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As has already been stated, the Tables are intended to give at once by inspection and without further calculation the data necessary to adjust the Planimeter for use in the solution of most of the more frequently occurring problems arising in the Engineer's practice. The Tables have been calculated and checked by both Logarithmic and Slide Rule methods, and contain all the factors necessary for adjustment of the Planimeter for any operation. Columns of factors by means of which the accuracy of the results obtained by the instrument can be readily checked have also been added to facilitate the work of the calculator, and to make the Tables as complete and useful as is possible. As the derivation and use of the factors contained in the Tables are fully explained in the following chapters it will be unnecessary to repeat it here, but a brief explanation of the general arrangement of the Tables will perhaps aid in facilitating their use and give a clearer idea of their arrangement.
As in many cases the scale of a drawing is given in the form or a ratio instead of being expressed as a definite number of parts to an inch, Column 2 has been added, so that the equivalent ratio of any scale expressed in inches and fractions of an inch is at once seen.
Columns 4, 7 and 8 in many of the Tables might perhaps have been omitted as not being strictly essential to the use of the Tables, but they serve a very useful purpose not only as a guide and aid to any desired extension of the Tables, but are also valuable both as a check on the accuracy of such additional calculations and on the position of the decimal point— a matter of the greatest importance where the numbers dealt with contain many figures.
It will be seen that in all of the Tables blank columns have been added (as Cols. Nos. 15 and 16 of Table 1). The reason for this is that although two Planimeters may have been made by the same maker and intended by him to be exactly alike in every detail, the delicacy of the instrument is such that there always exists between them a slight variation, due to process of manufacture and to the impossibility of two pieces of delicate mechanism being exactly alike in every respect. This variation, which is constant for any particular instrument, is shown by a slight difference in the numerical value of the Setting and Constant between the values of these factors given in the Table for any given scale and the values given for these same factors on the card usually furnished by the maker with each instrument. To illustrate: take the values given for the Setting and the Constant in Table 1 for the scale 1:1000 which are 302.8 and 17420 respectively. Let us suppose that the values of the Setting and Constant for that same scale given on the card enclosed in the case of any particular Planimeter to be 303.2 and 17430 respectively. The differences between these two sets of factors are 303.2 – 302.8 = 0.4 in the Setting and 17430 – 17420 = 10. for the Constants. These two values then, 0.4 and 10.0, represent the physical difference between the two instruments mentioned above, and if that instrument which has 302.8 for the value of the Setting and 17420 as the value of the Constant has been used in the preparation of the Tables, all that is necessary to be done to make the Tables applicable to the Planimeter having 303.2 for its Setting and 17430 for its Constant for the same scale will be to add .4 to the number given in the Tables for the Setting for each scale, and 10 to the number given for each Constant, placing these new values in the Columns left blank for this purpose. By this simple operation, which will take but a few moments to perform, we can make the entire set of Tables applicable to any particular Planimeter and the Tables will be in every respect the same as though that particular instrument had itself been solely used in their preparation.
In some forms of Planimeters the zero graduation of the Tracer Arm, instead of being at the Tracer end of the arm is at or near its other extremity. In this case it is evident that the values of the Settings are reversed, the higher values indicating a shorter Tracer Arm and the lower values a longer one. Formulae for finding the equivalent value of a Setting or Constant given for one form of graduation in terms of the other form are given later, but to facilitate the use of the Tables and increase their adaptability to every form of Planimeter, Columns Nos. 11 and 12 have been added, and the values given in these added columns are Nos. 9 and 10. Columns 9 and 10 being for those Planimeters having the zero of their graduation at the Tracer end and Columns 11 and 12 the equivalent value for those instruments in which the Zero of the graduation is at the opposite end of the Tracer arm.
As has already been stated, the Tables have been carefully computed and checked by both Logarithmic and Slide Rule methods, and are believed to be without error. Owing, however, to the large number of figures involved in the calculation of each factor, there may be a slight possibility or error, more especially in the position of the decimal point, but they are few if any, and easily discoverable by use.
Since there can be any number of Settings for any given Scale with their corresponding Vernier Units, that Setting has been selected which can be used for a number of different scales, making the Vernier Unit the variable, as by this method the same Setting can be used for various scales, thus lessening the necessity for frequent adjustments. While perhaps in a few cases a different Setting than that given in the Tables might allow of a simpler Vernier Unit, that advantage is more than offset in the method adopted by considering the Setting constant for a certain set of scales, and allowing the Vernier Unit to vary.
The value of Actual Vernier Unit for any given scale is given in Column 5, and will be found very convenient when the actual area of any figure drawn to that scale is desired: the actual area being in each case equal to the product of the Reading for that figure and the Actual Vernier Unit. In this way the actual area is at once obtained without the necessity of resetting the Planimeter to the setting for the Scale 1" = 1" and retracing the figure. The ease with which the actual area is thus obtained admits of a very easy check on the accuracy of the results obtained by the use of the Planimeter, since the check consists simply in multiplying the actual area thus found by the Unit of Area of the Scale to which the figure is drawn.
Column 4 of Table 7 and 8 gives the number of Cubic Yards represented by one actual Square Inch of cross-section area of any given prismoid for the given Scale and length, while Columns 7 and 8 of the same Table give the number of Cubic Yards in a prismoid whose length is the same, and the sum of whose end areas plus four times their mean area is an actual area of 15.5 Square Inches, or the area of the Test Plate. This will be more clearly understood after reading the description of the use of the Planimeter in Earthwork calculations which is given later.
As all of the Tables with method of using them together with detailed descriptions of the derivation and use of the factors entering into them are fully explained in their proper place further description need not be given here.
The diagram given on Plate II will be found very useful in many ways— and if accurately drawn and on a fairly large scale, with reference to the particular Planimeter with which it is to be used, will given the Reading for any given Setting or the Setting necessary to produce any desired Reading when tracing an actual area of 10,000 Square Millimeters with a degree of accuracy quite sufficient for any but the finest work. It can be prepared either by calculation, or the data for plotting the curve may be acquired by trial: the latter method being however preferable, as no instrument correction need then be applied to the factors obtained from it. In the diagram the curve BB is for the instrument we are using for illustration, while AA is the curve of Settings for an instrument having the Zero of the Tracer Arm graduation at the Tracer end of the Arm. Its method of construction and manner of use are evident, and will be referred to in a later Chapter.
As has already been stated the intent in the following demonstrations, descriptions and tables has been
Before beginning the actual measurement with the Planimeter in any problem, it is well to check the accuracy both of the Instrument and its Setting, and this is very easily accomplished for every operation by the arrangement of the factors given in the Table, the operations being as follows:
Having adjusted the Planimeter to the Setting given in the proper Table for the particular operation to be performed, the instrument is brought to a Zero reading and is then caused to trace the 15.5 Square Inch area of the Test Plate. At the end of this tracing, which should take but a moment to do, the Reading of the instrument is taken. If the Planimeter is in good adjustment and the tabulated value of the Setting is without error, this Reading should be exactly the same as that given in Column 8, under the heading “Reading” in the Table used. As the exact number of Vernier Units which the Planimeter should record when in accurate adjustment is given in this Column for every scale in all the Tables, the values of this method of checking is apparent.
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