the course were revised. Perhaps it would not be altogether wise to make the history distinctly commercial; at the same time it does seem that a readjustment might be made which would recognize the history of commerce as being of just as much interest to the ordinary student as it is to the commercial student. It has been said that the development of the colonies around the Mediterranean sea was a commercial development and that many of the crises in history in the Middle Ages were to be traced directly to the influence of commerce. Now, if history can be taught in that way, the students will have a correct idea of the part that commerce has played in the development of the world. W. C. STEVENSON.- While fully in accord with everything that Dr. Herrick has said regarding the advisability of the broadest courses in history, I desire to call attention to one difficulty in the way of the adoption of these courses in the high schools of this country, viz., that the teachers have not had the broad and comprehensive historical training that will enable them to sift the economic from the military and the purely literary features. The text-books which prevail are largely literary, military, and cultural, in which the economic phase of historical development is not emphasized. In fact, it is rarely referred to at all, and one must have a very broad and comprehensive training to sift the economic from the other phases of history.

I. R. GARBUTT.—I am very much pleased to know that the commercial course in the high schools of Cleveland coincides closely with the course recommended in Dr. Herrick's paper. The students in the commercial course are not taught alone by the commercial teachers, but by the teachers who have the subject of history in the other courses; so that they are not set apart in classes by themselves, but have the advantage of the regular teachers of the subject.

MATHEMATICS IN COMMERCIAL WORK

ERNEST LAWTON THURSTON, BUSINESS HIGH SCHOOL, WASHINGTON, D. c. There is need at intervals, in most general courses of study, to rediscover mathematics; to determine again and again, in the light of the best thought and method at the time, its educative, disciplinary, and purely utilitarian value, its relative place in the course, its proper subject-matter.

With a wealth of admirably developed new matter in pure and applied science, constantly demanding an increasing emphasis in already overburdened curricula, mathematics has been relegated at times to a less important position than formerly. No doubt this is right in some cases; in others, it follows from a failure to realize fully its value. In the larger scientific and engineering schools alone has it retained its relative position, because here it is regarded as the bed-rock on which to a large degree the superstructure of technical training must be built.

In our modern secondary schools of commerce, however, we need not to rediscover but to discover mathematics, in the light of the purpose of the school. We are only beginning to realize the possibilities of correlation of mathematics with other subjects, and of such selection and treatment of the subject-matter as shall be highly educative thruout, while strongly utilitarian in parts.

The subjects commonly required are not new, and no new ones need

be introduced, altho opportunities should be given in the final year for electives of a higher order. But these subjects should be commercialized, to a slight degree at least, where this is possible without decreasing their full educational and disciplinary value.

That which is distinctively utilitarian in the course must be thoroly practical and in accord with modern usage. Business arithmetic, especially, is undergoing marked changes in system and development. The arithmetic of today in method and application is not that of yesterday.

Educationally, the greater results will come from those parts which appeal to the pupil's life, activities, and interests. Mathematics should be live, and dead matter. The arithmetic we know is the outcome of daily needs in every phase of life. It touches life at many points, and in this close contact interest will start and flourish. As has been well said: "The mainspring of mathematical ability in a race is the attempt to adjust means accurately and economically to a given end."

As a whole, the course must give power, vigor, and strength to the mind, cultivating clear thinking and ability to see all sides of a question, and developing that individual capacity which is needed in every form of mental activity. A magnificent exercise in logic, it may sacrifice at times the teaching of facts, if only it gives power to prove facts.

In the light of these requirements, commercial arithmetic must be abridged by cutting off obsolete subjects and complicated methods, and enriched by increasing greatly the quantity of simple calculations and of modern, practical, concrete problems, especially those that deal with our active participation in life. Altho business grows steadily more complex, details of organization and methods of work tend steadily toward simplicity. This tendency results in increased demands for accuracy in fundamental processes of arithmetic, and for a working knowledge of the principles of percentage and of elementary business principles, with ability to apply them in an increasing number of ways. The arithmetic of actual business discloses :

1. That common fractions are uncommon; those with denominators of two, three, four, six, and eight alone finding extended use; for others, the nearest two-place decimal is the common substitute.

