The plans for the publication of Euler’s *Opera Omnia* began in 1910. The initial vision was for three series of volumes: series I would collect Euler’s published work on mathematics, series II the papers and books on mechanics and astronomy, and series III the remaining scientific work. These are now almost complete (the two last volumes in series II are due in the next couple of years). In 1967 a fourth series was added to the project, comprising two sub-series: IVA would publish the surviving correspondence and IVB would contain still-unpublished material. From what I understand, the plans for IVB have changed, with online publication being currently envisioned, but IVA is still going strong, as the publication of this set (IVA, volume 4, parts 1 and 2) makes clear.

Christian Goldbach was one of Euler’s most important correspondents, on at least two levels. First, he seemed to be the only person in Euler’s scientific circle to have a strong interest in number theory. While Goldbach was not in Euler’s league as a mathematician, he was an active and interested interlocutor, someone who asked good questions and knew how to applaud when applause was called for. Second, Goldbach and Euler became personal friends, so the correspondence contains lots of evidence of the two men’s personalities and life experiences.

The correspondence was initiated by Euler, still very young and having just arrived in St. Petersburg. Goldbach, though living in Moscow, was the Permanent Secretary of the St. Petersburg Academy. Euler knew Goldbach was interested in mathematics and had, in fact, just published a short mathematical paper. That gave Euler the chance to open a conversation, to show off how good he was, and to make a powerful friend.

It was Goldbach, in his very first letter to Euler (number 2 in this edition; Latin on page 104, English on page 590) who asked

Do you know about Fermat’s remark that all numbers of the form \(2^{2^{x-1}}+1\) (i.e., \(3\), \(5\), \(17\), and so on) are prime? He admitted however that he could not prove this, and as far as I know, nobody has proved it subsequently.

Euler’s response was dismissive:

With regard to Fermat’s remark I have not been able to find out anything at all; moreover, I am not yet quite convinced he could legitimately infer it by induction, as, in substituting for \(x\) in the formula \(2^{2^x}\), he certainly did not even reach the number \(6\). (p. 599)

But soon after we read

… after I had sent my last letter to you, I started to ponder Fermat’s theorem more carefully, and saw that it does not rest on as slight a foundation as I had thought at first… (p. 601)

Having discovered that Fermat should be taken seriously, Euler went on to read all he could find of Fermat’s works, and embarked on what must have been one of the most daunting intellectual projects in the history of mathematics: sorting out how to prove Fermat’s results. And since Goldbach was interested, much of that work is reflected in the correspondence. One can get hints of how Euler came to decide which results were correct, which needed to be proved first, and which were harder but still accessible.

One must mention, of course, letters 51 and 52, in which “Goldbach’s Conjecture” that every even number can be written as the sum of two primes is first mentioned. The most interesting aspect of that, perhaps, is that Euler tells Goldbach that he doesn’t know how to prove it, and the topic is then dropped, perhaps because Euler had the good sense to realize that he had no tools with which to attack the problem. Indeed, the conjecture remains unproved; we now do know that every odd number is the sum of three primes, but that result was established only in 2013.

Just as remarkable, however, is the insight into Euler’s personal life. For instance, one of the St. Petersburg letters:

Geography is fatal to me. As you know, Sir, I have lost an eye working on it; and just now I nearly risked the same thing again. This morning I was sent a lot of maps to examine, and at once I felt the repeated attacks. For as this work constrains one to survey a large area at the same time, it affects the eyesight much more violently than simple reading and writing. I therefore most humbly request you, Sir, to be so good as to persuade the President by a forceful intervention that I should be graciously exempted from this chore… (pp. 670–671)

Goldbach, the older and more influential man, perhaps can “persuade the President” of the St. Petersburg Academy to spare Euler’s eyes. We also see that even Euler was convinced that “eyestrain” can cause blindness. An editorial note then tells us that “No explicit order by President Brevern releasing Euler from his taks in the department of geography has been retrieved,” so we don’t know whether Euler’s estimate of Goldbach’s influence was correct. But it was 1740 and Euler was about to leave St. Petersburg in any case.

Because both Euler and Goldbach kept the letters they received and because for most of their lives they lived in different places, we have an almost complete record of their relationship. The only exception is the period between 1732 and 1741, when both men were living in St. Petersburg. The richest part of the correspondence began when Euler moved to Berlin in 1741. At this point there was no longer any professional reason to remain in touch: Euler was no longer in St. Petersburg, and Goldbach soon left the Academy to work for the Russian government. Nevertheless, the two friends continued to correspond and to discuss mathematics.

The new edition of the correspondence is, quite simply, excellent. The material is made even more valuable by the way it is presented here. Volume IVA/4 has been published in two parts. The first contains a very good introduction, followed by a careful transcription of all the surviving letters. (The transcription even takes care to distinguish between the two different kinds of script used for German and for Latin.) The second part contains an English translation of the letters, all the editors’ notes (also in English), and several excellent indices.

The translation is a welcome concession to modern frailty, particularly when one remembers that the other volumes of the *Opera Omnia* reprint only the original text. As the editors note, however, the strange mixture of German and Latin that Euler and Goldbach use would be tough going for most modern readers. Wisely, the transcriptions and the translations are put into separate volumes, so that the reader can have both open at one time.

I could go on much longer about the treasures to be found here. Let’s just say that this set is worth both the cost and the time it will take to read. To everyone involved: Bravo!

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He is particularly interested in the history of number theory.