Ursinus College Digital Commons @ Ursinus College Transforming Instruction in Undergraduate Calculus Mathematics via Primary Historical Sources (TRIUMPHS) Summer 2018 The Radius of Curvature According to Christiaan Huygens Jerry Lodder New Mexico State University, [email protected] Follow this and additional works at: https://digitalcommons.ursinus.edu/triumphs_calculus Click here to let us know how access to this document benefits ou.y Recommended Citation Lodder, Jerry, "The Radius of Curvature According to Christiaan Huygens" (2018). Calculus. 4. https://digitalcommons.ursinus.edu/triumphs_calculus/4 This Course Materials is brought to you for free and open access by the Transforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) at Digital Commons @ Ursinus College. It has been accepted for inclusion in Calculus by an authorized administrator of Digital Commons @ Ursinus College. For more information, please contact [email protected]. The Radius of Curvature According to Christiaan Huygens Jerry Lodder∗ November 9, 2020 1 The Longitude Problem An outstanding problem of navigation during the sixteenth, seventeenth and eighteenth centuries was determining longitude at sea. Countless lives were lost and ships were wrecked simply because the ship's captain did not know how far east or west the ship was with respect to an imaginary line (semi-circle) called the Prime Meridian, or with respect to any given meridian for that matter. This problem is chillingly described in The Illustrated Longitude [9], from which we relate one incident. In September, 1740, Captain George Anson (1697{1762) of the British Royal Navy set sail for the South Pacific aboard the Centurion, under orders to disrupt the Spanish trading monopoly in the Pacific. On March 7, 1741, the Centurion entered the Pacific Ocean from the Atlantic by passing through the Straits Le Maire at the southern end of South America. Then a violent storm blew in from the west, lasting some 58 days, unduly delaying the voyage and disorienting the Centurion. Many of the captain's men began to die of scurvy. Having rounded Cape Horn and now off the western coast of South America, the captain set sail for Juan Fern´andezIsland for fresh food and water. Sailing north, Anson reached the proper latitude of the island (about 35◦S) on May 24, 1741, but he could only guess whether the island lay east or west of his current position. The captain guessed west and sailed in that direction for four days only to find nothing. Changing course by 180◦, he then sailed east for about two days only to sight the steep cliffs of Spanish controlled Chile. Anson turned around again, sailed west and reached Juan Fern´andezIsland on June 9, 1741, with only about half of his men remaining [9, pp. 21{25]. What was the root cause of Captain Anson not being able to find the island? Anson knew his latitude, but he was unable to determine his longitude, namely how far east or west he was. The determination of latitude can be reduced to a simple trigonometrical calculation given the length of a shadow of an object (of known height) at high noon, when the sun is at its highest elevation for the day. ∗Mathematical Sciences; Dept. 3MB, Box 30001; New Mexico State University; Las Cruces, NM 88003; [email protected]. 1 Exercise 1.1. Suppose that on the day of Spring Equinox (or Autumn Equinox), a 1' pole casts a shadow of 0.4' at high noon. Determine the latitude of the observer at this location (in the Northern Hemisphere). Be sure to explain your answer. (For days other than an equinox, the tilt of the Earth could be looked up in a table and would be used to adjust the above calculation.) Hint: Since the sun is at such a great distance from the Earth, the rays of sun light are essentially parallel to each other when they reach the Earth. At high noon during an equinox, these rays are directly over head at the Equator. See Figure 1. Earth Rays of Sun Equator Figure 1: High Noon During an Equinox. Consider now the 1' pole perpendicular to the Earth, and imagine a line connecting the pole to the center of the Earth, O. The length of the pole is exaggerated in Figure 2, where angle α denotes the latitude of the observer in the Northern Hemisphere. Now draw a separate right triangle with legs being the pole and its shadow, and the hypotenuse formed by a ray of sun light. Finish Exercise (1.1). Ray upon the Top of the Pole α O Ray upon the Equator Figure 2: Determining Latitdue. Could the determination of longitude be reduced to a simple calculation as in Exercise (1.1)? Certainly the sun (in its apparent motion) travels from east to west across the sky. Could the position of the sun be used to determine longitude? How? When the sun reaches its highest