Pre computer calculations of π
| Mathematician | Date | Places | Comments | Notes | |
| 1 | Rhind papyrus | 2000 BC | 1 | 3.16045 (= 4(8/9)2) | Click for note 1 |
| 2 | Archimedes | 250 BC | 3 | 3.1418 (average of the bounds) | Click for note 2 |
| 3 | Vitruvius | 20 BC | 1 | 3.125 (= 25/8) | Click for note 3 |
| 4 | Chang Hong | 130 | 1 | 3.1622 (= √10) | Click for note 4 |
| 5 | Ptolemy | 150 | 3 | 3.14166 | Click for note 5 |
| 6 | Wang Fan | 250 | 1 | 3.155555 (= 142/45) | Click for note 6 |
| 7 | Liu Hui | 263 | 5 | 3.14159 | Click for note 7 |
| 8, | Zu Chongzhi | 480 | 7 | 3.141592920 (= 355/113) | Click for note 8 |
| 9 | Aryabhata | 499 | 4 | 3.1416 (= 62832/2000) | Click for note 9 |
| 10 | Brahmagupta | 640 | 1 | 3.1622 (= √10) | Click for note 10 |
| 11 | Al-Khwarizmi | 800 | 4 | 3.1416 | Click for note 11 |
| 12 | Fibonacci | 1220 | 3 | 3.141818 | Click for note 12 |
| 13 | Madhava | 1400 | 11 | 3.14159265359 | Click for note 13 |
| 14 | Al-Kashi | 1430 | 14 | 3.14159265358979 | Click for note 14 |
| 15 | Otho | 1573 | 6 | 3.1415929 | Click for note 15 |
| 16 | Viète | 1593 | 9 | 3.1415926536 | Click for note 16 |
| 17 | Romanus | 1593 | 15 | 3.141592653589793 | Click for note 17 |
| 18 | Van Ceulen | 1596 | 20 | 3.14159265358979323846 | Click for note 18 |
| 19 | Van Ceulen | 1596 | 35 | 3.1415926535897932384626433832795029 | Click for note 19 |
| 20 | Newton | 1665 | 16 | 3.1415926535897932 | Click for note 20 |
| 21 | Sharp | 1699 | 71 | Click for note 21 | |
| 22 | Seki Kowa | 1700 | 10 | ||
| 23 | Kamata | 1730 | 25 | ||
| 24 | Machin | 1706 | 100 | Click for note 24 | |
| 25 | De Lagny | 1719 | 127 | Only 112 correct | Click for note 25 |
| 26 | Takebe | 1723 | 41 | Click for note 26 | |
| 27 | Matsunaga | 1739 | 50 | Click for note 27 | |
| 28 | von Vega | 1794 | 140 | Only 136 correct | Click for note 28 |
| 29 | Rutherford | 1824 | 208 | Only 152 correct | Click for note 29 |
| 30 | Strassnitzky, Dase | 1844 | 200 | Click for note 30 | |
| 31 | Clausen | 1847 | 248 | Click for note 31 | |
| 32 | Lehmann | 1853 | 261 | Click for note 32 | |
| 33 | Rutherford | 1853 | 440 | Click for note 33 | |
| 34 | Shanks | 1874 | 707 | Only 527 correct | Click for note 34 |
| 35 | Ferguson | 1946 | 620 | Click for note 35 |
General Remarks:
A. In early work it was not known that the ratio of the area of a circle to the square of its radius and the ratio of the circumference to the diameter are the same. Some early texts use different approximations for these two "different" constants. For example, in the Indian text the Sulba Sutras the ratio for the area is given as 3.088 while the ratio for the circumference is given as 3.2.
B. Euclid gives in the Elements XII Proposition 2:
Circles are to one another as the squares on their diameters.
He makes no attempt to calculate the ratio.
