Nomographs (or nomograms, or alignment charts) are graphical representations of mathematical relationships (extending to empirical relationships of data) which are used by simply applying a straightedge across the plot through points on scales representing independent variables, which then crosses the corresponding datum point for the dependent variable; the choice among independent and dependent variable is arbitrary so that each variable may be determined in terms of the others. Examples of nomographs in common current use compute the lift available for a hot-air balloon, the boiling points of solvents under reduced pressure in the chemistry laboratory, and the relative forces in a centrifuge in a biochemical laboratory. Sundials represent another ancient yet widely-familiar example. The origins and geometry of the nomograph are seldom currently understood or taught. With the advent and ready accessibility of the computer, printed mathematical tables, slide rules and nomographs became generally redundant. However, there remains even today a place for nomographs since they provide insight into mathematical relationships, are useful for rapid and repeated application (as illustrated by the examples listed above), even in the absence of calculational facilities, and can reliably be used in the field even by nonspecialists. Many nomographs for various purposes may be found online. This paper describes the origins and development of nomographs, and illustrates their use with some relevant examples. A supplementary interactive Excel file demonstrates their application for some simple mathematical operations.

The helical spring is a widely used element in suspension systems. Traditionally, the springs have been wound from solid round wire. Significant weight savings can be achieved by fabricating helical springs from hollow tubing. For suspension systems, weight savings result in significant transportation fuel savings. This paper uses previously published equations to calculate the maximum shear stress and deflection of the hollow helical spring. Since the equations are complex, solving them on a computer or spreadsheet would require a trial-and-error method. As a design aid to avoid this problem, this paper gives nomograms for the design of lightweight hollow helical springs. The nomograms are graphical solutions to the maximum stress and deflection equations. Example suspension spring designs show that significant weight savings (of the order of 50% or more) can be achieved using hollow springs. Hollow springs could also be used in extreme temperature situations. Heating or cooling fluids can be circulated through the hollow spring.