"Math for Mystics: From the Fibonacci Sequence to Luna's Labyrinth to the Golden Section and Other Secrets of Sacred Geometry
by: Renna Shesso (Author)
Publisher:Red Wheel/Weiser
Publication Date: 27 Feb. 2007
Language:English
Print Length:208 pages
ISBN-10:1578633834
ISBN-13:9781578633838
Book Description
Much of math history comes to us from early astrologers who needed to be able to describe and record what they saw in the night sky. Whether you were the king's court astrologer or a farmer marking the best time for planting, timekeeping and numbers really mattered. Mistake a numerical pattern of petals and you could be poisoned. Lose the rhythm of a sacred dance or the meter of a ritually told story and the intricately woven threads that hold life together were spoiled. Ignore the celestial clock of equinoxes and solstices, and you'd risk being caught short of food for the winter.Shesso's friendly tone and clear grasp of the information make the math go down easy in this marvelous book.
About the Author
Review Math for MysticsFrom the Fibonacci Sequence to Luna's Labyrinth to the Golden Section and Other Secrets of Sacred GeometryBy Renna ShessoRed Wheel/Weiser, LLCCopyright © 2007 Renna ShessoAll rights reserved.ISBN: 978-1-57863-383-8ContentsIntroduction: ""Math?! Why?""CHAPTER 0 The Circle of CreationCHAPTER 1 CountingCHAPTER 2 The MoonCHAPTER 3 MeasurementsCHAPTER 4 The Days of the WeekCHAPTER 5 The Magical SquaresCHAPTER 6 The Knight's Tour and Templar Codes?CHAPTER 7 Shapes and Numbers MeditationCHAPTER 8 PythagorasCHAPTER 9 Fibonacci, the Golden Ratio, and the PentacleCHAPTER 10 Venus' PentacleCHAPTER 11 The Geometric SolidsCHAPTER 12 Individual NumbersCHAPTER 13 A Tale in Which Gods Do MathCHAPTER 14 Summing UpNotesBibliographyIndex CHAPTER 1Counting""He counts using his fingers."" Nowadays, that phrase is generally an unkind one,a snippy way of implying that a person isn't very bright. Once upon a time,though, counting on our fingers had sacred connotations, and to know the numberof things was itself an act of magic. Our word ritual comes from the Indo-Europeanroot ri, which means ""to count, to number."" The association of ri with""ritual"" comes from the use of rites to mark the seasons of the year, back inthe days when a comprehension of time and seasons could be crucial to survival.Clearly, our personal ancestors succeeded in their season-counting and winterfood storage, or we wouldn't be here now. We can take that to mean that we allhave some inherent aptitude for timekeeping and calculating. In other words, weeach have an aptitude for practical math.Even before the advent of written numbers, people had ways of enumeratingquantities. We could cut or scratch notches on a spare piece of bone (manyexamples of this have survived) or we could line up stones (which would then getscattered), or we could use our fingers. The Sanskrit word for ""finger-counting""is mÛdrâ, closely related to mudrâ, the word for the symbolic hand gestures seenin Hindu religious statuary and sacred dance. Maybe you do your own littlefinger dances while you drive, drumming out basslines in time with the radio, asrhythm divides time. Without consciously saying, ""One, two, three, and four ...,""you're counting, in the most primal way, as body-knowledge. Forgetmultiplication tables! Give me a turntable!Zero and NineIn 773 CE, a diplomatic mission from northern India arrived in Baghdad. FromBaghdad to northern India is roughly 1,500 miles, minimum. As modern humans, wetend to forget our most primal Road Trip roots. Chances are we had distantancestors who spent not hours, but days, weeks, months—or years—physicallygetting to a location that really mattered to them. The Indian delegation madewhat must have been an unimaginably arduous journey.This visiting Indian contingent included an astronomer/astrologer named Kanaka.Though the studies of astronomy and astrology are now firmly separated, theyoriginally evolved together, and the Indians were considered especially skilled.Caliph al-Mansur, the Arabian host, became so impressed with Kanaka's starskills that he had Arabic translations made of the Indian reference works Kanakahad brought along. These translations were avidly shared, copied and recopied(by hand, of course), studied, discussed and mentally digested among Arabianastrologers, and about 50 years later, an original work by Arab mathematicianal-Khuwarizmi appeared. Called Kitab al jam' wa'l tafriq bi hisab al hind(""Indian technique of addition and subtraction""), al-Khuwarizmi's text concernedthe still-novel Indian numbers that had so impressed Caliph al-Mansur. Al-Khuwarizmigave a