Grantville Gazette, Volume VIII Read online

  Without a doubt, the greatest living mathematician of the age was Pierre de Fermat (French, 1601-1665). He was a lawyer by vocation, and thus is often considered the greatest amateur mathematician in history, although the line between professional and amateur mathematician was not yet clearly defined. Among his numerous accomplishments, the most well-known must be his creation of the modern theory of numbers. A literal side-note of this was his famous "Last Theorem," that if an integer n is greater than two, the equation a^n + b^n = c^n has no solution for any positive integers a, b, c.

  The nature of Fermat's putative proof of this theorem has been a mystery to this day—and will almost certainly remain a mystery in the new timeline, as the marginal note concerning it was not written by Fermat until 1637 in our timeline, and is very unlikely to ever exist in this one. Fermat will be as puzzled as everybody else about his most famous theorem. Perhaps greater than his work on number theory was his discovery of methods of finding the greatest and smallest ordinates of curved lines, which prefigured much that became differential calculus, and his discovery of methods to evaluate the integral of general power functions. Given the context of the limited mathematical development of that time, this was just as impressive as Liebniz's and Newton's inventions of the full (integral and differential) calculus. By the time that Newton and Liebniz did their work, most of the foundations of the calculus had already been laid by others, but Fermat worked in an era when those foundations were much less developed. Fermat was among the most wide-ranging of mathematicians. His other work included inventing analytic geometry independently of the more well-known results by René Descartes, co-founding probability theory with Blaise Pascal, and inventing the proof technique of infinite descent. To this day, he is in the running for the title of "greatest mathematician in history."

  Fermat was one of a dozen mathematicians of considerable talent who were alive when the Ring of Fire occurred. The oldest of this group was Grégoire de Saint-Vincent (Belgian, 1584-1667), who introduced the method of exhaustions to proofs, produced results leading to the theory of logarithms, and described the Mercator series. Not as well-known as Fermat but just as original was Gérard Desargues (French, 1591-1661), who invented projective geometry, the first extension of geometry beyond the limits known to the ancient Greeks. His work was eventually virtually forgotten, only to be redeveloped by others over a century later. He is mentioned in several places in the 1911 Encyclopaedia Britannica, but the references to him are short—as was to be expected, since the only known copy of his book on projective geometry was not rediscovered until 1951. René Descartes (French, 1596-1650) was most famous for his invention of analytic geometry, but did other significant work, such as discovering the "rule of signs" for determining the zeros of a function, and creating modern exponential notation (as well as introducing x, y, z to represent unknowns in an equation). Bonaventura Cavalieri (Italian, 1598-1647) developed a method of indivisibles, which are a significant step on the way to modern infinitesimal calculus. He was four years older than Gilles Personne de Roberval (French, 1602-1675), who also invented the method of indivisibles (which he unfortunately did not publish, therefore losing credit), and developed a very general method of drawing tangents, thus contributing greatly to the foundations of the calculus. He and Descartes did not get along, since Descartes had criticized some of Roberval's mathematical methods, and Roberval returned the favor. Bernard Frénicle de Bessy (French, 1605-1675) produced many results in number theory and in combinatorics. He described all 880 essentially different magic squares of order 4.

  The final five people in this group were still under the age of fifteen on the day of the Ring of Fire. John Wallis (English, 1616-1703) in our world extended the methods of analysis of Descartes and Cavalieri. His 1655 book on conic sections was the first book to study them analytically, and his 1693 book on algebra contained the first systematic use of formulae. Blaise Pascal (French, 1623-1662) would discover Pascal's theorem and the "mystic hexagram" at the age of sixteen, and go on to co-found probability theory with Fermat, discover "Pascal's triangle" for binomial coefficients, and invent an early calculating machine. He is probably best known for his writings as a philosopher. Born three months later, Stefano Degli Angeli (Italian, 1623-1697) in our timeline used infinitesimals to study spirals, parabolas and hyperbolas. Johannes Hudde (Dutch, 1628-1704) worked on maxima and minima, and on the theory of equations. He was known for finding an ingenious method to find multiple roots of an equation. Finally, the infant Isaac Barrow (English, 1630-1677) would develop a method of determining tangents that closely approached the methods of calculus, and was the first to recognize that integration and differentiation are inverse operations.

  Another dozen mathematicians of some significance were also alive on the day of the Ring of Fire. Joost Burgi (Swiss, 1552-1632) invented logarithms independently of Napier, but by delaying publication, lost priority. He was a major contributor to prosthaphaeresis, a technique for computing products quickly using trigonometric identities, which predated logarithms. Thomas Fincke (Danish, 1561-1656) is also still alive, and remained so in our timeline for many more years. His 1583 book "Geometria rotundi" introduced the trigonometric functions tangent and secant. William Oughtred (English, 1574-1660) published his most important book in 1632. Circles of Proportion and the Horizontal Instrument described his invention of the circular and straight slide rules. Two years younger was Paul Guldin (Swiss, 1577-1643), who discovered the Guldinus theorem to determine the surface area and volume of a solid of revolution. Claude de Méziriac (French, 1581-1638) worked on the solution of indeterminate equations by means of continued fractions. He also did work on number theory, and was possibly the discoverer of Bézout's identity. His 1612 book, Problèmes plaisants, was the first book of mathematical puzzles and recreations.

