Many of the readers of this journal may like to participate in the discussion of the following proposition. The statement is this:
The space through which a body, near the surface of the earth, at mean latitude, in vacuo, descends by virtue of the accelerating force of gravity in 1/1000 of an hour is precisely 2,500 geometric inches = 100 geometric cubits = the side of a square geometric acre.
[The geometric inch is taken, in accordance with the view of Sir John Herschel, at 1/1,000,000,000 of twice the polar axis of the earth, and equals 1-1/1000 English inches very nearly.]
The strict decimal relation of the proposition is shown by the following table. It has been tested by Clairaut's theorem, and by other existing expressions, and has been found to agree, far within the probable limits of errors in observation, with the most approved values of the constant. In fact, it is contained in the existing expressions; but the decimal relation does not appear unless we state the unit of linear measure as a decimal of the earth's semi-polar axis, and, at the same time, divide the circle, both for time and for general purposes, geometrically, i.e., by strict decimalization upon the hour-angle. A mathematical reason underlies the proposition.
Time in Acquired Squares Total Ratio of Descent in Thousandths Velocity, of the Descent, Spaces, Each Successive of an Hour. Cubits. Time. Cubits. Interval of Intervals, Time. Cubits. 1 200 1 100 1 100 2 400 4 400 3 300 3 600 9 900 5 500 4 800 16 1,600 7 700 5 1,000 25 2,500 9 900 6 1,200 36 3,600 11 1,100 7 1,400 49 4,900 13 1,300 8 1,600 64 6,400 15 1,500 9 1,800 81 8,100 17 1,700 10 2,000 100 10,000 19 1,900
So that -
Cubits. Acre Sides. In 1/10,000 of an hour, the total descent = 1 = 1/100 In 1/1000 of an hour, the total descent = 100 = 1 In 1/100 of an hour, the total descent = 10,000 = 100
And so on, in strict decimal relation with the earth's semi-polar axis.
A two-fold reason why the constant for latitude 45° is vastly better than any other, is in its having this simple relation with the semi-axis, and at the same time a less complex way of applying the correction for latitude.
JACOB M. CLARK.
New York, February, 1885.