Military schools and courses of instruction in the science and art of war,
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Military Schools
and
COURSES OF INSTRUCTION
in the
SCIENCE AND ART OF WAR,
in
France, Prussia, Austria, Russia, Sweden, Switzerland, Sardinia,
England, and the United States.
Drawn from Recent Official Reports and Documents.
by HENRY BARNARD, LL.D.
REVISED EDITION.
New York:
Published By E. Steiger,
22 & 24 Frankfort Street.
1872.
REVISED EDITION.
The first edition of Military Schools in France and Prussia was issued
in 1862, as a number of the American Journal of Education; and
subsequently in the same year this portion was printed as Part I. of a
comprehensive survey of the whole field of Instruction in the Science
and Art of War in different countries. The circumstances under which the
publication was begun, are set forth in the Preface to the imperfect
edition of 1862. Now that the survey in the serial chapters of the
Journal is as complete as the material at the command of the Editor, and
the space which he can give to this special subject enable him to make
it, the several chapters have been revised and brought together in a
single volume, to present the actual condition of this important
department of national education in the principal states of Europe, as
well as in our own country.
It is due to the late Col. Samuel Colt, the inventor of the Colt
Revolver, and the founder of the Colt Patent Fire-Arms Factory--two
enterprises which have changed the character and the mode of
constructing fire-arms in every country--to state that the information
contained in the first edition of this Treatise, was collected and
prepared at his request, to assist him in maturing the plan of a School
of Mechanical Engineering, which he proposed to establish on his estate
at Hartford, and on which, after the breaking out of the War of
Secession, he decided to engraft both military drill, and military
history, and to give that scientific instruction which every graduate of
our national Military and Naval Academies ought to possess. Soon after
Col. Colt’s death (Jan. 10, 1862), Mrs. Elizabeth Jarvis Colt, learning
what had been done in the direction of her husband’s wishes, authorized
the use which has been made, of the material already collected, in the
preparation of this treatise, and of the volume already published on
Technical Schools in different countries, and of any more which might be
collected and prepared at her expense, to illustrate any department of
his plan of a scientific school at Hartford.
HENRY BARNARD.
HARTFORD, CONN., March, 1872.
CONTENTS.
PAGE.
INTRODUCTION, 3
I. FRANCE.
OUTLINE OF MILITARY SYSTEM, 9
System of Military Instruction, 10
I. Polytechnic School at Paris, 11
1. Subject and Methods of Instruction prescribed
for Admission, 13
2. Scientific Course in _Lycées_ and other Schools
in reference to, 49
3. History, Management, Studies, Examinations, 55
4. Public Services, Legal and Military,
provided for by, 88
5. Programmes of Lectures and Courses of Instruction, 91
II. The Artillery and Engineer School of Application
at Metz, 133
III. The Regimental Schools of Practice for Artillery
and Engineers, 221
IV. The Infantry and Cavalry School at St. Cyr, 225
V. The Cavalry School of Practice at Saumur, 241
VI. The Staff School at Paris, 245
VII. The Military Orphan School at La Fleche, 257
VIII. The School of Musketry at Vincennes, 259
IX. The Military and Naval Schools of Medicine
and Pharmacy, 261
X. The Naval School at Brest, 263
XI. The Military Gymnastic School at Vincennes, 265
Remarks on French Military Education, 273
II. PRUSSIA.
OUTLINE OF MILITARY SYSTEM AND MILITARY EDUCATION, 275
I. Outline of Military System, 281
II. Historical View of Military Education, 284
III. Present System of Military Education and Promotion, 293
IV. Examinations; General and Professional for
a Commission, 297
1. Preliminary or Ensign’s Examination, 297
2. Officers’ Examination, 302
V. Military Schools preparatory to the Officers’
Examination, 310
1. The Cadet Schools, or Cadet Houses, 310
2. The Division Schools, 321
3. The United Artillery and Engineers’ School, 325
VI. The School for Staff Officers at Berlin, 330
VII. Elementary Military Schools for Non-commissioned
Officers, 329
1. Military Orphan Houses, 339
Orphan-House at Potsdam, 340
Orphan-House at Annaburg, 345
2. The School Division or Non-commissioned Officers’
School, 348
3. Regimental Schools, 350
4. The Noble-School at Liegnitz, 350
VIII. Remarks on the System of Military Education
in Prussia, 351
APPENDIX, 351
The Artillery and Engineer School at Berlin, 353
The Staff School at Berlin, 395
III. AUSTRIA.
MILITARY SYSTEM AND INSTRUCTION 409-464
I. Schools of non-commissioned officers 411
II. School for officers 429
III. Special Military Schools 436
IV. Staff School at Vienna 447
V. Reorganization of Military Schools in 1868 453
VI. Cavalry Brigade School for officers 463
IV. BAVARIA, SAXONY, HOLLAND.
MILITARY SYSTEM AND SCHOOLS OF BAVARIA 465-480
I. Cadet Corps--War School--Artillery, Engineers, and
Staff Schools 467
II. MILITARY ACADEMY AT DRESDEN 471
III. MILITARY ACADEMY AT BREDA 477
V. ITALY.
MILITARY SYSTEM AND SCHOOLS 481-500
I. Military Academy at Turin 483
II. Artillery and Engineer School 489
III. Staff School and Staff Corps 492
IV. Regimental School for officers 494
V. School for Artillery officers 498
VI. Nautical School at Genoa 499
VI. RUSSIA.
MILITARY SYSTEM AND SCHOOLS 501-514
I. Imperial Staff School at St. Petersburg 505
VII. SWEDEN, &c.
MILITARY SYSTEM AND SCHOOLS 515-516
VIII. GREAT BRITAIN.
MILITARY SYSTEM AND SCHOOLS 511-686
I. Council of Military Education 535
II. Royal Military College at Sandhurst 557
III. Royal Military Academy at Woolwich 585
IV. Royal School of Military Engineering at Chatham 595
V. Professional Instruction for officers.
1. Survey Class at Aldershot.
2. Advanced Class of Artillery at Woolwich.
3. School of Gunnery at Shoeburyness 605
VI. Staff College and Staff appointments 619
VII. School of Musketry, and Army Schools 625
VIII. Naval and Navigation Schools 627
IX. English and other Naval Systems and Schools compared 655
1. French Naval and Navigation Schools 659
2. German Naval and Navigation Schools 681
IX. SWITZERLAND.
MILITARY SYSTEM AND MILITARY INSTRUCTION 687-714
I. Federal Militia--Cantonal Cadet System--
Target Shooting 689
II. Federal Instruction of officers--experience of 1870 710
X. UNITED STATES.
MILITARY SYSTEM AND SCHOOLS 713-940
A. Military Education for Land Service 715
I. National Military Academy at West Point 721
II. Special Artillery School at Fortress Monroe 819
III. Military element in State Schools 825
IV. Individual and Corporate Institutions 838
V. Military Drill in Public Schools 865
B. Naval and Navigation Schools 887
I. United States Naval Academy at Annapolis 897
II. School of Naval Construction and Marine Engineering 937
III. Instruction for the Mercantile Marine 939
GENERAL REVIEW OF MILITARY SYSTEM AND SCHOOLS 945
MILITARY SCHOOLS AND EDUCATION.
An account of the Military and Naval Schools of different countries,
with special reference to the extension and improvement, among
ourselves, of similar institutions and agencies, both national and
state, for the special training of officers and men for the exigencies
of war, was promised by the Editor in his original announcement of “_The
American Journal and Library of Education_.” Believing that the best
preparation for professional and official service of any kind, either of
peace or war, is to be made in the thorough culture of all manly
qualities, and that all special schools should rest on the basis, and
rise naturally out of a general system of education for the whole
community, we devoted our first efforts to the fullest exposition of the
best principles and methods of elementary instruction, and to
improvements in the organization, teaching, and discipline of schools,
of different grades, but all designed to give a proportionate culture of
all the faculties. We have from time to time introduced the subject of
Scientific Schools--or of institutions in which the principles of
mathematics, mechanics, physics, and chemistry are thoroughly mastered,
and their applications to the more common as well as higher arts of
construction, machinery, manufactures, and agriculture, are
experimentally taught. In this kind of instruction must we look for the
special training of our engineers, both civil and military; and schools
of this kind established in every state, should turn out every year a
certain number of candidates of suitable age to compete freely in open
examinations for admission to a great National School, like the
Polytechnic at Paris, or the purely scientific course of the Military
Academy at West Point, and then after two years of severe study, and
having been found qualified by repeated examinations, semi-annual and
final, by a board composed, not of honorary visitors, but of experts in
each science, should pass to schools of application or training for the
special service for which they have a natural aptitude and particular
preparation.
The terrible realities of our present situation as a people--the fact
that within a period of twelve months a million of able bodied men have
been summoned to arms from the peaceful occupations of the office, the
shop, and the field, and are now in hostile array, or in actual
conflict, within the limits of the United States, and the no less
alarming aspect of the future, arising not only from the delicate
position of our own relations with foreign governments, but from the
armed interference of the great Military Powers of Europe in the
internal affairs of a neighboring republic, have brought up the subject
of MILITARY SCHOOLS, AND MILITARY EDUCATION, for consideration and
action with an urgency which admits of no delay. Something must and will
be done at once. And in reply to numerous letters for information and
suggestions, and to enable those who are urging the National, State or
Municipal authorities to provide additional facilities for military
instruction, or who may propose to establish schools, or engraft on
existing schools exercises for this purpose,--to profit by the
experience of our own and other countries, in the work of training
officers and men for the ART OF WAR, we shall bring together into a
single volume, “_Papers on Military Education_,” which it was our
intention to publish in successive numbers of the NEW SERIES of the
“_American Journal of Education_.”
This volume, as will be seen by the Contents, presents a most
comprehensive survey of the Institutions and Courses of Instruction,
which the chief nations of Europe have matured from their own
experience, and the study of each other’s improvements, to perfect their
officers for every department of military and naval service which the
exigences of modern warfare require, and at the same time, furnishes
valuable hints for the final organization of our entire military
establishments, both national and state.
We shall publish in the Part devoted to the United States, an account of
the Military Academy at West Point, the Naval Academy at Newport, and
other Institutions and Agencies,--State, Associated, and Individual, for
Military instruction, now in existence in this country, together with
several communications and suggestions which we have received in
advocacy of Military Drill and Gymnastic exercises in Schools. We do not
object to a moderate amount of this Drill and these exercises, properly
regulated as to time and amount, and given by competent teachers. There
is much of great practical value in the military element, in respect
both to physical training, and moral and mental discipline. But we do
not believe in the physical degeneracy, or the lack of military aptitude
and spirit of the American people--at least to the extent asserted to
exist by many writers on the subject. And we do not believe that any
amount of juvenile military drill, any organization of cadet-corps, any
amount of rifle or musket practice, or target shooting, valuable as
these are, will be an adequate substitute for the severe scientific
study, or the special training which a well organized system of military
institutions provides for the training of officers both for the army and
navy.
Our old and abiding reliance for industrial progress, social well being,
internal peace, and security from foreign aggression rests on:--
I. The better Elementary education of the whole people--through better
homes and better schools--through homes, such as Christianity
establishes and recognizes, and schools, common because cheap enough for
the poorest, and good enough for the best,--made better by a more
intelligent public conviction of their necessity, and a more general
knowledge among adults of the most direct modes of effecting their
improvement, and by the joint action of more intelligent parents, better
qualified teachers, and more faithful school officers. This first great
point must be secured by the more vigorous prosecution of all the
agencies and measures now employed for the advancement of public
schools, and a more general appreciation of the enormous amount of
stolid ignorance and half education, or mis-education which now
prevails, even in states where the most attention has been paid to
popular education.
II. The establishment of a System of Public High Schools in every
state--far more complete than exists at this time, based on the system
of Elementary Schools, into which candidates shall gain admission only
after having been found qualified in certain studies by an open
examination. The studies of this class of schools should be preparatory
both in literature and science for what is now the College Course, and
for what is now also the requirements in mathematics in the Second
Year’s Course at the Military Academy at West Point.
III. A system of Special Schools, either in connection with existing
Colleges, or on an independent basis, in which the principles of science
shall be taught with special reference to their applications to the Arts
of Peace and War. Foremost in this class should stand a National School
of Science, organized and conducted on the plan of the Polytechnic
School of France, and preparatory to Special Military and Naval Schools.
IV. The Appointment to vacancies, in all higher Public Schools, either
among teachers or pupils, and in all departments of the Public Service
by Open Competitive Examination.
HENRY BARNARD.
HARTFORD, CONN., 1862.
[Errata for (first) Table of Contents:
VIII. GREAT BRITAIN.
VIII GREAT BRITAIN.
... V. 2. Advanced Class of Artillery at Woolwich.
Classs]
* * * * *
PART I
MILITARY SYSTEM AND SCHOOLS IN FRANCE.
* * * * *
AUTHORITIES.
The following account of the SYSTEM OF MILITARY EDUCATION IN FRANCE,
except in the case of three or four schools, where credit is given to
other authorities, is taken from an English Document entitled “_Report
of the Commissioners appointed_ (by the Secretary of War) _to consider
the best mode of reorganizing the system of Training Officers for the
Scientific Corps: together with an Account of Foreign and other Military
Education._” Reference has been had, especially in the Programmes and
Courses of Instruction to the original authorities referred to by the
Commissioners.
I. GENERAL MILITARY ORGANIZATION OF FRANCE.
Vauchelle’s Course d’ Administration Militaire, 3 vols.
II. THE POLYTECHNIC.
1. Fourcy’s Histoire de l’Ecole Polytechnique.
2. Décret portant l’Organisation, &c.
3. Règlement pour le Service Interieur.
4. Programme de l’Enseignement Interieur.
5. Programme des Connaissances Exigées pour Admission, &c.
6. Rapport de la Commission Mixte, 1850.
7. Répertoire de l’Ecole Polytechnique; by M. Marielle.
8. Calenders from 1833.
9. Pamphlets--by M. le Marquis de Chambray, 1836; by V. D.
Bugnot, 1837; by M. Arago, 1853.
III. SCHOOL OF APPLICATION AT METZ, AND St. CYR.
Décret Impérial, &c., 1854.
IV. SCHOOL FOR THE STAFF AT PARIS.
Manuel Réglementaire a l’Usage, &c.
V. ANNUAIRE DE L’INSTRUCTION PUBLIQUE, 1860.
MILITARY SYSTEM AND SCHOOLS OF FRANCE
I. MILITARY SYSTEM.
The French armies are composed of soldiers levied by yearly conscription
for a service of seven years. Substitutes are allowed, but in accordance
with a recent alteration, they are selected by the state. Private
arrangements are no longer permitted; a fixed sum is paid over to the
authorities, and the choice of the substitutes made by them.
The troops are officered partly from the military schools and partly by
promotion from the ranks. The proportions are established by law.
One-third of the commissions are reserved for the military schools, and
one-third left for the promotion from the ranks. The disposal of the
remaining third part is left to the Emperor.
The promotion is partly by seniority and partly by selection.
The following regulations exist as to the length of service in each rank
before promotion can be given, during a period of peace:--
A second Lieutenant can not be promoted to Lieutenant
under 2 years’ service.
A Lieutenant “ “ Captain “ 2 “
A Captain “ “ Major “ 4 “
A Major “ “ Lieut-Col. “ 3 “
A Lieutenant-Colonel “ “ Colonel “ 2 “
But in time of war these regulations are not in force.
Up to the rank of captain, two-thirds of the promotion takes place
according to seniority, and the other one-third by selection.
From the rank of captain to that of major (_chef de bataillon ou
d’escadron_) half of the promotion is by seniority and the other half by
selection, and from major upwards, it is entirely by selection.
The steps which lead to the selection are as follows:--The general
officers appointed by the minister at war to make the annual inspections
of the several divisions of the army of France, who are called
inspectors-general, as soon as they have completed their tours of
inspection, return to Paris and assemble together for the purpose of
comparing their notes respecting the officers they have each seen, and
thus prepare a list arranged in the order in which they recommend that
the selection for promotion should be made.
We were informed that the present minister of war almost invariably
promoted the officers from the head of this list, or, in other words,
followed the recommendation of the inspector-general.
II. MILITARY SCHOOLS.
The principal Military Schools at present existing in France are the
following:--
1. The Polytechnic School at Paris (_Ecole Impériale Polytechnique_,)
preparatory to--
2. The Artillery and Engineers School of Application at Metz (_Ecole
Impériale d’Application de l’Artillerie et du Génie_.)
3. The Military School at St. Cyr (_Ecole Impériale Spéciale
Militaire_,) for the Infantry and Cavalry, into which the Officers’
Department of the Cavalry School at Saumur has lately been absorbed.
4. The Staff School at Paris (_Ecole Impériale d’Application d’Etat
Major_.)
5. The Military Orphan School (_Prytanée Impériale Militaire_) at La
Flèche.
6. The Medical School (_Ecole Impériale de Médicine et de Pharmacie
Militaires_.) recently established in connection with the Hospital of
Val-de-Grâce.
7. The School of Musketry (_Ecole Normale de Tir_) at Vincennes,
founded in 1842.
8. The Gymnastic School (_Ecole Normale de Gymnastique_) near
Vincennes.
9. The Music School (_Gymnase Musical_.)
10. The Regimental Schools (_Ecoles Régimentaires_.)
The military schools are under the charge of the minister of war, with
whom the authorities of the schools are in direct communication.
The expenses to the state of the military schools, including the pay of
the military men who are employed in connection with them, for the year
1851, are as follows:--
For Polytechnic School at Paris, _fr._ 554,911. 91
“ Artillery and Engineers School at Metz, 187,352. 06
“ Infantry and Cavalry School at St. Cyr, 682,187. 35
“ Cavalry School at Saumur, 196,170. 27
“ Staff School at Paris, 145,349. 96
“ Gymnastic School of Musketry at Vincennes, 33,211. 33
“ Regimental Schools, 108,911½,30
From this sum, 2,224,542_fr._, should be deducted 421,372_fr._ secured
from paying pupils, leaving the total cost to the state to be
1,803,308_fr._, or about $360,000, for about 2,100 pupils. The cost to
the state for training an officer of Artillery and Engineers is about
$1,500, and that of an officer of the Staff is about $1,400.
SUBJECTS AND METHODS OF INSTRUCTION
IN MATHEMATICS AS PRESCRIBED FOR ADMISSION TO THE POLYTECHNIC SCHOOL OF
FRANCE.
“L’ÉCOLE POLYTECHNIQUE” is too well known, by name at least, to need
eulogy in this journal. Its course of instruction has long been famed
for its completeness, precision, and adaptation to its intended objects.
