Elementary arithmetic, with brief notices of its history... by Robert Potts.

INTRODUCTION.

21

arithmetic of integers had been known andi cultivated in Hindustan.
In the notation of integers it was well understood that each succeeding
figure placed on the left had a value ten times as great as the figure
next to it on the right, and if the converse of this had been perceived,
it would have been obvious that the figure on the right of any other
figure in the scale was one-tenth of the value of that adjacent to it on
the left. But the truth is, it had not occurred to any one to apply
this idea beyond the place of units, and thus to extend the scale to
express decimal fractions. The scale thus extended would have caused
no interruption of the law of continuity either in the ascending or
descending parts of the scale, but would have rendered the scheme of
notation complete for the expression of the smallest possible decimal
fraction as well as the greatest possible integral number.
The earliest notice of decimal fractions is found in a small tract
written by Simon Stevin, of Bruges, in Flemish, and published about
1590. He afterwards translated it into French, as he himself informs his readers. In the collection of his works published after his
death for the benefit of his widow and orphans, by his friend Albert
Girard, it will be found at the end of the treatise on Arithmetique.
IHis tract describes the advantages of this extension of the
descending scale of the denary notation, and calls decimal fractions "Nombres de disme." He designates the first, second,
third, &c., places of decimals by the numbers 1, 2, 3, &c., placed in
small circles,' reserving 0, included in the same manner for the
place of integers. These characteristic marks in his tract are
written after the figures whose places in the scale they severally
mark; thus, 8(0)9(1)3(2)7(3), will signify 8,,,, o, or 80. In
the same manner 3(4)7(5)8(6), mean, 7, or j-5. The
characteristic figures are also found written in operations both above
and below the figures they distinguish, according to convenience.
The notation of sexagesimals was continued to be used in astronomical calculations after the introduction of the Indian notation. The
mode, also, of marking degrees, minutes, seconds, &c., was retained.
It appears that Stifelius, in his "Arithmetica Integra," published in
1544, was the first who indicated minutes, seconds, &c., of the sexagesimal scale by the words minuta, prima, secunda, tertia, &c., and
employed the small figures 2, 3, 4, &c., with a circumflex to distingrard. min. 2 3 4
guishthem. Thus in page 65 he writes: " 1 1 1 1 1 instead of
1~ 1' 1t 1"11 1, which signifies 1 degree, 1 minute, 1 second, 1 third,
1 fourth. It is not improbable that Stevin, from this mode of noting
the orders of sexagesimals, was led to mark in a similar way the
orders of decimal fractions.
The dedication of Simon Stevin's tract, as translated by Mr.
Norton,2 begins with the following passage:-"Many seeing the
1 Parentheses are substituted in the text instead of the small circles employed by
Stevin, containing the figures which indicate the places of each decimal figure.
2 The tract of Stevin, in 1608, was translated by Robert Norton into English,
and published with the following title: —" Disme: The Art of Tenths; or Decimall
Arithmetike; teaching how to performe all computations whatsoever by whole
numbers without fractions, by the foure principles of common arithmcticke; namely,
addition, substraction, multiplication, and division. Invented by the excellent,mathematician, Simon Stevin. Published in English, with some additions, by
Robert Norton, gent. Imprinted at London by S. S. for Hugh Astley, and are
to be sold at his shop at Saint Magnus Corner. 1608." [4to, pp. 37.]