Computer calculations of π
| Mathematician | Date | Places | Type of computer | ||
| Ferguson | Jan 1947 | 710 | Desk calculator | ||
| Ferguson, Wrench | Sept 1947 | 808 | Desk calculator | ||
| Smith, Wrench | 1949 | 1120 | Desk calculator | ||
| Reitwiesner et al. | 1949 | 2037 | ENIAC | ||
| Nicholson, Jeenel | 1954 | 3092 | NORAC | ||
| Felton | 1957 | 7480 | PEGASUS | ||
| Genuys | Jan 1958 | 10000 | IBM 704 | ||
| Felton | May 1958 | 10021 | PEGASUS | ||
| Guilloud | 1959 | 16167 | IBM 704 | ||
| Shanks, Wrench | 1961 | 100265 | IBM 7090 | ||
| Guilloud, Filliatre | 1966 | 250000 | IBM 7030 | ||
| Guilloud, Dichampt | 1967 | 500000 | CDC 6600 | ||
| Guilloud, Bouyer | 1973 | 1001250 | CDC 7600 | ||
| Miyoshi, Kanada | 1981 | 2000036 | FACOM M-200 | ||
| Guilloud | 1982 | 2000050 | |||
| Tamura | 1982 | 2097144 | MELCOM 900II | ||
| Tamura, Kanada | 1982 | 4194288 | HITACHI M-280H | ||
| Tamura, Kanada | 1982 | 8388576 | HITACHI M-280H | ||
| Kanada, Yoshino, Tamura | 1982 | 16777206 | HITACHI M-280H | ||
| Ushiro, Kanada | Oct 1983 | 10013395 | HITACHI S-810/20 | ||
| Gosper | Oct 1985 | 17526200 | SYMBOLICS 3670 | ||
| Bailey | Jan 1986 | 29360111 | CRAY-2 | ||
| Kanada, Tamura | Sept 1986 | 33554414 | HITACHI S-810/20 | ||
| Kanada, Tamura | Oct 1986 | 67108839 | HITACHI S-810/20 | ||
| Kanada, Tamura, Kubo | Jan 1987 | 134217700 | NEC SX-2 | ||
| Kanada, Tamura | Jan 1988 | 201326551 | HITACHI S-820/80 | ||
| Chudnovskys | May 1989 | 480000000 | |||
| Chudnovskys | June 1989 | 525229270 | |||
| Kanada, Tamura | July 1989 | 536870898 | |||
| Chudnovskys | Aug 1989 | 1011196691 | |||
| Kanada, Tamura | Nov 1989 | 1073741799 | |||
| Chudnovskys | Aug 1991 | 2260000000 | |||
| Chudnovskys | May 1994 | 4044000000 | |||
| Kanada, Tamura | June 1995 | 3221225466 | |||
| Kanada | Aug 1995 | 4294967286 | |||
| Kanada | Oct 1995 | 6442450938 | |||
| Kanada, Takahashi | Aug 1997 | 51539600000 | HITACHI SR2201 | ||
| Kanada, Takahashi | Sept 1999 | 206158430000 | HITACHI SR8000 |
General Remarks:
A. Calculating π to many decimal places was used as a test for new computers in the early days.
B. There is an algorithm by Bailey, Borwein and Plouffe, published in 1996, which allows the nth hexadecimal digit of π to be computed without the preceeding n- 1 digits.
C. Plouffe discovered a new algorithm to compute the nth digit of π in any base in 1997.
Reference (One book/article)
Other Web sites:
- Astroseti (A Spanish translation of this article)
- Simon Fraser University (Tables of Computations)
A little known verse of the Bible reads
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23)
The same verse can be found in II Chronicles 4, 2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives π = 3. Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of 25/8 = 3.125 and √10 = 3.162 have been traced to much earlier dates: though in defence of Solomon's craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. There are some interpretations of this which lead to a much better value.
The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable. The earliest values of π including the 'Biblical' value of 3, were almost certainly found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4 
The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). He obtained the approximation
223/71 < π < 22/7.
Before giving an indication of his proof, notice that very considerable sophistication involved in the use of inequalities here. Archimedes knew, what so many people to this day do not, that π does not equal 22/7, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.
Here is Archimedes' argument.
Consider a circle of radius 1, in which we inscribe a regular polygon of 3
The diagram for the case n = 2 is on the right.
The effect of this procedure is to define an increasing sequence
b1 , b2 , b3 , ...
and a decreasing sequence
a1 , a2 , a3 , ...
such that both sequences have limit π.
Using trigonometrical notation, we see that the two semiperimeters are given by
an = K tan(π/K), bn = K sin(π/K),
where K = 3
an+1 = 2K tan(π/2K), bn+1 = 2K sin(π/2K),
and it is not a difficult exercise in trigonometry to show that
(1/an + 1/bn) = 2/an+1 . . . (1)
an+1bn = (bn+1)2 . . . (2)
Archimedes, starting from a1 = 3 tan(π/3) = 3√3 and b1 = 3 sin(π/3) = 3√3/2, calculated a2 using (1), then b2 using (2), then a3 using (1), then b3 using (2), and so on until he had calculated a6 and b6. His conclusion was that
b6 < π < a6 .
It is important to realise that the use of trigonometry here is unhistorical: Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a6 and b6 from (1) and (2) was by no means a trivial task. So it was a pretty stupendous feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.
For of course there is no reason in principle why one should not go on. Various people did, including:
Ptolemy (c. 150 AD) 3.1416 Zu Chongzhi (430-501 AD) 355/113 al-Khwarizmi (c. 800 ) 3.1416 al-Kashi (c. 1430) 14 places Viète (1540-1603) 9 places Roomen (1561-1615) 17 places Van Ceulen (c. 1600) 35 places
Except for Zu Chongzhi, about whom next to nothing is known and who is very unlikely to have known about Archimedes' work, there was no theoretical progress involved in these improvements, only greater stamina in calculation. Notice how the lead, in this as in all scientific matters, passed from Europe to the East for the millennium 400 to 1400 AD.
