# Why is distance sometimes abbreviated S?

While distance in physical formulas is often abbreviated as d (which is pretty intuitive), another common abbreviation is s, as seen e.g. here, here or here. It also seems to be used in optics to represent focal distances.

Does anyone happen to know what S stands for? A quick look at Latin and Greek terms for "distance" doesn't seem to reveal anything starting with an S.

• Perhaps it is from German for Strecke. Mar 12, 2018 at 16:37
• Good question. For example, we usually use $s$ for arc-length, see en.wikipedia.org/wiki/Frenet–Serret_formulas. Mar 12, 2018 at 21:23
• See also the post why is $s$ used for arc-length. Mar 13, 2018 at 13:55
• Wow Mauro, sorry I didn’t realize you had already zeroed in on the exact same excerpt of Euler elsewhere. Mar 13, 2018 at 16:46
• @Geremia: The etymological root is strecken, that means to stretch and alludes to straight line. But the current meaning is simply the "kilometers to do" or even only the route. Mar 24, 2018 at 8:44

I think it most likely stands for spatium. E.g. Euler’s first book Mechanica (1736) uses $$s$$ throughout and first introduces it as follows (p. 13):
Theorema. (...) oportet determinare tempus, quo arcus $$\mathrm{AM}$$ absolvitur.
Solutio. Sit spatium $$\mathrm{AM}$$, sive sit linea recta sive curva, $$=s$$, et celeritas, quam corpus habet in $$\mathrm M$$ sit $$c$$ (...). Integrando ergo habebitur tempus, quo totus arcus $$\mathrm{AM}$$ absolvitur $$=\int\frac{ds}c$$.
(I don’t know who started this. A letter of Leibniz to Huygens (1690) has “resistence $$r$$, vistesse $$v$$, temps $$t$$, espace $$s$$”. Johann Bernoulli’s work on caustics (1692) has spatium curvilineum (p. 57) but apparently not the letter $$s$$, which he uses in later papers (ibid. pp. 60, 309, 316, 409).)