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I have a few Latin sentences from very old mathematical works by Leonard Euler. There is no their translation in the net. I do have their rough interpretation but need more precise and careful one. This is because the translation (in English) will be quoted in a scientific paper. All the job is very easy to translate and I hope for your help. Thanks a lot in advance.

  1. Quaeris porro, Vir Amplissime, cum quantitates imaginariae neque nihil sint, neque majores, neque minores nihilo, quid turn sint? Ego respondeo, quod ideo sint imaginariae, nam si essent vel nihil, vel majores, vel minores nihilo, turn certe forent reales ideoque non imaginariae.

The case in point here is real and so-called imaginary numbers in mathematics.

  1. Si igitur eae res, quas decet numeris negativis exprimantur, additio et subtractio consueto more peracta nullis premitur difficultatibus.

This is about addition and subtraction of numbers.

  • Welcome to the site! This is a nice question. Are you sure you want to keep it deleted? May I ask what the topic of the paper is? If Euler's Latin text plays a bigger role, you might actually find a coauthor willing to help. – Joonas Ilmavirta May 17 '18 at 13:20
  • No. I've had some troubles with my account I've created right now. Thanks. Do not remove the question. The topic is mathematics. I've nowhere seen these phrases. No ideas who could help. – user2626 May 17 '18 at 13:48
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    A few questions: 1) Is it possible that it is TUM instead of TURN in both occurrences?, 2) do you know/can provide a link to the source, to get a bit more context? – Rafael May 17 '18 at 14:21
  • A picture/scan/screen capture also helps – Rafael May 17 '18 at 15:26
  • I had a meeting to attend so I had to disappear. As @Rafael points out, the original source (in picture, not just machine-recognized text) would be helpful. Can you tell more about the topic? I am a mathematician by profession myself, and I'm curious to hear if you are going to use this in a historical, philosophical, or some other context. – Joonas Ilmavirta May 17 '18 at 15:37
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Here are my suggested translations:

  1. You ask further, esteemed colleague, if imaginary quantities are neither nothing nor bigger or smaller than zero, what can they be? I reply that they are imaginary for just that reason; for if they were zero, or bigger or smaller than zero, they would be real and therefore not imaginary.

I translated the salutation vir amplissime as "esteemed colleague". There might be something more idiomatic, but I'm not sure what would fit. Perhaps "good man" would also work and be less dependent on context.

The turn should be tum; it is easy to mistake m for rn in old printed or handwritten text. Both humans and computers make this mistake. Checking the original indeed shows that it's tum.

I translated nihil as "zero". You can also use "nothing" which is more literal, but it is clear that here the zero of the field of real numbers (and by extension of complex numbers) is meant.

  1. If then there are those things that ought to be expressed by negative numbers, addition and subtraction carried out in the usual manner is hindered by no difficulties.

A verb seems to be missing. I supplied sunt (a form of esse is typical to leave out) in the meaning "there are". It would be useful if you could show the original printed text so we could take a closer look at some details. It is also possible that some other verb was intended, but for that we should look at the original text, including that surrounding this sentence.

The subjects of premitur are additio and subtractio. I would read it as additio premitur et subtractio premitur, "addition is pressed and subtraction is pressed". This is why I chose "is" instead of "are" in the translation.

More literally it would be "pressed" rather than "hindered". I will leave it for others to judge what would make the most idiomatic English translation. Mine is pretty faithful to the Latin original.

The point of this passage is clearly that addition and subtraction work in the usual manner with complex numbers — in the presence of things expressed by (square roots of) negative numbers. Upon seeing it in context, this passage is about arithmetic operations on positive and negative real numbers. Either way, the point is: Once a number system is extended (from positive reals to all reals or from reals to complex numbers), the operations stay "the same". He does discuss complex numbers too, but in my reading the quote you picked is not about them per se.

I am a mathematician myself and I would be interested in seeing how these quotes are used. Please share when you are ready to do so!

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    Note that neither in the sentences or in the question is there an explicit link between between the 2nd sentence and complex numbers (though it could be implied, depending on how you read). – Rafael May 17 '18 at 15:53
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    @Rafael True. I assumed the passages to be related in topic. That's yet another thing that would be easier to judge with the original source available. – Joonas Ilmavirta May 17 '18 at 16:36
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    I don't think that any verb is missing from no. 2. Between additio and subtractio I would expect not et, but a word for 'or' (sive, aut etc.). 'Whether it is addition or subtraction carried out in the usual way, it is not beleaguered by any difficulty.' As for vir amplissime, any polite formula would serve, perhaps in the formality of those times, 'My dear Sir'. – Tom Cotton May 17 '18 at 17:10
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(Partial answer, pending a better/more complete one)

Not a native English speaker, so you may want to correct my English:

  1. You further ask, great man, if imaginary numbers are neither zero, nor greater nor less than zero, what are they then? I respond that this is why they are imaginary. Were they either zero or greater than or less than zero, [then] certainly they would be real and therefore not imaginary.
  • Vir Amplissime seems to have been a common addressing formula at the time.

The second one is a bit more obscure, and I can't make complete sense of it yet. The best I've got so far is:

  1. Then, if those things that have to be represented by negative numbers, addition and subtraction [are] executed by usual practice, is held without difficulty [lit.: with no difficulties]
  • Some of the words chosen have a lot of meanings, and I have tried my best to make sense of them in a mathematical context.
  • The sentence is clearly missing a verb, or maybe it is intended to be read using the same verb twice, but I can't figure it out.
  • I'm puzzled by premitur: I think the subject is addition and subtraction, and hence it should be the plural premuntur. I can't see a singular subject either, but it is common (at least, it is a common mistake) in some languages to treat a copula of singulars as singular for the purpose of cojugation.
  • Maybe the word order is tricking me and si (if) may be seen as inside the relative clause: that if those things have to be expressed in negative numbers, but it could only be valid with more context (e.g. a previous sentence giving it).
  • Maybe I'm also tricked by a feature of Latin grammar that allows long sentences that are often untranslatable as such, and I should just split it, as I did in the first example.

If I'm getting the sense right, the main idea is that addition and subtraction of negative numbers offer no difficulty.

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    I posted a separate answer. I think the missing verb is sunt, so that "those things are around", eae res sunt. The subject of premitur is additio and subtractio (ellipsis from additio premitur et subtractio premitur). The structure becomes clearer when you know the point Euler must be trying to get across. – Joonas Ilmavirta May 17 '18 at 15:41
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    Thanks a million to Rafael and Joonas Ilmavirta! Nice to hear from mathematicians. You are right. There were distortions in my machine-scaned Euler; "turn" should really be "tum". Both the problems, 1 and 2, are from the 1736 and 1735 letters of Euler to Carl Ehler. You may get the file here: link and look for latin urtexts there. The sentences under question are on pages 363 and 322 respectively. PS. Is the phrase "is pressed by" good for native English speaker in the Joonas translation? – user2626 May 18 '18 at 11:38
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    @user2626 I updated my answer after taking a look at the originals. I rephrased "is pressed by" to "is hindered by" which sounds more idiomatic to me. I'm not a native English speaker, so others will have to judge that. – Joonas Ilmavirta May 18 '18 at 18:04

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