Will you suffer me to add some further remarks on the subject of the
Arabic numerals and cipher; as neither the querists nor respondents seem
to have duly appreciated the immense importance of the step taken by
introducing the use of a cipher. I would commence with observing, that
we know of no people tolerably advanced in civilisation, whose system of
notation had made such little progress, beyond that of the mere savage,
as the Romans. The rudest savages could make upright scratches on the
face of a rock, and set them in a row, to signify units; and as the
circumstance of having ten fingers has led the people of every nation to
give a distinct name to the number ten and its multiples, the savage
would have taken but a little step when he invented such a mode of
expressing tens as crossing his scratches, thus X. His ideas, however,
enlarge, and he makes three scratches, thus [C with square sides], to
express 100. Generations of such vagabonds as founded Rome pass away,
and at length some one discovers that, by using but half the figure for
X, the number 5 may be conjectured to be meant. Another calculator
follows {434} up this discovery, and by employing [C with square sides],
half the figure used for 100, he expresses 50. At length the rude man
procured a better knife, with which he was enabled to give a more
graceful form to his [C with square sides], by rounding it into C; then
two such, turned different ways, with a distinguishing cut between them,
made CD, to express a thousand; and as, by that time, the alphabet was
introduced, they recognised the similarity of the form at which they had
thus arrived to the first letter of _Mille_, and called it M, or 1000.
The half of this DC was adopted by a ready analogy for 500. With that
discovery the invention of the Romans stopped, though they had recourse
to various awkward expedients for making these forms express somewhat
higher numbers. On the other hand, the Hebrews seem to have been
provided with an alphabet as soon as they were to constitute a nation;
and they were taught to use the successive letters of that alphabet to
express the first ten numerals. In this way b and c might denote 2 and 3
just as well as those figures; and numbers might thus be expressed by
single letters to the end of the alphabet, but no further. They were
taught, however, and the Greeks learnt from them, to use the letters
which follow the ninth as indications of so many tens; and those which
follow the eighteenth as indicative of hundreds. This process was
exceedingly superior to the Roman; but at the end of the alphabet it
required supplementary signs. In this way bdecba might have expressed
245321 as concisely as our figures; but if 320 were to be taken from
this sum, the removal of the equivalent letters cb would leave bdea, or
apparently no more than 2451. The invention of a cipher at once
beautifully simplified the notation, and facilitated its indefinite
extension. It was then no longer necessary to have one character for
units and another for as many tens. The substitution of 00 for cb, so as
to write bdeooa, kept the d in its place, and therefore still indicating
40,000. It was thus that 27, 207, and 270 were made distinguishable at
once, without needing separate letters for tens and hundreds; and new
signs to express millions and their multiples became unnecessary.
I have been induced to trespass on your columns with this extended
notice of the difficulty which was never solved by either the Hebrews or
Greeks, from understanding your correspondent "T.S.D." p. 367, to say
that "the mode of obviating it would suggest itself at once." As to the
original query,--whence came the invention of the cipher, which was felt
to be so valuable as to be entitled to give its name to all the process
of arithmetic?--"T.S.D." has given the querist his best clue in sending
him to Mr. Strachey's Bija Ganita, and to Sir E. Colebrooke's Algebra of
the Hindus, from the Sanscrit of Brahmegupta. Perhaps a few sentences
may sufficiently point out where the difficulty lies. In the beginning
of the sixth century, the celebrated Boethius described the present
system as an invention of the Pythagoreans, meaning, probably, to
express some indistinct notion of its coming from the east. The figures
in MS. copies of Boethius are the same as our own for 1, 8, and 9; the
same, but inverted, for 2 and 5; and are not without vestiges of
resemblance in the remaining figures. In the ninth century we come to
the Arabian Al Sephadi, and derive some information from him; but his
figures have attracted most notice, because though nearly all of them
are different from those found in Boethius, they are the same as occur
in Planudes, a Greek monk of the fourteenth century, who says of his own
units, "These nine characters are Indian," and adds, "they have a tenth
character called [Greek: tziphra], which they express by an 0, and which
denotes the absence of any number." The date of Boethius is obviously
too early for the supposition of an Arabic origin; but it is doubted
whether the figures are of his time, as the copyists of a work in MS.
were wont to use the characters of their own age in letters, and might
do so in the case of figures also.