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Renna Shesso's friendly tone, delightful "math lore," meticulous research, and clear information makes math easy to understand. This marvelous book begins with the simplest lunary and planetary math and then tackles the most enigmatic of numerical esoterica such as Platonic Solids, the Golden Section, Luna's Labyrinth, and Benjamin Franklin's favorite way to pass the time, "Magical Squares," akin to the 17th century Sudoku. For anyone who tried to understand the Fibonacci Sequence of numbers from Dan Brown's (son...
Renna Shesso's friendly tone, delightful "math lore," meticulous research, and clear information makes math easy to understand. This marvelous book begins with the simplest lunary and planetary math and then tackles the most enigmatic of numerical esoterica such as Platonic Solids, the Golden Section, Luna's Labyrinth, and Benjamin Franklin's favorite way to pass the time, "Magical Squares," akin to the 17th century Sudoku. For anyone who tried to understand the Fibonacci Sequence of numbers from Dan Brown's (son of a mathematician) The DaVinci Code, this book is for you! Much of what we know as math comes to us directly from early astronomer magi who needed to be able to describe and record what they saw in the night sky. Everyone needed math: 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 poison yourself. 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.
"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 number of things was itself an act of magic. Our word ritual comes from the Indo-European root 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 in the days when a comprehension of time and seasons could be crucial to survival. Clearly, our personal ancestors succeeded in their season-counting and winter food storage, or we wouldn't be here now. We can take that to mean that we all have some inherent aptitude for timekeeping and calculating. In other words, we each have an aptitude for practical math.
Even before the advent of written numbers, people had ways of enumerating quantities. We could cut or scratch notches on a spare piece of bone (many examples of this have survived) or we could line up stones (which would then get scattered), 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 seen in Hindu religious statuary and sacred dance. Maybe you do your own little finger dances while you drive, drumming out basslines in time with the radio, as rhythm divides time. Without consciously saying, "One, two, three, and four ...," you're counting, in the most primal way, as body-knowledge. Forget multiplication tables! Give me a turntable!
Zero and Nine
In 773 CE, a diplomatic mission from northern India arrived in Baghdad. From Baghdad to northern India is roughly 1,500 miles, minimum. As modern humans, we tend to forget our most primal Road Trip roots. Chances are we had distant ancestors who spent not hours, but days, weeks, months—or years—physically getting to a location that really mattered to them. The Indian delegation made what 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, they originally evolved together, and the Indians were considered especially skilled. Caliph al-Mansur, the Arabian host, became so impressed with Kanaka's star skills that he had Arabic translations made of the Indian reference works Kanaka had brought along. These translations were avidly shared, copied and recopied (by hand, of course), studied, discussed and mentally digested among Arabian astrologers, and about 50 years later, an original work by Arab mathematician al-Khuwarizmi appeared. Called Kitab al jam' wa'l tafriq bi hisab al hind ("Indian technique of addition and subtraction"), al-Khuwarizmi's text concerned the still-novel Indian numbers that had so impressed Caliph al-Mansur. Al-Khuwarizmi gave a detailed explanation of decimal numeration, the nine Indian number symbols and "the tenth figure in the shape of a circle" that was used "so as 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' term for "higher computations" was dhuli-kharma, which actually means "sand-work." Let's put pebbles in rows to represent quantities. To subtract, we remove pebbles. What's left? Some pebbles, of course, as well as faint depressions in the sand. We check our math by looking at the dents left behind by the pebbles we removed. And the shape of each depression would be a soft-edged circle in the sand, now containing nothing.
But let's get back to ancient Arabia. The al-Khuwarizmi text became popular in the Arab world and quietly arrived in Europe during the long Moorish presence in Spain. Although the text seems not to have spread into the rest of Europe, its ideas spread readily in other lands, and by the early eleventh century the Indian numerals and the zero were in common use from the borders of central Asia into northern Africa and Egypt. Undoubtedly, variations on this numerical information migrated not just through Indian astrologers, but through other pragmatic folks as well, since what worked for scholars and astrologers would also be useful to merchants and accountants—to anyone making practical use of numbers. Finally, an abridged copy of al-Khuwarizmi's work, now simply called Arithmetic, was translated into Latin in 1126 CE, at which point it quickly became influential and controversial throughout Europe.
