114 
POPULAR SCIENCE NEWS. 
[August, 1890. 
the curious fact that, as the air rushes out 
aroiuitl the ball, a partial vacuum is pro- 
duced in the funnel-shaped end of the 
tube, and the ball is thus held in position by 
the excess of atmospheric pressure on the 
outside. 
If any reader of the Science News desires 
to try the experiment for himself, a similar 
apparatus can be made out of a piece of tin ; 
but a simpler and equally eHective illustration 
of the principle can be shown with an appa- 
ratus constructed from a piece of glass tubing, 
or even a pipe-stem, (B-C), to one end 
of which is .attached a circular piece of card- 
board (A), as shown in Fig. 2. A second 
Fig. a. 
disk of cardboard (D) is then placed on top 
of the first, and it will be found impossible to 
blow it off; and the apparatus may even be 
inverted without its falling, while the blast 
of air is kept up. To prevent the upper 
disk from sliding off, the edges of the lower 
one should be turned up as shown in the 
engraving, or a pin may simply be passed 
through the centre of the upper disk so as to 
project into the tube. 
The illustrations accompanying this article 
are reproduced from La Nature. 
[Original in PojmUtr Science Sewt.i 
ARITHMETICAL CALCULATIONS AMONG 
THE ANCIENT GREEKS AND ROMANS. 
BY JOHN C. ROLFK, PH. D. 
It is obvious from their system of figures that the 
methods used by the ancient Romans in performing 
arithmetical operations must have been essentially 
different from our own. They represented units not 
only by simple, but by compound and complex signs 
(V., VI., Villi.) ; two places in our decimal system 
were represented sometimes by two figures (XI.) 
and sometimes by nine (LXXXVIIIl.) ; and they 
had no zero. These peculiarities made mechanical 
assistance necessary in perf9rming operations in 
whole numbers, as well as in their system of duo- 
decimal fractions. Instruction in these operations 
was the most difficult work of their schools, and the 
calculator, wliose classes were attended by grown-up 
young men, was held in greater estimation than the 
litierator, and received higher pay. 
We learn from Horace that the simple operations 
were performed mentally, but the more important 
ones required the.aid of the fingers or of the abacus. 
Finger-reckoning was a natural development of the 
gesticulation common to the excitable Southern 
races, with which they always accompany speech, 
and by means of which they can carry on conversa- 
tions without words. It was common in Italy, as 
well as in the Orient and in Greek lands, and its 
use continued until the Middle Ages. By means 
of eighteen positions of the fingers of the left hand 
they represented the nine units and the nine tens, 
while an equal number of corresponding positions 
of the right hand gave the nine hundreds and the 
nine thousands. Ten thousand and higher numbers 
were indicated by touching the breast and different 
parts of tlie trunk wiili one or tlie otlier hand. It 
will readily be seen that long practice was necessary 
to give the hands the machine-like accuracy which 
this system of figuring demanded. A landlord, 
reckoning up the amount owed him by a departing 
guest, counted off the single items, indicated tlie 
amount of the first by a jjosition of the fingers, 
added to this the second and represented the sum 
by a second gesture, and so on until the calculation 
was complete. The same process was used by a 
lawyer who wished to reckon an account before 
a jury, and Quintilian tells us that it was highly 
important for an advocate to be able to perform 
such operations correctly and gracefully, as awkward 
gestures and hesitation stamped him as ignorant 
and uncultivated. 
The abacus (in the sense of a reckoning-table, for 
the word has other meanings) was a table of stone, 
wood, or metal, wliich was a common piece of house 
furniture, and was kept also in public treasuries and 
similar places. It was used in all payments which 
involved any but the simplest calculations. We 
find it represented in many works of art. In one 
two women sit before a reckoning-board, engaged 
in verifying a payment. In another we see the 
master of the house at his dinner, while before him 
stands a slave with a reckoning-board. In a third 
we see an abacus in the hands of a revenue collector. 
Moreover, several specimens have come down to us. 
They are of two kinds. In the»simpler form, which 
consisted merely of a flat table, reckoning-pebbles 
{calculi) were used. A form of this kind of abacus 
was in use as late as the seventeenth century, coins 
taking the place of pebbles. It was used for reck- 
oning small sums. If, for example, one wished to 
subtract 25 from 83, one put down S3 pebbles, took 
away 25, and counted the remainder. To divide 
969 by 26, one took from 969 pebbles 26 at a time 
until the remainder was less than 36. The 37 times 
which this IiAd to be done gave the quotient; and 
the 7 pebbles which were left, the remainder. 
