Calculating Machines

CALCULATING MACHINES.) To show graphically a relationship among three quantities, as, for instance, that connecting the pressure, volume and tempera ture of a gas, we assign a series of values to one of the three quan tities and plot on the same sheet the graphs of the resulting formulas connecting the other two quantities, each such graph being labelled to indicate the value of the first quantity to which it corresponds. This method may be varied by using logarithmic or other scales in constructing the field on which the graphs are plotted. The purpose in such cases is to facilitate plotting by re ducing the graphs to straight lines or at least to simple curves.

When a three-variable relationship has thus been plotted in a series of "contour lines," it is a simple and rapid process to read off approximately the value of one quantity corresponding to any specified values of the other two. The process is often more simple and rapid if a nomographic or alinement chart is constructed (see NOMOGRAPHY) and that method can also take care of relationships involving f our or more variables.

While in general the aim of graphic methods of this type is merely to do simply and rapidly what could otherwise be done at the cost of more time and labour by arithmetic, algebra or measurement, in some cases the alternatives are so laborious as to be prohibitive. For instance, graphic analysis makes clear the nature of the solution of certain differential equations which can not be solved in terms of elementary functions. In certain cases, moreover, mechanical methods whose basis is mainly graphic provide many of the numerical results which are the real reason for desiring to solve the differential equation.

Limitations of Graphic Methods.

The unquestionable value of graphic methods if properly handled has been sometimes ob scured by cases in which they have not accomplished the purpose intended as well as other methods, or more skilful use of graphic methods, would have done. If a proposed method of presenting the results of a statistical investigation does not make the results clearer to the audience than if the results had been presented merely in words or in a table, the method should be discarded or improved. If a proposed scheme for graphic computation is less accurate than is necessary or less rapid in actual use than other methods, the true friend of graphic methods will be the first to turn to some other mode of computation.

A serious difficulty with graphic computations is the existence in most charts of regions in which the results have a larger margin of error than is acceptable. If, for instance, a point is to be located by the intersection of two arcs, the location is accurate if the arcs are nearly perpendicular, but if they run in about the same direction, a slight error in one of the radii or one of the centres will shift the point of intersection to a much greater extent. This difficulty can sometimes be avoided by using a second chart modified by change of scale or otherwise so as to shift the region of inaccuracy. The careful maker and user of charts always bears in mind this possibility of serious error, and determines the probable size of such errors, either by experiment or by analysis of his methods and formulas. (See PROBABILITY AND ERROR; STATISTICS.) See J. Lipka, Graphical and Mechanical Computation (New York, 1918) ; R. W. Burgess, Introduction to the Mathematics of Statistics (Boston, 1927). (R. W. B.)