I had been exploring how Kepler originally discovered his first two laws and became fascinated by what he did in his Astronomia Nova (1609), as presented by a number of researchers. Among the writers was A. E. L. Davis. She mentioned that the characterization of the ellipse that Kepler was using was the idea of a “compressed circle,” that is, a circle all of whose points were shrunk vertically by a constant amount towards a fixed diameter of the circle. I did not recall ever hearing this idea before and tried to track down its origin together with a proof — futilely, Davis’s references notwithstanding. I then tried to prove it myself. It was easy to do with analytic geometry. But in the spirit of the Kepler era (before the advent of Fermat’s and Descartes’s beginnings at fusing algebra and geometry) I tried to prove it solely within Euclid’s plane geometry. Some critical steps seemed to come from the great work of Apollonius of Perga (262-190 BC) on Conics. But for me a final elegant proof was not evident until 1822 when Dandelin employed his inscribed spheres. See Kepler’s Ellipse.
In the process of exploring the compressed circle idea I acquired an immense appreciation and regard for Kepler and his perseverance in the face of the dominant paradigm of his era, namely, the 2000 year old idea that the celestial motions were all based on the most perfect motion of all, that of circles. The kinds of extremely laborious calculations he went through (just prior to the invention of logarithms by John Napier) were daunting, especially considering the trials he was undergoing in his personal life (trying to survive the religious destruction between Catholics and Protestants, along with defending his mother against charges of witchcraft).
Years ago (1963) I got the paperback The Calculus:A Genetic Approach, by Otto Toeplitz, which presented the basic ideas of the differential and integral calculus from a historical point of view. One thing Toeplitz did at the end of his book that I had not seen in other texts was to show the equivalence of Kepler’s Laws and Newton’s Law of Gravity. (Since 1963 David Bressoud has developed this theme in his excellent 1991 text.) I thought I would try to emulate Toeplitz’s approach with more modern notation (vectors) and arguments in hopes of extracting the essential ideas from the clutter.
A by-product of this effort was to reveal strongly the different paths that physics and mathematics follow in understanding physical reality. The mystery is that the mathematics ends up describing the physics so well. I will return to this theme a number of times in other posts. See Kepler’s Laws and Newton’s Laws.
This is another problem from the Futility Closet website. It turned out to be pretty simple. The idea is to show the length of BC remains the same no matter where A is chosen on its arc of C1. See the Keyhole Problem.
Years ago (1967) I read about an interesting solution to the three jugs problem in a book by Nathan Court which involved the idea of a billiard ball traversing a skew billiard table with distributions of the water between the jugs listed along the edges of the table. The ball bounced between solutions until it ended on the desired value. I thought it was very clever, but I really did not understand why it worked. Later I figured out an explanation, which I present here. See the Three Jugs Problem.
Another good source of problems is the Futility Closet site. This puzzle involved finding the line of maximal length passing through the intersection of two circles. I solved it before looking at the Futility Closet solution. Their solution of course was short, sweet, and elegant. Mine was more like the old adage of cracking a walnut with a sledge hammer. Still, I thought there were some unexplained parts to the elegant solution that justified the effort on mine. At least my solution provided an interesting, though convoluted, alternative. See the Two Circles Puzzle.
This is one of Alex Bellos’s Monday Puzzles in the Guardian. I basically found the same solution as Bellos and his commenters, but wrote it up with what I thought were more explanatory graphics. The idea is that there is a bunch of ants on a stick who all walk a the same speed of 1 centimeter per second. When an ant runs into another ant, they both turn around and go the opposite direction. “So here is the puzzle: Which ant is the last to fall off the stick? And how long will it be before he or she does fall off?” See the Ant Problem.
This was one of my more satisfying essays. Several years ago I gave some thought to what it meant for the earth to be considered a magnet. More recently in 2012 an article in the magazine BirdWatching brought it all back when I saw its diagram of the earth as a magnet for guiding migratory birds. Knowing that magnets have north and south poles, where should we expect to find the earth’s north and south magnetic poles? See Earth as Magnet.
This is a mildly pointless 2015 article about Leonardo Da Vinci’s famous drawing of the Vitruvian Man spread-eagled and inscribed in a circle and a square. I started wondering about the positions and whether they over-determined the circle and square. What hidden constraints were being assumed? One assumption turned out to be famous, namely, that the height of a man equaled the distance between his finger tips when he holds his arms straight out to either side of his body. I had been told this in childhood, and I never knew where it came from. Also, I don’t think it is true in every case (what about women?), though it does appear to be close (and is true in my case). See the Vitruvian Man Problem.
This 2011 article gives some thoughts I had after reading Michael Dirda’s review in the Washington Post of Larrie D. Ferreiro’s Measure of the Earth. The book described the 1735 Geodesic Mission, whose purpose was to resolve the question of the shape of the earth, that is, whether it was a sphere, or like an egg with the poles further from the center than the equator, or like an oblate spheroid with the equator further from the center than the poles, as Newton averred due to centrifugal force. In the review Dirda said, “A team, sympathetic to Newton’s view, would travel to what is now Ecuador and measure the exact length of a degree of latitude near the equator. This would then be compared with the same measurement taken in France. If the latter was larger, Newton was right.” I wondered at first if Dirda got it right. It turned out my confusion stemmed from a mistaken definition of a degree of latitude. See Degree of Latitude.
This light-weight problem arose from a newspaper article that had me looking at a Google map of the area near Sioux Falls, South Dakota. What I saw was an excellent example of the Taxicab Geometry, allegedly first considered by Hermann Minkowski, mathematical friend of Albert Einstein. The map configuration was perpetrated by the great Public Land Survey System (PLSS) that originated with Thomas Jefferson and spread from Ohio (more or less) west to the California coast. This scheme overlaid the country basically with a 1 mile x 1 mile square grid of roads, and South Dakota is a prime example.
I first confronted the PLSS doing genealogy research, where the grid became a main method for locating the farms of ancestors. I had no idea it was so extensive. A fabulous book about the system and the history of land surveying in America is Andro Linklater’s Measuring America: How the United States Was Shaped By the Greatest Land Sale in History (2002). I learned all about perches and 17th century English mathematician Edmund Gunter’s survey chain, which became an essential tool for so vast a survey undertaking. Regarding the implications of the PLSS in South Dakota, see the South Dakota Travel Problem.