Starting in May, many will have the opportunity to see for themselves how they did computing the old-fashioned way: with lots of gears, a big crank and some muscle.

The Computer History Museum, in the US, will unveil a new construction, the first in the United States, of the 19th century British mathematician Charles Babbage's Difference Engine No. 2, an improved version of his earlier mechanical digital calculator.

Babbage finalized the design in the late 1840's but it was not built during his lifetime, or for a long time afterward. Finally, in the late 1980's, London's Science Museum launched the first and until now only full-blown construction project, based on Babbage's original detailed drawings, and in 1991 unveiled the completed calculator, 11 feet long, 7 feet high, with 8,000 parts in bronze, cast iron and steel, weighing about 3 tons.

In operation, it looks rather like an industrial version of a street organ.

In 2000, the museum added the calculator's complex printer, almost as big and at 2.5 tons nearly as heavy.

The American model is the second built from Babbage's plans. This one was commissioned and paid for by Microsoft millionaire Nathan Myhrvold, and built by the Science Museum.

Myrhvold and Guest Curator Doron Swade, an authority on Babbage and the man who led the Science Museum's first construction effort, will speak on Babbage's creation Thursday, May 1 at the Computer History Museum (the event is sold out). The exhibit itself opens Saturday, May 10, with demonstrations of the Difference Engine.

The Difference Engine is not the only geared calculation marvel in the world. In late 2006, an international team of scientists revealed new details of a 2,100-year-old astronomical calculator, the Antikythera Device, which used a complex system of 37 finely cut bronze gears to accurately show the changing positions of the sun and moon, with its phases, and possibly predict solar and lunar eclipses.

So what is a "difference engine?" Not being a math wizard, I'll give this my best shot, drawing in part on the online expertise of Andrew Carol, an Apple software engineer who built a simpler difference engine, entirely of plastic LEGO pieces. You can find more details at Carol's Web site.

Carol points out that calculating the trigonometric and logarithmic tables used for an array of navigation, engineering and scientific purposes was all done by hand, with a skilled mathematician directing the efforts of a room filled with less-skilled people, called "computers," who could be trusted to do reliable arithmetic. Babbage was one of a number of people trying to automate this process with mechanical devices in the 19th Century.

In effect, the difference engine is a kind of shortcut to determine a series of successive mathematical values. It's based on something called the method of differences, developed by Sir Isaac Newton. Carol gives the example of multiplying 5 by successive numbers, such as 6, 7, 8. "In simple terms, the method of differences is based on the observation that if the work has already been done to multiply 5 by 5, [then] that work can be reused to multiple 5 by 7 with the addition of another 5 into the previous total," he writes.

Like this:

5 x 6 = 30

5 x 7 = 35 by adding 5 to the previous total

5 x 8 = 40 by adding 5 yet again to the previous total

As Carol puts it: "Successive multiplications have been reduced to an identical number of successive additions. As long as we are willing to calculate the table entries [remember this is all about tables of values] in order we can save an enormous amount of work."

Babbage's difference engine applies this same idea to solving polynomial equations, which are widely used in math and science. Polynomials are built from a combination of variables and constants; use only addition, subtraction and multiplication; and only exponents that are constant positive whole numbers.