  Among the younger men was Albert Girard (French, 1595-1632), who died very young. Perhaps he will live longer in the new timeline. He contributed to the early development of the fundamental theorem of algebra, and introduced the abbreviations sin, cos and tan (without a period) in a 1626 treatise. He spent his adult life in the United Provinces, and spent his last years as an engineer in the Dutch army. Jacques de Billy (French, 1602-1679) corresponded with Fermat and produced a number of results in number theory which are named after him. He was one of the first to decisively reject the role of astrology in science. Juan Caramuel y Lobkowitz (Spanish, 1606-1682) expounded the general principle of numbers to base n, pointing out the benefits of some other bases than ten. André Tacquet (Belgian, 1612-1660) published his most important work, Cylindricorum et Annularium (Cylinders and Rings) in 1651, introducing the idea, later elaborated by Barrow, that the tangent of a curve and the area under a curve were inverse to each other.

  Still children are William Brouncker (Irish, 1620-1684), who worked on continued fractions and calculated logarithms by means of infinite series; René François Walther de Sluze (French, 1622-1685), who discovered the curves called "the pearls of Sluze." Christiaan Huygens (Dutch, 1629-1695) is more famous as a physicist and astronomer but he discovered that the cycloid (the curve traced by a point on the edge of a circular wheel rolling along a straight line), when inverted, was a tautochrone curve (a curve for which the time taken by a frictionless particle sliding down it under uniform gravity to its lowest point is the same, regardless of its starting point on the curve). Huygens wrote the first book on probability theory, in 1657.

  A special mention must be made of Marin Mersenne (French, 1588-1648), a priest of the Order of the Minims, who while not particularly noted for his own discoveries (although he did work on the cycloid, among other topics including number theory), was important for acting as a central clearinghouse for mathematical discoveries, corresponding with most of the mathematicians active at the time. In the time before mathematics journals existed, he played much the same role. In Kim Mackey's story "Essen Steel: Crucibellus" (Grantville Gazette, Volume 7), he was one of the three people to whom Colette Mod
i started to send her trimonthly summaries of up-timer mathematics. Colette knew that he would spread the contents by writing to virtually every down-time mathematician of significance.

  What Came Through the Ring of Fire

  The mathematics texts that went through the Ring of Fire almost certainly formed a pyramid, with many basic and high school level books, fewer undergraduate texts, still fewer graduate texts, and possibly a very few advanced monographs.

  The number of advanced texts on mathematics that came through the Ring of Fire depends on how many up-timers had a reason to have them in their personal or professional libraries. Thirteen up-timers are known to have degrees in mathematics, three of them with master's degrees. The bachelor's degree in secondary mathematics education includes the same required mathematics courses as the bachelor's degree in mathematics, so I am including it as a degree in mathematics.

  There are at least 48 additional up-timers who have degrees involving significant amounts of mathematics. Many of these people would have attended West Virginia University. The current required courses for a bachelor's degree in physics, or in aeronautical, civil, chemical, electrical or mechanical engineering, from that institution include Calculus I and II, Multivariable Calculus, and Elementary Differential Equations. In addition, a bachelor's degree in civil or electrical engineering requires Probability and Statistics. The required courses for a bachelor's degree in computer science include Calculus I and II plus Discrete Mathematics, as well as Probability and Statistics. All of these courses would have been taken in the first three years of study. Two additional people are known to have taken the first three years of a degree in engineering, and another person has the first two years of an engineering degree. Fourteen additional up-timers are known to have taken the first two or three years of an unspecified bachelor's degree, and a few of those degrees are also likely to have been math-intensive. A list of up-timers known to have degrees (or the first two or three years toward a degree) in mathematics or mathematics-intensive subjects is included as Appendix One.

  Many of these people are likely to have kept all of their college textbooks. In addition, some of them probably had a great many additional books that they obtained beyond those necessary to get their degrees, either from a personal interest in the subject or from a desire to expand their knowledge for work-related reasons. Between them, these people are likely to have brought with them a good selection of undergraduate texts in all fields of mathematics, as well as a smaller but still significant number of graduate texts and monographs, and probably a few advanced research monographs on topics that had gained somebody's interest.

  Many of these people are also likely to have collected a number of popular books on such subjects as recreational mathematics or mathematical history and biography. This would be especially true of those people with degrees in mathematics education, who may have hopes of generating interest in mathematics among their students. Beyond this core group are people with no mathematics-related degree but who simply like mathematics, most of whom are also likely to have personal collections of mathematics-related books.

