Grantville Gazette, Volume VIII Read online

  The square root radical "√" was used surprisingly early, in Christoff Rudolff's 1525 book Die Coss, although the use of index numbers within the radical to indicate cube roots, etc., had to wait until Albert Girard's 1629 book Invention nouvelle.

  Square brackets "[ ]" had been introduced in Raphael Bombelli's 1550 book Algebra, while parentheses "( )" appeared soon after, in Nicolo Tartaglia's 1556 book General trattato di numeri e misure, and braces "{ }" were used in François Vieta's groundbreaking 1591 book In artem analyticem isogoge.

  A few functions had nearly modern abbreviations by the time of the Ring of Fire. The trigonometric functions sine, cosine and tangent had been abbreviated as "sin.", "cos." and "tan." in Thomas Fincke's 1583 book Geometria rotundi, although the period was not dropped until almost half a century later, and the newly introduced logarithm was abbreviated to "log." in Edward White's 1616 translation of Napier.

  Once the up-time mathematical symbolism was learned, a further problem would be faced. Modern mathematics is far more abstract and generalized than anything a down-time mathematician would have known. In fact, a major branch of modern mathematics called category theory is jokingly called "general abstract nonsense". Modern mathematics is also far more rigorous in its proofs than is commonly practiced down-time, and indeed proof theory is studied as a separate branch of mathematics. This is related to the fact that modern mathematics is heavily axiomized, meaning that in each formal system of study, the most basic concepts are explicitly stated as axioms, and the rest are logically derived from the set of axioms. This concept had been used by the ancient Greeks with Euclidean geometry, but had only been extended to up-time mathematics as a whole since the nineteenth century.

  The Growth of Mathematical Activity

  One of biggest long-term effects of the Ring of Fire will be on the number of qualified mathematicians in Europe. The list of down-time mathematicians given earlier contains 24 of the most significant names in the mathematics of the time, but eight of those people are still children, and another would presumably have still been an undergraduate in today's world. Another person was very elderly, due to expire of old age in 1632. This leaves 14 people, all men, who would be talented enough to have been employed in a modern university's department of mathematics once they had studied the up-time mathematics. These are the cream of down-time mathematicians, but there were certainly many more people who contributed in a lesser way, some of whom were also members of Marin Mersenne's circle of correspondents. Mersenne does not know that the author of the "Crucibellus Manuscripts" is a woman, but Colette Modi will be only the first of a great many female mathematicians to appear. In our timeline, the first modern woman to lecture in mathematics at a great university was Elena Lucrezia Cornaro Piscopia (1646-1684), who was also the first woman in Europe to receive a Ph.D., from the University of Padua. In this timeline, we may expect that roughly half of the bachelor's degrees in mathematics will eventually be awarded to women, as is the case today.

  How many people in Europe could potentially become top-ranked mathematicians? One way to answer this question would be to look at how many PhDs (as an indicator of people with outstanding mathematical talent) are awarded today. In the USA from 1990 to 1996, about eight thousand PhDs were produced by US mathematics departments, or about 1,140 per year. The percentage of those PhDs who were born in the USA has remained steady at 43 to 44 percent in that time span, so the USA, with about 3.7 million live births (that survived infancy) per year when those people were born, can produce about 135 PhD recipients per million surviving children. This rate should be regarded as a minimum, since many people who have the ability to earn a PhD in mathematics do not do so, for one reason or another.

  According to Roger Mols, "Population in Europe 1500-1700" (Economic History of Europe, ed. Carlo Cipola) the total population of Europe (including the Balkans) grew from about 100-110 million in 1600 to about 110-120 million in 1700. The number of live births per year would have been about 3.5 percent of that number per year, or about 3.5 to 4 million babies per year. At the time of the Ring of Fire, at least half of these infants would die before their tenth birthday, but given reasonable assumptions about the expected decline in infant mortality over the following several decades, it seems likely that by 1660, about 3 million Europeans will see their tenth birthdays that year (provided that the birth rate remains where it is). This suggests that by then, there would be at least 400 people having outstanding mathematical ability who turn ten each year. This number may be in excess of the capacity of the European educational system to provide with a high-level education in mathematics. Before students embark on a post-secondary education, they must first pass through primary and secondary school, so a system of universal education through high school must be set up across Europe.

