The Earliest Arithmetics in English
Anonymous
Rita Head's Group
The Earliest Arithmetics in English
by AnonymousTHE SUBJECT MATTER.Ancient and medieval writers observed a distinction between the Science
and the Art of Arithmetic. The classical treatises on the subject, those
of Euclid among the Greeks and Boethius among the Latins, are devoted to
the Science of Arithmetic, but it is obvious that coeval with practical
Astronomy the Art of Calculation must have existed and have made
considerable progress. If early treatises on this art existed at all
they must, almost of necessity, have been in Greek, which was the
language of science for the Romans as long as Latin civilisation
existed. But in their absence it is safe to say that no involved
operations were or could have been carried out by means of the
alphabetic notation of the Greeks and Romans. Specimen sums have indeed
been constructed by moderns which show its possibility, but it is absurd
to think that men of science, acquainted with Egyptian methods and in
possession of the abacus,[2*] were unable to devise methods for its use.THE PRE-MEDIEVAL INSTRUMENTS USED IN CALCULATION.The following are known:--(1) A flat polished surface or tablets, strewn with sand, on which
figures were inscribed with a stylus.(2) A polished tablet divided longitudinally into nine columns (or more)
grouped in threes, with which counters were used, either plain or marked
with signs denoting the nine numerals, etc.(3) Tablets or boxes containing nine grooves or wires, in or on which
ran beads.(4) Tablets on which nine (or more) horizontal lines were marked, each
third being marked off.The only Greek counting board we have is of the fourth class and was
discovered at Salamis. It was engraved on a block of marble, and
measures 5 feet by 2½. Its chief part consists of eleven parallel lines,
the 3rd, 6th, and 9th being marked with a cross. Another section
consists of five parallel lines, and there are three rows of
arithmetical symbols. This board could only have been used with counters
(_calculi_), preferably unmarked, as in our treatise of _Accomptynge by
Counters_.CLASSICAL ROMAN METHODS OF CALCULATION.We have proof of two methods of calculation in ancient Rome, one by the
first method, in which the surface of sand was divided into columns by a
stylus or the hand. Counters (_calculi_, or _lapilli_), which were kept
in boxes (_loculi_), were used in calculation, as we learn from Horace's
schoolboys (Sat. 1. vi. 74). For the sand see Persius I. 131, "Nec qui
abaco numeros et secto in pulvere metas scit risisse," Apul. Apolog. 16
(pulvisculo), Mart. Capella, lib. vii. 3, 4, etc. Cicero says of an
expert calculator "eruditum attigisse pulverem," (de nat. Deorum,
ii. 18). Tertullian calls a teacher of arithmetic "primus numerorum
arenarius" (de Pallio, _in fine_). The counters were made of various
materials, ivory principally, "Adeo nulla uncia nobis est eboris, etc."
(Juv. XI. 131), sometimes of precious metals, "Pro calculis albis et
nigris aureos argenteosque habebat denarios" (Pet. Arb. Satyricon, 33).There are, however, still in existence four Roman counting boards of a
kind which does not appear to come into literature. A typical one is of
the third class. It consists of a number of transverse wires, broken at
the middle. On the left hand portion four beads are strung, on the right
one (or two). The left hand beads signify units, the right hand one five
units. Thus any number up to nine can be represented. This instrument is
in all essentials the same as the Swanpan or Abacus in use throughout
the Far East. The Russian stchota in use throughout Eastern Europe is
simpler still. The method of using this system is exactly the same as
that of _Accomptynge by Counters_, the right-hand five bead replacing
the counter between the lines.THE BOETHIAN ABACUS.Between classical times and the tenth century we have little or no
guidance as to the art of calculation. Boethius (fifth century), at the
end of lib. II. of his _Geometria_ gives us a figure of an abacus of the
second class with a set of counters arranged within it. It has, however,
been contended with great probability that the whole passage is a tenth
century interpolation. As no rules are given for its use, the chief
value of the figure is that it gives the signs of the nine numbers,
known as the Boethian "apices" or "notae" (from whence our word
"notation"). To these we shall return later on.
