It is unworthy of excellent men to lose hours like slaves in the labour of calculation which could safely be relegated to anyone else if machines were used.

(Describing, in 1685, the value to astronomers of the hand-cranked calculating machine he had invented in 1673.)

Careers were made. Bletchley, meanwhile, also proved a haven for thinking about computation: Bombes were machines, and they ran programs to solve problems that humans, by themselves, could not.

As he got deeper into it he began to run into a lot of mathematics, which he had to work out with a pencil and paper—you couldn’t do magical equations with computers, they just spat out inconsistent answers before hanging completely. Magical math had to be thought through with a brain.

We needed a man to repair the machines, to keep them going and everything. And the army was always going to send this fellow they had, but he was always delayed. Now, we always were in a hurry. Everything we did, we tried to do as quickly as possible. In this particular case, we worked out all the numerical steps that the machines were supposed to do — multiply this, and then do this, and subtract that. Then we worked out the program, but we didn’t have any machine to test it on. So we set up this room with girls in it. Each one had a Marchant: one was the multiplier, another was the adder. This one cubed — all she did was cube a number on an index card and send it to the next girl. We went through our cycle this way until we got all the bugs out. It turned out that the speed at which we were able to do it was a hell of a lot faster than the other way, where every single person did all the steps. We got speed with this system that was the predicted speed for the IBM machine. The only difference is that the IBM machines didn’t get tired and could work three shifts. But the girls got tired after a while.

When the man was totting up the fare, all the de luxe supplements, he worked the sum out twenty times on the adding machine. The same sum, twenty times. Why? Did he think the machine was going to change its mind?

The miraculous powers of modern calculation are due to three inventions : the Arabic Notation, Decimal Fractions and Logarithms.

"Turing's 'Machines'. These machines are humans who calculate. And one might express what he says also in the form of games. And the interesting games would be such as brought one via certain rules to nonsensical instructions. I am thinking of games like the "racing game". One has received the order "Go on in the same way" when this makes no sense, say because one has got into a circle. For any order makes sense only in certain positions."

This was his mind, a storehouse, a computer programmed to life, minute by minute, hour by hour, day and night.

The total number of multiplications involved in the practical solution of our problem exceeds 450,000. This task alone would mean a two-year job, at 120 multiplications per hour. Fortunately, the recent invention of the Simultaneous Calculator by Professor Wilbur of the Massachusetts Institute of Technology has made it possible to perform all the necessary computations in a small fraction of the time they otherwise would have required. This apparatus solves nearly automatically a system of nine simultaneous linear equations.

God used beautiful mathematics in creating the world.

God is a mathematician of a very high order and He used advanced mathematics in constructing the universe.

Two centuries ago Leibnitz invented a calculating machine which embodied most of the essential features of recent keyboard devices, but it could not then come into use. The economics of the situation were against it: the labor involved in constructing it, before the days of mass production, exceeded the labor to be saved by its use, since all it could accomplish could be duplicated by sufficient use of pencil and paper. Moreover, it would have been subject to frequent breakdown, so that it could not have been depended upon; for at that time and long after, complexity and unreliability were synonymous.

A geometrician has learned to perform the most difficult demonstrations and calculations, as a monkey has learned to take his little hat off and on...

Some, of my unmathematical friends have incautiously urged me to include a note about the origin of modern calculating machines. This is the proper place to do so, as the Queen of queens has enslaved a few of these infernal things to do some of her more repulsive drudgery. What I shall say about these marvelous aids to the feeble human intelligence will be little indeed, for two reasons: I have always hated machinery, and the only machine I ever understood was a wheelbarrow, and that but imperfectly.