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When we consider how few of the details of professional, commercial or domestic life there are which do not to a greater or less extent require the employment of some one or more of the operations of mathematical computation for their solution, the importance of any instrument which will lessen in any degree the mental and physical labor involved in such computation is at once apparent.

In the case of the Engineer, the Scientist, Statistician and others of whose work mathematical calculations form by far the greatest part, the subject of mechanical aids in those labors assumes an importance which can not be over-estimated.

That this fact is now and has been in the past recognized is shown by the fact that the design and application of instruments for this purpose has in some one of their forms engaged the attention and thought of many of the most eminent scientists and mathematicians from the time of Briggs and Gunther to the present, and the results of the labors of these men is seen in a class of instruments which are marvels of mechanical skill and mathematical accuracy, performing the operations for which they were intended in a manner which leaves little to be desired.

While these instruments are in almost universal use abroad— France and Germany being notable examples— their introduction and use in this country has been inexplicably slow when their value and the short time necessary to acquire a working knowledge of them is considered. There is no possible explanation which can be given to account for the limited use of the instrument just referred to other than that it is due to a very limited or almost wholly lacking knowledge on the part of those most concerned both of the existence of these instruments themselves and of the invaluable aid they are capable of rendering in almost any and every form of mathematical computation. That this ignorance or lack of knowledge does exist and that it is almost unaccountably general is too evident to require proof— the limited use of the instrument being, as has already been stated, conclusive evidence of its existence.

There are several conditions which may be assigned as being the cause of this state of affairs. One cause, certainly a potent one, being the fact that in the countries named— more especially perhaps in Germany— the importance of these instruments is so fully recognized that a special course of lecture and instruction is often devoted to their theory and use, while in the crowded curriculum of our home technical institutions no attention whatever is given to them, a condition which is usually the case with any subject which has any chance whatever of being acquired in any other way. In a few— and that but very few— institutions in this country some instruction may be given in the subject, but in no case is it more than superficial, and always entirely inadequate when the great importance of the subject is considered.

Another assignable reason lies in the fact that there seems to exist with a great many people a vague and wholly unreasoning distrust of either the accuracy or the practical value of any instrument designed to perform a mental operation. The idea seeming to be prevalent that any result obtained in any such manner should be regarded with suspicion: that even granting such results to be of sufficient accuracy to be of use, that the instrument capable of performing it must of necessity be extremely complicated, requiring a large amount of time and study for its intelligent use, and that even when a working understanding of it had been acquired the saving of time and labor effected by its use would be much too small to warrant the time spent in its acquirement.

To the above two causes assigned as responsible for the limited use of mechanical aids to mathematical calculation may be added a third which is the almost total lack of literature dealing with the subject under discussion: a literature, if so it may be called, utterly lacking in both extent and quality when compared with the importance of the subject with which it is concerned. The instruments being for the most part of foreign design and make, it follows that most of the literature concerning them is also in a foreign language, and while the results of studies and investigations of many foreign mathematicians have appeared in book or pamphlet form their value to the student in another country must necessarily be limited by the linguistic attainment of the student who desires to avail himself of those results.

There of course remains to us the “Directions for Use” usually furnished by the maker, but unfortunately in the great majority of cases not only do these “directions” utterly fail to direct but they fail entirely to do justice to the instrument described and often do not even hint at many of its most valuable applications— thus doing an injustice both to instrument and owner.

The remarks thus far made while applicable to almost every form of instrument of the class under discussion are particularly true as applied to the particular instrument we are about to discuss, and much that can be said of the Planimeter in this connection will apply equally to other instruments of which it is a type.

Contrary to the usual belief and common with other instruments of its class, a knowledge of the theory or principle of the Planimeter is not an essential either to an ability to use it or to obtain accuracy in results when applied to the solution of those problems to which it is particularly adapted. An ability to use the Planimeter can easily be acquired in a very few moments, but at the same time the time and effort necessary to acquire the additional knowledge of the mathematical and mechanical operations involved in its theory and construction will be well repaid, not only in the increased appreciation of the instrument itself and greater confidence in the accuracy of the results obtained by its use, but also in the ability to apply it to problems involving unusual conditions and to pass intelligent judgment on instruments of like nature and design.

For this reason the demonstration of the theory of the Planimeter is given in as clear and concise a manner as possible and from it are deduced the formulae by which the data for the adjustment of the instrument to meet the varying conditions involved in its many applications is obtained. In all demonstrations the use of the higher mathematics has been purposely avoided and all mathematical operations are reduced to their simplest forms.

The examples given as illustrating the application of the Planimeter to the solution of the various problems of engineering have been selected with a view to having each such example a type of a class of problems requiring similar operations for their treatment, and the explanations are made as general as possible to enable the intelligent use of the instrument in every other problem belonging to that particular class.

In the treatment of each example given, the adjustment and method of use of the Planimeter as applied to the special problem involved is concisely and logically given: the particular operation involved being deduced directly from the general theory of the Planimeter. The adjustment and use of the instrument in each example is clearly described and is followed by a discussion of the relative and actual degree of accuracy attainable in each case, together with notes on the method of operating and the securing of a maximum efficiency.

The Tables have been made as complete as possible and the effort has been made to include in them the data for almost all of the more commonly occurring problems of engineering practice. A full description of the Tables with their arrangement and method of use is given under the head of Explanation of Tables and need not be repeated here.

Both in the treatment of the theory and use of the Planimeter and in the Tables the effort has been made to make all that follows not alone a Treatise on the Planimeter but also an Office Book for constant use, enabling the data for the adjustment of the Instrument for any operation to be at once obtained without calculation, and the entire arrangement has been with that end in view.

In conclusion it can be said not only of the Planimeter but equally of almost every instrument of the class to which it belongs, that a knowledge of its invaluable capabilities and of the enormous saving in time and labor effected by its use is the only requisite to make it the co-laborer of the Engineer in almost every detail of his professional life.

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