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As one such selection the following may be equally reprehensible seems to be the omission noted: Weber, "Lehrbuch der Algebra," three of the very useful Volume I., "Subject, Index, volumes; Goursat, “ Cours d'analyse mathé Pure Mathematics," Royal Society of London matique,” three volumes; Veblen and Young, Catalogue of Scientific Papers. Every student “Projective Geometry,” two volumes--the sec should have an opportunity to determine the ond by Veblen alone; Eisenhart, “Differential limits of our present knowledge along particGeometry," one volume. Those who do not ular lines which may interest him. read German might substitute for the three While the careful study of nine such volvolumes of Weber's algebra the following: umes as were noted above would serve as a Bôcher, "Introduction to Higher Algebra”; kind of admission card to the circle of pure Miller, Blichfeldt and Dickson, “Finite mathematicians it is necessary to emphasize Groups"; Ried, “Theory of Algebraic Num the fact that high standing in this circle would bers.” Fortunately the first two volumes of imply various other attainments. One of the Goursat's “ Cours” were translated into Eng- foremost of these is a comprehensive knowllish by members of the mathematical depart edge of the literature along at least one imporment of your state university.

tant line of mathematical work.

Such a It may be noted that this list of nine vol- knowledge could scarcely be acquired without umes contains three volumes on each of the using French, German and Italian literature. three broad fields of mathematics-algebra, Hence a reading knowledge of these languages, analysis and geometry. Moreover, the mastery especially the first two, is very important for of these nine volumes would usually be at the prospective mathematician. During the tended by considerable reference work since last two or three centuries the French have some subjects are treated therein too concisely contributed than any other nation for the average student. Unfortunately there towards the advancement of mathematics. exists at present no good mathematical dic We have thus far failed to mention what tionary in any language. It is to be hoped may appear to many as the foremost qualificathat the Mathematical Association of America tion for a high position in the circle of mathewill soon remedy this great drawback, espe maticians; viz., research ability of high order. cially for the private study of mathematics, It is true that the highest mathematical honors and it is interesting to note that the chairman are usually reserved for those who possess of its committee having this matter under con this ability in a high degree in addition to sideration belongs to your own state univer the attainments to which we referred. It is, sity.

however, equally true that the highest research Those who read French and German can not is usually spontaneous and takes care of itself be too strongly advised to provide themselves provided the proper foundations have been laid with the published parts of the large mathe and the necessary enthusiasm is present. matical encyclopedia, whose completion has

It is difficult to see how a man with high been so much delayed by the world war. The mathematical attainments and deep matheFrench edition of this work is especially com matical interest can help doing research work. plete, as far as it has been published, and our It is the most charming occupation in the chief objection to the list of 160 library books world even when the results appear unworthy noted above is that it makes no mention of of publication. When results are reached this superior work of reference. Almost which seem to be of permanent value and to

serve as rays of light to all future generations 4 This publication is not as far advanced as one

the investigator naturally experiences feelings would naturally infer from a reference thereto re

of delight that enrich his inner life as few cently made by the retiring chairman of the Chicago Section of the American Mathematical So things do. ciety on the opening page of an address published The view that all mankind has equal mathein volume 25 of its Bulletin.

matical opportunity in this world is not

strictly in accord with facts, but it is becom- have extended the mathematical advantages ing more and more nearly true. A little more of Paris to the whole world. It is still too than three and a half centuries ago Robert early to comprehend fully the marvelous Recorde, author of the first book in English mathematical transformation due to the addealing with algebra, remarked that “My for vantages of mathematical journals of various tune is not so good to have quite tyme to grades. This transformation has been gradual teache." Notwithstanding his valuable serv and hence it aroused little comment, but it ices to education he was compelled to spend the

has largely annihilated distances from mathelast days of his life in prison on account of his

matical centers, and mathematical research of debts.

high order may reasonably be expected to beSome of our modern teachers still feel that

come more and more cosmopolitan. their fortune is not so good as to give them

In recent years a new and important opporquiet time to study, and hence they are using

tunity for service has come to the high school their spare time to add to their financial in

teachers of mathematics. Public libraries comes. It is very unfortunate that the so

have increased in a most encouraging way, but called “Tangible Rewards of Teaching " are

useful mathematical literature is frequently still so meager, but these rewards have steadily

very inadequately represented therein. Teachincreased, especially during recent years.