But this course had gradually lost somewhat of its symmetrical
proportions by the introduction of some new subjects and the excessive
development of others. The same defects had crept into the programme of
the subjects of examination for admission to the school. Influenced by
these considerations, the Legislative Assembly of France, by the law of
June 5th, 1850, appointed a “_Commission_” to revise the programmes of
admission and of internal instruction. The President of the Commission
was THENARD, its “Reporter” was LE VERRIER, and the other nine members
were worthy to be their colleagues. They were charged to avoid the error
of giving to young students, subjects and methods of instruction “too
elevated, too abstract, and above their comprehension;” to see that the
course prescribed should be “adapted, not merely to a few select
spirits, but to average intelligences;” and to correct “the excessive
development of the preparatory studies, which had gone far beyond the
end desired.”
The Commission, by M. Le Verrier, prepared an elaborate report of 440
quarto pages, only two hundred copies of which were printed, and these
merely for the use of the authorities. A copy belonging to a deceased
member of the Commission (the lamented Professor _Theodore Olivier_),
having come into the hands of the present writer, he has thought that
some valuable hints for our use in this country might be drawn from it,
presenting as it does a precise and thorough course of mathematical
instruction, adapted to any latitude, and arranged in the most perfect
order by such competent authorities. He has accordingly here presented,
in a condensed form, the opinions of the Commission on _the proper
subjects for examination in mathematics, preparatory to admission to the
Polytechnic School, and the best methods of teaching them_.
The subjects which will be discussed are ARITHMETIC; GEOMETRY; ALGEBRA;
TRIGONOMETRY; ANALYTICAL GEOMETRY; DESCRIPTIVE GEOMETRY.
I. ARITHMETIC.
A knowledge of Arithmetic is indispensable to every one. The merchant,
the workman, the engineer, all need to know how to calculate with
rapidity and precision. The useful character of arithmetic indicates
that its methods should admit of great simplicity, and that its teaching
should be most carefully freed from all needless complication. When we
enter into the spirit of the methods of arithmetic, we perceive that
they all flow clearly and simply from the very principles of numeration,
from some precise definitions, and from certain ideas of relations
between numbers, which all minds easily perceive, and which they even
possessed in advance, before their teacher made them recognize them and
taught them to class them in a methodical and fruitful order. We
therefore believe that there is no one who is not capable of receiving,
of understanding, and of enjoying well-arranged and well-digested
arithmetical instruction.
But the great majority of those who have received a liberal education do
not possess this useful knowledge. Their minds, they say, are not suited
to the study of mathematics. They have found it impossible to bend
themselves to the study of those abstract sciences whose barrenness and
dryness form so striking a contrast to the attractions of history, and
the beauties of style and of thought in the great poets; and so on.
Now, without admitting entirely the justice of this language, we do not
hesitate to acknowledge, that the teaching of elementary mathematics has
lost its former simplicity, and assumed a complicated and pretentious
form, which possesses no advantages and is full of inconveniences. The
reproach which is cast upon the sciences in themselves, we out-and-out
repulse, and apply it only to the vicious manner in which they are now
taught.
Arithmetic especially is only an instrument, a tool, the theory of which
we certainly ought to know, but the practice of which it is above all
important most thoroughly to possess. The methods of analysis and of
mechanics, invariably lead to solutions whose applications require
reduction into numbers by arithmetical calculations. We may add that the
numerical determination of the final result is almost always
indispensable to the clear and complete comprehension of a method ever
so little complicated. Such an application, either by the more complete
condensation of the ideas which it requires, or by its fixing the mind
on the subject more precisely and clearly, develops a crowd of remarks
which otherwise would not have been made, and it thus contributes to
facilitate the comprehension of theories in such an efficacious manner
that the time given to the numerical work is more than regained by its
being no longer necessary to return incessantly to new explanations of
the same method.
The teaching of arithmetic will therefore have for its essential object,
to make the pupils acquire the habit of calculation, so that they may be
able to make an easy and continual use of it in the course of their
studies. The theory of the operations must be given to them with
clearness and precision; not only that they may understand the mechanism
of those operations, but because, in almost all questions, the
application of the methods calls for great attention and continual
discussion, if we would arrive at a result in which we can confide. But
at the same time every useless theory must be carefully removed, so as
not to distract the attention of the pupil, but to devote it entirely to
the essential objects of this instruction.
It may be objected that these theories are excellent exercises to form
the mind of the pupils. We answer that such an opinion may be doubted
for more than one reason, and that, in any case, exercises on useful
subjects not being wanting in the immense field embraced by mathematics,
it is quite superfluous to create, for the mere pleasure of it,
difficulties which will never have any useful application.
Another remark we think important. It is of no use to arrive at a
numerical result, if we cannot answer for its correctness. The teaching
of calculation should include, as an essential condition, that the
pupils should be shown how every result, deduced from a series of
arithmetical operations, may always be controlled in such a way that we
may have all desirable certainty of its correctness; so that, though a
pupil may and must often make mistakes, he may be able to discover them
himself, to correct them himself, and never to present, at last, any
other than an exact result.
* * *
The _Programme_ given below is made very minute to avoid the evils which
resulted from the brevity of the old one. In it, the limits of the
matter required not being clearly defined, each teacher preferred to
extend them excessively, rather than to expose his pupils to the risk of
being unable to answer certain questions. The examiners were then
naturally led to put the questions thus offered to them, so to say; and
thus the preparatory studies grew into excessive and extravagant
development. These abuses could be remedied only by the publication of
programmes so detailed, that the limits within which the branches
required for admission must be restricted should be so apparent to the
eyes of all, as to render it impossible for the examiners to go out of
them, and thus to permit teachers to confine their instruction within
them.
The new programme for arithmetic commences with the words Decimal
numeration. This is to indicate that the Duodecimal numeration will not
be required.
The only practical verification of Addition and Multiplication, is to
recommence these operations in a different order.
The Division of whole numbers is the first question considered at all
difficult. This difficulty arises from the complication of the methods
by which division is taught. In some books its explanation contains
twice as many reasons as is necessary. The mind becomes confused by such
instruction, and no longer understands what is a demonstration, when it
sees it continued at the moment when it appeared to be finished. In most
cases the demonstration is excessively complicated and does not follow
the same order as the practical rule, to which it is then necessary to
return. There lies the evil, and it is real and profound.
The phrase of the programme, Division of whole numbers, intends that the
pupil shall be required to explain the practical rule, and be able to
use it in a familiar and rapid manner. We do not present any particular
mode of demonstration, but, to explain our views, we will indicate how
we would treat the subject if we were making the detailed programme of a
_course_ of arithmetic, and not merely that of an _examination_. It
would be somewhat thus:
“The quotient may be found by addition, subtraction, multiplication;
“Division of a number by a number of one figure, when the quotient is
less than 10;
“Division of any number by a number less than 10;
“Division of any two numbers when the quotient has only one figure;
“Division in the most general case.
“_Note._--The practical rule may be entirely explained by this
consideration, that by multiplying the divisor by different numbers, we
see if the quotient is greater or less than the multiplier.”
The properties of the Divisors of numbers, and the decomposition of a
number into prime factors should be known by the student. But here also
we recommend simplicity. The theory of the greatest common divisor, for
example, has no need to be given with all the details with which it is
usually surrounded, for it is of no use in practice.
The calculation of Decimal numbers is especially that in which it is
indispensable to exercise students. Such are the numbers on which they
will generally have to operate. It is rare that the data of a question
are whole numbers; usually they are decimal numbers which are not even
known with rigor, but only with a given decimal approximation; and the
result which is sought is to deduce from these, other decimal numbers,
themselves exact to a certain degree of approximation, fixed by the
conditions of the problem. It is thus that this subject should be
taught. The pupil should not merely learn how, in one or two cases, he
can obtain a result to within 1/_n_, _n_ being any number, but how to
arrive by a practicable route to results which are exact to within a
required decimal, and on the correctness of which they can depend.
Let us take decimal multiplication for an example. Generally the pupils
do not know any other rule than “to multiply one factor by the other,
without noticing the decimal point, except to cut off on the right of
the product as many decimal figures as there are in the two factors.”
The rule thus enunciated is methodical, simple, and apparently easy.
But, in reality, it is practically of a repulsive length, and is most
generally inapplicable.
Let us suppose that we have to multiply together two numbers having each
six decimals, and that we wish to know the product also to the sixth
decimal. The above rule will give twelve decimals, the last six of
which, being useless, will have caused by their calculation the loss of
precious time. Still farther; when a factor of a product is given with
six decimals, it is because we have stopped in its determination at that
degree of approximation, neglecting the following decimals; whence it
results that several of the decimals situated on the right of the
calculated product are not those which would belong to the rigorous
product. What then is the use of taking the trouble of determining them?
We will remark lastly that if the factors of the product are
incommensurable, and if it is necessary to convert them into decimals
before effecting the multiplication, we should not know how far we
should carry the approximation of the factors before applying the above
rule. It will therefore be necessary to teach the pupils the abridged
methods by which we succeed, at the same time, in using fewer figures
and in knowing the real approximation of the result at which we arrive.
Periodical decimal fractions are of no use. The two elementary questions
of the programme are all that need be known about them.
The Extraction of the square root must be given very carefully,
especially that of decimal numbers. It is quite impossible here to
observe the rule of having in the square twice as many decimals as are
required in the root. That rule is in fact impracticable when a series
of operations is to be effected. “When a number N increases by a
comparatively small quantity _d_, the square of that number increases
very nearly as 2N_d_.” It is thus that we determine the approximation
with which a number must be calculated so that its square root may
afterwards be obtained with the necessary exactitude. This supposes that
before determining the square with all necessary precision, we have a
suitable lower limit of the value of the root, which can always be done
without difficulty.
The Cube root is included in the programme. The pupils should know this;
but while it will be necessary to exercise them on the extraction of the
square root by numerous examples, we should be very sparing of this in
the cube root, and not go far beyond the mere theory. The calculations
become too complicated and waste too much time. Logarithms are useful
even for the square root; and quite indispensable for the cube root, and
still more so for higher roots.
When a question contains only quantities which vary in the same ratio,
or in an inverse ratio, it is immediately resolved by a very simple
method, known under the name of _reduction to unity_. The result once
obtained, it is indispensable to make the pupils remark that it is
composed of the quantity which, among the data, is of the nature of that
which is sought, multiplied successively by a series of abstract ratios
between other quantities which also, taken two and two, are of the same
nature. Hence flows the rule for writing directly the required result,
without being obliged to take up again for each question the series of
reasonings. This has the advantage, not only of saving time, but of
better showing the spirit of the method, of making clearer the meaning
of the solution, and of preparing for the subsequent use of formulas.
The consideration of “homogeneity” conduces to these results.
We recommend teachers to abandon as much as possible the use of examples
in abstract numbers, and of insignificant problems, in which the data,
taken at random, have no connection with reality. Let the examples and
the exercises presented to students always relate to objects which are
found in the arts, in industry, in nature, in physics, in the system of
the world. This will have many advantages. The precise meaning of the
solutions will be better grasped. The pupils will thus acquire, without
any trouble, a stock of precise and precious knowledge of the world
which surrounds them. They will also more willingly engage in numerical
calculations, when their attention is thus incessantly aroused and
sustained, and when the result, instead of being merely a dry number,
embodies information which is real, useful, and interesting.
The former arithmetical programme included the theory of _progressions_
and _logarithms_; the latter being deduced from the former. But the
theory of logarithms is again deduced in algebra from exponents, much
the best method. This constitutes an objectionable “_double emploi_.”
There is finally no good reason for retaining these theories in
arithmetic.
The programme retains the questions which can be solved by making two
arbitrary and successive hypotheses on the desired result. It is true
that these questions can be directly resolved by means of a simple
equation of the first degree; but we have considered that, since the
resolution of problems by means of hypotheses, constitutes the most
fruitful method really used in practice, it is well to accustom students
to it the soonest possible. This is the more necessary, because teachers
have generally pursued the opposite course, aiming especially to give
their pupils direct solutions, without reflecting that the theory of
these is usually much more complicated, and that the mind of the learner
thus receives a direction exactly contrary to that which it will have to
take in the end.
“Proportions” remain to be noticed.
In most arithmetics problems are resolved first by the method of
“reduction to unity,” and then by the theory of proportions. But beside
the objection of the “_double emploi_,” it is very certain that the
method of reduction to unity presents, in their true light and in a
complete and simple manner, all the questions of ratio which are the
bases of arithmetical solutions; so that the subsequent introduction of
proportions teaches nothing new to the pupils, and only presents the
same thing in a more complicated manner. We therefore exclude from our
programme of examination the solution of questions of arithmetic,
presented under the special form which constitutes the theory of
proportions.
This special form we would be very careful not to invent, if it had not
already been employed. Why not say simply “The ratio of M to N is equal
to that of P to Q,” instead of hunting for this other form of
enunciating the same idea, “M _is to_ N _as_ P _is to_ Q”? It is in vain
to allege the necessities of geometry; if we consider all the questions
in which proportions are used, we shall see that the simple
consideration of the equality of ratios is equally well adapted to the
simplicity of the enunciation and the clearness of the demonstrations.
However, since all the old books of geometry make use of proportions, we
retain the properties of proportions at the end of our programme; but
with this express reserve, that the examiners shall limit themselves to
the simple properties which we indicate, and that they shall not demand
any application of proportions to the solution of arithmetical problems.
PROGRAMME OF ARITHMETIC.
Decimal numeration.
Addition and subtraction of whole numbers.
Multiplication of whole numbers.--Table of Pythagoras.--The product of
several whole numbers does not change its value, in whatever order the
multiplications are effected.--To multiply a number by the product of
several factors, it is sufficient to multiply successively by the
factors of the product.
Division of whole numbers.--To divide a number by the product of
several factors, it is sufficient to divide successively by the factors
of the product.
Remainders from dividing a whole number by 2, 3, 5, 9, and
11.--Applications to the characters of divisibility by one of those
numbers; to the verification of the product of several factors; and to
the verification of the quotient of two numbers.
Prime numbers. Numbers prime to one another.
To find the greatest common divisor of two numbers.--If a number
divides a product of two factors, and if it is prime to one of the
factors, it divides the other.--To decompose a number into its prime
factors.--To determine the smallest number divisible by given numbers.
_Vulgar fractions._
A fraction does not alter in value when its two terms are multiplied
or divided by the same number. Reduction of a fraction to its simplest
expression. Reduction of several fractions to the same denominator.
Reduction to the smallest common denominator.--To compare the relative
values of several fractions.
Addition and subtraction of fractions.--Multiplication. Fractions of
fractions.--Division.
Calculation of numbers composed of an entire part and a fraction.
_Decimal numbers._
Addition and subtraction.
Multiplication and division.--How to obtain the product of the
quotient to within a unit of any given decimal order.
To reduce a vulgar fraction to a decimal fraction.--When the
denominator of an irreducible fraction contains other factors than 2 and
5, the fraction cannot be exactly reduced to decimals; and the quotient,
which continues indefinitely, is periodical.
To find the vulgar fraction which generates a periodical decimal
fraction: 1º when the decimal fraction is simply periodical; 2º when it
contains a part not periodical.
_System of the new measures._
Linear Measures.--Measures of surface.--Measures of volume and
capacity.--Measures of weight.--Moneys.--Ratios of the principal foreign
measures (England, Germany, United States of America) to the measures of
France.
_Of ratios. Resolution of problems._
General notions on quantities which vary in the same ratio or in an
inverse ratio.--Solution, by the method called _Reduction to unity_, of
the simplest questions in which such quantities are considered.--To show
the homogeneity of the results which are arrived at; thence to deduce
the general rule for writing directly the expression of the required
solution.
Simple interest.--General formula, the consideration of which
furnishes the solution of questions relating to simple interest.--Of
discount, as practised in commerce.
To divide a sum into parts proportional to given numbers.
Of questions which can be solved by two arbitrary and successive
hypotheses made on the desired result.
_Of the square and of the square root. Of the cube and of the cube
root._
Formation of the square and the cube of the sum of two numbers.--Rules
for extracting the square root and the cube root of a whole number.--If
this root is not entire, it cannot be exactly expressed by any number,
and is called incommensurable.
Square and cube of a fraction.--Extraction of the square root and cube
root of vulgar fractions.
Any number being given, either directly, or by a series of operations
which permit only an approximation to its value by means of decimals,
how to extract the square root or cube root of that number, to within
any decimal unit.
_Of the proportions called geometrical._
In every proportion the product of the extremes is equal to the
product of the means.--Reciprocal proportion.--Knowing three terms of a
proportion to find the fourth.--Geometrical mean of two numbers.--How
the order of the terms of a proportion can be inverted without
disturbing the proportion.
When two proportions have a common ratio, the two other ratios form a
proportion.
In any proportion, each antecedent may be increased or diminished by
its consequent without destroying the proportion.
When the corresponding terms of several proportions are multiplied
together, the four products form a new proportion.--The same powers or
the same roots of four numbers in proportion form a new proportion.
In a series of equal ratios, the sum of any number of antecedents and
the sum of their consequents are still in the same ratio.
II. GEOMETRY
Some knowledge of Geometry is, next to arithmetic, most indispensable to
every one, and yet very few possess even its first principles. This is
the fault of the common system of instruction. We do not pay sufficient
regard to the natural notions about straight lines, angles, parallels,
circles, etc., which the young have acquired by looking around them, and
which their minds have unconsciously considered before making them a
regular study. We thus waste time in giving a dogmatic form to truths
which the mind seizes directly.
The illustrious _Clairaut_ complains of this, and of the instruction
commencing always with a great number of definitions, postulates,
axioms, and preliminary principles, dry and repulsive, and followed by
propositions equally uninteresting. He also condemns the profusion of
self-evident propositions, saying, “It is not surprising that Euclid
should give himself the trouble to demonstrate that two circles which
intersect have not the same centre; that a triangle situated within
another has the sum of its sides smaller than that of the sides of the
triangle which contains it; and so on. That geometer had to convince
obstinate sophists, who gloried in denying the most evident truths. It
was therefore necessary that geometry, like logic, should then have the
aid of formal reasonings, to close the mouths of cavillers; but in our
day things have changed face; all reasoning about what mere good sense
decides in advance is now a pure waste of time, and is fitted, only to
obscure the truth and to disgust the reader.”
_Bezout_ also condemns the multiplication of the number of theorems,
propositions, and corollaries; an array which makes the student dizzy,
and amid which he is lost. All that follows from a principle should be
given in natural language as far as possible, avoiding the dogmatic
form. It is true that some consider the works of Bezout deficient in
rigor, but he knew better than any one what really was a demonstration.
Nor do we find in the works of the great old masters less generality of
views, less precision, less clearness of conception than in modern
treatises. Quite the contrary indeed.
We see this in Bezout’s _definition of a right line_--that it tends
continually towards one and the same point; and in that of _a curved
line_--that it is the trace of a moving point, which turns aside
infinitely little at each step of its progress; definitions most
fruitful in consequences. When we define a right line as the shortest
path from one point to another, we enunciate a property of that line
which is of no use for demonstrations. When we define a curved line as
one which is neither straight nor composed of straight lines, we
enunciate two negations which can lead to no result, and which have no
connection with the peculiar nature of the curved line. Bezout’s
definition, on the contrary, enters into the nature of the object to be
defined, seizes its mode of being, its character, and puts the reader
immediately in possession of the general idea from which are afterwards
deduced the properties of curved lines and the construction of their
tangents.
