As, one after another, the effect upon the cutting speed of each of these variables was investigated, in order that practical use could be made of this knowledge, it was necessary to find a mathematical formula which expressed in concise form the laws which had been obtained.As examples of the twelve formulæ which were developed, the three following are given:
P = 45,000D 14/15 F 3/4
V = 90/T 1/8
V = 11.9/F 0.665 ((48/3)*D) 0.2373 + (2.4/(18 + 24D))After these laws had been investigated and the various formulæ which mathematically expressed them had been determined, there still remained the difficult task of how to solve one of these complicated mathematical problems quickly enough to make this knowledge available for every-day use.If a good mathematician who had these formulæ before him were to attempt to get the proper answer (i.e., to get the correct cutting speed and feed by working in the ordinary way) it would take him from two to six hours, say, to solve a single problem; far longer to solve the mathematical problem than would be taken in most cases by the workmen in doing the whole job in his machine.Thus a task of considerable magnitude which faced us was that of finding a quick solution of this problem, and as we made progress in its solution, the whole problem was from time to time presented by the writer to one after another of the noted mathematicians in this country.They were offered any reasonable fee for a rapid, practical method to be used in its solution.Some of these men merely glanced at it; others, for the sake of being courteous, kept it before them for some two or three weeks.They all gave us practically the same answer: that in many cases it was possible to solve mathematical problems which contained four variables, and in some cases problems with five or six variables, but that it was manifestly impossible to solve a problem containing twelve variables in any other way than by the slow process of "trial and error."A quick solution was, however, so much of a necessity in our every-day work of running machine-shops, that in spite of the small encouragement received from the mathematicians, we continued at irregular periods, through a term of fifteen years, to give a large amount of time searching for a simple solution.Four or five men at various periods gave practically their whole time to this work, and finally, while we were at the Bethlehem Steel Company, the slide-rule was developed which is illustrated on Folder No.11 of the paper "On the Art of Cutting Metals," and is described in detail in the paper presented by Mr Carl G.Barth to the American Society of Mechanical Engineers, entitled "Slide-rules for the Machine-shop, as a part of the Taylor System of Management" (Vol.XXV of The Transactions of the American Society of Mechanical Engineers).By means of this slide-rule, one of these intricate problems can be solved in less than a half minute by any good mechanic, whether he understands anything about mathematics or not, thus making available for every-day, practical use the years of experimenting on the art of cutting metals.
This is a good illustration of the fact that some way can always be found of making practical, everyday use of complicated scientific data, which appears to be beyond the experience and the range of the technical training of ordinary practical men.These slide-rules have been for years in constant daily use by machinists having no knowledge of mathematics.
A glance at the intricate mathematical formulæ (see page 109) which represent the laws of cutting metals should clearly show the reason why it is impossible for any machinist, without the aid of these laws, and who depends upon his personal experience, correctly to guess at the answer to the two questions, What speed shall I use? What feed shall I use? even though he may repeat the same piece of work many times.
To return to the case of the machinist who had been working for ten to twelve years in machining the same pieces over and over again, there was but a remote chance in any of the various kinds of work which this man did that he should hit upon the one best method of doing each piece of work out of the hundreds of possible methods which lay before him.In considering this typical case, it must also be remembered that the metal-cutting machines throughout our machine-shops have practically all been speeded by their makers by guesswork, and without the knowledge obtained through a study of the art of cutting metals.In the machine-shops systematized by us we have found that there is not one machine in a hundred which is speeded by its makers at anywhere near the correct cutting speed.So that, in order to compete with the science of cutting metals, the machinist, before he could use proper speeds, would first have to put new pulleys on the countershaft of his machine, and also make in most cases changes in the shapes and treatment of his tools, etc.Many of these changes are matters entirely beyond his control, even if he knows what ought to be done.
