The Tzemach Tzedek’s Maamer on √2
Translation and comments by Prof. Shimon Silman, RYAL Institute and Touro College
Throughout history Hashem has done—and continues to do—many wondrous things. Indeed, as we say every day in Shmoneh Esrei, “…and for Your wonders and acts of goodness that You do all the time.” But what if there was something hiding in plain view right before your eyes, something as simple as the diagonal of a square, in which you could see, after a moments reflection, Hashem’s wonders in creation—the infinite within the finite? What would that do for you? Well, in the Tzemach Tzedek’s own words, it would “give you an intellectual understanding of the wonders of the Creator.”
The Tzemach Tzedek wrote this Maamer around the year 1815 when he was about 26. It was published for the first time 4 years ago in 5772 and is translated here for the first time in honor of the Tzemach Tzedek’s birthday, 29 Elul, in this special Hakhel year which is also the year of the Tzemach Tzedek’s 150th yahrtzait.
THE MAAMER – להשכילך בינה נפלאות הבורא
To give you an intellectual understanding of the wonders of the Creator, Who set up the earth and founded it on wisdom (chochma), which is higher than comprehensible intellect—what have you discovered, is it not hidden? The earth and all of its physical creations are on a higher level than the spiritual heavens which He set up with understanding (tevuna)—a revelation of comprehension which is lower than wisdom. Do you not see that in the psalm of Barchi Nafshi (T’hillim 104), Dovid HaMelech speaks of the wonders of the Creator based on the lower creations [on the earth] such as the high mountains… the great sea… small animals… It was regarding these things that he said, “How manifold are your works, oh G-d!” He didn’t mention the wonders of the angels, the supernatural intellectual beings and the infinitely many spiritual worlds—which were not hidden from him; the paths of the heavens were clear to him.
It is only because of the fact that in the physical world Hashem’s infinite powers are more evident by His doing wonders in the lowest things. For this a greater power is required than for what He does in the higher worlds. This is why Hashem says [in foretelling the killing of the first born Egyptians], “I will do it, and not an angel” when He was going down to Egypt. The basis of this concept is explained elsewhere at length.
Now we will see the powerful amazing phenomenon which shows the wonders of the Creator even in the lowest aspects of His wisdom, a practical matter—measurement—which is not even termed “wisdom” but rather just a skill. It is well known from the books of the mathematicians that from any length one can construct a square. There are certain values of the sides of a square that we cannot know even though we know with a certainty that these values exist. The principal example of this is [the square root of] the number 50. We know with strong proofs that it can be verified visually how an actual square can be constructed with a side of exactly this value.
(For example, if you subdivide a 10 x 10 square into four squares which are each 5 x 5 and construct a square from the diagonals of each of these squares, the area of this new square will be 50 [half the area of the original square since these diagonals split each of the 4 smaller squares in half] as explained in Tosafos in Sukka 8a, where it is explained that the Gemara’s statement that “a 1 x 1 square has a diagonal of 1.4” is not exact. [See the diagram.] The truth is that the reason it’s not exact is not because the baalei Tosafos weren’t precise enough with their calculations but rather because it’s impossible to be exact, as we will explain.)
Nevertheless, it is impossible to know the value of the length [of the side of this inscribed square (see diagram)] under any circumstances. All the scholars who ever lived, from the time the earth was created until now, have already written that the impossibility of knowing this value is not because of a deficiency in our knowledge but rather because it is in the nature of this value [that it is unknowable]. See Tosafos Yom Tov, Chapter 5 of Kilayim and Rambam [Commentary on the Mishneh], Chapter 2 of Eruvin, Mishneh 5.
But these scholars never investigated how it could be that this value has the nature of being unknowable. Is it because it’s such a big number—several thousand adatzilions or is it in the heavens above, higher than the stars or far across the sea that it should be so far removed from us. Even though the philosophers of the nations have applied their human intellect—each one according to the level of his intellect—to investigate the heavens above and the earth below, they have come up with essentially nothing.
Yet they are not ashamed when they see that their intellect is inadequate to determine even such a simple calculation as this which is very close to us [it’s right in front of us] and its value is a little more than 7, but the exact value is impossible to know.
So what can we say about the nature of this value? It is something that can be measured—earth, wood and stone “which have no nature.” Rather the problem is with the nature of human intellect which is not complete and cannot know everything. G-d created a tangible physical object which is measurable before your eyes but you can’t get it exactly right.
If all the mathematicians in the world would get together and sit on this problem for 70 years—even if they would live 1,000 years—[they wouldn’t be able to solve it]. The essence of the impossibility of solving it is this: It is well known with proofs to those who enter the gates of this discipline [mathematics] that it is beyond all human intellect, as you can see in the Rambam referred to above.
What we have from this is the following reality: We see with our eyes a line of a certain length (the side of the inscribed square mentioned above—or the diagonal of any square) in existence right before our eyes, a physical entity like all other physical entities, whose [exact] measure is beyond human intellect. Contemplation of this amazes the eye of one who beholds it and confounds the ear of one who hears it. What is this and how could this be?
So one should make the following Kal V’chomer (an a fortiori logical argument) for himself:
If, regarding a physical object—an entity which is lower than intellect, a constructed square made out of whatever material, earth or paper or wood etc., that one can hold in his hand and make this shape himself—it has a measureable quantity right before you that is higher than your intellect, it is necessarily the case that, for a reason known to its Maker, it was made in a manner that is higher than your intellect…
Then certainly regarding things that stand above the world—the heavens and all that they contain, the movement of the celestial spheres and their creation from nothingness—there is something in them that is higher than intellect, and our intellect cannot comprehend it because it is limited, and every finite entity created from nothing [the celestial spheres] necessarily has in it the power of the One Who made it from nothingness—an unlimited power. It is because of this infinite power that it is beyond our comprehension.
