£98 
N U M 
This example will be found, on a flight infpeclion, to re- 
femble our compound divifion, or that fort of divifion 
that we mult necefiarily employ, if we were to divide 
feet, incites, and parts, by fintilar denominations; which, 
together with the number of different characters that they 
made life of, mull have rendered this rule extremely labo¬ 
rious. And that for the extraction of the fquare root was 
of collide equally difficult, the principle of which was 
the fame as ou r’s, except in the difference of the notation ; 
though it appears that they frequently, in dead of making 
tn'e of the rule, found the root by fucceffive trials, and 
then fquared it, in order to prove the truth of their aflurnp- 
tion. 
The introduction of the fexagefimal fyftem of notation, 
may therefore be regarded as the moftimportant improve¬ 
ment of the Greek arithmetic. Its procedure ref'embled 
clofely the method which we now praCtife in the manage¬ 
ment oi duodecimal fractions. There was only wanting, 
to facilitate its operations, a multiplication-table more ex- 
tenfive than ours, and comprehending the mutual pro¬ 
ducts of all the numbers inclufively, from i to 59. Such 
a table was aCfually conftruCied, long afterwards, by Phi¬ 
lip Lanfberg,an ingenious and learned Dutch clergyman. 
Purbach, one of the firft and mod ardent reiforers of ma¬ 
thematical fcience, had, about the middle of the fifteenth 
century, combined the fexigefimal with the decimal fyf- 
teni, which was lately fpread over Europe, having been 
introduced by the Moors into Spain. Inftead of 3600', or 
216,000", into which the ancient affrohomers divided the 
radius of the circle, Purbach made it to contain 600,000 
equal parts. His difciple and companion, Muller of 
Koningfberg, commonly fly led Regiomontanus, to whom 
trigonometry owes its preferrt form, completed the pro- 
grefs, by rejeCiing entirely that fexigefimal admixture, 
and adopting for the radius a divifion purely decimal. 
But this innovation had an influence (fill more extenfive, 
fince.it gave occafion, in the fequel, to the introduction 
of decimal fractions, the practice of which has fo mate¬ 
rially abridged and Amplified all our calculations. By 
finch gradual fleps have the mofl ufeful improvements 
been achieved ! The aflronomical divifion of the circle 
firft fuggefted the fexagefimal fcale ; the Aexagefimals were 
next blended with the decimal arrangement; and the de¬ 
cimal Htbdivi'fion, in its independent form, was finally 
reduced to vulgar praClice. 
The Greek arithmetic, then, as fucceffively moulded 
by the ingenuity of Archimedes, of Apollonius, and 
Ptolemy, had attained, on the whole, to a Angular de¬ 
gree of perfection, and was capable, notwithftanding its 
cumbrous ftruCture, of performing operations of very 
confiderable difficulty and magnitude. The great and ra¬ 
dical deleft of the fyftem confided in the want ofa gene¬ 
ral mark analagous to our cipher, and which, without 
having any value itfelf, fhould fierve to afeertain the rank 
or power of the other characters, by filling up the va¬ 
cant places in the fcale of numeration. Yet were the 
Greeks not altogether unacquainted with the ufeoffuch 
a fign ; for Ptolemy, in his Almageft, employs the final 1 
o, to occupy the accidental blanks which occurred in the 
notation of fexagefimals. This letter was perhaps chofen 
by him, becaule, immediately fucceeding to », which de¬ 
notes 60, it could not, in the fexagefimal arrangement, 
occafion any fort of ambiguity. But the advantage 
thence refulting was entirely confined to that particular 
cafe. The letters, being already fignificant, were gene¬ 
rally disqualified for the purpofe of a mere fupplementary 
notation ; and the feleftion of an alphabetic charafter to 
fiupply the place of the cipher, may be confidered as an 
unfortunate circumftance, which appears to have arrefted 
the progrefs towards a better and more complete fyftem. 
Plad Apollonius clafled the numerals by triads, inftead of 
tetrads, he would have greatly Amplified the arrangement, 
and avoided the confiifion arifing from the admixture of 
the punftuated letters expreffive of thoufands. It is by 
this method of proceeding with periods of three figures. 