Al-Khwarizmi lived in Baghdad, and incidentally gave his name to 'algorithm', while the words al jabr in the title of one of his books gave us the word 'algebra'. Al-Kashi lived still further east, in Samarkand, while Zu Chongzhi, one need hardly add, lived in China.
The European Renaissance brought about in due course a whole new mathematical world. Among the first effects of this reawakening was the emergence of mathematical formulae for π. One of the earliest was that of Wallis (1616-1703)
2/π = (1.3.3.5.5.7. ...)/(2.2.4.4.6.6. ...)
and one of the best-known is
π/4 = 1 - 1/3 + 1/5 - 1/7 + ....
This formula is sometimes attributed to Leibniz (1646-1716) but is seems to have been first discovered by James Gregory (1638- 1675).
These are both dramatic and astonishing formulae, for the expressions on the right are completely arithmetical in character, while π arises in the first instance from geometry. They show the surprising results that infinite processes can achieve and point the way to the wonderful richness of modern mathematics.
From the point of view of the calculation of π, however, neither is of any use at all. In Gregory's series, for example, to get 4 decimal places correct we require the error to be less than 0.00005 = 1/20000, and so we need about 10000 terms of the series. However, Gregory also showed the more general result
tan-1 x = x - x3/3 + x5/5 - ... (-1
x
from which the first series results if we put x = 1. So using the fact that
tan-1(1/√3) = π/6 we get
π/6 = (1/√3)(1 - 1/(3.3) + 1/(5.3.3) - 1/(7.3.3.3) + ...
which converges much more quickly. The 10th term is 1/(19
An even better idea is to take the formula
π/4 = tan-1(1/2) + tan-1(1/3) . . . (4)
and then calculate the two series obtained by putting first 1/2 and the 1/3 into (3).
Clearly we shall get very rapid convergence indeed if we can find a formula something like
π/4 = tan-1(1/a) + tan-1(1/b)
with a and b large. In 1706 Machin found such a formula:
π/4 = 4 tan-1(1/5) - tan-1(1/239) . . . (5)
Actually this is not at all hard to prove, if you know how to prove (4) then there is no real extra difficulty about (5), except that the arithmetic is worse. Thinking it up in the first place is, of course, quite another matter.
With a formula like this available the only difficulty in computing π is the sheer boredom of continuing the calculation. Needless to say, a few people were silly enough to devote vast amounts of time and effort to this tedious and wholly useless pursuit. One of them, an Englishman named Shanks, used Machin's formula to calculate π to 707 places, publishing the results of many years of labour in 1873. Shanks has achieved immortality for a very curious reason which we shall explain in a moment.
Here is a summary of how the improvement went:
1699: Sharp used Gregory's result to get 71 correct digits 1701: Machin used an improvement to get 100 digits and the following used his methods: 1719: de Lagny found 112 correct digits 1789: Vega got 126 places and in 1794 got 136 1841: Rutherford calculated 152 digits and in 1853 got 440 1873: Shanks calculated 707 places of which 527 were correct
A more detailed Chronology is available.
Shanks knew that π was irrational since this had been proved in 1761 by Lambert. Shortly after Shanks' calculation it was shown by Lindemann that π is transcendental, that is, π is not the solution of any polynomial equation with integer coefficients. In fact this result of Lindemann showed that 'squaring the circle' is impossible. The transcendentality of π implies that there is no ruler and compass construction to construct a square equal in area to a given circle.
Very soon after Shanks' calculation a curious statistical freak was noticed by De Morgan, who found that in the last of 707 digits there was a suspicious shortage of 7's. He mentions this in his Budget of Paradoxes of 1872 and a curiosity it remained until 1945 when Ferguson discovered that Shanks had made an error in the 528th place, after which all his digits were wrong. In 1949 a computer was used to calculate π to 2000 places. In this and all subsequent computer expansions the number of 7's does not differ significantly from its expectation, and indeed the sequence of digits has so far passed all statistical tests for randomness.
You can see 2000 places of π.
We should say a little of how the notation π arose. Oughtred in 1647 used the symbol d/π for the ratio of the diameter of a circle to its circumference. David Gregory (1697) used π/r for the ratio of the circumference of a circle to its radius. The first to use π with its present meaning was an Welsh mathematician William Jones in 1706 when he states "3.14159 andc. = π". Euler adopted the symbol in 1737 and it quickly became a standard notation.
