Numerical equations of higher degree, and so much of the theory of equations, with graphical methods, as is necessary for their treatment, including Descartes' rule of signs and Horner's method, but not Sturm's functions or multiple roots.

It may be of interest to compare the above formulations with those of the College Entrance Examination Board, based themselves on recommendations of a committee of the National Educational Association. The College Board presents elementary algebra in three divisions: algebra to quadratics, quadratics, etc., and progressions, etc.; advanced algebra in two divisions — series, etc., and theory of equations. The committee includes under elementary algebra all of the College Board elementary algebra, except permutations and combinations, and the use of the tables of logarithms. It omits, also, highest common factor by division and simultaneous quadratic equations. Its phraseology is somewhat more explicit, and it emphasizes problems depending upon algebraic equations. In advanced algebra the committee's formulation differs from that of the College Board in omitting undetermined coefficients, infinite series, binomial theorem for fractional and negative exponents, and the theory of logarithms. It includes, on the other hand, complex numbers, determinants, and a somewhat fuller statement of the theory of equations, the general aim being, on the one hand, to omit subjects which the secondary scholar is not prepared to appreciate fully, and, on the other hand, to secure a somewhat higher degree of unity.

The definitions of plane and solid geometry do not differ in principle from those of the College Board, but are stated with somewhat more fullness, while they still contain the general expression "the usual theorems and constructions of good text-books."

The definitions of plane trigonometry present no important differences in principle, but that of the committee is considerably fuller.

In presenting formulations of these five subjects, it is by no means the opinion of the committee that any college should require all for admission. The committee has undertaken to define those mathematical subjects which are at present required by any considerable number of colleges, and expresses no opinion of its own as to the appropriateness of the more advanced of these subjects for secondary-school work. The committee appreciates that the value of its definitions must depend to a great extent on their conformity to the needs and interests of secondary schools and teachers. It therefore begs to invite the freest criticism on the part of secondary teachers by correspondence with any of its members.

WILLIAM T. CAMPBELL, Boston Latin School, speaking of geometry in the grammar school, said that a great deal of the subject-matter taught to pupils of fifteen might be grasped by them at twelve, but that in general there was no particular disadvantage in deferring the instruction, because with increased maturity and logical power in the pupil the subject may be grasped with greater facility and thoroness. The case is different with geometry, because, if the pupil of fifteen thinks better, he sees worse. The character of the geometry suited to the boy of twelve is very different from that suited to the boy of fifteen. The latter begins with lines, points, angles, and ends with the cube, pyramid, sphere; the former begins with the pyramid, cube, sphere, and ends with lines and points. Demonstrative geometry exalts the proof and may neglect the application; with

observational geometry it is the reverse. Geometry in the grammar school should be taught with reference to the cultivation of the constructive imagination and keenness of observation, the power of drawing good diagrams, accuracy of measurement.

PROFESSOR OSGOOD very briefly summarized the work of the conference. These movements promise much of benefit and improvement in the teaching of mathematics: They look toward the organization ultimately of a national association of teachers of mathematics a step, however, which should not be taken until the movement shall make it inevitable. As to the relation of these associations to the American Mathematical Society, there should be the greatest interest on the part of the latter in the success of the former, and without doubt members of the American Mathematical Society will be influential in assisting in such associations. Any more close relationship is not, however, advisable.

IV. HISTORY CONFERENCE

LEADER, JAMES SULLIVAN, TEACHER OF HISTORY, HIGH SCHOOL OF COMMERCE, NEW YORK CITY

In opening the conference the leader said: In visiting schools I have been struck with the poor quality of teaching in such subjects as civics, and mediæval, and modern history. Such teaching is due, no doubt, in a certain degree to the poor quality of our texts, but in a far larger measure to the poor preparation of teachers in these subjects.

As the question of civics has already been considered in another section of this Association, it is our purpose here to take up the question of the teaching of mediæval and modern history in the secondary schools. In this subject poor teaching is due, aside from the question of poor texts, to an insufficient preparation of teachers in this line of history, and to a certain inability, due to lack of preparation, to present the subject in such a way as to make it possible for the pupils to seize the true meaning of medieval life and history. Teachers who fail most lamentably are those who attempt to teach the history of the Middle Ages as a series of histories of separate nations. It is no wonder that they fail, and that their pupils are incapable of retaining so many contemporaneous events in their heads as are given them to remember as important. Listening to the complaints of such teachers, the board of superintendents of New York recently withdrew mediæval and modern history from the list of required subjects in the high-school curriculum and placed it on the optional list.

This was felt by many teachers to be a serious mistake, because the high school is the only place where the larger percentage of our population have any opportunity of learning anything about European history and institutions-the lack of knowledge of which leads to a bigoted patriotism and a narrow provincialism on the part of our people. It was also felt that if the teachers in general had a more thoro preparation in medieval history and knew how to handle the subject, not as a series of national histories, but as the history of the great movements and institutions of Europe as a unit, they would have had such success in its teaching as to lead them to ask for its retention in the curriculum of the high school.

It is with the hope of interesting teachers in preparing themselves more thoroly in this special field of history that I have asked Professor Haskins, of Harvard, a former member of the Committee of Seven of the American Historical Association, to open the discussion today.

