Thinking in Numbers: How Maths Illuminates Our Lives (14 page)

Polyvalent talent like his is rare in any age. It likely led to jealousies, snide comments, upturned noses from certain quarters among his fellow countrymen. In one of several treatises on algebraic problems, Khayyám complained about the trials of the mathematician’s life.

I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles . . . which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes.

In 452 (1074
CE
by the Julian calendar), Sultan Jalal Al-Din invited Omar to the capital. His long Farsi texts, packed with numbers and equations, had preceded him. Even at the zenith of the golden age of Islamic mathematics, Khayyám’s talent marked him out. Anxiously, expectantly, he must have followed his guide through the halls of the turquoise-studded palace. Tessellating tiles ran the length of the floor; mirrors hanging on the walls returned to him every one of his features, down to the wrinkles of laughter around his eyes.

The Sultan that Khayyám
met hardly looked the part: he was very young, not yet twenty years of age. He was keen to make his mark. His illustrious guest, the Sultan’s Vizier reports, was promptly showered with the prince’s praises, and offered an annual pension of 1,200
mithkáls
of gold. For this, Khayyám
agreed to accept an important commission: he was to create – provided that God were willing – a new civil calendar in the young Sultan’s name.

Persia, geographically vast and intricately governed, had long lived on two times: while the imams practised the faith using the
hegira
calendar, the bureaucrats counted their days by looking to the sun. The old civil calendar consisted of twelve months each of thirty days, excepting the eighth month, which contained thirty-five days. Its total of 365 days (eleven days more than in a lunar year) helped fix administrative dates far more closely to the seasons: an essential requirement when the nation’s annual tax revenues relied heavily on the autumn harvests. But even these eleven extra days could not keep up precisely with the seasons: each year, as Al-Biruni had recorded, accumulated a lag of a quarter day. This was the problem that Khayyám set out to solve.

Day and night, Omar pondered how best to reform the old civil calendar. Astronomy had never before found itself so flattered and moneyed. A large observatory duly arose, from which Khayyám and his colleagues stalked the sky. He studied the sun’s pathway through the twelve constellations of stars, and compiled detailed statistical tables. With this data he mapped a calendar based on the seasons, beginning the year (which the Persians called
Nowruz
) on the spring equinox (March 20 or 21), with the fourth month falling on the summer solstice (June 21), the seventh month on the autumn equinox (September 21), and the tenth on the winter solstice (December 20 or 21).

To resolve the lagging of quarter days, Khayyám
devised an ingenious formula. His calendar interleaved eight extra days over each thirty-three-year span. The calculation: 365 + 8/33 = 365.2424 days, aligned almost perfectly with the actual year length (365.2423 days), and proved even more accurate than that of the later Gregorian calendar: 365 + ¼ – 1/100 + 1/400 = 365 + 97/400 = 365.2425 days.

The Sultan officially adopted Khayyám’s
reform on Friday March 15, 1079 (
Farvardin
1, 458 according to the new calendar). Drums and canon blasts across the nation proclaimed the first New Year.

Concerning Khayyám’s
later years, we can say very little. The Sultan’s premature death, just ten years after the calendar’s adoption, brought an end to his generous patronage. Khayyám
quit the royal court, only the ritual pilgrimage to Mecca delaying a return to his hometown. He continued to write poems.

Ah, but my computations, People say,

Reduced the year to better reckoning? Nay,

’Twas only striking from the Calendar

Unborn tomorrow and dead  Yesterday

One day, in his eighth decade, Khayyám tired and lay down to rest. His heavy head swaddled in sheets of turban, he gazed up toward the sky. The light around him slowly faded, and then went out. The year was around 500 of the
hegira
– the point Al-Tabari had long ago calculated as being the end of the world.

Counting by Elevens

‘Physicians say that thumbs are the master fingers of the hand,’ writes Michel de Montaigne, the Renaissance nobleman who invented the personal essay, in his short piece
On Thumbs
. So vital did Rome’s ancient rulers consider them, he explains, that war veterans who were missing thumbs received automatic exemption from all future military service.

