**
Bar-Ilan****
University's
**

**
Parashat
Va-Yakhel 5768/ March 1, 2008**

Lectures on the weekly Torah reading by the faculty of

*
*

*
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-yibbaqe ^{c}u*,
“the waters were split” (Ex. 14:21).

Medieval Jewish astronomers and translators,
first and foremost Rabbi Isaac ha-Yisraeli (14^{th} century *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

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.

[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,

[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

[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

[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