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Calculating Curves

Thomas Hakon Gronwall, Ron Doerfler, Alan Gluchoff, Scott Guthery, Paul Hamburg
Docent Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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Nomograms are a device for solving an equation graphically. Each nomogram is associated with an equation in three unknowns and typically consists of three ruled lines (straight or curved), one for each variable. It is constructed so that a straight edge intersecting two of the lines at given values for their variables will intersect the third variable’s line at the value that satisfies the equation. Thus nomograms provide a quick way to solve for any one of the variables given the other two. The heyday of nomograms was roughly 1890 to 1970, after which their position was eroded by pocket calculators and desktop computers.

T. H. Gronwall (1877–1932) was a Swedish-American mathematician who worked in several areas of mathematics and who wrote an important theoretical paper on nomograms in French in 1912. The present book consists of a new English translation of this paper, an extensive commentary on the paper with many examples, a historical essay about nomography and about Gronwall’s life, and a complete bibliography of Gronwall’s works. In some ways this is a companion book to Evesham’s The History and Development of Nomography, from the same publisher.

Gronwall’s paper starts out by developing a necessary and sufficient condition for an equation to be solvable by a nomogram. The rest of the paper deals with methods for characterizing whether one of more of the lines is straight (rectilinear). It also has some interesting examples where the nomogram can be drawn using only two lines, such as an elliptic curve and a straight line, where two of the points are on the same curve. The mathematical methods involved include differential equations, linear transformations, and projective geometry. The present-day interest in Gronwall’s paper is primarily in the techniques used here, which are applicable more generally in linearizabilty problems.

Bottom line: A very specialized work, but well-written and having many interesting examples.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

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