# When was the term “corollary” first used in proofs?

A dictionary search of the word "corollary" immediately yields the usual definition that all people involved with mathematics are used to dealing with.

However, this surely comes from the Latin "corollarium", which has arguably a completely unrelated meaning. Many Latin dictionaries define the word as follows:

1. Money paid for a garland of flowers.

How did a Latin word referring substantially to a tip become a word used for "secondary" theorems? When did this practice become common?

"Corollary" is similar to the word "bonus": a little extra (i.e. an extra proposition coming from a demonstration).

The term Euclid uses is πόρισμα "porism," which Liddell-Scott-Jones cite as akin to πορίζω in the sense of "to find (money)." For instance, after I.15:

Πόρισμα

ἐκ δὴ τούτου φανερὸν ὅτι, ἐὰν δύο εὐθεῖαι τέμνωσιν ἀλλήλας, τὰς πρὸς τῇ τομῇ γωνίας τέτρασιν ὀρθαῖς ἴσας ποιήσουσιν.

The earliest usage in English cited by the OED is Chaucer's translation of Boethius, De consolatione philosophiae, Bk. III, who proposes to call a porism a corollary. Boethius is making a theological argument, not a mathematical one, so he is using corollary as analogous to the mathematical term porism. Chaucer adds "or a mede of coroune", that is, "a reward of crown," as an explanation† of corollary:

ryȝt as þise geometriens whan þei han shewed her proposiciouns ben wont to bryngen in þinges þat þei clepen porismes or declaraciouns of forseide þinges. ryȝt so wil I ȝeue þe here as a corolarie or a mede of coroune....

Super haec, iniquit, veluti geometrae solent demonstratis propositis aliquid inferre, quae πορίσματα ispi vocant, ita ego quoque tibi veluti corollarium dabo.

LSJ also cites Pappus as using porism in the sense "a kind of proposition intermediate between a theorem and a problem."

Perhaps "meed of crown" was a phrase in the Middle Ages. I'm not familiar with it. Meed can mean reward or bribe or share (of money, honor, etc.). Possibly "crown" refers to king, lord, and by extension God in this case.

• Amazing answer @Michael E2! Thank you very much, extremely interesting. May I ask you how, as a point of clarification and for future reference, happened to find this? – Easymode44 Jan 14 '19 at 9:25
• @Easymode44 I've taught Euclid on and off for over twenty years, so I was aware that the idea of a corollary/porism is quite old. Euclid probably didn't invent the term; however, I have not run across an earlier use. In fact, I learned Greek to read Euclid among other authors. In reading Euclid, I had an interest in origins of technical terms in mathematics (and in etymology generally, since high school -- I got a compact OED as a graduation present). I had obtained a copy of LSJ when I learned Greek. Both the OED and LSJ cite usages, so I checked facts with them, which are online now. – Michael E2 Jan 14 '19 at 13:41

This is a complement to Michael's excellent answer. The source of Boethius and subsequent use is most likely Proclus's commentary on Euclid's Elements, where he organizes and names various concepts concerning demonstrations (given, lemma, case, reduction, enunciation, exposition, construction, proof, reductio, etc.), apparently extracting some from the Euclidean tradition and/or inventing his own. In particular, he explicitly distinguishes the two different senses of "porism":

"The term porism is used also of certain problems such as the Porisms written by Euclid. But it is specially used when from what has been demonstrated some other theorem is revealed at the same time without our propounding it, which theorem has on this very account been called a porism (corollary) as being a sort of incidental gain arising from the scientific demonstrations." (quoted by Heath in The Thirteen Books of the Elements, p.134)

Heath adds that, according to Tannery,

"Far from distinguishing his corollaries from the conclusions of his propositions, Euclid inserts them before the closing words "(being) what it was required to do" or "to prove." In fact the porism-corollary is with Euclid rather a modified form of the regular conclusion than a separate proposition."

Euclid's Porisms are not extant. Heath traces the "intermediate between problem and theorem" sense to Pappus's account of them (p.429).

• Thank you for your input @Conifold. Indeed this seems to give a more intuitive explanation of the logical passages in adopting this so omnipresent nomenclature. – Easymode44 Jan 14 '19 at 9:28