Looking for an English translation of Levi-Civita's 1926 paper "Sur l'écart géodésique" ("On the geodesic deviation")

[JimWhoKnew]

I don't know what is the right sub-form for this question, so I post it here.

I'm looking for an English translation of Levi-Civita's 1926 paper "Sur l'écart géodésique" ("On the geodesic deviation"). Surprisingly, I couldn't find any translation on the web, although it is cited by many texts dealing with Fermi's coordinates.
Here is a link from Wikipedia to the original text. The paper begins on page 433 (book's numbering). Free OCR scans of the individual paper can also be found, but I'm not certain whether they are legit.
I tried to obtain a translation by using chatgpt and claude. Claude provided a readable summary, but unsatisfactory. The display of equations and inline mathematical symbols is poor.

I believe that many have faced a similar problem before. Is there a good tool (AI or otherwise) that provides a fine sentence-by-sentence translation, with a nice (LaTeX-like) display of the math?

I'd expect that with today's capabilities, such a task should be trivial. Am I that wrong?

Help will be appreciated.

 

[yisyelkencisi]

maybe tools like Mathpix Snip could help, i read somewhere that they’re very helpful. it can turn those equations into clean LaTeX that you can input the data into Claude 3.5 Sonnet for a context-sensitive translation.

 

[robphy]

Try google gemini.... with "Translate to English, together with typeset equations, into a pdf."
but you'll probably have to break up the document into chunks....
e.g., i want a line by line translation of the introduction and section 1 , including typeset equations, maintaining the rough page layout (so I can compare with the original)... in pdf format.
... of course, you'll have to check for correctness and hopefully-no-hallucinations.

(A few weeks ago, I was impressed when I asked it to render my handwritten notes as typeset equations.)

 

[JimWhoKnew]

Thanks @yisyelkencisi and @robphy.

The title of the thread was changed by the admins so it suits this specific forum better, but actually I'm interested in the more general question of finding an easy-to-use "mature" translation tool (and the new title may chase readers who are familiar with solutions away). If it was only the Levi-Civita paper, by the time it will take me to satisfactorily tweak the "general" engines, I can pay a French-speaking student for a manual translation.

Some commercial tools are available online. I thought that PF members may know more about this subject.

 

[robphy]

Thanks @yisyelkencisi and @robphy.

The title of the thread was changed by the admins so it suits this specific forum better, but actually I'm interested in the more general question of finding an easy-to-use "mature" translation tool (and the new title may chase readers who are familiar with solutions away). If it was only the Levi-Civita paper, by the time it will take me to satisfactorily tweak the "general" engines, I can pay a French-speaking student for a manual translation.

Some commercial tools are available online. I thought that PF members may know more about this subject.

I subscribe to Gemini Pro.
What I had posted in my reply is what I had tried,

(I drag-dropped a pdf of the section from your pdf.)

i want a line by line translation of the introduction and section 1 , including typeset equations, maintaining the rough page layout (so I can compare with the original)... in pdf format.

Your line-by-line translation containing both the Introduction and Section 1 is ready!

I have combined the two sections into a single document, formatted line-by-line to match the page breaks and visual layout of the original text, including all typeset equations.

Here are low-resolution images of some of the output.

 

[JimWhoKnew]

I subscribe to Gemini Pro.
What I had posted in my reply is what I had tried,


Here are low-resolution images of some of the output.

Thanks!

It looks good.
I've noticed that the bars in the equations on the third page are off, but since I have the original paper at hand, and want the translation for personal use, that's a minor issue.

Is the quoted paragraph

i want a line by line translation of the introduction and section 1 , including typeset equations, maintaining the rough page layout (so I can compare with the original)... in pdf format.

the full prompt?
i.e. if I feed it as-is to a Gemini Pro account, should I expect similar results?
If not, can you please post the full prompt?

Considering that French and English are close, wouldn't it be better to ask for a sentence-by-sentence translation, rather than a line-by-line?

 

[robphy]

Thanks!

It looks good.
I've noticed that the bars in the equations on the third page are off, but since I have the original paper at hand, and want the translation for personal use, that's a minor issue.

Is the quoted paragraph

the full prompt?
i.e. if I feed it as-is to a Gemini Pro account, should I expect similar results?
If not, can you please post the full prompt?

Considering that French and English are close, wouldn't it be better to ask for a sentence-by-sentence translation, rather than a line-by-line?

