# The American Cyclopædia (1879)/Arithmetic

**ARITHMETIC** (Gr. ἀριθμητική, from ἀριθμεῖν, to
count), the science of the properties and
relations of numbers when expressed with figures
or relations of figures. The accepted opinion
is that we have derived this science from the
Greeks, who obtained it from the Phœnicians;
but if we consider that the Chaldeans, one of the
oldest nations, have given us the knowledge of
certain astronomical cycles or periods, of which
the determination required an advanced knowledge
of arithmetic, it is evident that its origin is
of much earlier date. The Hebrews and Greeks
used the first nine letters of their alphabet for
the numbers 1 to 9; the next nine letters for
10, 20, &c., to 90; and the others for hundreds;
while for thousands they recommenced the
alphabet and added to each letter a mark or
iota. The Romans followed a similar system,
of which our Roman numerals are a specimen.
But arithmetic did not reach its more modern
state of progress until the introduction of the
Arabic figures now used by all civilized nations.
The Arabs admit that they obtained these
figures from Hindostan in the 10th century. They
call them Indian figures, and arithmetic the
Indian science. Boëthius, in his work *De Geometria*,
informs us that the disciples of Pythagoras
used in their calculations nine peculiar figures,
while others used the letters of the alphabet;
and it is probable that this philosopher, who
had travelled considerably, had obtained this
knowledge in Hindostan, and communicating it
as a secret to his disciples, caused it to remain
sterile in their hands. The Greeks in the ordinary
way of writing expressed the fractions thus:
while β, γ, δ, &c., stood for 2, 3, 4, &c., β', γ', δ',
represented ½, ⅓, ¼, &c. The oldest text book
on arithmetic employing the Arabian or Indian
figures, and the decimal system, is undoubtedly
that of Avicenna, the Arabian physician, who
lived in Bokhara about A.D. 1000; it was
found in manuscript in the library at Cairo,
Egypt and contains, besides the rules for addition,
subtraction, multiplication, and division,
many peculiar properties of numbers. (For a
translation of a portion of this remarkable
manuscript by Marcel, see De Montfévrier,
*Dictionnaire des sciences mathématiques*, vol. i., p.
141 *et seq.*) It was not till the beginning of the
13th century that the science of arithmetic
began to be diffused in Europe. One of the earliest
writers on the subject was John Halifax,
better known as Sacro-Bosco, who in the
13th century composed an arithmetic in Latin
rhymes, in which the shapes of the figures are
nearly identical with those of the present day.
The monk Planudes, who flourished in the
early part of the 14th century, wrote a book
entitled “Indian Arithmetic, or the Manner of
Reckoning after the Indian Style,” of which
several manuscripts still exist. Contemporary
with him was Jordanus of Namur, author of
the *Algorithmus Demonstratus*, and also of a
treatise on arithmetic which Jacques Faber
published with commentaries immediately after
the invention of printing. A great development
of the science now took place. In the
16th century Clavius and Stifelius (Stiefel) in
Germany and Digges in England were conspicuous
for their services to this science, and
the Arabian or Indian figures came into
use among the learned; but it was not till
the 17th century that arithmetic began to
be a regular branch of common education.
—The value of our system of arithmetical
notation, as is well known, consists in the
adoption of a scale and of a system by which
the place of the figure in the order in which it
appears causes its value to increase in multiples
of that scale. The universally adopted scale is
the decimal, probably derived from the number
of fingers of the human hand, but other scales
might have been adopted as well; and the
advantages which some persons suppose might
have been derived from the adoption of a
different scale, as the duodecimal or twelve, the
tonal or sixteen, &c., are more apparent than
real. A smaller scale would, however, have
simplified arithmetical operations, as was forcibly
demonstrated by Leibnitz, who showed how
with the smallest possible scale, the binary,
and the consequent use of only two figures, 1
and 0, operations were so much simplified that
there might be even a saving of time in
reducing a decimal expression into a binary one,
performing the operation, and restoring it back
again into the decimal system. The regular
series of numbers, one, two, three, four, five,
six, seven, eight, nine, &c., is expressed in the
binary system thus: 1, 10, 11, 100, 101, 110,
111, 1000, 1001, &c.; in the ternary system, in
which three is adopted as the basis, it is 1, 2,
10, 11, 12, 20, 21, 22, 100, &c.—When
arithmetic goes beyond the practical calculations by
numbers, and treats of the properties of
numbers in general, it enters the field of algebra.
The properties of numbers are of two kinds:
some are general and inherent in the numbers
themselves, while others depend on the decimal
system adopted. Thus the law that the
sum of two numbers multiplied by their
difference is equal to the difference of their squares
is a general property; while the fact that if
the sum of the figures is divisible by 9, the
whole number is divisible by 9, is a property
depending on the adoption of the decimal system;
if we had adopted the duodecimal system,
11 would have that property.—Besides ordinary
arithmetic, we may distinguish a palpable
arithmetic performed by the sense of feeling
by the blind; an instrumental arithmetic, where
the solutions are obtained by peculiarly
contrived instruments; a tabular arithmetic, where
problems are solved by means of tables
computed for the purpose, &c.—Pestalozzi, the
great German pedagogue, applied his method
to instruction in arithmetic with the most
eminent success. It was introduced into the
United States by Warren Colburn of
Massachusetts, by the publication of treatises on
this subject which have largely influenced the
authors of arithmetical text books, a great variety
of excellent practical works having since
been published, to which we refer for further
information in regard to the practical details of
this science.—For many curious facts on the
properties of numbers, see Gauss, *Disquisitions*
*Arithmeticæ*, or Legendre, *Théorie des nombres*.