In

ers of mathematics everywhere should help to a large number of cases they are now sufficient

correct this situation. They should not only to permit complete devotion to the duties of

supply those in charge with lists of most suitthe position and the necessary study for self

able mathematical books and journals but they development. According to the report men

should also encourage their own students to tioned above, page 318, there was at least one

use the mathematical facilities offered by these school in this state about six years ago which

libraries. If I could encourage the teachers paid its teachers less than fifty cents a day.

of this state to make a strong effort to have It is to be hoped that such conditions do not mathematical literature properly represented exist to-day.

in their local public libraries I should feel The alleviation of the mathematics teachers

amply repaid for coming to this meeting. position is, however, more largely due to the

Few students can read such an elementary improvement in general library facilities and

book as “ Philosophy and Fun of Algebra," by the maintenance of good mathematical period- Mary E. Boole, without getting new light as icals than to improvements in salaries. It is

regards the real meaning of elementary alinteresting to note that our leading mathe gebra. The student of elementary geometry matical journal for teachers of the college

will not only take great delight in reading grade was started in this state, and was main

such books as E. A. Abbott's “ Flatland,” but tained for eighteen years (1894–1912) mainly he will also acquire from it new and important through the sacrifices of one of your college notions as regards the nature of geometric diprofessors. In 1916 it became the official mensions. Mathematical clubs show that genorgan of the Mathematical Association of

eral mathematical questions attract many of America.

our ablest young students and it seems reasonSuch periodicals have done much towards

able to suppose that this will always remain establishing closer contact between mathe

true. maticians, and are thus giving to people every

It is one of the mathematics teacher's great where a large number of the advantages for privileges to help to direct the thought of the merly enjoyed only by those living near the

younger generation towards a subject of susgreat centers of mathematical activity. They taining intellectual interest. One of the in3 A bulletin relating to teachers' salaries was

teresting experiences of my own student life published under this title by the U. S. Bureau of in Paris was to see two gentlemen beyond the Education as Bulletin, 1914, No. 16.

age of sixty follow regularly a course of

but a

lectures given at the Sorbonne by E. Picard. of an unreasonable statement which appeared Mathematics is not only for the young and on the first page of a recent publication of the those who make a living therefrom, but its Department of Commerce, U. S. Coast and study leads to an intellectual penetration with Geodetic Survey, No. 92, 1918, we cite the unlimited room for growth. Our interest in following: “It was his regular custom to this subject naturally grows with our knowl spend 17 hours per day in study and writing." edge thereof and the former is apt to grow An almost equivalent statement appears under much more rapidly than the latter.

the name of J. H. Lambert in the ninth In view of these facts it seems to me that edition of the “Encyclopedia Britannica," but all the larger city libraries should contain a fortunately it is not found in the later edition. considerable collection of modern mathemat As mathematics teachers, and perhaps as ical works, including current parts of the best teachers in general, our attitude towards salmodern mathematical reference work, viz., aries is often inconsistent. In choosing this “Encyclopédie des Sciences Mathématiques," profession we practically say that we are more so that new parts of this important work may interested in intellectual matters than in the become available soon after their publication. making of money. On the other hand, many High school teachers of mathematics can of our members sacrifice intellectual opporturender great assistance in this direction by nities for a little increase in salary. Positions familiarizing themselves with suitable mathe which offer a reasonable income together with matical collections and the needs of their local sufficient time and proper facilities for study libraries, and suggesting improvements to the should not be abandoned in favor of those proper authorities.

