inese[43] sources, yet their ancient literature testifies to a high state of civilization, and to a considerable advance in sciences, in philosophy, and along lite

rtly ritualistic (the Brāhma?as), and partly philosophical (the Upanishads). Our especial interest is in the Sūtras, versified abridgments of the ritual and of ceremonial rules, which contain considerable geometric material used in connection with altar construction, and also numerous examples of rational numbers the sum of whose squares is also a square,

0] Further fundamental mathematical notions such as the conception of irrationals and the use of gnomons, as well as the philosophical doctrine of the transmigration of souls,-all of these having long been attributed to the Greeks,-are shown in these works to be native to India

ns of Mesopotamia. We are quite in the dark as to these early steps; but as to their development in India, the approximate period of the rise of their essential feature of place value, their introduction into the Arab civilization, and their spread to the West, we have more or less definite information. When, ther

was the same attention to astronomy that was found in the earlier civilizations of Babylon, China, and Egypt, a fact attested by the Vedas themselves.[54] Such advance in science presupposes a fair knowledge of calculation, but of the manner of calculating we are quite ignorant and probably always shall be. One of the Buddhist sacred books, the Lalitavistara, relates that when the Bōdhisattva[55] was of age to marry, the father of Gopa, his intended bride, demanded an examination of the five hundred suitors, the subjects including arithmetic, writing, the lute, and archery.

tra said, 'I

umbers. Aft

n till we reac

, four, to ten,

housands.' Aft

decads, centur

reached, but so

he kōti, nah

skhamba, a

gundhikas,

īkas into

ow you count th

ground to fin

that a nu

d to count the

ha, for the

calculus of

a, by the wh

sands of Gung

lpas, where

e ten crore Gun

ive scale, th'

kya, which

ps that in ten

all the worlds

Maha Kalpas,

e their future

task, and asks to hear the "measure of the line" as far as yōj

master! if

te how many

end within

nstant skill, t

e total of th

tra heard i

e the boy; 'For

of thy teache

Gū

alitavistara relates of the number-series of the Buddha's time. While it extends beyond all reason, nevertheless it r

o be handed down by tradition. Of these the sixth is known as Jyoti?a (astronomy), a short treatise of only thirty-six verses, written not earlier than 300 B.C., and affording us some k

ammedan era (622 A.D.). About all that we know of the earlier civilization is what we glean from th

o much has been discovered within a century, than that we are so uncertain as to origins and dates and the early spread of the system. The probability being that writing was not introduced into India before the close of the fourth century B.C., and literature existing only in spoken form pr

eat groups, (1) the Kharo??hī, (2) the Brāhmī, and (3) the w

however, have been found in the earliest of these inscriptions, number-names probably having been written out in words as was the custom with many ancient peoples. Not until the time of the powerful King A?oka, in the third century B.C., do numerals appear in any inscriptions thus far discovered; and then only in the primitive form of marks, quite as they would be found in Egypt, Greece, Rome, or in various oth

d in more highly developed form, the right-to-left system appearing, together with evidences of thr

y a canceling of three marks as a workman does to-day for five, or a laying of one stick across three others. The ten has never been satisfactorily explained. It is similar to the A of the Kharo??hī alphabet, but we have no knowledge as to why it was chosen. The twenty is evidently a ligature of two tens, and this in turn suggested

or four, and the Kharo??hī form is employed for twenty. In addition to this there is a trace of an analogous use of a scale of twenty. While the symb

earliest historic times. There are various theories of its origin, none of which has as yet any wide acceptance,[72] although the problem offers hope of solution in due time. The numerals are not as old as the alphabet, or at least they have not as yet been found in inscrip

far as known, the only ones to

hey next appear in the second century B.C. in some inscriptions in the cave on the top of the Nānā Ghāt hill, about seventy-five miles from Poona in centra

he wall opposite the entrance are representations of the members of his family, much defaced, but with the names still legible. It would seem that the excavation was made by order of a king named Ved

e inscriptions, owing to the difficulty of deciphering them; but the follo

able Hindu numeral system connected with our own, is of sufficient inter

e seen, is found in certain other cave inscriptions dating back to the first or seco

he Brāhmī notation by adding the zero, the progress of these forms is well marked. It is therefore well to prese

Progress of Numb

me

ka[

ka

ka[

arī

ik[

rapa

ana

ta[

hab

al[

i?g

?aka

s are given by Bühler,

lace value, and the numbers like twenty, thirty, and other multiples of ten, one hundred, and so on, required separate symbols except where they were written out in words. The ancient Hindus had n

for

for

for

for

for

for

wing numerals below one hundred

] f

] fo

se are, however, so closely connected with the perfecting of the system by the invention of the zero that they are more appropriately

on. The or is simply one stroke, or one stick laid down by the computer. The or represents two strokes or two sticks, and so for the and . From some primitive

ogly

er

mo

inese symbol, which is practically identical with the symbols found commonly in India from 150 B.C. to 700 A.D. In the cursive form it becomes , a

did China get these forms? Surely not from India, for she had them, as her monuments and literature[102] show, long before the Hindus knew them. The tradition is that China brought her civilization around the north of Tibet, from Mongolia, the primitive habitat being Mesopotamia, or possibly the oases of Turkestan. Now what numerals did Mesopotamia use? The Babylonian system, simple in its g

ls of their ancient home, the first three, these being all that the people as a whole knew or needed. It is equally possible that these three horizontal forms represent primitive stick-laying, the most natural position of a stick placed in front of a calculator being the horizontal one. When, however, the cuneiform writing developed more fully, the vertical form may have been proved the easier to make, so that by the time the migrations to the West began the

d resemblance between the Hieratic five and seven and those of the Indian inscriptions. There have not, therefore, been wanting those who asserted an Egyptian origin for these numerals.[105] There has already been mentioned the fact that the Kharo??hī numerals were formerly known as Bactrian, Indo-Bactrian, and Aryan. Cunningham[106] was the first to suggest that these numerals were derived from t

ndu Bactri

atur, Lat

ancha,

s

?

erchanged as occasio

ng that in four cases (four, six, seven, a

esent the order of letters[113] in the ancient alphabet. From what we know of this order, however, there seems also no basis for this assumption. We have, therefore, to confess that we are not certain that the numerals were alphabetic at all, and if they were alphabetic we have no evidence at present as to the basis of selection. The later forms may

Chinese influence is to be seen in certai

eek alphabet.[117] This difficult feat is accomplished by twisting some of the letters, cutting off, adding on, and effecting other changes to m

work[120] on number mysticism. He quotes from Abenragel,[121] giving the Arabic and a Latin tra

, and if the Siamese, and the Singhalese, and the Burmese, and other peoples in the East, could have created alphabets of their own, why should not the numerals also have been fashioned by some temple school, or some king, or some merchant guild? By way of illustration, there are shown in the table on page 36 certain systems of the East, and while a few resemblances are evident, it is also evident that the creators of each system endeavored to find original forms that should not be found in other systems. This, then, would seem to be a fair

ten. Conjectures, however, without any historical evidence for support, have no plac

ertain Eas

i

ma[

bar[

et[

lon

yala

me priest or teacher or guild, by the order of some king, or as part of the mysticism of some temple. Whatever the origin, they were no better than scores of other ancient systems and no better than the present Chinese system when written without the zero, and there would