detailed explanation of decimal numeration, the nine Indiannumber symbols and ""the tenth figure in the shape of a circle"" that was used ""soas not to confuse the positions"" of the numbers.That ""tenth figure in the shape of a circle"" was Zero. One theory of its origin:People counted using pebbles laid in rows on a sandy surface. The Indians' termfor ""higher computations"" was dhuli-kharma, which actually means ""sand-work.""Let's put pebbles in rows to represent quantities. To subtract, we removepebbles. What's left? Some pebbles, of course, as well as faint depressions inthe sand. We check our math by looking at the dents left behind by the pebbleswe removed. And the shape of each depression would be a soft-edged circle in thesand, now containing nothing.But let's get back to ancient Arabia. The al-Khuwarizmi text became popular inthe Arab world and quietly arrived in Europe during the long Moorish presence inSpain. Although the text seems not to have spread into the rest of Europe, itsideas spread readily in other lands, and by the early eleventh century theIndian numerals and the zero were in common use from the borders of central Asiainto northern Africa and Egypt. Undoubtedly, variations on this numericalinformation migrated not just through Indian astrologers, but through otherpragmatic folks as well, since what worked for scholars and astrologers wouldalso be useful to merchants and accountants—to anyone making practical use ofnumbers. Finally, an abridged copy of al-Khuwarizmi's work, now simply calledArithmetic, was translated into Latin in 1126 CE, at which point it quicklybecame influential and controversial throughout Europe.Why did Arithmetic make such an impact? Because it presented some things Europedidn't have: a consistent and simple way to write the numbers 1 though 9, andthe radically innovative placeholder, zero. What we came to call the ""Arabicnumerals""—since they reached Europe through translations from the Arabic—in facthave their roots deeply in India. The legendary brilliance of Indian astronomer-astrologerslike Kanaka was credited to their superior skill in mathematics,skills made easier by their numerical system. Cuneiform and Roman numerals areokay for writing, and pebbles or fingers work fine for counting, but neither ismath-friendly. Astrologers needed writeable math formulae capable of greatercomplexity, and by creatively pushing to discover better and more detailed waysto express numbers, the ancient Indians moved to the forefront in astrology,astronomy, and math.Not everyone approved of the new-style written numbers. ""Quantities"" weren't—andaren't—the same as ""numbers."" The former are visible objects, like sheep orapples, while the latter, those ""numbers,"" are nothing more than weird shapesscrawled on a page. Zeros are especially suspect: Pen a tail on 0 and it becomes6 or 9. Tag on extra zeros, and that bogus 9 becomes 90, 900, 9,000, or worse.Small wonder that eleventh-century monk-historian William of Malmesburyconsidered the newfangled Indian-Arabic numerals, and especially that peskyzero, to be ""dangerous Saracen magic.""Back to trustworthy finger-counting. For the record, you can count to 9 on onehand using your five fingers and the spaces between them. The odd numbers landon the fingers, and the gaps get the even num-bers—5 odd, 4 even—and it worksperfectly. The Chinese believed that even numbers were bad luck and odd numberswere lucky. Perhaps this stemmed from even numbers landing on between-fingersnothingness. The Pythagoreans simply believed that even numbers were female andodd numbers were male, without the more pejorative good-or-bad-luckconnotation. That gender distinction could have been based on how these numbersare counted on the human hand: The ""male"" odd numbers land on the projectingfingers, and the ""female"" even numbers nestle in the open crevices betweenfingers. Pretty graphic, pretty basic.Nines have their own category of math tricks. For instance, any numbermultiplied by 9 ""reduces"" to 9. Try this with 18, 27, 36, 45, and any othermultiplied-by-9 sum, and in each case you'll get 9, a good memory trick for 9'smultiplication table. When written together, those multiplied-by-9 sums create anumerical palindrome— 9, 18, 27, 36, 45, 54, 63, 72, 81, 99—which (except forthe doubled 99) reads the same backward and forward.There's another old trick, dating back to at least the tenth century, called""casting out nines."" Flip back to the ""Using This Book"" section in theIntroduction and look at the total for December 31, 1950: 1993. Reduced, the sumwas 4. That's the same sum we can get immediately from 1993 by just