The law of sines ensures that all three interior angles
match, and so the SAS rule applies.
E
UCLIDEAN GEOMETRY
takes the SAS rule as an
AXIOM
, that is, a basic assumption that does not
require proof. It is then possible to justify the validity
of the remaining rules by making use of this rule solely,
and to also justify the law of sines. (The fact that the
sine of an angle is the same for all right triangles con-
taining that angle relies on SAS being true.)
See also
CONGRUENT FIGURES
;
SIMILAR FIGURES
.
abacus Any counting board with beads laid in paral-
lel grooves, or strung on parallel rods. Typically each
bead represents a counting unit, and each groove a
place value. Such simple devices can be powerful aids
in performing arithmetic computations.
The fingers on each hand provide the simplest “set
of beads” for manual counting, and the sand at one’s
feet an obvious place for writing results. It is not sur-
prising then that every known culture from the time of
antiquity developed, independently, some form of
counting board to assist complex arithmetical compu-
tations. Early boards were simple sun-baked clay
tablets, coated with a thin layer of fine sand in which
symbols and marks were traced. The Greeks used trays
made of marble, and the Romans trays of bronze, and
both recorded counting units with pebbles or beads.
The Romans were the first to provide grooves to repre-
sent fixed place-values, an innovation that proved to be
extremely useful. Boards of this type remained the stan-
dard tool of European merchants and businessmen up
through the Renaissance.
The origin of the word abacus can be traced back
to the Arabic word abq for “dust” or “fine sand.” The
Greeks used the word abax for “sand tray,” and the
Romans adopted the word abacus.
The form of the abacus we know today was devel-
oped in the 11th century in China and, later, in the 14th
century in Japan. (There the device was called a
soroban.) It has beads strung on wires mounted in a
wooden frame, with five beads per wire that can be
pushed up or down. Four beads are used to count the
units one through four, and the fifth bead, painted a dif-
ferent color or separated by a bar, represents a group of
five. This provides the means to represent all digits from
zero to nine. Each wire itself represents a different power
of ten. The diagram at left depicts the number 35,078.
Addition is performed by sliding beads upward
(“carrying digits” as needed when values greater than
10 occur on a single wire), and subtraction by sliding
beads downward. Multiplication and division can be
computed as repeated addition and subtraction. Histo-
rians have discovered that the Chinese and Japanese
scholars also devised effective techniques for computing
square and cube roots with the aid of the abacus.
The abacus is still the popular tool of choice in
many Asian countries—preferred even over electronic
2 abacus
A simple abacus
A Chinese abacus from before 1600. Notice that two beads, each
representing five units, are placed in each column above the bar.
(Photo courtesy of the Science Museum, London/Topham-HIP/The
Image Works)