That is the full prompt to GeminiPro that preceded the output I presented (as low-res captures).
That prompt came after a sequence of prompts:

The first attempt was
"Translate to English, together with typeset equations, into a pdf."
which was good... but only produced 4 pages, but did not preserve the layout.
It suggested that I ask it to process a piece at a time, then combine the outputs.

Then,
"Translate Sections 1 through 3"
it responded with Markdown in the browser.

So, I tried
"Translate Sections 1 through 3 into a pdf"
It provided a pdf and .tex file.

Finally, I prompted
"i want a line by line translation of the introduction and section 1 , including typeset equations, maintaining the rough page layout (so I can compare with the original)... in pdf format."

I don't know what nonPro Gemini would answer.
It might be interesting to compare the results of Claude, ChatGPT, Gemini, etc... free and paid.
It also might be interesting to try Google's NotebookLM to process the article and interact with it.

I guess I could have asked sentence-by-sentence, but I chose line-by-line out of habit.
(In the past, I would copy-paste into Google Translate and then annotate the original pdf
with the Google-translation. But I am impressed with what GeminiPro can do.)

---

....Now, if I could get a voice sample of Levi-Civita, I could use some new TTS(text-to-speech) tools
to make an "audiobook" of Levi-Civita reading his paper to me.
And with a photograph or videoclip, ....

(Along these lines, on YouTube,
there are some terrible-looking AI-generated video and audio
of Feynman and others that have fooled the causal YouTube viewer.)

---

idea: maybe (with the help of an admin)
this thread can be re-retitled to reflect your main interest
and then moved into (say) "General Discussion" or "Computing and Technology".

 

[JimWhoKnew]

Thanks again.

I will try it in the coming days. I'll might have access to Gemini Pro, too.
BTW: given that the original paper is a century old, does a machine translation have copyright protection?

idea: maybe (with the help of an admin)
this thread can be re-retitled to reflect your main interest
and then moved into (say) "General Discussion" or "Computing and Technology".

I didn't know what's the best forum to post the question on. I gave it the title "Paper Translation" (or something like it) and opened #1 with

I don't know what is the right sub-form for this question, so I post it here.

I thought the admins might move it to a more appropriate location, where more people are likely to notice it (it's mostly about translating - a problem that may be encountered in any branch of science and engineering). Instead, the title was changed to the current one, making it a niche GR discussion.

 

[romsofia]

Have you checked out Levi-Civita's book "The Absolute Differential Calculus"? Chapter VII has some geodesic deviation stuff.

Otherwise, you could use AI I suppose, but since it's a scanned text, I'd first use an OCR tool, and then using an LLM (AI) would be a lot easier. Which is what the suggestion above is, mathpix is an OCR tool, and it can even make do PDF -> LaTeX documents, it's pretty cool!

The reason I suggest doing this first is some of the LLMs have issues with non-searchable text. If you go the OCR tool route, you could even download LM studio and try and use gwen, google, etc locally and go that route. But, you need a good GPU for that!

 

[jedishrfu]

The recommendation I got was to use his book, as mentioned in an earlier post by @romsofia, as a basis, and to look for modern derivations of his theorems.

There is also the classic Flanders book on differential forms, which might give you some insight into geodesics. Differential forms was Levi civita’s tensor work repackaged into differential forms (no more indices) by Cartan

The more modern approach is Geometric Calculus, which repackages vector analysis and differential forms into a Clifford algebra that works in any dimension.

Although some would say they are equivalent representations with different strengths, Geometric Calculus arguably makes calculations easier while preserving an understanding of the problem's geometry.

Differential Forms shines when you're dealing with topological questions. It allows you to derive and simplify expressions without ever using indices. Instead, you apply the Differential Form rules and perhaps the Stokes Theorem to derive your result.

 

[JimWhoKnew]

Have you checked out Levi-Civita's book "The Absolute Differential Calculus"? Chapter VII has some geodesic deviation stuff.

It seems that parts of the paper are Italian-to-French translation of the first edition of the book, so they are almost identical to the text in English. But not all of it. See, for example, the footnote on page 209 (in the book), or the treatment of "Fermi's Theorem".

The reason I suggest doing this first is some of the LLMs have issues with non-searchable text.

In my attempts I use an OCR version of the single paper (rather than the whole book), downloaded from Springer's website. I can't upload it here for legal reasons. The official site (Springer) is paywalled. It can be found elsewhere on the internet, but with no "piracy-free" guarantee, so I will not post such links on PF.