no less than 34 hours of class and laboratory work per week 
 and no limit as to the number of hours expected in preparing for 
 the classes! 
 
 When first I looked over the programme it seemed a well nigh 
 impossible performance for one year. Surely no other country 
 can show anything to compare with it. 
 
 Did time permit it would be inter,esting to reproduce in full the 
 mathematical programme as given at the end of the plan d' etude, 
 but I shall hastily refer to only a few of the subjects treated: 
 In Algebra and Analysis we find developed, the fundamental 
 ideas concerning irrational numbers, convergency and divergency 
 
 8It is only for mathematical or scientific students that such a break occurs, as no 
 special classes are provided in other subjects except in the case of half a dozen Paris 
 lycees which have classes in "letters" preparatory for entry into the Ecole Normale 
 Superieure. 
 
MATHEMATICS IN THE FRENCH LYCES 55 
 
 of series, the elements of the theory of functions of a real vari- 
 able, power series, their multiplication and division, their dif- 
 ferentiation and integration term by term. Taylor's formula, the 
 theory of algebraic equations, including symmetric functions, but 
 omitting the discussion of infinite roots. The latter part of the 
 course treats of differentials of several variables, elementary ideas 
 concerning definite integrals, integration of such functions as are 
 considered in a first calculus course of the best American col- 
 leges, rectification of curves, calculation of volumes, plane areas, 
 moments of inertia, centres of gravity, differential equations of 
 the first order, solutions of simpler differential equations of the 
 second order, which occur in connection with problems of me- 
 chanics and physics. Whenever possible in the discussion of 
 these topics the power to work numerical examples is emphasized. 
 
 Plane Trigonometry and the discussion of spherical trigonom- 
 etry through the law of cosines are treated in class and five- 
 place tables are used. 
 
 In the course on Analytical Geometry is given a thorough dis- 
 cussion of equations of the second degree, of homography and 
 anharmonic ratios as they enter into the discussion of curves 
 and surfaces of the second degree, of points at infinity, asymptotes, 
 foci, trilinear coordinates, curvature, concavity and convexity, 
 envelopes, evolutes. The professor also discusses thoroughly the 
 various questions connected with the treatment of quadric sur- 
 faces and less completely, the theory of surfaces in general, of 
 space curves, osculating planes, curvature of surfaces. The ele- 
 ments of the theory of unicursal curves and surfaces and of 
 anallagmatic curves and surfaces are also taken up. 
 
 So also, we find broadly arranged programmes mapped out 
 in mechanics and descriptive geometry. The whole number of 
 class hours per week is broken up as follows : 
 
 Mathematics, 15 ; physics, 7 (2 in laboratory) ; chemistry, 2 ; 
 descriptive geometry, 4; drawing, 4; German, 2; French, 1. The 
 scope of the mathematical work may be judged from some books 
 which were prepared with the needs of such a class especially in 
 view. 
 
 B. Niewenglozvski, Cours d' algebre, I, 382 p. ; II, 508 p. ; 
 Supplement G. Papelier Precis de geometrie analytique, 696 p. 
 Girod Trigonometric, 495 p. P. Appell Cours de mecanique, 
 650 p. X. Antomari Cours de geometrie descriptive, 619 p. 
 
 If anything, this list underestimates the work actually covered 
 
 9That is, much more than what is called for by examination questions is studied. 
 The eleves find truth in the adage: Qui pent Ic plus pent le mains. 
 
56 SCHOOL SCIENCE AND MATHEMATICS 
 
 by those who finally go out from the class. Tannery's Legons 
 d'algebre et d'analys,e (I, 423 p., II, 636 p.), might well replace 
 Niewenglowski's work while Niewenglowski's Cours de geometrie 
 analytique <I, 483 p.; II, 292 p.; Ill, 569 p.), represents the 
 standard almost as nearly as Papelier's volume. Another treat- 
 ise on mechanics widely used is that of Humbert and Antomari. 
 
 When we further realize that the main parts of the books in 
 this list, which represents the work for only one of a half dozen 
 courses, are covered by the professor in about fifteen months 
 the last three months of the second year are given over to drill 
 in review and detail we begin to get some conception of what 
 the Classe de Mathematiques Speciales really stands for. In his 
 instruction the professor is officially ''recommended" ''not to 
 overload the courses, to make consicierable use of books, not to 
 abuse general theories, to expound no theory without numerous 
 applications dealt with in detail, to commence invariably with 
 the more simple cases, those most easy to understand, for leading 
 up finally to the general theorems. Among the applications of 
 mathematical theory, those which present themselves in math- 
 ematical physics should be giyen the preference, those which the 
 young people will meet later in the course of their studies either 
 theoretical or practical. Thus in the construction of curves, 
 choose as examples those curves which present themselves in 
 Physics and Mechanics, as the curves of Van der Waals, the 
 Cycloid, the Catenary, etc. in the theory of envelopes choose 
 those examples of envelopes which are met in the theory of 
 cylindrical gearing and so on. The pupils should be trained 
 to reason directly on the particular cases and not to apply the 
 formulae. To sum up, one ^ ought to develop 'their judgment and 
 their initiative not their memory." 
 
 {Continued in February Issue.) 
 
MATHEMATICS IN THE FRENCH LYCSES 105 
 
 MATHEMATICAL INSTRUCTION AND THE PROFESSORS 
 
 OF MATHEMATICS IN THE FRENCH LYCEES FOR 
 
 BOYS. 
 
 By R. C. Archibald, 
 Professor of Mathematics at Brown University. 
 
 (Continued from the January Number). 
 
 In France, as everywhere else, the success of the system de- 
 pends much on the personality of the professor. A renowned Paris 
 lycee instructor who had a genius for getting hold of his boys 
 has recently died. No less than 35 of his pupils were admitted to 
 the cole Polytechnique in a single year. The ordinary professor 
 has to be content with a half or a third of this number. But 
 the success of a class is, by happy arrangement, not left to de^ 
 pend wholly upon a single man. Take, for example, lycee Saint 
 Louis, which is the greatest preparatory school for the cole 
 Normale Superieure and the ficole Polytechnique. There are 
 four Classes de Mathematiques Speciales and for all the members 
 of these classes, conferences, interrogations and individual ex- 
 amination are organized. These exercises, which complete the 
 daily instruction, are conducted by one of the professors in the 
 lycee itself, or by one of those from the College de France, the 
 Sorbonne, the cole Polytechnique, the cole Normale, from 
 other lycees or from the colleges. Incapable students are thus 
 speedily weeded out. Of perhaps greater value than the solidity 
 of the training got in this way is the fact that the interest of the 
 pupil is sustained. 
 
 With the end of the year the el eve has his first experience of a 
 concours. Previously he has found that it was necessary only 
 to make a certain percentage in order to mount to the next stage 
 in his scholastic career; but now it is quite different. In 1908, 
 1.078 pupils tried for admission into the cole Polytechnique, 
 but only 200, or 19.5 per cent, were received ; for the department 
 of science in the ficole Normale Superieure, 22 out of 274, or 8 
 per cent, succeeded. In each case the number was fixed in ad- 
 vance by the Government according to the capacity of the School ; 
 the fortunate ones were those who stood highest in the examina- 
 tions, written and oral. In the case of the ficole Polytechnique, 
 the written examinations were held in all the lycees which had 
 a Classe de Mathematiques Speciales. The 387 candidates de- 
 clared admissible were then examined orally at Paris, and from 
 
106 SCHOOL SCIENCE AND MATHEMATICS 
 
 them the 200 were chosen. Similarly for the cole Normale, 
 the written examinations are conducted at the seats of the 
 various academies and the oral at Paris. Since 1904 the con- 
 cours passed by the ficole Normalians has been that for the 
 bourses de licence, open to candidates of at least 18 years of age 
 and not more than 24. Certain dispensations in the matter of age 
 are sometimes granted. The value of the bourse, for the section 
 of science, is from 600-1,200 francs a year and is intended to 
 help the student to prepare for the licence and other examinations 
 required of prospective professors in the lycees and universities. 
 The candidates leading the list in the concours are sent to the 
 cole Normale Superieure for from three to four years. It is 
 necessary for the six or seven other boursiers to prepare for 
 future examinations at the various universities of the provinces. 
 Their bourses last regularly for two, and exceptionally for three, 
 years. 
 
