Pathfinders
Page 16
Here is one type of problem that I have set many times when teaching basic algebra. It can be found in Euclid’s Elements (Book Two, Proposition Eleven): Divide the straight line AC, which is of known length, into two unequal segments: AB and BC. What are the lengths of these segments such that a square of side AB will have the same area as a rectangle of sides AC and BC? (see diagram on opposite page)
Euclid solved this problem geometrically by dividing up the shapes in the diagram into smaller parts and comparing the different areas. There is no doubt that the Greeks were both the founders and masters of geometry, with Euclid’s Elements representing their crowning glory. In fact, it was still the standard text taught at schools all over the world until well into the twentieth century. However, far neater and aesthetically more pleasing is to set this problem up algebraically. Call one of the unknown lengths (AB, say) x. Then BC will be the full length, L, less this length: L − x. We therefore know that the area of the square is x2, and that of the rectangle, L times L − x and we have the equation: x2 = L(L − x), which is another quadratic equation with a solution for x that depends on the value of L.
An example of a geometric problem requiring the solution of a quadratic equation, taken from Euclid’s Elements (Book Two, Proposition Eleven). See text for details.
If Greek and Babylonian mathematicians really were solving quadratic equations long before al-Khwārizmi – and he was certainly not doing anything more complicated than that – then surely he cannot be credited with founding the field of algebra. And what of the contribution of Hindu mathematicians such as Brahmagupta? Most importantly, who was this man Diophantus that my detractor argued had the worthier claim to the title?
Little is known about the life of Diophantus other than that he flourished in Alexandria in the third century CE. His most famous work, the Arithmetica, in which he solves a large number of mathematical problems, is essentially a book about numbers. But as in modern algebra, he uses a symbol for the unknown quantity as well as for certain arithmetic operations like subtraction. Diophantus explains how to multiply positive and negative terms and different powers of the unknown quantity, and then goes on to show how to simplify a collection of quantities to a more compact form. All this does suggest that Diophantus was doing algebra.
Most famously, Diophantus deals with a certain class of problems in which there is more than one unknown quantity and for which the solution is always an integer, or whole number. A simple example of this might be cast symbolically as x + 1 = y. Here, one possible solution is that x = 1, and so y = 2 or, alternatively, x = 7 and y = 8 and so on – there is no unique answer, and such an equation is referred to as indeterminate. It is also an example of what is called a linear equation since neither x nor y is raised to a power greater than 1. More generally, any equation involving two or more unknown quantities raised to any power, such that the solutions are always integer numbers, is called a Diophantine equation, even though Diophantus himself did not appreciate the richness of this field nor provide any general methods for solving such equations. The most famous Diophantine equation of all is the one highlighted by the great Pierre de Fermat (1601–65), the founder of modern number theory. In the margins of his personal copy of the Arithmetica, Fermat wrote various comments proposing solutions, corrections and generalizations of Diophantus’ methods. The most famous of these remarks was the following: ‘It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.’2 This is a statement of one of the most famous problems in the whole of mathematics and is known as Fermat’s Last Theorem. In fact, for three and a half centuries it should have more correctly been called a ‘conjecture’ rather than a theorem. Stated mathematically, it says that there are no whole number values for x, y and z such that xn + yn = zn, when n is greater than 2. For instance, there are no whole numbers for which the sum of the cubes of two integers equals the cube of another (unless they are all equal to zero, of course). A proof was finally found by the British mathematician Andrew Wiles in 1995, and I for one have no intention of checking his method, since it runs to more than a hundred pages and took him seven years to complete.
None of this should be credited to Diophantus, of course, but what it is meant to show is that his interest, like Fermat’s, was more in the properties of numbers than in the algebraic manipulation of symbols.
In the seventh century the great Hindu mathematician Brahmagupta took up the challenge of tackling another Diophantine equation – what is known today as the Pell equation, which has the general form x2 − ay2 = 1. Brahmagupta posed the challenge of finding a value for each of x and y if a = 92. He suggested that anyone who could solve this problem within a year had earned the right to be called a mathematician. His solution was x = 1151 and y = 120. These days of course it is a simple matter, if you know how, to program a computer to search for the answer very quickly.
