Mathematical Geography by Willis Ernest Johnson

November 27, 2010

Before the advent of coordinated universal time and internationally-recognized time zone boundaries, practical, day-to-day considerations of time measurement quickly became entangled in geographical, political, historical, and legal problems of surprising recalcitrance.

If you live in the northern hemisphere, does the sun pass due south of you every 24 hours? Perhaps surprisingly, no! An hour is 1/24 of a sidereal day, the time required for the earth to rotate once with respect to the stars. Because the earth orbits the sun, the sun’s position on the celestial sphere traverses a circle once per year. Consequently, the mean solar day is shorter than the sidereal day by a factor of roughly 1/365.25: The sun therefore passes due south on average every 23 hours, 56 minutes, 4 seconds. The actual length of the solar day varies seasonally because the earth’s orbit is not circular and the dates of perihelion and aphelion (when the earth is closest or furthest from the sun) do not coincide with the solstices (when the north pole is tipped most nearly toward or away from the sun).

A related phenomenon is said to have confounded Ferdinand Magellan’s crew briefly upon return to Spain after circumnavigating the globe. Their careful reckoning gave the date of September 6, 1522, but the Spaniards assured them the date was the 7th. Eventually the crew realized they had lost a day over their long voyage by traveling west, with the sun. Today we would say Magellan lost a day crossing the date line.

Civilizations successfully used local (solar) time for millennia. Only when consistent time measurements were required over extended longitudes did standard time become important. Of course, time standards were not immediately or universally adopted. Consequences ranged from minor confusion for travellers to legal cases with substantial financial stakes.

During the late Nineteenth Century in the United States, railroads and cities used their own time conventions for many years. When Mathematical Geography was written, the vicinity of El Paso Texas used four systems of time, due to the confluence of three U.S. time zones and an entirely separate Mexican standard time used in Juarez, across the Rio Grande.

In 1902, thirteen lawsuits were brought in the courts of Kentucky over the wording of fire insurance policies. In one case, the policy expired at noon on April 1, 1902, but left unspecified whether solar or standard time was to be reckoned. On the date in question, a warehouse fire began at 11:45 A.M. standard time, namely at 12:02:30 P.M. local time. Nearly $20,000 of insurance money hung in the balance of the court’s decision. Such improbable occurrences were not as rare as one might expect. Willis’s book does not divulge the outcome of these cases, but does highlight the need for pedantic attention to issues we might all regard as too obvious to warrant detailed consideration.

In the modern world of the global positioning system, universal time, and largely undisputed geopolitical boundaries, we easily forget that divisions of time and space are nothing more than social agreements. The earth has no intrinsic clocks convenient for international travel and commerce, nor many geographical features that unambiguously separate neighboring states, provinces, territories, or countries.

Mathematical Geography is not a mere catalogue of amusing events and curious factoids, but a clear, engaging, and systematic exposition of the shape and movement of the earth as an astronomical body, and the consequences for time-keeping, map making, and geodesy. Written in 1907 for use in American secondary schools and for teacher preparation, the book intertwines purely scientific issues of astronomy and geography with the historical growth of the United States in the 1800s as the political entity expanded across the North American continent, and with then-current legal and practical issues related to time and place. Much of the factual content is up-to-date, and even the remainder should be of historical interest.

Despite the book’s imposing title, the mathematical content is light, entailing only trigonometry, plane geometry, and basic algebra. Mathematical Geography is easily accessible to a modern reader with a good high school education. A curious and intelligent younger reader can also learn much, skipping brief mathematical derivations as needed or even learning useful mathematics in a realistic context.

This enjoyable book deserves, and earns, the attention of anyone who wants to understand more about the planet we inhabit and how the earth’s shape and motion affect our daily lives.

This review was contributed by DP-volunteer adhere.

Mathematical Recreations and Essays

November 13, 2010

W. W. Rouse Ball‘s “Mathematical Recreations and Essays” contains an odd but decidedly interesting collection of essays about a range of different subjects. The 4th edition dating from 1905 was recently posted to Project Gutenberg. Far from being interesting to mathematicians only, this book has something for everybody who’s interested in puzzles and number games or in the history of science.