 But to return to our eleves of the Classes de Mathematiques 
 Speciales. At the end of the first year, when 18 or 19 years 
 old, they usually present themselves for the concours of both 
 the bourse de licence and the cole Polytechnique, the examina- 
 tions in the former being more strenuous and searching. Only 
 from 2 to 5 per cent, succeed on the first trial. The others then 
 go back to the lycee and take another year in the Classe de 
 Mathematiques Speciales. Many points not fully understood 
 before are now clear, and at the end of the second year from 
 25 to 28 per cent are successful. 'The persevering again return 
 to their Classe and try yet a third time .(the last permitted for 
 the bourse de licence) ; but it is a matter of record that less than 
 one-half of those who enter the Classe de Mathematiques Spec- 
 iales succeed even with this trial. This is usually the last trial 
 possible for entry into the ficole Polytechnique, as the young 
 man who has passed the age of 21 on the first of January pre- 
 ceding the concours may not present himself. The remainder of 
 the students either seek for entrance into government schools 
 with less severe admission requirements, and thus give up their 
 aspirations to become mathematicians, or else continue their 
 studies at the Sorbonne. The candidate who heads the list in 
 each of these concours has his name widely published. In the 
 case of the bourse de licence he is called the cacique, and he very 
 frequently tops also the ficole Polytechnique list. 
 
 If the work in the Classe de Mathematiques Speciales is so 
 enormously difficult that only 2 to 5 per cent of its members can. 
 
MATHEMATICS IN THE FRENCH LYCES 107 
 
 at the end of one year, meet the standard of requirements of the 
 examinations for which it prepares, why is not the instruction 
 spread over two? Since nearly all the mathematical savants who 
 now shed lustre on France's fair fame have passed from this re- 
 markable class on the first trial, there can be no doubt that the 
 answer to this question may be found in the fact that the govern- 
 ment ever seeks her servants among the elite of the nation's 
 intellectuals. 
 
 Those who pass from the Classe de Mathematiques Speciales 
 at the early age of 18 years are not numerous, but Borel and 
 Picard are such men while <jOursat entered the cole Normale 
 Superieure at 17 years of age. For the average boy the lycee 
 course is heavy and more than once he may have to halt in 
 order to repeat a year. The system of training is largely form- 
 ulated to develop to the full the powers of the brilliant boy and 
 to promote his rapid advancement. For such youths poverty 
 is no detriment. Every lycee has a number of bursaries (cover- 
 ing all expenses) which it distributes to just such boys coming 
 with distinguished records from the primary schools. If the 
 boy's record is sustained, renewal of his bursary from year to 
 year is inevitable. 
 
 We have now seen something of the nature of the remarkable 
 mathematical training which the French boy may receive in the 
 lycee and have incidentally remarked that the men who have 
 given this training are also exceptional. In conclusion I propose 
 to describe very briefly the preparation necessary to become a 
 lycee professor and the inducements offered by the state to the 
 youth of the country to enter upon this preparation. 
 
 I have had occasion to point out the strong influence 
 which the cole Normale Superieure and ficole Polytechnique 
 exert on the careers of the flower of the French youth ; how that 
 instead of entering the university on passing the baccalaureat, as 
 in America or in Germany, "they seek to enter these schools. 
 The reason for this is not difficult to find. The ficole Poly- 
 technique which prepares its pupils as military and naval en- 
 gineers, artillery officers, civil engineers, in government em- 
 ploy, telegraphists and officials of the government tobacco manu- 
 factories, offers all of its graduates a career which is at once 
 rapid, brilliant and certain. The cole Normale practically as- 
 sures its graduates at least a professorship in a lycee and pre- 
 pares its eleves for this, or for a university career better and 
 more rapidly than the university can do it." 
 
108 SCHOOL SCIENCE AND MATHEMATICS 
 
 Let us suppose that our future mathematical professor in the 
 lycee is one of the eleven mathematical students who is success- 
 ful in getting into the ficole Normale in a given year. He studies 
 there for three years and receives special drill in pedagogy and 
 in connection with courses of lectures which he hears at the 
 Sorbonne and at the College de France. Almost the whole pur- 
 pose of the drill and instruction is to prepare for two examina- 
 tions, the licence and the agregation. 
 
 The diploma Licence es Science, which is necessary for all 
 those who take up secondary teaching is granted to those who 
 have 3 certiUcats in any one of three groups of subjects. Our 
 mathematician is examined in the following subjects: (1) Dif- 
 ferential and Integral Calculus; (2) Rational Mechanics; (3) 
 General Physics or some advanced topic in mathematics. The 
 examinations may be taken singly in July or in November; each 
 examination successfully passed entitles the student to a certiHcat 
 for that subject. The examination consists of three parts, epreuve 
 ccrite, epreuve pratique, epreuve orale. The first two are writ- 
 ten examinations of about four hours each. Theoretical consider- 
 ations abound in the ecrite while numerical calculation is char- 
 acteristic of the pratique. The orale lasts for 15-20 minutes and 
 is held before a jury of those professors who have the whole 
 examination in charge. The pass mark is fifty per cent. 
 
 Unlike the haccalaureat and the licence, the agregation is a 
 competitive examination and is conducted by the state. The 
 number who become agreges each year is fixed in advance by the 
 Minister of Public Instruction according to the needs of the 
 lycees in the country. This number in recent years has been 
 about 14; the number of candidates is usually about 80. Our 
 candidate for this examination must have four certiUcats, (1) 
 Differential and Integral Calculus; (2) Rational Mechanics; 
 (3) General Physics'; and (4) a subject chosen at pleasure in 
 the advanced mathematical fields in which courses are offered. 
 Just what is implied by the possession of one of these certiUcats 
 we may not pause to consider further than to say that no one 
 graduate course in any American university gives the pupil such 
 a comprehensive grasp and mastery of the subject. 
 
 To pass the agregation our future professor disposes of his 
 three years as follows : During each of the first two years he 
 passes the examinations for two of the four certiUcats. With 
 these off his hands he turns his whole attention to preparing 
 for the agregation proper. This examination is unique in its 
 
MATHEMATICS IN THE FRENCH LYCES 109 
 
 difficulty and exactions. As it is organized for selecting the 
 most efficient young men in the country to take charge of the 
 mathematical classes in the lycees, the examination turns largely 
 on the subjects there taught. It consists of epreiives prepar- 
 atoires and epreiives definitives. The former are four written 
 examinations each of seven consecutive hours in length (7 a. m. 
 to 2 p. m.) ! The first two of these are on subjects chosen from 
 the programme of the lycee in mathematiqu^s elementaires and 
 mathematiques speciales. The last two, based on the work of the 
 candidates in the universities, are a composition on Analysis and 
 its geometrical applications and a composition on Rational Me- 
 chanics. The epreiives are held at the seats of the various acade- 
 mies of France. Those who have reached a sufficiently high stand- 
 ard are declared admissible. Their number is usually a little less 
 than twice the possible number to be finally received. They must 
 present themselves at Paris for the epreuves definitives. These 
 consist of two written examinations and two lecons. The written 
 tests are an epreuve de geometrie descriptive, and a calcul num- 
 erique. Their duration is fixed by the jury, but it is usually four 
 hours for each. The lecons which are supposed to be such as 
 a professor might give (during %-l hour) in a lycee, are on 
 subjects from the programmes of the classes: (a) Mathemat- 
 iques Speciales; (b) Seconde, Premiere C. D. and Mathemat- 
 iques A. B. The subjects are drawn by lot, and for each lesson 
 the candidate has four hours to think over what he is going 
 to say. No help from book or other source is permitted. The 
 unfortunate who has little to say is speedily "adjourned." 
 
