Recently, I came across this article and wondered if there really is a definitive answer to the question of who invented the number line?


Not mentioned in the linked thread is Bombelli, who although not widely published until after cited “candidates” Stevin and Wallis, came before them per Bourbaki’s historical notes (my emphasis):

(...) up to the end of the Middle Ages [the] “ratios” of Euclid were customarily described as “numbers”, and the rules for calculating with integers were applied to them without any attempt to analyse the reasons for the success of these methods.

Nevertheless we see R. Bombelli, as early as the middle of the 16th century, expounding a point of view on this subject in his Algebra [1572] (*), which is essentially correct (provided that the results of Book V of Euclid are assumed to be known); having realized that once the unit of length has been chosen there is a one-to-one correspondence between lengths and ratios of magnitudes, he defines the various algebraic operations on lengths (assuming of course that the unit has been fixed) and, representing numbers by lengths, obtains the geometrical definition of the field of real numbers (a point of view which is usually credited to Descartes) and thus gives his algebra a solid geometrical foundation (**).

(*) We are concerned here with Book IV of his Algebra, which remained unpublished until modern times; for our purposes it matters little whether or not the ideas of Bombelli on this subject were known to his contemporaries.

(**) (...) Bombelli, in the same context, gives with perfect clarity the purely formal definition (such as one would find in modern algebra) not only of negative numbers, but also of complex numbers.

  • $\begingroup$ Thanks for your answer, this also explains the terminology "a is to b as c is to d." $\endgroup$ – user 726941 Feb 18 '18 at 7:08
  • $\begingroup$ I don't really follow how replacing the word "number" by the word "length" solves the problem here. As far as Euclid and geometry are concerned, it suffices to work with constructible numbers, which are of course not enough. $\endgroup$ – Mikhail Katz Feb 18 '18 at 15:16
  • $\begingroup$ @MikhailKatz Which problem do you mean? They say he is in effect defining all numbers geometrically as all lengths. (You may prefer Barrow, who they next say “gave a brilliant exposition” of “a point of view which differed little from that of Bombelli”, and “obtain[ed] the field of real numbers in terms which Newton took up again in his Arithmetic, and which his successors up to Dedekind and Cantor did not change”?) $\endgroup$ – Francois Ziegler Feb 18 '18 at 18:50
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    $\begingroup$ Right :-) Bombelli identified his numbers with the line, or all lengths. I think the question of who (if anyone) started doing that — whether for rationals, constructible numbers, or any extension — is still meaningful. $\endgroup$ – Francois Ziegler Feb 19 '18 at 5:52
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    $\begingroup$ @FrancoisZiegler, thanks for the clarification. Now I understand better what you were trying to say. I would point out however that the system envisioned by Stevin (namely, unending decimals representing all numbers, whether rational or not) seems to have been the first to enable what is generally considered the modern number line. $\endgroup$ – Mikhail Katz Feb 19 '18 at 10:34

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