A QUEER MISTAKE.
By G. A. MILLER,

University of Illinois.

Recently a teacher who was using the Stone-Millis Geometry as a textbook asked me where she could get some information in regard to the Egyptian unit of measure called "ruth," stating that her students became interested in a problem relating to the area of a triangle the lengths of the sides being expressed in terms of "ruths." I naturally did not recall having seen the term but we examined the tables of units of measure found in Eisenlohr's translation of the work of Ahmes without meeting any word which could reasonably be translated "ruths." The indexes of mathematical histories did not supply us with any clue relating to this term.

On looking over the problems in Eisenlohr's Commentary on Ahmes we found on page 125 the following problem: "Wenn dir gegeben ist ein Dreieck von Ruthen 10 an seinem Schenkel, Ruthen 4 an seiner Mündung (Basis). Was ist sein Flächeninhalt?" This problem furnished the key to the solution of the mystery. The authors of the Stone-Millis textbook had evidently failed to see, or followed one who failed to see that the German word "Ruthen" or "Ruten" means rods, and had written "ruths" instead of rods in their translation of this problem, found on page 199 of their Elementary Geometry, 1910. This note aims to save other teachers a similar embarrassment. The Egyptians had no unit of measure known as a "ruth."

It should be added that the same incorrect translation of the German word "Ruthen" appeared also in Cajori's History of Elementary Mathematics, 1896, page 44, and is repeated in the so-called "enlarged and revised edition" which appeared in 1917. In view of the popularity of this history it seems the more important that this correction should be given publicity, and the fact that the mistake involved is so obvious may perhaps be an element of interest to those who study the history of errors, especially since this error appeared also in other histories of mathematics.

SHORT METHODS IN MULTIPLICATION.

By ROBERT C. COLWELL,

Geneva College, Beaver Falls, Pa.

The multiplication of 111 by 111 gives at once 12,321; of 1111 by 1111 gives 1,234,321; 121 by 121 also gives a symmetrical form 14,641. The 1 of the unit's column comes immediately by multiplication; the 4 of the ten's columns by adding 2 plus 2. The 6 is the result of 2 times 2 plus 1 plus 1. The rest comes from symmetry. The product of 131 by 131 equals 17,161 is just as symmetrical, but the symmetry is hidden because 1 has been added to the 6 in the fourth place. Written 16161 +1000 = 17161 the symmetry is plain. All such numbers as 141, 151, 161, 191 may be squared in this fashion. This is a particular application of a much more general method of multiplication which might be called the algebraic method.

The addition may be performed mentally and the result written down at once. The position of the digit tells whether it is units, tens, hundreds, etc., so the multiplication may be carried on without the use of t or u. (With practice, this too becomes a mental operation.)

It will be noticed that: Where there are equal angles in congruent triangles, there are equal angles in similar triangles; where there are equal sides in congruent triangles, there are sides in proportion in similar triangles, except when only one pair of sides are equal in congruent triangles, and then there is no proportion in similar triangles.

The basic theorems in each group are 1 and 2 (although 2 can be made to depend upon 1, and in that case, only one is basic). These are proved by superposition. The other theorems in each group are proved by getting the conditions of one of the basic theorems, 1 or 2.

The method of proof for 3 in each group is 1.

The method of proof for 4 in each group is 2.

The method of proof for 5 in each group is adjacent position and 1, and in the proof of B, A is used.

The method of proof for 6 in each group is adjacent position and 1 (although the method of 6A can be 2); and in the proof of B, A is used. The summary in each group consists of 1, 2, 6, and 7.

The principal use of congruent triangles is to prove lines or angles equal.

The principal use of similar triangles is to prove lines in proportion or angles equal.

THE CHEMISTRY TEACHER'S OPPORTUNITY.
By FRANK B. WADE,

Shortridge High School, Indianapolis.

My title might well have been "The Science Teacher's Opportunity" and it is only because I have perhaps considered more specifically the connection between the conditions brought about by the great war and the teaching of chemistry that I have taken the above as my title. You will all doubtless speedily apply to your own fields such general suggestions as I may be able to bring to you and the specific illustrations that I may select from chemical material will, I hope, serve to point the way to concrete application of these general principles.

In the present public emergency, the welfare of the country depends more than at any time in our past history upon the successful application of the principles of science to the solving of the problems thrust upon us by the war. It thus comes about that there falls upon the teachers of science a tremendous duty and a great opportunity.

