Paper on the Length of the Pendulum, 3S9 
The vibrations of the pendulum having been made in diffe- 
rent arcs, it became necessary to apply a correction, to determine 
what they would have been in an arc infinitely small. For this 
correction, I might have used a formula depending on the de- 
crease of the arcs in geometrical progression, whilst the times 
decreased in arithmetical ; but as there is an uncertainty in ob- 
serving the arcs, amounting to one or two hundredths of a de- 
gree, this method, though preferable in theory, would have 
been an unnecessary refinement in practice.” And again, ‘‘ The 
error arising from the greater length of the vibration in a circu- 
lar arc, being nearly as the square of the arc, if the mean of the 
arcs at the commencement and end of each interval be taken, 
and its square multiplied by 1.635, the difference between the 
number of vibrations made by the pendulum in twenty-four 
hours in a cycloid, and in a circular arc of one degree, the re- 
quired correction will be obtained, to be added to the number 
of vibrations computed.” 
Now, with regard to the first of these extracts, I observe, 
that the alleged uncertainty of one or two hundredths of a de- 
gree in the determination of the arcs of vibration, forms no just 
objection against the employment of the appropriate formula ; and 
that, with Captain KateFs means, namely, a telescope, micro- 
' scope, barometer, thermometer, and micrometer, together with 
a horizontal scale, divided into equal parts, and placed before 
the pendulum at a given distance from the point of suspension, 
the deflection of the pendulum on each side of the vertical line, 
might have been assigned to a great degree of exactness ; and 
hence the angle which it described might have been readi- 
ly deduced even to seconds of a degree, which would have been 
sufficiently exact for the purpose of determining the requisite 
correction, which ought to be applied to any finite arc, in order 
to reduce it to the case of an infinitely small arc of vibration. 
And, with respect to the second extract, I remark, that the 
number 1.635, which is stated to be the difference between 
the number of vibrations made by the pendulum in twenty-four 
hours in an arc of a cycloid, and in a circular arc of one degree, 
is incorrect. It should have been 1.645 nearly ; and this er- 
ror runs through the whole set of experiments. 
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