I read in a book of popularization of Mathematics that in 1830 Mikhail Ostrogradsky wrote, about non-Euclidean Geometry, that he did not see why anyone would care about that, since the world is obviously Euclidean. Has anyone a good source for this quotation (assuming that it is real)? The closest thing that I was able to find was this sentence, from Jeremy Gray's Worlds out of Nothing: “Russian authorities like Ostrogradsky denigrated Lobachevsky's work, which they simply did not understand.”


2 Answers 2


Ostrogradsky is well-known for his negative reaction to Lobachevsky's work, but in his signed reviews, at least, his complaints were different. Lobachevsky, in contrast to Ostrogradsky, was not a good expositor, and the paper he submitted to the Academy was obscurely written. Apparently, Ostrogradsky was only able to make sense of two integrals computed using hyperbolic geometry, in one of which he saw an error (the manuscript was generally full of typos, and some authors say that Ostrogradsky himself made a mistake). The text of Ostrogradsky's review does not seem to be available online, but Prasolov-Skopenkov and Papadopoulos quote Rosenfeld's summary of it. Here is edited Google translation:

"Pointing out that of the two definite integrals, on the calculation of which, using his new method, Mr. Lobachevsky laid claim, one is already known and easily derived using integral calculus and the other is wrong, Mr. Ostrogradsky remarks, moreover, that the work is done with such little diligence that most of it is incomprehensible. Therefore, he believes that this work of Mr. Lobachevsky did not merit the attention of the Academy."

What comes closest to "since the world is obviously Euclidean" (and I am not sure it is that close) is not Ostrogradsky's 1832 review, but a review published two years later in Bulgarin's journal The Son of the Fatherland, which many call a smear, and some associate with Ostrogradsky. It does touch on the same themes as his review (obscurity, error in the integral), but is written in an exquisitely condescending and mocking tone aimed at public humiliation. The lengthiest quotes are given in Smilga's In the Search for Beauty: Unravelling Non-Euclidean Geometry, along with detailed analysis of its rhetorical style:

"Many of our first-rate mathematicians have read it, thought about it and still do not see the point. After that I hardly need say that I, having thought over this book for some time, could think of nothing to say; in other words, I hardIy understood a single idea... It is oven difficult to understand how Mr. Lobachecvsky was able to concoct out of the simplest and clearest chapter of mathematics that we know geometry to be - how he could build such an abstruse, murky and impenetrable theory, if it were not that he himself helped us by saying that his Geometry differs from the common kind that we all studied and which, most likely, we cannot unlearn, and is only an imaginary geometry. Yes, that makes things clear indeed.

Just try to picture what a lively, yet monstrous, imagination can conjure up! Why, for instance, not try to imagine black to be white, round to be quadrangular, the sum of all side angles in a right triangle to be less than two right angles and one and the same definite integral to be equal first to $\pi/4$, then to $\infty$. Very, very possible, yet to normal reason it is meaningless.

[...] It would seem that after a few definitions, composed with the same art and the same precision as the preceding ones, the author says something about triangles, about the dependence of the angles in them upon two sides- therein lies the difference between his geometry and ours - he then proposes a new theory of parallels, which - and he admits as much - nobody is capable of proving whether it exists in nature or not; finally, this is followed by a consideration of how, in this imaginary geometry, one determines the magnitude of curved lines, of areas, of curved surfaces and volumes of solids. And all of this, I must repeat once again, is written so that nothing at all can be understood... I cannot refrain from blaming him slightly for not giving his book a proper title and compelling us to cogitate so wastelully for such a long time. Why, instead of the title The Principles of Geometry, did he not have named it, say, A Satire On Geometry, A Caricature On Geometry, or something of that nature?"

As Smilga says, "after reviews of that kind, people take to their beds, give up work altogether, or even commit suicide". Some Russian sources associate the "review" (signed SS) with Zeleny and Burachek, who taught at the same Navy college as Ostrogradsky, Burachek worked for The Son of the Fatherland. But the evidence is circumstantial, and there are alternative speculations involving the Academy's secretary Fuss, for example. Even if Ostrogradsky was the inspiration, apparently the matter was complicated by Lobachevsky's terse style, and another Lobachevsky (distant relative) submitting a cranky paper on the circle quadrature that Ostrogradsky also happened to review.


No, Ostrogradski did not say this in his report on Lobachevski's work. His main argument was that this is a "futile phantasy", with no applications. He did not dispute the logical consistency of the theory, but at that time the common opinion was that mathematics has to describe the so-called "real world". He criticized the absence of applications, or more precisely that there are too few of them and that they are insignificant. (I remember he mentioned that Lobachevski just "computed one new integral", and this does not justify the development of such a theory).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.