 The agreges are those specially prepared by the State for the 
 positions of professeurs titulaires in the lycees. Although this 
 title is not conferred regularly till the agrege has completed his 
 twenty-fifth year, those who are younger receive temporary ap- 
 pointment for every agrege may demand a position as his right. 
 The salaries vary according to the classe of the professor. At 
 Paris the lowest salary is 6,000 francs per year, and the highest, 
 9,500. In this range seven classes are represented; six, each 
 differing from the one before by 500 francs, and the hors classe 
 for which the salary is 9,500 francs. Promotion from one class 
 to another takes place by selection and by seniority. From the 
 sixth (the lowest classe) to the third, the number of those who 
 can be advanced each year by selection is equal to the number 
 which can be advanced by seniority. In the second and first 
 classes two advancements may be made by selection to one by 
 
no SCHOOL SCIENCE AND MATHEMATICS 
 
 seniority. In choosing those for the hors classe, selection alone is 
 taken into account. The promotions are made at the end of 
 each calendar year, and take place so that there are always 20 per 
 cent of them in the sixth classe, 18 in the fifth, 18 in the fourth, 
 16 in the third, 14 in the second, and 14 in the first. This ar- 
 rangement is obviously a happy one, both by way of recognition of 
 the merits of the unusually successful teacher, as well as those of 
 him w4iose service is rather characterized by faithfulness. 
 
 In addition to the professeurs tihilaires there are professeurs 
 charge de coins, who are usually selected from those ficole Nor- 
 malians and those admissible to the agregation, who fail to be- 
 come agreges. After 20 years of service they may become profes- 
 seurs titulaires and receive the salaries we have indicated above. 
 The government has, however, recently passed a law which 
 gives the higher reward to the agrege. It is to the effect that 
 500 francs per year shall be added to the regular salary of every 
 agrege. The real range of salaries mentioned above is then 6,500- 
 10,000 ; in the provinces this reduces to 4,700-6,700. For the 
 professeurs charge de cours, the salaries at Paris vary from 4,500 
 to 6,000 francs ; in the provinces, from 3,200 to 5,200.^0 In the 
 Premier Cycle the professors have 12 hours of teaching per week, 
 in the second cycle and the Classe de Mathematiques Speciales, 
 14-15 hours. Except for correcting exercises and filling out 
 reports the professors have absolutely no obligations outside of 
 class hours. They do not live in the lycees. The superintendence 
 of the study of the eleves is carried on by repetiteurs, the more 
 advanced of whom receive at Paris 2,600-4,600 francs for 36 
 hours service per week. 
 
 Attractions connected with a professorship in a lycee are, that 
 the remuneration is ample to live on comfortably, that the work 
 is not onerous but often not a little inspiring, that colleagues are 
 brilliant specialists in the same or other lines of study, that 
 the professorships are positions of honor and prominence in the 
 community and that the incumbents are in demand in many ways 
 which frequently materially increase their regular income. In 
 general, only a professor in the first or hors classe has charge of 
 the Classe de Mathematiques Speciale with its members the pick 
 of the French youth, but to this position all may aspire. 
 
 With inducements such as these it U no surprise to learn that, CnAA. 
 in marked contrast to America, France draws to the development 
 
 lOThe tendency of recent legislation is to exclude from the lycees all professors 
 who are not agreges. 
 
MATHEMATICS IN THE FRENCH LYC&ES ill 
 
 of her system of secondary education much of the best math- 
 ematical talent in the country. 
 
 In the John Hay Library a few days ago, I came across an 
 old Latin work which turned out to be the course of lectures on 
 Greek Geometry, delivered at Oxford University in 1620,^^ by 
 the erudite Sir Henry Savile. In the course of his concluding 
 remarks he used the following language: **By the grace of God, 
 gentlemen hearers, I have performed my promise ; I have re- 
 deemed my pledge. I have explained, according to my ability, 
 the definitions, postulates, axioms, and the first eight propositions 
 of the Elements of Euchd. Here sinking under the weight of 
 years, I lay down my art and my instruments."^^ 
 
 It is interesting to speculate on the thoughts which would 
 likely pass through the minds of these "gentlemen hearers" were 
 they privileged to listen for a time to the discussion of questions 
 in geometry, and in other parts of mathematics, as carried on by 
 master and pupil in a Class e de Mathematiques Speciales of a 
 French lycee. 
 
 Bibliography. 
 
 Of prime importance in the study of French Secondary Educa- 
 tion for boys in the ''plan df etudes et programmes d'enseignement 
 dans les lycees et colleges de gargons [Arretes du 31 mai 1902, 
 des 27, 28 juillet et 8 Septembre, 1905, 6 Janvier, 26, 30 juillet et 
 5 aout 1909]." Paris (Librairie Delalain), pp. 264 Next in 
 importance should be mentioned Tomes II and III (Librairie 
 Hachette) of the report of the French sub-committee of the 
 International Commission on Mathematical Teaching. Tome 
 II (159 pages), devoted to secondary education and edited by 
 M. Bioche contains special articles by MM. Bioche, Blutel, Levy, 
 Guitton, Th. Rousseau, Beghin, Muxart, Lombard. In Tome III 
 (123 pages), Superior Education is discussed and articles by 
 various writers are edited by M. Albert de Saint-Germain. A 
 perusal of this volume will give a good idea of the course of 
 training for lycee teachers. Each year about 100 pages of M. 
 H. Vuibert's Annuaire de la Jeunesse, Paris (Librairie Vuibert), 
 is devoted to "Enseignement secondaire des gargons." Informa- 
 tion as to the cost of the various classes in the different lycees 
 of France, bursaries offered, regulations with regard to uniform 
 
 '^'^Praelectiones tresdecim in principium elementorum Euclidis, Oxonii habitae 
 MDCXX. Oxonii, . . . 1621. 
 
 i2Cajori represents (^History of Elementary Mathematics, New York, 1897, p. 281) 
 that the above mentioned lectures were delivered "about 1570." According to this 
 statement Savile was "sinking under the weight of years" about the time that he 
 attained to his majority for he wae born in 1549. 
 
112 SCHOOL SCIENCE AND MATHEMATICS 
 
 for the manner in which the boys dress is determined in part 
 by the State etc., are here set forth. 
 
 In German, references may be given to Klein's sketch of the 
 Teaching of Geometry in France, Elementarmathematik vom 
 hoheren Standpunkte cms, Teil II, Geometrie, Leipzig, 1909, pp. 
 456-476. 
 
 In English, the most recent article dealing with our subject 
 is Professor G. W. Myers' "Report on the Teaching of Math- 
 ematics in France" (with its curiously inaccurate statements 
 concerning the baccalaureat). School Review, September, 1911, 
 XIX, 433-453 Professor F. E. Farrington's book French Sec- 
 ondary Schools (Longmans, 1910) contains a chapter, pages 
 257-287, "Mathematics and Science"; the first six pages form 
 an historical introduction A pleasant sketch of "Life at the 
 Sorbonne," was given by H. Jones in the Nation, XCI, p. 576-577, 
 December 15, 1910 The second edition of Barrett Wendell's 
 ''France of To rfa/' (Constable, 1908) contains a chapter (p. 1- 
 46) on "The Universities" ; attractive in style and treatment, its 
 value is marred by a number of misleading and incorrect state- 
 ments In March, 1900, Professor James Pierpont -published in 
 Bulletin of the American Mathematical Society (VI2, 225-249) a 
 very interesting article entitled "Mathematical Instruction in 
 France." Although secondary education as it was before the 
 great reform of 1902-5, is analyzed to a certain extent, the article 
 was doubtless intended to be read more particularly by the grad- 
 uate students or professors in American universities In 1910 
 I presented a similar paper to the Royal Society of Canada ; it is 
 printed in the Transactions and Proceedings for 1910, IV3, 89- 
 152. Appendix A of this paper contains the programme for the 
 Concours of the Agregation des Sciences Mathematiques for 
 1910 and copies of the examination papers set in 1909. The 
 answers expected (unofficial) to the questions in the annual 
 agregation examination papers, appear a few months later in 
 Nouvelles AnncUes de Mathematiques, Revue de Mathematiques 
 Speciales and Journal de Mathematiques lementaires, A study 
 of a series of such examination papers and answers soon leads 
 to adequate appreciation of the mathematical capabilities of the 
 agrege or lycee professor The programmes and examination 
 papers for the agregation from year to year may be purchased 
 at Librairie Croville-Morant, Paris. In a new publication, 
 Annales du Baccalaureat (Librairie Vuibert) containing 9 fas- 
 icules a year, the first and ninth fasicules give full information 
 
MATHEMATICS IN THE FRENCH LYC&ES 113 
 
 with regard to the mathematical examination papers of can- 
 didates for the Baccalaurcat. 
 