Addition and multiplication were done in a similar 
way. This form was adapted for larger operations 
by ruling on it seven horizontal lines, representing, 
respectively, 1000, 500, 100, 50, 10, 5, and 1. Then 
two pebbles on the 1000 line represented 2000, etc. 
The operations were performed in the way just 
described. 
A more elaborate form of the abacus is represented 
in the figure. It had vertical grooves, in which 
were inserted buttons which could be moved back 
and forth. There were eight long and eight short 
grooves opposite to each other, and a ninth long 
groove which had no corresponding small one. In 
each of the first seven long grooves were four 
buttons, while the short ones had one each. The 
grooves were marked with signs for i, 10, 100, 
1000, lo.ooo, 100,000, 1,000,000. Thus every groove 
represented a decimal place, while the buttons, 
which represented the nine units, were divided like 
the Roman nine itself into V. (in the short groove) 
and IIII. (in the long groove.) Thus in groove 7, 
each one of the four buttons represented o/te, and 
the single button ^ce ; in 6, each of the four buttons 
ten, the single button Ji/ti/ : and so on. Thus to 
represent 587,615, one would push up the V. in 
groove 2, the V. and three ones in 3, tlie V. and two 
ones in 4, the V. and one in 5, one in 6, and V. in 7. 
Grooves 8 and 9 were used for reckoning in frac- 
tions. Groove 8 was used for duodecimal fraction-, 
having the denominator 12, while the ninth was 
used for smaller fractions. The four buttons 
of groove 9 were distinguished from one another 
by three difterent colors; or in some forms of the 
abacus they were divided between three separate 
grooves, which represented 1-24, 1-48, and 2-72 (two 
buttons.) The mode of reckoning with this abacus 
was the same in principle as that used in the 
simpler form. A special difficulty in its use — which, 
of course, was overcome by practice — was that not 
only the buttons which were in use in any calcula- 
tion, but also all the others, were visible, so that the 
eye had to be trained to take account only of the 
former. 
Among the Greeks the designation of figures was 
at first the same in principle as among the Romaii'^, 
and the same aids in reckoning were used, namclj , 
the fingers and the abacus. In comparatively late 
times, probably not long before the Christian era, 
the use of the Greek letters to designate numbers 
on a decimal system superseded the old method. 
By this system, as each letter represented a decimal 
place, arithmetical operations could be considerably 
simplified. An example in multiplication given by 
Eutokius, who lived during the si.xth century of our 
era, illustrates this : 
a65=3oo-|-6o-|-5 
265=2oo4-6o-{-5 
40,000 
12,000 
1,000 
12,000 
3,600 
300 
1, 000 
300 
25 
70,225 
This process became rather complicated when 
larger numbers were multiplied, and where fractions 
were involved. In the example given, of course, 
the two numbers and the several multiplications 
were represented by Greek letters. The several 
multiplications were done mentally, or, if this was 
impracticable, with the abacus. 
[Original in Popular Science News.] 
THE SIXTH CENTENNIAL OF THE UNIVER- 
SITY OF MONTPELLIER. 
The last days of the month of May have wit- 
nessed a series of interesting ceremonies which 
have been held in the town of Montpellier, in 
France, the seat of one of the oldest ktTown univer- 
sities. Montpellier is a town of the south of France, 
situated in one of the richest vineyards of the coun- 
try, at half an hour's distance from the Mediterra- 
nean, close to the prosperous harbor of Cette, and 
in sight of the first beginnings of tlie Ceicunes 
mountain range. For over six centuries — and, as 
far as one can know, for nearly seven centuries — 
Montpellier has been an important scientific center. 
While the ancient professors of Paris were more 
especially proficient in theological studies, and 
while those of the medi;eval schools of Orleans, . 
Bourges, and Poitiers were principally concerned 1 
with law studies, Montpellier was and has remained 
an important medical school, and for a long period 
this school has been the glory not only of the town, 
but of the whole country and of entire Europe. 
Montpellier entertained numerous commercial rela- ( 
tions with Egyptian and Arab merchants, and it is j 
supposed that in the nth and 12th centuries the ' 
medical works of Avicenne, Averroes, and others 
were imported into Montpellier, and became the 
