There is a well-known fact that the Vilna Ga'on studied math while in the bathroom since Torah was out of the question in an unclean place. It is also somewhat well-known that he produced the material for a book on geometry/הנדסה(?) during his time in the bathroom called איל משולש.

There is (or at least was in 1993) some speculation over the verity of these facts, as evidenced by the discussions on Mail Jewish here, here, and here, but my question specifically concerns the book and its origins.

According to the book's introduction, written by the compiler and editor Sh'mu'el ben Yosef from Loknik(?), the book is a collection of the Vilna Ga'on's notes. There is no indication that he did or did not intend them for publication.

  1. Is anything known about the man who actually published the book and what his intentions were?
  2. Was it supposed to simply exhibit the genius of the Vilna Ga'on?
  3. Was it intended as a guide to students of math?
  4. Students of math and Torah, perhaps?
  5. Is there information in it that pertains to Torah at all?
  6. Does it have value to the modern Torah- or math-learning audience?
  7. Finally, how is the title pronounced?
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    Can't claim to remember what I meant by "well-known fact", but there is an interesting insight into the Vilna Ga'on's views on the study of science in this post onthemainline: "...to the extent that one lacks in understanding the sciences, he will lack 100 measures in understanding the Torah, because the Torah and science are intertwined."
    – WAF
    Jun 14, 2011 at 2:49

2 Answers 2


Okay, I'll address part of part ("Does it have value to the modern... math-learning audience?") of question 6, and part of question 5, for now.

I've read chapter 1 only (and the main text only, not the marginal notes) so far, and it has definitions, postulates, and theorems from elementary plane geometry, lumping postulates and theorems together (i.e., not caring which are which; indeed, not dealing at all with the fact that some things are definitions and some are postulates and some are theorems). For example, the wider angle of a triangle is opposite the longer side; a quadrilateral with parallel opposite sides has equal opposite sides; etc. (The only misstatement AFAICT in that chapter, in #25, is that if an angle and two sides of one triangle equal an angle and two sides of another triangle, then the remaining angles and side of the triangles are also equal.) This is a reasonable basic-high-school-geometry text, I suppose, but not the way high school geometry is taught nowadays (where a lot of attention is paid to the difference between postulates and theorems, and to proofs). It may also be of some interest to students of the history of mathematics (I don't know), but not to mathematicians.

I'll bl"n get to the remaining chapters (and perhaps the introductions) at a later date.

A later date:

I've now read chapter 2. Most of it deals with proportions. He discusses how if you add the first to the second term in a proportion, and add the third to the fourth, the result is still a valid proportion, and other such manipulations: rather more manipulations than I would have expected (he adds the first to the second, subtracts it, adds the second to the first, subtracts it, etc., etc.), but nothing that's not covered — though some of them as exercises — in a prealgebra class (or so). The last two paragraphs are on another topic: they state, first, that if you do the same thing to both sides of an equation then the result is still a valid equation, and, second, that if a=b and b=c then a=c. Again, this is covered in prealgebra (or so). Nothing of interest to mathematicians here.

An even later date:

I've now read chapter 3. It deals with several topics, all of which seem to be therein order to build up to its two main results, the formula for the area of a circle and the Pythagorean theorem. Along the way he mentions and proves various results from high-school geometry, including that a triangle with two (or three) equal sides (or angles) has two (or three) equal angles (or sides), that corresponding angles formed by two parallel lines with a common transversal are equal, that the area of a triangle is half the base times the altitude, and the like. His two proofs of the Pythagorean theorem are standard, including Euclid's. He proves also that the circumference of a circle is more than three times its diameter, using a proof I haven't seen before, personally, but strongly suspect is well known. Finally, he proves that the area of a circle is half its circumference times its radius, using a method of proof that is by far not rigorous enough for a modern mathematician, but which approximates a standard proof from calculus: take concentric infinitesimal shells and sum their infinitesimal area. Again, there is nothing here that won't be found in many other places (except perhaps his proof that the circumference of a circle is more than three times its diameter, but, again, I doubt it).

Also, re question 5: In the first three chapters, there's no Torah content.

  • He "proves" that the area of a circle is half its circumference times its radius? That's new to me... May 29, 2011 at 5:57
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    @PeterOfTheCorn: .5 * c * r = .5 * pi * d * r = .5 * pi * 2r * r = pi * r^2
    – Isaac Moses
    May 31, 2011 at 3:03
  • @Isaac I see it now. I'm sorry I was so snappy before I even tried to think of a counterexample. May 31, 2011 at 3:05
  • @ba How do you know it isn't SAS and then it's valid in all triangles?
    – Double AA
    Dec 19, 2012 at 4:20
  • @DoubleAA I tried; it doesn't work with his language: hebrewbooks.org/pdfpager.aspx?req=20713&st=&pgnum=17 (I also tried looking at it as if he were saying it only about right triangles, but that doesn't fit into his language either)
    – b a
    Dec 19, 2012 at 4:22


this link shows the Haskamos on the Sefer איל משולש. In the haskomos it mentions how it is worthy to publish this Sefer to show that the Jews are the true Chachomim.

The name of the Sefer is Ayil Meshulash

  • How do you know it's ayil and not ayal or el?
    – msh210
    Feb 25, 2011 at 6:56
  • from reading through the aurthors introduction it seems like Ayil to me. Willing to be corrected if wrong. Feb 25, 2011 at 14:24
  • 2
    Also, the name is presumably taken from Gen. 15:9, where the word is pronounced ayil.
    – Alex
    Feb 25, 2011 at 15:37
  • 1
    The full quote is: וקראתי שם הספר הזה איל משולש להיותו מקורו ברוך מהגאון שהוא האיל בעל הקרנים מנגח עד אפסי ארץ. The image is taken from Dan. 8:3-4 where it's metaphorically referring to the Persian empire's having the strength of a ram; the author probably also had in mind Sifri, Chukas 33, where this verse is applied to R' Akiva's prowess in Torah discussion. (Though in both cases the "ram" is beaten by a stronger power...). And yes, he goes on to say that it's meant as a play on the name אליהו, and also that it's an anagram of שלי שמואל to represent his own contribution (the notes).
    – Alex
    Feb 25, 2011 at 22:08
  • 6
    Incidentally, yet another proof that the name is pronounced "ayil meshulash" is that it's transliterated that way in the Russian title (at the bottom of the title page).
    – Alex
    Feb 25, 2011 at 22:08

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