 A List of Mathematical Text Books Used in French 
 Secondary Education. 
 
 No student of Secondary Education in France should omit the 
 study of the text-books. Similar preparation for professors in 
 lycees and universities, and the intimate relations between these 
 institutions has for many years inspired French university pro- 
 fessors to write school texts. Legendre, Clairaut, Bertrand, Bo- 
 billier and Catalan are among others in earlier days, while Gour- 
 sat, Tannery, Darboux, Hadamard, Meray and Borel are of 
 our time. On the other hand those authors who are lycee 
 professors are brilliant and excellently equipped mathemati- 
 cians. 
 
 The ignorance of the mathematician concerning the books used 
 in connection with elementary or secondary education of a 
 foreign country is very general, and yet, in the case of America, 
 at least, familiarity with works used in France would be a 
 source of great inspiration and help not only to the vast body 
 of text-book writers and teachers in high schools but also to 
 professors in colleges. For, as we have seen, the mathematical 
 student in the lycee is thoroughly conversant with those topics 
 usually treated well on in the undergraduate course of the better 
 American colleges, while those leaving the Classes de Mathemat- 
 iques Specicdes at the age of 18 or 19 years are many times better 
 equipped mathematicians than our sophomore or junior college 
 student of 20 or 21 who has specialized in mathematics. 
 
 To refer to the books in the following list as "text-books" will 
 not lead to any misconception when it is recalled that French 
 professors make sparing use of books in classes, since the 
 subject matter of the courses is usually presented by dictation 
 or lecture." But these books are such as the pupils have used 
 in working up the lectures, or such as a professor has encouraged 
 the pupil to consult in the lycee library. 
 
 My selection Is limited to about 35 works or courses which 
 I have thought would be representative. Many others which 
 would have been of interest to the American mathematician are 
 omitted on account of space limitations. An analysis of the 
 contents and characteristics of a number of the geometrical texts 
 
 iSThe reasons for this are clearly set forth in Rapports, French Sub-Committee, 
 /. c. III. 90. 
 
114 SCHOOL SCIENCE AND MATHEMATICS 
 
 may be found in M. Th. Rousseau's report (pages 88-110) re- 
 ferred to above. 
 
 The arrangement is alphabetical according to authors. The 
 date and place of publication, the name of the publisher, the 
 number of pages, and in most cases the names of the classes for 
 which the books were especially designed, are given. 
 
 At the end of the list I have added the names of four defunct 
 and three current periodicals, published in the interests of French 
 elementary and secondary mathematical education. Much of in- 
 terest in this department may be gleaned from these sets.^* 
 
 A number of French books (such as those by Gelin and Neu- 
 berg) by Belgian authors are purposely excluded from this list, 
 although they are widely used in France. Translations of books 
 originally written in a language other than French are also ex- 
 cluded ; otherwise I might have listed several popular works 
 (such as those by Alexandroff, Faifofer and Petersen). 
 
 The editions indicated in the following list are not necessarily the last 
 published but simply the last I have seen. The size of the works may 
 not be judged by the number of pages only as the formats vary from 
 16mo. to royal 8vo. ; e. g. Borel's course is in 12mo. Darboux's in large 8vo. 
 
 Antomari, X. Cours de geometric descriptive (Mathematiques Spe- 
 ciales) 3e ed. Paris (Vuibert)." 1906, 619 p. 
 
 Appell, p. Cours de mecanique (Mathematiques Speciales) 3e ed., 
 entierement refondue Paris (G. V.)." 1912, 527 p. 
 
 Appell, P. and Chappuis, J. Legons de mecanique (Mathematiques 
 A. B.). Paris (G. V.), 2 volumes. 
 
 Tome i : Notions geometrique. Cinematique. 3e ed. 1909, 178 p. 
 Tome ii : Dynamique et statique du point. Statique des corps solides, 
 machines simples. 2e ed. 1907, 240 p. 
 Appell, P. See also Briot and Bouquet. 
 
 AuBERT, P. and Papelier, G. Exercices d'algebre d'analyse et de trig- 
 onometric (Mathematiques Speciales). Paris (Vuibert), Tome i, 1908, 362 
 p.; tome ii (deuxieme annee), 1910, 359 p. 
 
 Borel, fi AND Royer, M. "Cours mile Borel." Paris (Colin) : - 
 Arithmetique (I Cycle), 1907, 220 p. 
 Geometrie (I, II, Cycle), 2e ed. 1908, 383 p. 
 
 Note: Klein (1. c) speaks of this work as "ein sehr interes- 
 santes Buch." 
 Algebra (I Cycle), 2e ed., 1905, 256 p. 
 
 Trigonometric (II Cycle), 2e ed., 1905, 198 ^.-[-Fasicule, 1908, 11 p. 
 Geometrie cotee par R. Danelle, 1908, 64 p. 
 
 Algebre (II Cycle), 3e ed., 1905, 401 p. + Fasicule, 1908, 30 p. 
 Bourdon, P. L. M. Application de I'algebre a la geometrie, com- 
 prenant la geometrie analytique a deux dimensions. 9e ed., revue et 
 annotee par G. Darboux. Paris (G. V.), 1906, 648 p. +10 pi. 
 
 '^^L'Enseignement Mathematique (XlVe Annee, 1912), published in Geneva, is 
 also of value. 
 
 isimmediately after the place of publication follows the name of the publisher. 
 The publishing house Vuibert was formerly known as "Vuibert' & Nony." 
 
 16G. v. Gauthier Villars. 
 
MATHEMATICS IN THE FRENCH LYC&ES 115 
 
 BouRLET, C. Legons d'algebre elementaire (Mathematiques A. B.), 
 5e ed., Paris (Colin), 1907, 566 p. Cours de Darboux. 
 
 BouRLET, C (i) filements de geometrie, plane et dans I'espace (I et 
 
 II Cycles A), Paris (Hachette), 1908, 3?8 p. [2e ed., 1910, vi+383 p.]" 
 
 (ii) Corriges des 773 exercices et problemes dans les Elements" 
 
 avec collaboration de Paul Baudoin, Paris (Hachette), 1908, 348 p. 
 
 BouRLET, C. Cours abrege de geometrie avec nombreux exercices 
 
 theoriques et pratiques et des applications au dessin geometriques avec 
 
 la collaboration de M. P. Baudoin, Paris (Hachette) : 
 
 (i) Geometrie plane (VI, V, IV, B), 4e ed., 1909, 408 p. 
 (ia) Corriges des exercices theoriques, 1908, 302 p. 
 (ii) Geometrie dans I'espace (III B), 3e ed., 1911, viii+239 p. 
 (iia) Corriges des exercices theoriques, 1908, 172 p. 
 BouRLET, C. LeQons de trigonometric rectiligne (Mathematiques A. B.), 
 3e ed., Paris (Colin), 1908, 322 p. Cours de Darboux. 
 
 Briot, Ch. Legons d'algebre. 2e partie revue par . Goursat, 18^ ed. 
 (Mathematiques Speciales), Paris (Delagrave), [1906], 635 p. 
 
 Briot, Ch. and Bouquet, J. C Legons de geometrie analytique, 15e ed., 
 revue et annotee par M. Appell. Paris (Delagrave), 1893, 758 p. 
 
