Bibliography#

Nomography dates from the nineteenth century and has a long textbook tradition. This page lists the works this documentation draws on, and where to read more.

The founding text#

Maurice d'Ocagne. Traité de nomographie. Théorie des abaques, applications pratiques. Paris: Gauthier-Villars, 1899. A public-domain scan is on the Internet Archive.

The alignment method, in which three points lie on one straight line and the condition is written as a vanishing determinant, is d'Ocagne's. Every chart in this package is based on it.

Textbooks#

L. Ivan Epstein. Nomography. New York: Interscience, 1958.
Edward Otto. Nomography. New York: Macmillan, 1963. Discusses the criterion of Saint-Robert and Massau's method for deciding when an equation can be charted.
Douglas P. Adams. Nomography: Theory and Application. Hamden, CT: Archon, 1964.

A modern overview#

Ron Doerfler. "On Jargon: The Lost Art of Nomography." The UMAP Journal 30(4), 2009, pp. 457–493. The author posts a free PDF.

This is the best single introduction. It derives the parallel-scale chart, the N-chart and the determinant method, works the curved-scale chart for the quadratic w² + uw + v = 0 (after Adams and Otto), builds grid nomograms from determinants, and reshapes charts by projective transformation.

When a formula has a chart#

P. de Saint-Robert. "De la résolution de certaines équations à trois variables par le moyen d'une règle glissante." Memorie della Reale Accademia delle Scienze di Torino XXV, 1871, pp. 53–72. Presented in 1867.

auto_nomogram applies the test that $$z = F(x, y)$$ can be put in the form $$h(z) = f(x) + g(y)$$ exactly when $\partial^2 \ln(F_x / F_y) / \partial x \, \partial y = 0$ . Two accounts attribute this test to Saint-Robert: V. A. Esaulov, "The Art of Nomography", and Lucien Pirio, "Géométrie anamorphique II" in Images des mathématiques (CNRS), 16 April 2011. Pirio notes that the test has been rediscovered many times since. The link is to a copy archived in 2023, because the original address no longer responds reliably. We follow those two accounts; the 1871 paper itself was not consulted.

The more general problem, of which equations can be brought into a chartable form at all, was taken up by J. Massau, "Mémoire sur l'intégration graphique et ses applications", Annales de l'Association des ingénieurs sortis des écoles spéciales de Gand 7, 1884, pp. 53–132, and independently by Léon Lecornu, "Sur le problème de l'anamorphose", Comptes rendus de l'Académie des sciences 102, 1886, pp. 813–816.

PyNomo, by Leif Roschier, builds nomograms in Python from a description of each scale. P. Martínez-Pagán and L. Roschier collect PyNomo scripts for common engineering formulas in Data in Brief, 2022, doi:10.1016/j.dib.2022.108661. Nomograms.jl is an independent implementation and borrows no code from it.

What is original here#

The derivations in these pages were written for this package, starting from the collinearity condition, and are not copied from the sources above. Where a result is classical, it comes from those sources. The following are specific to this package:

the test that a formula of three or more inputs splits into a chain (it depends on a pair of inputs only through one combination of them exactly when $$F_x / F_y$$ does not depend on the rest), and the numerical recovery of every transform from the formula itself;
how a chain's stages are paired, mirrored and sized so that no axis lands on another;
how ticks are chosen and thinned, and how labels are kept clear of each other and of the alignment lines;
the isopleth palette, whose colour-difference figures come from the package's own check against colour-vision deficiency rather than from a published source.