The Ten Commandments in Our Parshah
The first verse of this week’s parshah reads: “Speak to the entire congregation of Israel and tell them: ‘You shall be holy, for I, Havayah your God, am holy.’”
On the words “Speak to the entire congregation of Israel,” Rashi writes: “This teaches us that this parshah was spoken at Hakhel [a great gathering of the entire Jewish people], because most of the corpus of the Torah depends on it.” Rashi’s commentary is based on the Midrash Rabbah, which also compares the content of our parshah with the Ten Commandments. Various other commentaries (like the Ibn Ezra, Nachmanides, and the Chizkunee) bring the full correspondence between the commandments in our parshah and the Ten Commandments, though each has his own variation (apparently based on the reading of the Midrash Rabbah that they had before them).
Actually, the Ten Commandments appear twice in the Pentateuch. The first time is when the Torah narrates the receiving of the Ten Commandments at Mt. Sinai in parshat Yitro, in the Book of Exodus. The second time is in parshat Va’etchanan, in the Book of Deuteronomy, when Moses retells the story of the giving of the Ten Commandments. Since the sages tell us that our parshah also contains direct parallels to the Ten Commandments, the difference being that the commandments as they are worded in the Ten Commandments (both inparshat Yitro and parshat Va’etchanan) are considered general rules that allude to all 613 commandments, while in our parshah they are worded in a more particular manner.
Thus, we have in all 3 times that the commandments of the Ten Commandments are stated in the Torah, in parshat Yitro, in our parshah, Kedoshim, and in parshat Va’etchanan. But, as noted, the scope of the commandments in all three instances follows the order of general, particular, and general.
General, Particular, General
One of the 13 homiletic principles by which the Torah is analyzed is “When there is a general statement, then a particular statement, followed by another general statement, you should learn from the particular statement.” So in a certain sense, the manner in which the (ten) commandments appear in parshat Kedoshim, this week’s parshah, is the most important.
In addition, from a Kabbalistic perspective, the general statement followed by a particular statement and then another generalization, corresponds to the three modes of thought exhibited by the three intellectual sefirot: wisdom, understanding, and knowledge. This explains why there are variations between the two instances of the Ten Commandments in Exodus and Deuteronomy. The generalization that precedes the particularization is a generalization that is clear, but fleeting, similar to the clarity that one gets when lightning lights up a dark night. This is the hallmark of the sefirah of wisdom. The second generalization is far more steady and has greater scope (it can include more in its general principles), but it is not as enlightening as the first generalization. This is how we describe the intellectual mode of the sefirah of knowledge. The Ten Commandments in Exodus offered a rarified, intense experience of clear wisdom, which was then particularized by understanding in our parshah,parshat Kedoshim, and then re-generalized by the knowing heart1 that the Jewish people possessed on the eve of their entry into the Holy Land and restated by Moses in Deuteronomy.
Doing the Math
Now, let us look at these 3 instances of the Ten Commandments from a mathematical perspective.
The first instance, as stated is in parshat Yitro, which is the 17th parshah of the Pentateuch. 17 is the numerical value of טוב , meaning “good,” which alludes to the sefirah of wisdom, as in the verse “And God saw the light that it was good,” where light represents wisdom (as in enlightenment, and in the parable of lightning mentioned earlier).
Our parshah, parshat Kedoshim, where the second particularized restatement of the Ten Commandments appears is the 30th parshah of the Pentateuch. 30 is the numerical value of the letter lamed (ל ), described as “a tower that flies high,” one of the connotations of thesefirah of understanding. In addition, 30 is the numerical value of כי , meaning “for.” Together with טוב , this word forms the idiom כי טוב , meaning, “for it is good,” an idiom that repeats a number of times in the first account of Genesis. That together these two words form an idiom alludes to the fact that wisdom and understanding always go together.
The third instance is in parshat Va’etchanan, which is the 45th parshah. 45 is the numerical value of מאד , meaning “very.” In the story of creation, the word very is found in the verse: “And it was all very good,” about which the sages say that “good” is the angel of life, while “very” refers to the angel of death. The beginning of death, as explained in the Arizal’s teachings is in the sefirah of knowledge, the first sefirah to be shattered in the breaking of the vessels. Likewise, the beginning of life is when the sefirah of knowledge is rectified through the essential holiness of the Jewish soul, returning the verse to its original meaning “all very good,” as explained in length in Chassidic teachings.
But, now let us look at these three numbers, 17, 30, and 45 a little more carefully. As we know, the triangle of an integer n is defined as the sum of integers from 1 to n. We denote this as: (n). Now, if we sum the triangles of 17, 30, and 45, we get:
17 ┴ 30 ┴ 45 = 153 ┴ 465 ┴ 1035 = 1653
But, 1653 is itself the triangle of 57. So we have here a case where the sum of the triangle of three numbers (a, b, and c) is equal itself to a triangular number.
Now, we would like to ask, for what other numbers (if any) a, b, and c does this hold? In other words, for what integers a, b, and c does:
(a) ┴ (b) ┴ (c) = (d)
Let us note in clarification of our query that we are looking for a definite class of number triplets that are similar in nature (obey a well definable mathematical law) to the example that we discovered in our above analysis of a Torah phenomenon.
This question is a beautiful example of how the study of the properties of numbers (the essence of number theory) can be motivated by relationships found in our learning of Torah.
If we try our hand a little we will find that 11, 20, and 30 also work, because:
11 ┴ 20 ┴ 30 = 66 ┴ 210 ┴ 465 = 741 = 38
And before that, we have that this holds for 5, 10, and 15:
5 ┴ 10 ┴ 15 = 15 ┴ 55 ┴ 130 = 190 = 19
Now, let us try to generalize from these three cases for the general case.2 First, let us look at the relationship between the sums, meaning the right hand side of our equations. The first sum was 57; the second was 38; and the third 19. So our d’s are 57, 38, and 19. Of course, all three are multiples of 19. In the Torah, 19 is the value of חוה , “Eve,” the first woman. So, we can generalize that the variable on the right hand side will always be a multiple of 19!