 Note: The sections of this work devoted to plane analytical geom- 
 etry (about two-thirds of the whole work) were translated into' 
 English, by J. H. Boyd in 1896 (Werner School Book Co., Chicago 
 and New York). 
 Darboux, G. Cours complet pour la classe de mathematique A. B., 
 public sous la direction de G. Darboux ^See Bourlet (2), Hadamard, Tan- 
 nery, Tisserand. 
 
 Note: The title of this course is somewhat of a misnomer as the 
 various topics are treated with an elaboration scarce possible in 
 any lycee class outside of the Classe de Mathematiques 
 Speciales. 
 Darboux, G. See also Bourdon. 
 
 DupoRCQ, E. Premier principes de geometrie moderne a I'usage des 
 eleves de Mathematiques Speciales et des candidate a la licence et 
 a I'agregation. Paris (G. V.), 1890, 160 p. [2e ed., 1912, viii+174 p.] 
 
 F. J.^^ Elements de geometrie. Tours (Mame) et Paris (Poussielgue) 
 1909, 523 p. 
 
 F. G. M. Exercices de geometrie comprenant I'expose des methodes 
 geometriques et 2000 questions resolues. 5 ed Tours (Mame) et Paris 
 (Gigord), 1912, xxiv+1298 p. 
 
 Note: A livre du niaitre for the "Elements." Historical notes 
 abound and 39 pages are devoted to Indexes of various kinds. 
 F. J. Elements de geometrie descriptive. Tours (Mame) et Paris 
 (Poussielgue), 1910, 458 p. 
 
 F. G. M. Exercices de geometrie descriptive. 4e ed. Tours (Mame) et 
 Paris (Poussielgue), 1909, X+1099 p. Complements, idem, 1912, pp. 
 1100-1161. 
 
 Note: A livre du maitre for the "filements." Indexes, pp. 1073-1099. 
 Gomtrie, Cours superieure, par une reunion de professeurs. Tours 
 (Mame) et Paris (Poussielgue), [1908], 323 p. 
 
 Note: Preparatory for F. J. Elements de geometrie. 
 Grew, A. Traite d'algebre (Mathematiques A. B.), 4e ed., Paris 
 (Vuibert), 1908, 498 p. 
 
 ^"Information concerning editions I have not seen is added in this way. 
 iSThe two authors of this and the three following works are Freres of the 
 Ecoles Chretiennes. 
 
116 SCHOOL SCIENCE AND MATHEMATICS 
 
 Hadamard, J. Legons de geometric elementaire (Mathematiques A, B), 
 Paris (Colin) Cours de Darhoux. 
 
 Tome i : Geometric plane, 2e ed., 1906, 308 p. 
 Tome ii : Geometric de I'espacc, 1901, 582 p. 
 
 Humbert, . Traite d'arithmetique (Baccalaureat et ecoles du gou- 
 vernemcnt). Avec unc preface de Jules Tannery, 4e ed., Paris (Vuibert), 
 1908, 496 p. 
 
 KoEHLER, J. Exerciccs de geometric analytiqucs et de geometric su- 
 perieure a I'usage des candidats aux ficole Polytechnique et Normale et 
 a I'agregation. Questions et solutions. Paris (G. V.). Tome i, 1886, 
 "349 p. ; tome ii, 1888, 469 p. 
 
 Mray, Ch. Nouveaux elements de geometric. Nouv. ed rcfondue et 
 augmentee. Dijon (Jobard), 1903, 450 p.+23 pi. 3e ed., 1906, 300 p. 
 Note: Klein (1. c.) devotes more thai> a page to this highly inter- 
 esting work of which perhaps the most prominent characteristics 
 are treatment based on the idea of motion and the fusion of 
 planimetry and stereometry from the very first. M. Rousseau 
 gives up over five pages of his report to the discussion of 
 Meray's book. 
 
 NiEWENGLOWSKi, B. Cours de geometric analytique (Mathematiques 
 Speciales). Paris (G. V.), 3 tomes. Tome i: 2e ed., 1911, vi-|-496 p.; 
 tome ii : Constructions des courbes planes, complement rclatifs aux 
 coniques, 2e ed., 1911, iv+324 p. ; tome iii : geometric dans I'espacc 
 avec unc note de . Borel sur les transformations en geometric, 1896, 572 p. 
 
 NiEWENGLOWSKi, B. Cours d'algebre (Mathematiques Speciales), Paris 
 (Colin), 2 tomes et supplement. Tome i, 5e ed., 1902, 391 p.; tome ii, 
 5e ed., 1902, 488 p. ; supplement, 1904, 43 p. 
 
 Papelier, G. Precis de geometric analytique (Mathematiques 
 Speciales), Paris (Vuibert), 1907, 696 p. 
 Note: Pages 431-696, 3 dimensions. 
 
 Rouch, . et Comberousse, Ch. de. Elements de geometric suivis d'un 
 complement a Fusage des elevcs de mathematiques elementaircs et de math- 
 ematique speciales, etc., 7^ ed. Paris (G. V.), 1904, 651 p. 
 
 Rouch, fi. et Comberousse, Ch. de. Traite de geometric. 7^ ed., Paris 
 (G. v.), 1900. Tome i: geometric plane, 548 p.; tome ii : geometric 
 dans Tcspace, 664 p. 8e ed., 1912. 
 
 RoYER, M. See Borel, . 
 
 Serret, J. A. Traite de trigonometric. 9e ed., Paris (G. V.), 1908, 
 336 p. 
 
 Tannery, Jules. Legons d' arithmetique theorique et pratique (Mathe- 
 matiques A, B), 2e ed., Paris (Colin), 1911, xvi+545 p. Cours de Darboux. 
 
 Tannery, J. Notions de mathematiques avec notions historiqucs par 
 Paul Tannery, 3e ed, augmentee de notions d'astronomie (Programmes 
 du 1902 et 1905 Classc de philosophic), Paris (Delagrave), 1905, 370 p. 
 
 Tannery, J. Legons d'algebre et d'analyse (Mathematiques Speciales), 
 Paris (G. V.), 1906; tome i, 423 p.; tome ii, 636 p. 
 
 Tannery, J. See also Humbert, . 
 
 Tannery, P. See Tannery J. 
 
 Tisserand and Andoyer. Legons de Cosmographie (Mathematics A. 
 B.), Paris (Colin), 1909, 371 p.+12 pi. Cours de Darboux. 
 
 Vacquant, Ch. and Mac de Lepinay. Cours de trigonometric. Paris 
 (Masson). Premiere Partie (Classes C, D et Mathematiques, A. B). Nouv. 
 ed., 1909, 294 p. Deuxieme Partie (Mathematiques Speciales). Nouv. 
 ed., 1909, 172 p. 
 
MATHEMATICS IN THE FRENCH LYC&ES 117 
 
 L'Education Mathematique public par A. Durand et H. Vuibert XIV^ 
 annee, 1911-1912 (Vuibert), 20 numbers a year. 
 Note: Very elementary. 
 Journal de Mathematiques lementaires public par H. Vuibert, XXXVI^ 
 Annee, 1911-1912 (Vuibert), 20 numbers a year. 
 
 Revue de Mathematiques Speciales redigee par Humbert, G. Pape- 
 lier, P. Aubert, P. Lemaire, C. Riviere, H. Vuibert, XXIIe annee, 1911- 
 1912 (Vuibert), 10 numbers a year. 
 
 Note: The solutions of the more elementary portions of the ex- 
 aminations for the agregation are published each year in these 
 last two mentioned periodicals. 
 
 Bulletin de Mathematiques Elementaires dirigee par M. Ch. Michel. 
 Octobre, 1895 Juillet, 1910 (Lamarre). 
 
 Bulletin de Mathematiques Speciales redigee par Niewenglowski at de 
 Longchamps, Octobre, 1894 Juillet, 1900 (Lamarre). 
 