So too when Bezout says that, in order to form an exact idea of an
angle, it is necessary to consider the movement of a line turning around
one of its points, he gives an idea at once more just and more fruitful
in consequences, both mathematical and mechanical, than that which is
limited to saying, that the indefinite space comprised between two
straight lines which meet in a point, and which may be regarded as
prolonged indefinitely, is called an _angle_; a definition not very
easily comprehended and absolutely useless for ulterior explanations,
while that of Bezout is of continual service.
We therefore urge teachers to return, in their demonstrations, to the
simplest ideas, which are also the most general; to consider a
demonstration as finished and complete when it has evidently caused the
truth to enter into the mind of the pupil, and to add nothing merely for
the sake of silencing sophists.
* * *
Referring to our Programme of Geometry, given below, our first comments
relate to the “Theory of parallels.” This is a subject on which all
students fear to be examined; and this being a general feeling, it is
plain that it is not their fault, but that of the manner in which this
subject is taught. The omission of the natural idea of the constant
direction of the right line (as defined by Bezout) causes the
complication of the first elements; makes it necessary for Legendre to
demonstrate that all right angles are equal (a proposition whose meaning
is rarely understood); and is the real source of all the pretended
difficulties of the theory of parallels. These difficulties are now
usually avoided by the admission of a _postulate_, after the example of
Euclid, and to regulate the practice in that matter, we have thought
proper to prescribe that this proposition--_Through a given point only a
single parallel to a right line can be drawn_--should be admitted purely
and simply, without demonstration, and as a direct consequence of our
idea of the nature of the right line.
We should remark that the order of ideas in our programme supposes the
properties of lines established without any use of the properties of
surfaces. We think that, in this respect, it is better to follow Lacroix
than Legendre.
When we prove thus that three parallels always divide two right lines
into proportional parts, this proposition can be extended to the case in
which the ratio of the parts is incommensurable, either by the method
called _Reductio ad absurdum_, or by the method of _Limits_. We
especially recommend the use of the latter method. The former has in
fact nothing which satisfies the mind, and we should never have recourse
to it, for it is always possible to do without it. When we have proved
to the pupil that a desired quantity, X, cannot be either larger or
smaller than A, the pupil is indeed forced to admit that X and A are
equal; but that does not make him understand or feel why that equality
exists. Now those demonstrations which are of such a nature that, once
given, they disappear, as it were, so as to leave to the proposition
demonstrated the character of a truth evident _à priori_, are those
which should be carefully sought for, not only because they make that
truth better felt, but because they better prepare the mind for
conceptions of a more elevated order. The method of limits, is, for a
certain number of questions, the only one which possesses this
characteristic--that the demonstration is closely connected with the
essential nature of the proposition to be established.
In reference to the relations which exist between the sides of a
triangle and the segments formed by perpendiculars let fall from the
summits, we will, once for all, recommend to the teacher, to exercise
his students in making numerical applications of relations of that kind,
as often as they shall present themselves in the course of geometry.
This is the way to cause their meaning to be well understood, to fix
them in the mind of students, and to give these the exercise in
numerical calculation to which we positively require them to be
habituated.
The theory of similar figures has a direct application in the art of
surveying for plans (_Lever des plans_). We wish that this application
should be given to the pupils in detail; that they should be taught to
range out and measure a straight line on the ground; that a graphometer
should be placed in their hands; and that they should use it and the
chain to obtain on the ground, for themselves, all the data necessary
for the construction of a map, which they will present to the examiners
with the calculations in the margins.
It is true that a more complete study of this subject will have to be
subsequently made by means of trigonometry, in which calculation will
give more precision than these graphical operations. But some pupils may
fail to extend their studies to trigonometry (the course given for the
Polytechnic school having become the model for general instruction in
France), and those who do will thus learn that trigonometry merely gives
means of more precise calculation. This application will also be an
encouragement to the study of a science whose utility the pupil will
thus begin to comprehend.
It is common to say that an angle is measured by the arc of a circle,
described from its summit or centre, and intercepted between its sides.
It is true that teachers add, that since a quantity cannot be measured
except by one of the same nature, and since the arc of a circle is of a
different nature from an angle, the preceding enunciation is only an
abridgment of the proposition by which we find the ratio of an angle to
a right angle. Despite this precaution, the unqualified enunciation
which precedes, causes uncertainty in the mind of the pupil, and
produces in it a lamentable confusion. We will say as much of the
following enunciations: “A dihedral angle is measured by the plane angle
included between its sides;” “The surface of a spherical triangle is
measured by the excess of the sum of its three angles above two right
angles,” etc.; enunciations which have no meaning in themselves, and
from which every trace of homogeneity has disappeared. Now that
everybody is requiring that the students of the Polytechnic school
should better understand the meaning of the formulas which they are
taught, which requires that their homogeneity should always be apparent,
this should be attended to from the beginning of their studies, in
geometry as well as in arithmetic. The examiners must therefore insist
that the pupils shall never give them any enunciations in which
homogeneity is not preserved.
The proportionality of the circumferences of circles to their radii must
be inferred _directly_ from the proportionality of the perimeters of
regular polygons, of the same number of sides, to their apothems. In
like manner, from the area of a regular polygon being measured by half
of the product of its perimeter by the radius of the inscribed circle,
it must be _directly_ inferred that the area of a circle is measured by
half of the product of its circumference by its radius. For a long time,
these properties of the circle were differently demonstrated by proving,
for example, with Legendre, that the measure of the circle could not be
either smaller or greater than that which we have just given, whence it
had to be inferred that it must be equal to it. The “Council of
improvement” finally decided that this method should be abandoned, and
that the method of limits should alone be admitted, in the examinations,
for demonstrations of this kind. This was a true advance, but it was not
sufficient. It did not, as it should, go on to consider the circle,
purely and simply, as the limit of a series of regular polygons, the
number of whose sides goes on increasing to infinity, and to regard the
circle as possessing every property demonstrated for polygons. Instead
of this, they inscribed and circumscribed to the circle two polygons of
the same number of sides, and proved that, by the multiplication of the
number of the sides of these polygons, the difference of their areas
might become smaller than any given quantity, and thence, finally,
deduced the measure of the area of the circle; that is to say, they took
away from the method of limits all its advantage as to simplicity, by
not applying it _frankly_.
We now ask that this shall cease; and that we shall no longer reproach
for want of rigor, the Lagranges, the Laplaces, the Poissons, and
Leibnitz, who has given us this principle: that “A curvilinear figure
may be regarded as equivalent to a polygon of an infinite number of
sides; whence it follows that whatsoever can be demonstrated of such a
polygon, no regard being paid to the number of its sides, the same may
be asserted of the curve.” This is the principle for _the most simple_
application of which to the measure of the circle and of the round
bodies we appeal.
Whatever may be the formulas which may be given to the pupils for the
determination of the ratio of the circumference to the diameter (the
“Method of isoperimeters” is to be recommended for its simplicity), they
must be required to perform the calculation, so as to obtain at least
two or three exact decimals. These calculations, made with logarithms,
must be methodically arranged and presented at the examination. It may
be known whether the candidate is really the author of the papers, by
calling for explanations on some of the steps, or making him calculate
some points afresh.
The enunciations relating to the measurement of areas too often leave
indistinctness in the minds of students, doubtless because of their
form. We desire to make them better comprehended, by insisting on their
application by means of a great number of examples.
As one application, we require the knowledge of the methods of surveying
for content (_arpentage_), differing somewhat from the method of
triangulation, used in the surveying for plans (_lever des plans_). To
make this application more fruitful, the ground should be bounded on one
side by an irregular curve. The pupils will not only thus learn how to
overcome this practical difficulty, but they will find, in the
calculation of the surface by means of trapezoids, the first application
of the method of quadratures, with which it is important that they
should very early become familiar. This application will constitute a
new sheet of drawing and calculations to be presented at the
examination.
Most of our remarks on plane geometry apply to geometry of three
dimensions. Care should be taken always to leave homogeneity apparent
and to make numerous applications to the measurement of volumes.
The theory of similar polyhedrons often gives rise in the examination of
the students to serious difficulties on their part. These difficulties
belong rather to the form than to the substance, and to the manner in
which each individual mind seizes relations of position; relations
always easier to feel than to express. The examiners should be content
with arriving at the results enunciated in our programme, by the
shortest and easiest road.
The simplicity desired cannot however be attained unless all have a
common starting-point, in the definition of similar polyhedrons. The
best course is assuredly to consider that theory in the point of view in
which it is employed in the arts, especially in sculpture; i.e. to
conceive the given system of points, M, N, P, . . . . to have lines
passing from them through a point S, the _pole of similitude_, and
prolonged beyond it to M’, N’, P’, . . . . so that SM’, SN’, SP’,
. . . . are proportional to SM, SN, SP, . . . . . Then the points M’,
N’, P’, . . . . form a system _similar_ to M, N, P, . . . . .
The areas and volumes of the cylinder, of the cone, and of the sphere
must be deduced from the areas and from the volumes of the prism, of the
pyramid, and of the polygonal sector, with the same simplicity which we
have required for the measure of the surface of the circle, and for the
same reasons. It is, besides, the only means of easily extending to
cones and cylinders with any bases whatever, right or oblique, those
properties of cones and cylinders,--right and with circular
bases,--which are applicable to them.
Numerical examples of the calculations, by logarithms, of these areas
and volumes, including the area of a spherical triangle, will make
another sheet to be presented to the examiners.
PROGRAMME OF GEOMETRY.
1. OF PLANE FIGURES.
Measure of the distance of two points.--Two finite right lines being
given, to find their common measure, or at least their approximate
ratio.
_Of angles._--Right, acute, obtuse angles.--Angles vertically opposite
are equal.
_Of triangles._--Angles and sides.--The simplest cases of
equality.--Elementary problems on the construction of angles and of
triangles.
_Of perpendiculars and of oblique lines._
Among all the lines that can be drawn from a given point to a given
right line, the perpendicular is the shortest, and the oblique lines are
longer in proportion to their divergence from the foot of the
perpendicular.
_Properties of the isosceles triangle._--Problems on tracing
perpendiculars.--Division of a given straight line into equal parts.
Cases of equality of right-angled triangles.
_Of parallel lines._
Properties of the angles formed by two parallels and a
secant.--Reciprocally, when these properties exist for two right lines
and a common secant, the two lines are parallel.[1]--Through a given
point, to draw a right line parallel to a given right line, or cutting
it at a given angle.--Equality of angles having their sides parallel and
their openings placed in the same direction.
[Footnote 1: It will be admitted, as a postulate, that only one
parallel to a given right line can pass through a given point.]
Sum of the angles of a triangle.
The parts of parallels intercepted between parallels are equal, and
reciprocally. Three parallels always divide any two right lines into
proportional parts. The ratio of these parts may be incommensurable.--
Application to the case in which a right line is drawn, in a triangle,
parallel to one of its sides.
To find a fourth proportional to three given lines.
The right line, which bisects one of the angles of a triangle, divides
the opposite side into two segments proportional to the adjacent sides.
_Of similar triangles._
Conditions of similitude.--To construct on a given right line, a
triangle similar to a given triangle.
Any number of right lines, passing through the same point and met by
two parallels, are divided by these parallels into proportional parts,
and divide them also into proportional parts.--To divide a given right
line in the same manner as another is divided.--Division of a right line
into equal parts.
If from the right angle of a right-angled triangle a perpendicular is
let fall upon the hypothenuse, 1º this perpendicular will divide the
triangle into two others which will be similar to it, and therefore to
each other; 2º it will divide the hypothenuse into two segments, such
that each side of the right angle will be a mean proportional between
the adjacent segment and the entire hypothenuse; 3º the perpendicular
will be a mean proportional between the two segments of the hypothenuse.
In a right-angled triangle, the square of the number which expresses
the length of the hypothenuse is equal to the sum of the squares of the
numbers which express the lengths of the other two sides.
The three sides of any triangle being expressed in numbers, if from
the extremity of one of the sides a perpendicular is let fall on one of
the other sides, the square of the first side will be equal to the sum
of the squares of the other two, _minus_ twice the product of the side
on which the perpendicular is let fall by the distance of that
perpendicular from the angle opposite to the first side, if the angle is
_acute_, and _plus_ twice the same product, if this angle is _obtuse_.
_Of polygons._
Parallelograms.--Properties of their angles and of their diagonals.
Division of polygons into triangles.--Sum of their interior
angles.--Equality and construction of polygons.
Similar polygons.--Their decomposition into similar triangles.--The
right lines similarly situated in the two polygons are proportional to
the homologous sides of the polygons.--To construct, on a given line, a
polygon similar to a given polygon.--The perimeters of two similar
polygons are to each other as the homologous sides of these polygons.
_Of the right line and the circumference of the circle._
Simultaneous equality of arcs and chords in the same circle.--The
greatest arc has the greatest chord, and reciprocally.--Two arcs being
given in the same circle or in equal-circles, to find the ratio of their
lengths.
Every right line drawn perpendicular to a chord at its middle, passes
through the centre of the circle and through the middle of the arc
subtended by the chord.--Division of an arc into two equal parts.--To
pass the circumference of a circle through three points not in the same
right line.
The tangent at any point of a circumference is perpendicular to the
radius passing through that point.
The arcs intercepted in the same circle between two parallel chords,
or between a tangent and a parallel chord, are equal.
_Measure of angles._
If from the summits of two angles two arcs of circles be described
with the same radius, the ratio of the arcs included between the sides
of each angle will be the same as that of these angles.--Division of the
circumference into degrees, minutes, and seconds.--Use of the
protractor.
An angle having its summit placed, 1º at the centre of a circle; 2º on
the circumference of that circle; 3º within the circle between the
centre and the circumference; 4º without the circle, but so that its
sides cut the circumference; to determine the ratio of that angle to the
right angle, by the consideration of the arc included between its sides.
From a given point without a circle, to draw a tangent to that circle.
To describe, on a given line, a segment of a circle capable of
containing a given angle.
_To make surveys for plans._ (_Lever des plans._)
Tracing a straight line on the ground.--Measuring that line with the
chain.
Measuring angles with the graphometer.--Description of it.
Drawing the plan on paper.--Scale of reduction.--Use of the rule, the
triangle, and the protractor.
To determine the distance of an inaccessible object, with or without
the graphometer.
Three points, A, B, C, being situated on a smooth surface and
represented on a map, to find thereon the point P from which the
distances AB and AC have been seen under given angles. “The problem of
the three points.” “The _Trilinear_ problem.”
_Of the contact and of the intersection of circles._
Two circles which pass through the same point of the right line which
joins their centres have in common only that point in which they touch;
and reciprocally, if two circles touch, their centres and the point of
contact lie in the same right line.
Conditions which must exist in order that two circles may intersect.
_Properties of the secants of the circle._
Two secants which start from the same point without the circle, being
prolonged to the most distant part of the circumference, are
reciprocally proportional to their exterior segments.--The tangent is a
mean proportional between the secant and its exterior segment.
Two chords intersecting within a circle divide each other into parts
reciprocally proportional.--The line perpendicular to a diameter and
terminated by the circumference, is a mean proportional between the two
segments of the diameter.
A chord, passing through the extremity of the diameter, is a mean
proportional between the diameter and the segment formed by the
perpendicular let fall from the other extremity of that chord.--To find
a mean proportional between two given lines.
To divide a line in extreme and mean ratio.--The length of the line
being given numerically, to calculate the numerical value of each of the
segments.
_Of polygons inscribed and circumscribed to the circle._
To inscribe or circumscribe a circle to a given triangle.
Every regular polygon can be inscribed and circumscribed to the
circle.
A regular polygon being inscribed in a circle, 1º to inscribe in the
same circle a polygon of twice as many sides, and to find the length of
one of the sides of the second polygon; 2º to circumscribe about the
circle a regular polygon of the same number of sides, and to express the
side of the circumscribed polygon by means of the side of the
corresponding inscribed polygon.
To inscribe in a circle polygons of 4, 8, 16, 32, sides.
To inscribe in a circle polygons of 3, 6, 12, 24, sides.
To inscribe in a circle polygons of 5, 10, 20, 40, sides.
To inscribe in a circle polygons of 15, 30, 60, sides.
Regular polygons of the same number of sides are similar, and their
perimeters are to each other as the radii of the circles to which they
are inscribed or circumscribed.--The circumferences of circles are to
each other as their radii.
To find the approximate ratio of the circumference to the diameter.
_Of the area of polygons and of that of the circle._
Two parallelograms of the same base and of the same height are
equivalent.--Two triangles of the same base and height are equivalent.
The area of a rectangle and that of a parallelogram are equal to the
product of the base by the height.--What must be understood by that
enunciation.--The area of a triangle is measured by half of the product
of the base by the height.
To transform any polygon into an equivalent square.--Measure of the
area of a polygon.--Measure of the area of a trapezoid.
The square constructed on the hypothenuse of a right-angled triangle
is equivalent to the sum of the squares constructed on the other two
sides.--The squares constructed on the two sides of the right angle of a
right-angled triangle and on the hypothenuse are to each other as the
adjacent segments and entire hypothenuse.
The areas of similar polygons are to each other as the squares of the
homologous sides of the polygons.
Notions on surveying for content (_arpentage_).--Method of
decomposition into triangles.--Simpler method of decomposition into
trapezoids.--Surveyor’s cross.--Practical solution, when the ground is
bounded, in one or more parts, by a curved line.
The area of a regular polygon is measured by half of the product of
its perimeter by the radius of the inscribed circle.--The area of a
circle is measured by half of the product of the circumference by the
radius.--The areas of circles are to each other as the squares of the
radii.
The area of a sector of a circle is measured by half of the product of
the arc by the radius.--Measure of the area of a segment of a circle.
2. OF PLANES AND BODIES TERMINATED BY PLANE SURFACES.
Conditions required to render a right line and a plane respectively
perpendicular.
Of all the lines which can be drawn from a given point to a given
plane, the perpendicular is the shortest, and the oblique lines are
longer in proportion to their divergence from the foot of the
perpendicular.
Parallel right lines and planes.--Angles which have their sides
parallel, and their openings turned in the same direction, are equal,
although situated in different planes.
Dihedral angle.--How to measure the ratio of any dihedral angle to the
right dihedral angle.
Planes perpendicular to each other.--The intersection of two planes
perpendicular to a third plane, is perpendicular to this third plane.
Parallel planes.--when two parallel planes are cut by a third plane
the intersections are parallel.--Two parallel planes have their
perpendiculars common to both.
The shortest distance between two right lines, not intersecting and
not parallel.
Two right lines comprised between two parallel planes are always
divided into proportional parts by a third plane parallel to the first
two.
Trihedral angle.--The sum of any two of the plane angles which compose
a trihedral angle is always greater than the third.
