‘Non-Integer’ Parashat Teruma 5785

The Portion of Teruma largely concerns the design and construction of the Tabernacle (Mishkan) and its utensils. Over the years, many questions have been posed regarding certain curiosities in their structure. Paraphrasing Rabbi J.B. Soloveichik[1], we may not ask questions such as, “Why does the candelabrum (Menorah) have seven lamps and not twelve?” or “Why is the Ark of the Covenant (Aron) constructed from three boxes that fit one inside the other, like a Matryoshka doll?”. We can only ask questions like, “What can I learn from the structure of the Menorah and the Aron?”. With this clarification in hand, let us address a peculiarity pertaining to the dimensions of the Aron [Shemot 25:10]: “Two and a half cubits long, a cubit and a half wide, and a cubit and a half high.” Why are its dimensions all non-integer[2] while the dimensions of the Mishkan and all of its other utensils are integers[3]?

The Baal HaTurim[4] explains that the Aron, which houses the Torah, is symbolic of a Torah learning. The lesson being taught here is that no matter how much one has learnt, he must always maintain his humility and see himself in half measures, as if he is not yet living up to his full potential. Other commentators continue down this path, asserting that one must always think that his Torah learning is incomplete. Even when one finishes the Talmud or some other benchmark, he should see his job as only “half done”. In the words of Rabbi Dr. Abraham Twerski[5], “Self-esteem comes from knowing what one has accomplished. Humility comes from knowing what one has not yet accomplished, although he is able to do so”.

In this essay, we will propose an alternate explanation, one that is more physical than metaphysical. How long is a cubit? In the Bronze Age, around the time the Mishkan was being built, the “common cubit” was the length of the forearm from the elbow to the tip of the middle finger. The “royal cubit” was slightly longer, equal to one common cubit plus the width of the palm of the Pharaoh ruling at the time. A single royal cubit master, the “Cubit Standard”, was a rod carved from a block of black granite. Surviving cubit rods are between 52.2 and 52.9 cm in length. Workers were supplied with copies of the cubit rod, made of wood or granite. At every full moon, the cubit rod copies had to be brought to the royal cubit master and compared to it. Failure to do so was punishable by death. This mechanism led to a high level of standardization and uniformity of length. The problem with this method of standardization is that it is time-dependent: Each time a new Pharaoh was appointed, the length of the royal cubit required updating. Further, during the building of the Mishkan, whose arm determined the length of the cubit? Was it Moshe’s arm? Or, perhaps, did G-d show Moshe a cubit of fire? The answer to this question is critical because even if Moshe knew the precise length of the cubit, how was it possible to perform maintenance on the Mishkan during the four hundred or so years that it was in existence? How can we one day build a Holy Temple (Beit HaMikdash) without knowing the precise length of a cubit?

Perhaps we can gain more precision if we start from the bottom. According to normative Jewish Law, one cubit is equal to six handbreadths (tefachim). If the variability in the width of the human palm is less than the variability in the human cubit, then perhaps we can standardize the length of the cubit with better precision. No such luck. Both the cubit and the handbreadth have a variability of about 7%. No gain in precision, here. The truth is that trying to standardize any measurement using nature is doomed to fail. According to urban legend, the carat used to weigh diamonds is the same weight as one carob seed because all carob seeds have the exact same weight. This is patently false. In a paper written in 2006, five European scientists show that the weight of a carob seed has a variability of 23%, about the same as the average variability of seeds from the sixty-three other species they measured. The precise weight of the carat is defined as 200 mg, the same as the weight of one aspirin.

One might assume that in modern times, when we no longer measure using arms, palms, and carob seeds, we can measure with infinite precision. Unfortunately, no. How long is 1 metre? Since 2019, the metre has been defined as the length of the path travelled by light in a vacuum during the time interval of 1/299,792,458 of a second. Fair enough, but how long is one second? One second is equal to 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. Of course. So doing the math, to measure one metre, all you have to do is measure the length of the path travelled by light in a vacuum during 9,192,631,770 / 299,792,458 radiations of a caesium-133 atom, or about 30.66 radiations. This definition is problematic because particle radiation is a binary process – either something radiates or it does not. There is no such thing as “half a sunbeam”. What, then, is 0.66 of a radiation? It is undefined.

Weight is no better. How much does a kilogram weigh? Until 2019, the definition was based on the International Prototype of the Kilogram, a platinum-iridium alloy cylinder, maintained at the International Bureau of Weights and Measures in Sèvres, near Paris. In 2019, the definition was modified such that a kilogram is now defined by the Planck constant, a fundamental constant of nature. Specifically, the kilogram is defined by setting the fixed numerical value of the Planck constant to 6.62607015×10^-34 Joule seconds (J·s), where the second and the meter are defined in terms of the speed of light and caesium radiation from the previous paragraph. Of course. But if the weight of the kilogram is based on the undefined “second”, then that, too, should be undefined. Professional timekeepers address this issue by claiming that, in essence, the definitions are based on precise constants and relationships, and the non-integer number of radiations is a consequence of these precise definitions rather than a physical measurement of radiations. When translated from “Scientist” to “Non-scientist”, this means that the whole thing is a hypothetical construct that has no physical application.

The fact that Aron, the first utensil described in the Mishkan, has dimensions that are incomplete teaches a critical lesson. We live in world that is, by its nature, messy and imprecise. But it is in this imprecise world that we are commanded to keep an infinitely precise Torah. Rabbi Shmuel Rabinowitz, the Rabbi of the Western Wall, refers to the Talmud in Tractate Bechorot [17a] that refutes the assertion that since the Torah gives precise measurements for the Mishkan and its utensils, it must therefore be possible to measure them precisely. Rabbi Rabinowitz teaches that G-d does not expect man to do things that he cannot do. If there is any instruction, commandment or prohibition directed to man, this is proof positive that the instruction is within the scope of his ability. A person must carefully consider his ability, seek advice, and look for implementable solutions. However, when a person is truly prevented from fulfilling the directive, he is exempt from it. “Even when [the Torah] gives us exact instructions regarding a particular act, if we have tried and succeeded in it only partially, our actions are desirable before G-d. There is no perfectionist expectation in Judaism that man will always be able to do things accurately”.

G-d seeks man’s heart, his will, and his willingness to make sacrifices and to invest effort. Even partial results are sufficient when they are based on a sincere desire to do the right thing. When we build the next Aron, speedily in our days, we will use the best available science to measure the length of a cubit. It will not be a perfect cubit – it will be the next best thing. And that will be more than good enough.

Ari Sacher, Moreshet, 5785

Please daven for a Refu’a Shelema for Shlomo ben Esther, Sheindel Devorah bat Rina, Esther Sharon bat Chana Raizel, and Meir ben Drora.

[1] Rabbi Soloveichik was the leader of Modern Orthodox Jewry in North America during the second half of the previous century.

[2] An integer is a whole number: 1,3,9… A non-integer is an number with a decimal point: 1.5, 3.14, 9.81…

[3] Other than the Table of the Showbread (Shulchan), which was 1.5 cubits high.

[4] Rabbi Jacob ben Asher, also known as “Baal HaTurim” and “Yaakov ben HaRosh” lived in Germany and in Spain at the turn of the 14^th century.

[5] Rabbi Dr. Twerski lived in Pittsburgh until his death in 2021. He was a learned Rabbi and psychiatrist specializing in substance abuse.