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In Measurement by Paul Lockhart (Harvard Press), he says (p.351):

the classical geometers (as far as I know) never even conceived of four-dimensional space, whereas adding another variable is an obvious and natural analytic extension.

Q. When was 4D space "conceived of"?

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In his A First Course in Calculus Serge Lang mentions on pages 525-526 that d'Alembert wrote in an article on dimension in Diderot's Encyclopédie in 1754 that "duration" could be considered a fourth dimension so the product of time by space has four dimensions, and that whether or not this is useful it at least has the merit of being new.

The page here also mentions d'Alembert in 1754 and gives the year of Lagrange's work as 1797. If credit needs to be given to one person, it should be d'Alembert before Lagrange.

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According to Klein, the first mathematician who considered space-time as a 4-dimensional manifold was Lagrange, but these ideas were not immediately developed by others. Then he mentions Cauchy and Cayley (Cayley published in 1844 "Chapters on the analytic geometry in $n$ dimensions"), and credits Grassmann (1844) with the first systematic exposition.

While geometers discussed $4$ dimensions, people doing statistical mechanics (Maxwell, Bolzmann and others) were inevitably lead to consideration of $N$-dimensional manifolds where $N$ is of the order of Avogadro number.

Ref. F. Klein, Vorlesungen uber die Entwicklung der Mathematik im 19 Jahrhundert (there is an English transl.), Ch. IV, 3-d part: "Spaces of n dimensions".

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    $\begingroup$ (feel free to add this to your answer) This first 11 pages of Geometry of Four Dimensions by Henry Parker Manning (1914) gives a very nice discussion of the historical development of higher dimensions. $\endgroup$ – Dave L Renfro May 7 at 8:45
  • $\begingroup$ @Dave L Renfro, Yes, he gives more detail and more references than Klein. $\endgroup$ – Alexandre Eremenko May 7 at 12:05
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19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes (including the six that occur in four dimensions). SOURCE

See the mini-biography of Schläfli at St Andrews. His major treatise was written in 1850-1852, parts were published, but the whole thing was not published until 1901 after his death.

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  • $\begingroup$ This was much later then the work I was referring to. $\endgroup$ – Alexandre Eremenko May 7 at 16:39

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