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.

Early in my professional life, I found that many areas of economics attracted me. I started working and publishing in price theory by 1938. In 1946, I published an early work on linear programming (The Cost of Subsistence) which solved the problem only approximately; George Dantzig soon presented the exact solution. In the 1940s, I began empirical work on price theory, starting with a test of the kinked oligopoly demand curve theory of rigid prices.

"He studied the composition of food-stuffs, and knew exactly how many proteids and carbohydrates his body needed; and by scientific chewing he said that he tripled the value of all he ate, so that it cost him eleven cents a day. About the first of July he would leave Chicago for his vacation, on foot; and when he struck the harvest fields he would set to work for two dollars and a half a day, and come home when he had another year's supply — a hundred and twenty-five dollars. That was the nearest approach to independence a man could make "under capitalism," he explained; he would never marry, for no sane man would allow himself to fall in love until after the revolution."

Vieta was the first algebraist after Ferrari to make any noteworthy advance in the solution of the biquadratic. He began with the type <math>x^4 + 2gx^2 + bx = c,</math> wrote it as <math>x^4 + 2gx^2 = c - bx,</math> added <math>gx^2 + \frac{1}{4}y^2 + yx^2 + gy</math> to both sides, and then made the right side a square after the manner of Ferrari. This method... requires the solution of a cubic resolvent.
Descartes (1637) next took up the question and succeeded in effecting a simple solution... a method considerably improved (1649) by his commentator Van Schooten. The method was brought to its final form by Simpson (1745).

I am sorry not to have known the mathematician who first made use of this method because I would have cited him. Regarding the researches of d'Alembert and Euler could one not add that if they knew this expansion they made but a very imperfect use of it. They were both persuaded that an arbitrary and discontinuous function could never be resolved in series of this kind, and it does not seem that anyone had developed a constant in cosines of multiple arcs, the first problem which I had to solve in the theory of heat.

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.

In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months' time, was achieved in three days by Euler with aid of improved methods of his own... With still superior methods this same problem was solved by the illustrious Gauss in one hour.

The method of exhaustion was not discovered all at once; we find traces of gropings after such a method before it was actually evolved. It was perhaps Antiphon. the sophist, of Athens, a contemporary of Socrates, who took the first step. He inscribed a square (or, according to another account, a triangle) in a circle, then bisected the arcs subtended by the sides, and so inscribed a polygon of double the number of sides; he then repeated the process, and maintained that, by continuing it, we should at last arrive at a polygon with sides so small as to make the polygon coincident with the circle. Thought this was formally incorrect, it nevertheless contained the germ of the method of exhaustion.

These computing machines had already been designed, and some built, by Vannevar Bush, Norbert Wiener, and others, and were almost ready-made for the job. These scientists, as well as others such as von Neumann, Shannon and Bigelow, were in a position to see that machines of an electronic kind were ideally suited to carry out the whole of the operations of range-finding and location without any human intervention whatever. These electronic computing machines were already developed to a very high degree of efficiency for the solution of mathematical equations, and some technical difficulties had led to the suggestion that a process of scanning, similar to that used in television, might be incorporated into the computer. Another innovation was the use of binary notation rather than decimal notation as in the early computer.

In the 1950s, I proposed the survivor method of determining the efficient sizes of enterprises, and worked on delivered price systems, vertical integration, and similar topics.

Nevertheless his prodigious intellectual powers persisted unabated. In 1696, the Swiss mathematician Johann Bernoulli challenged his colleagues to solve an unresolved issue called the brachistochrone problem, specifying the curve connecting two points displaced from each other laterally, along which a body, acted upon only by gravity, would fall in the shortest time.

The first paper I ever wrote was "Gestalt Programming" and that was in 1955. The whole idea there was to replace the laborious writing out of detailed programs and all those steps by having analyzed a problem area well enough so that you had what I later came to call a "systematized solution." Then you could compose different problems of this class by just plugging together pieces of program, and they would in turn be controlled by a pushbutton language. The user would make a number of discreet selections. It's just like nowadays it's done with menus, and when you had indicated all the pieces that you wanted to put together--by these mnemonic names and words for things associated with buttons, switches, with one meaning "period," essentially, for that sentence, you see--all these things would be brought together and that would be the man/machine, manual-intervention mode of problem-solving. I took over the term from studying Gestalt psychology, meaning that everything was brought together at once, as a unit, instead of this laborious step-by-step build-up.

There is nothing more to this than a simple iterative formula. It is so simple that most children can program their home computers to produce the Mandelbrot set. … Its astounding complication was completely out of proportion with what I was expecting. Here is the curious thing: the first night I saw the set, it was just wild. The second night, I became used to it. After a few nights, I became familiar with it. It was as if somehow I had seen it before. Of course I hadn't. No one had seen it. No one had described it. The fact that a certain aspect of its mathematical nature remains mysterious, despite hundreds of brilliant people working on it, is the icing on the cake to me.

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.