The sum of the plane angles which form a convex polyhedral angle is
always less than four right angles.
If two trihedral angles are formed by the same plane angles, the
dihedral angles comprised between the equal plane angles are
equal.--There may be absolute equality or simple symmetry between the
two trihedral angles.
_Of polyhedrons._
If two tetrahedrons have each a trihedral angle composed of equal and
similarly arranged triangles, these tetrahedrons are equal. They are
also equal if two faces of the one are equal to two faces of the other,
are arranged in the same manner, and form with each other the same
dihedral angle.
When the triangles which form two homologous trihedral angles of two
tetrahedrons are similar, each to each, and similarly disposed, these
tetrahedrons are similar. They are also similar if two faces of the one,
making with each other the same angle as two faces of the other, are
also similar to these latter, and are united by homologous sides and
summits.
Similar pyramids.--A plane parallel to the base of a pyramid cuts off
from it a pyramid similar to it.--To find the height of a pyramid when
we know the dimension of its trunk with parallel bases.
Sections made in any two pyramids at the same distance from these
summits are in a constant ratio.
Parallelopipedon.--Its diagonals.
Any polyhedron can always be divided into triangular pyramids.--Two
bodies composed of the same number of equal and similarly disposed
triangular pyramids, are equal.
_Similar polyhedrons._
The homologous edges of similar polyhedrons are proportional; as are
also the diagonals of the homologous faces and the interior diagonals of
the polyhedrons.--The areas of similar polyhedrons are as the squares of
the homologous edges.
_Measure of volumes._
Two parallelopipedons of the same base and of the same height are
equivalent in volume.
If a parallelogram be constructed on the base of a triangular prism,
and on that parallelogram, taken as a base, there be constructed a
parallelopipedon of the same height as the triangular prism, the volume
of this prism will be half of the volume of the parallelopipedon.--Two
triangular prisms of the same base and the same height are equivalent.
Two tetrahedrons of the same base and the same height are equivalent.
A tetrahedron is equivalent to the third of the triangular prism of
the same base and the same height.
The volume of any parallelopipedon is equal to the product of its base
by its height.--What must be understood by that enunciation.--The volume
of any prism is equal to the product of its base by its height.
The volume of a tetrahedron and that of any pyramid are measured by
the third of the product of the base by the height.
Volume of the truncated oblique triangular prism.
The volumes of two similar polyhedrons are to each other as the cubes
of the homologous edges.
3. OF ROUND BODIES.
_Of the right cone with circular base._
Sections parallel to the base.--Having the dimensions of the trunk of
a cone with parallel bases, to find the height of the entire cone.
The area of a right cone is measured by half of the product of the
circumference of its circular base by its side.--Area of a trunk of a
right cone with parallel bases.
Volume of a pyramid inscribed in the cone.--The volume of a cone is
measured by the third of the product of the area of its base by its
height.[2]
[Footnote 2: The volume of the cone is derived from that of the
pyramid; and it is to be noted that the demonstration applies to
the cone with closed base, whatever the figure of that base.]
Which of the preceding properties belong to the cone of any base
whatever?
_Of the right cylinder with circular base._
Sections parallel to the base.
The area of the convex surface of the right cylinder is measured by
the product of the circumference of its base by its height.--This is
also true of the right cylinder of any base.
Measure of the volume of a prism inscribed in the cylinder.--The
volume of a right cylinder is measured by the product of the area of its
base by its height.--This is also true of any cylinder, right or
oblique, of any base whatever.
_Of the sphere._
Every section of the sphere, made by a plane, is a circle.--Great
circles and small circles.
In every spherical triangle any one side is less than the sum of the
other two. The shortest path from one point to another, on the surface
of the sphere, is the arc of a great circle which joins the two given
points.
The sum of the sides of a spherical triangle, or of any spherical
polygon, is less than the circumference of a great circle.
Poles of an arc of a great or small circle.--They serve to trace arcs
of circles on the sphere.
Every plane perpendicular to the extremity of a radius is tangent to
the sphere.
Measure of the angle of two arcs of great circles.
Properties of the polar or supplementary triangle.
Two spherical triangles situated on the same sphere, or on equal
spheres, are equal in all their parts, 1º when they have an equal angle
included between sides respectively equal; 2º when they have an equal
side adjacent to two angles respectively equal; 3º when they are
mutually equilateral; 4º when they are mutually equiangular. In these
different cases the triangles may be equal, or merely symmetrical.
The sum of the angles of any spherical triangle is less than six, and
greater than two, right angles.
The lune is to the surface of the sphere as the angle of that lune is
to four right angles.
Two symmetrical spherical triangles are equivalent in surface.
The area of a spherical triangle is to that of the whole sphere as the
excess of the sum of its angles above two right angles is to eight right
angles.
When a portion of a regular polygon, inscribed in the generating
circle of the sphere, turns around the diameter of that circle, the
convex area engendered is measured by the product of its height by the
circumference of the circle inscribed in the generating polygon.--The
volume of the corresponding polygonal sector is measured by the area
thus described, multiplied by the third of the radius of the inscribed
circle.
The surface of a spherical zone is equal to the height of that zone
multiplied by the circumference of a great circle.--The surface of the
sphere is quadruple that of a great circle.
Every spherical sector is measured by the zone which forms its base,
multiplied by the third of the radius. The whole sphere is measured by
its surface multiplied by the third of its radius.[3]
[Footnote 3: Numerical examples on the areas and volumes of the
round bodies, including the area of a spherical triangle, will be
required by the examiners. The calculations will be made by
logarithms.]
III. ALGEBRA.
Algebra[4] is not, as are Arithmetic and Geometry, indispensable to
every one. It should be very sparingly introduced into the general
education of youth, and we would there willingly dispense with it
entirely, excepting logarithms, if this would benefit the study of
arithmetic and geometry. The programme of it which we are now to give,
considers it purely in view of its utility to engineers, and we will
carefully eliminate every thing not necessary for them.
[Footnote 4: The true distinction between ALGEBRA and ARITHMETIC
is so commonly overlooked that it maybe well to present it here,
in the words of Comte. “The complete solution of every question of
calculation is necessarily composed of two successive parts, which
have essentially distinct natures. In the first, the object is to
_transform_ the proposed equations, so as to make apparent the
manner in which the unknown quantities are formed by the known
ones; it is this which constitutes the _Algebraic_ question. In
the second, our object is _to find the value_ of the formulas thus
obtained; that is, to determine directly the values of the numbers
sought, which are already represented by certain explicit
functions of given numbers; this is the _Arithmetical_ question.
Thus the stopping-point of the algebraic part of the solution
becomes the starting-point of the arithmetical part.
“ALGEBRA may therefore be defined as having for its object the
_resolution of equations_; taking this expression in its full
logical meaning, which signifies the transformation of _implicit_
functions into equivalent _explicit_ ones. In the same way
ARITHMETIC may be defined as intended for _the determination of
the values of functions_. Henceforth, therefore, we may call
ALGEBRA _the Calculus of Functions_, and ARITHMETIC _the Calculus
of Values_.”]
Algebraical calculation presents no serious difficulty, when its
students become well impressed with this idea, that every letter
represents a number; and particularly when the consideration of negative
quantities is not brought in at the outset and in an absolute manner.
These quantities and their properties should not be introduced except as
the solution of questions by means of equations causes their necessity
to be felt, either for generalizing the rules of calculation, or for
extending the meaning of the formulas to which it leads. CLAIRAUT
pursues this course. He says, “I treat of the multiplication of negative
quantities, that dangerous shoal for both scholars and teachers, only
after having shown its necessity to the learner, by giving him a problem
in which he has to consider negative quantities independently of any
positive quantities from which they are subtracted. When I have arrived
at that point in the problem where I have to multiply or divide negative
quantities by one another, I take the course which was undoubtedly taken
by the first analysts who have had those operations to perform and who
have wished to follow a perfectly sure route: I seek for a solution of
the problem which does not involve these operations; I thus arrive at
the result by reasonings which admit of no doubt, and I thus see what
those products or quotients of negative quantities, which had given me
the first solution, must be.” BEZOUT proceeds in the same way.
We recommend to teachers to follow these examples; not to speak to their
pupils about negative quantities till the necessity of it is felt, and
when they have become familiar with algebraic calculation; and above all
not to lose precious time in obscure discussions and demonstrations,
which the best theory will never teach students so well as numerous
applications.
It has been customary to take up again, in algebra, the calculus of
fractions, so as to generalize the explanations given in arithmetic,
since the terms of literal fractions may be any quantities whatsoever.
Rigorously, this may be well, but to save time we omit this, thinking it
better to employ this time in advancing and exercising the mind on new
truths, rather than in returning continually to rules already given, in
order to imprint a new degree of rigor on their demonstration, or to
give them an extension of which no one doubts.
The study of numerical equations of the first degree, with one or
several unknown quantities, must be made with great care. We have
required the solution of these equations to be made by the method of
_substitution_. We have done this, not only because this method really
comprehends the others, particularly that of _comparison_, but for this
farther reason. In treatises on algebra, those equations alone are
considered whose numerical coefficients and solutions are very simple
numbers. It then makes very little difference what method is used, or in
what order the unknown quantities are eliminated. But it is a very
different thing in practice, where the coefficients are complicated
numbers, given with decimal parts, and where the numerical values of
these coefficients may be very different in the same equation, some
being very great and some very small. In such cases the method of
_substitution_ can alone be employed to advantage, and that with the
precaution of taking the value of the unknown quantity to be eliminated
from that equation in which it has relatively the greatest, coefficient.
Now the method of _comparison_ is only the method of substitution put in
a form in which these precautions cannot be observed, so that in
practice it will give bad results with much labor.
The candidates must present to the examiners the complete calculations
of the resolution of four equations with four unknown quantities, made
with all the precision permitted by the logarithmic tables of Callet,
and the proof that that precision has been obtained. The coefficients
must contain decimals and be very different from one another, and the
elimination must be effected with the above precautions.
The teaching of the present day disregards too much the applicability of
the methods given, provided only that they be elegant in their form; so
that they have to be abandoned and changed when the pupils enter on
practice. This disdain of practical utility was not felt by our great
mathematicians, who incessantly turned their attention towards
applications. Thus the illustrious Lagrange made suggestions like those
just given; and Laplace recommended the drawing of curves for solving
directly all kinds of numerical equations.
As to literal equations of the first degree, we call for formulas
sufficient for the resolution of equations of two or three unknown
quantities. Bezout’s method of elimination must be given as a first
application of that fruitful method of indeterminates. The general
discussion of formulas will be confined to the case of two unknown
quantities. The discussion of three equations with three unknown
quantities, _x_, _y_, and _z_, in which the terms independent of the
unknown quantities are null, will be made directly, by this simple
consideration that the system then really includes only two unknown
quantities, to wit, the ratios of _x_ and _y_, for example, to _z_.
The resolution of inequalities of the first degree with one or more
unknown quantities, was added to equations of the first degree some
years ago. We do not retain that addition.
The equations of the second degree, like the first, must be very
carefully given. In dwelling on the case where the coefficient of
_x_^{2} converges towards zero, it will be remarked that, when the
coefficient is very small, the ordinary formula would give one of the
roots by the difference of two numbers almost equal; so that sufficient
exactness could not be obtained without much labor. It must be shown how
that inconvenience may be avoided.
It is common to meet with expressions of which the maximum or the
minimum can be determined by the consideration of an equation of the
second degree. We retain the study of them, especially for the benefit
of those who will not have the opportunity of advancing to the general
theory of maxima and minima.
The theory of the algebraic calculation of imaginary quantities, given
_à priori_, may, on the contrary, be set aside without inconvenience.
It is enough that the pupils know that the different powers of √-1
continually reproduce in turn one of these four values, ±1, ±√-1. We
will say as much of the calculation of the algebraic values of radicals,
which is of no use. The calculation of their _arithmetical_ values will
alone be demanded. In this connection will be taught the notation of
fractional exponents and that of negative exponents.
The theory of numbers has taken by degrees a disproportionate
development in the examinations for admission; it is of no use in
practice, and, besides, constitutes in the pure mathematics a science
apart.
The theory of continued fractions at first seems more useful. It is
employed in the resolution of algebraic equations, and in that of the
exponential equation _a_^{_x_}=_b_. But these methods are entirely
unsuited to practice, and we therefore omit this theory.
The theory of series, on the contrary, claims some farther developments.
Series are continually met with in practice; they give the best
solutions of many questions, and it is indispensable to know in what
circumstances they can be safely employed.
We have so often insisted on the necessity of teaching students to
calculate, as to justify the extent of the part of the programme
relating to logarithms. We have suppressed the inapplicable method of
determining logarithms by continued fractions, and have substituted the
employment of the series which gives the logarithm of _n_+1, knowing
that of _n_. To exercise the students in the calculation of the series,
they should be made to determine the logarithms of the numbers from 1 to
10, from 101 to 110, and from 10,000 to 10,010, the object of these last
being to show them with what rapidity the calculation proceeds when the
numbers are large; the first term of the series is then sufficient, the
variations of the logarithms being sensibly proportional to the
variations of the numbers, within the limits of the necessary exactness.
In the logarithmic calculations, the pupils will be exercised in judging
of the exactness which they may have been able to obtain: the
consideration of the numerical values of the proportional parts given in
the tables is quite sufficient for this purpose, and is beside the only
one which can be employed in practice.
The use of the sliding rule, which is merely an application of
logarithms, gives a rapid and portable means of executing approximately
a great number of calculations which do not require great exactness. We
desire that the use of this little instrument should be made familiar to
the candidates. This is asked for by all the professors of the “School
of application,” particularly those of Topography, of Artillery, of
Construction, and of Applied Mechanics, who have been convinced by
experience of the utility of this instrument, which has the greatest
possible analogy with tables of logarithms.
Before entering on the subjects of higher algebra, it should be
remembered that the reductions of the course which we have found to be
so urgent, will be made chiefly on it. The general theory of equations
has taken in the examinations an abnormal and improper development, not
worth the time which it costs the students. We may add, that it is very
rare to meet a numerical equation of a high degree requiring to be
resolved, and that those who have to do this, take care not to seek its
roots by the methods which they have been taught. These methods moreover
are not applicable to transcendental equations, which are much more
frequently found in practice.
The theory of the greatest common algebraic divisor, in its entire
generality, is of no use, even in pure science, unless in the
elimination between equations of any degree whatever. But this last
subject being omitted, the greatest common divisor is likewise dispensed
with.
It is usual in the general theory of algebraic equations to consider the
derived polynomials of entire functions of _x_. These polynomials are in
fact useful in several circumstances, and particularly in the theory of
equal roots; and in analytical geometry, they serve for the discussion
of curves and the determination of their tangents. But since
transcendental curves are very often encountered in practice, we give in
our programme the calculation of the derivatives of algebraic and
fractional functions, and transcendental functions, logarithmic,
exponential, and circular. This has been long called for, not only
because it must be of great assistance in the teaching of analytical
geometry, but also because it will facilitate the elementary study of
the infinitesimal calculus.
We have not retrenched any of the general ideas on the composition of an
entire polynomial by means of factors corresponding to its roots. We
retain several theorems rather because they contain the germs of useful
ideas than because of their practical utility, and therefore wish the
examiners to restrict themselves scrupulously to the programme.
The essential point in practice is to be able to determine conveniently
an incommensurable root of an algebraic or transcendental equation, when
encountered. Let us consider first an algebraic equation.
All the methods which have for their object to separate the roots, or to
approximate to them, begin with the substitution of the series of
consecutive whole numbers, in the first member of the equation. The
direct substitution becomes exceedingly complicated, when the numbers
substituted become large. It may be much shortened, however, by deducing
the results from one another by means of their differences, and guarding
against any possibility of error, by verifying some of those results,
those corresponding to the numbers easiest to substitute, such as ±10,
±20. The teacher should not fail to explain this to his pupils.
Still farther: let us suppose that we have to resolve an equation of the
third degree, and that we have recognized by the preceding calculations
the necessity of substituting, between the numbers 2 and 3, numbers
differing by a tenth, either for the purpose of continuing to effect the
separation of the roots, or to approximate nearer to a root comprised
between 2 and 3. If we knew, for the result corresponding to the
substitution of 2, the first, second, and third differences of the
results of the new substitutions, we could thence deduce those results
themselves with as much simplicity, as in the case of the whole numbers.
The new third difference, for example, will be simply the thousandth
part of the old third difference. We may also remark that there is no
possibility of error, since, the numbers being deduced from one another,
when we in this way arrive at the result of the substitution of 3, which
has already been calculated, the whole work will thus be verified.
Let us suppose again that we have thus recognized that the equation has
a root comprised between 2.3 and 2.4; we will approximate still nearer
by substituting intermediate numbers, differing by 0.01, and employing
the course just prescribed. As soon as the third differences can be
neglected, the calculation will be finished at once, by the
consideration of an equation of the second degree; or, if it is
preferred to continue the approximations till the second differences in
their turn may be neglected, the calculation will then be finished by a
simple proportion.
When, in a transcendental equation _f_(X) = 0, we have substituted in
_f_(X) equidistant numbers, sufficiently near to each other to allow the
differences of the results to be neglected, commencing with a certain
order, the 4th, for example, we may, within certain limits of _x_,
replace the transcendental function by an algebraic and entire function
of _x_, and thus reduce the search for the roots of _f_(X) = 0 to the
preceding theory.
Whether the proposed equation be algebraic or transcendental, we can
thus, when we have obtained one root of it with a suitable degree of
exactness, continue the approximation by the method of Newton.
PROGRAMME OF ALGEBRA.
_Algebraic calculation._
Addition and subtraction of polynomials.--Reduction of similar terms.
Multiplication of monomials.--Use of exponents.--Multiplication of
polynomials. Rule of the signs.--To arrange a polynomial.--Homogeneous
polynomials.
Division of monomials. Exponent _zero_.--Division of polynomials. How
to know if the operation will not terminate.--Division of polynomials
when the dividend contains a letter which is not found in the divisor.
_Equations of the first degree._
Resolution of numerical equations of the first degree with one or
several unknown quantities by the method of substitution.--Verification
of the values of the unknown quantities and of the degree of their
exactness.
Of cases of impossibility or of indetermination.
Interpretation of negative values.--Use and calculation of negative
quantities.
Investigation of general formulas for obtaining the values of the
unknown quantities in a system of equations of the first degree with two
or three unknown quantities.--Method of Bezout.--Complete discussion of
these formulas for the case of two unknown quantities.--Symbols m/o and
o/o.
Discussion of three equations with three unknown quantities, in which
the terms independent of the unknown quantities are null.
_Equations of the second degree with one unknown quantity._
Calculus of radicals of the second degree.
Resolution of an equation of the second degree with one unknown
quantity.--Double solution.--Imaginary values.
When, in the equation _ax^{2} + bx + c = 0_, _a_ converges towards 0,
one of the roots increases indefinitely.--Numerical calculation of the
two roots, when _a_ is very small.
