Indian Mathematics/Arithmetical notations etc.

by George Rusby Kaye

VI.

22. According to the Hindus the modern place-value system of arithmetical notation is of divine origin. This led the early orientalists to believe that, at any rate, the system had been in use in India from time immemorial; but an examination of the facts shows that the early notations in use were not place-value ones and that the modern place-value system was not introduced until comparatively modern times. The early systems employed may be conveniently termed (a) the Kharoshthī, (b) the Brāhmī, (c) Āryabhata's alphabetic notation, (d) the word-symbol notation.

(a) The Kharoshthī script is written from right to left and was in use in the north-west of India and Central Asia at the beginning of the Christian era. The notation is shown in the accompanying table. It was, apparently, derived from the Aramaic system and has little direct connection with the other Indian notations. The smaller elements are written on the left.

(b) The Brāhmī notation is the most important of the old notations of India. It might appropriately be termed the Indian notation for it occurs in early inscriptions and was in fairly common use throughout India for many centuries, and even to the present day is occasionally used. The symbols employed varied somewhat in form according to time and place, but on the whole the consistency of form exhibited is remarkable. They are written from left to right with the smaller elements on the right. Several false theories as to the origin of these symbols have been published, some of which still continue to be recorded. The earliest orientalists gave them place-value, but this error soon disproved itself; it was then suggested that they were initial letters of numerical words; then it was propounded that the symbols were aksharas or syllables; then it was again claimed that the symbols were initial letters (this time Kharoshthī) of the corresponding numerals. These theories have been severally disproved.

The notation was possibly developed on different principles at different times. The first three symbols are natural and only differ from those of many other systems in consisting of horizontal instead of vertical strokes. No principle of formation of the symbols for "four" to "thirty" is now evident but possibly the "forty" was formed from the thirty by the addition of a stroke and the "sixty" and "seventy" and "eighty" and "ninety" appear to be connected in this way. The hundreds are (to a limited extent) evidently built upon such a plan, which, as Bayley pointed out, is the same as that employed in the Egyptian hieratic forms; but after the "three hundred" the Indian system forms the "four hundred" from the elements of "a hundred" and "four," and so on. The notation is exhibited in the table annexed.

(c) Āryabhata's alphabetic notation also had no place-value and differed from the Brāhmī notation in having the smaller elements on the left. It was, of course, written and read from left to right. It may be exhibited thus:

${\begin{array}{l c r r r r r r r r r r}\scriptstyle {\text{Letters}}&\scriptstyle {..}&\scriptstyle {k}&\scriptstyle {kh}&\scriptstyle {g}&\scriptstyle {gh}&\scriptstyle {\dot {n}}&\scriptstyle {c}&\scriptstyle {ch}&\scriptstyle {j}&\scriptstyle {jh}&\scriptstyle {\tilde {n}}\\\scriptstyle {\text{Values}}&\scriptstyle {..}&\scriptstyle {1}&\scriptstyle {2}&\scriptstyle {3}&\scriptstyle {4}&\scriptstyle {5}&\scriptstyle {6}&\scriptstyle {7}&\scriptstyle {8}&\scriptstyle {9}&\scriptstyle {10.}\\\end{array}}$

${\begin{array}{l c r r r r r r r r r r}\scriptstyle {\text{Letters}}&\scriptstyle {..}&\scriptstyle {\overset {t}{.}}&\scriptstyle {{\overset {t}{.}}h}&\scriptstyle {\overset {d}{.}}&\scriptstyle {{\overset {d}{.}}h}&\scriptstyle {\overset {n}{.}}&\scriptstyle {t}&\scriptstyle {th}&\scriptstyle {d}&\scriptstyle {dh}&\scriptstyle {n}\\\scriptstyle {\text{Values}}&\scriptstyle {..}&\scriptstyle {11}&\scriptstyle {12}&\scriptstyle {13}&\scriptstyle {14}&\scriptstyle {15}&\scriptstyle {16}&\scriptstyle {17}&\scriptstyle {18}&\scriptstyle {19}&\scriptstyle {20.}\\\end{array}}$

