Bar-Ilan University's Parashat Hashavua Study Center

Parashat VaYaqhel 5763/ March 1, 2003

Lectures on the weekly Torah reading by the faculty of Bar-Ilan University in Ramat Gan, Israel. A project of the Faculty of Jewish Studies, Paul and Helene Shulman Basic Jewish Studies Center, and the Office of the Campus Rabbi. Published on the Internet under the sponsorship of Bar-Ilan University's International Center for Jewish Identity.
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Parashat VaYaqhel 5763/ March 1, 2003

The Angle of the Ramp to the Altar

Boaz Zaban and David Garber
Department of Mathematics

In today's parasha we read: "He made the altar for burnt offering" (Ex. 38:1). As is well known, the altar had a ramp with which to ascend, whose details are provided by the Mishnah. Tractate Middot (3.1) describes the structure of the altar:
The altar measured thirty-two by thirty-two. It went up one cubit and in one cubit, forming the base, leaving thirty by thirty. It went up five and in one, forming the surround, leaving twenty-eight by twenty-eight.
Further on (mishnah 3) the Mishnah adds: "At the south end of the altar was a ramp, thirty-two by sixteen wide."
If so, the length of the altar was thirty-two cubits, and the length of the ramp was thirty-two cubits as well. Elsewhere in this tractate (5.1), however, it says, "the altar and the ramp totaled sixty-two cubits," seemingly a contradiction, since 32 +32 = 64, not 62! The gemara (Zevahim 62b) attempts to explain this by saying, "It turns out that it extended one cubit on the base and one cubit on the surround." In other words, there were two cubits overlap between the altar and the ramp:

Further on in the same gemara (Zevahim, loc. sit.), Rami Bar Hama says, "All ramps are three cubits [length] per cubit [height], except for the ramp of the altar, which was three and a half cubits and one and a third etzba (finger) in its principle dimension (Heb. be-zikhruta(. In other words, there was a difference between the slope of the ramp to the altar and the slope of other ramps. Rashi explains Bar Hama's words as follows:
All the large and small ramps there had a slope of three cubits per cubit height, save for the large ramp of the altar, which was ascended while carrying heavy body parts, and which was slippery and therefore had to be more gently graded and easier to ascend. Hence it was extended to a slope of 32 [cubits length] for 9 cubits [height].
Afterwards Rashi proceeds to prove Rami Bar Hama's computation:

(An etzba is a fourth of a tefah, and a tefah is a sixth of a cubit, therefore the ratio of a cubit to an etzba is 1:24). Rashi's approach has a problem in that common sense would say the thirty-two cubits are measured along the slope of the ramp (the diagonal), so that the measurement along the ground is approximately 30.7 cubits by Pythagoras' theorem):
Accordingly, the question raised by the gemara about the seeming contradiction between 62 and 64 cubits loses its poignancy, since the sum of the lengths of the altar and the ramp, measured along the ground, is 30.7 + 32 = 62.7, which is quite close to 62.

Birkat ha-Zevah (on the gemara in question) suggests a way of explaining this. The measurement of 32 cubits given in the Mishnah pertains to the base of the ramp, not its slope. Accordingly, for each cubit that one ascends, one proceeds forward three and a half cubits plus one and a third etzba (i.e. 3.555... cubits). The length of the diagonal, however, which is the actual distance that one traverses, is somewhat greater: it is the diagonal of a right-triangle with one leg being a cubit and the other leg being three and a half cubits plus one and a third etzba. The total length of the ramp's slope, therefore, would be .
A supporting argument for this explanation can be found in the word zikhruta, which appears at the end of Rami Bar Hama's comment. Responsa Me'il Zedaka (par. 28) says that the word zikhruta alludes to this matter, since the meaning of this word is "in its principle, main [dimension]" (as found also in the gemara in Tractate Bekhorot 55a), and the main dimension of the ramp is its foundation on the ground. Thus the ratio given by Rami Bar Hama would pertain to the ramp's extent over the ground and not to the length of its diagonal.
Panim Me'irot (loc. sit.) offers a different explanation: the last two cubits of the ramp were level, not at an angle (since that is where the priests used to stand as they threw the animal's parts on the altar):

According to this theory, there were thirty-two cubits not only along the ground under the ramp, but also along the diagonal ascending the altar, since the ramp was constructed of two segments: approximately 30.7 cubits horizontally measured beneath the sloping part of the ramp, so that the ramp was thirty-two cubits along the sloping part plus approximately another two cubits along its level section.
A parallel passage in the Jerusalem Talmud (Eruvin 7.2) gives a completely different measurement:
It is taught: all ramps ascend one cubit for every three cubits that they draw inward, save for the ramp of the altar, which increased 10 tefahs out of three [cubits] and a third etzba; for the altar was ten cubits, and its ramp was thirty-two.
The difficulty with the Jerusalem Talmud's approach is that the height of the altar itself was nine cubits, and only the horns of the altar reached ten cubits. Clearly the ramp had to reach the height of the altar's surface, not the height of the horns.
Korban ha-Edah, in a gloss on the Jerusalem Talmud, compares the version there with the text of the Babylonian Talmud. The difficulty in such a gloss is that the version of the Jerusalem Talmud which we have gives a different reading (see Sheyarei Korban, loc. sit.). Consequently, he suggests another solution: As we know, the Mishkan had cubits of five tefahs and cubits of six tefahs. According to his approach, the cubits of the ramp were five-tefah cubits. The total height of the altar was 53 tefahs, since it consisted of one cubit of five tefahs (the foundation) and another 8 cubits of 6 tefahs. Translating this into cubits of five tefahs yields a height of ten cubits and approximately another half cubit. For these ten cubits the ramp was 32 cubits (also 5-tefah units). Thus , so for every (5-tefah) cubit that the ramp rose, it extended approximately three cubits plus a third of an etzba in slope (therefore he emends the text of the Jerusalem Talmud to read "five tefahs " instead of "ten tefahs"). Sheyarei Korban notes that one should not make an issue of the difference between ten and a half cubits which we computed and the measurement of "ten" cubits given by the Jerusalem Talmud, since one could argue that the Jerusalem Talmud did not wish to give fractions of cubits.