by AnonymousTHE SUBJECT MATTER.Ancient and medieval writers observed a distinction between the Science
and the Art of Arithmetic. The classical treatises on the subject, those
of Euclid among the Greeks and Boethius among the Latins, are devoted to
the Science of Arithmetic, but it is obvious that coeval with practical
Astronomy the Art of Calculation must have existed and have made
considerable progress. If early treatises on this art existed at all
they must, almost of necessity, have been in Greek, which was the
language of science for the Romans as long as Latin civilisation
existed. But in their absence it is safe to say that no involved
operations were or could have been carried out by means of the
alphabetic notation of the Greeks and Romans. Specimen sums have indeed
been constructed by moderns which show its possibility, but it is absurd
to think that men of science, acquainted with Egyptian methods and in
possession of the abacus,[2*] were unable to devise methods for its use.THE PRE-MEDIEVAL INSTRUMENTS USED IN CALCULATION.The following are known:--(1) A flat polished surface or tablets, strewn with sand, on which
figures were inscribed with a stylus.(2) A polished tablet divided longitudinally into nine columns (or more)
grouped in threes, with which counters were used, either plain or marked
with signs denoting the nine numerals, etc.(3) Tablets or boxes containing nine grooves or wires, in or on which
ran beads.(4) Tablets on which nine (or more) horizontal lines were marked, each
third being marked off.The only Greek counting board we have is of the fourth class and was
discovered at Salamis. It was engraved on a block of marble, and
measures 5 feet by 2½. Its chief part consists of eleven parallel lines,
the 3rd, 6th, and 9th being marked with a cross. Another section
consists of five parallel lines, and there are three rows of
arithmetical symbols. This board could only have been used with counters
(_calculi_), preferably unmarked, as in our treatise of _Accomptynge by
Counters_.CLASSICAL ROMAN METHODS OF CALCULATION.We have proof of two methods of calculation in ancient Rome, one by the
first method, in which the surface of sand was divided into columns by a
stylus or the hand. Counters (_calculi_, or _lapilli_), which were kept
in boxes (_loculi_), were used in calculation, as we learn from Horace's
schoolboys (Sat. 1. vi. 74). For the sand see Persius I. 131, "Nec qui
abaco numeros et secto in pulvere metas scit risisse," Apul. Apolog. 16
(pulvisculo), Mart. Capella, lib. vii. 3, 4, etc. Cicero says of an
expert calculator "eruditum attigisse pulverem," (de nat. Deorum,
ii. 18). Tertullian calls a teacher of arithmetic "primus numerorum
arenarius" (de Pallio, _in fine_). The counters were made of various
materials, ivory principally, "Adeo nulla uncia nobis est eboris, etc."
(Juv. XI. 131), sometimes of precious metals, "Pro calculis albis et
nigris aureos argenteosque habebat denarios" (Pet. Arb. Satyricon, 33).There are, however, still in existence four Roman counting boards of a
kind which does not appear to come into literature. A typical one is of
the third class. It consists of a number of transverse wires, broken at
the middle. On the left hand portion four beads are strung, on the right
one (or two). The left hand beads signify units, the right hand one five
units. Thus any number up to nine can be represented. This instrument is
in all essentials the same as the Swanpan or Abacus in use throughout
the Far East. The Russian stchota in use throughout Eastern Europe is
simpler still. The method of using this system is exactly the same as
that of _Accomptynge by Counters_, the right-hand five bead replacing
the counter between the lines.THE BOETHIAN ABACUS.Between classical times and the tenth century we have little or no
guidance as to the art of calculation. Boethius (fifth century), at the
end of lib. II. of his _Geometria_ gives us a figure of an abacus of the
second class with a set of counters arranged within it. It has, however,
been contended with great probability that the whole passage is a tenth
century interpolation. As no rules are given for its use, the chief
value of the figure is that it gives the signs of the nine numbers,
known as the Boethian "apices" or "notae" (from whence our word
"notation"). To these we shall return later on.