And so, back to ninth-century Baghdad. One thing we can be quite sure of is that al-Khwārizmi would not have been aware of Diophantus or his Arithmetica, for the first Arabic translation of this book was not made until several decades after al-Khwārizmi wrote al-Jebr. So where did he gain his mathematical education from?
One question, which seems not to have a clear answer, is whether al-Khwārizmi was familiar with Euclid’s Elements. We know that he could have been – perhaps it is even fair to say that he ‘should have been’. An early translation of the Elements was made by al-Hajjāj ibn Yūsuf, a contemporary of al-Khwārizmi, in the first years of the ninth century during the reign of al-Rashīd, and he would later produce an improved translation for al-Ma’mūn. It is not known which version of al-Hajjāj’s translations, if either, would have been available to al-Khwārizmi. Some modern historians believe that al-Khwārizmi’s use of geometric figures to supplement and justify his algebraic proofs suggests that he was familiar with the Elements and Euclid’s geometrical methods for solving problems.3
Whether al-Khwārizmi had studied the Elements or not, the consensus now is that he was influenced by both Greek geometry and Hindu arithmetic. However, one twentieth-century scholar did not agree. Solomon Gandz was an Austrian-American historian of mathematics who claimed in a paper published in 1936 that al-Khwārizmi’s al-Jebr was essentially a translation of an old Hebrew book on geometry called the Mishnat ha Middot, dating to around 150 CE.4 Gandz argued that there was no trace or flavour of Euclid’s Elements in al-Khwārizmi’s work; that he must have been completely unaware of it since his text contained none of the definitions, axioms, postulates or demonstrations of proof that are such an integral part of the writing of Euclid. In fact, Gandz goes so far as to claim that al-Khwārizmi’s work was a reaction against Greek mathematics. Many historians have disagreed with Gandz and some have explained any similarities between Kitab al-Jebr and Mishnat ha Middot as being because the Mishnat was actually written after al-Khwārizmi’s time.5
Whatever the truth, those aspects of al-Jebr that are not original are not its important features. In particular, the geometrical diagrams showing a technique called ‘completing the square’ had been known since Babylonian times and were used by al-Khwārizmi only as a means of justifying the answers he arrived at algebraically.
I should clarify then what is meant by the two key words in the title of his book: jebr and muqābala. The former means ‘completion’ or ‘restoration’ – for instance the fixing, or setting, of broken bones. In mathematical terms it means moving a negative quantity from one side of an equation to the other and ‘restoring’ its sign to positive. Thus, if we have the equation 5x − 2 = 8 then we can move the ‘2’ that is subtracted from the 5x on the left over to the other side of the ‘equals’ sign so that it is added to the 8 instead: 5x = 8 + 2, or 5x = 10. The second word, muqābala, is the Arabic noun meaning the placing of something face to face with, or across from, or to compare. In ma
thematical terms, it means to balance an equation, or to do the same thing to both sides. Thus, if we have the expression 3x + 1 = y + 1 then we can subtract the ‘1’ from both sides to simplify the relation to 3x = y.
These are two of the basic techniques in algebra and, along with several other rules al-Khwārizmi describes early on in his book, make it clear that this was intended as an instruction manual for manipulating quantities algebraically. But his motivation went beyond this. He describes at the start that the purpose of his book is to teach ‘what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.’6 The book is divided into two halves. The first is of more interest to us, for this is where al-Khwārizmi lays down the rules of algebra and the sequences of steps (the algorithms) needed for solving different kinds of quadratic equations, each followed by a diagrammatic proof of his answer. The second half of the book is full of the applications of his methods and is where he tackles a wide range of everyday problems, as he describes in the quotation above.
He defines three kinds of quantities: the unknown (or what he calls the shay’, meaning ‘the thing’), squares of the unknown (what he calls the māl) and numbers. He describes how a relationship involving these three types of quantities can be manipulated and rearranged so that we can find the value of the shay’. But it is the way he describes these steps and procedures that is remarkable. For this book is unlike any book on algebra one would find today. Rather than fill his pages with symbols and notations, he instead wrote entirely in prose. This did of course mean that it took him two pages to explain the steps necessary to compute a quantity, a process that could be stated in just a few lines using symbols in equations.