${\begin{array}{l c r r r r r r r r r r r r r}\scriptstyle {\text{Letters}}&\scriptstyle {..}&\scriptstyle {\overset {p}{.}}&\scriptstyle {ph}&\scriptstyle {b}&\scriptstyle {bh}&\scriptstyle {m}&\scriptstyle {y}&\scriptstyle {r}&\scriptstyle {l}&\scriptstyle {v}&\scriptstyle {\acute {s}}&\scriptstyle {sh}&\scriptstyle {s}&\scriptstyle {h}\\\scriptstyle {\text{Values}}&\scriptstyle {..}&\scriptstyle {21}&\scriptstyle {22}&\scriptstyle {23}&\scriptstyle {24}&\scriptstyle {25}&\scriptstyle {30}&\scriptstyle {40}&\scriptstyle {50}&\scriptstyle {60}&\scriptstyle {70}&\scriptstyle {80}&\scriptstyle {90}&\scriptstyle {100.}\\\end{array}}$

The vowels indicate multiplication by powers of one hundred. The first vowel a may be considered as equivalent to $\scriptstyle {100^{0}}$, the second vowel $\scriptstyle {i=100^{1}}$ and so on. The values of the vowels may therefore be shown thus:

${\begin{array}{l c r r r r r r r r r}\scriptstyle {\text{Vowels}}&\scriptstyle {..}&\scriptstyle {a}&\scriptstyle {i}&\scriptstyle {u}&\scriptstyle {{\overset {r}{.}}i}&\scriptstyle {{\overset {l}{.}}i}&\scriptstyle {e}&\scriptstyle {ai}&\scriptstyle {o}&\scriptstyle {au}\\\scriptstyle {\text{Values}}&\scriptstyle {..}&\scriptstyle {1}&\scriptstyle {10^{2}}&\scriptstyle {10^{4}}&\scriptstyle {10^{6}}&\scriptstyle {10^{8}}&\scriptstyle {10^{10}}&\scriptstyle {10^{12}}&\scriptstyle {10^{14}}&\scriptstyle {10^{16}}\\\end{array}}$

The following examples taken from Āryabhata's Gītikā illustrate the application of the system:

${\begin{aligned}&\scriptstyle {khyughri=(2+30).10^{4}+4.10^{6}=4320000}\\&{\begin{aligned}\scriptstyle {cayagiyi{\dot {n}}u{\acute {s}}ulchli=6}&\scriptstyle {+30+3.10^{2}+30.10^{2}+5.10^{4}+70.10^{4}}\\&\scriptstyle {(50+7).10^{8}=57,753,336}\end{aligned}}\end{aligned}}$

The notation could thus be used for expressing large numbers in a sort of mnemonic form. The table of sines referred to in paragraph 9 above was expressed by Āryabhata in this notation which, by the way, he uses only for astronomical purposes. It did not come into ordinary use in India, but some centuries later it appears occasionally in a form modified by the place-value idea with the following values:

${\begin{array}{r r r r r r r r r r}\scriptstyle {1}&\scriptstyle {2}&\scriptstyle {3}&\scriptstyle {4}&\scriptstyle {5}&\scriptstyle {6}&\scriptstyle {7}&\scriptstyle {8}&\scriptstyle {9}&\scriptstyle {0}\\\scriptstyle {k}&\scriptstyle {kh}&\scriptstyle {g}&\scriptstyle {gh}&\scriptstyle {\dot {n}}&\scriptstyle {c}&\scriptstyle {ch}&\scriptstyle {j}&\scriptstyle {jh}&\scriptstyle {\tilde {n}}\\\scriptstyle {\overset {t}{.}}&\scriptstyle {{\overset {t}{.}}h}&\scriptstyle {\overset {d}{.}}&\scriptstyle {{\overset {d}{.}}h}&\scriptstyle {\overset {n}{.}}&\scriptstyle {t}&\scriptstyle {th}&\scriptstyle {d}&\scriptstyle {dh}&\scriptstyle {n}\\\scriptstyle {p}&\scriptstyle {ph}&\scriptstyle {b}&\scriptstyle {bh}&\scriptstyle {m}&&&&&\\\scriptstyle {y}&\scriptstyle {r}&\scriptstyle {l}&\scriptstyle {v}&\scriptstyle {\acute {s}}&\scriptstyle {sh}&\scriptstyle {s}&\scriptstyle {h}&\scriptstyle {l}&\\\end{array}}$

