I am indebted to Mr. Archibald Smith for this very ingenious notion, which I have worked into the following Tables.  In Table I., I give the appropriate number of seconds corresponding to paces of various lengths.  I find, however, that the pace of neither man nor horse is constant in length during all rates of walking; consequently, where precision is sought, it is better to use this Table on a method of approximation.  That is to say, the traveller should find his approximate rate by using the number of seconds appropriate to his estimated speed.  Then, knowing the length of pace due to that approximate rate, he will proceed afresh by adopting a revised number of seconds, and will obtain a result much nearer to the truth than the first.  Table I. could of course be employed for finding the rate of a carriage, when the circumference of one of its wheels was known; but it is troublesome to make such a measurement.  I therefore have calculated Table ii., in terms of the radius of the wheel.  The formulae by which the two Tables have been calculated are, m=l x 0.5682 for Table I., and m=r x 3.570 for Table ii., where m is the appropriate number of seconds; l is the length of the pace, or circumference of the wheel; and r is the radius of the wheel.

The Tables will be found on the next page.

[Tables I and ii appear on p 34].

b.  When the length of Pace is unknown till after observation.—­In this case, the following plan gives the rate of travel per hour, with the smallest amount of arithmetic.

For statute miles per hour—­Observe the number of paces (n) taken in 5.7 seconds:  let i be the number of inches (to be subsequently determined at leisure) in a single pace; then ni/100 is the rate per hour.

For geographical miles per hour—­The number of seconds to be employed is 5.  This formula is therefore very simple, and it is a useful one. (A statute mile is 1760 yards, and a geographical mile is 2025 yards.)

For finding the rate in statute miles per hour in a carriage—­Observe the number of revolutions (n) made by the wheel in 18 seconds:  let d be the number of inches in the diameter of the wheel; then n d/200 is the rate per hour.

The above method is convenient for measuring the rate at which an animal gallops.  After counting its paces it may be through a telescope, during the prescribed number of seconds, you walk to the track, and measure the length of its pace.  If you have no measuring tape, stride in yards alongside its track, to find the number of yards that are covered by 36 of its paces.  This is, of course, identical with the number of inches in one of its paces.

Convenient Equivalents.—­The rate of 1 mile per hour, is the equivalent to each of the rates in the following list:—­

   Yards.  Feet.  Inches.
   29.333, or 88.000, or 1056.000, in one minute
 or 0.488, or 1.466, or 17.600, in one second