Decomposition of the trinomial _x^{2} + px + q_ into factors of the
first degree.--Relations between the coefficients and the roots of the
equation _x^{2} + px + q = 0_.
Trinomial equations reducible to the second degree.
Of the maxima and minima which can be determined by equations of the
second degree.
Calculation of the _arithmetical_ values of radicals.
Fractional exponents.--Negative exponents.
_Of series._
Geometrical progressions.--Summation of the terms.
What we call a series.--Convergence and divergence.
A geometrical progression is convergent, when the ratio is smaller
than unity; diverging, when it is greater.
The terms of a series may decrease indefinitely and the series not be
converging.
A series, all the terms of which are positive, is converging, when the
ratio of one term to the preceding one tends towards a _limit_ smaller
than unity, in proportion as the index of the rank of that term
increases indefinitely.--The series is diverging when this _limit_ is
greater than unity. There is uncertainty when it is equal to unity.
In general, when the terms of a series decrease indefinitely, and are
alternately positive and negative, the series is converging.
* * *
Combinations, arrangements, and permutations of _m_ letters, when each
combination must not contain the same letter twice.
Development of the entire and positive powers of a binomial.--General
terms.
Development of _(a + b √-1)^{m}_.
Limit towards which _(1 + 1/m)^{m}_ tends, when _m_ increases
indefinitely.
Summation of piles of balls.
_Of logarithms and of their uses._
All numbers can be produced by forming all the powers of any positive
number, greater or less than _one_.
General properties of logarithms.
When numbers are in geometrical progression, their logarithms are in
arithmetical progression.
How to pass from one system of logarithms to another system.
Calculation of logarithms by means of the series which gives the
logarithm of _n + 1_, knowing that of _n_.--Calculation of Napierian
logarithms.--To deduce from them those of Briggs. Modulus.
Use of logarithms whose base is 10.--Characteristics.--Negative
characteristics. Logarithms entirely negative are not used in
calculation.
A number being given, how to find its logarithm in the tables of
Callet. A logarithm being given, how to find the number to which it
belongs.--Use of the proportional parts.--Their application to
appreciate the exactness for which we can answer.
Employment of the sliding rule.
Resolution of exponential equations by means of logarithms.
Compound interest. Annuities.
_Derived functions._
Development of an entire function F(_x + h_) of the binomial
(_x + h_).--Derivative of an entire function.--To return from the
derivative to the function.
The derivative of a function of _x_ is the limit towards which tends
the ratio of the increment of the function to the increment _h_ of the
variable, in proportion as _h_ tends towards zero.
Derivatives of trigonometric functions.
Derivatives of exponentials and of logarithms.
Rules to find the derivative of a sum, of a product, of a power, of a
quotient of functions of _x_, the derivatives of which are known.
_Of the numerical resolution of equations._
Changes experienced by an entire function _f(x)_ when _x_ varies in a
continuous manner.--When two numbers _a_ and _b_ substituted in an
entire function _f(x)_ give results with contrary signs, the equation
_f(x) = 0_ has at least one real root not comprised between _a_ and _b_.
This property subsists for every species of function which remains
continuous for all the values of _x_ comprised between _a_ and _b_.
An algebraic equation of uneven degree has at least one real root.--An
algebraic equation of even degree, whose last term is negative, has at
least two real roots.
Every equation _f(x) = 0_, with coefficients either real or imaginary
of the form _a + b √-1_, admits of a real or imaginary root of the same
form. [Only the enunciation, and not the demonstration of this theorem,
is required.]
If _a_ is a root of an algebraic equation, the first member is
divisible by _x - a_. An algebraic equation of the _m_^{th} degree has
always _m_ roots real or imaginary, and it cannot admit
more.--Decomposition of the first members into factors of the first
degree. Relations between the coefficients of an algebraic equation and
its roots.
When an algebraic equation whose coefficients are real, admits an
imaginary root of the form _a + b √-1_, it has also for a root the
conjugate expression _a - b √-1_.
In an algebraic expression, complete or incomplete, the number of the
positive roots cannot surpass the number of the variations; consequence,
for negative roots.
Investigation of the product of the factors of the first degree common
to two entire functions of _x_.--Determination of the roots common to
two equations, the first members of which are entire functions of the
unknown quantity.
By what character to recognize that an algebraic equation has equal
roots.--How we then bring its resolution to that of several others of
lower degree and of unequal roots.
* * *
Investigation of the commensurable roots of an algebraic equation with
entire coefficients.
When a series of equidistant numbers is substituted in an entire
function of the _m_^{th} degree, and differences of different orders
between the results are formed, the differences of the _m_^{th} order
are constant.
Application to the separation of the roots of an equation of the third
degree.--Having the results of the substitution of -1, 0, and +1, to
deduce therefrom, by means of differences, those of all other whole
numbers, positive or negative.--The progress of the calculation leads of
itself to the limits of the roots.--Graphical representation of this
method.
Substitution of numbers equidistant _by a tenth_, between two
consecutive whole numbers, when the inspection of the first results has
shown its necessity.--This substitution is effected directly, or by
means of new differences deduced from the preceding.
How to determine, in continuing the approximation towards a root, at
what moment the consideration of the first difference is sufficient to
give that root with all desirable exactness, by a simple proportion.
The preceding method becomes applicable to the investigation of the
roots of a transcendental equation X = 0, when there have been
substituted in the first member, numbers equidistant and sufficiently
near to allow the differences of the results to be considered as
constant, starting from a certain order.--Formulas of interpolation.
Having obtained a root of an algebraic or transcendental equation,
with a certain degree of approximation, to approximate still farther by
the method of Newton.
* * *
Resolution of two numerical equations of the second degree with two
unknown quantities.
Decomposition of rational fractions into simple fractions.
IV. TRIGONOMETRY.
In explaining the use of trigonometrical tables, the pupil must be able
to tell with what degree of exactness an angle can be determined by the
logarithms of any of its trigonometrical lines. The consideration of the
proportional parts will be sufficient for this. It will thus be seen
that if the _sine_ determines perfectly a small angle, the degree of
exactness, which may be expected from the use of that line, diminishes
as the angle increases, and becomes quite insufficient in the
neighborhood of 90 degrees. It is the reverse for the _cosine_, which
may serve very well to represent an angle near 90 degrees, while it
would be very inexact for small angles. We see, then, that in our
applications, we should distrust those formulas which give an angle by
its sine or cosine. The _tangent_ being alone exempt from these
difficulties, we should seek, as far as possible, to resolve all
questions by means of it. Thus, let us suppose that we know the
hypothenuse and one of the sides of a right-angled triangle, the direct
determination of the included angle will be given by a cosine, which
will be wanting in exactness if the hypothenuse of the triangle does not
differ much from the given side. In that case we should begin by
calculating the third side, and then use it with the first side to
determine the desired angle by means of its tangent. When two sides of a
triangle and the included angle are given, the tangent of the half
difference of the desired angles may be calculated with advantage; but
we may also separately determine the tangent of each of them. When the
three sides of a triangle are given, the best formula for calculating an
angle, and the only one never at fault, is that which gives the tangent
of half of it.
The surveying for plans, taught in the course of Geometry, employing
only graphical methods of calculation, did not need any more accurate
instruments than the chain and the graphometer; but now that
trigonometry furnishes more accurate methods of calculation, the
measurements on the ground require more precision. Hence the requirement
for the pupil to measure carefully a base, to use telescopes, verniers,
etc., and to make the necessary calculations, the ground being still
considered as plane. But as these slow and laborious methods can be
employed for only the principal points of the survey, the more
expeditious means of the plane-table and compass will be used for the
details.
In spherical trigonometry, all that will be needed in geodesy should be
learned before admission to the school, so that the subject will not
need to be again taken up. We have specially inscribed in the programme
the relations between the angles and sides of a right-angled triangle,
which must be known by the students; they are those which occur in
practice. In tracing the course to be pursued in the resolution of the
three cases of any triangles, we have indicated that which is in fact
employed in the applications, and which is the most convenient. As to
the rest, ambiguous cases never occur in practice, and therefore we
should take care not to speak of them to learners.
In surveying, spherical trigonometry will now allow us to consider cases
in which the signals are not all in the same plane, and to operate on
uneven ground, obtain its projection on the plane of the horizon, and at
the same time determine differences of level.
It may be remarked that Descriptive Geometry might supply the place of
spherical trigonometry by a graphical construction, but the degree of
exactitude of the differences of level thus obtained would be
insufficient.
PROGRAMME OF TRIGONOMETRY.
1. PLANE TRIGONOMETRY.
Trigonometrical lines.--Their ratios to the radius are alone
considered.--Relations of the trigonometric lines of the same
angle.--Expressions of the sine and of the cosine in functions of the
tangent.
Knowing the sines and the cosines of two arcs _a_ and _b_, to find the
sine and the cosine of their sum and of their difference.--To find the
tangent of the sum or of the difference of two arcs, knowing the
tangents of those arcs.
Expressions for sin.2_a_ and sin.3_a_; cos.2_a_ and cos. 3_a_;
tang.2_a_ and tang.3_a_.
Knowing sin._a_ or cos._a_, to calculate sin.½_a_ and cos.½_a_.
Knowing tang._a_, to calculate tang.½_a_.
Knowing sin._a_, to calculate sin.⅓_a_--Knowing cos._a_, to
calculate cos.⅓_a_.
Use of the formula cos._p_+cos._q_ = 2cos.½(_p + q_)cos.½(_p - q_), to
render logarithms applicable to the sum of two trigonometrical lines,
sines or cosines.--To render logarithms applicable to the sum of two
tangents.
Construction of the trigonometric tables.
Use in detail of the tables of Callet.--Appreciation, by the
proportional parts, of the degree of exactness in the calculation of the
angles.--Superiority of the tangent formulas.
_Resolution of triangles._
Relations between the angles and the sides of a right-angled triangle,
or of any triangle whatever.--When the three angles of a triangle are
given, these relations determine only the ratios of the sides.
Resolution of right-angled triangles.--Of the case in which the
hypothenuse and a side nearly equal to it are given.
Knowing a side and two angles of any triangle, to find the other
parts, and also the surface of the triangle.
Knowing two sides _a_ and _b_ of a triangle and the included angle C,
to find the other parts and also the surface of the triangle.--The
tang.½(A - B) may be determined; or tang.A and tang.B directly.
Knowing the three sides _a_, _b_, _c_, to find the angles and the
surface of the triangle.--Employment of the formula which gives
tang.½A.
_Application to surveying for plans._
Measurement of bases with rods.
Measurement of angles.--Description and use of the circle.--Use of the
telescope to render the line of sight more precise.--Division of the
circle.--Verniers.
Measurement and calculation of a system of triangles.--Reduction of
angles to the centres of stations.
How to connect the secondary points to the principal system.--Use of
the plane table and of the compass.
2. SPHERICAL TRIGONOMETRY.
Fundamental relations (cos._a_ = cos._b_ cos._c_ + sin._b_ sin._c_
cos.A) between the sides and the angles of a spherical triangle.
To deduce thence the relations sin.A : sin.B = sin._a_ : sin._b_;
cot._a_ sin._b_ - cot.A sin.C = cos._b_ cos.C, and by the consideration
of the supplementary triangle cos.A = -cos.B cos.C + sin.B sin.C
cos._a_.
Right-angled triangles.--Formulas cos._a_ = cos._b_ cos._c_; sin._b_ =
sin._a_ sin.B; tang._c_ = tang._a_ cos.B, and tang._b_ = sin._c_ tang.B.
In a right-angled triangle the three sides are less than 90°, or else
two of the sides are greater than 90°, and the third is less. An angle
and the side opposite to it are both less than 90°, or both greater.
Resolution of any triangles whatever:
1º Having given their three sides _a_, _b_, _c_, or their three angles
A, B, C.--Formulas tang.½_a_ and tang.1/2A, calculable by logarithms:
2º Having given two sides and the included angle, or two angles and
the included side.--Formulas of Delambre:
3º Having given two sides and an angle opposite to one of them, or two
angles and a side opposite to one of them. Employment of an auxiliary
angle to render the formulas calculable by logarithms.
Applications.--Survey of a mountainous country.--Reduction of the base
and of the angles to the horizon.--Determination of differences of
level.
Knowing the latitude and the longitude of two points on the surface of
the earth, to find the distance of those points.
V. ANALYTICAL GEOMETRY.
The important property of homogeneity must be given with clearness and
simplicity.
The transformation of co-ordinates must receive some numerical
applications, which are indispensable to make the student clearly see
the meaning of the formulas.
The determination of tangents will be effected in the most general
manner by means of the derivatives of the various functions, which we
inserted in the programme of algebra. After having shown that this
determination depends on the calculation of the derivative of the
ordinate with respect to the abscissa, this will be used to simplify the
investigation of the tangent to curves of the second degree and to
curves whose equations contain transcendental functions. The discussion
of these, formerly pursued by laborious indirect methods, will now
become easy; and as curves with transcendental equations are frequently
encountered, it will be well to exercise students in their discussion.
The properties of foci and of the directrices of curves of the second
degree will be established directly, for each of the three curves, by
means of the simplest equations of these curves, and without any
consideration of the analytical properties of foci, with respect to the
general equation of the second degree. With even greater reason will we
dispense with examining whether curves of higher degree have foci, a
question whose meaning even is not well defined.
We retained in algebra the elimination between two equations of the
second degree with two unknown quantities, a problem which corresponds
to the purely analytical investigation of the co-ordinates of the points
of intersection of two curves of the second degree. The final equation
is in general of the fourth degree, but we may sometimes dispense with
calculating that equation. A graphical construction of the curves,
carefully made, will in fact be sufficient to make known, approximately,
the co-ordinates of each of the points of intersection; and when we
shall have thus obtained an approximate solution, we will often be able
to give it all the numerical rigor desirable, by successive
approximations, deduced from the equations. These considerations will be
extended to the investigation of the real roots of equations of any form
whatever with one unknown quantity.
Analytical geometry of three dimensions was formerly entirely taught
within the Polytechnic school, none of it being reserved for the course
of admission. For some years past, however, candidates were required to
know the equations of the right line in space, the equation of the
plane, the solution of the problems which relate to it and the
transformation of co-ordinates. But the consideration of surfaces of the
second order was reserved for the interior teaching. We think it well to
place this also among the studies to be mastered before admission, in
accordance with the general principle now sought to be realized, of
classing with them that double instruction which does not exact a
previous knowledge of the differential calculus.
We have not, however, inserted here all the properties of surfaces of
the second order, but have retained only those which it is indispensable
to know and to retain. The transformation of rectilinear co-ordinates,
for example, must be executed with simplicity, and the teacher must
restrict himself to giving his pupils a succinct explanation of the
course to be pursued; this will suffice to them for the very rare cases
in which they may happen to have need of them. No questions will be
asked relating to the general considerations, which require very
complicated theoretical discussions, and especially that of the general
reduction of the equation of the second degree with three variables. We
have omitted from the problems relating to the right line and to the
plane, the determination of the shortest distance of two right lines.
The properties of surfaces of the second order will be deduced from the
equations of those surfaces, taken directly in the simplest forms. Among
these properties, we place in the first rank, for their valuable
applications, those of the surfaces which can be generated by the
movement of a right line.
PROGRAMME OF ANALYTICAL GEOMETRY.
1. GEOMETRY OF TWO DIMENSIONS.
Rectilinear co-ordinates.--Position of a point on a plane.
Representation of geometric loci by equations.
Homogeneity of equations and of formulas.--Construction of algebraic
expressions.
Transformation of rectilinear co-ordinates.
Construction of equations of the first degree.--Problems on the right
line.
Construction of equations of the second degree.--Division of the
curves which they represent into three classes.--Reduction of the
equation to its simplest form by the change of co-ordinates.[5]
[Footnote 5: The students will apply these reductions to a
numerical equation of the second degree, and will determine the
situation of the new axes with respect to the original axes, by
means of trigonometrical tables. They will show to the examiner
the complete calculations of this reduction and the trace of the
two systems of axes and of the curves.]
Problem of tangents.--The coefficient of inclination of the tangent to
the curve, to the axis of the abscissas, is equal to the derivative of
the ordinate with respect to the abscissa.
_Of the ellipse._
Centre and axes.--The squares of the ordinates perpendicular to one of
the axes are to each other as the products of the corresponding segments
formed on that axis.
The ordinates perpendicular to the major axis are to the corresponding
ordinates of the circle described on that axis as a diameter, in the
constant ratio of the minor axis to the major.--Construction of the
curve by points, by means of this property.
Foci; eccentricity of the ellipse.--The sum of the radii vectors drawn
to any point of the ellipse is constant and equal to the major
axis.--Description of the ellipse by means of this property.
Directrices.--The distance from each point of the ellipse to one of
the foci, and to the directrix adjacent to that focus, are to each other
as the eccentricity is to the major axis.
Equations of the tangent and of the normal at any point of the
ellipse.[6]--The point in which the tangent meets one of the axes
prolonged is independent of the length of the other axis.--Construction
of the tangent at any point of the ellipse by means of this property.
[Footnote 6: They will be deduced from the property, previously
demonstrated, of the derivative of the ordinate with respect to
the abscissa.]
The radii vectores, drawn from the foci to any point of the ellipse,
make equal angles with the tangent at that point or the same side of
it.--The normal bisects the angle made by the radii vectores with each
other.--This property may serve to draw a tangent to the ellipse through
a point on the curve, or through a point exterior to it.
The diameters of the ellipse are right lines passing through the
centre of the curve.--The chords which a diameter bisects are parallel
to the tangent drawn through the extremity of that diameter.--
Supplementary chords. By means of them a tangent to the ellipse can be
drawn through a given point on that curve or parallel to a given right
line.
Conjugate diameters.--Two conjugate diameters are always parallel to
supplementary chords, and reciprocally.--Limit of the angle of two
conjugate diameters.--An ellipse always contains two equal conjugate
diameters.--The sum of the squares of two conjugate diameters is
constant.--The area of the parallelogram constructed on two conjugate
diameters is constant.--To construct an ellipse, knowing two conjugate
diameters and the angle which they make with each other.
Expression of the area of an ellipse in function of its axes.
_Of the hyperbola._
Centre and axes.--Ratio of the squares of the ordinates perpendicular
to the transverse axes.
Of foci and of directrices; of the tangent and of the normal; of
diameters and of supplementary chords.--Properties of these points and
of these lines, analogous to those which they possess in the ellipse.
Asymptotes of the hyperbola.--The asymptotes coincide with the
diagonals of the parallelogram formed on any two conjugate
diameters.--The portions of a secant comprised between the hyperbola and
its asymptotes are equal.--Application to the tangent and to its
construction.
The rectangle of the parts of a secant, comprised between a point of
the curve and the asymptotes, is equal to the square of half of the
diameter to which the secant is parallel.
Form of the equation of the hyperbola referred to its asymptotes.
_Of the parabola._
Axis of the parabola.--Ratio of the squares of the ordinates
perpendicular to the axis.