  The best library collection of mathematics books in Mannington (the model for Grantville) is the one at North Marion High School. These books are written at a more basic level than the ones discussed above, but books at this level are essential to help bridge the gap between down-time mathematics and the more advanced up-time texts. In addition, there are the high school textbooks. Books of particular interest known to be at the high school library include the following:

  A four-volume set entitled World of Mathematics: A Small Library of the Literature of Mathematics from Ahmose the Scribe to Albert Einstein (1956, 2535 pages) is a collection of the writings of historical mathematicians. Books that describe the history of mathematics include Mathematics and the Physical World (1959, 482 pages) and Mathematics for the Million (1965, 697 pages), among others. Men of Mathematics (1965, 590 pages) by E. T. Bell is entertaining, if unreliable. The library has five different dictionaries of mathematics, ranging from 223 to 509 pages. One of these dictionaries is indexed in French, German, Russian and Spanish. One of many copies of CRC Standard Mathematic Tables is in the library. It may well end up being sold to a university for their own collection, where it would no doubt be heavily used.

  What is Mathematics? An Elementary Approach to Ideas and Symbols (1996, 556 pages) has the following catalog description: "discusses the history and philosophy of mathematics and presents its principles, covering the number system, geometry, algebra, topology, functions and limits, calculus, and other related topics such as the Four-Color Theorem and Fermat's Last Theorem." This sounds like a good general introduction to up-timer mathematics. More basic texts include Realm of Numbers (1959, 200 pages) by Isaac Asimov, aimed at younger readers. The catalog description of this book says: "Explanations of mathematical techniques and principles are combined with the history of mathematics. Includes simple arithmetic, square root, logarithms, and even imaginary numbers."

  The Mannington Public Library mathematics collection is much smaller, and appears to include little of interest, although it is known that some of its overall holdings are not listed in the online catalog.

  Other sources of information on mathematics will be found in such places as encyclopedias and some books on science, especially physics and astronomy. For example, the 1911 Encyclopaedia Britannica, which is known to exist in Grantville, has the following major mathematics-related articles, each containing more than ten thousand words:

  Algebra (30K words)

  Algebraic Forms (32K words)

  Arithmetic (29K words)

  Curve (18K words)

  René Descartes (16K words)

  Differential Equation (24K words)

  Dynamics (11K words)

  Energetics (12K words)

  Equation (20K words)

  Function (51K words)

  Geometry (81K words)

  Hydromechanics (29K words)

  Infinitesimal Calculus (26K words)

  Logarithm (15K words)

  Mensuration (20K words)

  Isaac Newton (15K words)

  Probability (48K words)

  Projection (12K words)

  Spherical Harmonics (16K words)

  Surface (13K words)

  Theory of Groups (19K words)

  Thermodynamics (11K words)

  Trigonometry (19K words)

  These articles are admittedly almost a century old, but the information contained in them would still be accurate.

  Difficulties Faced by Down-time Mathematicians

  A down-timer reading an up-time mathematics text would be faced by several difficulties. First of all, the book would be written in English, not Latin. Latin was the language of down-time scholars for a very practical reason. Scholars could write to each other, in letters or via books, and be understood all across Europe. In addition, a single printing of a book written in Latin would be sufficient to reach all of its intended audience, instead of needing to be translated into a multitude of languages. This problem can be remedied, but it will take some time for the scholars wishing to study the up-time mathematics to learn to read English, and an even longer time for the texts to be translated into Latin.

  We see this process under way in Jack Carroll's story "Stepping Up" (Grantville Gazette, Volume 14), where a group of down-timers, who know Latin, are told that once they have learned electrodynamics, they will be the first scholars in Europe to be able to write electrodynamics texts in Latin. Whether this is actually true is debatable, since many people across Europe are no doubt also studying electrodynamics, because of the obvious benefits that radio would provide. A number of these people are likely to also be literate in Latin.

  Another obvious difference from the down-time texts is the greater use of symbols instead of verbal argumentation, some of which would be completely unfamiliar. Among the symbols which a down-timer would know are the symbols for addition "+" and subtraction "-". O
riginating as marks to indicate full and underweight barrels, they first appeared in print in Johann Widman's 1489 book Und hüpsche Rechenung auff allen Kauffmanschafft, but did not go into common use until the second half of the sixteenth century. The use of "x" for multiplication had just been introduced in William Oughtred's 1631 book Clavis Mathematica (which also saw the first use of plus-or-minus "±") but was otherwise unknown. However, the common symbol for division "÷" would not be invented until 1659 (Johann Rahn's book Teuche Algebra), and the use of "/" to indicate fractions is first attested in 1718.

  The symbol for equality "=" was known but rare. It was first used in Robert Recorde's 1557 book Whetstone of Witte (which also popularized the "+" and "-" symbols) but not used again until 1616 in an anonymous appendix—probably written by William Oughtred—to Edward White's translation into English of John Napier's 1614 book Mirifici logarithmorum canonis descriptio.

  Decimal fractions were slowly gaining in popularity. They had been used by Arab mathematicians for centuries, but were not used by Europeans until 1530, when Christoff Rudolff in his Exempel Büchlin used a vertical bar "|" as a decimal separator. These fractions were popularized by Simon Stevin in his 1585 book La Thiende (The Tenth) and La Disme (The Decimal), using his own notation. The modern notation, using a period or comma for the separator, was first used in G. A. Magini's 1592 text De planis triangulus, and popularized by Napier in his 1614 book.