Throughout history, mathematicians devised geometric and mechanical ways to speed up calculations. From abacuses and Napier’s bones (see the article Mirifiques Logarithmes in Tangente 196, 2020, or the feature “Les logarithmes” in Tangente 177, 2017), through Pascal’s and Leibniz’s calculating machines (see our feature “Calculer plus vite,” Tangente 184, 2018), all the way to slide rules, a great many methods were developed.
Geometric figures with scales
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Nomograms are geometric diagrams with scales that, with a simple procedure such as drawing a line between two points, yield the results of arithmetic operations or allow an equation to be solved. During the 19th century, various authors proposed graphical methods for solving problems using geometric figures.
To make the new units of measurement introduced by the revolutionaries easier to use, the French manufacturer Louis Ézechiel de Pouchet (1748–1809), a specialist in metrology, devised graphical tables in 1795 linking the different units of measurement; they became known as Pouchet abacuses.
In a more mathematical vein, the German mathematician August Ferdinand Möbius (1790–1868), best known for the Möbius strip, constructed a family of parabolas in 1841 that could be used to find the prime divisors of an integer. Around the same time, a French engineer, Léon Lalanne (1811–1892), invented and simplified numerous nomograms. He regarded straight-line constructions as the easiest and most reliable way to obtain a result, so he altered the scales to transform curves into straight lines. His aim was to help engineers with their calculations as the Industrial Revolution gathered pace, creating many new needs. He called these new objects abacuses, a term already used for various calculating devices, including what we now call counting frames (see Histoire des mathématiques, de l’Antiquité à l’an mil, Bibliothèque Tangente 30, 2007).