offering poorer facilities for intellectual growth Above all let us try to instil in our students

little more salary. School officials a desire for more mathematical knowledge, should be impressed by the fact that their and encourage them to utilize the facilities of teachers appreciate advantages for developlocal libraries along mathematical lines. Our ment and that the best teachers can be secured large general dictionaries and encyclopedias and held only by furnishing advantages for contain much that can be used to advantage their development, especially in the form of during mathematical recitation periods. It is good library facilities. scarcely necessary to say that such outside The great war for justice and democracy contact should not take the place of penetra should tend to dignify our high calling since tion into the subject in hand, but this penetra it directs so forcibly attention to the facts that tion is more likely to become attractive if it is sometimes necessary to make great sacribroad contact is kept in mind.

fices for the opportunities for higher developIt should be noted that many of our best ment and the rights of nations and of indigeneral works of reference are weak along viduals do not depend upon their sizes. We mathematical lines. As an illustration we as teachers should be especially impressed by may note an entirely senseless definition of the fact that curtailments of rights must be regular group appearing under the word group based on other considerations. With the imin the 1917 edition of the “New Standard provement in world ideals as a result of this Dictionary.” This definition is as follows: war there should come a keener appreciation “a transitive group whose order is the same of thorough preparation for the various duties as that of the letter on which it is made." of life. The appointment of an athlete to a Such weaknesses are, however, not always chemical position in Washington for which he harmful to the young student since they may was wholly unprepared should be regarded as serve to promote the important attitude of close to treason even if it may have been due mind of not accepting statements without to the ignorance of politicians. Thorough study and verification. As another instance preparation for our various duties should be

our motto as teachers and our own practise facts. Mathematical growth is not based so should convince the world of our sincerity. much on the number of facts as on the kind

We have thus far considered only the ex of facts. The facts must be general and far isting means for the scientific improvements reaching. A formula involving a parameter of mathematics teachers. It may be desirable is more general than a large logarithmic table to consider also possible new means, for our because the former contains potentially an science is one of infinite progress and hence infinite number of special values while the we naturally look for new things. Possibly latter represents only a finite number of such the new

means for scientific development values. It is, however, necessary to exercise which I shall outline briefly will appear to care in regard to the use of the word general you as too idealistic, but high ideals are es in mathematics, for, what is often called gensential for great progress. Hence I venture eral is really very special. to propose that high school teachers should be If one established theorem includes another required to give evidence at the end of every it is evidently proper to speak of the former seventh year, until they are forty years old, as the more general, but if one undeveloped of having made during the preceding seven theory embraces another it is not so clear that years scientific progress equivalent to at least the former should always be regarded the more one year of university work.

general. It may be that the generality of the In fact, this might commonly be in the form principles underlying this theory is too great of a sabbatical year spent in study at some to permit of much progress. A theory ought university. In special cases it might be in to be regarded as general in proportion to its the form of attendance at summer sessions, or possible development and not in proportion to the publication of scientific work. In all the generality of the definitions underlying it. cases is should be understood that the proper It is evident that such a use of the word authorities would go over the records care general is attended by great difficulties, but it fully every seventh year and would insist on is hard to see how this word can maintain its such progress as a necessary condition for re position of respect in the mathematical literaappointment. If the young teacher does not ture unless we do make an effort to restrict its grow scientifically at least at the rate of one use to the potentially larger things. My seventh of the normal growth of the university thought may become clearer if I note the student he does not possess the type of mind fact that the most general definition of the that inspires his pupils properly.