Why did Arithmetic make such an impact? Because it presented some things Europe didn't have: a consistent and simple way to write the numbers 1 though 9, and the radically innovative placeholder, zero. What we came to call the "Arabic numerals"—since they reached Europe through translations from the Arabic—in fact have their roots deeply in India. The legendary brilliance of Indian astronomer-astrologers like Kanaka was credited to their superior skill in mathematics, skills made easier by their numerical system. Cuneiform and Roman numerals are okay for writing, and pebbles or fingers work fine for counting, but neither is math-friendly. Astrologers needed writeable math formulae capable of greater complexity, and by creatively pushing to discover better and more detailed ways to 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—and aren't—the same as "numbers." The former are visible objects, like sheep or apples, while the latter, those "numbers," are nothing more than weird shapes scrawled on a page. Zeros are especially suspect: Pen a tail on 0 and it becomes 6 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 Malmesbury considered the newfangled Indian-Arabic numerals, and especially that pesky zero, to be "dangerous Saracen magic."
Back to trustworthy finger-counting. For the record, you can count to 9 on one hand using your five fingers and the spaces between them. The odd numbers land on the fingers, and the gaps get the even num-bers—5 odd, 4 even—and it works perfectly. The Chinese believed that even numbers were bad luck and odd numbers were lucky. Perhaps this stemmed from even numbers landing on between-fingers nothingness. The Pythagoreans simply believed that even numbers were female and odd numbers were male, without the more pejorative good-or-bad-luck connotation. That gender distinction could have been based on how these numbers are counted on the human hand: The "male" odd numbers land on the projecting fingers, and the "female" even numbers nestle in the open crevices between fingers. Pretty graphic, pretty basic.
Nines have their own category of math tricks. For instance, any number multiplied by 9 "reduces" to 9. Try this with 18, 27, 36, 45, and any other multiplied-by-9 sum, and in each case you'll get 9, a good memory trick for 9's multiplication table. When written together, those multiplied-by-9 sums create a numerical palindrome— 9, 18, 27, 36, 45, 54, 63, 72, 81, 99—which (except for the 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 the Introduction and look at the total for December 31, 1950: 1993. Reduced, the sum was 4. That's the same sum we can get immediately from 1993 by just adding 1 + 3 and "casting out"—that is, ignoring—the two 9s, which will cancel themselves out in the next step anyway. Nines void themselves out—at least they have every time I've tested this. Why does this work? I understand it—sort of—but happily it works whether or not I can explain it to myself.
Look at one of your hands. The index finger is also called the "pointer," and point it does, at least when we're looking at something beyond that hand. But when we're counting on a single hand, the thumb is generally the built-in pointer. It's an easy and automatic gesture, useful when enumerating small amounts, but only small amounts.
Or maybe not. Each of the four fingers has three easily seen joints. Our pointer-thumbs can reach each finger joint and count to 12 using them. Did the Babylonians, who had a 60-based number system, use one hand's twelve finger joints to count to 12 and use the other hand's five fingers for tracking how many times they'd done so—12, 24, 36, 48, 60?
We still buy eggs by twelves, and other things, too, like fresh cookies and flowers. 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 word dozen as coming from the Latin for "two" and "ten"—duo-decem—shortened gradually to dozen. Others trace it back to an ancient Sumerian word that meant "a fifth of 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 at circles more closely later.
Now 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 two readily visible joints. While 14 isn't a number that readily springs to mind like "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 something weighed "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 it has a longtime magical correlation that would have made it important to many ancient people: the Moon. See Chapter 2 for a look at this connection.
Fifteen and More
Other handy hand-counts of yore include an Indian and Bengali count of 15 (the 14-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-day year. A Muslim version of 15 used both hands to count to 30, then added the tips of three fingers to reach 33, repeated thrice when reciting the 99 attributes of Allah.
The Venerable Bede (673–735 CE, a monk who wrote De ratione temporum—Of the Division of Time) tracked an important lunar cycle by counting 19 years on finger joints and tips, thumb joints and thumb pad of the left hand. Called the "Metonic cycle" (for Meton, a fifth-century BCE Athenian astronomer), this tracks the Moon's motions as they repeat every 19 years, with Earth's natural satellite in the same phase, sign, degree, and declination on the same day of the month.
For a dramatic example, here are Full Moon eclipses in 1991 and 2010:
1991, December 21: Moon at 29° Gemini, 24° N declination
2010, December 21: Moon at 29° Gemini, 24° N declination
Dice: The Fickle Finger of Fate?
Gambling with dice or "knuckles bones" has some loose but logical ties back to finger-counting. The earliest dice were real knuckle bones, roughly cubical bones from the toes of various critters. Called astragali (a reference to Astraea, goddess of justice?), they were used for games of chance, but also came into play for making legal decisions, such as dividing inheritances or sharing out temple income, even for selecting government officials. In the geometric "Platonic" solids, the cube symbolizes Earth. That's fitting, since in some ancient cases, the cube-shaped dice influenced earthly, practical matters.