2. That quantities are generally expressed in one or two denominations. The merchant sells 134 yards, not I yard, 2 feet, 3 inches; the grocer, 11⁄2 pounds, not I pound, 8 ounces; the engineer measures in feet and hundreds of feet, not in rods and feet.

3. That the majority of numbers expressing quantity and value are exceedingly simple. It follows naturally that ability to work mentally should be cultivated, even if the volume of modern business did not demand it. Employees waste time, energy, and frequently costly stationery in unnecessary paper calculations; yet mental calculation, once a habit, is always easier.

4. That in actual business there is little recognition of text-book case or subject. A single real-estate problem may involve simple percentage, taxes, commission, insurance, interest. Solutions must rest on the bed-rock of fundamental principles, not on the shifting sands of arbitrary cases.

5. That actual problems are frequently so expressed as to make essential the ability to understand them as well as to solve them. A book says: "I bought 40 chairs: $8.40, less 15 per cent. discount, paying freight of $11.20. Terms: 30 days; 2 per cent. cash. I paid cash. Find the marked price to gain 15 per cent." A similar problem I overheard expressed in these words from dealer to clerk: "John, we want to clear 15 per cent. on this invoice!" (handing him a bill). And John noted terms, discounts, prices, allowed for freight and store burden, and marked his chairs. The employer says: "Do this." The clerk must make the problem and find or select the values necessary for its solution.

6. That calculation tables for interest, discount, insurance, taxes, wages, earthwork, etc., are commonly used to save time and insure accuracy.

7. That the use of ruled forms, many requiring extensions and calculations for which text-book courses do not prepare, is increasing rapidly.

The course in arithmetic now, to meet business and educational requirements, must be woven together by mental exercises. These may to advantage cover one-half of each recitation period. Mental calculation finds its first field in rapid reviews of fundamental processes in whole numbers, and in common and decimal fractions; its second, in exercises in numbers under 100, continued thruout the course, and in percentage and interest all intended to develop speed, accuracy, and knowledge of number combinations. It is the tool for systematic review and for developing shorthand arithmetic. Ready-made short methods must be handled. with exceeding care. There is danger that they will "go off" the wrong way, or at the wrong time, or-not go off at all. But those developed instinctively by the pupil, thru increasing knowledge of number in combination, remain with him -a valuable business capital.

Moreover, every practical topic may be introduced and developed thru brief, pointed mental problems, and drill secured by a series of related mental problems a series with the same central idea, a statement on the board around which exercises may be woven.

In close co-ordination with the mental work, the written exercises and test problems "clinch" the subject. These should be brief, practical, living questions, at times expressed in memoranda or bill form, in order that the problem may be determined as well as solved; at times grouped to relate to the same business or business condition, for related problems have far more educational value than those having simply the arbitrary connections of the text-book case.

From the principles and terms of arithmetic, in combination with. business terms and forms, has been evolved a business language in which business transactions are expressed and business records written. Its literature consists of notes, drafts, bills, estimates, books of record; its phraseology, of symbols, business expressions, terms, forms of tabulation, etc. Some knowledge of it the pupil gains from his bookkeeping, but it is acquired more effectively thru applied arithmetic, which should form the next step in the mathematical course.

Commence with sales and order sheets, requiring horizontal and vertical addition; follow with carefully graded bills of different kinds of business, reading and solving the problems involved, studying the mean. ing and relative value of "terms" and the essentials of form. One wholesale bill, with discounts and choice of terms, contains several pages worth of text-book problems. Master, then, commission forms; use actual notes for interest, discount, and partial payments. Solve office paper, pay-rolls, requisitions, inventories-the field of arithmetic as recorded in business paper. At every step, too, require the preparation of original paper, having it checked and audited by the class.

