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History of Hindu Mathematics: A Source Book v1.c8.s2: 2. Addition

History of Hindu Mathematics: A Source Book v1.c8.s2

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2. Addition

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**Terminology.** Âryabhaṭa II (950) defines addition thus:

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“The making into one of several numbers is addition.”³

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³ *MSi*, p. 143.

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The Hindu name for addition is *saṁkalita* (made together). Other equivalent terms commonly used are *saṁkalana* (making together), *miśraṇa* (mixing), *sammelana* (mingling together), *prakṣepaṇa* (throwing together), *saṁyojana* (joining together), *ekîkaraṇa* (making into one), *yukti*, *yoga* (addition) and *abhyâsa*,⁴ etc. The word *saṁkalita* has been used by some writers in the general sense of the sum of a series.⁵

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⁴ This word has been used in the sense of addition in the *Śulba* only. It is used for multiplication in later works.

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⁵ *E.g.*, *Triś*, p. 2; *GSS*, p. 17.

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**The Operation.** In all mathematical and astronomical works, a knowledge of the process of addition is taken for granted. Very brief mention of it is made in some later works of elementary character. Thus Bhâskara II says in the *Lîlâvatî*:

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"Add the figures in the same places in the direct or the inverse order."¹

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¹ *L*, p. 2; direct (*krama*), *i.e.*, beginning from the units place; inverse (*utkrama*), *i.e.*, beginning from the last place on the left. The commentator Gaṅgâdhara says: *aṅkânâṃ vâmatogatiriti vitarkeṇa ekasthânâdi yojanam kramaḥ utkramastu antyasthânâdi yojanam*, *i.e.*, "According to the rule 'the numerals increase (in value) towards the left', the addition of units first is the direct method, the addition of figures in the last place first is the inverse method."

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**Direct Process.** In the direct process of addition referred to above, the numbers to be added are written down, one below the other, just as at present, and a line is drawn at the bottom below which the sum is written. At first the sum of the numbers standing in the units place is written down, thus giving the first figure of the sum. The numbers in the tens place are then added together and their sum is added to the figure in the tens place of the partial sum standing below the line and the result substituted in its place. Thus the figure in the tens place of the sum is obtained; and so on. An alternative method used was to write the biggest addend at the top, and to write the digits of the sum by rubbing out corresponding digits of this addend.²

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² Dvivedi, *History of Mathematics*, Benares, 1910, p. 60.

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**Inverse Process.** In the inverse process, the numbers standing in the last place (extreme left) are added together and the result is placed below this last place. The numbers in the next place are then added and so on. The numbers of the partial sum are corrected, if necessary, when the figures in the next vertical line are added. For instance, if 12 be the sum of the numbers in the last place, 12 is put below the bottom line, 2 being directly below the numbers added; then, if the sum of the numbers in the next place is 13 (say), 3 is placed below the figures added and 1 is carried to the left. Thus the figure 2 of the partial sum 12 is rubbed out and substituted by 3.¹

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¹ The *Manorañjana* explains the process of addition thus: *Example.* Add 2, 5, 32, 193, 18, 10 and 100.

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The Arabs used to separate the places by vertical lines, but this was not done by the Hindus.²

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² *Cf.* Taylor, *Lîlâwatî*, Bombay, 1816, Introduction, p. 14.

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