 Journal de Mathematiques lementaires redigee par Bourlet, de Long- 
 champs, G. Mariaud. Octobre, 1876 Mai, 1901 (Delagrave). 
 
 Journal de Mathematiques Speciales redigee par de Longchamps Mari- 
 aud. Octobre, 1879 Mai, 1901 (Delagrave). 
 
GITT 
 %uy 99 1914 
 
 SOME MATHEMATICAL BOOKLET SERIES. 
 
 U. C. ARCHIBALD 
 
 Reprinted from the 
 
 lUTLLKTIN OF THE AMERICAN MATHEMATICAL SOCIETY 
 
 2d Series, Vol. XX., No. 5, pp. 238-243. 
 
 New York, February, 1914 
 
[Reprinted from Bull. Amer. Math. Society, Vol. 20, No. 5, Feb., 1914 | 
 
 SOME MATHEMATICAL BOOKLET SERIES. 
 
 Matematica dilettevole e curiosa. Di Italo Ghersi. Con 693 
 figure originali dell'Autore. Milano, Ulrico Hoepli, 1913. 
 viii+730 pp. Price L. 9.50. 
 Wo steckt der Fehlerf Trugschlusse und Schillerfehler . Ge- 
 sammelt von Dr. W. Lietzmann und V. Trier. Mathe- 
 matische Bibliothek, Nr. 10. Leipzig and Berlin, B. G. 
 Teubner, 1913. 57 pp. Price M. 0.80. 
 English and French mathematical literature is entirely- 
 lacking in such admirable booklets dealing with elementary 
 topics, as those which have wide circulation in Germany and 
 Italy.f I refer to the Mathematische Bibliothek of the 
 
 t It may be suggested that the volumes on Elimination by Laurent and 
 on Geometrography by Lemoine, of the excellent " Scientia " series 
 (Gauthier-Villars, Paris) are elementary, but these are only two of a dozen 
 volumes by Appell, Gibbs, Hadamard, Poincar^, etc., which certainly may 
 not be classed in this way. And even these two brochures are more 
 
239 SOME MATHEMATICAL BOOKLET SEKIES. [Feb., 
 
 Sammlung Goschen,* the Lietzmann- Witting Mathematische 
 Bibliothek f and the mathematical volumes of the Biblioteca 
 degli studentijt and the Manuali Hoepli. 
 
 The Mathematische Bibliothek contains about 35 volumes 
 (4J X 6 J inches; uniform price, 22| cents), each neatly bound 
 in cloth and containing from 130 to 230 pages. A. Sturm, 
 H. Schubert, M. Simon, O. Th. Biirklen, K. Doehlemann 
 and E. Beutel are among the authors and the volumes treat 
 of History of mathematics, Plane geometry. Descriptive 
 geometry (2 volumes), Determinants, Analytical geometry 
 of the plane. Analytical geometry of space (notably fine 
 figures). Projective geometry. Algebraic curves (2 volumes), 
 Insurance mathematics. Vector analysis. Geodesy, Surveying, 
 Astronomy, etc. 
 
 Of the Mathematische Bibliothek herausgegeben von W. 
 Lietzmann und A. Witting a dozen volumes have already 
 appeared. They are bound in boards, contain 41 to 93 pages 
 (4f X 7j inches) each, and are of the same uniform price as the 
 Goschen Sammlung before 1913. In this series Wieleitner has 
 written on the Idea of number in its logical and historical de- 
 velopment; 0. Meissner is author of Theory of probabilities 
 Vvith applications; M. Zacharias wrote the Introduction to 
 projective geometry; Ziihlke, Geometrical constructions in a 
 I in. 1 ted plane; Beutel, Squaring the circle. 
 
 In the Biblioteca degli Studenti are nearly a score of vol- 
 umes (4x6j inches; limp covers; single numbers of about 85 
 pages, 10 cents, double numbers of about 170 pages, 20 cents). 
 They include. Manual of plane trigonometry. Manual of 
 spherical trigonometry. Exercises of elementary geometry, 
 Guide to the resolution of problems in algebra, Principles of 
 perspective, Repertorium of mathematics and elementary 
 physics, etc., and treat of very elementary topics. 
 
 Some 40 of the 1,200 odd volumes (4 J x 6 inches) in the 
 Manuali Hoepli series are of mathematical, content. Per- 
 haps the two best known works are the volumes (658+950 
 
 advanced in character than any of those in three, and than many of those 
 of the fourth series, about to be considered. The same may be remarked 
 concerning the Cambridge Tracts in Mathematics and Mathematical 
 Physics. 
 
 * G. J. Goschen'sche Verlagshandlung, BerUn und Leipzig. 
 
 t B. G. Teubner, Leipzig und Berhn, 1912-1913. 
 
 t Raffaello Giusti, editore, Livorno. 
 
 The second volume was reviewed by Professor White in this Bulletin, 
 vol. 19, pp. 417^19, May, 1913. 
 
1914.] SOME MATHEMATICAL BOOKLET SERIES. 240 
 
 pages) of Pascal's Repertorio di matematiche superiori* since 
 translated into German and enlarged, t and Pascal's Deter- 
 minanti e applicazioni, 1897, which three years later was 
 elaborated into a volume of the Sammlung von Lehrbiichern 
 auf dem Gebiete der mathematischen Wissenschaften. Then 
 there are 4 volumes on Algebra, 4 on Arithmetic, 2 on Astron- 
 omy, 4 on the Calculus, including volumes on Calculus of 
 variations and Finite differences,t and Critical exercises on the 
 differential and integral calculus; 1 on Mathematical formulae ; 
 Saccheri's Euclide emendato; 3 volumes on Functions (analytic, 
 elliptic, polyhedral and modular||); 13 on Geometry, 1 on the 
 Mathematics of economics, 1[ 1 on Groups, 4 on Mechanics, and 
 1 by G. Loria on Exact science in ancient Greece.** The 
 volume of Ghersi under review is the second he has written for 
 this mathematical series, the earlier one having dealt with 
 Methods for resolving problems of elementary geometry. 
 
 In recent times English, French, and German writers have 
 published popular works for recreation hours of those who are 
 in any wise interested in mathematics. Ball's Mathemat- 
 ical Recreations and Essays, which has recently reached 
 a fifth edition, tt is almost a classic in its special field. The 
 older works of Lucas, " Recreations mathematiques "tt and 
 ** L'Arithmetique amusante " are frequently referred to, 
 
 * Milano, 1898 and 1900. The first volume is reviewed by E. O. Lovett 
 in this Bulletin, vol. 5, pp. 357-362, April, 1899. 
 
 t Two volumes, Leipzig, 1900, 1901. New edition to be completed in 
 4 volumes; vols, li, 2i, 1910, reviewed in this Bulletin by C. H. Sisam, 
 vol. 19, pp. 372-374, April, 1913. 
 
 X Translated into German by A. Schepp. Leipzig, 1899. Reviewed 
 by J. K. Whittemore in this Bulletin, vol. 6, pp. 352-4, May, 1900. 
 
 This volume by Rossotti was reviewed by E. O. Lovett in this Bulletin, 
 vol. 5, pp. 261-2, Feb., 1899. 
 
 II This volume by Vivanti was reviewed by J. I. Hutchinson in this 
 Bulletin, vol. 14, pp. 144-5, Dec, 1907. The French edition by A. Cahen 
 was reviewed by G. A. Miller in this Bulletin, vol. 19, pp. 534-5, July, 
 1913. 
 
 U This volume by Virgilii was reviewed by J. M. Gaines in this Bulletin, 
 vol. 5, pp. 488-9, July, 1899. 
 
 ** This work, which has just been published, 1914, contains about 1000 
 pages. The title page with "seconda edizione totalmente riveduta" is 
 misleading, as the original work of over 900 pages of quarto format was a 
 reprint of memoirs (in five books) in Mem. Ace. Modena (2), vol. 10, pp. 
 3-168; vol. 11, pp. 3-237; vol. 122, pp. 3-411; (1893-1902). Futhermore, 
 "Libro II, II period aureo della geometria Greca " appeared in still another 
 form in Mem. Reale Ace. d. Sc. di Torino (2), vol. 40, pp. 369-445; (1890). 
 
 tt London, 1911. 
 