adding 1 + 3and ""casting out""—that is, ignoring—the two 9s, which will cancel themselves outin the next step anyway. Nines void themselves out—at least they have every timeI've tested this. Why does this work? I understand it—sort of—but happily itworks whether or not I can explain it to myself.TwelveLook at one of your hands. The index finger is also called the ""pointer,"" andpoint it does, at least when we're looking at something beyond that hand. Butwhen we're counting on a single hand, the thumb is generally the built-inpointer. It's an easy and automatic gesture, useful when enumerating smallamounts, but only small amounts.Or maybe not. Each of the four fingers has three easily seen joints. Ourpointer-thumbs can reach each finger joint and count to 12 using them. Did theBabylonians, who had a 60-based number system, use one hand's twelve fingerjoints to count to 12 and use the other hand's five fingers for tracking howmany times they'd done so—12, 24, 36, 48, 60?We still buy eggs by twelves, and other things, too, like fresh cookies andflowers. We have 12 months and 12 astrological signs, and 12 inches in a foot,so the number has some practical applications. Some sources credit the worddozen as coming from the Latin for ""two"" and ""ten""—duo-decem—shortened graduallyto dozen. Others trace it back to an ancient Sumerian word that meant ""a fifthof sixty."" Twelve is wonderfully useful. It can be evenly divided by 2, 3, 4,and 6, and its multiples include significant numbers like 24 (hours in a day),60 (number systems, seconds, minutes), 108 (the Buddhist mala or prayer beads),144 (a gross), and 360 (the number of degrees in a circle). We'll look atcircles more closely later.FourteenNow let's go a step—two steps—further. If we include the thumb in our joint-counting,our number becomes 14. Unlike the fingers, the thumb has only tworeadily visible joints. While 14 isn't a number that readily springs to mindlike ""dozen,"" it's had its uses. A fortnight is 2 weeks, literally 14 nights,shortened from fourteen-night. In pre-metric Britain, if you said somethingweighed ""a stone,"" you meant 14 pounds. The ancient Chinese, Assyrians,Babylonians, and Sumerians all counted to 14 on the finger-and-thumb joints.The number 14 doesn't do as many tricks as 9 or crop up as often as 12, but ithas a longtime magical correlation that would have made it important to manyancient people: the Moon. See Chapter 2 for a look at this connection.Fifteen and MoreOther handy hand-counts of yore include an Indian and Bengali count of 15 (the14-count plus the pad at the base of the thumb) to track their 15-day ""months,""each half a lunar cycle in length. Twenty-four of these made up their 360-dayyear. A Muslim version of 15 used both hands to count to 30, then added the tipsof three fingers to reach 33, repeated thrice when reciting the 99 attributes ofAllah.The Venerable Bede (673–735 CE, a monk who wrote De ratione temporum—Of theDivision of Time) tracked an important lunar cycle by counting 19 years onfinger joints and tips, thumb joints and thumb pad of the left hand. Calledthe ""Metonic cycle"" (for Meton, a fifth-century BCE Athenian astronomer), thistracks the Moon's motions as they repeat every 19 years, with Earth's naturalsatellite in the same phase, sign, degree, and declination on the same day ofthe month.For a dramatic example, here are Full Moon eclipses in 1991 and 2010:1991, December 21: Moon at 29° Gemini, 24° N declination2010, December 21: Moon at 29° Gemini, 24° N declinationDice: The Fickle Finger of Fate?Gambling with dice or ""knuckles bones"" has some loose but logical ties back tofinger-counting. The earliest dice were real knuckle bones, roughly cubicalbones from the toes of various critters. Called astragali (a reference toAstraea, goddess of justice?), they were used for games of chance, but also cameinto play for making legal decisions, such as dividing inheritances or sharingout temple income, even for selecting government officials. In the geometric""Platonic"" solids, the cube symbolizes Earth. That's fitting, since in someancient cases, the cube-shaped dice influenced earthly, practical matters.The Assyrians were the first to make clay dice, more evenly shaped than theirregular bones. In northern Europe, the invention of dice was credited toclever Woden, deity of wisdom and prophecy. Our words lot (as in parcel ofland), lottery, and allotment all have their roots in the throwing of lots withdice. To say you ""cast your lot"" with someone meant you were gambling your ownfortune and luck with theirs.The words—singular die, plural dice— come from the Low Latin