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So, I'm a PhD student working on the history of algebraic number theory and algebraic geometry. To a great extent that involves me having to read copious amounts of text in German and French. Now I'd lying if I claimed fluency in either, but when it comes to reading mathematical texts in either language, my basic knowledge is more than sufficient. Now, with Latin, the case is thoroughly different, there I have no knowledge whatsoever, and this has caused me some problems as I try making my way through Ernst Eduard Kummer's De Numeris Complexis, seeing Google Translate really isn't that good for translations to and from Latin. For the most part, though, even when the output is a complete word salad, I am able to infer what is being conveyed from the mathematical formulae accompanying the prose. Still, I do every once in a while run into some really puzzling sentences, and it is on account of one of those that I'm appealing to you for some help today!

In the passage in question, Kummer first writes:

Quia numeri primi reales formae mλ+1 non semper tanquam producta λ-1 factorum complexorum representari possunt, multis etiam numeroram integrorum realium proprietatibus simplicibus numeri complexi carent. Pro iis generaliter non valet propositio fundamentalis ut quilibet numerus sit productum factorum complexorum simplicium, qui neglectis unitatibus complexis semper iidem sint, re enim vera nonnunquam idem numerus compositus pluribus modis diversis in factores simplices complexos diffindi potest.

Which Google Translate renders as:

Because real prime numbers of the form mλ+1 cannot always be represented as products of λ-1 complex factors, complex numbers also lack many of the properties of simple real integers. For them, the basic proposition that every number is the product of simple complex factors is not valid in general, which, disregarding of the complex units, are always the same, for in truth sometimes the same composite number can be broken down into complex simple factors in several different ways.

So far, so good. Then comes the confusing sentence. Kummer writes as follows:

Si numerus complexus per alium numerum complexum ita dividi potest, ut quotiens sit integer complexus, factores simplices divisoris non ubique cum factoribus simplicibus dividendi compensari possunt.

Which Google Translate renders as follows:

If a complex number can be divided by another complex number in such a way that every time it is a complex integer, then the prime factors of the divisor cannot everywhere be compensated with the prime factors of the dividend.

Now, mathematically this statement makes no sense for reasons that even third graders will have no problems understanding. One number can only ever be divided by another number in a single way, and the answer is always the same. It's not like there exist several different ways in which 21 may be divided by 3, and every single time, the answer is 7.

The only way that I can make the sentence make sense is to suppose that it is meant to say:

It is not always the case when one complex number divided by another complex number is a complex integer that the prime factors of the divisor are all compensated for by prime factors of the divided.

Would such a reading of the original Latin be legitimate?

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    It worth noting the context (as revealed by the text before ). If I get this right, Kummer says that the fundamental theorem of arithmetic does not hold in the complex world. That is: it is the case that the same complex number can be written as the product of different "simple factors" (which I guess it the prime equivalent in the complex?); then appears the Q's quoted part in another wording. Google Translate superficially looks okay. I think it is better to replace that every time it is with that every time that it is. i.e., however often that happens that...
    – d_e
    Mar 27 at 20:33
  • You are entirely correct, the "simple factors" in question are prime elements. I shall update the question to include the previous sentence. Are you some kind of mathematician yourself, or are you just a well-read layman? :) Mar 27 at 20:43
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    Is it possible that quotiens is a noun meaning 'quotient' here, not the adverb, so that it means 'If a complex number can be divided by another complex number in such a way that the quotient is a complex integer'? Would that make mathematical sense? Otherwise, the sentence seems to be missing a clause somewhere.
    – cnread
    Mar 27 at 21:12
  • @cnread That would actually make perfect sense. If one then translates ubique a bit more liberally to mean "always" as opposed to "everywhere", more akin to the English ubiquitous, in fact, then the sentence could be translated as "If a complex number can be divided by another complex number in such a way that the quotient is a complex integer, then it is not ubiquitous that the prime factors of the divisor can be compensated for by the prime factors of the dividend." At first glance, the sentence would look more like a blind idiot's translation, but mathematically it makes perfect sense. Mar 27 at 21:50
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    Even though Google quite reasonably translates "compensari" as "to be compensated", I think the mathematical intent is rather "to be matched up with" or "to be found among". The point of the whole passage is that the Fundamental Theorem of Arithmetic" can fail in the complex case, and this last sentence should therefore be about the uniqueness of the prime factorization (up to order and unit factors). Mar 28 at 17:41

2 Answers 2

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I understand "ut quotiens sit integer complexus" as "in such a way that the quotient is a complex integer" (presumably what we now call a Gaussian integer).


Si numerus complexus per alium numerum complexum ita dividi potest,
If a complex number can be divided by another complex number

ut quotiens sit integer complexus,
in such a way that the quotient is a complex integer

factores simplices divisoris non ubique compensari possunt
the simple factors of the divisor cannot everywhere/always be compensated

cum factoribus simplicibus dividendi
with the simple factors of the number to be divided/the dividend

All in all, I would translate the paragraph as:

If a complex number can be divided by another complex number in such a way that the quotient is a complex integer, the simple factors of the divisor cannot everywhere be compensated with the simple factors of the number to be divided.


Nota bene. I happen to be a fellow PhD student (in mathematical logic), and French is my mother tongue, so don't hesitate if you have any further questions about Latin or French translations.

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    Anserin, you're either a true gentleman or a true gentlewoman! I might have some further questions later down the line, so I'm very happy to have you around. My university webpage is here: math.hhu.de/lehrstuehle-/-personen-/-ansprechpartner/innen/… in case you want to send me a quick e-mail so I can properly credit you for the translation when it comes to writing the thesis! Mar 27 at 23:13
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    Best todays approximation to 'non ubique' in the mathematical universe is 'not in the general case', meaning, the trick often works, but one cannot prove it for all possible cases. Kummer references the logical fault in Cauchys (and others) attempts to find a proof of Fermats famous theorem by extending natural integer divisibilty arguments into the domain of complex integers, a field absolutely not understood before the 1920ties without modern algebra.
    – Roland F
    Mar 28 at 8:52
  • @RolandF I don't know whether Kummer means "in each position" (if you consider that the factors are written one above the other, for example) or "in all cases" (I would have expected non semper), which is why I've kept the English ambiguous. One probably have to look at the rest of the article to find out
    – Anserin
    Mar 28 at 9:15
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    @Anserin Kummer uses ubiquitious in the sense of his German Latin slang of mathematicians, coined by Euler and Gauss as "willkürlich, beliebig, frei wählbar" to denote a symbol as a free variable in a given set of numbers. Wiktionary yields arbitrary, random, haphazard, any, whichever as english synonyms. For the specialists see math.stackexchange.com/questions/85520/…
    – Roland F
    Mar 28 at 13:18
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I would advocate for a freer translation: I think the English word "compensate" is likely to be misleading here. If I understand correctly, the intent of the sentence is that the set of simple factors of the divisor is not necessarily a subset of the set of simple factors of the dividend (because of failure of unique factorisation). How to render this in a way that's more or less faithful to the Latin but makes sense to a modern-day reader of English?

From wiktionary: one sense of "compensate" is "to make up for; to do something in place of something else; to correct, satisfy; to reach an agreement such that the scales are literally or (metaphorically) balanced..."

My suggestion:

If a complex number can be divided by another complex number so that the quotient is a complex integer, the simple factors of the divisor can not always be equated with simple factors of the dividend.

Or perhaps "matched" or "paired" instead of "equated".

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