Measurement and Limits in the Principia, Section 10
Chris Smeenk and George Smith

This paper has two related aims: first, to elucidate the methodological
sophistication displayed in Newton’s treatment of constrained motion in
Section 10 of the Principia, and second, to discuss measurement and limitcase
reasoning more generally, drawing on our reading of Newton.
Section 10 of Book 1 of the Principia includes 10 theorems regarding
constrained motion – motion on inclined planes, more general constraint
surfaces, and the oscillating motion of pendulums – that significantly extend
earlier results obtained by Galileo, Huygens, and others. These topics
were of central importance to the history of mechanics before and after the
Principia. Newton’s treatment differs from the earlier mechanical tradition
in two striking ways: first, pathwise independence of acquired velocity
follows directly from the Laws of Motion, rather than being assumed as
a separate principle; second, Galilean gravity, a force with constant magnitude
directed along parallel lines, is replaced with gravity treated as a
centripetal force. Newton proved a more general version of the key result
of Huygens’s Horologium: he stated a necessary and sufficient condition
for isochrony, and established that oscillations along a (generalized) cycloid
are isochronous for a force law varying as f (r )/r .
The results of Section 10 are particularly striking when one considers
why Newton pursued them and how they relate to the Principia as a whole.
This section is where Newton considers limiting cases most carefully, and
thus it illustrates an under-appreciated aspect of his methodology. We will
argue that Newton’s work in Section 10 establishes that the central results
of the Horologium survive the conceptual transition from Huygensian mechanics
to Newton’s more general framework. The fact that Huygens’s
results can be recovered as well-defined limit cases justifies Newton’s reliance
on these results, most prominently Huygens’s measurement of surface
gravity. In addition, Newton’s careful treatment insures that evidence
in favor of Galilean mechanics carries over to the more general theoretical
framework of the Principia, and there will be no grounds for objecting to
the new physics on the basis of earlier results regarding constrained motion.
Turning to more general questions, our discussion of Newton’s method
addresses two philosophical issues. First, Newton’s approach to measurement
requires that a real system used to measure a quantity can approach
1
ideal precision in specific circumstances. Assessing whether the requirement
holds depends on the theory itself, not on external stipulations regarding
observability and measurement. In this case, Newton showed
that the actual motions of a cycloidal pendulum bob approximate a motion
that would be exactly periodic in specific circumstances, and he further
quantified departures from isochronism. We will illustrate the importance
of considerations along these lines by contrasting this case with
an example in which this requirement does not hold. Second, Newton’s
results in Section 10 make it possible to characterize the relationship between
Galilean and Newtonian mechanic quite clearly. Galileo’s theory is
a suitable approximation of Newtonian theory in a specific circumstance,
namely near the surface of the earth (on the assumption that it has uniform
density). The limiting relationship between theories is not simply
characterized as holding between fundamental equations, but instead depends
on the descriptions of particular situations. Furthermore, various
lawlike relationships in Galilean mechanics retain their lawlike force, as
Newton’s limit-case reasoning shows. More generally, we will argue that
such limit-case reasoning provides a convincing reply to (one aspect of)
Kuhnian worries regarding incommensurability.