dadus, meaning""given,"" as in ""given by the gods."" Luck at dice wasn't viewed as pure randomchance: Winning was a cosmic sign that the gods were smiling upon you.These pagan associations were a perfect setup for the church to condemn dice asyet another of the devil's innumerable playthings. Besides, why leave templeprofits, inheritances, and government jobs to the dicey whims of pagan deities?Incidentally, dice needn't be six-sided cubes. Eight-sided octahedron dice havebeen discovered in Egyptian tombs (interesting: an octahedron is the shape oftwo pyramids joined at their bases), while dodecahedron (twelve-sided) andicosahedron (twenty-sided) dice were sometimes used by early fortune-tellers.Perhaps this was due to their respective Platonic associations with etherealSpirit and emotional Water, or maybe it was simply the fortune-tellers' bid formore repeat business, by having even more possible answers available.There are twenty-one possible combinations when rolling a pair of cubical dice:1-1, 1-2, 1-3, 1-4, 1-5, 1-62-2, 2-3, 2-4, 2-5, 2-63-3, 3-4, 3-5, 3-64-4, 4-5, 4-65-5, 5-66-6Add up all the dots on a single die—1 + 2 + 3 + 4 + 5 + 6—and the total is 21.CHAPTER 2The MoonHow many neopagans does it take to tell what phase the Moon is in? A surprisingnumber of us depend on printed calendars rather than our own eyes to tell us ifthe Moon is waxing or waning. Tsk, tsk! This primal piece of cosmic timekeepingcan help us easily tap into a deeper cyclical awareness of time.Some of this happens at a physiological level, in our own bodies. Young womenare often told at puberty that a ""normal"" menstrual cycle is 28 days long. Whatis seldom mentioned (at least in mundane environs) is that the Moon and themenstrual cycle can move together. Like the tides of the ocean, drawn higher bythe Moon's gravitational pull, the small ocean of the human body responds to theMoon, too. Women are said to be more likely to begin labor around the Full Moon,and once upon a time, women may have ovulated near the Full Moon and bled nearthe New Moon.The menstrual cycle is still called the ""moon time,"" but according to sometheories, women in industrialized nations have been knocked out of harmony withthe lunar cycle by all the artificial lights that illuminate the modern night.Few of us now sleep in such total darkness that the cyclical waxing and waninglight of the Moon is noticeable to the responsive physiology of the light-sensitivepineal gland in our ""third eye"" area. There may be ways of gettingback in sync (first, sleep in total darkness; then, use a night light on the 3nights when the Moon is fullest), but one tiny miscalculation remains in theseproceedings:The Moon's cycle isn't really 28 days. (See Figure 2-1.) From New Moon to NewMoon, the Moon's cycle in relation to the Sun is between 29 and 30 days. This iscalled the synodic month, and its precise average is 29 days, 12 hours, 44minutes and 3 seconds—29.53 days. Our earliest ancestors were already trackingit roughly 30,000–35,000 years ago, making clusters of scratch-marks on bones,antlers, and cavern walls, sometimes showing 29 scratches, sometimes 30.Meanwhile, Luna's monthly journey through the twelve constellations of thezodiac is on a slightly different schedule. Based on the relation to the starsrather than to the Sun, and called the sidereal month, the Moon's star-orientedcircuit averages 27 days, 7 hours, 43 minutes, and 12 seconds. If the synodicand sidereal months were exactly synchronized, the Full Moon would be in thesame sign of the zodiac every month. But it isn't (and how boring if it were).Instead, just as the Sun progresses one zodiac sign per month, so does the FullMoon. Full Moons happen when the Sun and Moon are opposite each other in oursky, so whatever sign the Sun is in, the Full Moon will be—must be—in theopposite sign of the zodiac. Since it's based on the backdrop of stars behindthe Moon, this Moon cycle isn't nearly as obvious as her cycle of shape-changingphases.So where did the association of the Moon with the number 28 come from? The NewMoon—shown as a solid black circle on calendars—occurs when the Moon and Sun areconjunct, aligned with each other in the same astrological sign as seen fromEarth. Except, of course, the Moon isn't seen at that time. (See Figure 2-2.)It's in between the Earth and the Sun, so we'd have to look directly into theSun to find the Moon at all. Even then, the Moon's lit ""face"" is facing the Sunwhile the shadowed side faces us. We can't see it. (Continues...)Excerpted from Math for Mystics by Renna Shesso. Copyright © 2007 Renna Shesso. Excerpted by permission of Red Wheel/Weiser, LLC. