oot of the matter, and perhaps the
fact questioned by him should be more clearly placed beyond dispute.

I will dismiss the introductory part of his letter, merely observing
that his "logical inference" is quite gratuitous and unwarranted. He
says himself that its absurdity is obvious, in which I quite agree
with him.

The real question is this: What is the figure representing the
acceleration of the motion of a piston, controlled by a crank which
revolves with a uniform velocity? I stated it to be a right-angled
triangle, and indicated, as I supposed, clearly enough, a simple
method by which this could be shown. Your correspondent claims that
the calculation, according to my own rule, gives a figure of a totally
different form, and one that shows the acceleration, as well as the
motion, to be reduced to zero at the commencement of the stroke. Let
us see. Let the straight line, AJ, in the following figure, represent
half the stroke of the piston, and let the distances, AB, AC, etc., on
this line, represent the versed sines of 10 deg., 20 deg., etc., up to 90 deg., or
the motion of the piston while the crank is moving through these arcs.
At the points A, B, C, etc., erect the perpendiculars, Aa, Bb, Cc,
etc., and let the length of each of these ordinates represent the
acceleration imparted in a given time at that point of the stroke.
Then will AJ be to Aa as IJ is to Ii, as HJ is to Hh, etc., showing
that the straight line, aJ, connects the extremities of all the
ordinates, and that the triangle, AJa, represents the acceleration of
the motion of the piston, from the commencement to the middle of the
stroke.

[Illustration]

The following table will enable any one to make the calculations
proving the truth of the above proposition:

Degrees. Versed sine. Motion for 10 deg. Acceleration during 1 deg..
0 deg. .0000000 _Aa_ .0003046
10 deg. _AB_ .0151922 _AB_ .0151922 _Bb_ .0003001
20 deg. _AC_ .0603074 _BC_ .0451152 _Cc_ .0002862
30 deg. _AD_ .1339746 _CD_ .0736672 _Dd_ .0002638
40 deg. _AE_ .2339556 _DE_ .0999810 _Ee_ .0002332
50 deg. _AF_ .3572124 _EF_ .1232568 _Ff_ .0001958
60 deg. _AG_ .5000000 _FG_ .1427876 _Gg_ .0001523
70 deg. _AH_ .6579799 _GH_ .1579799 _Hh_ .0001041
80 deg. _AI_ .8263518 _HI_ .1683719 _Ii_ .0000529
90 deg. _AJ_ 1.0000000 _IJ_ .1736482 _Jj_