CHARLES H. HASKINS, Harvard University.--- As matters stand at present, the question of the teaching of the history of medieval and modern Europe in schools seems to relate not so much to the desirability of such instruction as to the practicability of giving

it effectively in the average school. It would be hard to maintain either that a systematic course of study ought to omit subjects of such importance, or that they cannot be made interesting and profitable to the ordinary school pupil. Medieval history in particular, with its wealth of color and movements, and its vital relations to the origins of the modern world, can easily be made attractive. At the same time, we should not conceal from ourselves the considerable difficulties which this field presents. Some of these are temporary, such as the scarcity of good text-books and the inadequate training of teachers, and are fast being met. But there is also a permanent difficulty, inherent in the nature of the subject—namely, the great extent and complicated nature of the field, and its remoteness in time and place from the experience of American boys and girls. In no other of the subjects ordinarily taught in schools is there so great need of careful selection of material and firm grasp of essentials on the part of the teacher. Any attempt to cover the whole field in the time usually allotted is almost sure to result either in vague generalities or in a confused and lifeless mass of names and dates. In selecting topics for treatment it is possible either to take only certain culminating episodes or epochs, or to make a single country, such as France, the center of study, or to pay less attention to continuous narrative and emphasize the principal underlying institutions. Particularly for the Middle Ages, the teacher will find it desirable to make clear such topics as the ecclesiastical system, feudalism, the crusades, the medieval empire, and the rise of France, by extensive use of concrete description, emphasizing continuous and typical conditions rather than individual events.

RAY GREENE HULING, Cambridge, Mass.-I think that there is something better than medieval history. The field of medieval history is not in general interesting to boys and girls of high-school age. Moreover, the study of that period does not conduce to the cultivation of a desire to read history later in life, with some knowledge of the books to be read and a good method of reading. For the first year in the high school he suggested English history; for the second, ancient history; for the third, American history in its constitutional development; and for the fourth, a careful and detailed study of the discovery and colonization of the eastern part of the United States. This would furnish good training for the imagination, and also for the budding powers of the reason and the judgment. Such a course would have three good results: the first is an interest in reading; the second, an ability to reason intelligently and record the results clearly; and the third, something of an expert's knowledge of the means of research to find out what the student wishes to know.

W. J. S. BRYANT, principal of the St. Louis High School, said that a large percentage of high-school pupils do not go beyond the high school. Medieval history should be studied in the high school, or it will not be studied at all. The difficulty of a subject is not a good ground for leaving it out of the curriculum. Modern institutions cannot be understood unless we trace them back to their beginnings in the past. The pupils need a well-connected view of the whole scope of world-movements and forces. Pupils need preparation for minute study later.

MISS HARRIET G. KING, Oak Park, Ill.

Which of the various studies are to be preferred, if some must be left out? Colonial history is not so profitable in the middle West as it is in New England. What shall be done with all the books sent to us? Have not we too much to read and too much to do?

PROFESSOR HASKINS,- The date 800 was suggested by the Committee of Seven, not because it was considered a fixed dividing-line in history, but because it offered the most obvious and convenient break in a two-years' course covering ancient, mediæval, and modern history. No doubt it will often happen that the work of the first year cannot well be carried so far, and the newer text-books are so arranged that the four or five centuries preceding 800 may be placed with the work of either year. At the same time, by carrying the study of Roman history thru the later empire and into the early Middle Ages pupils

get a much more adequate idea of the place of Rome in the general history of the world than if the course breaks off somewhere in the early empire and fails to make proper connection with what followed.

E. W. LYTTLE, Albany, N. Y.-Mediaval history does not lack interest. We are farther away in experience from medieval history than from the history of Greece and Rome. We therefore know the history of Greece and Rome better than we know mediæval history. Hence arises the difficulty of teaching mediæval history. For that very reason it should be taught, and it can be made interesting. Youth is intensely interested in religious questions. The great facts of the church in the Middle Ages, of the feudal system, of chivalry, and of the crusades, if rightly handled, will be full of interest and profit.

DEPARTMENT OF HIGHER EDUCATION

SECRETARY'S MINUTES

FIRST SESSION. - TUESDAY, JULY 7, 1903

The Department of Higher Education met at 9:30 A. M. in the Central Congregational Church of Boston, with the president, Benjamin Ide Wheeler, president of the University of California, in the chair. About twelve hundred persons were present.

It was moved and carried that the president appoint a committee on nominations for the ensuing year.

The topic for the morning was "The Length of the Baccalaureate Course and the Preparation for the Professional Schools."

Elmer E. Brown, professor of the theory and practice of education, University of California, introduced the subject in a paper setting forth the history and evolution of the American college and the bachelor's degree.

Papers dealing with the general subject from various standpoints were read by the following: Charles W. Eliot, president of Harvard University; Nicholas Murray Butler, president of Columbia University; William R. Harper, president of the University of Chicago; Andrew F. West, dean of the Graduate School, Princeton University.

Discussion was opened with a paper by Daniel W. Hering, dean of the Graduate School, New York University. A general discussion followed, which was participated in by G. Stanley Hall, president of Clark University; W. H. Hadow, dean of Worcester College, Oxford, England; Rev. G. A. Kratzer, pastor of the Universalist Church, Fitchburg, Mass.; James H. Baker, president of the University of Colorado; Charles W. Eliot, president of Harvard University; William DeWitt Hyde, president of Bowdoin College; W. H. P. Faunce, president of Brown University.

At the close of the discussion, President Wheeler announced the Committee on Nominations as follows:

Andrew S. Draper, president of the University of Illinois. W. H. P. Faunce, president of Brown University. Andrew F. West, dean of the Graduate School, Princeton University.

The session then adjourned to meet in the same place on Wednesday morning at 9:30 o'clock.

SECOND SESSION. - WEDNESDAY, JULY 8

The second session of the Department of Higher Education was held Wednesday morning, July 8, at 9:30 o'clock, in the Central Congregational Church. President Wheeler presided.

About nine hundred persons were present.

President W. H. P. Faunce of Brown University presented the report of the

Committee on Nominations, as follows:

For President - Benjamin Ide Wheeler, University of California.

For Vice-President — George Harris, Amherst College.

For Secretary-John Henry MacCracken, New York University.

On motion, the report was unanimously adopted, and the officers were declared elected for the ensuing year.