term group is too broad to serve as the basis While our young university instructors are of a theory. The most general group theory not formally subjected to such a rule they are is therefore of zero extent and will probably alpractically subject to a more severe scientific ways be of this extent. There is a type of test in our better universities by means of a definitions which give rise to the most genconsiderable series of grades, such as instruc eral theory, but it is practically impossible to tor, associate, assistant professor, associate pro fix the limitations imposed by such definitions. fessor, professor. In the better institutions As an instance of a tendency to generalize each higher grade normally implies scientific unduly for the pedagogical purposes in eleattainments which are superior to those re mentary mathematics we may refer to one of quired for the next lower grade. It is, of the oldest among the somewhat complicated course, difficult to enforce high standards in mathematical formulas, viz., the Heron forthese times of scarcity of teachers, but with mula expressing the area of a triangle in the return of peace we may naturally look for terms of its three sides. This formula is found greater competition and higher standards. in the majority of our text-books on trigonom

To meet these higher standards it is not etry but it is questionable whether it can be sufficient that we learn more mathematical regarded as a useful formula for the ordinary

to those recently inaugurated along financial lines to protect the ever too gullible public. The scientific development securities to which I directed your attention above do not promise the largest returns but they have withstood the severest test of the ages and hence they should be regarded as the soundest of all intellectual investments. Our students need to be trained to enjoy ideals as well as to utilize the real. Mathematics is the ideal science and there is more moving than improving in reforms.

G. A. MILLER . UNIVERSITY OF ILLINOIS

student of trigonometry. It seems easier to solve such a triangle by dividing it into two right triangles and I understand this method is commonly pursued by the engineer. The love for generalization on the part of the teacher seems to have led him in this case to commit a serious pedagogical blunder.

In closing, I desire to urge you to do your own thinking and not to allow yourself to waste energies on the many modern fads appearing under the high sounding term of reform. The very rapid modern transformations have made us unduly vulnerable to the darts of the faddist whose audacity has outstripped that of the mine and oil promoters of the last few decades. A few mines and oil wells have paid handsomely but most of those which have been advertised extensively proved to be disastrous to the too credulous investor. A similar fate has come to those who are too credulous about educational reforms, whether they appear in the form of the function rattle popularized by F. Klein, vocational training, transfer of training, ability tests, or simply the emphasis on methods above knowledge.

As a result of the many wildcat propositions the universities used to avoid pedagogical investments altogether and they used to be fearless in warning the public against investing their hard-earned money in this way. During recent years, however, our American universities have abandoned this policy, under the leadership of Columbia, and have invested heavily in this line of securities. At first they selected the best class only but recently they seem also to invest heavily, again under the leadership of our largest university, in the more doubtful class. This is done even in the graduate schools.

Hence the public has become more and more unwary, and wildcat pedagogical promotions are thriving as never before. The richness of a few reputable pedagogical mines has served to inspire hope as regards others whose only asset is proximity to the former. Hence the grave need of caution at the present time. The educational public would seem to need some public educational commissions similar

BANDED STRUCTURES OF THE ADIRONDACK SYENITE-GRANITE

SERIES THE syenite-granite series constitutes the greatest bulk of Adirondack rock. It is younger than both the Grenville metamorphosed sedimentary series and the anorthosite, the former especially having been broken up and badly cut to pieces by the syenite-granite intrusion. In mineral composition the range is from syenite rich in microperthite, orthoclase, and hornblende or augite, together with some plagioclase; to granite rich in microperthite, quartz, orthoclase and microcline, together with some plagioclase, hornblende and biotite; to monzonitic and dioritic facies rich in plagioclase, orthoclase, pyroxene and hornblende. Medium grained rocks greatly predominate but there are many variations to fine and coarse grained and even porphyritic facies. Granulation is common, the feldspars especially being most notably crushed. In structure the syenite-granite series exhibits all sorts of variations from non-gneissoid to excessively gneissoid types, with a moderate degree of foliation prevalent. The color of the typical fresh syenite is greenish-gray, while the fresh granite varies from greenishgray, to light gray, to light red.

In this paper the features of special interest in connection with the syenite-granite series are the comparatively sharp transitions from acidic to basic facies; from greenish-gray or gray to pink or red varieties; from coarser to

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