  At the start, it is likely that only a few major institutes capable of conducting significant new mathematical research will exist. Mathematicians need daily feedback to produce their best work, both as a source of new ideas and as an incentive to keep improving on their own past work. The existing body of up-time and down-time mathematicians is probably sufficient to populate two or maybe three such institutions.

  Almost certainly, the list of locations for such an organization will include Essen, where Colette Modi of Crucibellus fame now lives. Her patron and uncle Louis de Geer is interested in modernizing the new Republic of Essen, and is likely to encourage the development of a "technology research and development center" built around such an institute. Given that Colette Modi is in Essen, this site may become a magnet for women seeking to do mathematics, once Colette's identity is revealed. It is quite possible that the institute in Essen will be the first (but not the last) such institution in modern Europe to have a female chief researcher, and might be named for an historical (Hypatia) or "would-have-been" (Sophie Germain, Emma Noether, Ada Lovelace) woman in mathematics.

  Magdeburg, or possibly Grantville, would be another obvious location. Grantville would have the advantage of being the source of the up-time texts (although the more important books are likely to all have been reprinted in Latin within a few decades at most), while Magdeburg has the new Imperial College of Science, Engineering and Technology. Jena is another possibility. By 1633 it already had a professor of statistics in Carol Koch, who came through the Ring of Fire with a bachelor's degree in mathematics and statistics. The center of the mathematical universe before the Ring of Fire was Paris, as can be seen by the fact that 10 of the 24 top down-time mathematicians were born in France. Cardinal Richelieu is more than intelligent enough to see that creating such an institute is an economical method of attracting the people he needs to ensure the economic and technological future of France. Any government interested in keeping up with the USE will want to establish at least one department of advanced mathematics within their universities. A start on such a department already exists in the circle of Mersenne's correspondents. Fermat and Descartes may not be as familiar with up-time mathematics as those up-timers with degrees in mathematics (at least not until they have had a chance to catch up), but they are still almost certainly the greatest mathematical minds in Europe. It is probable that if they do decide to stay with Mersenne, many others will also wish to study with them.

  Progress in mathematics is helped by frequent, rapid communication of new results. To accomplish this on a continent-wide basis, mathematical journals are needed. As noted concerning Marin Mersenne, none of these existed at the time of the Ring of Fire. This is likely to change very quickly. A start on this was already being made in 1632, with Colette Modi's "Crucibellus Manuscripts." To quote from Kim Mackey's story "Essen Steel: Crucibellus" (Grantville Gazette, Volume 7):

  To say that the Crucibellus Manuscripts took the European mathematical community by storm would be a vast understatement. In early 1632 many Europeans were still unaware that something unusual had happened to their universe. Even those who had heard the tales of a community of Englishmen in Thuringia tended to discredit the id
ea unless they had actually traveled to Grantville themselves. But when the Crucibellus Manuscripts began circulating in 1632, people's minds began to change. It was not that all of the concepts were totally new and different. But it was the style and the breadth and the mystery which set intellectual circles abuzz. For Crucibellus had outlined the topics of future manuscripts and promised that each would appear at approximately three month intervals. Mathematical Symbology of the Future. Analytical Geometry. Differential Calculus. Integral Calculus. Differential Equations. Matrix Algebra. Probability. Statistics. Fractals. Special and General Relativity. Quantum Mechanics.

  These trimonthly manuscripts could be considered the first mathematical journal. Others will surely follow.

  Once the available texts have been studied and digested, the mathematicians of the post Ring of Fire world will be faced with the enormous task of reconstructing all the mathematics that did not make it down-time. Many areas of higher mathematics will have most or all of their major results known, but without proofs, since the up-time texts will have skipped most of the proofs to save page space. Given what tools are available, it should be time-consuming but feasible to eventually redo the missing proofs. This will be a lengthy process, as some proofs are very long. For example, the classification of the finite simple groups took almost 15,000 pages in around 500 journal articles. Other areas will be known of in passing, but with only a few scattered references to results, they will have to be essentially redone from scratch. In any case, the mathematicians of the day will be aware that generations or even centuries are likely to pass before they reach the boundaries of what had been reached up-time.