Now if I mention that the Hindus long before al-Khwārizmi, and even Diophantus, were using rudimentary symbols to describe their equations; that al-Khwārizmi in his al-Jebr never solved problems beyond the quadratic (x2); that Diophantus tackled more complex problems; and that even the techniques al-Khwārizmi used, such as the method of ‘completing the square’ to solve a quadratic equation, were not new, then surely in the light of all this the argument championing his claim evaporates.
I have heard it said that the reason for al-Khwārizmi’s reputation is simply because his was the first book that popularized the subject and set it in a form that could be followed by many people. But this is a feeble argument. One might just as well say that Stephen Hawking’s reputation as one of the greatest scientists of the modern era is due to his best-selling Brief History of Time, rather than his pioneering work in cosmology and the theories of black holes.
So to settle the matter once and for all, I spoke to a mathematician friend of mine from the University of Warwick, Ian Stewart, who has a long-standing interest in the history of algebra. And, at last, the penny dropped. It turns out that it really has nothing to do with whether or not symbols are used, whether or not there are geometric proofs, the level of complexity of the equations, or the accessibility of the writing. What al-Khwārizmi did for the very first time and what sets him apart from all other mathematicians before him is subtle but crucial. He abandons the practice of solving particular problems, and instead provides a general series of principles and rules for dealing with (quadratic) equations, solving them in a set of steps: the algorithm. He thus made it possible for algebra to exist as a subject in its own right, rather than just a technique for manipulating numbers. In Stewart’s words:
It is the difference between on the one hand supplying lots of specific examples and leaving the reader to conclude that the same procedure works on similar ones and, on the other hand, explaining the procedure in general terms in its own right. So, when al-Khwārizmi says ‘māl’ (meaning x2), he doesn’t refer to a specific square like 16, say. He means the square of his unknown, his shay’, which does not represent any number in particular at all. He may use specific numbers that illustrate the method later, but the method itself is conceived as a general procedure.
Thus, although al-Khwārizmi uses words rather than symbols as Diophantus did, he is in fact much closer to the sort of algebra we do today than the Greek is, because, for al-Khwārizmi, the unknown quantity (the shay’) is a new kind of object which can be manipulated in its own right. For him, 2x + 3x = 5x is a statement about how to combine multiples of the unknown and not just a formula that works for certain values of x. The unknown has become a thing, not a placeholder for numbers. That is the true power of what al-Khwārizmi did, and that is real algebra.
Ultimately, Diophantus was more interested in the theory of numbers and the relationships between them, just as Euclid’s interest was geometry, whereas al-Khwārizmi’s text teaches algebra for the very first time as a separate discipline from either arithmetic or geometry.
Many may still hold reservations about al-Khwārizmi’s use of prose rather than symbols, so how relevant was this? One can categorize algebra broadly into three types. First, there is what is known as rhetorical algebra, which contains no symbols at all and each step is described in words rather than equations. This was the tradition that al-Khwārizmi inherited and systematized. Next, there is syncopated algebra, which is mainly rhetorical but with a few notations and abbreviations included. This is the form of algebra used by Diophantus and it might appear at first sight to be more advanced than al-Khwārizmi’s prose, but this is not the case, as I shall explain shortly. Finally, we have symbolic algebra, which was first developed in a crude form by the Hindus but would only develop into the modern form used in every language and culture around the world today in the hands of sixteenth-century European mathematicians.
Consider the following equation and how it might be presented in each of the three ways. Today we might write it symbolically as 3x2 + 5x = 22, but rhetorically this becomes ‘three squares of the unknown added to five of the unknown equals twenty-two’. In syncopated algebra I will follow the notation that would have been used by Diophantus himself. He would have written it as , where ζ is the symbol he used for the unknown (his x) and Δν is its square. There is no symbol for ‘+’ and the ‘=’ sign is written as ισ (the first two letters of the Greek word for ‘equal’). The symbol defines the constant number. So Diophantus’ notation was often nothing more than the first letter or two of the word describing the quantity or operation. For the numerals, he followed the standard tradition of the time in using letters of the Greek alphabet, with bars on top to distinguish them from the letters representing operations.
