Bar-Ilan University's Parashat Hashavua Study Center

Parashat Va-Yakhel 5768/ March 1, 2008

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. Prepared for Internet Publication by the Computer Center Staff at Bar-Ilan University. Inquiries and comments to: Dr. Isaac Gottlieb, Department of Bible, gottlii@mail.biu.ac.il

 

 

Beqa as the term for the sine function

From Rabbi Isaac ha-Yisraeli through Joseph Delmedigo to Hazon Ish*

 

Yaakov Lewinger, Eng. 

 

Tel Aviv

 

From the phrase in this week’s reading, “a beqa a head, half a shekel by the sanctuary weight” (Ex. 38:26), we see that beqa means the weight of half a shekel.

This term is derived from the root b-q-‘, which in the kal conjugation means to cut in half, [1] to break, divide, crack, or slice, as illustrated by the word va-yibbaqecu, “the waters were split” (Ex. 14:21).

Medieval Jewish astronomers and translators, first and foremost Rabbi Isaac ha-Yisraeli (14th century Toledo, Spain), translated beqa as jaib in Arabic.   This is the term that ancient Arab scientists used for the trigonometric function known today as the sine function. [2]   It was borrowed from Sanskrit mathematical terminology, but the Arabs understood it according to its meaning in their own tongue:  a crack or slit (in a garment, as for a pocket or for putting in one’s head), money-bag, lap or bosom, and more.

Choosing beqa as the Hebrew word for the sine function was most successful, because the term beqa signifies dividing in half another known quantity; in the matter at hand:   the segment which is half of the known chord BE (see sketch in note below). [3]   Thus, the term beqa is an accurate Hebrew translation both of the Sanskrit origins of the word (= half a chord or string) and of the (erroneous) Arabic translation (= a slit), as well as the mathematical meaning of the word (= fraction).

The ancient Greeks were familiar with trigonometric functions, but expressed them in a slightly different way from the usual formulation in modern trigonometry.   They drew up tables for the size of the chord BE as a function of the angle at apex A (see sketch in note 3), and called this function a chord.  Such tables can be seen in the Almagest of the famous Greek astronomer Ptolemy of Alexandria, Egypt, written circa 150 C.E.   Examining the sketch we clearly see that the beqa (BC) = half of the chord (BE).

The Arab astronomer Al-Battani, who greatly influenced Maimonides, drafted the most ancient sine table which we know of (circa 880), and called the new function by its Sanskrit name, as explained above.

Many Jewish astronomers, mathematicians and scholars drafted sine tables and called them beqa tables; to wit, Rabbi Joseph Solomon Delmedigo, [4] in Sefer Elim – Gevurot Hashem, [5] and more recently, the Hazon Ish, in his Kuntres al Kiddush ha-Hodesh. [6]   The ancient sine tables for the most part represented the sines as decimal fractions, and therefore the sine values with five places after the decimal point are multiplied in these tables by 100,000 and are written as whole numbers.   The same was done in the table drawn up by Rabbi Joseph Solomon Delmedigo, as well as the table of the Hazon Ish, who copied his table primarily from the aforementioned one. [7]

An unfortunate error entered the beqa table of the Hazon Ish:  it was copied from a carelessly written edition of Sefer Elim and about one quarter of the values in it are inaccurate. [8] Nevertheless it has been reproduced as such in several editions of the works of Hazon Ish to this day.

One wonders why the Hazon Ish did not copy the sine values from the current tables that were readily available even in his day, [9] preferring to copy them out of a three-hundred-year-old book in which the sine values were not precise in modern terms. Even the original edition of the work which was generally correct, lacked precision, all the more so the corrupted version of the book he had at hand.  However, we must admit that it was innovative of the Hazon Ish to present in a book of Jewish learning such as his a sine table and other computations in modern numerical notation, as Rabbi Joseph Solomon Delmedigo had also done before him.

Recently, Rabbi Hayyim Kanievsky printed a sine table in his work Shekel ha-Kodesh, [10] following the format used by Hazon Ish, but according to an emended edition, without the earlier errors.

                                                                                                                                         



* This article appeared in fuller form, including the tables discussed here, in Ha-Ma’ayan, Pesah, 2006.

[1] According to the Even Shoshan dictionary, beqa means, among other things, to cut in half (khatzah).   Also see under khatzah = divided into two equal parts.

[2] See the exhaustive discussion by Prof. Gad Ben-Ami Tzarfati, Munahei ha-Mathematika ba-Sifrut ha-Mada’it ha-Ivrit shel Yemei ha-Beinayim, Jerusalem 1969, section 152, pp. 107-108, and section 273, pp. 218-220.

[3]

     

[4] Rabbi Joseph Solomon Delmedigo (1591-1655), born in Crete, Greece, at the age of 15 studied mathematics and astronomy under Galilei Galileo (1564-1642) at the University of Padua, Italy.  His tombstone stands to this day in the ancient Jewish cemetery of Prague, the city where he died.

[5] First edition:   Amsterdam 1629, printed by Manasseh ben Israel.  The beqa table appears on page 191.  Incorrect values appear for sine 46, 79, and 86, in addition to some twenty mistakes in rounding, yielding a total of 23 mistakes.   From the modern standpoint, all the ancient tables share the shortcoming of not following the rules of rounding that are in use today, and therefore one might say that the errors of rounding are not really mistakes.  The name Elim was derived from the verse (Ex. 15:27):  “And they came to Elim, where there were twelve springs of water and seventy palm trees; and they encamped there beside the water.”   The book opens with twelve deep questions, like twelve springs.  They are followed by seventy more sharply focused, paradoxical questions, like the seventy palms.

[6] Rabbi Abraham Isaiah Karelitz, Vilna and Bnei Brak (1878-1953), Kuntres al Kiddush ha-Hodesh, in: Hazon Ish, Yoreh De‘ah, Hilkhot Avodah Zarah, Likkutim, par. 41, Jerusalem 1954, and, beginning wih the 1973 edition of Hazon Ish, Orah Hayyim, par. 115.  Copied, inter alia, into the anthology, Po‘al Hashem, vol. 3.

[7] Hazon Ish’s assistants copied the beqa table from Sefer Elim (referenced in the next note), or perhaps from the Amsterdam edition.

[8] Sefer Elim, Odessa edition, 1864-1868.  The beqa table appears on page 350.  The sine values of 11, 46, 79, and 86 are incorrect, in addition to some 20 mistakes in rounding the last digit that appear from the very first edition of the book; this comes to a total of some 24 mistakes out of the 92 values in the table.

[9] Hazon Ish, apparently having noticed the inaccuracies and contradictions between the old tables, wrote in his Kuntres al Kiddush ha-Hodesh, par. 17:   “… it seems that the table of beqa values was not the same among the scholars of measurements, since it is not precise; that is, those who made the tables were not precise in their computations and therefore the values in the old tables are not alike.”   One should not ascribe to Hazon Ish the idea that there is any dispute among scientists regarding the values of the sine function, as implied by the remarks of Abraham Kolonymus (Kalman) Klickstein in his critique of Hazon Ish, He‘arot Mathematiyot al Hilkhot Kiddush ha-Hodesh, New York 1978, pp. 72-73(c-d), which stem from his misunderstanding Hazon Ish.

[10] Rabbi Joseph Hayyim Kanievsky, Shekel ha-Kodesh – ve-Hu Perush u-Ve’ur al Sefer Mishneh Torah le-ha-Rambam, Z”l, Bnai Brak 1997.  The sine table appears on page 124.   The author is the nephew of Hazon Ish.