P. HISTORY OF ' AATHEMATICAL - School of Mathematics

INDIVIDUAL WRITERS 100
but exclusively with the powers z0, x, x2, . . . . , 2291 he is not quite
accurate, for besides the foregoing symbols placed on the margin of
the page, he gives on the margin also the following: "Rx. Radici;
R R. Radici de Radici; Rv. Radici vniuersale. Ouer radici legata. 0
voi dire radici vnita; R. cu. Radici cuba; $? quantita." These expressions
are uxd by Pacioli in dealing with roots as well as with
powers, except that Rv. is employed with roots only; as we have seen,
it signifies the root of a binomial or polynomial. In the foregoing two
uses of R, how did Pacioli distinguish between roots and powers? The
ordinal number, prima, secunda, terza, etc., placed after the R, always
signifies a "power," or a dignita. If a root was intended, the number
affected was written after the R; for example, R.200. for /a. In
folio 143AB Pacioli dwells more fully on the use of R in the designation
of powers and explains the multiplication of such expressions as
R. 5P via. R. 1 l? fa R. 15a, i.e., x4X x1° = x14. In this notation one looks
in vain for indications of the exponential concepts and recognition of
the simple formula am-an= am+". Pacioli's results are in accordance
with the formula am.an=am+n-l. The ordinal numbers in R lla, etc.,
exceed by unity the power they represent. This clumsy designation
made it seem necessary to Pacioli to prepare a table of products,
occupying one and one-half pages, and containing over two hundred
and sixty entries; the tables give the various combinations of factors
whose products do not exceed xZ9. While Enestrom and Rey Pastor
have pointed out that expressions like R.28~ mark powers and not
roots, they have failed to observe that Pacioli makes no use whatever
of this curious notation in the working of problems. Apparently his
aim in inserting it was encyclopedial.
137. In working examples in the second part of the Summa,
Pacioli exhibits a third use of the sign R not previously noted by
historians. There R is used to indicate powers of numbers, but in a
manner different from the notation just explained. We quote from
the Summa a passage1 in which R refers to powers as well as to roots.
Which is meant appears from the mode of phrasing: 'l... 8.108. e
questo m2a con laxis ch' R.16. fa. R.1728 piglit5 el .Q. cioe recca .3. a. R.
fa .9. parti -1728 in. 9. neuien. 192. e. R. 192. . . ." (:.
and mul-
tiplying this with the axis which is /16 gives /r28. Take Q, i.e.,
raising 3 to the second power gives 9; dividing 1,728 by 9 gives 192,
and the m . . . .) Here "recca. 3. a. R. fa. 9." identifies R with
a power. In Part I, folio 186A, one reads, "quando fia recata prima. 1.
Ibid., Part 11, fol. 72 B.