## Theory of the Polar Planimeter.

### Theory of Planimeters in General.

While a full discussion of the general theory of Planimeters and of the special theoretical considerations involved in the various particular forms of the instrument would be of much interest and especially so to the mathematician and mechanical expert, lack of space will necessitate the limiting of such detail discussion to the particular type we have selected for description and illustration in the following chapters, confining reference to other forms to description only.

The general theoretical principles involved in the design and operation of the Planimeter class of instruments while differing widely in some forms of the instrument— notably in the higher types— are essentially the same in most of the simpler forms and in the particular Polar form to which the Planimeter we are discussing belongs.

For this reason, while particular instruments may vary in some one or more points of mechanical detail or arrangement of parts, the following demonstration and statement of the theory of the Polar Planimeter will be found to apply to almost every form of the Polar type of the instrument.

Various attempts to classify the different forms of Planimeters, according to their mechanical action or some other standard of comparison, have been made, but have all proved unsatisfactory, owing to the fact that in may instances a given instrument while placed in one class on account of its possessing certain characteristics common to that class, may with equal reason be included in some other class, one or more of whose characteristics it may also possess.

For this reason, the only actual and logical basis of classification or comparison is really one of relative degree of accuracy attainable by any given instrument and the range of operation of which the particular instrument may be capable.

It has been already stated in the introductory chapter that a knowledge of the theory of the design or operation of the Planimeter is not an essential either to an ability to use it or to apply it in the solution of many practical problems for which its peculiar properties so eminently fit it.

It will readily be seen from the following discussion that what we shall term the “General Principle” of the Polar Planimeter is very simple, and can be states in a very few words, while at the same time clearly expressing the fundamental theoretical considerations involved not only in the design and operation of the instrument, but also in all of its application to special conditions.

In fact, the first requisite for the intelligent successful use of the Planimeter and an ability to apply its principles to every form of practical application of which the instrument is capable, lies in a clear understanding of this “General Principle” rather than in a knowledge of how to mathematically demonstrate the truth of that Principle.

For this reason, while the mathematical proof of the truth of the “General Principle of the Planimeter” will be given as being necessary and proper to a clear and full theoretical treatment of the instrument, further mathematical discussion will be limited to showing how this General Principle can be utilized and adapted to various conditions so as to make the Planimeter the efficient mechanical aid to the Engineer which it is so capable of being.

The theory of the Polar Planimeter has been demonstrated in a variety of ways by writers on the subject and by others who have made a study of the instrument.

In some of these demonstrations use has been made of the Calcubus and other methods of the higher mathematics, while in others the desired results have been reached without going beyond the more simple of mathematical operations. While perhaps the use of the Calcubus would shorten the demonstrations, and in some cases render the mathematical treatment clearer, it is for our purpose that the discussion be confined to the more ordinary processes of mathematical computation.