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Before the stored program computer became widely available the most common means of performing computations was by some type of mechanical or electromechanical desk calculator, although punched card equipment was used extensively when it was available. Moreover desk calculators were a useful and even necessary adjunct to the stored program computer as they provided a means of performing test calculations to be used during the developing and checking of computer programs. Starting in the early 1970s they were gradually replaced by the electronic pocket calculator and soon they disappeared almost completely from offices and laboratories. Almost all of them were discarded and now only a few remain on display in various places. Possibly a very few remarks about desk calculators might be useful here to remind some readers that such machines actually existed.

I have already mentioned the octal calculator in the the Computation Centre at the University of Toronto. I can still remember the effort involved in carrying a conventional decimal machine across the campus from the Compuation Centre to my office in University College. In 1949 the statistical laboratory at Toronto was in a large room on the top floor of University College adjacent to the office of the Chairman of the Department of Mathematics. The room had a skylight which leaked when it rained. Amongst the machines were a few old Millionaire calculators which were first introduced in 1893 and became one of the most important machines for scientific calculation for about thirty years. It was this machine that was used by the astronomer Percival Lowell for the calculations that resulted in the discovery of the planet Pluto.

Statistical calculations at the University of Alberta too depended for a surprisingly long time on desk calculators as evidenced by the following remarks in Professor E. S. Keeping's history of the Department of Mathematics:

Equipment is another item which has increased significantly in cost in recent years. For a long time the laboratories in elementary statistics used small Monroe calculators which were cranked by hand, and there were only one or two electric desk-calculators in the whole department, of rather old-fashioned type. By 1962 many of the old calculators were almost worn out, and some newer types of hand calculators were purchased. [These were probably the Swedish-made Odhner calculators which weighed thirteen pounds and which accomplished multiplication and division by addition and subtraction with repeated shifting.] A little later some improved electric calculators, such as the Friden square-root type, came along and then some desk electronic models were purchased.

The Friden calculator referred to by Professor Keeping was the only type of calculator I have used that had the facility to calculate square roots automatically. With other calculators square roots had to be found iteratively or from tables. To illustrate just how onerous was the calculation of square roots on most desk calculators we shall give a brief discussion of method recommended by the makers of the Marchant calculator.

The Marchant Calculating Machine Company produced in 1941 a copyrighted one-page table of "Square Root Divisors" a portion of which is given here. The "Marchant Method" of using these tables may be illustrated by the following example:

Suppose we wish to find the square root of 1234.56, which we can nowfind immediately with the aid of a pocket calculator to be 35.136306 to six places of decimals.
First we select from Column A the three-digit number nearest to the three left significant digits of the given number. Add this number, which is 124, to the given number, aligning the first significant digits, giving 1240 + 1234.56 = 2474.56. Now, since there is an even number of digits before the decimal point, divide this sum by the number in Column 2 giving

2474.56 / 7042727

or 0.00035136390. (If there had been an odd number of digits before the decimal point, the divisor would have been taken from Column 1.) Finally we locate the decimal point by inspection of the original number, and we have that the square root of 1234.56 is 35.13639 which is correct to almost four decimal places. The discussion on the reverse side of the tables shows that this method is based on the well-known Newton-Ralphson iterative method of finding square roots.

Some of the early statistics texts have discussions of the use of desk calculators which appear quaint today. A good example is the seven-page Appendix C of the well-known
*Applied General Statistics* which was first published in the 1940s. A discussion is given of the use of adding machines, hand-operated and electric calculating machines, and slide rules. The first part of the discussion of calculating machines, which begins with an illustration of a hand-operated Monroe, is representative:

Calculating machine. A machine that will both multiply and divide (as well as add and subtract) is shown in Figure 2.

Addition and subtraction. To set the machine for addition or subtraction, release the repeat key. To add a number put it on the keyboard the same as with an adding machine and turn the handle forward (clockwise); to subtract it, turn the handle backward (counterclockwise). Turning the handle registers the result in the lower dial and clears the keyboard.

Multiplication. Multiplication can be thought of as repeated addition. ...

As a final note I might mention that a couple of years ago I put together a small display of calculators at the request of the the Dean of Science, Dr. R. E. (Dick) Peter, who generously supplied the funds. The design and assembly of the exhibit was done by staff of Museums and Collections Services with me providing the artifacts and the accompanying text. A discussion of the completed exhibit, "Computation: From Abacus to Silicon Chip", is given at Calculators.

There is a marvellous display of old and new calculators and information about them at The X-Number World.

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