elevation on a given day, shadows are at their shortest, marking high noon. If a sailor knew when high noon occurred (or will occur) at the port of embarkment, then the difference between when noon occurs aboard ship and at the port of embarkment could be converted into a longitude reading, indicating how far west (or east) the ship is from its port. 2 Exercise 1.2. Suppose that a ship sets sail from England with a clock set to give time in Greenwich, England, through which the Prime Meridian passes. After sailing for several days on the open sea, suppose further that noon occurs on ship when the clock giving time in Greenwich reads 1:00 p.m. (a) Is the ship sailing west or east of Greenwich? Justify your answer. (b) How many degrees is the ship west of Greenwich? Explain your answer. Hint: There are 360◦ in a circle and the sun circles the Earth once every 24 hours (in its apparent motion). The reader should now verify that one minute discrepancy between the time aboard ship and the time at the port of embarkment (or any given, fixed location) corresponds to 0:25◦ of longitude. Using R = 3960 miles as the radius of the Earth, then 0:25◦ of longitude at the Equator (a great circle on the Earth) corresponds to (0:25)(π=180)R ≈ 17 miles, a non-trivial distance. However, a clock that is one minute off, even during a lengthy voyage, could result in a navigational error as large as 17 miles. One solution to the longitude problem would be to construct a very accurate clock, set the clock to give time at Greenwich, England, and then send the clock to sea aboard ship. This project discusses Christiaan Huygens's (1629{1695) work on constructing a pendulum clock that theoretically keeps perfect time. Huygens described the path of the pendulum bob as one \whose curvature is marvelously and quite rationally suited to give the required equality to the pendulum" [3, p.11]. With these words, Huygens has identified a key concept, curvature, used to this day in physics and calculus to describe motion along a curved path. Alas, a pendulum clock behaved erratically on the high seas, and did not solve the longitude problem for naval navigation, although it remained the standard for terrestrial time keeping until the development of spring balance timepieces. So acute was the need to determine longitude that the British government had already issued the Longitude Act in 1714 that offered a first prize of $20,000 for a method to determine longitude to an accuracy of half a degree of a great circle [9, p.66]. There were also second and third prizes. Using the idea of a spring balance, John Harrison (1693{1776) perfected a sea clock that did solve the longitude problem. Due to prejudice against a clock method and in favor of a lunar method to solve the longitude problem, Harrison received marginal credit for his work during his lifetime, much of it spent perfecting his various timekeepers [9]. In the next section we turn to a detailed study of Huygens's work on the isochronous pendulum, a pendulum clock that theoretically keeps perfect time. To describe the path of a pendulum bob in terms of its curvature, Huygens first offers a geometric construction for what is known as the radius of curvature (see below). Since his work predates the development of calculus by Issac Newton (1642{1727) and Gottfried Wilhelm Leibniz (1646{1716), Huygens relies on known techniques from Euclidean geometry, such as similar triangles, to capture the ratios of certain side lengths that play the role of certain derivatives. The geometric idea of a tangent line had been known well before Huygens, which he uses adroitly. The goal of this project is to follow Huygens's clever geometric arguments and then, through guided exercises, rewrite the expression for the radius of curvature in terms of derivatives, and finally to reconcile this with the modern equation for curvature found in many calculus 3 textbooks. For further details on the construction of an isochronous pendulum, see Huygens's Horologium oscillatorium (The Pendulum Clock) [4, pp. 86{365] originally published in 1673, translated into English [3], and presented as part of a book chapter on curvature [6, pp. 174{ 178]. 2 Huygens and The Radius of Curvature Holland during the seventeenth century was a center of culture, art, trade and religious tolerance, nurturing the likes of Harmenszoon van Rijn Rembrandt (1606{1669), Johannes Vermeer (1632{1675), Benedict de Spinoza (1632{1677) and Ren´eDescartes (1596{1650). Moreover, the country was the premier center of book publishing in Europe during this time with printing presses in Amsterdam, Rotterdam, Leiden, The Hague and Utrecht, all publishing in various languages, classical and contemporary [2, p.