P. HISTORY OF ' AATHEMATICAL - School of Mathematics

90 A HISTORY OF MATHEMATICAL NOTATIONS
I I
sit numerus .a.e.c. quaedam coniunctio quae uocetur prima, numeri
vero .d.b.f. sit coniunctio se~unda,"~ is interpreted by him as a product,
the word wniunclio being taken to mean "product." On the other
hand, Enestrom conjectures that n u m should be numeri, and trans-
lates the passage as meaning, "Let the numbers a, el c be the first, the
numbers d, b, f the second combination." If Enestrom's interpretation
is correct, then a.e.c and d.b.f are not products. Leonardo used in
his Liber abbaci the word res for x, as well as the word radix. Thus,
he speaks, "et intellige pro re summam aliquam ignotam, quam
inuenire uis."' The following passage from the Liber abbaci contains
the words numerus (for a given number), radix for x, and census for 22:
"Prirnus enim modus est, quando census et radices equantur numero.
. . . . Verbi gratia: duo census, et decem radices equantur denariis
30,"3 i.e., 2x2+10x=30. The use of res for x is found also in a Latin
translation of al-Khowirizmi's algebra,4 due perhaps to Gerard of
Cremona, where we find, "res in rem fit census," i.e., x.x=x2. The
word radix for x as well as res, and substantia for x2, are found in
Robert of Chester's Latin translation of al-Khowbrizmi's algebra.5
Leonardo of Pisa calls 9 cubus, census census, x6 cubus cubus, or else
census census census; he says, " . . . . est, multiplicare per cubum cubi,
sicut multiplicare per censum census cen~us."~ He goes even farther
and lets x8 be census census census census. Observe that this phraseology
is based on - the additive principle x2.x2. x2. x2 = x8. Leonardo
speaks also of radix census cens~s.~
The first appearance of the abbreviation R or R for radix is in his
Pradica geumetriae (1220),8 where one finds the R meaning "square
root" in an expression "et minus R. 78125 dragme, et diminuta radice
28125 dragme." A few years later, in Leonardo's Floslg one finds
marginal notes which are abbreviations of passages in the text relating
to square root, as follows:
Op. cit., Vol. I (3d ed.), p. 132.
Ibid., Vol. I, p. 191.
a Ibid., Vol. I, p. 407.
4 Libri, Histoire des seiences mathhatiques en Ilalie, Vol. I (Park, 1838)) p. 268.
6L. C. Karpinski, op. cil., p. 68, 82.
"1 liber abbaci, Vol. I, p. 447.
IM., Vol. I, p. 448.
8 Scdti di Leonalzlo Pisano (ed. B. Boncompagni), Vol. I1 (Rome, 1862), p.
209.
9 Op. cit., Vol. 11, p. 231. For further particulars of the notations of Leone
of Pka, eee our #i 219, 220, 235, 271-73, 290, 292, 318, Vol. 11, #389.