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site. --Excerpt. © Reprinted by permission. All rights reserved.""As thoughtful as it is readable, Renna Shesso's Math for Mystics is the book I wish I had when I first started trying to make sense of the mathematics that underlie so much of modern magic and traditional occult lore. Not the least of its virtues is the way it makes magical number theory accessible even to those who think they don't like or can't handle math. It provides a firstrate introduction to a fairly neglected branch of magical lore."" --John Michael Greer, Grand Archdruid, Ancient Order of Druids in America and author of The Druidry Handbook""In times past, math was seen as magic for its power and associations. It was even banned by authorities who thought it a threata power that no one else should hold. In this book, that ancient magic is relived, and the power yours."" --Jeff Hoke, author of The Museum of Lost Wonder
About the Author Renna Shesso has been a student of mystical traditions and spiritual selfdiscovery since the late 1960s. She is the author of Math for Mystics. A longtime resident of Colorado, Shesso follows her calling as shamanic healing practitioner and teacher, professional tarot reader, and priestess of the Craft. Visit her at www.rennashesso.com. Excerpt. © Reprinted by permission. All rights reserved. Math for MysticsFrom the Fibonacci Sequence to Luna's Labyrinth to the Golden Section and Other Secrets of Sacred GeometryBy Renna ShessoRed Wheel/Weiser, LLCCopyright © 2007 Renna ShessoAll rights reserved.ISBN: 978-1-57863-383-8ContentsIntroduction: ""Math?! Why?""CHAPTER 0 The Circle of CreationCHAPTER 1 CountingCHAPTER 2 The MoonCHAPTER 3 MeasurementsCHAPTER 4 The Days of the WeekCHAPTER 5 The Magical SquaresCHAPTER 6 The Knight's Tour and Templar Codes?CHAPTER 7 Shapes and Numbers MeditationCHAPTER 8 PythagorasCHAPTER 9 Fibonacci, the Golden Ratio, and the PentacleCHAPTER 10 Venus' PentacleCHAPTER 11 The Geometric SolidsCHAPTER 12 Individual NumbersCHAPTER 13 A Tale in Which Gods Do MathCHAPTER 14 Summing UpNotesBibliographyIndex CHAPTER 1Counting""He counts using his fingers."" Nowadays, that phrase is generally an unkind one,a snippy way of implying that a person isn't very bright. Once upon a time,though, counting on our fingers had sacred connotations, and to know the numberof things was itself an act of magic. Our word ritual comes from the Indo-Europeanroot ri, which means ""to count, to number."" The association of ri with""ritual"" comes from the use of rites to mark the seasons of the year, back inthe days when a comprehension of time and seasons could be crucial to survival.Clearly, our personal ancestors succeeded in their season-counting and winterfood storage, or we wouldn't be here now. We can take that to mean that we allhave some inherent aptitude for timekeeping and calculating. In other words, weeach have an aptitude for practical math.Even before the advent of written numbers, people had ways of enumeratingquantities. We could cut or scratch notches on a spare piece of bone (manyexamples of this have survived) or we could line up stones (which would then getscattered), or we could use our fingers. The Sanskrit word for ""finger-counting""is mÛdrâ, closely related to mudrâ, the word for the symbolic hand gestures seenin Hindu religious statuary and sacred dance. Maybe you do your own littlefinger dances while you drive, drumming out basslines in time with the radio, asrhythm divides time. Without consciously saying, ""One, two, three, and four ...,""you're counting, in the most primal way, as body-knowledge. Forgetmultiplication tables! Give me a turntable!Zero and NineIn 773 CE, a diplomatic mission from northern India arrived in Baghdad. FromBaghdad to northern India is roughly 1,500 miles, minimum. As modern humans, wetend to forget our most primal Road Trip roots. Chances are we had distantancestors who spent not hours, but days, weeks, months—or years—physicallygetting to a location that really mattered to them. The Indian delegation madewhat must have been an unimaginably arduous journey.This visiting Indian contingent included an astronomer/astrologer named Kanaka.Though the studies of astronomy and astrology are now firmly separated, theyoriginally evolved together, and the Indians were considered especially skilled.Caliph al-Mansur, the Arabian host, became so impressed with Kanaka's starskills that he had Arabic translations made of the Indian reference works Kanakahad brought along. These translations were avidly shared, copied and recopied(by hand, of course), studied, discussed and mentally digested among