The book is divided into two parts of quite different character. The first part, titled “Mathematical Recreations,” ranges from simple number games of the “guess the number” kind to magical squares and mazes, discussing topics such as mathematical and geometrical fallacies, the “Eight Queens” problem on a chessboard, map colourings and many more. The problems presented are not exactly new or original and don’t pretend to be, but I like the systematic treatment given to many of them.

Part II of the book, titled “Miscellaneous Essays and Problems,” contains a wealth of historical information about mathematics-related topics made even more fascinating by the fact that it was written more than a century ago. It starts with a description of the development of the Mathematical Tripos at Cambridge, giving a very interesting glimpse into the history of mathematics education at one of Britain’s most prestigious universities. The next chapters give a history of classical geometrical problems, the quadrature of the circle the most prominent of them, followed by an introduction to Mersenne’s numbers. After that comes a short description of the “scientific” aspect of astrology, which the author himself wasn’t too sure whether to include. There’s a chapter introducing early cryptography, one on hyper-space, including space with more than three dimensions as well as non-Euclidean geometry, and one on time measurements.

But my absolute favourite is the last chapter on matter and ether theories. At the time this book was written, the internal workings of atoms were not yet known and the subject of the wildest speculations. The author gives an account of the different theories proposed and how they explain the way atoms interact with each other. Rather than reporting scientific developments from a historical standpoint, this chapter provides some valuable insights into science in action, which makes it really fun to read.

Tucked away behind the index are advertisements for the W. W. Rouse Ball’s other works, together with blurbs from probably every review that was ever printed. Let me cite from one of the reviews for this book, which I have to heartily agree with:

… A great deal of the information is hardly accessible in any English books; and Mr. Ball would deserve the gratitude of mathematicians for having merely collected the facts. But he has presented them with such lucidity and vivacity of style that there is not a dull page in the book; and he has added minute and full bibliographical references which greatly enhance the value of his work.–The Cambridge Review.

I thoroughly enjoyed reading this book and would really like to see the other works by this author on PG: they are surely worth a closer look.

Calculus Made Easy

October 13, 2010

The mathematical study of rates of change and total change, also known as “the differential and integral calculus”, has frightened generations of students. Ironically, this scholastic trauma is often unnecessary: Many formal rules of calculus are nearly trivial to carry out. The mathematical difficulties lie in understanding and using the rigorous logical framework of the calculus—what mathematicians nowadays term “elementary real analysis”—and in establishing the correctness of the formal manipulations and their connections with mathematical interpretations.

Sylvanus Phillips Thompson’s “Calculus Made Easy” gives cogent yet entertaining and irreverent explanations of these easy calculational rules. Justifications are conceptual, but beneficially simplified and intuitive. Popularized and later updated by the recreational mathematics author Martin Gardner, “Calculus Made Easy” has remained a widely-read introductory text for the past century.

With the appearance of Project Gutenberg’s public domain version on July 28, 2010, the second British edition of this classic, originally published in 1914, is freely available over the Internet. In its first weeks, the book has proved wildly popular for a mathematics text.

Thompson’s choice of material is both varied and selective. Most topics will be familiar to the modern student: Differentials as minute quantities; derivatives as relative rates of change; rules for differentiating sums, products, quotients, and compositions of functions; the geometric meaning of the first and second derivatives; finding maxima and minima; the natural exponential and logarithm functions; circular trig functions; partial derivatives; integration and antidifferentiation; the fundamental theorems; integration by parts, and by partial fractions; elementary differential equations, including exact first-order equations and d’Alembert’s solution of the one-dimensional wave equation.

If you’ve had unpleasant experiences with calculus, if your knowledge has grown rusty, or if you’ve simply never encountered this powerful and intriguing branch of mathematics, Thompson’s gem of a textbook should prove a pleasant, thought-provoking, and ultimately rewarding read.

This review was contributed by DP-volunteer adhere.

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