# What does “per se praeclarissima videtur” mean when talking about a difficult problem?

I am translating De numeris primis valde magnis by Leonhard Euler and I am somewhat puzzled by the following phrase on the second page: “per se praeclarissima videtur”.

Ac profecto natura numerorum primorum, cum ex iis modo tam admirabili omnes numeri componantur, per se praeclarissima videtur, et quo magis adhuc in proprietates, quibus sunt praeditae, penetrare licuit, eo magis haec doctrina digna censeri debet, cui excolendae plus operae tribuatur, quam nunc quidem plerumque fieri solet.

In the preceding paragraphs, Euler compares the search for the law of the progression of prime numbers to that of squaring the circle: they have little practical applications but the solving method itself (“methodus ipsa”) will no doubt be fruitful since great obstacles must be overcome. Therefore, the translation I would normally have chosen (“seems very clear by itself”) does not make sense.

And certainly the nature of prime numbers, since it is from them that all numbers are composed in such an admirable way, seems by itself very clear, etc.

“Illustrious” might be more appropriate but Euler already mentions a little earlier that the problem has a great reputation.

• Welcome to the site! I think this is a fine question, and I as a Latinist and a professional mathematician would be happy to see it undeleted and answered. Sep 7, 2022 at 17:02

The clause of interest, with all decorations removed, is:

Natura numerorum primorum per se praeclarissima videtur.

It is not unusual for a cum clause to be inserted in the middle of another clause. The feminine form of the adjective and the verb shows that the subject must be a singular feminine, and the most obvious candidate is natura [numerorum primorum]. I read the first two pages of the paper and found no other obvious candidate for the subject.

I think the adjective praeclarus (from clarus) is being used here in the sense "very clear", not "very famous". I would thus translate the simplified clause above as follows:

The nature of prime numbers is very clear in itself.

I am a professional mathematician but not a number theorist. The way I understand this is that the definition of a prime number is very simple. This is being contrasted with the fact that there does not appear to be any kind of recursion formula or other law of progression for computing them, making them appear somewhat random. (The seeming randomness of prime numbers can actually be made use of, but this is a digression the users of this site might not find enjoyable.) My interpretation based on the surrounding text and the topic (and what was known at the time) clearly points to Euler saying that the fine details of this simply defined concepts are very odd and worth deep study.

Your suggested translation is good. I think the issue is that Euler does not communicate very clearly (!) here that the simplicity is in the definition and the complication is in the behaviour.

• Thank you for your answer, I think you are right, the nature of prime numbers is clear, but the law they obey is obscure. It could probably have been more explicit but so be it.
– user11510
Sep 7, 2022 at 17:56
• @Victor I agree, it could have been more explicit. Euler could have written less and made the point more transparent. I think I wouldn't have arrived at my conclusion without earlier subject knowledge. Oddities like this make great questions! Sep 7, 2022 at 18:07