B E R. 
oradvancing at once by thousands inftead of tens, that 
we are enabled rnoft expeditioufly to read off the larged 
numbers. The extent of the alphabet was favourable to 
the firft attempts at numeration; fince, with the help of 
three intercalations, it furniftied characters for the whole 
range below athoufand; but that very circumftance, in 
the end, proved a bar to future improvements. It would 
have been a moft important ftride, to have next exchanged 
thofe triads into monads, by difearding the letters ex¬ 
preffive of tens and hundreds, and retaining only the find 
clafs, which, with its inferted epi/emon, fhould denote the 
nine digits. The iota, which fignified ten, now lofing its 
force, might have been employed as a convenient fubfti- 
tute for the cipher. By fuch progreffive ciianges, the 
arithmetical notation of the Greeks would at iaft have 
reached its utmoft perfection, and have exaftly ref’embled 
our own. A wide interval, no doubt, did ftili remain; yet 
the genius of that acute people, had it continued unfet¬ 
tered, would in time, we may prefume, have triumphantly 
palled the intervening boundaries. But the deajdi of 
Ptolemy was f’ucceeded by ages of languor and decline; 
and the fpirit of difeovery infenfibly evaporated in rnife- 
rable polemical difputes, till the fair eftablifhment of 
Alexandria was finally overwhelmed under the irrefiftible 
arms of the Arabs, lately roufed to viftory and conqueft 
by the enthufiafm of a new religion. 
It fhould, however, be remarked, that the Greeks dif- 
tinguiflied the theory from the praftice of numbers, by 
feparate names. The term arithmetic itfelf was reftrifted 
by them to the fcience which treats of flip nature and 
general properties of numbers; while the appellation 
logijiic, was appropriated to the colieftion of rules framed 
to direft and facilitate the common operations of calcula¬ 
tion. The ancient fyftems of arithmetic, accordingly, 
from the books of Euclid to the treatife of Boethius, and 
the verfes and commentaries of Capelia, are merely fpe- 
culative, and often abound with fanciful analogies. Py¬ 
thagoras had brought from the Eafl a paffion for the myf- 
tical properties of numbers, under the veil of which he 
probably concealed fome of his f’ecret or efoteric doctrines. 
He regarded numbers as of divine origin, the fountain of 
exiftence, and the model and archetype of all things. He 
divided them into a variety of different clafles, to each of 
which wereaffigned diltinft properties. They were prime 
or compofite, perreft or imperfeft, redundant or deficient, 
plane or folid ; they were triangular, fquare, cubic, or 
pyramidal. Even numbers were held by that vifionary 
philofopher as feminine, and ailiecfto earth ; but the odd 
numbers were confidered by him as endued with mafeu- 
line virtue, and partaking of the celeftial nature. He 
efteemed the unit, or monad , as the moft eminently facred, 
and as the parent of all fcientific numbers; lie viewed 
two, or the duad, as the afi’oeiate of the monad, and the 
mother of the elements; and he regarded three, or the 
triad, as perfeft, being the firft of the mafeuline numbers, 
comprehending the beginning, middle, and end ; and 
hence fitted to regulate by its combinations the repetition 
of prayers and libations. As the monad reprefented the 
Divinity, or the Creative Power, fo the duad was the image 
of Matter; and the triad, refuiting from their mutual 
conjunftion, became the emblem of Ideal Forms. 
But th t tetrad, or four, was the number which Pytha¬ 
goras nffefted to venerate the moft. It is a fquare, and 
contains within itfelf all the mufical proportions, and ex¬ 
hibits by fummation all the digits as far as ten, the root 
of the univerfal fcale of numeration ; it marks the fea- 
fons, the elements, and the fucceffive ages of man ; and 
it like wife reprefen ts the cardinal virtues, and the oppo- 
fite vices. The ancient divifion of mathematical fcience 
into arithmetic, geometry, aftronomy, and mu fie, was 
four-fold, and the courfe was therefore termed a letrach/s, 
or quaternion. Hence Dr. Barrow would explain the oath 
familiar to the difciples of Pythagoras: “ I lwear by him 
who communicated the tetraSlys." 
Five, or the pentad, being compofed of the firft male 
4 and 
