Seeing there is nothing, (right well beloved students of mathematics,) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculations, that the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expence of time, are for the most part subject to many slippery errors, I began, therefore, to consider in my mind, by what certain and ready art I might remove these hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of perhaps hereafter: But amongst all, none more profitable than this, which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away even the very numbers themselves that are to be multiplied, divided, and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and substraction, division by two, or division by three. Which secret invention being, (as all other good things are,) so much the better as it shall be the more common, I thought good heretofore, to set forth in Latin for the public use of mathematicians.

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

The speed and density of computation have been doubling every three years (at the beginning of the twentieth century) to one year (at the end of the twentieth century), regardless of the type of hardware used. ...Despite many decades of progress since the first calculating equipment was used in the 1890 census, it was not until the mid-1960s that this phenomenon was even noticed (although Alan Turing had an inkling of it in 1950).

Even Magnifico was put to work on the calculating machine for routine computations, a type of work, which, once explained, was a source of great amusement to him and at which he was surprisingly proficient.

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.

One of the first applications of the simplex algorithm was to the determination of an adequate diet that was of least cost. In the fall of 1947, Jack Laderman of the Mathematical Tables Project of the National Bureau of Standards undertook, as a test of the newly proposed simplex method, the first large-scale computation in this field. It was a system with nine equations in seventy-seven unknowns. Using hand-operated desk calculators, approximately 120 man-days were required to obtain a solution. … The particular problem solved was one which had been studied earlier by George Stigler (who later became a Nobel Laureate) who proposed a solution based on the substitution of certain foods by others which gave more nutrition per dollar. He then examined a "handful" of the possible 510 ways to combine the selected foods. He did not claim the solution to be the cheapest but gave his reasons for believing that the cost per annum could not be reduced by more than a few dollars. Indeed, it turned out that Stigler's solution (expressed in 1945 dollars) was only 24 cents higher than the true minimum per year $39.69.

Multiplication of whole Numbers ...Note, that for the more easie solution of this proposition, it were necessary to have in memory the multiplication of the 9 simple Characters among themselves, learning them by rote out of the Table here placed...

There is no problem in all mathematics that cannot be solved by direct counting. But with the present implements of mathematics many operations can be performed in a few minutes which without mathematical methods would take a lifetime.

When we make our own calculations, we need so many numbers and factors that any mistake is possible. The Lord's calculation boils down to love.

A man, by working 24 hours a day, could multiply himself 3 times. To multiply himself more than 3 times the only recourse is to train others to take over some of his work.

The question you raise “how can such a formulation lead to computations” doesn’t bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand – and it always turned out that understanding was all that mattered.

Since 1954, the raw speed of computers, as measured by the time it takes to do an addition, increased by a factor of 10,000. That means an algorithm that once took 10 minutes to perform can now be done 15 times a second. Students sometimes ask my advice on how to get rich. The best advice I can give them is to dig up some old algorithm that once took forever, program it for a modern workstation, form a startup to market it and then get rich.

The desire to economize time and mental effort in arithmetical computations, and to eliminate human liability to error is probably as old as the science of arithmetic itself.

The simpler, the better. Complications lead to multiplicative chains of unanticipated effects.