We conclude with one further statistical curiosity about the calculation of π, namely Buffon's needle experiment. If we have a uniform grid of parallel lines, unit distance apart and if we drop a needle of length k <>k/π. Various people have tried to calculate π by throwing needles. The most remarkable result was that of Lazzerini (1901), who made 34080 tosses and got
π = 355/113 = 3.1415929
which, incidentally, is the value found by Zu Chongzhi. This outcome is suspiciously good, and the game is given away by the strange number 34080 of tosses. Kendall and Moran comment that a good value can be obtained by stopping the experiment at an optimal moment. If you set in advance how many throws there are to be then this is a very inaccurate way of computing π. Kendall and Moran comment that you would do better to cut out a large circle of wood and use a tape measure to find its circumference and diameter.
Still on the theme of phoney experiments, Gridgeman, in a paper which pours scorn on Lazzerini and others, created some amusement by using a needle of carefully chosen length k = 0.7857, throwing it twice, and hitting a line once. His estimate for π was thus given by
2
from which he got the highly creditable value of π = 3.1428. He was not being serious!
It is almost unbelievable that a definition of π was used, at least as an excuse, for a racial attack on the eminent mathematician Edmund Landau in 1934. Landau had defined π in this textbook published in Göttingen in that year by the, now fairly usual, method of saying that π/2 is the value of x between 1 and 2 for which cos x vanishes. This unleashed an academic dispute which was to end in Landau's dismissal from his chair at Göttingen. Bieberbach, an eminent number theorist who disgraced himself by his racist views, explains the reasons for Landau's dismissal:-
Thus the valiant rejection by the Göttingen student body which a great mathematician, Edmund Landau, has experienced is due in the final analysis to the fact that the un-German style of this man in his research and teaching is unbearable to German feelings. A people who have perceived how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture.
G H Hardy replied immediately to Bieberbach in a published note about the consequences of this un-German definition of π
There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all cost not to be outdone, may be natural if not particularly heroic excuses. Professor Bieberbach's reputation excludes such explanations of his utterances, and I find myself driven to the more uncharitable conclusion that he really believes them true.
Not only in Germany did π present problems. In the USA the value of π gave rise to heated political debate. In the State of Indiana in 1897 the House of Representatives unanimously passed a Bill introducing a new mathematical truth.
Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square of one side.
(Section I, House Bill No. 246, 1897)
The Senate of Indiana showed a little more sense and postponed indefinitely the adoption of the Act!
Open questions about the number π
- Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in π?
- Brouwer's question: In the decimal expansion of π, is there a place where a thousand consecutive digits are all zero?
- Is π simply normal to base 10? That is does every digit appear equally often in its decimal expansion in an asymptotic sense?
- Is π normal to base 10? That is does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense?
- Is π normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.
- Another normal question! We know that π is not rational so there is no point from which the digits will repeat. However, if π is normal then the first million digits 314159265358979... will occur from some point. Even if π is not normal this might hold! Does it? If so from what point? Note: Up to 200 million the longest to appear is 31415926 and this appears twice.
As a postscript, here is a mnemonic for the decimal expansion of π. Each successive digit is the number of letters in the corresponding word.
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard...:
3.14159265358979323846264...
You can see more about the history of π in the History topic: Squaring the circle and you can see a Chronology of how calculations of π have developed over the years.
References (30 books/articles)
Other Web sites:
The Indian numerals discussed in our article Indian numerals form the basis of the European number systems which are now widely used. However they were not transmitted directly from India to Europe but rather came first to the Arabic/Islamic peoples and from them to Europe. The story of this transmission is not, however, a simple one. The eastern and western parts of the Arabic world both saw separate developments of Indian numerals with relatively little interaction between the two. By the western part of the Arabic world we mean the regions comprising mainly North Africa and Spain. Transmission to Europe came through this western Arabic route, coming into Europe first through Spain.
There are other complications in the story, however, for it was not simply that the Arabs took over the Indian number system. Rather different number systems were used simultaneously in the Arabic world over a long period of time. For example there were at least three different types of arithmetic used in Arab countries in the eleventh century: a system derived from counting on the fingers with the numerals written entirely in words, this finger-reckoning arithmetic was the system used for by the business community; the sexagesimal system with numerals denoted by letters of the Arabic alphabet; and the arithmetic of the Indian numerals and fractions with the decimal place-value system.
The first sign that the Indian numerals were moving west comes from a source which predates the rise of the Arab nations. In 662 AD Severus Sebokht, a Nestorian bishop who lived in Keneshra on the Euphrates river, wrote:-
I will omit all discussion of the science of the Indians, ... , of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value.
This passage clearly indicates that knowledge of the Indian number system was known in lands soon to become part of the Arab world as early as the seventh century. The passage itself, of course, would certainly suggest that few people in that part of the world knew anything of the system. Severus Sebokht as a Christian bishop would have been interested in calculating the date of Easter (a problem to Christian churches for many hundreds of years). This may have encouraged him to find out about the astronomy works of the Indians and in these, of course, he would find the arithmetic of the nine symbols.