  Another area of great activity will be using the up-time mathematics to advance science and technology. The available up-time texts in engineering will be useful, but are unlikely to include the full mathematical development of the theories behind the described technologies, which will need to be redeveloped. More pressingly, for some considerable time the number of people able to understand mathematics well enough to efficiently design electronic or other advanced technology will be very low. Most mathematicians will be likely to spend most of their working time either teaching, or helping with various technology development projects as time allows, instead of working on mathematical research.

  Appendix One: Known Up-timers with degrees in Mathematics of Math-intensive subjects

  PEOPLE WITH DEGREES IN MATHEMATICS

  Emmanuel Onofrio (1930) M.A. in mathematics

  Viola (Petrini) Saluzzo (1942) B.A. in secondary mathematics education

  Allan Sebastian (1954) A.B. in secondary mathematics education

  Jennie Lee (Song) Cheng (1958) B.A. in mathematics

  Carol Elizabeth (Unruh) Koch (1959) A.B. in mathematics and statistics plus a couple of graduate courses in sampling

  Horace Bolender (1961) B.A. in statistics

  John Lobkowitz (1961) B.A. in mathematics, M.Ed. in mathematics education

  Kimberly Jane (Collins) Glazer (1963) M.A. in applied mathematics

  Lennon "Lenny" Washaw (1966) B.A. in mathematics; A.B. in mathematics education; most course work for an M.Ed.completed

  Anselm Gerard (1967) B.A. in mathematics, .Ed. in mathematics education

  Johnny Lee Horton (1967-1633) A.B. in mathematics; M.Ed. in mathematics education

  Kelley Josefina Bonnaro (1972) A.B. in mathematics

  Jerome Vincent "Jerry" Calafano (1972) M.A. in mathematics, M.Ed. in mathematics education

  PEOPLE WITH OTHER MATH-INTENSIVE DEGREES

  Alvin Pierce (1927-1638) A.B. in mathematics education

  Asa McDonald, (1930) B.A. in mechanical engineering

  Charnock David "Chuck" Fielder (1931-November 1634) M.A. in physics

  Henry "Hal" Smith, Sr. (1932) M.E. in aeronautical engineering

  Marshall Ambler (1944) B.S. in engineering

  Garland Franklin (1944) B.S. in civil engineering

  John Chandler Simpson, (1945/1950) B.S. in engineering

  Otto Kubala, (1946) B.S. in mechanical engineering

  Kyle Fleming (1947) A.B. in mathematics education

  James D. "Darry" Kip (1947) A.B. in mathematics education, with coursework towards M.A.

  Joseph Jesse Wood, "der Adler" (1950) B.S. in engineering

  Claude Yardley (1950) A.S. in electrical engineering

  Elaine (Mockbee) Pierce (1951) B.S. in mechanical engineering

  James Alvin Pierce (1951) B.S. in mechanical engineering

  Norris Patton (1953) B.S. in electrical engineering

  Bill Porter (1953) B.S. in electrical engineering

  Peter Rush (1954) M.S. in computer science

  Bob Kelly (1955/1960) M.S. in civil engineering

  Marshall Kitt (1956) B.S. in mechanical engineering

  Sara Lynn (Larson) Shaver (1956) B.S. in engineering

  Natalie (Fritz) Bellamy (1957) A.B. in mathematics education

  Jason Cheng (1957) B.S. in mechanical engineering

  Matthew Difabri (1957) A.A. plus course work toward B.A. in engineering

  Ronaldus "Ron" Koch (1957) B.S. in civil engineering; M.E. in mining engineering

  Simon Koudsi (1957) B.S. in mechanical engineering

  Peter Barclay (1958) B.S. in mechanical engineering

  Vanessa (Holcomb) Kitt (1958) B.S. in computer science

  Matthew Shaver (1958) M.S. in engineering

  Kay (Doxtader) Kelly (1959) B.S. in mechanical engineering

  Jacob Bruner, (1961) BS in civil engineering

  Farris Clinter (1961) B.S. in civil engineering

  Jere Haygood (1963) B.S. in civil engineering

  James Michael "Jim" McNally (1964) B.S. in physics

  Ripley Cunningham (1965) B.S. in computer science

  Joseph Hayes Daniels (1965) B.S. in computer science

  Lewis Hunsaker (1965) A.A. in chemical engineering

  Mason Chaffin (1966) B.S. in civil engineering

  Allen Lydick (1971) B.S. in civil engineering

  Derek Modi (1971) M.A. in civil engineering

  Laban Trumble (1971-1633) B.S. in engineering

  John McDougal "Mac" Clements (1972) M.S. in physics

  Jerry Trainer (1973) B.S. in chemical engineering; graduate student in chemical engineering