The Assyrians were the first to make clay dice, more evenly shaped than the irregular bones. In northern Europe, the invention of dice was credited to clever Woden, deity of wisdom and prophecy. Our words lot (as in parcel of land), lottery, and allotment all have their roots in the throwing of lots with dice. To say you "cast your lot" with someone meant you were gambling your own fortune 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 random chance: 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 as yet another of the devil's innumerable playthings. Besides, why leave temple profits, inheritances, and government jobs to the dicey whims of pagan deities?
Incidentally, dice needn't be six-sided cubes. Eight-sided octahedron dice have been discovered in Egyptian tombs (interesting: an octahedron is the shape of two pyramids joined at their bases), while dodecahedron (twelve-sided) and icosahedron (twenty-sided) dice were sometimes used by early fortune-tellers. Perhaps this was due to their respective Platonic associations with ethereal Spirit and emotional Water, or maybe it was simply the fortune-tellers' bid for more 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, 1-6
2-2, 2-3, 2-4, 2-5, 2-6
3-3, 3-4, 3-5, 3-6
4-4, 4-5, 4-6
Add up all the dots on a single die—1 + 2 + 3 + 4 + 5 + 6—and the total is 21.
How many neopagans does it take to tell what phase the Moon is in? A surprising number of us depend on printed calendars rather than our own eyes to tell us if the Moon is waxing or waning. Tsk, tsk! This primal piece of cosmic timekeeping can help us easily tap into a deeper cyclical awareness of time.
Some of this happens at a physiological level, in our own bodies. Young women are often told at puberty that a "normal" menstrual cycle is 28 days long. What is seldom mentioned (at least in mundane environs) is that the Moon and the menstrual cycle can move together. Like the tides of the ocean, drawn higher by the Moon's gravitational pull, the small ocean of the human body responds to the Moon, 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 near the New Moon.
The menstrual cycle is still called the "moon time," but according to some theories, women in industrialized nations have been knocked out of harmony with the 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 waning light of the Moon is noticeable to the responsive physiology of the light-sensitive pineal gland in our "third eye" area. There may be ways of getting back in sync (first, sleep in total darkness; then, use a night light on the 3 nights when the Moon is fullest), but one tiny miscalculation remains in these proceedings:
The Moon's cycle isn't really 28 days. (See Figure 2-1.) From New Moon to New Moon, the Moon's cycle in relation to the Sun is between 29 and 30 days. This is called the synodic month, and its precise average is 29 days, 12 hours, 44 minutes and 3 seconds—29.53 days. Our earliest ancestors were already tracking it 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 the zodiac is on a slightly different schedule. Based on the relation to the stars rather than to the Sun, and called the sidereal month, the Moon's star-oriented circuit averages 27 days, 7 hours, 43 minutes, and 12 seconds. If the synodic and sidereal months were exactly synchronized, the Full Moon would be in the same 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 Full Moon. Full Moons happen when the Sun and Moon are opposite each other in our sky, so whatever sign the Sun is in, the Full Moon will be—must be—in the opposite sign of the zodiac. Since it's based on the backdrop of stars behind the Moon, this Moon cycle isn't nearly as obvious as her cycle of shape-changing phases.
So where did the association of the Moon with the number 28 come from? The New Moon—shown as a solid black circle on calendars—occurs when the Moon and Sun are conjunct, aligned with each other in the same astrological sign as seen from Earth. 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 the Sun to find the Moon at all. Even then, the Moon's lit "face" is facing the Sun while the shadowed side faces us. We can't see it.
Excerpted from Math for Mystics by Renna Shesso. Copyright © 2007 Renna Shesso. Excerpted by permission of Red Wheel/Weiser, LLC.
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Introduction: "Math?! Why?"
CHAPTER 0 The Circle of Creation
CHAPTER 1 Counting
CHAPTER 2 The Moon
CHAPTER 3 Measurements
CHAPTER 4 The Days of the Week
CHAPTER 5 The Magical Squares
CHAPTER 6 The Knight's Tour and Templar Codes?
CHAPTER 7 Shapes and Numbers Meditation
CHAPTER 8 Pythagoras
CHAPTER 9 Fibonacci, the Golden Ratio, and the Pentacle
CHAPTER 10 Venus' Pentacle
CHAPTER 11 The Geometric Solids
CHAPTER 12 Individual Numbers
CHAPTER 13 A Tale in Which Gods Do Math
CHAPTER 14 Summing Up
Posted March 19, 2012