61 5 4 5 5 4 5 1 6 61 5 4, &c.
17 6 3 9 2 9 3 6 7 17 6 3, &c.
61 1 3 1 5 1 3 1 1 1 21 1 3, &c.

Set down the number whose square root is wanted, say 43. This square root is 6 and a fraction. Set down the integer 6 in the first and third row, and 1 in the second row always. Form the successive rows each from the one before, in the following manner:

One row
being		The next row has b′, a′, c′, formed
in this order, thus,
aa′ = excess of b′c′, already formed, over a.
bb′ = quotient of 43 - a² divided by b.
cc′ = integer in the quotient of 6 + a divided by b′.
Thus the second row is formed from the first, as under:
61 = excess of 7 × 1 (both just found) over 6.
17 = 43 - 6 × 6 divided by 1.
61 = integer of 6 + 6 divided by 7 (just found).
The third row is formed from the second, thus:
1 5 = excess of 1 × 6 over 1.
7 6 = 43 - 1 × 1 divided by 7.
1 1 = integer of 6 + 1 divided by 6;

and so on. In process of time the second column, 1, 7, 1, occurs again, after which the several columns are repeated in the same order. As a final process, take the set in the lowest line (excluding the first, 6), namely, 1, 1, 3, 1, 5, 1, 3, &c. and use them by the rule given at the beginning of this article, as follows:

113151311, &c.
11452934131165296
1 2 7 9 52 61 235 296 531

Hence, 6¹⁶⁵/₂₉₆ is very near the square root of 43, not erring by so much as

1 .
296 × 531

If we try it, we shall find (⁶¹⁶⁵/₂₉₆) to be ¹⁹⁴¹/₂₉₆, the square of which is ³⁷⁶⁷⁴⁸¹/₈₇₆₁₆, or 43⁷/₈₇₆₁₆.

This rule is of use when it is frequently wanted to use one square root, and therefore desirable to ascertain whether any easy approximation exists by means of a common fraction. For example, √2 is often used.