By 776 AD the Arab empire was beginning to take shape and we have another reference to the transmission of Indian numerals. We quote from a work of al-Qifti Chronology of the scholars written around the end the 12th century but quoting much earlier sources:-
... a person from India presented himself before the Caliph al-Mansur in the year [ 776 AD] who was well versed in the siddhanta method of calculation related to the movment of the heavenly bodies, and having ways of calculating equations based on the half-chord [essentially the sine] calculated in half-degrees ... This is all contained in a work ... from which he claimed to have taken the half-chord calculated for one minute. Al-Mansur ordered this book to be translated into Arabic, and a work to be written, based on the translation, to give the Arabs a solid base for calculating the movements of the planets ...
Now in [A universal history of numbers : From prehistory to the invention of the computer (London, 1998).',1)">1] (where a longer quote is given) Ifrah tries to determine which Indian work is referred to. He concludes that the work was most likely to have been Brahmagupta's Brahmasphutasiddhanta (The Opening of the Universe) which was written in 628. Irrespective of whether Ifrah is right, since all Indian texts after Aryabhata I's Aryabhatiya used the Indian number system of the nine signs, certainly from this time the Arabs had a translation into Arabic of a text written in the Indian number system.
It is often claimed that the first Arabic text written to explain the Indian number system was written by al-Khwarizmi. However there are difficulties here which many authors tend to ignore. The Arabic text is lost but a twelfth century Latin translation, Algoritmi de numero Indorum (in English Al-Khwarizmi on the Hindu Art of Reckoning) gave rise to the word algorithm deriving from his name in the title. Unfortunately the Latin translation is known to be much changed from al-Khwarizmi's original text (of which even the title is unknown). The Latin text certainly describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. The first use of zero as a place holder in positional base notation is considered by some to be due to al-Khwarizmi in this work. The difficulty which arises is that al-Baghdadi refers to the Arabic original which, contrary to what was originally thought, seems not to be a work on Indian numerals but rather a work on finger counting methods. This becomes clear from the references by al-Baghdadi to the lost work. However the numerous references to al-Khwarizmi's book on the Indian nine symbols must mean that he did write such a work. Some degree of mystery still remains.
At first the Indian methods were used by the Arabs with a dust board. In fact in the western part of the Arabic world the Indian numerals came to be known as Guba (or Gubar or Ghubar) numerals from the Arabic word meaning "dust". A dust board was used because the arithmetical methods required the moving of numbers around in the calculation and rubbing some out some of them as the calculation proceeded. The dust board allowed this in the same sort of way that one can use a blackboard, chalk and a blackboard eraser. Any student who has attended lectures where the lecturer continually changes and replaces parts of the mathematics as the demonstration progresses will understand the disadvantage of the dust board!
Around the middle of the tenth century al-Uqlidisi wrote Kitab al-fusul fi al-hisab al-Hindi which is the earliest surviving book that presents the Indian system. In it al-Uqlidisi argues that the system is of practical value:-
Most arithmeticians are obliged to use it in their work: since it is easy and immediate, requires little memorisation, provides quick answers, demands little thought ... Therefore, we say that it is a science and practice that requires a tool, such as a writer, an artisan, a knight needs to conduct their affairs; since if the artisan has difficulty in finding what he needs for his trade, he will never succeed; to grasp it there is no difficulty, impossibility or preparation.
In the fourth part of this book al-Uqlidisi showed how to modify the methods of calculating with Indian symbols, which had required a dust board, to methods which could be carried out with pen and paper. Certainly the fact that the Indian system required a dust board had been one of the main obstacles to its acceptance. For example As-Suli, after praising the Indian system for its great simplicity, wrote in the first half of the tenth century:-
Official scribes nevertheless avoid using [the Indian system] because it requires equipment [like a dust board] and they consider that a system that requires nothing but the members of the body is more secure and more fitting to the dignity of a leader.
Al-Uqlidisi's work is therefore important in attempting to remove one of the obstacles to acceptance of the Indian nine symbols. It is also historically important as it is the earliest known text offering a direct treatment of decimal fractions.
Despite many scholars finding calculating with Indian symbols helpful in their work, the business community continued to use their finger arithmetic throughout the tenth century. Abu'l-Wafa, who was himself an expert in the use of Indian numerals, nevertheless wrote a text on how to use finger-reckoning arithmetic since this was the system used by the business community and teaching material aimed at these people had to be written using the appropriate system. Let us give a little information about the Arab letter numerals which are contained in Abu'l-Wafa's work.
The numbers were represented by letters but not in the dictionary order. The system was known as huruf al jumal which meant "letters for calculating" and also sometimes as abjad which is just the first four numbers (1 = a, 2 = b, j = 3, d = 4). The numbers from 1 to 9 were represented by letters, then the numbers 10, 20, 30, ..., 90 by the next nine letters (10 = y, 20 = k, 30 = l, 40 = m, ...), then 100, 200, 300, ... , 900 by the next letters (100 = q, 200 = r, 300 = sh, 400 = ta, ...). There were 28 Arabic letters and so one was left over which was used to represent 1000.