  Thomas Holcomb (1976) B.S. in computer science

  Landon Reardon, (1977) B.A. in physics

  Eve Zibarth (1977) enough courses to make a B.S. major in physics

  Jason Gotkin (1978) A.B.-6 in computer science

  James Victor Saluzzo (1978) A.B.-6 in physics

  Danny Song (1978) B.S. in computer science

  PEOPLE WITH THREE OR TWO YEARS OF MATH-RELATED UNDERGRADUATE SCHOOLING

  Mark Johnson Ellis (1979) three years of college in civil engineering

  Dane Marshall Kitt (1979) three years of college in mechanical engineering

  Matt Carson (1963) two years of college, engineering major

  Safety First: Industrial Safety in 1632,

  Part Two, Technical Aspects

  Written by Iver P. Cooper

  Ambrose Bierce, in The Devil's Dictionary, defined an "accident" as "an inevitable occurrence due to the action of immutable natural laws." But some industrial accidents are avoidable, and the secret to minimizing them is to know what the hazards of the job are, and to reduce those hazards by a combination of engineering controls (e.g., safer machinery), administrative controls (e.g., worker training), and personal protective equipment.

  As a result of the Ring of Fire, the Industrial Revolution is starting at least a century ahead of schedule, and will occur at a much accelerated pace. This will dramatically increase the risk of workplace accidents and occupational diseases. Fortunately, the up-timers can also educate the down-timers as to how to improve occupational safety.

  Hazard Monitoring

  We need to have ways of quantifying conditions in order to determine which workplaces are unsafe, and how good a particular safety technology
is at mediating the hazard. (A hazard measuring device, when coupled to some kind of signaling means, becomes a warning device.)

  Heat and Cold. Simple thermometers are already known (in 1629, Delmedigo described a sealed glass thermometer with brandy as the expandable liquid), and temperature scales appeared by 1613) but as of the RoF, scales weren't standardized, and the thermometers were pressure sensitive. These problems will be solved quickly.

  Humidity. Crude hygrometers were known before the RoF. Two thermometers, one with a wet bulb and the other a dry bulb, can be combined to make a hygrometer.

  Air movement. The modern cup anemometer is a simple mechanical design and should be duplicated fairly quickly. In fact, I think I put one into "Stretching Out, part 4."

  Dust. Collect dust on filter, then weigh.

  Noxious gases. Quantitative gas analyzers are largely beyond early post-RoF capabilities. We are essentially back to the "miner's canary" level of monitoring. Old mining books speak of detecting "fire damp" (mixture of methane and air) by observation of how it affects the operation of the Davy "safety lamp"; of "white damp" (carbon monoxide) by brightening of the flame of an ordinary lamp; "black damp" (carbon dioxide) by the reduction of such a flame, or by reaction with lime water or litmus paper; and "stink damp" (hydrogen sulfide) by its smell. (Treatise on Coal Mining, 1900).

  Noise. Most common sound measuring devices measure the pressure exerted by a sound and convert it into an electrical signal. I am not aware of any pre-RoF sound monitoring devices. The first practical device was the carbon button microphone (1870s) connected to a current meter. These should be fairly straightforward post-RoF introductions.

  There is enough up-time audio equipment in Grantville to set up a testing lab for noise protection devices. Nothing fancy; we hook up a microphone to an amplifier or other device with a VU meter. That's our detector. Next we need a reference sound source. This might be a cassette player, with a speaker, playing a "standard" tape. We separate the speaker and mike by a "standard" distance, and check the VU meter readings with and without the test material covering the speaker. If the sound is so muffled that it doesn't stir the meter, we can move the speaker closer and then apply the rule that the sound intensity is inversely proportional to the square of the distance.