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(100,000 x 365.25) In n days, the accumulated loss of time in seconds would be 5T = 2.738x 10-8 x[ 1 + 2 + 3 +... + n] = ^ x(n + n2) x 2.738 x 10-8 = 1.369 x 10-8 x n2, (4.25)

where we make use of a series summation. The approximation sign arises because we have ignored the much smaller quantity n/2 term in the sum of the series 1 + 2 + 3 +... + n. In our example, n = 4000 x 365.25 = 1.461 x 106. Therefore, 8T ~ 2.92 x 104 seconds = 8.12 hours, over the span of 4000 years.

The cause of the long-term deceleration of the Earth's rotation is due to a number of circumstances, but principally due to (lunar) tidal friction. Before we discuss the effects of tidal friction, we must describe the origin of the tides, which were important to ancient navigators, as they are to navigators today.

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