A History of Mathematical Notations/Volume 1/Hebrews

by Florian Cajori

HEBREWS

29. The Hebrews used their alphabet of twenty-two letters for the designation of numbers, on the decimal plan, up to 400. Figure 11 shows three forms of characters: the Samaritan, Hebrew, and Rabbinic or cursive. The Rabbinic was used by commentators of the Sacred Writings. In the Hebrew forms, at first, the hundreds from 500 to 800 were represented by juxtaposition of the sign for 400 and a second number sign. Thus, תק stood for 500, תר for 600, תש for 700, תת for 800.

30. Later the end forms of five letters of the Hebrew alphabet came to be used to represent the hundreds 500–900. The five letters representing 20, 40, 50, 80, 90, respectively, had two forms; one of

LETTRES			NOMS ET TRANSCRIPTION DES LETTRES.		VALEURS.	NOMS DE NOMBRE.
SAMARITAINES.	HÉBRAÏQUES.	RABBINIQUES.
ࠀ	א		aleph,	a	1	ekhâd.
ࠁ	ב		bet,	b	2	chenaïm.
ࠂ	ג		ghimel,	gh	3	chelochâh.
ࠃ	ד		dalet,	d	4	arbâ‛âh.
ࠄ	ה		hé,	h	5	khamichâh.
ࠅ	ו		waw,	w	6	chichâh.
ࠆ	ז		zaïn,	z	7	chib‛âh.
ࠇ	ח		khet,	kh	8	chemonâh.
ࠈ	ט		t′et′,	t′	9	tich‛âh.
ࠉ	י		iod,	i	10	‛asârâh.
ࠊ	כ		kaph,	k	20	‛esrîm.
ࠋ	ל		lamed,	l	30	chelochîm.
ࠌ	מ		mem,	m	40	arbâ‛îm.
ࠍ	נ		noun,	n	50	khamichîm.
ࠎ	ס		s′amek,	s′	60	chichîm.
ࠏ	ע		‛aïn,	‛a	70	chib‛îm.
ࠐ	פ		phé,	ph	80	chemonîm.
ࠑ	צ		tsadé,	ts	90	tich‛îm.
ࠒ	ק		qoph,	q	100	mêâh.
ࠓ	ר		rech,	r	200	mâtaïm.
ࠔ	ש		chin,	ch	300	châlôch mêôt.
ࠕ	ת		tau,	t	400	arba‛ mêôt.

Fig. 11.—Hebrew numerals. (Taken from A. P. Pihan, Exposé des signes de numération [Paris, 1860], p. 172, 173.)

the forms occurred when the letter was a terminal letter of a word. These end forms were used as follows:

ץ	ף	ן	ם	ך
900	800	700	600	500.

To represent thousands the Hebrews went back to the beginning of their alphabet and placed two dots over each letter. Thereby its value was magnified a thousand fold. Accordingly, א߳ represented 1,000. Thus any number less than a million could be represented by their system.

31. As indicated above, the Hebrews wrote from right to left. Hence, in writing numbers, the numeral of highest value appeared on the right; ה߳א meant 5,001, א߳ה meant 1,005. But 1,005 could be written also אה, where the two dots were omitted, for when א meant unity, it was always placed to the left of another numeral. Hence when appearing on the right it was interpreted as meaning 1,000. With a similar understanding for other signs, one observes here the beginning of an imperfect application in Hebrew notation of the principle of local value. By about the eighth century A.D., one finds that the signs הףמה signify 5,845, the number of verses in the laws as given in the Masora. Here the sign on the extreme right means 5,000; the next to the left is an 8 and must stand for a value less than 5,000, yet greater than the third sign representing 40. Hence the sign for 8 is taken here as 800.^[1]

 

1. G. H. F. Nesselmann, Die Algebra der Griechen (Berlin, 1842), p. 72, 494; M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol. I (3d ed.), p. 126, 127.