BOOK REVIEW / SCIENCE : Recognition Eluded Him--Until Now : LONGITUDE: The True Story of a Lone Genius Who Solved the Greatest Scientific Problem of His Time <i> by Dava Sobel</i> ; Walker & Co. $19, 184 pages

Here’s a swell little book that tells an amazing story that is largely forgotten today but that deserves to be remembered.

It is the story of the problem of navigation at sea--which plagued ocean-going mariners for centuries--and how it was finally solved.

It is the story of how an unknown, uneducated and unheralded clockmaker solved the problem that had stumped some of the greatest scientific minds. And it is the story of how the Establishment of the 18th Century tried to block his solution.

The essential problem is this: In the middle of the ocean, how can you tell where you are? That is, how can you tell how far east or west of your starting point you have gone?

Determining one’s latitude at sea is easy. You can tell how far north or south of the Equator you are by observing the stars at night. But longitude is a much more difficult matter.

By the 17th Century, England was a great sea power, but sailors were lost as soon as they got out of sight of land. There were many shipwrecks, and thousands of lives were lost because captains did not know where they were.

The longitude problem, as it was called, was so important that in 1714, the British Parliament offered a prize of 20,000 pounds--equivalent to millions today--to anyone who could solve it.

For several centuries, astronomers and mathematicians thought that the answer to the question could be found in observing the heavens. Many years of work went into this approach.

A completely different approach was to have a clock aboard the ship that would tell what time it was where you started--in, say, London. Then you could observe local noon wherever you were by the sun and compare it to what time it was at that moment in London.

Since one hour equals 15 degrees of longitude (360 degrees divided by 24 hours), the difference in time would tell you where you were. Thus, if it was noon where you were when the clock said it was 2 p.m. in London, the two-hour difference meant you were 30 degrees west.

But in the era of pendulum clocks, no timepiece was reliable at sea. Isaac Newton himself wrote: “By reason of the motion of the Ship, the Variation of Heat and Cold, Wet and Dry, and the Difference of Gravity in different Latitudes, such a watch hath not yet been made.”

Enter John Harrison of Yorkshire. He built a clock and then a watch that could keep time at sea, and Dava Sobel, a science writer, tells his story with grace, clarity and affection.

Actually, as Sobel tells us, Harrison devoted his life to the problem and built several sea-going clocks over a period of decades. In 1761 and 1762, one of his instruments--called H-4--was given a sea trial from England to the West Indies and back, a six-month voyage.

“Upon its arrival home on March 26,” Sobel writes, “H-4 was still ticking. And its adjusted total error, outbound and homebound combined, amounted to just under two minutes.”

But the scientific Establishment, which still thought that the answer lay in observing the stars, conspired to keep Harrison from getting the prize. It put up many barriers and many objections.

After years of frustration, in 1772, Harrison’s son, William, appealed directly to King George III. The king is reported to have said, “These people have been cruelly treated,” and to have told William, “By God, Harrison, I will see you righted.”

The money--but not the prize--was subsequently awarded to John Harrison. The Longitude Prize itself was never awarded.

Harrison’s clocks--the actual clocks, some still ticking--can be seen today at the Maritime Museum in London. Sobel went to see them. She writes:

“Coming face-to-face with these machines at last--after having read countless accounts of their construction and trial, after having seen every detail of their insides and outsides in still and moving pictures--reduced me to tears.”

And small wonder. Navigation today is no problem at all. Using satellites, you can effortlessly fix your position at sea within a few feet.

But it was not always so. And Dava Sobel tells of a brilliant craftsman who made a stunning contribution to knowledge and to human affairs.