As we can see, using the notation of Diophantus does not make the task of solving equations any easier and this is very far from full symbolic algebra. In fact, it is no more powerful than the rhetorical algebra of al-Khwārizmi and the only advantage of such abbreviations would have been to save on Alexandrian papyrus. Furthermore, al-Khwārizmi’s prose would have been far easier to follow, certainly for medieval translators.
The transition to symbolic notation in algebra was a slow and tortuous process, with many mathematicians inventing their own notation. There was even a concerted effort in Europe to resist the symbolization of mathematics on principle. One of the first books on algebra to contain equations and formulae was written by the English mathematician John Wallis in 1693. But another Englishman, the philosopher Thomas Hobbes (1588–1679), was highly critical of Wallis and referred to the ‘scab of symbols’. Needless to say, Wallis emerged the victor.
Al-Khwārizmi died around 850, but his status as the man who created a new sub-discipline of mathematics to sit alongside arithmetic and geometry had already been secured. Countless mathematicians throughout the Islamic world were inspired by his text and took on the subject’s further development; men such as al-Mahani, Thābit ibn Qurra (the translator of Diophantus’ Arithmetica into Arabic) and Abū Kāmil (known as ‘the Egyptian Calculator’
) all extended al-Khwārizmi’s work in the second half of the ninth century. Later, al-Karkhi, Ibn Tāhir and the great Ibn al-Haytham in the tenth/eleventh century took it further by considering cubic and quartic equations, followed by the Persian mathematician and poet Omar Khayyām in the eleventh century, the astronomer al-Tūsi and mathematician al-Fārisi in the thirteenth and al-Qalasādi in the fifteenth. All these men made wonderful contributions long before European Renaissance mathematicians really got going in the sixteenth and seventeenth centuries.
Before I leave the subject of algebra, I wish to pick out one man in particular from the illustrious list of mathematicians just mentioned. This is partly because I shall not return to him later in the book and he deserves mention. Better known for his poetry, it often surprises many to learn that Omar Khayyām (1048–1131) was one of the greatest of all medieval mathematicians. Unlike other Abbāsid scholars, he wrote in his native Persian rather than in Arabic and it is often suggested that he was a Sufi mystic, although it is far more likely that he was an agnostic Muslim.
Khayyām’s greatest contribution was his work on cubic equations (in which the unknown quantity, x, can appear in powers up to x3). In his famous Treatise on Demonstration of Problems of Algebra, he classifies thirteen different types of cubic equation and provides a general theory for their solution. He also developed both algebraic and geometric methods for solving them systematically and elegantly, using the method of conical sections (which involves slicing through a cone at different angles to produce different types of curves such as circles, ellipses, parabolas and hyperbolas).
While in his mid-twenties, Omar Khayyām made use of basic instruments such as a sundial, a water clock and an astrolabe to measure the length of the solar year to a quite incredibly precise 365.24219858156 days, which is in agreement with the modern value to six decimal places. The difference, however, is not necessarily due to any inaccuracy on Khayyām’s part, but rather because the earth’s spin on its axis is gradually slowing down, so the length of a day is increasing (by 2 milliseconds every century). This means that the exact number of days in a year is decreasing. Compared with Khayyām’s time, the length of the year will be ‘out’ in the sixth decimal place, or by two-hundredths of a second. Several years later Khayyām used this measurement to help devise a calendar7 that was even more precise than today’s Gregorian calendar, which is itself accurate to one day in just 3,330 years. The project, completed in 1079, was carried out by Khayyām and a group of astronomers who introduced several reforms to the existing Persian calendar, which was largely based on ideas from the Hindu calendar. It was named the Jalali calendar, after the Seljuk (in Arabic, Seljūq) sultan who had commissioned the work, and remained in use throughout Persia until the early decades of the twentieth century. In his Treatise on Demonstration of Problems of Algebra, Khayyām gives a remarkable insight into the prevailing attitude towards science among the wider public. One cannot help but wonder about the extent to which such attitudes have changed in a thousand years: