# Why did Euclid define “a unit” instead of “the unit”?

I know Euclid's Definition VII.1 of a unit only from English and German translations:

A unit is (that) according to which each existing (thing) is said (to be) one.
[translation by Fitzpatrick]

A unit is that by virtue of which each of the things that exist is called one.
[translation by Heath]

Eine Einheit ist etwas, das zu einer solchen bestimmt ist.
[translated by Haller: A unit is something that is destined as such. ]

But it seems so that what Euclid defines is the unit instead of a unit. At least Joyce in his commentary on Definition VII.1 says:

Definition 1 is meant to define the unit as 1.

From his commentary on Definition VII.2:

Definition 2 says that a number is a multitude of units.

one could conclude that a number was thought of as a multitude of "copies" of the unit.

Can one reconstruct how Euclid actually did think about the concept of "unit": Did he think, there is one distinguished unit, and numbers were multitudes of copies of the unit? If so: Why did he define "a unit"?

If he not thought of one distinguished unit: How did he conceive other units, what did distinguish different units for him?

Note that it's definitely not a matter of translation, because $$\mu o\nu\acute{\alpha}\varsigma$$ (monas) has no article which means is deliberately indefinite: Note, that in modern terms, "unit" is defined as an invertible element of a (unital) ring, as opposed to "unity" (as in "root of unity") or just $$1$$. Of course, the unity is a unit, but in general not every unit is the unity (only in some rings).

• – Mauro ALLEGRANZA Nov 8 '18 at 9:55
• IMO the definitions reflect the "counting" origins of number : an individual cow is a unit, while an herd of ten cows is a multitude of individuals. – Mauro ALLEGRANZA Nov 8 '18 at 9:58
• Ah, that's great! My question would not have arised when $\mu o\nu\acute{\alpha}\varsigma$ had been translated by "indiviual". But note, that in this case Joyce' comment on Definition VII.1 would not make much sense. – Hans-Peter Stricker Nov 8 '18 at 10:12
• It seems to me that a unit of production still has the same meaning. – Michael E2 Dec 29 '18 at 0:07

The problem again is that the modern framing of mathematical concepts was completely different from ancient Greek one. Today mathematical objects and entities are seen as abstract, to Greeks they were much more concrete. The abstract idea of a single unit did not even arise in their context, and hence there was nothing to make "copies" of. There is a unit for each sort of thing, "How many pairs of socks?' has a different answer from 'how many socks?'", as Stein puts it, but they are abstracted from what is already there, not copied from a single template. In modern terms, it is better to think of Greek units as set elements of a pre-given concrete set than universal abstractions that "count" them.

Even today's notion of "abstraction" itself is far more abstract (pun intended) than it was to Greeks. To Aristotle, Euclid, et al., it meant selective attention to concrete things, and even Plato's forms were forms of concrete things with causal powers. Not disembodied causally inert ghosts, like modern numbers or equivalence classes, at the service of mathematicians by the stroke of a definition. One could say that Greek units are individuals on which the mind did some work of selective attention (disregarding texture and color, for example). It is for the same reason that the idea of assigning numbers (multitudes of units) to lines or figures did not make sense either. Here are some excerpts from Stein's Eudoxos and Dedekind that discusses the differences at length:

"According to Aristotle, the objects studied by mathematics have no independent existence, but are separated in thought from the substrate in which they exist, and treated as separable - i.e., are "abstracted" by the mathematician. In particular, numerical attributives or predicates (which answer the question 'how many?') have for "substrate" multitudes with a designated unit. 'How many pairs of socks?' has a different answer from 'how many socks?'. (Cf. Metaph. XIV i 1088a5ff.: "One signifies that it is a measure of a multitude, and number that it is a measured multitude and a multitude of measures".)... There is perhaps some ambiguity in the quoted passage: the statement, "Number signifies that it is a measured multitude", might be taken either to identify numbers with finite sets, or to imply that the subjects numbers are predicated of are finite sets. Euclid's definition - "a number is a multitude composed of units" - points to the former reading (which implies, for example, that there are many two's - a particular knife and fork being one of them).

[...] And indeed one finds, in Euclid's arithmetic and geometry, that "sameness" is never predicated of numbers, lengths, areas, volumes, or angles: ratios, for example, of two areas on the one hand, two lengths on the other, are (in appropriate circumstances) said to be "the same" - but never "equal"; on the other hand, the areas of two figures are said to be "equal", but never "the same" (indeed, most often it is simply said that "the two figures are equal" - that area is the appropriate magnitude-kind is taken to be understood). This difference is quite alien to our present way of thinking about such matters: for us, to say that two distinct triangles are equal in area is to say that they have "the same area". But on the suggested reading of the Greek terminology, it would be incorrect to speak of "the area of this triangle": a triangle does not have an area, it is an area - that is, a finite surface; this area means this figure, and the two distinct triangles are two different, but equal, areas.

[...] Thus, we may say that each species of quantity (whether discrete or continuous) is distinguished in Greek mathematics by its own proper equivalence-relation, called in each case just "equality"; and that where our own practice is to proceed to the corresponding equivalence classes, regarding these as particulars (numbers, lengths, etc.), the Greeks did not, in principle, make this abstraction."

• Thanks a lot, slowly I'm delving deeper into ancient thinking - with your help. – Hans-Peter Stricker Nov 8 '18 at 21:18
• "Greek units are individuals" - that's what I believe to have understood. That's a very good slogan - so I wonder why the translation of "monas" still is "unit" instead of "individual". – Hans-Peter Stricker Nov 8 '18 at 21:24
• @HansStricker Online Etymology Dictionary:"1560s, "single number regarded as an undivided whole... Popularized in John Dee's English translation of Euclid, to express Greek monas (Dee says unity formerly was used in this sense). Meaning "single thing regarded as a member of a group" is attested from 1640s. Extended sense of "a quantity adopted as a standard of measure" is from 1738." – Conifold Nov 8 '18 at 21:53