Focus and directrix of the parabola.--Every point of the curve is
equally distant from the focus and from the directrix.--Construction of
the parabola.
The parabola may be considered as an ellipse, in which the major axis
is indefinitely increased while the distance from one focus to the
adjacent summit remains constant.
Equations of the tangent and of the normal.--Sub-tangent and
sub-normal. They furnish means of drawing a tangent at any point of the
curve.
The tangent makes equal angles with the axis and with the radius
vector drawn to the point of contact.--To draw, by means of this
property, a tangent to the parabola, 1º through a point on the curve; 2º
through an exterior point.
All the diameters of the parabola are right lines parallel to the
axis, and reciprocally.--The chords which a diameter bisects are
parallel to the tangent drawn at the extremity of that diameter.
Expression of the area of a parabolic segment.
* * *
Polar co-ordinates.--To pass from a system of rectilinear and
rectangular co-ordinates to a system of polar co-ordinates, and
reciprocally.
Polar equations of the three curves of the second order, the pole
being situated at a focus, and the angles being reckoned from the axis
which passes through that focus.
Summary discussion of some transcendental curves.--Determination of
the tangent at one of their points.
Construction of the real roots of equations of any form with one
unknown quantity.--Investigation of the intersections of two curves of
the second degree.--Numerical applications of these formulas.
2. GEOMETRY OF THREE DIMENSIONS.
The sum of the projections of several consecutive right lines upon an
axis is equal to the projection of the resulting line.--The sum of the
projections of a right line on three rectangular axes is equal to the
square of the right line.--The sum of the squares of the cosines of the
angles which a right line makes with three rectangular right lines is
equal to unity.
The projection of a plane area on a plane is equal to the product of
that area by the cosine of the angle of the two planes.
Representation of a point by its co-ordinates.--Equations of lines and
of surfaces.
Transformation of rectilinear co-ordinates.
_Of the right line and of the plane._
Equations of the right line.--Equation of the plane.
To find the equations of a right line, 1º which passes through two
given points, 2º which passes through a given point and which is
parallel to a given line.
To determine the point of intersection of two right lines whose
equations are known.
To pass a plane, 1º through three given points; 2º through a given
point and parallel to a given plane; 3º through a point and through a
given right line.
Knowing the equations of two planes, to find the projections of their
intersection.
To find the intersection of a right line and of a plane, their
equations being known.
Knowing the co-ordinates of two points, to find their distance.
From a given point to let fall a perpendicular on a plane; to find the
foot and the length of that perpendicular (rectangular co-ordinates).
Through a given point to pass a plane perpendicular to a given right
line (rectangular co-ordinates).
Through a given point, to pass a perpendicular to a given right line;
to determine the foot and the length of that perpendicular (rectangular
co-ordinates).
Knowing the equations of a right line, to determine the angles which
that line makes with the axes of the co-ordinates (rectangular
co-ordinates).
To find the angle of two right lines whose equations are known
(rectangular co-ordinates).
Knowing the equation of a plane, to find the angles which it makes
with the co-ordinate planes (rectangular co-ordinates).
To determine the angle of two planes (rectangular co-ordinates).
To find the angle of a right line and of a plane (rectangular
co-ordinates).
_Surfaces of the second degree._
They are divided into two classes; one class having a centre, the
other not having any. Co-ordinates of the centre.
Of diametric planes.
Simplification of the general equation of the second degree by the
transformation of co-ordinates.
The simplest equations of the ellipsoid, of the hyperboloid of one
sheet and of two sheets, of the elliptical and the hyperbolic
paraboloid, of cones and of cylinders of the second order.
Nature of the plane sections of surfaces of the second order.--Plane
sections of the cone, and of the right cylinder with circular
base.--Anti-parallel section of the oblique cone with circular base.
Cone asymptote to an hyperboloid.
Right-lined sections of the hyperboloid of one sheet.--Through each
point of a hyperboloid of one sheet two right lines can be drawn, whence
result two systems of right-lined generatrices of the hyperboloid.--Two
right lines taken in the same system do not meet, and two right lines of
different systems always meet.--All the right lines situated on the
hyperboloid being transported to the centre, remaining parallel to
themselves, coincide with the surface of the asymptote cone.--Three
right lines of the same system are never parallel to the same
plane.--The hyperboloid of one sheet may be generated by a right line
which moves along three fixed right lines, not parallel to the same
plane; and, reciprocally, when a right line slides on three fixed lines,
not parallel to the same plane, it generates a hyperboloid of one sheet.
Right-lined sections of the hyperbolic paraboloid.--Through each point
of the surface of the hyperbolic paraboloid two right lines may be
traced, whence results the generation of the paraboloid by two systems
of right lines.--Two right lines of the same system do not meet, but two
right lines of different systems always meet.--All the right lines of
the same system are parallel to the same plane.--The hyperbolic
paraboloid may be generated by the movement of a right line which slides
on three fixed right lines which are parallel to the same plane; or by a
right line which slides on two fixed right lines, itself remaining
always parallel to a given plane. Reciprocally, every surface resulting
from one of these two modes of generation is a hyperbolic paraboloid.
General equations of conical surfaces and of cylindrical surfaces.
VI. DESCRIPTIVE GEOMETRY.
The general methods of Descriptive Geometry,--their uses in
Stone-cutting and Carpentry, in Linear Perspective, and in the
determination of the Shadows of bodies,--constitute one of the most
fruitful branches of the applications of mathematics. The course has
always been given at the Polytechnic School with particular care,
according to the plans traced by the illustrious _Monge_, but no part of
the subject has heretofore been required for admission. The time given
to it in the school, being however complained of on all sides as
insufficient for its great extent and important applications, the
general methods of Descriptive Geometry will henceforth be retrenched
from the internal course, and be required of all candidates for
admission.
As to the programme itself, it is needless to say any thing, for it was
established by _Monge_, and the extent which he gave to it, as well as
the methods which he had created, have thus far been maintained. We
merely suppress the construction of the shortest distance between two
right-lines, which presents a disagreeable and useless complication.
Candidates will have to present to the examiner a collection of their
graphical constructions (_épures_) of all the questions of the
programme, signed by their teacher. They are farther required to make
free-hand sketches of five of their _épures_.
PROGRAMME OF DESCRIPTIVE GEOMETRY.
_Problems relating to the point, to the straight line, and to the
plane._[7]
[Footnote 7: The method of the change of the planes of projection
will be used for the resolution of these problems.]
Through a point given in space, to pass a right line parallel to a
given right line, and to find the length of a part of that right line.
Through a given point, to pass a plane parallel to a given plane.
To construct the plane which passes through three points given in
space.
Two planes being given, to find the projections of their
intersection.
A right line and a plane being given, to find the projections of the
point in which the right line meets the plane.
Through a given point, to pass a perpendicular to a given plane, and
to construct the projections of the point of meeting of the right line
and of the plane.
Through a given point, to pass a right line perpendicular to a given
right line, and to construct the projections of the point of meeting of
the two right lines.
A plane being given, to find the angles which it forms with the
planes of projection.
Two planes being given, to construct the angle which they form
between them.
Two right lines which cut each other being given, to construct the
angle which they form between them.
To construct the angle formed by a right line and by a plane given
in position in space.
_Problems relating to tangent planes._
To draw a plane tangent to a cylindrical surface or to a conical
surface, 1º through a point taken on the surface; 2º through a point
taken out of the surface; 3º parallel to a given right line.
Through a point taken on a surface of revolution, whose meridian is
known, to pass a plane tangent to that surface.
_Problems relating to the intersection of surfaces._
To construct the section made, on the surface of a right and vertical
cylinder, by a plane perpendicular to one of the planes of
projection.--To draw the tangent to the curve of intersection.--To make
the development of the cylindrical surface, and to refer to it the curve
of intersection, and also the tangent.
To construct the intersection of a right cone by a plane perpendicular
to one of the planes of projection. Development and tangent.
To construct the right section of an oblique cylinder.--To draw the
tangent to the curve of intersection. To make the development of the
cylindrical surface, and to refer to it the curve which served as its
base, and also its tangents.
To construct the intersection of a surface of revolution by a plane,
and the tangents to the curve of intersection.--To resolve this
question, when the generating line is a right line which does not meet
the axis.
To construct the intersection of two cylindrical surfaces, and the
tangents to that curve.
To construct the intersection of two oblique cones, and the tangents
to that curve.
To construct the intersection of two surfaces of revolution whose axes
meet.
VII. OTHER REQUIREMENTS.
The preceding six heads complete the outline of the elementary course of
mathematical instruction which it was the object of this article to
present; but a few more lines may well be given to a mere enumeration of
the other requirements for admission to the school.
MECHANICS comes next. The programme is arranged under these heads:
Simple motion and compound motion; Inertia; Forces applied to a free
material point; Work of forces applied to a movable point; Forces
applied to a solid body; Machines.
PHYSICS comprises these topics: General properties of bodies;
Hydrostatics and hydraulics; Densities of solids and liquids; Properties
of gases; Heat; Steam; Electricity; Magnetism; Acoustics; Light.
CHEMISTRY treats of Oxygen; Hydrogen; Combinations of hydrogen with
oxygen; Azote or nitrogen; Combinations of azote with oxygen;
Combination of azote with hydrogen, or ammonia; Sulphur; Chlorine;
Phosphorus; Carbon.
COSMOGRAPHY describes the Stars; the Earth; the Sun; the Moon; the
Planets; Comets; the Tides.
HISTORY and GEOGRAPHY treat of Europe from the Roman Empire to the
accession of Louis XVI.
GERMAN must be known sufficiently for it to be translated, spoken a
little, and written in its own characters.
DRAWING, besides the _épures_ of descriptive geometry, must have been
acquired sufficiently for copying an academic study, and shading in
pencil and in India ink.
Will not our readers agree with M. Coriolis, that “_There are very few
learned mathematicians who could answer perfectly well at an examination
for admission to the Polytechnic School_”?
SCHOOLS OF PREPARATION FOR THE POLYTECHNIC SCHOOL
There are strictly speaking no Junior Military Schools preparatory to
the Polytechnic School, or to the Special Military School at St. Cyr.
These schools are recruited in general from the _Lycées_ and other
schools for secondary instruction, upon which they exert a most powerful
influence. Until 1852 there was no special provision made in the courses
of instruction in the _Lycées_ for the mathematical preparation required
for admission into the Polytechnic, and the Bachelor’s degree in science
could not be obtained without being able to meet the requirements in
Latin, rhetoric, and logic for graduation in the arts, which was
necessary to the profession of law, medicine, and theology. In
consequence, young men who prepared to be candidates for the preliminary
examinations at the Polytechnic and the St. Cyr, left the _Lycées_
before graduation in order to acquire more geometry and less literature
in the private schools, or under private tuition.
A new arrangement, popularly called the _Bifurcation_, was introduced by
the Decrees of the 10th of April, 1852; and has now come into operation.
The conditions demanded for the degree in science were adapted to the
requirements of the Military Schools; and in return for this concession
it is henceforth to be exacted from candidates for the Military Schools.
The diploma of arts is no longer required before the diploma of science
can be given. The instruction, which in the upper classes of the
_Lycées_ had hitherto been mainly preparatory for the former, takes
henceforth at a certain point (called that of _Bifurcation_) two
different routes, conducting separately, the one to the baccalaureate of
arts, the other to that of science. The whole system of teaching has
accordingly been altered. Boys wanting to study algebra are no longer
carried through a long course of Latin; mathematics are raised to an
equality with literature; and thus military pupils--pupils desirous of
admission at the Polytechnic and St. Cyr, may henceforth, it is hoped,
obtain in the _Lycées_ all the preparation which they had latterly
sought elsewhere.
Under this new system the usual course for a boy seems to be the
following:--
He enters the _Lycée_, in the Elementary Classes; or, a little later, in
the Grammar Classes, where he learns Latin and begins Greek. At the age
of about fourteen, he is called upon to pass an examination for
admission into the Upper Division, and here, in accordance with the new
regulations, he makes his choice for mathematics or for literature, the
studies henceforth being divided, one course leading to the bachelorship
of science, the other to that of arts.
In either case he has before him three yearly courses, three
classes--the Third, the Second, and what is called the Rhetoric. At the
close of this, or after passing, if he pleases, another year in what is
called the Logic, he may go up for his bachelor’s degree. The boy who
wants to go to St. Cyr or the Polytechnic chooses, of course, the
mathematical division leading to the diploma he will want, that of a
bachelor of _science_. He accordingly begins algebra, goes on to
trigonometry, to conic sections, and to mechanics, and through
corresponding stages in natural philosophy, and the like. If he chooses
to spend a fourth year in the Logic, he will be chiefly employed in
going over his subjects again. He may take his bachelor’s degree at any
time after finishing his third year; and he may, if he pleases, having
taken that, remain during a fifth or even a sixth year, in the class of
Special Mathematics.
If he be intended for St. Cyr, he may very well leave at the end of his
year in Rhetoric, taking of course his degree. One year in the course of
Special Mathematics will be required before he can have a chance for the
Polytechnic. Usually the number of students admitted at the latter, who
have not passed more than one year in the _mathematiques spéciales_ is
very small. Very probably the young aspirant would try at the end of his
first year in this class, and would learn by practice to do better at
the end of the second.
The following are the studies of the mathematical section of the upper
division as laid down by the ordinance of 30th August, 1854.
THE THIRD CLASS (_Troisième_,) at fourteen years old.
Arithmetic and first notions of Algebra. Plane Geometry and its
applications. First notions of Chemistry and Physics. General notions of
Natural History; Principles of classification. Linear and imitative
Drawing.
THE SECOND CLASS (_Seconde_,) at fifteen years old.
Algebra; Geometry, figures in space, recapitulation; Applications of
Geometry, notions of the geometrical representations of bodies by
projections; Rectilineal Trigonometry; Chemistry; Physics; and Drawing.
THE RHETORIC, at sixteen years old.
Exercises in Arithmetic and Algebra; Geometry; notions on some common
curves; and general recapitulation; Applications of Geometry; notions of
leveling and its processes; recapitulation of Trigonometry; Cosmography;
Mechanics; Chemistry concluded and reviewed; Zoölogy and Animal
Physiology; Botany and Vegetable Physiology; Geology; Drawing. (The
pupil may now be ready for the Degree and for St. Cyr.)
THE LOGIC, at seventeen years old.
Six lessons a week are employed in preparation for the bachelorship of
science, and in a methodical recapitulation of the courses of the three
preceding years according to the state of the pupil’s knowledge.
Two lessons a week are allowed for reviewing the literary instruction;
evening lessons in Latin, French, English, and German, and in History
and Geography, having been given through the whole previous time.
THE SPECIAL MATHEMATICS, at eighteen and nineteen years old.
Five lessons a week are devoted to these studies; in the other lessons
the pupils join those of the Logic class for reviewing all their
previous subjects, whether for the bachelorship in science or for
competition for admission at the _Ecole Normale_ or the Polytechnic.
It will only be necessary to add a few sentences in explanation of the
methods pursued in the upper classes of the _Lycées_. The classes are
large--from 80 to above 100; the lessons strictly professorial lectures,
with occasional questions, as at the Polytechnic itself. In large
establishments the class is divided, and two professors are employed,
giving two parallel courses on the same subject. To correct and fortify
this general teaching, we find, corresponding to the interrogations of
the Polytechnic, what are here called conferences. The members of the
large class are examined first of all in small detachments of five or
six by their own professors once a week; and, secondly, a matter of yet
greater importance, by the professor who is conducting the parallel
course, and by professors who are engaged for this purpose from other
_Lycées_ and preparatory schools, and from among the _répétiteurs_ of
the Polytechnic and the Ecole Normale themselves. It appeared by the
table of the examinations of this latter kind which had been passed by
the pupils of the class of Special Mathematics at the _Lycée_ St. Louis,
that the first pupil on the list had in the interval between the opening
of the school and the date of our visit (February 16th) gone through as
many as twenty-four.
The assistants, who bear the name of _répétiteurs_ at the _Lycées_, do
not correspond in any sense to those whom we shall hereafter notice at
the Ecole Polytechnique. They are in the _Lycées_ mere superintendents
in the _salles d’étude_, who attend to order and discipline, who give
some slight occasional help to the pupils, and may be employed in
certain cases, where the parents wish for it, in giving private tuition
to the less proficient. The system of _salles d’étude_ appears to
prevail universally; the number of the pupils placed in each probably
varying greatly. At the Polytechnic we found eight or ten pupils in
each; at St. Cyr as many as 200. The number considered most desirable at
the _Lycée_ of St. Louis was stated to be thirty.
It thus appears that in France not only do private establishments
succeed in giving preparation for the military schools, but that even in
the first-class public schools, which educate for the learned
professions, it has been considered possible to conduct a series of
military or science classes by the side of the usual literary or arts
classes. The common upper schools are not, as they used to be, and as
with us they are, _Grammar_ schools, they are also _Science_ schools. In
every _Lycée_ there is, so to say, a sort of elementary polytechnic
department, giving a kind of instruction which will be useful to the
future soldier, and at the same time to others, to those who may have to
do with mines, manufactures, or any description of civil engineering.
There is thus no occasion for Junior Military Schools in France, for all
the schools of this class are more or less of a military character in
their studies.
The conditions of admission to the examination for the degree of
Bachelor of Science are simply, sixteen years of age, and the payment of
fees amounting to about 200 fr. (10_l._) Examinations are held three
times a year by the Faculties at Paris, Besançon, Bordeaux, Caen,
Clermont, Dijon, Grenoble, Lille, Lyons, Marseilles, Montpellier, Nancy,
Poitiers, Rennes, Strasburg, and Toulouse, and once a year at Ajaccio,
Algiers, and nineteen other towns. There is a written examination of six
hours, and a viva voce examination of an hour and a quarter. It is, of
course, only a _pass_ examination, and is said to be much less difficult
than the competitive examination for admission to St. Cyr.--_Report of
English Commissioners_, 1856.
THE POLYTECHNIC SCHOOL OF FRANCE.
CONTENTS.
PAGE.
