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Related to a previous question of mine, I'm working through the first paragraph of E025, Euler's Methodus Generalis Summandi Progressiones (available for download here). A translation has already been completed by Ian Bruce (available as a PDF here), but as I learn Latin (almost 3 months in now and having a great time!) I'm translating it on my own.

The first five sentences are written below:

Proposui anno praeterito methodum innumeras progressiones summandi, quae non solum se ad series algebraicam summam habentes extendit, sed earum etiam, quae algebraice summari nequeunt summas a quadraturies curvarum pendentes exhibet. Synthetica tum usus sum methodo; generalibus enim assumtis formulis quaesivi series, quarum summae iis formulis exprimerentur. Hocque modo plurimas series generales adeptus sum, quarum summas poteram assignare. Proposita igitur quapiam progressione summanda, necesse erat eam cum illis formulis comparare, et indagare, num in aliqua earum contineatur. Potuissem autem numerum earum generalium serierum in infinitum multiplicare, et propterea saepius mihi series occurrerunt, quae etiamsi in datis generalibus non comprehenderentur, ipsa tamen methodo poterant summari.

I've worked through each sentence and need some help with some sticky points. My translation at the moment is:

I proposed in the last year a method of summing innumerable progressions, which not only extends itself to series having algebraic sums, but to those as well which are not algebraic sums, depending rather on the quadrature of curves. Thereupon I have used a synthetic method; in fact I sought series with assumed general formulas, whose sums are expressed by those general formulas. And with this method, I have obtained several more general series, whose sums I can assign. Proposed any sort of progression to be summed, therefore, it used to be necessary to compare it with those formulas, and to [determine], whether it is contained in any one of them. I would have been able to extend the number of general series infinitely, and more often for this reason series suggest themselves to me, for which even if they had not been expressed in the general given [terms], with this method they are able to be summed.

There are some differences between my translation and Mr. Bruce's, in sort of significant ways (tenses, implied meanings). His translation is below:

In the past year, I have proposed a method of summing innumerable progressions [see E20], which not only extended to the series themselves having an algebraic sum, but also to establishing sums depending on the quadrature of curves, and which cannot be summed algebraically. Thereupon, by using a synthetic method, I have indeed searched for the sum of series that can be expressed from assumed general formulas. And in this way, I have come upon many more general series to which I can assign the sum. Therefore for some proposed progression requiring to be summed, whereas [before] it was necessary for it to be compared to see whether it was contained in any of these [summation formulas]. Moreover, I have been able to increase the number of these series indefinitely, and more often therefore series have occurred to me, which even if they could not be summed from the given general terms, yet these could still be summed by this method.

I worry this actually is several questions, because of the open nature of "is this correct" questions, but if I had to point out the issues I'm having I would list them as:

  1. "..., quae non solum se ad series algebraicam summam habentes extendit, ..." Does "se" refer to the method of summing, or to the sums?

  2. "...; generalibus enim assumtis formulis quaesivi series, quarum summae iis formulis exprimerentur." I'm not sure how exactly this should be translated, the two translations differ slightly.

  3. The verbs indagare, multiplicare, occurrere, and comprehendere are used in ways I don't understand. Help with any of them would be great.

  4. Picking up on that last point, in the last sentence, comprehenderentur is used in Bruce's translation to mean "summed", but I'm not familiar enough with the verb to give it such a meaning, so I went with "expressed". What's more common/likely New Latin usage?

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  • Note: There are some typos in your version of the Latin original which I have not corrected so as not to obscure the question; but I noticed, for example: progressione summanda; necesse erat. Jan 28 at 23:58
  • Youre right, thanks, fixed! Jan 29 at 2:49
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To address your points:

  1. Se is a reflexive pronoun, because extendere is used reflexively here. As such it refers to the subject of the sentence, which is quae. This relative pronoun obviously refers to methodus. (Summam is not only not the subject of this sentence, it's the object of habentes, so miles away, logically speaking.)

  2. I would interpret Generalibus assum[p]tis formulis as an ablative absolute, yielding somewhat literally: “For, general formulas having been taken up, I sought series whose sums are expressed by those formulas” or more freely: “for, given general formulas etc.” (Note that the verb of the quarum summae part is in the subjunctive. Euler is is stressing that this is not just any old property of the series he was looking for, this property was specifically what he wanted to establish.) Bruce's translation seems wrong to me, Euler did not look for the sum, he looked for the series.

  3. indagare, L&S give me, inter alia, “to investigate, explore,” which seems appropriate (given the context, you might translate this as “to check”); multiplicare does not have to refer to the arithmetic operation and can mean “make greater in number,” or simply “increase, enlarge”; occurrere, the basic meaning is something like “to run up to someone, something,” in English I would say: “I came across.” By the way, translating saepius as the comparative “more often” seems nonsensical to me, it should in my opinion be translated as “quite often” or so, as there is nothing to compare it against.

  4. The truth is that comprehendere is really not much more than plain old prehendere with a fancy prefix that does not mean much (except perhaps “from all sides” or some such). So I would say: could not be grasped. Bruce's translation “summed from” seems rather free.

I am not sure where you are getting “it used to be necessary,” Euler is simply saying “it was necessary.” I think Bruce's translation here is confused, his “whereas” leads nowhere. In my understanding Euler describes the state of affairs given that he had obtained an inventory of series with their assigned closed formulas. But he could have done away with that, because there is a general method, and he's about to tell us how that's done.

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  • Macte decem milia punctorum cumulando!
    – Ben Kovitz
    Jan 29 at 2:25
  • Thanks for such a detailed answer! This helps a lot. One point though, I don't think 'used to be necessary' is as off-base as you think. Not that it's the best, but in general (you probably know) it's a perfectly acceptable way to translate the imperfect tense, and it's probably better without it, but I think it's clear "where I got it from". The question for me was, is Euler talking about the state of things before this new method (used to be, 'we used to have to do this...'), or the state of things with a less refined version of the same method (was, '...but now it's even better') Jan 29 at 2:39
  • That being my reasoning, I think I am leaning more towards your translation, which works a fair bit better in the paragraph. It's most likely he's talking about the way he came upon a general method, not that he's comparing with previous methods Jan 29 at 2:44
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    @Ben Kovitz gratias cupio agere suffragantibus, parentibus meis, deis denique inmortalibus, quorum ope et auxilio multo magis haec situs interretialis quam ratione hominum gubernatur ;-) Jan 29 at 8:12

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