Diirer's mechanical drawing methods—does not really work any better than the
standard methods.^21
Referring to Fig. 12, in which a line is drawn from one peak of a cycloid to
the next and an identical cycloid then drawn atop that line, he showed that BE is
perpendicular to the cycloid. But the curve that cuts all the tangents to another
curve at right angles is precisely the "curve generated by evolution."
3.2. Newton. In his Fluxions, which was first published in 1736, after his death,
even though it appears to have been written in 1671, Newton found the circle that
best fits a curve. Struik (1933, 19, p. 99) doubted that this material was really in the
1671 manuscript. Be that as it may, the topic occurs as Problem 5 in the Fluxions:
At any given Point of a given Curve, to find the Quantity of Curvature. Newton
needed to find a circle tangent to the curve at a given point, which meant finding
its center. However, Newton wanted not just any tangent circle. He assumed that
if a circle was tangent to a curve at a point and "no other circle can be interscribed
in the angles of contact near that point,... that circle will be of the same curvature
as the curve is of, in that point of contact." In this connection he introduced terms
center of curvature and radius of curvature still used today. His construction is
shown in Fig. 13, in which one unnecessary letter has been removed and the figure
has been rotated through a right angle to make it fit the page. The weak point
of Newton's argument was his claim that, "If CD be conceived to move, while it
insists [remains] perpendicularly on the Curve, that point of it C (if you except the
motion of approaching to or receding from the Point of Insistence C,) will be least
moved, but will be as it were the Center of Motion." Huygens had had this same
problem with clarity. Where Huygens had referred to points that can be treated as
coincident, Newton used the phrase will be as it were.
Newton also treated the problem of the cycloidal pendulum in his Principia
Mathematica, published in 1687. Huygens had found the evolute of a complete
arch of a cyloid. That is, the complete arch is the involute of the portion of two
half-arches starting at the halfway point on the arch. In Proposition 50, Problem 33
of Book 1, Newton found the evolute for an arbitrary piece of the arch, which was
(^21) The master's thesis of Robert W. Katsma at California State University at Sacramento in the
year 2000 was entitled "An analysis of the failure of Huygens' cycloidal pendulum and the design
and testing of a new cycloidal pendulum." Katsma was granted patent 1992-08-18 in Walla Walla
County for a cycloidal pendulum. However, the theoretical consensus is that "in every case, such
devices would introduce greater errors into the going of a good clock than the errors they are
supposed to eliminate." (See the website http: //www.ubr. com/clocks/navec/hsc/hsn95a.html.)