Lenses and Waves

96 CHAPTER 3
During the final weeks of 1659, Huygens took up and solved a problem
that Mersenne had discussed 12 years earlier in Reflexiones physico-mathematicae
(1647). The problem was to determine the distance traversed by a body in its
first second of free fall, which amounts to determining half the value of the
constant of gravitational acceleration. After having tried Mersenne’s
experimental approach, Huygens abandoned it in favor of a theoretical
consideration of gravitational acceleration. He began a study of circular
motion which in his view was closely connected to gravity: “The weight of a
body is the same as the conatus of matter, equal to it and moved very swiftly,
to recede from a center.” 189 Circular motion had been discussed by both
Descartes and Galileo, but only in qualitative and fairly rough terms. 190
Huygens set out to analyze circular motion mathematically. He derived an
expression for the tension on a chord exerted by a body moving in a circle,
by equating it with the tension exerted by the weight of the body. 191 He then
considered the situation in which a body revolves on a chord in such a way
that a stable situation is created and centrifugal and gravitational tension are
counterbalanced. With the conical pendulum thus procured and reversing his
calculations, Huygens found an improved value for gravitational acceleration
and dismissed Mersenne’s original experiment. 192 Analyzing the experiment
mathematically and comparing the time of vertical fall to the time of fall
along an arc, he derived a theory of pendulum motion eventually resulting in
the discovery of the isochronity of the cycloid. 193
The aim of Huygens’ studies of curvilinear fall and circular motion was to
render these motions with the same exactness Galileo had achieved with free
fall. 194 In the case of curvilinear fall this meant to solve the tricky
mathematical problem of relating the times with which curved and straight
paths are traversed. In the case of circular motion, he quantitatively
compared centrifugal and gravitational acceleration. Huygens’ success came
from his proficiency in using infinitesimal analysis and his control of
geometrical reasoning. 195 He conceptualized the forces he was studying in a
way that could be geometrically represented, which in his view meant to treat
free fall and centrifugal force in terms of velocities. 196 He considered, for
example, gravity as mere weight, and acceleration as continuous alteration of
inertial motion. 197 In other words, rather than mathematizing these
189
Yoder, Unrolling time, 16-17.
190
Yoder, Unrolling time, 33-34.
191
Yoder, Unrolling time, 19-23. This expression for centrifugal tendency amounts to the modern formula:
F = mv2/r. 192
Yoder, Unrolling time, 27-32.
193
Yoder, Unrolling time, 48-59.
194
The first draft of De vi centrifuga opened with a quotation of Horace: “Freely I stepped into the void, the
first”, above his discovery of the isochronicity of the cycloid he wrote: “Great matters not investigated by
the men of genius among our forefathers; Yoder, Unrolling time, 42 and 61.
195
Yoder, Unrolling time, 62-64.
196
The same goes for his earlier study of impact, to be discussed in section 4.2.2.
197 Westfall, Force, 160-165.