Finally, later in the course, when the pupil has gained strength of mind, breadth of outlook, and a knowledge of business conditions, study in detail some of the greater problems based on arithmetic: those of banking and finance, of insurance, annuities, and endowments, of taxation and duties; the use and proper design of working tables; the effective preparation of statistics; the great problems of "cost-keeping" and factory mathematics. Here, in its highest phase, arithmetic may touch and interpret the work of most other departments of the school.

The course, thus outlined in salient points, is highly utilitarian; yet, when one has taught it, he finds it also highly educational. The principle of fair settlement, which underlies so many business arithmetic processes, and other business and ethical principles, are constantly emphasized. The unusual opportunities for individual and original work bring breadth of mind and training in system, form, and arrangement; while class discussions and rigid analyses give ability to judge before solving, to reason accurately, and to do away with that inaccuracy of statement which is the parent of inaccuracy of thought.

Algebra is not taught distinctively for its utility to the coming merchant, altho to the mathematician and to the engineer it is indispensable. Its greater value is as an exercise in applied logic, where it gives character to the teacher's work and raises it to the plane of true education. It developes capacity to master subjects of kindred, or of totally different,

nature.

Algebra in part is distinctively universal arithmetic, and the two subjects work well in double harness. Elementary algebra and arithmetic, in combination, should precede commercial arithmetic, for the methods. of algebraic reasoning aid in mastering arithmetical problems-the method of the equation often solving easily what is otherwise difficult. This suggests, too, the substitution of practical business-arithmetic problems for the many exercises in algebra now used.

A scientific treatment of the subject should lead from the beginning. to the equation, which should be introduced early, and emphasized until the pupil is familiar with the principles on which the processes of operation are based. Factoring and its relation to equations and fractions

should also be a strong feature. cises should develop the same accuracy and facility in handling the literal as later the numerical. In all stages of the work methods of checking solutions are important in cultivating a valuable business habit and in encouraging independence as well.

In work of this class the mental exer

That part of higher algebra, less distinctively universal arithmetic, covering the theories of combinations and probabilities has also sufficient value, from practical and disciplinary standpoints, to warrant its rigid treatment. On problems of life insurance and in studies of various business conditions it will be found to have direct bearing.

Geometry claims place, especially because of its value as an exercise in formal logic, altho in parts-in mensuration, for example—it has high utilitarian value. In class it is often effectively taught as a combination of the inventional and the demonstrative. The inventional, leading to a right conception of the truths to be established, introduces naturally the deductive method of establishing them. Elementary ideas of logic, however, may be introduced from the beginning, and demonstrations made exceedingly rigid-with the rigor contributing to soundness of logical development, as well as to clearness and effectiveness of expression. The field of demonstrative work should include plane geometry and the principal theorems of solid geometry--many of the latter having unusual disciplinary value.

The field of applied work should be as broad as time allows, for here is possible correlation with other subjects and contact with actual life. The practical problems of mensuration, the preparation of plans and estimates, designing, pattern-making, the geometrical representation of statistics, suggest lines of development.

The value of geometry is measured to an unusual degree in terms of the teacher. His insistence on rigid demonstration and clear statement, especially in oral work, and on neat, accurate, effective figures; his method and expression before the class; his choice of original exercises for assignment at every stage of the work; his methods of review, measure the value of geometry to discipline the mind, to arouse interest, and to inculcate habits of neatness, order, diligence, and honesty.

The final year should offer opportunities for advanced elective work, consisting possibly of trigonometry, or of problems relating to heat, light, and power, with which many business men need familiarity; but preferably of descriptive geometry. From experience with classes in this latter subject, I regard it as the most attractive subject-matter mathematics has to offer. As a theoretical subject it has no mathematical equal in arousing general class interest, while it develops a high degree of mind-power. Its applications in practice, also, cover an exceedingly broad and interesting field. The Committee of Ten, speaking of projective geometry, which includes descriptive, says: