ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

208 BRAHMAGUPTA AS AN ALGEBRAIST
controversy regarding the date of the Bakhaśālī Manuscript
where we have definitely a method of solving linear equations
by the Rule of False Position. It would be interesting to give
an account of this rule here by taking an illustration from the
Bakhaśālī Manuscript.
Problem :
The amount given to the first is not known.
The second is given twice as much as the first; the third
thrice as much as the second; and the fourth four times
as much as the third. The total amount distributed is
132. What is the amount of the first ?
(BMS. Folio 23, recto)
In modern algebraic language, the solution of the problem
would be given by the equation
x + 2x + 6x + 24x = 132
where x is the amount given to the first.
The solution of this equation is given as follows in the
Bakhaśālī Manuscript :
‘Putting any desired quantity in the vacant place’; any
desired quantity is || 1 ||, ‘then construct the series’
| 1 | 2 | 2 | 3 | 6 | 4 |
| 1 | 1 | 1 | 1 | 1 | 1 |
‘multiplied’ || 1 | 2 | 6 | 24 | ; ‘added’ 33. ‘Divide the
visible quantity’ | 132 / 33 ; which) on reduction be-
comes | 4 / 1 |. (This is) the amount given (to the first)
(BMS. Folio 23, recto)
The Rule of False Position may be regarded as an early
stage of the development of the science of algebra, since no
symbol could have been evolved for an unknown quantity. As
soon as the system of notations was introduced, the application
of this Rule was no longer considered as necessary. Thus we
find that Āryabhaṭa I does not mention of this Rule.
Āryabhaṭa I states as follows regarding the solution of
linear equations :
The difference of the known ‘amounts’ (rūpaka) relat-
ing to two persons should be divided by the difference