The times. [volume] (Richmond, Va.) 1890-1903, 1899-04-14
FINE LECTURE
BY PROF. GAINES.
lie Talks Interestingly on Number
Systems.
The ART OF CALCULATION.
The Speaker Traced the Development
OF Numbers From Their Early Use,
aud showed Their Infinite Val
nc To the World
THE ART OF CALCULATION.

Professor r. L. Gaines. OF the School
81 Mathematics Oh IaonmCnd College
delivered thc third OF the series Of lee
tuIos bv members OF the faculty last
I.sh, la the College chapel There was
q very gratifying audience, and the lee
turer was given close attention. lie took
- his subject Number systems
Professor Galucs said ia part.
Ia studying the history OF the art OF
ejaculation. the question very naturally
arises. What IL the origin OF the number
c0nccpt--0f our notion OF numbers? But
81 seems that there never have lived
people who are entirely ignorant of num-
bcrs. In South America. Australia, and
elsewhere there are saVagos who have
made little progress IN numbers In
fact. 'Cw tribes have been found who
have no distinct idea of numbers above
two. of any collection of things above
Two they have simply the vague idea Of
many Others have no names for num-
bers above two, but are able TO count "a
little further 10y saying two and one
more etc.
ON the other hand. some savaEes can
by using artificial counters, carry the
process of counting into surprisingly
large numbers. We wonder why any could
not count as high as ten, since We carry
around with us such a handy set Of
counters. The fingers. toes finger joints
etc. are by far the most convenient arti-
nCial counters. and hence most number
systems are thus based on human anat-
omy. After learning how 10 count, the
savage seeks some way of keeping re-
cord Of his counting. and so We nnd him
using pebbles, or sticks. or notches on a
stick.
An interesting fact in this connection
is the way i'm which the savage invents
names for his numbers. lie usually can
count as far as three with artificial aid.
but from this point on the very names
which ne employs shows that he II using
his fingers. His word for four Will be ''l
lack one" or "a hand nearly. for five
~ hand for six zone on the next
Ten he will call two hands, and for
eleven he will very likely say zone on the
foot"
NUMBER SYSTEM
I'm building up a number system. Of
course the first thing is 10 decide upon
a base. At first this is only resting
flace In the counting 1t Is very well
known that our own system Is by no
means the only one that has been de-
Vsed Among the very low orders Of
savages the binary system-if it can he
called a system a all--is quite common.
This amounts 20 little more than count
ing in pairs Ternary and uuarternary
systems have been found but they are
very rare
BY far the most common system
among the savages Is the culinary. arts.
ing from the fact that We have i've mn
gers on one hand. But they frequently
use mixed systems For example. when
they call fifteen three hands" they use
the culinary: but when they reach twenty
they are very apt 10 say "a man doth
hands and feeO and then they are using
the vieessimaJ system.
Among civilized peoples the decimal
system is by far the most common.
From theoretical considerations ii would
seem that twelve would have been more
fortunate base but the decimal system
is now too firmly fixed to be In any dan
ger of being displaced.
SYSTEMS or NOTATION
But the choice of base-that is
whether we shall use five or ten. or
twclve-is of far less consequence than
the question as 10 how We snail write
numbers.
The lecturer made use of the black
board 10 exhibit the methods OF notation
employed by the Egyptians, Babyionidns
Greeks and Romans. but the characters
would be troublesome TO reproduce lie
showed that by means Of hIeroslyphics.
characters or letters of the alphabet they
devised schemes which enable them 10
record large numbers. but which gave lit
tie or no aid in computation. fhey lack
ed the all-impOrtant principle OF local
value and were consequently very IN
ferior 10 our methods With the Greeks
and Romans this difficulty was in some
measure overcome by means OF me-
cnanic-l devise-the abacus. But when
ever they undertook 10 write numbers
.f...r ma nd
unsuggestive. and progress IN arithmetic
was practically impossible
Early IN the Christian era the Roman
notation. with which We are all familiar,
was carried into Europe where lt held
compct sway for centuries. I'm the mean
time the HindOos were displaying Won
derful mathematical power and il was
reserved for them 10 invent system OF
notation which was infinitely sup rior To
10 any which man had hitherto been
able 10 devise. For centuries they had
used the characters which we have TO-
day. except the zero. but had not hit
upon the principle of local value In
fact. this had 20 wait upon the invention
of a symbol for the absence of value
he zero This they did about the sixth
century, and thus made perfect system
10 Was through the Arabs that this
splendid system found its way into
Europe. and hence is commonly called the
Arabic notation.
But the new system did not find imme-
diatc and universal acceptance a. might
have been expected. II was not unul
about loco more than four centuries atier
is appearance IN Europe. that n receiV-
cd the important addition, the invention
OF decimal fractions Bot there was stil
another invention which was necessary
20 make our modern computation posi-
ble-our system Of common loearthms
This came a few years after the invention
III decimal fractions and was the first
important contribution 90 mathematics
made by The English speaking people
n was the invention OF John Napier. II
was admirably timed for as Professor
Cacori says: "Kepler was then examining
the planetary orbits ana Calico had just
turned his telescope 10 the stars Inc
invention o. logarth ma. as Laplace aptly
said. by lessening the labor shortened
the life ~ the astronomer