Throughout his writings, the Frenchman marvelled at the extent to which we depend upon our hands. A gesture of thumbs-up or down, an index finger pressed to the mouth, the palms held flat open toward the skies: delivered at the right instant all can say more than any word. In another essay Montaigne describes the case of a ‘native of Nantes, born without arms’ whose feet did double duty so well that the man ‘cuts anything, charges and discharges a pistol, threads a needle, sews, writes, takes off his hat, combs his head, plays at cards and dice.’ Elsewhere he observed that hands sometimes seem almost to possess a life of their own, as when his idle fingers would tap during a period of daydreaming, free of conscious effort or instruction.

Montaigne neglects to mention counting as being among the many useful tasks undertaken by our hands (biographers have pointed out that arithmetic was not his strong suit). Of course, there is much to be said for the idea that our decimal number system originated from the practice of counting with the fingers. In our Latin-derived word ‘digit’ the meanings ‘number’ and ‘finger’ coincide. The Homeric Greek term for counting,
pempathai
, translates literally: ‘to count by fives’.

People the world over, numbers at their fingertips, count to ten and by tens (twenty, thirty . . . fifty . . . one hundred . . .). But the manner of their getting to ten varies from hand to hand and from culture to culture. Like many Europeans, I count ‘one’ beginning on the thumb of my left hand and continue past five with the digits of my right till I arrive at the other thumb. Americans, on the other hand, will often start with their left index finger and count five on their left thumb – repeating the procedure on their right hand for the numbers six through ten. Where people read from right to left, as in the countries of the Middle East, counting generally begins on the little finger of the right hand. In Asia, the counter employs a single hand, folding the fingers thumb first to reach five, before unfolding them from the little finger (six) returning to the thumb (ten).

What, I wonder, would it be like for a person who had not too few fingers – like Montaigne’s Roman veterans – but too many? Would such a person learn to count like you or me? How would it be to count by elevens?

According to tradition, Henry VIII’s second wife, Anne Boleyn, carried a sixth finger on one of her hands – a medical condition known as polydactyly. Girls of high birth in Tudor society had the attention of tutors and learned to read, write and do sums. Counting on her eleven fingers, however, would have posed Anne certain difficulties.

To start with, she would have found herself a number word short. Between the ninth finger and the last (which would still be the tenth – 10 meaning ‘1 set and 0 remainders’) her surplus digit would have needed naming. Given that she spent part of her childhood in France, she might have come up with the label
dix
. So Anne would have counted: one, two, three, four, five, six, seven, eight, nine,
dix
, ten. Though the numbers’ music sounds a little strange to our ears, in her mind the sequence would have come to feel like second nature. A year before the age of twenty, she would have celebrated (if only mentally) her dixteenth birthday. Counting higher, she would have taken the nineties down a peg, to make room for the dixties, and by way of dixty-dix arrive finally at one hundred.

Counting in this way produces some funny-looking sums. Subtracting seven from ten (where, given the extra
dix
, ‘ten’ would to our way of thinking be eleven), for example, would have left Anne not with three, but four. Half of thirteen equals seven. Six squared (6 × 6) is thirty-three (three ‘tens’ of eleven, plus three units): a pretty result.

Fractions would have proven particularly tricky. Unlike ten, eleven is a prime number: divisible only by itself and one. No precise midway point exists between one and eleven, or between Anne’s ten (equivalent to our eleven) and her one hundred (11 × 11). What then might a half
of something mean to someone with eleven fingers? And what about a fifth
or a quarter or two
-
thirds? Possibly such concepts would remain as airy and intangible as a
dixth
of something might seem to you or me. (We can nonetheless imagine that rote learning would have sufficed to acquire them.)