Arabianastrologers, and about 50 years later, an original work by Arab mathematicianal-Khuwarizmi appeared. Called Kitab al jam' wa'l tafriq bi hisab al hind(""Indian technique of addition and subtraction""), al-Khuwarizmi's text concernedthe still-novel Indian numbers that had so impressed Caliph al-Mansur. Al-Khuwarizmigave a detailed explanation of decimal numeration, the nine Indiannumber symbols and ""the tenth figure in the shape of a circle"" that was used ""soas not to confuse the positions"" of the numbers.That ""tenth figure in the shape of a circle"" was Zero. One theory of its origin:People counted using pebbles laid in rows on a sandy surface. The Indians' termfor ""higher computations"" was dhuli-kharma, which actually means ""sand-work.""Let's put pebbles in rows to represent quantities. To subtract, we removepebbles. What's left? Some pebbles, of course, as well as faint depressions inthe sand. We check our math by looking at the dents left behind by the pebbleswe removed. And the shape of each depression would be a soft-edged circle in thesand, now containing nothing.But let's get back to ancient Arabia. The al-Khuwarizmi text became popular inthe Arab world and quietly arrived in Europe during the long Moorish presence inSpain. Although the text seems not to have spread into the rest of Europe, itsideas spread readily in other lands, and by the early eleventh century theIndian numerals and the zero were in common use from the borders of central Asiainto northern Africa and Egypt. Undoubtedly, variations on this numericalinformation migrated not just through Indian astrologers, but through otherpragmatic folks as well, since what worked for scholars and astrologers wouldalso be useful to merchants and accountants—to anyone making practical use ofnumbers. Finally, an abridged copy of al-Khuwarizmi's work, now simply calledArithmetic, was translated into Latin in 1126 CE, at which point it quicklybecame influential and controversial throughout Europe.Why did Arithmetic make such an impact? Because it presented some things Europedidn't have: a consistent and simple way to write the numbers 1 though 9, andthe radically innovative placeholder, zero. What we came to call the ""Arabicnumerals""—since they reached Europe through translations from the Arabic—in facthave their roots deeply in India. The legendary brilliance of Indian astronomer-astrologerslike Kanaka was credited to their superior skill in mathematics,skills made easier by their numerical system. Cuneiform and Roman numerals areokay for writing, and pebbles or fingers work fine for counting, but neither ismath-friendly. Astrologers needed writeable math formulae capable of greatercomplexity, and by creatively pushing to discover better and more detailed waysto express numbers, the ancient Indians moved to the forefront in astrology,astronomy, and math.Not everyone approved of the new-style written numbers. ""Quantities"" weren't—andaren't—the same as ""numbers."" The former are visible objects, like sheep orapples, while the latter, those ""numbers,"" are nothing more than weird shapesscrawled on a page. Zeros are especially suspect: Pen a tail on 0 and it becomes6 or 9. Tag on extra zeros, and that bogus 9 becomes 90, 900, 9,000, or worse.Small wonder that eleventh-century monk-historian William of Malmesburyconsidered the newfangled Indian-Arabic numerals, and especially that peskyzero, to be ""dangerous Saracen magic.""Back to trustworthy finger-counting. For the record, you can count to 9 on onehand using your five fingers and the spaces between them. The odd numbers landon the fingers, and the gaps get the even num-bers—5 odd, 4 even—and it worksperfectly. The Chinese believed that even numbers were bad luck and odd numberswere lucky. Perhaps this stemmed from even numbers landing on between-fingersnothingness. The Pythagoreans simply believed that even numbers were female andodd numbers were male, without the more pejorative good-or-bad-luckconnotation. That gender distinction could have been based on how these numbersare counted on the human hand: The ""male"" odd numbers land on the projectingfingers, and the ""female"" even numbers nestle in the open crevices betweenfingers. Pretty graphic, pretty basic.Nines have their own category of math tricks. For instance, any numbermultiplied by 9 ""reduces"" to 9. Try this with 18, 27, 36, 45, and any othermultiplied-by-9 sum, and in each case you'll get 9, a good memory trick for 9'smultiplication table. When written together, those multiplied-by-9 sums create anumerical palindrome— 9, 18, 27, 36, 45, 54, 63, 72, 81, 99—which (except forthe doubled 