 $t Paris, T. I, 2 6d., 1891; T. II, 2^ d., 1896; T. Ill, 1893; T. IV, 1894. 
 
 Paris, 1895. 
 
241 SOME MATHEMATICAL BOOKLET SERIES. [Feb., 
 
 while the circulation of Ahren's Mathematische Unter- 
 haltungen iind Spiele* and Schubert's Mathematische 
 Musestundent is confined more to Germany. Each work 
 has its own peculiar ideals, but Ball is perhaps the most 
 comprehensive in range, while he and Ahrens alone introduce, 
 to an appreciable extent, references to the widely scattered 
 literature of the subject. E. Fourrey's " Curiosites geome- 
 triques "t is also notably full in exact statement of authorities. 
 
 From works such as these, from books like Blythe's on 
 Models of cubic surfaces, Catalan's Theoremes et problemes 
 de geometric elementaire, Cremona's Elementi di geometria 
 proiettiva, Enriques' Questioni riguardanti la geometria elemen- 
 tare, de Longchanms' Essai sur la geometric de la regie et de 
 I'equerre, Loria's Spezielle algebraische und transzendente 
 ebene Kurven, and from various periodicals, Ghersi has 
 compiled the present little work on Matematica dilettevole 
 e curiosa. 
 
 The first 74 pages are taken up with " Curious and bizarre 
 problems " such as: Euler's problem of the Konigsberg 
 bridges, the Hampton Court maze and other unicursal 
 problems, map-coloring problem, and chess problems. Of 
 course little more than the statement of a problem is frequently 
 given. 
 
 In the next 100 pages various curious properties of numbers, 
 and problems of arithmetic and arithmetic geometry are set 
 forth. For example, we have properties of perfect and amic- 
 able numbers, of the triangle of Pascal, of Lucas's singular 
 products, as well as problems of Benedetti (Speculationes 
 diversse, 1585) and of Leonardo Pisano (Liber Abaci, 1202). 
 
 Fermat's equation and other problems of the theory of 
 numbers are treated in the next 15 pages, then follows a 
 collection of miscellaneous algebraic problems which conclude 
 with graphical solutions of equations of the second, third and 
 fourth degrees and with a sketch of Demanet's and Meslin's 
 hydrauUc,t and Lucas's electric solution of equations. 
 
 Magic squares, magic polygons, and magic polyhedra are 
 illustrated on pages 251-326. 
 
 * Leipzig, 1901. 
 
 t Grosse Ausgabe, Leipzig, Bd. I, 3. Aufl., 1907; Bd. II, 2. Aufl., 1900. 
 
 t2ed., Paris, 1906. 
 
 Cf. " Two hydraulic methods to extract the nth root of any number " 
 and " HydrauHc solution of an algebraic equation of the nth degree," 
 by A. Emch, American Mathematical Monthly, January and March, 1901, 
 vol. 8, pp. 10-12 and 58-9. 
 
1914.] SOME MATHEMATICAL BOOKLET SERIES. 242 
 
 Then follow 350 pages treating of miscellaneous questions 
 in geometry. On pages 329-367 we find definitions and deri- 
 vation of properties, of notable transcendental, and cubic, 
 quartic, and other algebraic curves. The next dozen pages 
 contain instruments for tracing by continuous motion such 
 curves as the conic sections, cissoid, and conchoid. Some 20 
 pages given over to discussion of the solution of problems 
 in elementary geometry, by ruler and compass, and then 
 (pages 407-422) cyclotomy is touched upon. Then come 
 100 pages occupied with the problems of trisection of an 
 angle, squaring the circle, duplication of the cube. Dissection 
 of figures, geometrical pavements, star-polyhedra, and hyper- 
 space are some of the concluding topics under the head 
 geometry. 
 
 In the final sections are paradoxes and other recreations in 
 mechanics. 
 
 It will be remarked, as indeed the title implies, that the 
 volume is not confined to so-called recreations, although 
 these occupy the major part of the volume. It is written with 
 light touch and anyone unacquainted with books on mathe- 
 matical recreations may pass a few pleasant hours in turning 
 over the pages and find some things not met with in other 
 books of the kind. The reader who wishes to learn more of 
 the underlying theory will then naturally turn for guidance 
 to such a book as Ball's or to the article in the Encyklopadie* 
 or to such works in fields other than those of recreations, as 
 mentioned above. 
 
 In the Lietzmann-Trier Bandchen, which may be classed as 
 a small addition to the literature of mathematical recreations, 
 Lietzmann collected the 36 fallacies (Trugschliisse) and Trier 
 the 50 pupils' mistakes. Arithmetic, algebra, elementary 
 geometry (synthetic and analytic), trigonometry are the only 
 subjects illustrated. The errors in the reasoning are not 
 indicated. 
 
 Among the fallacies are (1) numerous examples depending 
 for their results upon division of each side of an equality by 
 zero or neglect of consideration of double sign before a radical; 
 (2) a series of geometrical paradoxes, several of which are 
 already familiar through Ball's book. 
 
 * Mathematische Spiele, von W. Ahrens, vol. I, 2, pp. 1080-1093, Leip- 
 zig, 1902. 
 
243 SOME MATHEMATICAL BOOKLET SERIES. 1914.] 
 
 Here is an example of a different kind, which appears to be 
 new: "Consider 
 
 (1) loge2= 1-1/2+1/3-1/4+1/5 ; 
 
 multiplying through by 2 we get 
 
 2 loge 2 = 2-1 + 2/3-1/2 + 2/5- 1/3 + 2/7 - -. 
 
 Collecting terms with common denominators and arranging 
 according to increasing denominators, we get 
 
 (2) 2 loge 2=1-1/2 + 1/3-1/4+1/5 . 
 
 This is, however, the same as (1). Therefore 
 
 loge 2 = 2 log. 2." 
 
 The examples of Schiilerfehler are taken from the exercises^ 
 of Danish pupils. The vagaries of American youth suggest 
 that an equally interesting collection could be made on this 
 side of the water. The error in No. 32 is not evident. But 
 here is No. 36 : " Given two circles which cut one another in 
 P and Q and touch the sides of an angle, on the same side of 
 the vertex, at the points A, Ai for one circle and B, Bi for the 
 other. Prove (1) that PQ produced passes through the middle 
 points of AB and AiBi] (2) that AA^ BBi and PQ are parallel 
 to one another." Solution: " PQ cuts AB in C, AiBi in Ci. 
 Then by the power theorem, CA^ = CP ^ CQ = CB\ There- 
 fore C is the middle point of AB. In the same way Ci is the 
 middle point of AiBi. AAi, PQ and BBi are parallel to one 
 another because they cut off the equal segments on the lines AB 
 and AiBi." 
 
 Finally, No. 47: " The sides of a triangle are a, b, c. To 
 express sin A in terms of the given quantities.'* Solution: 
 "Of course the following relations hold good: 
 
 a h c 
 
 sin A sin B sin C * 
 
 In a proportion it is allowable to interchange the means; hence 
 
 a b sin B . > ^c 
 
 sm ^ = -r . 
 
 sin ^ c sin C * * * b 
 
 R. C. Archiba ld. 
 
 Brown University, 
 Providence, R. I. 
 
/ 29 1914 
 
 MATHEMATICAL MODELS. 
 
 R. C. ARCHIBALD 
 
 Reprinted from the 
 
 BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY 
 
 2d Series, Vol. XX., No. 5, pp. 244-247 
 
 New York, Febrnary, 1914 
 
[Reprinted from Bull. Amer. Math. Society, Vol. 20, No. 5, Feb., 1914.] 
 
 MATHEMATICAL MODELS. 
 
