### Christopher Wren and the Cycloid

The remarkable polymath Christopher Wren died in March 1723, just 300 years ago. Sarah Hart, Professor of Geometry at Gresham College, recently presented a lecture, The Mathematical Life of Sir Christopher Wren; a video of her presentation in available online (see sources below). The illustration above is from the Gresham College website.

Christopher Wren

In their History of Mathematics, Boyer and Merzbacher remark that “Had not the great fire of 1666 destroyed much of London, Wren might now be known as a mathematician, rather than as the architect of St. Paul’s Cathedral.”  Before he flourished as an architect, Wren’s research resulted in several substantial accomplishments in mathematics. He also made significant meteorology, anatomy, navigation and astronomy.

Wren was educated at Oxford. He was a professor at Gresham College London and later held the Savilian professorship in astronomy in Oxford. He was a founder member of the Royal Society, and was President of the society for a period. It is reported that Wren was mentioned seven times in Newton’s Principia.

The Cycloid

Galileo noted that the cycloid is the curve traced out by a point on the rim of a wheel as it rolls along a horizontal path. He tried to find the area under an arch of the cycloid. He traced out the curve on paper, cut out an arch, and weighed it. He concluded that the area was approximately three times the area of the generating circle.

Blaise Pascal also studied the cycloid. Indeed, there was a kind of `cycloid fever’ running amongst mathematicians at this time, just before the emergence of the calculus. Having found some areas, volumes, and centres of gravity associated with the cycloid, Pascal proposed half a dozen such questions to the mathematicians of his day, offering prizes for their solution.

Christopher Wren sent Pascal his rectification of the cycloid. Wren used an idea of John Wallis in his Arithmetica Infinitorum, that a small arc is approximately the hypotenuse of a right-angled triangle whose sides are, in modern terms, the increments in ${x}$ and ${y}$, that is, ${ds=\sqrt{dx^2+dy^2}}$. In 1658, he found an expression for the arc length of a cycloid.

In Modern Terms

The cycloid is the locus of a point fixed to the rim of a circular disk that is rolling along a straight line (Figure above). The parametric equations for the cycloid are $\displaystyle x = r (\theta - \sin\theta)\,, \qquad y = r (1 - \cos\theta ) \ \ \ \ \ (1)$

where ${\theta}$ is the angle through which the disk has rotated. The centre of the disk is at ${(x_0,y_0) = (r\theta, r)}$. The differentials of the coordinates (1) are $\displaystyle \mathrm{d}x = r (1 - \cos\theta)\mathrm{d}\theta\,, \qquad \mathrm{d}y = r \sin\theta\,\mathrm{d}\theta \,. \ \ \ \ \ (2)$

From these we find the increment of arc-length ${\mathrm{d}\ell}$, $\displaystyle \mathrm{d}\ell = \sqrt{ \mathrm{d}x^2 + \mathrm{d}y^2 } = 2r\sin\textstyle{\frac{1}{2}}\theta\,\mathrm{d}\theta \,, \ \ \ \ \ (3)$

and the increment of area ${\mathrm{d}A}$, $\displaystyle \mathrm{d}A = y\,\mathrm{d}x = r^2(1-\cos\theta)^2\mathrm{d}\theta \,. \ \ \ \ \ (4)$

The length ${L}$ of an arch is easily computed by integrating (3): $\displaystyle L = \int_{0}^{2\pi} 2r\sin\textstyle{\frac{1}{2}}\theta\,\mathrm{d}\theta = 8 r \,.$

We see that the arc-length does not depend on ${\pi}$. The area under an arch is the integral of (4) over the interval ${\theta\in[0,2\pi]}$, which is slightly more tricky: $\displaystyle A = \int_{0}^{2\pi} r^2(1-\cos\theta)^2\mathrm{d}\theta = 3\pi r^2 \,. \ \ \ \ \ (5)$

Thus, the area under an arch is three times the area of the generating circle or three-quarters of the area of the surrounding rectangle.

Sources ${\bullet}$ Hart, Sarah, 2023: The Mathematical Life of Sir Christopher Wren Gresham College Video. ${\bullet}$ Carl Boyer and Uta Merzbach, 2011: A History of Mathematics, Third Edn., John Wiley & Sons.