Arabic astronomers used a base 60 version of Arabic letter system. Although Arabic is written from right to left, we shall give an example writing in the left to right style that we use in writing English. A number, say 43
A contemporary of al-Baghdadi, writing near the beginning of the eleventh century, was ibn Sina (better known in the West as Avicenna). We know many details of his life for he wrote an autobiography. Certainly ibn Sina was a remarkable child, with a memory and an ability to learn which amazed the scholars who met in his father's home. A group of scholars from Egypt came to his father's house in about 997 when ibn Sina was ten years old and they taught him Indian arithmetic. He also tells of being taught Indian calculation and algebra by a seller of vegetables. All this shows that by the beginning of the eleventh century calculation with the Indian symbols was fairly widespread and, quite significantly, was know to a vegetable trader.
What of the numerals themselves. We have seen in the article Indian numerals that the form of the numerals themselves varied in different regions and changed over time. Exactly the same happened in the Arabic world.
Here is an example of an early form of Indian numerals being used in the eastern part of the Arabic empire. It comes from a work of al-Sijzi, not an original work by him but rather the work of another mathematician which al-Sijzi copied at Shiraz and dated his copy 969.
The numerals from al-Sizji's treatise of 969
The numerals had changed their form somewhat 100 years later when this copy of one of al-Biruni's astronomical texts was made. Here are the numerals as they appear in a 1082 copy.
The numerals from al-Biruni's treatise copied in 1082
In fact a closer look will show that between 969 and 1082 the biggest change in the numerals was the fact that the 2 and the 3 have been rotated through 90
Here is an example of how the text was written
Perhaps because scribes did not have much experience at writing Indian numerals, they wrote 2 and 3 the correct way round instead of writing them rotated by 90
Here is an example of what the scribe should write
and here is what the scribe actually wrote
The form of the numerals in the west of the Arabic empire look more familiar to those using European numerals today which is not surprising since it is from these numerals that the Indian number system reach Europe.
al-Banna al-Marrakushi's form of the numerals
He gave this form of the numerals in his practical arithmetic book written around the beginning of the fourteenth century. He lived most of his life in Morocco which was in close contact with al-Andalus, or Andalusia, which was the Arab controlled region in the south of Spain.
The first surviving example of the Indian numerals in a document in Europe was, however, long before the time of al-Banna. The numerals appear in the Codex Vigilanus copied by a monk in Spain in 976. However the main part of Europe was not ready at this time to accept new ideas of any kind. Acceptance was slow, even as late as the fifteenth century when European mathematics began its rapid development which continues today. We will not examine the many contributions to bringing the Indian number system to Europe in this article but we will end with just one example which, however, is a very important one. Fibonacci writes in his famous book Liber abaci published in Pisa in 1202:-
When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.
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Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.
There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century. The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.
That such views should be generally held is of no surprise. Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem in [Le système du monde (Paris, 1965).',3)">3]:-
... Arabic science only reproduced the teachings received from Greek science.
Before we proceed it is worth trying to define the period that this article covers and give an overall description to cover the mathematicians who contributed. The period we cover is easy to describe: it stretches from the end of the eighth century to about the middle of the fifteenth century. Giving a description to cover the mathematicians who contributed, however, is much harder. The works [Studies in the Islamic Exact Sciences (1983).',6)">6] and [Encyclopaedia Britannica.',17)">17] are on "Islamic mathematics", similar to [The Muslim contribution to mathematics (London, 1978).',1)">1] which uses the title the "Muslim contribution to mathematics". Other authors try the description "Arabic mathematics", see for example [Entre arithmétique et algèbre: Recherches sur l\'histoire des mathématiques arabes (Paris, 1984).',10)">10] and [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',11)">11]. However, certainly not all the mathematicians we wish to include were Muslims; some were Jews, some Christians, some of other faiths. Nor were all these mathematicians Arabs, but for convenience we will call our topic "Arab mathematics".
The regions from which the "Arab mathematicians" came was centred on Iran/Iraq but varied with military conquest during the period. At its greatest extent it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China.
The background to the mathematical developments which began in Baghdad around 800 is not well understood. Certainly there was an important influence which came from the Hindu mathematicians whose earlier development of the decimal system and numerals was important. There began a remarkable period of mathematical progress with al-Khwarizmi's work and the translations of Greek texts.