Initial vowels are sometimes used as ciphers also. The earliest example of this modified system is of the twelfth century. Slight variations occur.

(d) The word-symbol notation.—A notation that became extraordinarily popular in India and is still in use was introduced about the ninth century, possibly from the East. In this notation any word that connotes the idea of a number may be used to denote that number: e.g. Two may be expressed by nayana, the eyes, or karņa, the ears, etc.; seven by aśva, the horses (of the sun); fifteen by tithi, the lunar days (of the half month); twenty by nakha, the nails (of the hands and feet); twenty-seven by nakshatra the lunar mansions; thirty-two by danta, the teeth; etc., etc.

(e) The modern place-value notation.—The orthodox view is that the modern place-value notation that is now universal was invented in India and until recently it was thought to have been in use in India at a very early date. Hindu tradition ascribes the invention to God! According to Maçoudi a congress of sages, gathered together by order of king Brahma (who reigned 366 years), invented the nine figures! An inscription of A.D. 595 is supposed to contain a genuine example of the system. According to M. Nau, the "Indian figures" were known in Syria in A.D. 662; but his authority makes such erroneous statements about "Indian" astronomy that we have no faith in what he says about other "Indian" matters. Certain other mediæval works refer to 'Indian numbers' and so on.

On the other hand it is held that there is no sound evidence of the employment in India of a place-value system earlier than about the ninth century. The suggestion of 'divine origin' indicates nothing but historical ignorance; Maçoudi is obviously wildly erratic; the inscription of A.D. 595 is not above suspicion^[1] and the next inscription with an example of the place-value system is nearly three centuries later, while there are hundreds intervening with examples of the old non-place value system. The references in mediæval works to India do not necessarily indicate India proper but often simply refer to 'the East' and the use of the term with regard to numbers has been further confused by the misreading by Wœpcke and others of the Arabic term hindasi (geometrical, having to do with numeration, etc.) which has nothing to do with India. Again, it has been assumed that the use of the abacus "has been universal in India from time immemorial," but this assumption is not based upon fact, there being actually no evidence of its use in India until quite modern times. Further, there is evidence that indicates that the notation was introduced into India, as it was into Europe, from a right to left script.

23. In paragraph 7 above certain attempts at squaring the circle are briefly described and it has been pointed out (in § 10) that Āryabhata gives an extremely accurate value of $\scriptstyle {\pi }$. The topic is perhaps of sufficient interest to deserve some special mention. The Indian values given and used are not altogether consistent and the subject is wrapped in some mystery. Briefly put—the Indians record an extremely accurate value at a very early date but seldom or never actually use it. The following table roughly exhibits how the matter stands:—

