Toepler's method for square root on example

Statement of the problem

We evaluate square root of 1984 (the number from Orwell's book). Using modern tools we know that $\sqrt{1984}\approx 44.5421149$.

We show how to obtain this result on mechanical calculator.

Toepler's method

The numbers from the first column are subtracted from 1984, The number 900 would give negative result (1984-100-300-500-700-900<0), therefore the first digit is 4 (4 digits can be subtracted from 1984: 100, 300, 500, 700) and we continue with the next column with the number 81.

	1-st digit	2-nd digit	3-rd digit	4-th digit	5-th digit	6-th digit
	100	81	8.81	0.8901	0.089081	0.00890841
	300	83	8.83	0.8903	0.089083
	500	85	8.85	0.8905
	700	87	8.87	0.8907
			8.89
Count of rows	4	4	5	4	2	1

Conclusion: $\sqrt{1984}\approx 44.5421$ (with 26 subtractions and 6 additions - these additions are used to cancel subtraction which sends the total to negative number)

Toepler's method multplied by 5

Like the previous method, but the numbers are subtracted from $5\times 1984=9920$. If negative number is obtained by subtraction, cancel this subtraction and continue in the next column. Note, that the number which is subtracted converges to the square root (neglecting the position of the separator of decimal places).

	1-st digit	2-nd digit	3-rd digit	4-th digit	5-th digit	6-th digit
	500	405	44.05	4.4505	0.445405	0.04454205
	1500	415	44.15	4.4515	0.445415
	2500	425	44.25	4.4525
	3500	435	44.35	4.4535
			44.45
Count of rows	4	4	5	4	2	1

Conclusion: $\sqrt{1984}\approx 44.5421$ (with 26 subtractions and 10 additions - multiplication by 5 is 4-times addition)