# Who was the first individual that used the word “torus” to refer to $\mathbb{S}^{1} \times \mathbb{S}^{1}$?

Further, I believe that the idea to call it thus had to do with its resemblance to the "torus" in the base of some Greek columns of old:

What do you think of this hypothesis of mine?

Hero mentions the work of Dionysodorus in his "Metrica" this way: "διονυσωδώρω έν τω περί τησ σπείρασ έπιγραφομένω", namely "Dionysodorus in his work On σπείρασ". In modern Greek σπείρασ stays for "spirals", but was used in ancient Greek for the Latin torus. You can find the original text here at page 128, line 3 and 4.

• Great. What is the Greek term for the torus in architecture? Since "torus" comes from Latin, it would be nice to know if the Greek names (geometric and architectural) were the same or different. – Gerald Edgar Nov 10 '17 at 17:08
• GeraldEdgar: According to user6530, the ancient greeks used the same word ("σπείρασ") to refer to both types of tori. @user6530: How do you know that "σπείρασ" was the word in Old Greek to refer to what Vitruvius et al called a "torus"? – José Hdz. Stgo. Nov 10 '17 at 19:34
• @JoséHdz.Stgo. Because sections of torus were called "spiric sections" by Hellenistic geometers even before Vitruvius. According to online etymology dictionary, Latin "torus" did mean a bulge at the base of a column in 1560s, but the causality is reversed: σπείρα generically referred to winding and coiling, and "torus" to swelling and bulging, and it is the latter that gave the name to the decoration. – Conifold Nov 11 '17 at 0:28

TORUS. Hero mentions a mathematician named Dionysodorus as the author of On the Tore, in which a formula for the volume of the torus is given [DSB].

An early use of torus as a mathematical term in English is in 1860 in The Practical Draughtsman’s Book of Industrial Design by William Johnson: “In geometry, the torus is a solid, generated by a circle, revolving about an axis, continuing constantly in the plane of this axis, in such a manner, that all sections made by planes passing through the axis are equal circles, and all sections by planes perpendicular to the axis will also be circles, but of variable diameters.” [Google print search]

So, do you think the term used for Greek columns was named for the geometric term as in Dionysodorus? Or the other way around?

• The other way round. I suppose Dionysodorus wrote his "On the tore" in some variant of Koine Greek. If the (Koine) Greek for "torus/tore" has to do with the word with which architects in Ancient Greece referred to that component in their columns, then my hypothesis would not be that off the mark... For the time being, I can't say much more: alas, it's all Greek to me... – José Hdz. Stgo. Nov 10 '17 at 4:04

The mathematical usage comes from the usage in architecture. The word is of latin origin "torus" and not of greek origin, and has a number of different meanings. It means muscle but also pillow (probably circular one) and it means also a strong rope. In this last meaning it was used to describe circular decoration made of strong ropes and afterwards it started being used to describe the part of columns you're referring to. With this meaning it is used already by Vitruvio Pollione (80ac-15 ac).

Nothing strange that the word passed from architecture to geometry, the two subjects being strongly related throught Renaissance and Baroque periods. I easily found Italian citations much pre-dating Cayley:

Vincenzo Antonio Rossi "Intorno ad una superficie anulare..." 1844 (Architecture - "About an annular surface..." in which he explicitely discuss toric surface, with the usual $\mathbb R^3$ equations). but also

Francesco Paolo Tucci Elementi di Calcolo Differenziale e Integrale 1850 which is a textbook for students of a Military school (and where you see the expression "superficie torica o toro" together with its equations).

while in V. Gambardella "Istituzioni di Architettura secondo il metodo matematico" - 1798 I've found many reference to the torus surface but no equations. This is just half an hour search on Googlebooks in Italian so I guess many more instances can be found easily.

I guess that the word easily migrated from architecture to Geometry and it will be difficult to trace back a "first usage" of it.

The ancient origin of the word is unclear. It should be related to the suffix Tor-Tur that is connected to the idea of revolving around like in "turning" or in the Italian "tornare" (which means "coming back") and appears in many Indo-European languages, so that it is supposed to be a very old lemma.

ADDED For what concerns "modern" usage it has to be remarked that Kepler in his Nova Stereometria 1615 is credited for having computed the volume of a torus. Since the text is in Latin it is plausible he is using the word torus but it would be nice to check it.

• The work by Dionysodorus mentioned above predates the "De architectura" of Vitruvio, right? Further, do you think it is possible to find out what was the exact (Greek) word that Dionysodorus used to refer to tori? – José Hdz. Stgo. Nov 10 '17 at 10:53
• Yes. But I guessed the word used by Dionysodorus is different from torus which, as a word, as no analogue in Greek (that's why I referred to the Indo-European roots). Here you can find in Greek characters the title of the work of Dyonisodouros.Just let me remember that we do not have such work, only some authors citing it. – Nicola Ciccoli Nov 10 '17 at 15:17

The OED gives an 1870 Cayley citation.