# Levi's book on Leibnizian calculus

Raphael Levi learned from Leibniz at a late stage in Leibniz's career. This might be a definite advantage for understanding Leibniz. Leibniz did not elaborate some of the philosophical principles behind the calculus until quite late in his career. If he conveyed them to Levi the latter might have included some interesting explanations in this text. Some examples:

1. Does Levi comment on the nature of infinitesimals: fictions? ideal entities?

2. Does Levi mention the idea that equality is a generalized relation of equality "up to", in line with the Leibnizian "transcendental law of homogeneity" which he did not elaborate explicitly until 1710?

3. Does Levi say anything about the law of continuity along the lines of Leibniz's fairly late text "Cum prodiisset" (1701)?

Two books on Leibnizian calculus were published by his secretary Rafael Levi, see http://opac.tib.uni-hannover.de/DB=1/LNG=EN/CHARSET=iso-8859-1/CMD?ACT=SRCHA&IKT=1016&SRT=YOP&TRM=Raphael+Levi&Submit=go Does anyone know anything about those books? What's in them?

The main title of the two books seems to be:

Calculus differentialis oder Rechnung des Unendlichen des Herrn von Leibnitz

The books date from 1747 and 1776. Thus they appear to be different from the ones mentioned in Beenakker's answer in the name of Schwarzschild. I haven't been able to get a pdf to see what the introduction says. If anyone has the pdf I would appreciate it.

The 1776 entry: http://opac.tib.uni-hannover.de/DB=1/XMLPRS=N/PPN?PPN=031333311 is in manuscript. Whoever has access please let me know.

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Why do you know (or assume) that these were books by Leibniz pusblished posthumoulsly as opposed to books by Levi? From a quick look, but I could be wrong of course, I get the latter impression, and this is in line (the second entry especially) with Wikipedia (in German) mentioning that Levi published in 1747/48 logarithmic tables. (Also the first seems handwritten.) –  quid May 12 '14 at 18:36
"...so I am changing the title." Changing a lot more than the title, methinks. –  Gerry Myerson May 13 '14 at 7:43
@Gerry, I take this to mean that you regret your closing vote. No hard feelings, pal. –  katz May 13 '14 at 8:05
No, it means I regret that you changed the question in such a way that the one answer currently posted is no longer as relevant as it was when it was posted. This I find to be a somewhat rude way to treat Carlo for showing an interest in your problem. I also note for the record that you have no way of knowing who voted to close. –  Gerry Myerson May 13 '14 at 10:11
@quid, Thanks for pointing out my error. –  katz May 13 '14 at 14:40

This is what Steven and Henry Schwarzschild write here about these works of Rafael Levi, which indeed seem to be his own research, based no doubt on what he had learned from Leibniz:

In his scholarly capacity Levi published several books over the decades. They are essentially pedestrian in content and reflect his two areas of expertise - in German on commercial arithmetic, and in Hebrew on Jewish astronomical calendration. By thus putting into literary practice his commercial philosophico-mathematical training as well as his Jewish learning, Levi served in a double-barreled way the Jewish mercantile class and the religious leadership (typically combined in personal union).

The mathematical works of Rafael Levi include:

• Vorbericht vom Gebrauch der neuerfundenen Logarithmischen Wechseltafeln + Supplement (Frankfurt-Hannover 1747/1748/1749)

• Vorbericht zum Gebrauch der neuerfundenen logarithmische Wechsel-Tabellen, wodurch ein Cours aus 2. 3. oder mehr gegebene Coursen bloß durch addiren und subtrahiren kan gefunden werden (Frankfurt-Leipzig 1749).

This has been scanned and can be accessed from here.

• Rechnungsmethode (Hannover 1783, edited posthumously by Meyer Aaron)
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Thanks, but the books mentioned by Schwarzschild don't seem to be the same as those appearing in the Hanover listing. –  katz May 13 '14 at 6:59
well, "Vorbericht zum Gebrauch" is the second one in your Hannover listing. The fist one in your listing is a manuscript, not a book. –  Carlo Beenakker May 13 '14 at 7:59
What's odd is that the second book also lists "Calculus differentialis" as the main title. Is this a cataloguing mistake? –  katz May 13 '14 at 8:04
the catalogue says "Title: Calculus differentialis" and "Part 1: Vorbericht zum Gebrauch", so it seems not to be a mistake, just a more precise determination. –  Carlo Beenakker May 13 '14 at 8:36
Thanks. There seems to be nothing but tables there. Does this include both books? I was expecting to find a calculus book but did not find any mention of "Calculus differentialis" either in the title or the contents. –  katz May 13 '14 at 11:52