``Complex numbers'' and the problem of multiplication between quantities (Q2145720)

The article is not about complex numbers in the modern sense: \(a+bi\), but is about what were called \(^*\)complex numbers in 18th-century France (well, and elsewhere), namely, quantities measured according to various non-metric units such as length, time, or money. (We here use the asteriksed word \(^*\)complex to refer to those mixed quantities.) Charlemagne, for example, had adopted the ancient measure of a man's `wingspan' (i.e., the distance between the fingertips of a man's outstretched arms), the \textit{toise}, as a measure of length. One \textit{toise} (English: fathom; German: \textit{Klafter}; ca. two meter) was divided into 6 \textit{pieds} (feet); each \textit{pied} was divided into 12 \textit{pouces} (inches); each \textit{pouce} was divided into 12 \textit{lignes} (lines); and each \textit{ligne} was 12 points. Common monetary units at the time were \textit{livre}, \textit{sou} (or \textit{sol}), and \textit{denier}, where one \textit{livre} was worth 20 \textit{sous} or 240 \textit{deniers}. Quantities expressed in more than one unit, such as a price tag of ``34 livres 10 sous 2 deniers,'' were called \(^*\)complex numbers. These numbers raised two challenges for textbook authors. (1) Devise algorithms for the four basic operations of arithmetic but defined on \(^*\)complex numbers. In doing so, commutativity of multiplication was often lost and therefore doubted to be a characteristic algebraic feature of this operation. (2) Get a firm conceptual grip on handling combined dimensions such as ``price/length \(\times\) length'' that would yield ``price'' as the dimension of the result. According to our two authors, mathematicians of the time managed (1) reasonably well, but failed (2) miserably. In the conclusion, they argue that mathematicians did not start to tackle (2) head-on before the 1950s, with heavyweights such as Hassler Whitney or Hans Freudenthal joining the effort. The fact that the discussion around \(^*\)complex numbers as described above was absent from German textbooks of the time, make it plausible to assume (p.~121) that this gave Gauß permission to adopt the same name for `his' complex numbers. The article is divided into the following eight sections: 1.~Introduction (2~p.) -- 2.~Major units of measurement used in the article (2~p.) -- 3.~The notion of multiplication (2~p.) -- 4.~The first appearances of ``complex numbers''; here they focus on Charles-Étienne Camus (1699--1768) and Étienne Bézout (1730--1783) (7~p.) -- 5.~Searching for changes in France after the Revolution and the introduction of the metric system; here they discuss S.-F.~Lacroix, F.~Peyrard, P.\, H.~Caillet, and P.-L.~Bourdon (4~p.) -- 6.~The practice in the Brazilian reception; here Cristiano B. Ottoni (1811--1896) is the main figure (4~p.) -- 7.~An outlook on the practice in other countries; here they include Portugal, England, Italy, and Spain (3~p.) -- 8. Conclusion -- solving the riddle (2~p.) -- followed by a two-page bibliography. The article deals with a niche few will have strong feelings about but, being well-organized and well-written, it is a pleasure to read.