 Catalog mathematischer Modelle filr den hoheren mathematischen 
 
 Unterricht. Veroffentlicht durch die Verlagshandlung von 
 
 Martin Schilling in Leipzig, mit 106 Abbildungen. Siebente 
 
 Auflage, Leipzig, 1911, xiv + 172 pp. 
 Verzeichnis von H. Wieners und P. Treutleins Sammlung 
 
 mathematischer Modelle filr Hochschulenf hohere LehranstaUen 
 
 und technische Fachschulen. Zweite Ausgabe mit 6 Tafeln. 
 
 Leipzig und Berlin, Verlag von B. G. Teubner, 1912, 64 pp. 
 Ahhandlungen zur Sammlung mathematischer Modelle. In 
 
 zwanglosen Heften herausgegeben von Hermann Wiener. 
 
 Leipzig, Verlag von B. G. Teubner. 1. Heft von H. 
 
 Wiener, 1907, 90 pp.; 2. Heft von P. Treutlein, 1911, 20 pp. 
 Illustrierter Spezialkatalog mathematischer Modelle und Ap- 
 
 parate. Entworfen von G. Koepp und anderen bewahrten 
 
 Fachmannern. New York City, Eimer and Amend, 
 
 128 pp. 
 
 For fifteen years the bulk of the models used by advanced 
 mathematical students all over the country has been procured 
 from Schilling of I^eipzig. This firm was developed from the 
 Firma L. Brill of Darmstadt, the foundation of which reaches 
 back some 35 years. Klein and A. von Brill, in those early years 
 professors in the Technische Hochschule at Munich, had more 
 than two score of models made for the Hochschule under 
 their direction. Copies of these (for example: the tractrix 
 of revolution, geodetic lines on an ellipsoid of rotation, 
 Kummer's surface, forms of Dupin's cyclide, the spherical 
 catenary, twisted cubics) in gypsum, wire, and brass form a 
 portion of the great Schilling collection. In more recent times 
 construction of new models has been carried on by the as- 
 sistance of many other mathematicians. Among them are 
 Professors Dyck, Finsterwalder, Kummer, Schoenflies, H. A. 
 Schwarz, C. und H. Wiener. 
 
 As nearly ten years had passed since the sixth edition of the 
 catalogue, the seventh editionf fills a long felt want. It 
 
 t Descriptions of models which have been manufactured since this edition 
 was published, have appeared in Jahreshericht d. Deutsch. Math.- Ver., 1913, 
 vol. 22, pp. 75-76, 134-137. 
 
245 MATHEMATICAL MODELS. [Feb., 
 
 describes some 400 models. Nothing more than an indication 
 of the subjects illustrated can be given here: surfaces of the 
 second order, algebraic surfaces of the third order, algebraic 
 surfaces of the fourth and higher orders, line geometry, screw 
 surfaces, space curves and developable surfaces, descriptive 
 and projective geometry, analysis situs, algebra, function 
 theory, mechanics and kinematics, mathematical physics, and 
 structure of crystals. 
 
 Shortly after the third International Mathematical Congress 
 at Heidelberg in 1904, Teubner offered to the public a selection 
 of about 60 of the mathematical models for Hochschule in- 
 struction which had been exhibited at the Congress by the 
 mathematical Institut of the Technische Hochschule of 
 Darmstadt. The construction of the models in the selection 
 was inspired by Professor H. Wiener.* In the new catalogue 
 now before us we find that Professor Wiener has increased 
 his collection by 50 models, while the late Professor Treutlein 
 has contributed about 200 more.f All of the models are 
 designed as aids to instruction in German secondary schools 
 and Hochschulen. For students of higher mathematics the 
 models of twisted curves and deformable quadric surfaces 
 will probably be the only ones of especial interest. 
 
 The Abhandlungen are intended to be of value for those using 
 the models. In Heft 1 are 9 Abhandlungen by Wiener: (1) 
 Mathematical models and their use in instruction (pages 3-8) ; 
 (2) On the projection of some plane figures (9-10); (3) The 
 regular Platonic polyhedra, Regularity in a group (11-14); 
 
 (4) Regular polygons and closed reflective systems (15-18); 
 
 (5) The building up of the regular polyhedra (19-51); (6) How 
 shall surfaces, especially those of the second order, be drawn? 
 (52-54) ; (7) On surfaces of the second order (55-84) ; (8) De- 
 formable thread models of ruled surfaces of the second order 
 with fixed thread lengths (85-87); (9) Deformable metal-bar 
 models for transforming a surface of the second order into 
 confocal surfaces (88-91). 
 
 These Abhandlungen are similar to those which Schilling 
 
 * The catalogue (Verzeichnis mathematischer Modelle, 28 pp.) was pub- 
 lished in 1905. 
 
 t An interesting account of these models written by Prof. H. Wiener, 
 may be seen in Jahresbericht d. Deutsch. Math.-Ver., Nov., 1913, vol. 22, 
 pp. 297-304. 
 
1914.] MATHEMATICAL MODELS. 246 
 
 distributes* with his models and some of them are of consider- 
 able interest; on the one hand because of the developments of 
 the theory of the surfaces, on the other through the applica- 
 tion of the theory to construction of the models. Numerous 
 bibliographical references are given. In illustration of these 
 characteristics note, for example, (5) and (9). 
 
 In (5) the first five pages contain an historical review of the 
 subject, then the theoretical considerations are treated under 
 the headings: The notion of a polyhedron (e. g.. Idea of a side, 
 of "Vielkant," of " Vielflach," of " Vielzell") ; First and 
 second definitions of the regular polyhedron by the group; 
 Transformation of an angle into a neighboring angle (by 
 rotation); Range of the different suppositions; Third definition 
 of the regular polyhedra; Construction of a regular poly- 
 hedron from its group. 
 
 In (9) we find that the construction of the model was 
 made possible through theorems of Henrici and Greenhill. 
 Among other studies in this connection, those of Mannheim, 
 Darboux, and Schur are also considered. 
 
 The second Heft, written by Treutlein, contains Abhand- 
 lungen on the following subjects: " On the intuitive method 
 of mathematical instruction" (pages 3-6) ; " On mathematical 
 models and their use in teaching" (7-9); "Explanations in 
 connection with the series, and the single models, of the 
 Treutlein collection " (10-20). 
 
 In all of the above mentioned publications, Dyck's Katalogf 
 is frequently referred to. 
 
 The Eimer and Amend collection is of use more particularly 
 in connection with elementary work in planimetry, stereom- 
 
 *The "Erste Folge, Abhandlungen zu den Serien I-XXIII, mit 71 
 Figuren auf 6 Tafeln und im Text " have also been published in a single 
 volume, in connected form. In the " Neue Folge " Hefte 1-9 have been 
 already issued between 1899 and 1912. The authors are: Fr. Schilling, H. 
 Wiener, W. Ludwig, H. Grassmann, W. Boy, E. Estanave, R. Hartenstein. 
 and F. Pfeififer. 
 
 t Katalog mathematischer und mathematisch-physikalischer Modelle, 
 Apparate und Instrumente. Unter Mitwirkung zahlreicher Fachgenossen 
 herausgegeben im Auftrage des Vorstandes der Deutschen Mathematiker- 
 Vereinigung von geh. Hofrat Dr. Walther v. Dyck, Professor an der 
 Technischen Hochschule zu Miinchen. Teubner, Leipzig, 1892, xvi+ 
 430 pp. 
 
 Nachtrag, Leipzig, 1893, x+135 pp. 
 
 These volumes contain papers by Klein, Voss, Brill, Hauck, v. Braun- 
 miihl, Boltzmann, Amsler, and Henrici, beside descriptions of the vari- 
 ous models by their respective designers. 
 
247 MATHEMATICAL MODELS. 1914.] 
 
 etry, trigonometry, and related branches. The models of star- 
 polyhedra, Poinsot polyhedra, and the so-called Archimedian 
 semi-regular solids may be mentioned as desirable for more 
 advanced mathematical considerations. 
 
 R. C. Archibald. 
 
 Brown University, 
 Providence, R. I. 
 
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