Indeed, it is the case with everything that its essence and its coming into being is beyond our comprehension. Its presence and physical measurements, however, can be measured and are clear to our intellect because it has entered the physical world in a finite object.
But to have a situation where the infinite appears in the finite and as a result of this the finite becomes unknown—this is like the space [occupied by] the Ark which does not take up space even though, in fact, it was 2½ amos. This was because of the revelation of the Infinite from the [limitless] world of Atzilus which is higher than the limited physical world.
The same was true of the grave of Moshe Rabbeinu: To those standing above it appeared that it was located below; to those standing below it appeared that it was above. Similar to this is what the Midrash says about the lower level of Gan Eden: “It touches and it doesn’t touch.” There are other similar things explained elsewhere.
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MATHEMATICAL COMMENTS
The whole story starts with the statement in the Gemara Sukka that כל אמתא ברבוע אמתא ותרי חומשי באלכסונה which means that if you have a 1 x 1 square, the diagonal will be 1.4. But, as Tosafos explains, this value is not precise; it is only an approximation.
What we will attempt to do here is:
• Explain the proof that Tosafos gives;
• Give a brief mathematical analysis of the 1 x 1 triangle;
• Explain why the Gemara gives only an approximate value.
The proof in the Tosafos is based on the 10 x 10 square which the Tzemach Tzedek discusses, as in the diagram. This diagram is found in Tosafos itself. The area of this square is clearly 100 square units. Now, this square can be subdivided into 4 squares, each 5 x 5. If you draw the line AB it will split the 5 x 5 square in the upper left exactly in half. Similarly, the lines BC, CD and DA will split the remaining 3 squares in half. This produces a new square, ABCD, inscribed inside the original square. Since it split each of the 5 x 5 squares in half, its area is half of the area of the original square; its area then is 50 square units.
But if we use the Gemara’s value of 1.4 for the diagonal of a 1 x 1 square then, proportionally, the diagonal of a 5 x 5 square will be 5 x 1.4 = 7. So AB will have a length of 7. Similarly the length of line BC will also be 7. Then the area in the new inscribed square will be 7 x 7 = 49, not 50; we will be off by 1 square unit.
(You don’t need a 10 x 10 square to do this; you can do it with a smaller square. But to get the result of being off by 1, a whole number, you need at least a 10 x 10 square. This is because 5 x 1.4 = 7, a whole number, and no smaller multiple of 1.4 gives you a whole number. I think this is why the Tzemach Tzedek says that 50 is the principal example.)
Mathematically speaking, splitting a 1 x 1 square along its diagonal gives you a 1 x 1 right triangle with 2 equal sides. We will apply the well-known formula for a right triangle, quoted by the Tosafos Yom Tov in Kilayim: If the 2 perpendicular sides of the triangle are denoted a and b, and the hypotenuse (diagonal of the square) is denoted by c, the formula is a² + b² = c². If the 2 sides a and b are equal, a = b, this becomes a² + a² = c² or 2a² = c².
Now, in our 1 x 1 square, a = 1, so the above formula becomes 2 = c² so c = √2 (square root of 2). For any other square, say our 5 x 5 square, where a = 5, the formula 2a² = c² becomes 2 x 5² = c² or 50 = c². Solving this for c gives c = √50 which simplifies to 5√2. So it always comes back to √2 and this is essentially the value that the Tzemach Tzedek is discussing here.
Now √2 means a number that you can multiply by itself to equal 2. Such a number is known in mathematics as an irrational number which, practically speaking, means that if you calculated it out in decimals, the decimals would go on forever with no pattern—so you can never know exactly what it is! This is the basis of the entire Maamer. In the case of √2 the value is 1.41421356… going on forever with no pattern. That’s why the Gemara uses the approximation of 1.4.
The mathematical proof of the fact that √2 is an irrational number (not given here) is an example of proving through intellect the existence of a level which is higher than intellect, an important concept in Chassidus.
EPILOGUE
This diagram of a square oriented as a diamond within a square twice its size is evidently of great significance in demonstrating the wonders of the Creator within nature. It is interesting to note that just a few weeks ago, as this translation was being prepared for publication, Arutz 7, the Israel National News website, reported that an archeologist with a background in mathematics succeeded in reconstructing the floor of the Har HaBayis from stones that were found there and discovered a beautiful pattern of colored stones—in the shape of our diagram! Follow this link for the full story: http://www.israelnationalnews.com/News/News.aspx/217411
So there may be a message for us here—that from the Beis HaMikdash, which radiates the light of G-dliness to the whole world, we are taught to open our eyes and see the wonders of the Creator in every aspect of creation throughout the world.
Now that we got the message, we are certainly ready for the long overdue Third Beis HaMikdash, speaking of which, it is well known from the Rebbe MH”M’s sicha “Kuntres Beis Rabbeinu Sh’B’Bavel” that the Third Beis HaMikdash first appears in “770.” Take a look at the pattern of the tiles on the floor of “770.” Do you see our נפלאות הבורא – diagram?
[For questions, comments or further explanation, please contact Prof. Silman at Moshiach.science@gmail.com]