of Edinburgh, Session 1870-71. 3G3 
proportional to its own cosine; or, writing A for this angle, we 
have 
dAcc cos A . dt , dt <x sec A . dA 
and, therefore, the time occupied in passing over some fixed 
minute portion of the arc at A is proportional to the secant of the 
angle NZA. 
In Mercator’s Projection of the Sphere, the differences of the 
meridional parts are proportional to the secants of the latitudes, 
wherefore the time of describing 
the arc 1ST A must he proportional 
to the meridional part correspond¬ 
ing to the angle NZA, that is, 
must be proportional to the 
logarithmic tangent of 45° + ^A. 
Measure off then some distance 
ZE horizontally to represent the 
linear unit, and bisect the angle 
AZE by the line ZT meeting the 
plumb-line from E in T, the time 
of passing along NA is propor¬ 
tional to the logarithm of ET, 
or rather to the logarithm of the 
ratio of ET to EZ. Hence, when 
the angle is given we can readily 
compute the time, or when the time is given we can as readily 
compute the angle; and thus for this particular case the problem 
is completely resolved. 
Making El equal to EZ, if we make a series of continued pro¬ 
portionals El, EK, EL, ET, EU, &c., and, joining Z with the 
several points, make angles doubles of EIK, EIL, &c., we shall 
obtain the positions of the moving body after equal intervals of 
time. The time of its reaching Z is thus infinite. 
The relation of the continuous to the reciprocating motion may 
be exhibited by a simple contrivance. Let two straight rods 
AO, CE be jointed at the point C, and let the two ends A, B be con¬ 
nected by a straight line, say an elastic thread. 
If the rods be turned so as to lessen the angle ACE, the angles 
Fig. 1. 