Still, I am curious about the intuitions that an eleven-fingered girl might have brought to such ideas. From our two hands, equally endowed, we understand immediately that halves are clean and precise, leaving no remainder. Half of eight is four, not three or five. A right-angle triangle is exactly one half of a square. Prime numbers, by definition, cannot be halved in this way. Might Anne’s hands – with six fingers on one hand, and ‘only’ five on the other – have given her a more approximate, fuzzier feeling for the concept of a half? To a casual remark like, ‘I’ll be along in half an hour’, would the impatient thought have occurred to her, ‘Which half do you mean – the lesser or the greater?’

At her secret wedding to the King of England, Boleyn would have well understood which half of the couple she represented. Her triumph was infamously short-lived. Within months, the Archbishop of Canterbury had declared the marriage invalid. Roman Catholics denounced her as a scarlet woman.

The charge of treason brought against her only three years after the wedding, makes no mention of witchcraft. All the same, her enemies alleged that the Queen’s body showed strange warts and growths – the same body that had produced only stillborn foetuses for a male heir. It was in the role of a conspiring adulteress that she knelt in a black damask dress before the executioner’s block on the morning of 19 May 1536.

It is certainly conceivable that the story of an eleventh finger was a fabrication of Anne’s enemies. A famous portrait by an unknown artist, which now hangs in Hever Castle (Boleyn’s childhood home) in Kent, reveals a striking young woman clutching a rose. The hands peek shyly from long sleeves; the fingers – ten in all – appear, in contrast to the smooth oval face, somewhat ill formed. A slander, or a secret, Anne’s eleven fingers have become an integral part of her legend.

I was reminded of it recently when a fascinating newspaper article caught my attention. It was about one Yoandri Hernandez Garrido, a thirty-seven-year-old Cuban man with twelve fingers and twelve toes. Apparently, Fidel Castro’s doctor once paid the extra-dextrous boy a visit and declared his hands and feet the most beautiful that he had ever seen.

In keeping with the Latin American taste for nicknames based on physical appearance, schoolmates gave Garrido the name ‘Veinticuatro’ (twenty-four). He learned to count in class like his friends but one day, he recalls, his primary school teacher asked him for the answer to the sum 5 + 5. In his confusion he answered, ‘Twelve’.

Garrido tells the reporter that he makes a fine living with his hands. American tourists regularly tender their dollars to have a photo taken with the twelve-fingered Cuban. A few extra bucks can go a long way in Castro’s republic. Proudly, Garrido holds his hands up to the camera. There is generosity in his smile.

The article makes no mention of the adult Garrido’s arithmetic. Counting time, to give one example, should be simpler with twelve fingers – one digit for every hour on the clock. To find what time is nine hours before four o’clock in the afternoon, he need simply hold down five fingers (9 – 4) on his right hand to reveal the answer: one hand (six fingers) plus the right hand’s thumb: seven a.m. One digit for every month as well: so half-year periods jump from one hand’s finger to the same finger on the other hand.

We know that the Romans preferred to perform certain calculations using a base of twelve. In his
Ars Poetica
the poet Horace offers us a vignette of Roman boys learning their fractions.

Suppose Albinus’s son says: if one-twelfth is taken from five-twelfths, what is left? You might have answered by now: One-third. Well done. You will be able to manage your money. Now add a twelfth: what happens? One-half.

The Venerable Bede taught his fellow monks to quantify the various periods of time in the Biblical stories using the Roman fractions. One-twelfth, he noted, went by the name of an
uncia
(from which comes the word ‘ounce’), while the remaining part of 11/12 was called
deunx
. Dividing into six, the sixth part (1/6) was called
sextans
, and
dextans
for the rest (5/6).
Quadrans
was the Roman name for a quarter (1/4); three-quarters they called
dodrans
.

What do we get if we add one
sextans
to a
dodrans
(1/6 + ¾)? As quick as Horace’s schoolboys, or the Venerable Bede’s monks, Garrido’s fingers (or toes) would tell him: one-sixth equals two digits and three-quarters equals nine digits, so 1/6 + ¾ = 11/12 (a
deunx
).