This period begins under the Caliph Harun al-Rashid, the fifth Caliph of the Abbasid dynasty, whose reign began in 786. He encouraged scholarship and the first translations of Greek texts into Arabic, such as Euclid's Elements by al-Hajjaj, were made during al-Rashid's reign. The next Caliph, al-Ma'mun, encouraged learning even more strongly than his father al-Rashid, and he set up the House of Wisdom in Baghdad which became the centre for both the work of translating and of of research. Al-Kindi (born 801) and the three Banu Musa brothers worked there, as did the famous translator Hunayn ibn Ishaq.
We should emphasise that the translations into Arabic at this time were made by scientists and mathematicians such as those named above, not by language experts ignorant of mathematics, and the need for the translations was stimulated by the most advanced research of the time. It is important to realise that the translating was not done for its own sake, but was done as part of the current research effort. The most important Greek mathematical texts which were translated are listed in [Encyclopaedia Britannica.',17)">17]:-
Of Euclid's works, the Elements, the Data, the Optics, the Phaenomena, and On Divisions were translated. Of Archimedes' works only two - Sphere and Cylinder and Measurement of the Circle - are known to have been translated, but these were sufficient to stimulate independent researches from the 9th to the 15th century. On the other hand, virtually all of Apollonius's works were translated, and of Diophantus and Menelaus one book each, the Arithmetica and the Sphaerica, respectively, were translated into Arabic. Finally, the translation of Ptolemy's Almagest furnished important astronomical material.
The more minor Greek mathematical texts which were translated are also given in [Encyclopaedia Britannica.',17)">17]:-
... Diocles' treatise on mirrors, Theodosius's Spherics, Pappus's work on mechanics, Ptolemy's Planisphaerium, and Hypsicles' treatises on regular polyhedra (the so-called Books XIV and XV of Euclid's Elements) ...
Perhaps one of the most significant advances made by Arabic mathematics began at this time with the work of al-Khwarizmi, namely the beginnings of algebra. It is important to understand just how significant this new idea was. It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry.
Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject. Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. As Rashed writes in [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',11)">11] (see also [Entre arithmétique et algèbre: Recherches sur l\'histoire des mathématiques arabes (Paris, 1984).',10)">10]):-
Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.
Let us follow the development of algebra for a moment and look at al-Khwarizmi's successors. About forty years after al-Khwarizmi is the work of al-Mahani (born 820), who conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Abu Kamil (born 850) forms an important link in the development of algebra between al-Khwarizmi and al-Karaji. Despite not using symbols, but writing powers of x in words, he had begun to understand what we would write in symbols as xn.xm = xm+n. Let us remark that symbols did not appear in Arabic mathematics until much later. Ibn al-Banna and al-Qalasadi used symbols in the 15th century and, although we do not know exactly when their use began, we know that symbols were used at least a century before this.
Al-Karaji (born 953) is seen by many as the first person to completely free algebra from geometrical operations and to replace them with the arithmetical type of operations which are at the core of algebra today. He was first to define the monomials x, x2, x3, ... and 1/x, 1/x2, 1/x3, ... and to give rules for products of any two of these. He started a school of algebra which flourished for several hundreds of years. Al-Samawal, nearly 200 years later, was an important member of al-Karaji's school. Al-Samawal (born 1130) was the first to give the new topic of algebra a precise description when he wrote that it was concerned:-
... with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.
Omar Khayyam (born 1048) gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work [Arch. History Exact Sci. 18 (3) (1977/78), 191-243.',18)">18]:-
If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared.
Sharaf al-Din al-Tusi (born 1135), although almost exactly the same age as al-Samawal, does not follow the general development that came through al-Karaji's school of algebra but rather follows Khayyam's application of algebra to geometry. He wrote a treatise on cubic equations, which [The development of Arabic mathematics : between arithmetic and algebra (London, 1994).',11)">11]:-
... represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.
Let us give other examples of the development of Arabic mathematics. Returning to the House of Wisdom in Baghdad in the 9th century, one mathematician who was educated there by the Banu Musa brothers was Thabit ibn Qurra (born 836). He made many contributions to mathematics, but let us consider for the moment consider his contributions to number theory. He discovered a beautiful theorem which allowed pairs of amicable numbers to be found, that is two numbers such that each is the sum of the proper divisors of the other. Al-Baghdadi (born 980) looked at a slight variant of Thabit ibn Qurra's theorem, while al-Haytham (born 965) seems to have been the first to attempt to classify all even perfect numbers (numbers equal to the sum of their proper divisors) as those of the form 2k-1(2k - 1) where 2k - 1 is prime.
Al-Haytham, is also the first person that we know to state Wilson's theorem, namely that if p is prime then 1+(p-1)! is divisible by p. It is unclear whether he knew how to prove this result. It is called Wilson's theorem because of a comment made by Waring in 1770 that John Wilson had noticed the result. There is no evidence that John Wilson knew how to prove it and most certainly Waring did not. Lagrange gave the first proof in 1771 and it should be noticed that it is more than 750 years after al-Haytham before number theory surpasses this achievement of Arabic mathematics.