99) reads the same backward and forward.There's another old trick, dating back to at least the tenth century, called""casting out nines."" Flip back to the ""Using This Book"" section in theIntroduction and look at the total for December 31, 1950: 1993. Reduced, the sumwas 4. That's the same sum we can get immediately from 1993 by just adding 1 + 3and ""casting out""—that is, ignoring—the two 9s, which will cancel themselves outin the next step anyway. Nines void themselves out—at least they have every timeI've tested this. Why does this work? I understand it—sort of—but happily itworks whether or not I can explain it to myself.TwelveLook at one of your hands. The index finger is also called the ""pointer,"" andpoint it does, at least when we're looking at something beyond that hand. Butwhen we're counting on a single hand, the thumb is generally the built-inpointer. It's an easy and automatic gesture, useful when enumerating smallamounts, but only small amounts.Or maybe not. Each of the four fingers has three easily seen joints. Ourpointer-thumbs can reach each finger joint and count to 12 using them. Did theBabylonians, who had a 60-based number system, use one hand's twelve fingerjoints to count to 12 and use the other hand's five fingers for tracking howmany times they'd done so—12, 24, 36, 48, 60?We still buy eggs by twelves, and other things, too, like fresh cookies andflowers. We have 12 months and 12 astrological signs, and 12 inches in a foot,so the number has some practical applications. Some sources credit the worddozen as coming from the Latin for ""two"" and ""ten""—duo-decem—shortened graduallyto dozen. Others trace it back to an ancient Sumerian word that meant ""a fifthof sixty."" Twelve is wonderfully useful. It can be evenly divided by 2, 3, 4,and 6, and its multiples include significant numbers like 24 (hours in a day),60 (number systems, seconds, minutes), 108 (the Buddhist mala or prayer beads),144 (a gross), and 360 (the number of degrees in a circle). We'll look atcircles more closely later.FourteenNow let's go a step—two steps—further. If we include the thumb in our joint-counting,our number becomes 14. Unlike the fingers, the thumb has only tworeadily visible joints. While 14 isn't a number that readily springs to mindlike ""dozen,"" it's had its uses. A fortnight is 2 weeks, literally 14 nights,shortened from fourteen-night. In pre-metric Britain, if you said somethingweighed ""a stone,"" you meant 14 pounds. The ancient Chinese, Assyrians,Babylonians, and Sumerians all counted to 14 on the finger-and-thumb joints.The number 14 doesn't do as many tricks as 9 or crop up as often as 12, but ithas a longtime magical correlation that would have made it important to manyancient people: the Moon. See Chapter 2 for a look at this connection.Fifteen and MoreOther handy hand-counts of yore include an Indian and Bengali count of 15 (the14-count plus the pad at the base of the thumb) to track their 15-day ""months,""each half a lunar cycle in length. Twenty-four of these made up their 360-dayyear. A Muslim version of 15 used both hands to count to 30, then added the tipsof three fingers to reach 33, repeated thrice when reciting the 99 attributes ofAllah.The Venerable Bede (673–735 CE, a monk who wrote De ratione temporum—Of theDivision of Time) tracked an important lunar cycle by counting 19 years onfinger joints and tips, thumb joints and thumb pad of the left hand. Calledthe ""Metonic cycle"" (for Meton, a fifth-century BCE Athenian astronomer), thistracks the Moon's motions as they repeat every 19 years, with Earth's naturalsatellite in the same phase, sign, degree, and declination on the same day ofthe month.For a dramatic example, here are Full Moon eclipses in 1991 and 2010:1991, December 21: Moon at 29° Gemini, 24° N declination2010, December 21: Moon at 29° Gemini, 24° N declinationDice: The Fickle Finger of Fate?Gambling with dice or ""knuckles bones"" has some loose but logical ties back tofinger-counting. The earliest dice were real knuckle bones, roughly cubicalbones from the toes of various critters. Called astragali (a reference toAstraea, goddess of justice?), they were used for games of chance, but also cameinto play for making legal decisions, such as dividing inheritances or sharingout temple income, even for selecting government officials. In the geometric""Platonic"" solids, the cube symbolizes Earth. That's fitting, since in someancient cases, the cube-shaped dice influenced earthly, practical matters.The Assyrians were the first to make clay dice, more evenly shaped than theirregular
bones."
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