perience would get it out of us. We are not only prejudiced against our fellows, but out of touch with the Infinite. We seem to be unable to comprehend Him and His relations to us. A friend of mine once said: "The pupil may discover through his geography lessons that adjustment to nature is valuable, that harmony with nature is as essential in the physical world as agreement with nature's God is in the spiritual world."21 I wonder too if he might not learn the other truth, that if adjustment and harmony are so essential in the natural world how much more so they must be in the spiritual.

Stimulates Desire for Travel and Travel Reading.-It has been shown that good geography teaching gives a pupil the use of maps and atlases. It may also teach him the library habit for magazines, papers, reports, books and bibliographies; sources of information. It certainly will make reading more satisfying and intelligible. Even place geography can help in this item. Many a time my students have stopped after class or called at other periods to talk something over and have exclaimed, "How I would like to see this or that!" With the geographic knowledge, they want to see the country and people as well as the canvas and pillars. Some have to do their best to satisfy the wanderlust by reading travel books and descriptive matter; and several have actually found the means to gratify the desire for travel kindled in the geography classes.

Makes for Good Citizenship.-It has been said that "a knowledge of place and regional geography is what the average person, no matter in what walk in life, most needs." A committee on secondary school geography reported the above statement to the National Educational Association in 1909. The report continues: "Farmer, fruit grower, stockman, miner, promoter of railroads and waterways, manufacturer, shipper, fisherman, artisan, health-seeker and traveler all feel its daily need. One cannot read with intelligence the daily papers and periodicals, not to mention history, literature, politics and religion, without some considerable knowledge of regional geography. Familiarity with the geography of our own country and of Europe is a muchto-be-desired factor in good citizenship. Geography touches and broadens the daily life of every individual." However sweeping the above may be, it is certainly true that geography gives a more enlightened citizenship. Goode,22 in speaking

21Sutherland, W. J., Jour. Geog., Vol. 5 (1906), p. 126. "Goode, J. P., Jour. Geog., Vol. 15, p. 274.

of this regional knowledge of the resources and industrial achievements of peoples, said: "No other study offered in school or college is better adapted, when properly presented, to give an extensive acquaintance with the current history of the industrial world, and this in itself is a large element in a liberal education." I firmly believe there is no one subject so full of possibilities in promoting a healthy spirit among men, in opening one's eyes to the real situations in the regions of the world, in other words, in broadening, enlightening and vitalizing one's citizenship as geography."

If a student goes to college, I suppose one can excuse his high school work by saying "He will get it in college." But what about the ninety in a hundred who never attain to college, and the nine in ten going to college who, under an elective system, do not get geography there? This makes the call for good, clear, sane, abundant geography in the secondary school more imperative. Most of the young people must get it there if anywhere.

Professor Richard E. Dodge, President of the Association of American Geographers, says "Geography will come into its own when it is proved to be worth while."23

I trust my effort today will add a little to the proof that has been accumulating in recent years.

23 School and Society, Vol. 3, pp. 253-4.

WAR SAVINGS.

"War Savings Stamps mark an epoch in our national life."-[Secretary of the Treasury McAdoo.

Many a successful business man has said that the saving of his first dollar was the most important single act of his life; that it marked the beginning of a habit and a course of conduct to which he attributed his

success.

Something very analogous to this, it is believed, is going to be the effect on the American nation of the War Savings campaign. Not only are millions of individual citizens going to begin to save, but this habit of economy and saving is going to be a collective movement, a movement not of individuals alone but of the nation.

The habit of saving formed now has a deeper incentive than ordinary. We are saving now not alone for selfish reasons, we are saving now from patriotism, saving not alone for ourselves but for our country. The combination of patriotism and thrift is, indeed, going to make the War Savings campaign an epoch in our national life. It is not only going to be a thing of tremendous benefit to millions of citizens, it is going to be a thing of tremendous advantage to the nation as a whole, and affect our whole national life. It marks the beginning of a new era in American life, an era of economy, good sense, and patriotism.

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.