valuable to the research worker and inventor, and equally valuable, if not infinitely more so, in mental training.

Mathematics has been called the most important tool of the engineer, and my only plea is to keep the tool a practical one, and not a fancy plaything taught at the expense of more deserving subjects. Opinions differ as to the wisest methods of teaching mathematics. Subcommittee IX of the International Commission on the Teaching of Mathematics reports "that it appears that mathematics teachers are generally agreed that mathematics should be taught as a science by professional mathematicians, and not as a tool by engineers." This is, of course, all right if the professional mathematicians do not lose their heads and overdo it by insisting on teaching the poor engineer to blow, so to speak, all kinds of useless fancy rings of smoke. While the study of advanced mathematics quickens a very useful type of intelligence, it does not necessarily develop resourcefulness or self-reliance except on paper.

It is often a boast that our Naval Academy and many of our engineering schools are based upon a wonderful mathematical curriculum, but just for fun, let the reader sometime test out an Annapolis graduate or an engineer after he has been away from the academy or his engineering school for a few years or even a few

months, and it will be found that this Annapolis graduate or the engineer has considerable difficulty in using his mathematics as a rough and ready tool. He will, however, remember, to a large degree, his working principles of chemistry and physics, his practical foundations of navigation and engineering and the more concrete things and be very hazy on the abstractions he learned. I asked a well-known mechanical engineer very recently if he remembered his calculus. Let me quote his reply: "Oh, the Devil, no!" he said.

Should I be overstrenuous in my protest against the continued persistence in the teaching of unnecessary mathematics, it may be due in part to the fact that one of my mathematical professors said when I was a student in the university," that I would never make an engineer or be successful in original research because I did not know mathematics enough."

I have managed to hobble along with my old slide-rule and pocketbooks of physical and chemical data, and I do not expect to change my tactics in the future. I did not like mathematics and I was loath to be absorbed by its abstraction to the necessary exclusion of other subjects I loved best. It has too often come under my notice that a fellow with a natural passion for creative work will become discouraged when he looks through text-books with the pages all

covered with mathematical idiosyncrasies and say: "It is all beyond me, I do not feel that I can ever succeed." He becomes disheartened in his reading, and the rich field of experimental research very often loses many good men to work it through this cause.

But there is a great field also for the mathematically inclined. Let them specialize in it as a language, for a complex language it is, with its rules, its grammar, and its signs and symbols. The specialist in mathematics becomes proficient in expressing through formulae and equations verbally stated facts in this condensed mathematical language, but its value as a working tool depends absolutely upon the great foundation of the laboratory and its deductive methods. Let the mathematically inclined specialize and come in for conferences when essential. I have called these fellows in quite frequently in my engineering work to help out my old slide-rule, but up to the present writing with only fair success. A distinguished consulting steam engineer, a friend of mine, recently sought data on the static and dynamic balance of a turbine rotor and shaft. These rotors and shafts have various vibratory periods in their performance and go through nodes and harmonics at the different speeds at which they are driven. My friend had a famous mathematician work out the dynamic balance and when the

machine was put into operation the maximum racking vibration took place at the normal operating speed! Nothing could have been worse. Mathematicians had it conclusively demonstrated that mechanical flight was impossible and the helicopter, with its vertical shaft and horizontally operated propellers, conformed to their abstractions by proving a failure. But all of the wise mathematician's endeavor was completely upset by the simple little trick of turning the vertical shaft with its horizontal propellers into a horizontal shaft with vertical propellers and equipping the machine with planes.

Dr. Charles P. Steinmetz, one of the ablest mathematicians and electrical engineers today, told me recently during a friendly discussion of the value of mathematics, that the steamer Persia in 1832 brought over to America in the mail a mathematical calculation setting forth the impossibility of a coal-burning steamer to cross the Atlantic! Of course mathematicians will claim that the physical constants were in error, and I suppose in this instance some of them were.

Enough experimental deductive work had evidently not been done to construct the formulae. I take the position that the two great basic sciences of physics and chemistry originated experimentally in the laboratory and

that deductive work is the most basic and vital and by far the most fertile field for the great majority who read these lines.

But enough of mathematics. Let us now try to set down an ideal mental equipment for those without the benefit of a college education as projected by an analysis of the governing conditions, and endeavor to fill in the gaps, so to speak, wherever in the light of them we may consider ourselves deficient.

The foundation of the practical inventor and research worker should most naturally consist of a good working knowledge of the principles of chemistry and physics, sufficient mathematics, possibly limited to advanced arithmetic, and enough manual training to give him "the feel of the machines and tools" with which he deals.

Professor Thorndike made practical tests on the Freshmen in an engineering school:

"Out of one hundred and three engineering Freshmen who reported on the matter of boyish activities, writes Dr. Charles Riborg Mann, in his splendid' Study of Engineering Education,' ninetyone had constructed on their own initiative mechanical or scientific devices, such as cannon, telegraph lines, telephones, electric motors, arc lights, gasoline motors, lathe, steam engine, water wheels, boats, etc. None of the engineering schools at present record this type of information, or make