Maths in Torah

(23) Then he made the tank of cast metal, 10 cubits across from brim to brim, completely round; it was 5 cubits high, and it measured 30 cubits in circumference.

(5) If [the beam] is [made of] stubble or reeds, we view it, etc.: You should know that the ratio of the diameter of a circle to its circumference is not known and it is always impossible to speak about it in truth. And this lack of comprehension is not from us like the thought of the group called Gahlia, but (rather) it is in the nature of this thing that it is not known. And it is not [part of] its reality that it should be comprehended but it is known by approximation and the scholars of Mathematics have already written essays about this to know the approximate ratio of the diameter to the circumference. And the approximation that is the guideline that is relied upon by the scholars of applied wisdoms is that the ratio is one to three and one seventh (3.1428571429), such that any circle that has a diameter of one ell would have in its circumference approximately three and one seventh ells. And since this will never be comprehended except via approximation, [the Jewish sages] took a larger (coarser) calculation and said, "Anything that has three handbreadths in its circumference has one handbreadth in its width (diameter)." And they relied upon this [ratio] for that which they required [such a] measurement in the Torah.

(1) Two are holding a garment. One says, "I found it," and the other says, "I found it." [If] one says, "all of it is mine" and the other says, "all of it is mine." This one shall swear that he owns no less than half of it, and this one shall swear that he owns no less than half of it, and they shall divide it [between them.] [If] one says, "all of it is mine" and the other says "half of it is mine." The one who says "all of it is mine" shall swear that he owns no less than three quarters of it; and the one who says "half of it is mine" shall swear that he owns no less than one quarter of it. He [the former] takes three quarters and he [the latter] takes one quarter.

(4) [With regard to] one who was married to three wives and he died, [and] the ketubah of this one was a maneh [a specific unit of money equal to one hundred dinar], of this [other] one was two hundred, and of this [third] one was three hundred, and there is only [property worth] one hundred, they divide it equally. If there is [property worth] two hundred, [she] of [the ketubah worth] a maneh receives fifty, while [she] of [the ketubah worth] two hundred and [she] of [the ketubah worth] three hundred each receive three gold [coins worth twenty-five dinar each]. If there is [property worth] three hundred dinar, [she] of a maneh receives fifty, [she] of two hundred [receives] a maneh, and she of three hundred [receives] six gold [coins]. And similarly, three [businessmen] who cast [their money] into [one] pot [i.e., form a partnership, albeit with differing amounts of monetary interest], whether they diminished or increased [their capital], this is how they [too] divide [the funds].

(1) If a Se'ah [specific unit of volume] of impure Terumah [produce consecrated for priestly consumption] fell into less than a hundred [Se'ah] of Chulin [non-sacred produce], or into Ma'aser Rishon [first tithe, which must be given to the Levite], or into Ma'aser Sheni [second tithe, which must be eaten in Jerusalem], or into Hekdesh [consecrated donation], whether these were impure or pure, they must all [be left to] rot. If the Se'ah [of Terumah] was pure, [the mixture] should be sold to priests at the price of Terumah, excluding the value of that Se'ah [itself]. If it fell into Ma'aser Rishon, he should declare Terumat Ma'aser [a portion of the tithes that is given to the priest]. And if it fell into Ma'aser Sheni or into Hekdesh, they must be redeemed. If the Chulin was impure, it may be eaten in small quantities, or roasted, or kneaded with fruit juice, or divided into pieces of dough so that there is not in any one place [a piece of dough] the size of an egg.

(2) If a Se'ah of impure Terumah fell into a hundred [Se'ah] of pure Chulin, Rabbi Eliezer says: [a Se'ah] must be set aside as Terumah and burned, for I say that the Se'ah that fell in is the Se'ah that was brought up. But the Sages say: it may be brought up and eaten in small quantities, or roasted, or kneaded with fruit juice, or divided into pieces of dough so that there is not in any one place [a piece of dough] the size of an egg.

(3) If a Se'ah of pure Terumah fell into a hundred [Se'ah] of impure Chulin, it may be eaten in small quantities, or roasted, or kneaded with fruit juice, or divided into pieces of dough so that there is not in any one place [a piece of dough] the size of an egg.

(4) If a Se'ah of impure Terumah fell into a hundred Se'ah of pure Terumah: Beit Shammai prohibits, but Beit Hillel permits. Beit Hillel said to Beit Shammai: since pure [Terumah] is forbidden to non-priests and impure [Terumah is forbidden] to priests, then just as pure [Terumah] may be brought up, so too may impure [Terumah] be brought up. Beit Shammai answered them: No! If lenient Chulin, which is permitted to non-priests, may allow us to bring up pure [Terumah that fell into it], does stringent Terumah, which is forbidden to non-priests, allow us to bring up impure [Terumah that fell into it]? After they agreed, Rabbi Eliezer said: it should be set aside as Terumah and burned. But the Sages say: it has disappeared on account of its being a minor [portion of the mixture].

(5) If a Se'ah of Terumah fell into a hundred [of Chulin], and he lifted it out and it fell into [Chulin] elsewhere: Rabbi Eliezer says: it renders [the mixture] Demai [produce from which it is uncertain if tithes were already taken] as though it were certainly Terumah. But the Sages say: it render Demai only according to proportion.

(6) If a Se'ah of Terumah fell into less than a hundred [of Chulin], and rendered [the mixture] Demai, and part of the Demai mixture [then] fell into another place: Rabbi Eliezer says: it renders [that place] Demai as if certain Terumah [had fallen in]. But the Sages say: the Demai mixture renders [another mixture] Demai only according to proportion. And that which is leavened [with Terumah] renders other dough leavened [as if with Terumah] only according to proportion. And drawn water disqualifies a Mikvah [ritual bath] also only according to proportion.

(7) If a Se'ah of Terumah fell into a hundred [of Chulin] and he lifted [a Se'ah] out and then another fell in, and he lifted another out and another fell in, this is permitted until the amount of Terumah exceeds that of the Chulin.

(8) If a Se'ah of Terumah fell into a hundred [of Chulin], and before he could take it out another fell in, this is forbidden. Rabbi Shimon permits it.

(9) If a Se'ah of Terumah fell into a hundred [of Chulin], and they were ground up and reduced [in bulk], just as the Chulin was reduced so too the Terumah was reduced, and it is permitted. If a Se'ah of Terumah fell into less than a hundred [of Chulin] and they were ground up and increased in bulk, just as the Chulin increased so too the Terumah increased, and it is forbidden. If it is known that the kernels of Chulin were better than those of Terumah, it is permitted. If a Se'ah of Terumah fell into less than a hundred [of Chulin], and [more] Chulin fell in afterwards, if it was unintentional it is permissible, but if intentional it is forbidden.