As the public begins to realize and to appreciate more and more how dependent our modern civilization is upon the proper application of scientific knowledge, especially in time of war, there will come to us a larger and larger demand for efficient instruction in scientific subjects. We will be expected to give a proper start to the scientists of the future. With this demand, however, we may expect to find a certain lack of understanding of the length of time required to produce a real chemist or other scientist, and it will be one of our new duties to stand tactfully, but firmly, against any attempt to have us teach applications of chemical or other scientific principles before we have had time to teach the principles themselves.

We secondary school teachers will have all we can do to properly start our young scientists in the elementary principles of our sciences. If we can lay sure foundations, the colleges and technical schools will attend to the further progress of the student. Some of us have perhaps devoted rather too much time in the past to what we thought were practical applications, and too little time to the real solid foundations; and in consequence it is undeniable that many college teachers would rather themselves start students in the special sciences than have us do it for them.

We must correct this condition by giving our pupils thorough drill in all the fundamentals of our subjects and by training them to think chemically or physically, as the case may be; and then these college men will gladly accept the help we can give them in getting science students rightly started. We have more time than the college teacher and a better chance to get at the individual pupil and if we go about our task intelligently, we should be better able than the college man to teach the beginnings of the science courses.

The early recognition of the pupil who is adapted to follow a scientific career and the encouragement of that type of pupil to elect further science courses will be, to a greater extent than ever before, one of our duties and privileges.

Up to this point in my discussion I have spoken as though the preparation of pupils for actual scientific work were the all important matter in secondary science teaching. Really, however, while we will probably be called upon to start more pupils than formerly in scientific careers, the actual numbers so started by any of us will still be very small when compared to the numbers of pupils who will take our science courses with other ends in view. All men cannot be scientists. While this would be a pretty poor world were it without any scientists and a rather better one if we had a few more of them who could get what we now know into general use, yet I suspect that too many scientists might prove almost as bad as too few.

It will thus be necessary for us to provide suitable instruction for the ninety per cent or so of our pupils who will probably never become scientific men. Our principal business in life ought to be to find out just what we can and ought to give these pupils in the present crisis in order to best fit them to cope with the many problems that are upon us as men and women rather than as scientists.

The crying need of the times is clearly brought out by a recent letter from President Wilson's pen. The letter was addressed to "School Officers." It reads as follows:

The war is bringing to the minds of our people a new appreciation of the problems of national life, and a deeper understanding of the meaning and aims of democracy. Matters which heretofore have seemed commonplace and trivial are seen in a truer light. The urgent demand for the production and proper distribution of food and other national resources has made us aware of the close dependence of individual on individual and nation on nation. The effort to keep up social and industrial organizations in spite of the withdrawal of men for the army has revealed the extent to which modern life has become complex and specialized.

These and other lessons of the war must be learned quickly if we are intelligently and successfully to defend our institutions. When the war is over we must apply the wisdom which we have acquired in purging and ennobling the life of the world.

In these vital tasks of acquiring a broader view of human possibilities the common school must have a large part. I urge that teachers and other school officers increase materially the time and attention devoted to instruction bearing directly on the problems of community and national life.

Such a plea is in no way foreign to the spirit of American public education or of existing practices. Nor is it a plea for a temporary enlargement of the school program appropriate merely to the period of the war. It is a plea for a realization in public education of the new emphasis which the war has given to the ideals of democracy and to the broader conceptions of national life.

In order that there may be definite material at hand with which the schools may at once expand their teaching I have asked Mr. Hoover and Commissioner Claxton to organize the proper agencies for the preparation and distribution of suitable lessons for the elementary grades and for the high school classes.

Lessons thus suggested will serve the double purpose of illustrating in a concrete way what can be undertaken in the schools and of stimulating teachers in all parts of the country to formulate new and appropriate materials drawn directly from the communities in which they live.

WOODROW WILSON.

It is the need for the formulation of such new and appropriate materials drawn from the subject matter of chemistry that chiefly interests me at this time. This material must have a bearing on current problems of individual, of community and of national life. We must first teach the underlying principles to the mass of the class as well as to those who are to become scientists; then we must hunt out and bring before them the relations of these principles to current problems. In so far as it is possible, our illustrations and our experiments should be derived from or applied to such problems. We should also bring to our classes information as to what has been done and is being done by practical scientists toward solving the real problems of the country and of the people.

If we can thus make our pupils intelligent as to the need for and the uses of science, we shall be doing them a real service of a very practical nature, far more practical indeed than any service we can render them in the way of so-called practical applications of half-learned principles.

Having now stated my thesis, may I not take a few more minutes of your time to suggest in çoncrete fashion how a chemistry teacher can act upon my suggestions in a few specific instances.

Let us consider the problem of fuel conservation. While our fuel administration battles as best it can with the problems of supply and demand, and of transportation, what can we chemistry teachers do toward helping our pupils and their