Square root on mechanical calculator

The first video shows evaluation of the square root of 1952 to five digits (three decimal places) using Monroe multiplication tables. The computations is done on Monroe calculator Model K from 1920. The video starts with introduction the parts of the calculator, application of Monroe tables follows and the whole computation is also shown. The final part of the video explains mathematical background - linear approximation.

 

  • Monroe tables on Monroe calculator
  • Why the square root of 1952 has bee choosen for the video? The year 1952 is the year when the first mechanical calculator with automatic extraction of square root has been introduced to market (Friden SQRT).
  • Thanks to M. Monroe and Naslouchej mandarinkám for a pleasant music

The three other videos describe the evaluation of $\sqrt{1896}$ (the year when Christel Hamann established his company). Videos are in Czech with English subtitles.

 

  • part 1 - short introduction to the Hamann's calculator and short description of three methods
  • part 2 - performing the methods
  • part 3 - performing the methods (no comments - just raw computation, normal speed of video, nice sound of the computing machine)
  • Thanks to Naslouchej mandarinkám for a pleasant music

Hamann's calculator

The computations will be demonstrated on the calculator Hamann Manus, model R. This calclator is not pinwheel calculator, but it enjoys the switching latch-wheel technology. As a big advantage, this calculator allows automatic division! See some comments concerning this calculator.

You may try these methods on vintage computers. You may use also online and gratis Curta simulator or Schubert simulator

Töpler's method

This method is also referred as Toepler's method. Following this method, the square root can be evaluated by subtracting odd numbers and counting, how many numbers have been subtracted. It is based on the fact that the sum of the first $k$ odd numbers is $k^2$ (as follows from the formula for the sum of arithmetic sequence).

Advantages : no initial guess necessary, all numbers obtained are correct

Disadvantages : slow convergence (but you get the first digits first, i.e. you obtain the first one or two digits relatively fast), needs a lot of work in input register (skip even numbers, change inputs in two columns), the input register is total mess (almost -- actually it is a double of the counter register, i.e. double of the square root)

Improved Töpler's method

Töpler's method can be simplified by subtrating five-multiples of odd numbers from five-multiple of the number from which we extract the square root. You may see the table which shows the computation of $\sqrt{1984}$ using both methods (plain Töpler and 5-multiple Töpler).

Advantages : like Töpler's method, but it is easier to change the numbers being subtracted and the result is in both input and counter register, i.e. both registers contain the final square root

Disadvantages : slow

This method does not need initial guess, can be performed automatically and is implemented in advanced versions of Friden - the fastest mechanical calculator of the world. On older Friden you have to evaluate square root manually as on this video - see also the discussion and the comment by PopeLando.

Linear approximation

Classical mathematical methods which replaces locally(!) a function by its tangent. The evaluation of square root is replaced by one operation of addition and one multiplication (or division).

Given function $f(x)=\sqrt{x}$ and the value $x_0$ with known value of $\sqrt{x_0}$ we find the linear approximation of the function in the neighborhood of the point $x_0$.

Advantages : fast and comfortable

Disadvantages : Needs to look into the table for coefficients and does not allow iteration. Limited precision (to get more significant digits you can use one of the following formulations of Newton-Raphson method).

Newton-Raphson method

Given value $a$ we find $\sqrt{a}$ as the positive zero of the function $f(x)=x^2-a$.

Application for mechanical calculator: Heron formula (division required), Herrmann's method, also see below

Advantages : fast convergence (quadratic), allows iterations

Disadvantages : Needs initial guess. Different implementations depending on the properties of each mechanical calculator (presence of carry mechanism in counter, ability to do automatic division, ...)

Possible implementations of Newton-Raphson on mechanical calculator

We have to evaluate $\sqrt{a}$. The initial guess is $\sqrt{a}\approx x_0$.

Literature

  1. Mechanical Calculator web site by Wim Hasselo
  2. Arthur's blog by Arthur Vause
  3. Computarium lcd by Francis Massen
  4. Rechenmaschinen by Prof. Dr.-Ing. Christian-M. Hamann
  5. The national museum of american history
  6. www.curtamania.com by Alfredo Logioia