Continuing the story of amicable numbers, from which we have taken a diversion, it is worth noting that they play a large role in Arabic mathematics. Al-Farisi (born 1260) gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorisation and combinatorial methods. He also gave the pair of amicable numbers 17296, 18416 which have been attributed to Euler, but we know that these were known earlier than al-Farisi, perhaps even by Thabit ibn Qurra himself. Although outside our time range for Arabic mathematics in this article, it is worth noting that in the 17th century the Arabic mathematician Mohammed Baqir Yazdi gave the pair of amicable number 9,363,584 and 9,437,056 still many years before Euler's contribution.
Let us turn to the different systems of counting which were is use around the 10th century in Arabic countries. There were three different types of arithmetic used around this period and, by the end of the 10th century, authors such as al-Baghdadi were writing texts comparing the three systems.
1. Finger-reckoning arithmetic.
This system derived from counting on the fingers with the numerals written entirely in words; this finger-reckoning arithmetic was the system used by the business community. Mathematicians such as Abu'l-Wafa (born 940) wrote several treatises using this system. Abu'l-Wafa himself was an expert in the use of Indian numerals but these:-
... did not find application in business circles and among the population of the Eastern Caliphate for a long time.
Hence he wrote his text using finger-reckoning arithmetic since this was the system used by the business community.
2. Sexagesimal system.
The second of the three systems was the sexagesimal system, with numerals denoted by letters of the Arabic alphabet. It came originally from the Babylonians and was most frequently used by the Arabic mathematicians in astronomical work.
3. Indian numeral system.
The third system was the arithmetic of the Indian numerals and fractions with the decimal place-value system. The numerals used were taken over from India, but there was not a standard set of symbols. Different parts of the Arabic world used slightly different forms of the numerals. At first the Indian methods were used by the Arabs with a dust board. A dust board was needed because the methods required the moving of numbers around in the calculation and rubbing some out as the calculation proceeded. The dust board allowed this to be done in the same sort of way that one can use a blackboard, chalk and a blackboard eraser. However, al-Uqlidisi (born 920) showed how to modify the methods for pen and paper use. Al-Baghdadi also contributed to improvements in the decimal system.
It was this third system of calculating which allowed most of the advances in numerical methods by the Arabs. It allowed the extraction of roots by mathematicians such as Abu'l-Wafa and Omar Khayyam (born 1048). The discovery of the binomial theorem for integer exponents by al-Karaji (born 953) was a major factor in the development of numerical analysis based on the decimal system. Al-Kashi (born1380) contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots which is a special case of the methods given many centuries later by Ruffini and Horner.
Although the Arabic mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy. Ibrahim ibn Sinan (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particular al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections. Omar Khayyam combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means.
Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials. Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy and also used formulas involving sin and tan. Al-Biruni (born 973) used the sin formula in both astronomy and in the calculation of longitudes and latitudes of many cities. Again both astronomy and geography motivated al-Biruni's extensive studies of projecting a hemisphere onto the plane.
Thabit ibn Qurra undertook both theoretical and observational work in astronomy. Al-Battani (born 850) made accurate observations which allowed him to improve on Ptolemy's data for the sun and the moon. Nasir al-Din al-Tusi (born 1201), like many other Arabic mathematicians, based his theoretical astronomy on Ptolemy's work but al-Tusi made the most significant development of Ptolemy's model of the planetary system up to the development of the heliocentric model in the time of Copernicus.
Many of the Arabic mathematicians produced tables of trigonometric functions as part of their studies of astronomy. These include Ulugh Beg (born 1393) and al-Kashi. The construction of astronomical instruments such as the astrolabe was also a speciality of the Arabs. Al-Mahani used an astrolabe while Ahmed (born 835), al-Khazin (born 900), Ibrahim ibn Sinan, al-Quhi, Abu Nasr Mansur (born 965), al-Biruni, and others, all wrote important treatises on the astrolabe. Sharaf al-Din al-Tusi (born 1201) invented the linear astrolabe. The importance of the Arabic mathematicians in the development of the astrolabe is described in [Encyclopaedia Britannica.',17)">17]:-
The astrolabe, whose mathematical theory is based on the stereographic projection of the sphere, was invented in late antiquity, but its extensive development in Islam made it the pocket watch of the medievals. In its original form, it required a different plate of horizon coordinates for each latitude, but in the 11th century the Spanish Muslim astronomer az-Zarqallu invented a single plate that worked for all latitudes. Slightly earlier, astronomers in the East had experimented with plane projections of the sphere, and al-Biruni invented such a projection that could be used to produce a map of a hemisphere. The culminating masterpiece was the astrolabe of the Syrian Ibn ash-Shatir (1305-75), a mathematical tool that could be used to solve all the standard problems of spherical astronomy in five different ways.
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