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a position which those in the first line of attack feel to be reactionary

The position of these two classes now being understood, what are the prospects? In the first place we may safely say that there is to be no radical change in our problems. No such birthright of the race is up for sale for some uncertain mess of pottage. Teachers are not going to give up those unapplied problems which furnish the abstract drill work, because they know perfectly well that the necessary facility of operation cannot be secured thru the limited number of applied problems possible. We may therefore look to a continuance of the traditional balance between the applied and the unapplied. The only change in the latter class is likely to be in the replacing of exercises that have no future significance by those that have.

In considering the question of applied problems it must be recognized that there are four well defined subgroups:

1. Problems that represent genuine conditions, real problems of the present. Such examples are easily secured in arithmetic, their usual lack being due to the fact that compilers find it easier to copy inherited exercises than to search for new ones. In algebra, however, they are not found save with great difficulty. Physics furnishes only a very few types, the recent efforts to find others being disappointing even to the most enthusiastic. There are a number of types of business problems, involving only a single unknown in a linear equation, but we have practically no good modern applications of general quadratics that are within the comprehension of the pupils. In geometry there are exceedingly few propositions that admit of practical application. The parallelogram of forces in physics, and the pythagorean and a dozen other propositions in mensuration, are all that admit of being called genuinely utilitarian. In trigonometry, the surge of analysis which began to become prominent in the 18th century, has crowded out a large range of excellent applications to be found in early text-books, and here we may hope for some reform. The calculus has seemed in a hopeless state until recently, the applications being as unreal as those of algebra. A reform is now in progress, however, and it is reasonable tɔ expect that this branch, which has

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such a large range of applications, may soon claim its utilitarian as well as its culture rights. Our analytic geometry, which most text-books present as nothing but Apollonius of Perga. treated in the cartesian manner, has many possibilities from the standpoint of applied problems, but the field has hardly yet been touched.

2. There is a second and well known class of applied problems which have had an honorable career in the teaching of mathematics. These may be characterized as mathematical puzzles, making no pretense to being utilitarian, and having their chief value in the interest which they excite and in the exercise which they give in the translation of questions into algebraic symbols. There is a feeling in some quarters that these problems have fulfilled their mission, but it is rather probable that we shall see more of them in the future than we have in the past. Rightly used, they serve an excellent purpose; they excite the interest, they give the drill, they make no pretense to be what they are not, and they help to fill the void that exists, and seems liable to exist, in the range of applied problems of algebra. We have a large but ill-arranged literature on mathematical recreations at the present time, and it is to be hoped that this may be so digested as to furnish some valuable material for text-book makers.

3. There is a third class of applied problems that is also. liable to maintain its place. Problems of this class once set forth genuine applications of the day, but changed conditions have left them as mere relics of the past. Such are the well known cases of the cistern pipes and the couriers, serving the twofold purpose of showing the historical development of mathematics, and acting as recreative exercises. It is unfortunate that more is not made of these antiquities, treating them as such, and thru them telling some of the story of the science.

4. A fourth class of applied problems consists of those which make some false pretense. In arithmetic these problems give wrong ideas of business, being tolerated because of a mistaken notion that some kind of mental discipline is secured from a question that tells a falsehood about actual life. In algebra they are tolerated, as in certain problems under quadratic equa

tions, because of the paucity of genuine applications. But their use cannot be justified, and we may reasonably expect the common sense of teachers to banish them, as it has banished the absurd problems of alligation from our arithmetics.

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To summarize the matter the following seems within the range of probability:

1. No radical change will be made in the balancing of the abstract and the concrete, since each serves a purpose that the other does not.

2. The abstract problems will be limited to those usable in future work.

3. There will be a recognition of the fact that physics, or more generally science, furnishes but few distinct types for mathematical problems, and is not a panacea for the ills of elementary education.

4. There will be a more complete recognition of the value of the mathematical puzzle thruout the course.

5. There is liable to be a recognition of the value of problems. of historical interest, as recreations, as drill, and as the basis for the fascinating story of the development of mathematics.

6. Teachers will not be led away by impractical, ill-considered, dangerous suggestions that any one of the several types of problems mentioned is a panacea for all the ills of mathematics, to the exclusion of all others.

7. The greatest improvement of all, however, will be in the earnest effort of teachers to secure genuine applications, not so manifestly beyond the interests of the pupils as are many that have been suggested, but such as shall make the mathematics of the high school and college as real as arithmetic is coming to be. At the same time it will be recognized that there will probably never be enough such problems, particularly in algebra, to supply all the exercise needed in translating applied problems into algebraic language, and recourse will be had to the puzzle, either in its traditional or in some modern form..



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The nations of the world are at school. None has yet reached such perfection either in the administration of its government or in social and economic advancement that it can regard its education as completed. At the present moment France is learning some peculiarly interesting lessons about alliances and the dangers which attend them. Norway and Sweden are learning mutual respect and, by the exercise of candor and self-restraint, are showing how, in these latter days, revo lutions may be accomplished without war. The republics of South America, some of which have scarcely yet learned the alphabet of the science and art of good government, are painfully acquiring lessons of order, industry, and intelligence under the sane example and tutelage of the great nation lying to the north of them. Japan is gaining a national self-consciousness without undue conceit, and is training herself in all those virtues that make a people great either in peace or war. Russia, alas! is learning too late that a nation's strength in modern times rests upon the intelligence, the loyalty, and the national pride of her people. She is being so severely disciplined in the hard school of adversity that there is good hope that under the impulse of this self-revelation she may learn the lessons of honesty and fair dealing and may adopt the golden rule as a national motto.

Not only are nations educable as regards their corporate life, but all those elements that go to make up citizenship and the joy and comfort of living are capable of continuous culture and improvement. It is the fact that standards of character are ever being raised and that thru education men and women can be brought up to those standards. I do not by this mean simply the education of the schools but the process which,

begun in school and home, is carried on thru life, enriching, sweetening, and elevating both work and leisure. Nations learn much from each other. No people is able for any great length of time to preserve secrets, whether in respect of the composition of smokeless powder or the best kind of armorplate for battleships. The most ingenious machines and the most subtle discoveries in the arts are studied and copied.

The international conferences now held so frequently and in such variety are economical means of short-circuiting the world's knowledge. During the past year there have been several of these conferences, as, for example, that of Drawing and Art at Bern, one on Mathematics at Geneva, one on Psychology at Rome, the Congress of Orientalists at Algiers, of Archäology at Athens, of Agriculture at Rome. The most developed and the most firmly established conferences appear to be composed of groups of scholars engaged in research. Especially successful and valuable have been those in the various fields of applied science and medicine.

Some of the greatest and most beneficent discoveries have first been made public at these international conferences and so have been placed at the service of the world.

Not only do international conferences bring scholars and scientific workers of different countries into friendly and sympathetic co-operation, and tend to disseminate the fruits of painstaking research for the use of all mankind, but they are an important factor in bringing the leaders of thought of every nation together, thus sowing the seeds of international fraternity and good will. The Scripture dictum, “ Be not forgetful to entertain strangers,” has been practically applied over and over again when municipalities have opened their doors to the world's representatives of some great cause, as Prison Reform, Religious Federation, or International Arbitration. The great Peace Congress in Boston, in 1904, was a fine illustration of the faith and high optimism which are engendered when prophets and seers from remote nations come together and join hands at the shrine of human brotherhood and the higher destiny of man. A great city opens its doors most generously, and such is the compelling power of the moral issues involved in the thought of universal peace that the whole nation receives an

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