Date Circa.	Authority.	Value of $\scriptstyle {\pi }$
B.C. 1700	Ahmes the Egyptian	$\scriptstyle {({\frac {16}{9}})^{2}}$	$\scriptstyle {=3\cdot 160.}$
„  250	Archimedes	$\scriptstyle {<3{\frac {1}{7}}{\text{ and }}>3{\frac {10}{71}}}$
„    ?	The S'ulvasūtras	$\scriptstyle {({\frac {26}{15}})^{2}}$	$\scriptstyle {=3\cdot 004.}$
	"	$\scriptstyle {({\frac {9785}{5568}})^{2}}$	$\scriptstyle {=3\cdot 097.}$
„  230	Apollonius	$\scriptstyle {3{\frac {17}{120}}}$	$\scriptstyle {=3\cdot 14166.}$
„  120	Heron	$\scriptstyle {3}$	$\scriptstyle {=3\cdot }$
	"	$\scriptstyle {3{\frac {1}{7}}}$	$\scriptstyle {=3\cdot 14286.}$
A.D.  150	Ptolemy	$\scriptstyle {3{\frac {17}{120}}}$	$\scriptstyle {=3\cdot 14166.}$
„   263	Liu Hiu	$\scriptstyle {3{\frac {7}{50}}}$	$\scriptstyle {=3\cdot 14.}$
„     ?	Puliśa	$\scriptstyle {3{\frac {177}{1250}}}$	$\scriptstyle {=3\cdot 1416.}$
„   450	Tsu Ch'ung-chi	$\scriptstyle {3{\frac {1}{7}}}$	$\scriptstyle {=3\cdot 14286.}$
„   500	Aryabhata	$\scriptstyle {3}$	$\scriptstyle {=3\cdot }$
	"	$\scriptstyle {\frac {62832}{20000}}$	$\scriptstyle {=3\cdot 1416.}$
	"	$\scriptstyle {\frac {3393}{1080}}$	$\scriptstyle {=3\cdot 14166.}$
„   628	Brahmagupta	$\scriptstyle {3}$	$\scriptstyle {=3\cdot }$
	"	$\scriptstyle {{\sqrt {10}}=3{\frac {1}{7}}}$	$\scriptstyle {=3\cdot 14286.}$
	"	$\scriptstyle {{\sqrt {10}}={\frac {721}{228}}}$	$\scriptstyle {=3\cdot 16228.}$
„   800	M. ibn Musa	$\scriptstyle {\sqrt {10}}$	$\scriptstyle {={\text{?}}}$
„   ?	Māhavīra	$\scriptstyle {3}$	$\scriptstyle {=3\cdot }$
	"	$\scriptstyle {\sqrt {10}}$	$\scriptstyle {={\text{?}}}$
„  1020	Srīdhara	$\scriptstyle {3}$	$\scriptstyle {=3\cdot }$
	"	$\scriptstyle {\sqrt {10}}$	$\scriptstyle {={\text{?}}}$
	"	$\scriptstyle {3{\frac {1}{6}}}$	$\scriptstyle {=3\cdot 1666.}$
„  1150	Bhāskara	$\scriptstyle {3}$	$\scriptstyle {=3\cdot }$
	"	$\scriptstyle {3{\frac {1}{7}}}$	$\scriptstyle {=3\cdot 14286.}$
	"	$\scriptstyle {3{\frac {17}{120}}}$	$\scriptstyle {=3\cdot 14166.}$
	"	$\scriptstyle {3{\frac {177}{1250}}}$	$\scriptstyle {=3\cdot 1416.}$
	Approximately correct value ..		$\scriptstyle {\ 3\cdot 14159.}$

24. The mistakes made by the early orientalists have naturally misled the historians of mathematics, and the opinions of Chasles, Wœpcke, Hankel and others founded upon such mistakes are now no longer authoritative. In spite, however, of the progress made in historical research there are still many errors current, of which, besides those already touched upon, the following may be cited as examples: (a) The proof by "casting out nines" is not of Indian origin and occurs in no Indian work before the 12th century; (b) The scheme of multiplication, of which the following is

	1		3		5
1
		1		3		5
2					1
		2		6		0
1	6		2		0

an Indian example of the 16th century, was known much earlier to the Arabs and there is no evidence that it is of Indian origin; (c) The Regula duorum falsorum occurs in no Indian work; (d) The Indians were not the first to give double solutions of quadratic equations; Bhaskāra was not the discoverer of the "principle of the differential calculus," etc., etc.

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1. The figures were obviously added at a later date.