POLYTECHNIC SCHOOL AT PARIS, 11
SUBJECTS OF INSTRUCTION AS PRESCRIBED FOR ADMISSION IN 1850, 13
PREPARATORY COURSE IN THE LYCEES, 49
HISTORY, MANAGEMENT, CONDITIONS OF ADMISSION, COURSE OF STUDY,
EXAMINATIONAL SYSTEM, AND RESULTS, 55
I. FOUNDATION AND HISTORY, 55
Out growth of the Necessities of the Public Service
in 1794, 56
High Scientific Ability of its first Teachers, 58
Peculiar Method of Scientific Teaching, 59
Characteristic features of the _Répétitorial_ System, 59
The _Casernement_, or Barrack Residence of the Pupils, 60
Permanent Organization in 1809, 60
Commission of 1850, 62
II. OUTLINE OF THE PLAN, OBJECTS, AND MANAGEMENT, 63
Public Services provided for in its General Scientific
Course, 63
Admission by Competition in an Open Examination, 63
Annual Charge for Board and Instruction, 64
Exhibitions, (or _bourses, demi-bourses_,) and Outfits
(_trousseaux_,) 64
Length of Course of Study, 64
Number of Professors and Teachers, besides its Military
Staff, 64
Military Establishment, 65
Civil Establishment, 65
General Control and Supervision, 65
1. Board of Administration, 65
2. Board of Discipline, 65
3. Board of Instruction, 65
4. Board of Improvement, 66
III. CONDITIONS AND EXAMINATIONS FOR ADMISSION, 66
Who may be Candidates for Admission, 66
Subjects of Entrance Examination, 66
Preliminary Examination, 67
Written Examination, 67
Oral Examination, 68
Scale of Merit, and Latitude in Amount of Credit given, 68
Reports of Examiners to Minister of War, 69
Co-efficients of Influence, varying with the Study and
Mode of Examination, 69
Decision of Jury on all the Documents of each Candidate, 70
Final Action of the Minister of War, 70
IV. SCHOOL BUILDINGS, COURSE AND METHOD OF STUDY, 70
Situation, Number, and Purposes of Buildings, 70
Daily Routine of Exercises, 72
Method of Teaching and Study, 73
Professorial and Répétitorial, 74
Interrogations, _Général_, 74
“ _Particulieres_ by the _Répétiteurs_, 74
One Répétiteur to every eight Pupils, 74
System of Credits for every Lecture, every Interrogation,
and Exercise, 75
Final Admission to Public Service, depends on daily and
hourly fidelity, 76
Division of First Year’s Work into three portions, 76
First portion--Analysis and Descriptive Geometry, 76
Second “ Mechanics, Geodesy, Physics, &c., 76
Third “ General Private Study, 76
Number and Subjects of Lectures in Second Course, 78
V. EXAMINATIONAL SYSTEM, 78
Ordinary Examinations, 78
1. By Professors on their own Lectures, both Written
and Oral, 78
2. By Examiners on the Manipulations of the Pupils, 78
3. By Répétiteurs every ten or fourteen days, 78
4. By Professors and Répétiteurs at the close of
each Course, 79
First Annual Examination, 79
Table--Co-efficient of Influence in Second Division of
First Year’s Course, 79
Specimen of Credits gained by one Student in First
Year’s Course, 80
Persons excluded from the Second Year’s Course, 81
Second Annual or Great Final Examination, 81
Conducted by the same Examiners as the First, 81
Oral, and extends over the whole Two Years’ Course, 81
Results based on each Day’s Study’s, Year’s, and
Examination’s results, 82
Tables--Co-efficients of Influence in Final
Classification, &c., 82
Order in which the Public Services are Selected, 83
VI. GENERAL REMARKS ON CHARACTER AND RESULTS OF THE
POLYTECHNIC SCHOOL, 84
APPENDIX, 88
PUBLIC SERVICES BESIDE THE ARMY SUPPLIED BY THIS SCHOOL, 88
1. Gunpowder and Saltpetre, 88
2. Navy, 88
3. Marine Artillery and Foundries, 88
4. Naval Architects. School of Application at L’Orient, 88
5. Hydrographers, 88
6. Roads and Bridges. School of Application at Paris, 89
7. Mining Engineers. School of Mines at Paris and
St. Etienne, 89
8. Tobacco Department 90
9. Telegraphs, 90
PROGRAMMES OF INTERNAL INSTRUCTION DURING THE TWO YEARS
OF STUDY, 91
1. Analysis, 91
First Year--Calculus, Differential, 91
Calculus, Integral, 93
Second Year--Calculus, Integral, (continuation,) 94
2. Descriptive Geometry and Stereotomy, 97
First Year--Descriptive Geometry, Geometrical
Drawing, 97
Second Year--Stereotomy: Wood-work, 103
Masonry, 103
3. Mechanics and Machines, 104
First Year--Kinematics, 105
Equilibrium of Forces, 105
Second Year--Dynamics, 112
Hydrostatics, 115
Hydraulics, 115
Machines in Motion, 116
4. Physics, 116
First Year--General Properties of Bodies,
Hydrostatics, Hydrodynamics, 117
Heat, 119
Statical Electricity, 123
Second Year--Dynamical Electricity, 124
Acoustics, 125
Optics, 126
5. Manipulations in Physics, 129
First and Second Year, 130
Distribution of Time, 131
THE POLYTECHNIC SCHOOL AT PARIS[8]
[Footnote 8: Compiled from “_Report and Appendix of English
Commissioners on Military Education_.” 1857.]
I. FOUNDATION AND HISTORY.
The origin of the _Ecole Polytechnique_ dates from a period of disorder
and distress in the history of France which might seem alien to all
intellectual pursuits, if we did not remember that the general stimulus
of a revolutionary period often acts powerfully upon thought and
education. It is, perhaps, even more than the Institute, the chief
scientific creation of the first French Revolution. It was during the
government of the committee of public safety, when Carnot, as war
minister, was gradually driving back the invading armies, and
reorganizing victory out of defeat and confusion, that the first steps
were taken for its establishment. A law, dating the 1st Ventose, year
II., the 12th of March 1794, created a “Commission des Travaux Publics,”
charged with the duty of establishing a regular system for carrying on
public works; and this commission ultimately founded a central school
for public works, and drew up a plan for the competitive examination of
candidates for admission to the service. It was intended at first to
give a complete education for some of the public services, but it was
soon changed into a preparatory school, to be succeeded by special
schools of application. This was the Ecole Polytechnique.
The school and its plan were both owing to an immediate and pressing
want. It was to be partly military and partly civil. Military, as well
as civil education had been destroyed by the revolutionists. The
committee of public safety had, indeed, formed a provisional school for
engineers at Metz, to supply the immediate wants of the army on the
frontier, and at this school young men were hastily taught the elements
of fortification, and were sent direct to the troops, to learn as they
best could, the practice of their art. “But such a method,” says the
report accompanying the law which founded the school, “does not form
engineers _in any true sense of the term_, and can only be justified by
the emergency of the time. The young men should be recalled to the new
school to complete their studies.” Indeed no one knew better than
Carnot, to use the language of the report, “that patriotism and courage
can not always supply the want of knowledge;” and in the critical
campaigns of 1793-4, he must often have felt the need of the institution
which he was then contributing to set on foot. Such was the immediate
motive for the creation of this school. At first, it only included the
engineers amongst its pupils. But the artillery were added within a
year.
We must not, however, omit to notice its civil character, the
combination of which with its military object forms its peculiar
feature, and has greatly contributed to its reputation. Amongst its
founders were men, who though ardent revolutionists, were thirsting for
the restoration of schools and learning, which for a time had been
totally extinguished. The chief of these, besides Carnot, were Monge and
Fourcroy, Berthollet and Lagrange. Of Carnot and Lagrange, one amongst
the first of war ministers, the other one of the greatest of
mathematicians, we need not say more. Berthollet, a man of science and
practical skill, first suggested the school; Monge, the founder of
Descriptive Geometry, a favorite _savant_ of Napoleon though a zealous
republican, united to real genius that passion for teaching and for his
pupils, which makes the _beau idéal_ of the founder of a school; and
Fourcroy was a man of equal practical tact and science, who at the time
had great influence with the convention, and was afterwards intrusted by
Napoleon with much of the reorganization of education in France.
When the school first started there was scarcely another of any
description in the country. For nearly three years the revolution had
destroyed every kind of teaching. The attack upon the old schools, in
France, as elsewhere, chiefly in the hands of the clergy, had been begun
by a famous report of Talleyrand’s, presented to the legislative
assembly in 1791, which recommended to suppress all the existing
academies within Paris and the provinces, and to replace them by an
entirely new system of national education through the country. In this
plan a considerable number of military schools were proposed, where boys
were to be educated from a very early age. When the violent
revolutionists were in power, they adopted the destructive part of
Talleyrand’s suggestions without the other. All schools, from the
university downwards, were destroyed; the large exhibitions or
_Bourses_, numbering nearly 40,000, were confiscated or plundered by
individuals, and even the military schools and those for the public
works (which were absolutely necessary for the very roads and the
defense of the country) were suppressed or disorganized. The school of
engineers at Mézières (an excellent one, where Monge had been a
professor,) and that of the artillery at La Fère, were both broken up,
whilst the murder of Lavoisier, and the well known saying in respect to
it, that “the Republic had no need of chemists,” gave currency to a
belief, which Fourcroy expressed in proposing the Polytechnic, “that the
late conspirators had formed a deliberate plan to destroy the arts and
sciences, and to establish their tyranny on the ruins of human reason.”
Thus it was on the ruin of all the old teaching, that the new
institution was erected; a truly _revolutionary_ school, as its founders
delighted to call it, using the term as it was then commonly used, as a
synonym for all that was excellent. And then for the first time avowing
the principle of public competition, its founders, Monge and Fourcroy,
began their work with an energy and enthusiasm which they seem to have
left as a traditional inheritance to their school. It is curious to see
the difficulties which the bankruptcy of the country threw in their way,
and the vigor with which, assisted by the summary powers of the
republican government, they overcame them. They begged the old Palais
Bourbon for their building; were supplied with pictures from the Louvre;
the fortunate capture of an English ship gave them some uncut diamonds
for their first experiments; presents of military instruments were sent
from the arsenals of Havre; and even the hospitals contributed some
chemical substances. In fine, having set their school in motion, the
government and its professors worked at it with such zeal and effect,
that within five months after their project was announced, they had held
their first entrance examination, open to the competition of all France,
and started with three hundred and seventy-nine pupils.
The account of one of these first pupils, who is among the most
distinguished still surviving ornaments of the Polytechnic, will convey
a far better idea of the spirit of the young institution than could be
given by a more lengthy description. M. Biot described to us vividly the
zeal of the earliest teachers, and the thirst for knowledge which,
repressed for awhile by the horrors of the period, burst forth with
fresh ardor amongst the French youth of the time. Many of them, he said,
like himself, had been carried away by the enthusiasm of the revolution,
and had entered the army. “My father had sent me,” he added, “to a
mercantile house, and indeed I never felt any great vocation to be a
soldier, but _Que voulez vous? les Prussiens etaient en Champagne._” He
joined the army, served two years under Dumouriez, and returned to Paris
in the reign of terror, “to see from his lodgings in the Rue St. Honore
the very generals who had led us to victory, Custine and Biron, carried
by in the carts to the guillotine. “Imagine what it was when we heard
that Robespierre was dead, and that we might return safely to study
after all this misery, and then to have for our teachers La Place,
Lagrange, and Monge. We felt like men brought to life again after
suffocation. Lagrange said, modestly, “Let me teach them arithmetic.”
Monge was more like our father than our teacher; he would come to us in
the evening, and assist us in our work till midnight, and when he
explained a difficulty to one of our _chefs de brigade_, it ran like an
electric spark through the party.” The pupils were not then, he told us,
as they have since been, shut up in barracks, they were left free, but
there was no idleness or dissipation amongst them. They were united in
zealous work and in good _camaraderie_, and any one known as a bad
character was avoided. This account may be a little tinged by
enthusiastic recollections, but it agreed almost entirely with that of
M. de Barante, who bore similar testimony to the early devotion of the
pupils, and the unique excellence of the teaching of Monge.
We are not, however, writing a history of this school, and must confine
ourselves to such points as directly illustrate its system of teaching
and its organization. These may be roughly enumerated in the following
order:
1. Its early history is completed by the law of its organization, given
it by La Place in his short ministry of the interior. This occurred in
the last month of 1799, a memorable era in French history, for it was
immediately after the revolution of the 18th of Brumaire, when Napoleon
overthrew the Directory and made himself First Consul. One of his
earliest acts was to sign the charter of his great civil and military
school. This charter or decree deserves some attention, because it is
always referred to as the law of the foundation of the school. It
determined the composition of the two councils of instruction and
improvement, the bodies to which the direction of the school was to be,
and still is, intrusted; some of its marked peculiarities in the mode
and subject of teaching. It is important to notice each of the two
points.
The direction of the school was at first almost entirely in the hands of
its professors, who formed what is still called its Council of
Instruction. Each of them presided over the school alternately for one
month, a plan copied from the revolutionary government of the
Convention. In the course of a few years, however, another body was
added, which has now the real management of the school. This is called
the “Council of Improvement” (_Conseil de perfectionnement_,) and a part
of its business is to see that the studies form a good preparation for
those of the more special schools (_écoles d’application_) for the civil
and military service. It consists of eminent men belonging to the
various public departments supplied by the school, and some of the
professors. It has had, as far as we could judge, an useful influence;
_first_, as a body not liable to be prejudiced in its proposals by the
feelings of the school, and yet interested in its welfare and
understanding it; _secondly_, as having shown much skill in the
difficult task of making the theoretical teaching of the Polytechnic a
good introduction to the practical studies of the public service;
_thirdly_, as being sufficiently influential, from the character of its
members, to shield the school from occasional ill-judged interference.
It should be added that hardly any year has passed without the Council
making a full report on the studies of the school, with particular
reference to their bearing on the Special Schools of Application.
The method of scientific teaching has been peculiar from the beginning.
It is the most energetic form of what may be called the _repetitorial_
system, a method of teaching almost peculiar to France, and which may be
described as a very able combination of professional and tutorial
teaching. The object of the _répétiteur_, or private tutor, is to second
every lecture of the professor, to explain and fix it by ocular
demonstration, explanations, or examination. This was a peculiarity in
the scheme of Monge and Foureroy. The latter said, in the first
programme, “Our pupils must not only learn, they must at once carry out
their theory. We must distribute them into small rooms, where they shall
practice the plans of descriptive geometry, which the professors have
just shown them in their public lectures. And in the same manner they
must go over in practice (_répéteront_) in separate laboratories the
principal operations of chemistry.” To carry out this system the twenty
best pupils, of whom M. Biot was one, were selected as _répétiteurs_
soon after the school had started. Since then the vacancies have always
been filled by young but competent men, aspiring themselves to become in
turn professors. They form a class of teachers more like the highest
style of private tutors in our universities, or what are called in
Germany _Privat-docenten_, than any other body--with this difference,
that they do not give their own lectures, but breaking up the
professor’s large class into small classes of five and six pupils,
examine these in _his_ lecture. The success of this attempt we shall
describe hereafter.
2. A change may be noticed which was effected very early by the Council
of Improvement--the union of pupils for artillery and engineers in a
single school of application. The first report in December 1800, speaks
of the identity in extent and character of the studies required for
these two services; and in conformity with its recommendation, the law
of the 3rd of October 1802, (12th Vendémiaire, XI.) dissolved the
separate artillery school at Châlons, and established the united school
for both arms in the form which it still retains at Metz.
3. In 1805 a curious change was made, and one very characteristic of the
school. The pupils have always been somewhat turbulent, and generally on
the side of opposition. In the earliest times they were constantly
charged with _incivisme_, and the aristocracy was said to have “taken
refuge within its walls.” In fact, one of its earliest and of its few
great _literary_ pupils, M. de Barante, confirmed this statement,
adding, as a reason, that the school gave for a while the only good
instruction in France. It was in consequence of some of these changes
that the pupils who had hitherto lived in their own private houses or
lodgings in Paris, were collected in the school building. This
“_casernement_” said to be immediately owing to a burst of anger of
Napoleon, naturally tended to give the school a more military character;
but it was regarded as an unfortunate change by its chief scientific
friends. “_Ah! ma pauvre école!_” M. Biot told us he had exclaimed, when
he saw their knapsacks on their beds. He felt, he said, that the
enthusiasm of free study was gone, and that now they would chiefly work
by routine and compulsion.
4. The year 1809 may be called the epoch at which the school attained
its final character. By this time the functions, both of boards and
teachers, were accurately fixed, some alterations in the studies had
taken place, and the plan of a final examination had been drawn up,
according to which the pupils were to obtain their choice of the branch
of the public service they preferred. In fact, the school may be said to
have preserved ever since the form it then assumed, under a variety of
governments and through various revolutions, in most of which, indeed,
its pupils have borne some share; and one of which, the restoration of
1816, was attended with its temporary dissolution.
Thus, during the first years after its foundation the Polytechnic grew
and flourished in the general dearth of public teaching, being indeed
not merely the only great school, but, until the Institute was founded,
the only scientific body in France. Working on its first idea of high
professorial lectures, practically applied and explained by
_répétiteurs_, its success in its own purely scientific line was, and
has continued to be, astonishing. Out of its sixteen earliest
professors, ten still retain an European name. Lagrange, Monge,
Fourcroy, La Place, Guyton de Morveau were connected with it. Malus,
Hauy, Biot, Poisson, and De Barante, were among its earliest pupils.
Arago, Cauchy, Cavaignac, Lamoricière, with many more modern names, came
later. All the great engineers and artillerymen of the empire belonged
to it, and the long pages in its calendar of distinguished men are the
measure of its influence on the civil and military services of France.
In fact its pupils, at a time of enormous demands, supplied all the
scientific offices of the army, and directed all the chief public works,
fortresses, arsenals, the improvement of cities, the great lines of
roads, shipbuilding, mining--carried out, in a word, most of the great
improvements of Napoleon. He knew the value of his school, “the hen” as
he called it, “that laid him golden eggs”--and perhaps its young pupils
were not improved by the excessive official patronage bestowed by him
upon “the envy of Europe,” “the first school in the world.” It can not,
however, be matter of surprise, that its vigor and success should have
caused Frenchmen, even those who criticise its influence severely, to
regard it with pride as an institution unrivaled for scientific
purposes.
It is not necessary to give any detailed account of the later history of
the school, but we must remark that disputes have frequently arisen with
regard to the best mode of harmonizing its teaching with that of the
special schools of application to which it conducts. These disputes have
been no doubt increased by the union of a civil and military object in
the same school. The scientific teaching desirable for some of the
higher civil professions has appeared of doubtful advantage to those
destined for the more practical work of war. There has been always a
desire on the one side to qualify pure mathematics by application, a
strong feeling on the other that mathematical study sharpens the mind
most keenly for some of the practical pursuits of after life. We should
add, perhaps, that there has been some protest in France (though little
heard among the scientific men who have been the chief directors of the
school) against the _esprit faux_, the exclusive pursuit of mathematics
to the utter neglect of literature, and the indifference to moral and
historical studies. Some one or other of these complaints any one who
studies the _literature_, the pamphlets, and history of the school will
find often reproduced in the letters of war ministers, of artillery and
engineer officers commanding the school of application at Metz, or of
committees from the similar schools for the mines and the roads and
bridges. The last of these occasions illustrates the present position of
the school.
On the 5th of June 1850, the legislative assembly appointed a mixed
commission of military men and civilians, who were charged to revise all
the programs of instruction, and to recommend all needful changes in the
studies of the pupils, both those preparatory to entrance[9] and those
actually pursued in the school. The commission was composed as
follows:--
M. Thenard, Member of the Academy of Sciences, and of the Board of
Improvement of the Polytechnic School, President.
