"EAM room also had a particular keypunch machine called the 407. Not only could it punch cards, but it could also read cards, sort them, and print them on listings. No one seemed to be guarding these machines, which were computers, sort of. Of course, using them would be no picnic: one needed to actually wire up what was called a plug board, a two-inch-by-two-inch plastic square with a mass of holes in it."

This really dates me, but I've used and wired plugboards for an IBM 407 tabulator. The plugboards are about two feet square, not two inches.[1]

A 407 cannot sort cards; it has one card input hopper, one output hopper, a printer, about twenty mechanical decimal counters, and relay logic. It can add and subtract, but not multiply or divide. (With enough plugboard wiring, you can, with great difficulty, do repeated adds to get a multiply, maybe at one multiply per minute. Multiplication was usually done on a separate machine, the 602A, which could do it much faster.)

Internally, it's not just relays. The counters are driven from rotating shafts through clutches. The machine is full of shafts and bevel gears to distribute the power from the motor. There's a built-in lubrication system, with tiny tubes fed by a very slow oil pump.

A 407 itself cannot punch cards, but it has a connector into which a cable from an IBM 514 Reproducing Punch can be plugged. The 407 can then send data to the 514, which can punch cards. The connector alone is about 3" x 6", with a 2 inch diameter cable.

This and the 602A Multiplier were IBM's last pre-electronic machines. Then IBM started using tubes. The 407 remained in the IBM product line until 1976, and many continued to grind on into the 1980s.

The 602A, the multiplier, is interesting because it's the foundation of IBM's computer line. The 602A was electromechanical, programmed with a plugboard. Then IBM introduced the IBM 603 electronic multiplier in 1946, which was like a 602A, but used tube electronics instead of multiplying mechanically. Only 100 were made, to find out what it was like to sell and service machines with tube electronics. Next was the 604, which was a bigger and better 603 and was produced in quantity. It still had a plugboard program. This was followed by the 605, another incremental improvement, and the Card Programmed Calculator, which was a 605 plugged into a 407 and a 527 punch, which was a modified 514, plus some additional memory units (10 numbers each, mechanical!). This was almost a general purpose computer, but it still used plugboards. Then came the 607, which integrated all those units into a more coherent system.

There were a few more plugboard-wired machines, but the big step was the IBM 650, in 1954. This reused many of the parts from the 610, but was organized like a modern computer. Finally, the plugboard disappeared. Programs were stored on the magnetic drum. This was a useful workhorse machine; thousands were made. Knuth learned to program on one. The 650 was very profitable for IBM; it was the first electronic computer produced in volume.

Meanwhile, IBM had also produced the IBM 701 (1953) and the IBM 703 (1954), which were full-blown huge tube computers with tape drives and drums. But those were originally special purpose DoD projects; the 701 was called the Defense Calculator. They were too expensive to be volume products. The 6xx machines made money.

The point of all this is that there was no one big breakthough. IBM just kept developing improved models. If there hadn't been ENIAC or UNIVAC or Bletchley Park, the 6xx line would have been improved year by year into modern computers. In fact, it was; the 6xx line led to the IBM 1401, released in 1959, the transistorized business computer that powered accounting for big business for much of the 1960s. The 1401's architecture is weird, partly due to the punched card legacy, but it got the job done.

[1] http://www.columbia.edu/cu/computinghistory/407.html