Noting this point, we can now proceed to generalize the left hand side. The relationship between the c’s is very easy to see: they are all consecutive multiples of 15. The same is true for the b’s, which are all consecutive multiples of 10. The a’s are just a little bit trickier, but all they are is multiples of 6 less 1. So, now let us write the general rule:
For every integer k: (6k – 1) ┴ 10k ┴ 15k = 19k
Let us apply our rule for n = 7 to check it:
(6 · 7 – 1) ┴ (10 · 7) ┴ (15 · 7) = (19 · 7)
41 ┴ 70 ┴ 105 = 133
861 ┴ 2485 ┴ 5565 = 8911
So, we have formulized the correct rule.
Now, let us prove this rule. To do this we need to know that: (n) = n(n ┴ 1)/2. If we substitute the k’s in our rule with this equation we will get:
(6n – 1)(6n – 1 ┴ 1)/2 ┴ 10n(10n ┴ 1)/2 ┴ 15n(15n ┴ 1)/2 = 19n(19n ┴ 1)/2
Multiplying by 2 (to get rid of the fraction) and solving we get:
36n2 – 6n ┴ 100n2 ┴ 10n ┴ 225n2 ┴ 15n = 361n2 ┴ 19n
361n2 ┴ 19n = 361n2 ┴ 19n
And indeed both sides are equal!
Creation and Kingdom
So much for the pure number theory. Now, what about the significance of this rule. We already mentioned that 19 (the coefficient on the right side of our rule) is the numerical value of חוה , “Eve.” Indeed, all 613 commandments of the Torah, which as we noted are alluded to in the Ten Commandments, whose appearance we have been studying, are explained in Kabbalah to be geared towards בנין המלכות , “the construction of kingdom,” an idiom, whose value is also 613!
As taught by the Arizal, God began creating the world from the midpoint of His Infinite light, a metaphor for the sefirah of kingdom of the Infinite (מלכות דאין סוף ), meaning that His essential reason for creating our reality was in order that His kingdom, i.e., His sovereignty, be revealed. Within the kingdom of the Infinite, all 10 sefirot are concealed.
Mathematically, 10 is the midpoint of 19 (“Eve”), the first representative of the sefirah of kingdom in the Torah. As explained this year in parshat Noach, every even number starts an infinite series of numbers called its midpoint series. 10 is the midpoint of 19. 19 is the midpoint of 37, the numerical value of הבל , meaning either “vanity” or “vapor,” and also the name of the son of Eve, Abel. 37 is the midpoint of 73, the numerical value ofחכמה , meaning “wisdom.” As is well known, the value of the first verse of Genesis is 2701, or 37 times 73 (and the triangle of 73, 37 times 73). 73 is the midpoint of 145, which is the numerical value of the holy Name קמה , the Name associated with the appointment of a king (the digits of 145 can be permuted to 541, the value of ישראל , “Israel”). 145 is the midpoint of 289, or 172, where 17 is the gematria of טוב , or “good.” 289 is also the value of ברא א־להים , the second and third words of Genesis, which mean, “God created….” Thus, all of the numbers in the midpoint series that begins with 10 (corresponding to the 10 sefirot of kingdom of the Infinite) are intrinsically linked with creation.
Now, looking at the rule we found one more time let us discuss the significance of the coefficients on the left side of our rule: 6, 10, and 15. If we translate these three coefficients into letters we get ו (=6), י (=10), and הי (=15), which together spell the word ויהי , meaning “and there was.” The sages tell us that whenever this word appears in the Bible it signifies sorrow. However, the first time that the word appears in the Bible is in the third verse: “And God said: ‘Let there be light!’; and there was light.” Superficially, at least, it does not seem that the creation of light should be associated with sorrow.
[Incidentally, this word ויהי appears 817 times in the entire Bible. 817 is equal to the primordial value of אור , “light,” written:א אבגדהו אבגדהוזחטיכלמנסעפצקר ! 817 is also equal to 43 times 19, meaning that it is a multiple of חוה , “Eve,” the coefficient on the right hand side of our general rule!]
But, note that this word ויהי in the verse “And there was light,” is the 26th word of the Bible and 26 is the value of God’s essential Name, Havayah, which permutes to spell the word והיה (also meaning “and there was”). About the word והיה , the sages tell us that whenever this word appears in the Bible it signifies joy.
[The combined numerical value of ויהי and והיה , 31 and 26, is 57, or 3 times 19 (Eve), the triangular root of the sum of the three triangles observed above.] In other words, the first ויהי (corresponding to our coefficients) alludes to the paradoxical statement of the sages: “The reward fits the level of difficulty” (לפום צערא אגרא ). Indeed, the numerical value of this statement is 722, or 2 times 192, where 19 is once again the value of “Eve” (חוה = 19)! Indeed, it is Eve, who represents the feminine aspect of reality, is always living through cycles of sorrow and joy (difficulty and reward).
2. The astute reader will have probably noticed that our mathematical analysis followed a general-particularized-general order. The first equation of three triangles equaling a triangle that was derived directly from the order of the parashot which include the Ten Commandments provided the lightning strike of insight—our flash of wisdom. Then we particularized by finding two more examples that fit the pattern—the task of the sefirah of understanding. Finally, we generalized once more to find the general rule—and, thus gained knowledge. Later, we will rectify this knowledge by explaining the spiritual significance of the general rule.