Le Verrier, Member of the Academy of Sciences and of the Legislative
Assembly, Reporter.
Noizet, General of Brigade of Engineers.
Poncelet, General of Brigade of Engineers, Commandant of the
Polytechnic School, Member of the Academy of Sciences.
Piobert, General of Brigade of Artillery, Member of the Academy of
Sciences.
Mathieu, Rear Admiral.
Duhamel, Member of the Academy of Sciences, Director of Studies at the
Polytechnic School.
Mary, Divisional Inspector of Roads and Bridges.
Morin, Colonel of Artillery, Member of the Academy of Sciences.
Regnault, Engineer of Mines, Member of the Academy of Sciences.
Olivier, Professor at the _Conservatoire des Arts et Metiers_.
Debacq, Secretary for Military Schools at the Ministry of War,
Secretary.
[Footnote 9: In an Analysis of the Report of this Commission, see
page 97.]
A chronic dispute which has gone on from the very first year of the
school’s existence, between the exclusive study of abstract mathematics
on the one hand, and their early practical application on the other, was
brought to a head (though it has scarcely been set at rest) by this
commission. All the alterations effected have been in the direction of
eliminating a portion of the pure mathematics, and of reducing abstract
study to the limits within which it was believed to be most directly
applicable to practice. The results, however, are still a subject of
vehement dispute, in which most of the old scientific pupils of the
Polytechnic, and many of what may be styled its most practical members,
the officers of the artillery and engineers, are ranged on the side of
“early and deep scientific study _versus_ early practical applications.”
It is, indeed, a question which touches the military pupils nearly,
since it is in their case particularly that the proposed abstract
studies of the Polytechnic might be thought of the most doubtful
advantage. We do not try to solve the problem here, though the facts
elsewhere stated will afford some materials for judgment. We incline to
the opinion of those who think that the ancient _genius loci_, the
traditional teaching of the school, will be too strong for legislative
interference, and that, in spite of recent enactments, abstract science
and analysis will reign in the lecture-rooms and halls of study of the
Polytechnic, now as in the days of Monge.
II. AN OUTLINE OF THE MANAGEMENT AND OF THE ESTABLISHMENT OF THE SCHOOL,
ETC.
The Polytechnic, as we have said, is a preparatory and general
scientific school; its studies are not exclusively adapted for any one
of the departments to which at the close of its course the scholars will
find themselves assigned; and on quitting it they have, before entering
on the actual discharge of their duties of whatever kind, to pass
through a further term of teaching in some one of the schools of
application specially devoted to particular professions.
The public services for which it thus gives a general preparation are
the following:
_Military: Under the Minister at War._
Artillery (_Artillerie de terre_.)
Engineers (_Génie_.)
The Staff Corps (_Corps d’Etat Major_.)
The Department of Powder and Saltpetre (_Poudres et Salpétres_.)
_Under the Minister of Marine._
Navy, (_Marine_.)
Marine Artillery (_Artillerie de mer_.)
Naval Architects (_Génie maritime_.)
The Hydrographical Department (_Corps des Ingénieurs Hydrographes_.)
_Civil: Under the Minister of Public Works._
The Department of Roads and Bridges (_Ponts-et-chaussées_.)
The Department of Mines (_Mines_.)
_Under the Minister of the Interior._
The Telegraph Department (_Lignes Télégraphiques_.)
_Under the minister of Finance._
The Tobacco Department (_Administration des Tabacs_.)
To these may be added at any time, by a decree on the part of the
government, any other departments, the duties of which appear to require
an extensive knowledge of mathematics, physics, or chemistry.
Admission to the school is, and has been since its first commencement in
1794, obtained by competition in a general examination, held yearly, and
open to all. Every French youth, between the age of sixteen and twenty,
(or if in the army up to the age of twenty-five,) may offer himself as a
candidate.
A board of examiners passes through France once every year, and examines
all who present themselves, that have complied with the conditions,
which are fully detailed in the decree given in the appendix. It
commences at Paris.
A list of such of the candidates as are found eligible for admittance to
the Polytechnic is drawn up from the proceedings of the board, and
submitted to the minister at war; the number of places likely to be
vacant has already been determined, and the minister fixes the number of
admissions accordingly. The candidates admitted are invariably taken in
the order of merit.
The annual charge for board and instruction is 40_l._ (1,000 fr.,)
payable in advance in four installments. In addition there is the cost
of outfit, varying from 20_l._ to 24_l._ Exhibitions, however, for the
discharge of the whole or of one-half of the expense (_bourses_ and
_demi-bourses_,) are awarded by the state in favor of _all_ the
successful candidates, whose parents can prove themselves to be too poor
to maintain their children in the school. Outfits and half outfits
(_trousseaux_) and _demi-trousseaux_) are also granted in these cases,
on the entrance of the student into the school; and the number of these
_boursiers_ and _demi-boursiers_ amounts at the present time to
one-third of the whole.
The course of study is completed in two years. On its successful
termination which is preceded by a final examination, the students are
distributed into the different services, the choice being offered them
in the order of their merit, and laid down in the classified list drawn
up after the examination. If it so happen that the number of places or
the services which can be offered is not sufficient for the number of
qualified students, those at the bottom of the list are offered service
in the infantry or cavalry, and those who do not enter the public
service, are supplied with certificates of having passed successfully
through the school. Students who have been admitted into the school from
the army, are obliged to re-enter the army.
All others, as has been said, have the right of choosing, according to
their position on the list, the service which they prefer, so far, that
is, as the number of vacancies in that service will allow; or they may
if they please decline to enter the public service at all.
Such is a general outline of the plan and object of the school. We may
add that, besides its military staff, it employs no less than
thirty-nine professors and teachers; that it has four boards of
management, and that ten scientific men unconnected with the school, and
amongst the most distinguished in France, conduct its examinations. The
magnitude of this establishment for teaching may be estimated by the
fact, that the number of pupils rarely exceeds three hundred and fifty,
and is often much less.
A fuller enumeration of these bodies will complete our present sketch.
I. The military establishment consists of:--
The Commandant, a General Officer, usually of the Artillery or the
Engineers, at present a General of Artillery.
A Second in Command, a Colonel or Lieutenant-Colonel, chosen from
former pupils of the school; at present a Colonel of Engineers.
Three Captains of Artillery and Three Captains of Engineers, as
Inspectors of Studies, chosen also from former pupils of the school.
Six Adjutants (_adjoints_,) non-commissioned officers, usually such as
have been recommended for promotion.
II. The civil establishment consists of:--
1. A Director of Studies, who has generally been a civilian, but is at
present a Lieutenant-Colonel of Engineers.
2. Fifteen Professors, viz.:--Two of Mathematical Analysis. Two of
Mechanics and Machinery. One of Descriptive Geometry. Two of Physics.
Two of Chemistry. One of Military Art and Fortification. One of Geodesy.
One of Architecture. One of French Composition. One of German. One of
Drawing. Of these one is an officer of the Staff, another of the
Artillery, and a third of the Navy; two are Engineers in Chief of the
Roads and Bridges; nine are civilians, of whom two are Members of the
Academy of Sciences.
3. Three Drawing Masters for Landscape and Figure Drawing; one for
Machine Drawing, and one for Topographical Drawing.
4. Nineteen Assistant and Extra Assistant Teachers, (_répétiteurs_ and
_répétiteurs adjoints_) whose name and functions are both peculiar.
5. Five Examiners for Admission, consisting at present of one Colonel
of Artillery, as President, and four civilians.
6. Five Examiners of Students (civilians,) four of them belonging to
the Academy of Sciences.
7. There is also a separate Department for the ordinary Management of
Administration of the affairs of the school, the charge of the fabric
and of the library and museums; and a Medical Staff.
III. The general control or supervision of the school is vested, under
the war department, in four great boards of councils, viz.:--
1. A board of administration, composed of the commandant, the second in
command, the director of studies, two professors, two captains, and two
members of the administrative staff. This board has the superintendence
of all the financial business and all the minutiae of the internal
administration of the school.
2. A board of discipline, consisting of the second in command, the
director, two professors, three captains (of the school,) and two
captains of the army, chosen from former pupils. The duty of this board
is to decide upon cases of misconduct.
3. A board of instruction, whose members are, the commandant, the second
in command, the director, the examiners of students, and the professors;
and whose chief duty is to make recommendations relating to
ameliorations in the studies, the programmes of admission and of
instruction in the school, to--
4. A board of improvement, charged with the general control of the
studies, formed of--
The Commandant, as President.
The Second in Command.
The Director of Studies.
Two Delegates from the Department of Public Works.
One Delegate from the Naval Department.
One Delegate from the Home Department.
Three Delegates from the War Department.
Two Delegates from the Academy of Sciences.
Two Examiners of Students.
Three Professors of the School.
III. CONDITIONS AND EXAMINATIONS FOR ADMISSION.
The entrance examination is held yearly in August; the most important
conditions for admission to it are always inserted in the _Moniteur_
early in the year, and are--
1st. All candidates must be bachelors of science.
2nd. All candidates (unless they have served in the army) must have been
as much as sixteen and not more than twenty years old on the 1st of
January preceding.
3rd. Privates and non-commissioned officers of the army must be above
twenty and under twenty-five years of age; must have served two years,
and have certificates of good conduct.
4th. Candidates who propose to claim pecuniary assistance (a _bourse_ or
_demi-bourse_) must present formal proofs of their need of it.
The subjects of the entrance examination are the following:--
_Arithmetic_, including Vulgar and Decimal Fractions, Weights and
Measures, Involution and Evolution; Simple Interest.
_Geometry_ of Planes and Solids; application of Geometry to Surveying;
Properties of Spherical Triangles.
_Algebra_, including Quadratic Equations with one unknown quantity,
Series and Progressions in general; Binomial Theorem and its
applications; Logarithms and their use; on Derived Functions; on the
Theory of Equations; on Differences; application of the Theory of
Differences to the Numerical Solution of Equations.
_Plane and Spherical Trigonometry_; Solution of Triangles; application
of Trigonometry to Surveying.
_Analytical Geometry_, including Geometry of two dimensions;
Co-ordinates; Equations of the first and second degree, with two
variables; Tangents and Asymptotes; on the Ellipse, Hyperbola, and
Parabola; Polar Co-ordinates; Curved Lines in general.
_Geometry of three dimensions_, including the Theory of Projections;
Co-ordinates; the Right Line and Plane; Surfaces of the second degree;
Conical and Cylindrical Surfaces.
_Descriptive Geometry_; Problems relative to a Point, Right Line and
Plane; Tangent Planes; Intersection of Surfaces.
_Mechanics_; on the Movement of a Point considered geometrically; on
the Effect of Forces applied to points and bodies at rest and moving; on
the Mechanical Powers.
_Natural Philosophy_, including the Equilibrium of Liquids and Gasses;
Heat; Electricity; Magnetism; Galvanism; Electro-magnetism and Light;
Cosmography.
_Chemistry_, the Elements; _French_; _German_; _Drawing_, and
(optionally) _Latin_.
This examination is partly written and partly oral. It is not public,
but conducted in the following manner:--
Five examiners are appointed by the minister of war to examine the
candidates at Paris, and at the several towns named for the purpose
throughout France.
Two of these examiners conduct what may be called a preliminary
examination (_du premier degré_,) and the other three a second
examination (_du second degré_.) The preliminary examiners precede by a
few days in their journey through France those who conduct the second
examination. The written compositions come before either.
The preliminary examination (_du premier degré_) is made solely for the
purpose of ascertaining whether the candidates possess sufficient
knowledge to warrant their being admitted to the second examination; and
the second examination serves, in conjunction with the written
compositions, for their classification in the order of merit.
Prior to the examination, each candidate is called upon to give in
certain written sheets containing calculations, sketches, plans and
drawings, executed by him at school during the year, certified and dated
by the professor under whom he has studied. Care is taken to ascertain
whether these are the pupils’ own work, and any deception in this
matter, if discovered, excludes at once from the competition of the
school.
This done, the candidates are required to reply in writing to written or
printed questions, and to write out French and German exercises; great
care being taken to prevent copying. This written examination occupies
about twenty-four hours during three and a half separate days, as shown
in the following table. It usually takes place in the presence of
certain official authorities, the examiners not being present.
_First Sitting._
Hours.
Arithmetic, 1
Geometry, 1
Latin, 1
--
3
_Second Sitting._
Algebra, 1
History, geography, and French, 3
--
4
_Third Sitting._
Descriptive geometry, and }
diagram, or sketch, } 4
_Fourth Sitting._
Mechanics, 1
Physics, chemistry, and cosmography, 2
--
3
_Fifth Sitting._
Hours.
Applied analysis, 1½
German exercise, 1½
-----
3
_Sixth Sitting._
Solution of a triangle by logarithms 3
_Seventh Sitting._
Drawing 4
--------------
Total 24 hours
Next, each candidate is examined orally for three-quarters of an hour,
on two successive days, by each of the two examiners separately, and
each examiner makes a note of the admissibility or non-admissibility of
the candidate.
At the close of this oral examination, the notes relating to the various
candidates are compared, and if the examiners differ as to the
admissibility of any candidate, he is recalled, further orally examined,
and his written exercises carefully referred to, both examiners being
present. A final decision is then made.
The preliminary examiners then supply the others with a list of the
candidates who are entitled to be admitted to the second oral
examination. On this occasion each candidate is separately examined for
one hour and a half by each examiner, but care is taken that in all the
principal subjects of study the candidate is examined by at least two
out of the three examiners.
Each examiner records his opinion of the merits of every candidate in
replying, orally and in writing, by awarding him a credit varying
between O and 20, the highest number indicating a very superior result.
This scale of merit is employed to express the value of the oral
replies, written answers, or drawings. It has the following
signification, and appears to be generally in use in the French military
schools:--
20 denotes perfect.
19}
18} “ very good.
17}
16} “ good.
15}
14}
13} “ passable.
12}
11}
10} “ middling.
9}
8}
7} “ bad.
6}
5}
4} “ very bad.
3}
2}
1} “ almost nothing.
0} “ nothing.
Considerable latitude is granted to the examiner engaged in deciding
upon the amount of credit to be allowed to the student, for the manner
in which he replies to the various questions. He is expected to bear in
mind the temperament of the candidate, his confidence or timidity, as
well as the difficulty of the questions, when judging of the quality of
the reply, more value being given for an imperfect answer to a difficult
question than for a more perfect reply to an easy one.
The reports of the examiners, together with the various documents
belonging to each candidate, are sent from each town to the minister at
war, who transmits them to the commandant of the Polytechnic School to
make out a classified list.
Very different value of course is attached to the importance of some of
the subjects, when compared with others; and the measure of the
importance is represented in French examinations by what are termed
_co-efficients of influence_, varying for the several subjects of study
and kind of examination. The particular co-efficients of influence for
each subject in these written and oral examinations, are as follows:--
Co-efficients of Influence.
Oral examination--analytical mathematics, 20}
“ “ geometrical ditto, 14} 52
“ “ physics and mechanics, 16}
“ “ German language, 2}
Written compositions on mathematical subjects, 5}
“ “ descriptive geometry, drawing, and
description, 5}
“ “ logarithmic calculations of a triangle, 2}
“ “ mechanics, 2}
“ “ physics or chemistry, 4} 34
German exercise, 1}
French composition, 5}
Latin translation, 5}
Copy of a drawing, 5}
--
Total, 86
--
In order to make out the above mentioned classified list, the respective
credits awarded by the examiners to each candidate are multiplied by the
co-efficients representing the weight or importance attached to each
subject; and the sum of their products furnishes a numerical result,
representing the degree of merit of each candidate.
A comparison of these numerical results is then made, and a general list
of all the candidates is arranged in order of merit.
This list, and the whole of the documents from which it has been drawn
up, are then submitted to a jury composed of the
Commandant of the School.
The Second in Command.
The Director of Studies.
Two Members of the Board of Improvement.
The Five Examiners.
It is the special business of this jury carefully to scrutinize the
whole of the candidates’ documents, drawings, &c., and they further take
care that a failure in any one branch of study is duly noted, as such
failure is a sufficient reason for the exclusion of the candidate from
the general list.
As soon as this general list has been thoroughly verified, it is
submitted to the minister of war, who is empowered to add one-tenth to
the number actually required for the public services; and thus it may
happen that one-tenth of the pupils may annually be disappointed.
IV. THE SCHOOL BUILDINGS AND THE COURSE AND METHOD OF STUDY.
A brief description of the buildings may be a suitable introduction to
an account of the studies that are pursued, and the life that is led in
them.
The Polytechnic School stands near the Pantheon, and consists of two
main buildings, one for the official rooms and the residence of the
commandant and director of studies, the other, and larger one, for the
pupils. Detached buildings contain the chemical lecture room and
laboratory, the laboratory of natural philosophy, the library, fencing
and billiard rooms.
The basement floor of the larger building contains the kitchen and
refectories. On the first floor, are the two amphitheaters or great
lecture rooms, assigned respectively to the pupils of the two years or
divisions, in which the ordinary lectures are given. The rooms are large
and well arranged; the seats fixed, the students’ names attached to
them. The students are admitted by doors behind the upper tier of seats;
at the foot of all is a platform for the professor, with a blackboard
facing his audience, and with sufficient room for a pupil to stand and
work questions beside him. Room also is provided for one of the
captains, inspectors of studies, whose duty it is to be present, for the
director of studies, whose occasional presence is expected, and for the
assistant teachers or _répétiteurs_, who in the first year of their
appointment are called upon to attend the course upon which they will
have to give their subsequent questions and explanations. On this floor
are also the museums, or repositories of models, instruments, machines,
&c., needed for use in the amphitheaters, or elsewhere. The museum
provided for the lecturer on Physics (or Natural Philosophy) appeared in
particular to be well supplied.
The whole of the second floor is taken up with what are called the
_salles d’interrogation_, a long series of small cabinets or studies,
plainly furnished with six or eight stools and a table, devoted to the
_interrogations particulières_, which will presently be described.
The third floor contains the halls of study, _salles d’étude_, or
studying rooms, in which the greater part of the student’s time during
the day is passed--where he studies, draws, keeps his papers and
instruments, writes his exercises, and prepares his lectures. These are
small chambers, containing eight or, exceptionally, eleven occupants. A
double desk runs down the middle from the window to the door, with a
little shelf and drawers for each student. There is a blackboard for the
common use, and various objects are furnished through the senior
student, the sergeant, a selected pupil, more advanced than the rest,
who is placed in charge of the room, and is responsible for whatever is
handed in for the use of the students. He collects the exercises, and
generally gives a great deal of assistance to the less proficient. “When
I was sergeant,” said an old pupil, “I was always at the board.” The
spirit of _camaraderie_, said to exist so strongly among the Polytechnic
students, displays itself in this particular form very beneficially.
Young men of all classes work heartily and zealously together in the
_salles d’étude_, and no feeling of rivalry prevents them from assisting
one another. The serg
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