Egyptian numerals

The Egyptians had a writing system based on hieroglyphs from around 3000 BC. Hieroglyphs are little pictures representing words. It is easy to see how they would denote the word "bird" by a little picture of a bird but clearly without further development this system of writing cannot represent many words. The way round this problem adopted by the ancient Egyptians was to use the spoken sounds of words. For example, to illustrate the idea with an English sentence, we can see how "I hear a barking dog" might be represented by:

"an eye", "an ear", "bark of tree" + "head with crown", "a dog".

Of course the same symbols might mean something different in a different context, so "an eye" might mean "see" while "an ear" might signify "sound".

The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we mean that they has separate symbols for one unit, one ten, one hundred, one thousand, one ten thousand, one hundred thousand, and one million.

Here are the numeral hieroglyphs.

To make up the number 276, for example, fifteen symbols were required: two "hundred" symbols, seven "ten" symbols, and six "unit" symbols. The numbers appeared thus:

276 in hieroglyphs.

Here is another example:

4622 in hieroglyphs.

Note that the examples of 276 and 4622 in hieroglyphs are seen on a stone carving from Karnak, dating from around 1500 BC, and now displayed in the Louvre in Paris.

As can easily be seen, adding numeral hieroglyphs is easy. One just adds the individual symbols, but replacing ten copies of a symbol by a single symbol of the next higher value. Fractions to the ancient Egyptians were limited to unit fractions (with the exception of the frequently used ²/₃ and less frequently used ³/₄). A unit fraction is of the form 1/n where n is an integer and these were represented in numeral hieroglyphs by placing the symbol representing a "mouth", which meant "part", above the number. Here are some examples:

Notice that when the number contained too many symbols for the "part" sign to be placed over the whole number, as in ¹/₂₄₉ , then the "part" symbol was just placed over the "first part" of the number. [It was the first part for here the number is read from right to left.]

We should point out that the hieroglyphs did not remain the same throughout the two thousand or so years of the ancient Egyptian civilisation. This civilisation is often broken down into three distinct periods:

Old Kingdom - around 2700 BC to 2200 BC
Middle Kingdom - around 2100 BC to 1700 BC
New Kingdom - around 1600 BC to 1000 BC

Numeral hieroglyphs were somewhat different in these different periods, yet retained a broadly similar style.

Another number system, which the Egyptians used after the invention of writing on papyrus, was composed of hieratic numerals. These numerals allowed numbers to be written in a far more compact form yet using the system required many more symbols to be memorised. There were separate symbols for

1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 20, 30, 40, 50, 60, 70, 80, 90,
100, 200, 300, 400, 500, 600, 700, 800, 900,
1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000

Here are versions of the hieratic numerals

With this system numbers could be formed of a few symbols. The number 9999 had just 4 hieratic symbols instead of 36 hieroglyphs. One major difference between the hieratic numerals and our own number system was the hieratic numerals did not form a positional system so the particular numerals could be written in any order.

Here is one way the Egyptians wrote 2765 in hieratic numerals

Here is a second way of writing 2765 in hieratic numerals with the order reversed

Like the hieroglyphs, the hieratic symbols changed over time but they underwent more changes with six distinct periods. Initially the symbols that were used were quite close to the corresponding hieroglyph but their form diverged over time. The versions we give of the hieratic numerals date from around 1800 BC. The two systems ran in parallel for around 2000 years with the hieratic symbols being used in writing on papyrus, as for example in the Rhind papyrus and the Moscow papyrus, while the hieroglyphs continued to be used when carved on stone.

An overview of Indian mathematics

It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them.

We shall examine the contributions of Indian mathematics in this article, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested. Laplace put this with great clarity:-

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

We shall look briefly at the Indian development of the place-value decimal system of numbers later in this article and in somewhat more detail in the separate article Indian numerals. First, however, we go back to the first evidence of mathematics developing in India.

Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita. Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.

The first mathematics which we shall describe in this article developed in the Indus valley. The earliest known urban Indian culture was first identified in 1921 at Harappa in the Punjab and then, one year later, at Mohenjo-daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by our term "Indian mathematics" which, in this article, refers to mathematics developed in the Indian subcontinent. The Indus civilisation (or Harappan civilisation as it is sometimes known) was based in these two cities and also in over a hundred small towns and villages. It was a civilisation which began around 2500 BC and survived until 1700 BC or later. The people were literate and used a written script containing around 500 characters which some have claimed to have deciphered but, being far from clear that this is the case, much research remains to be done before a full appreciation of the mathematical achievements of this ancient civilisation can be fully assessed.

We often think of Egyptians and Babylonians as being the height of civilisation and of mathematical skills around the period of the Indus civilisation, yet V G Childe in New Light on the Most Ancient East (1952) wrote:-

India confronts Egypt and Babylonia by the 3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture.

We do know that the Harappans had adopted a uniform system of weights and measures. An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several scales for the measurement of length were also discovered during excavations. One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch". Of course ten units is then 13.2 inches which is quite believable as the measure of a "foot". A similar measure based on the length of a foot is present in other parts of Asia and beyond. Another scale was discovered when a bronze rod was found which was marked in lengths of 0.367 inches. It is certainly surprising the accuracy with which these scales are marked. Now 100 units of this measure is 36.7 inches which is the measure of a stride. Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in construction.

It is unclear exactly what caused the decline in the Harappan civilisation. Historians have suggested four possible causes: a change in climatic patterns and a consequent agricultural crisis; a climatic disaster such flooding or severe drought; disease spread by epidemic; or the invasion of Indo-Aryans peoples from the north. The favourite theory used to be the last of the four, but recent opinions favour one of the first three. What is certainly true is that eventually the Indo-Aryans peoples from the north did spread over the region. This brings us to the earliest literary record of Indian culture, the Vedas which were composed in Vedic Sanskrit, between 1500 BC and 800 BC. At first these texts, consisting of hymns, spells, and ritual observations, were transmitted orally. Later the texts became written works for use of those practicing the Vedic religion.

The next mathematics of importance on the Indian subcontinent was associated with these religious texts. It consisted of the Sulbasutras which were appendices to the Vedas giving rules for constructing altars. They contained quite an amount of geometrical knowledge, but the mathematics was being developed, not for its own sake, but purely for practical religious purposes. The mathematics contained in the these texts is studied in some detail in the separate article on the Sulbasutras.

The main Sulbasutras were composed by Baudhayana (about 800 BC), Manava (about 750 BC), Apastamba (about 600 BC), and Katyayana (about 200 BC). These men were both priests and scholars but they were not mathematicians in the modern sense. Although we have no information on these men other than the texts they wrote, we have included them in our biographies of mathematicians. There is another scholar, who again was not a mathematician in the usual sense, who lived around this period. That was Panini who achieved remarkable results in his studies of Sanskrit grammar. Now one might reasonably ask what Sanskrit grammar has to do with mathematics. It certainly has something to do with modern theoretical computer science, for a mathematician or computer scientist working with formal language theory will recognise just how modern some of Panini's ideas are.

Before the end of the period of the Sulbasutras, around the middle of the third century BC, the Brahmi numerals had begun to appear.

Here is one style of the Brahmi numerals..


These are the earliest numerals which, after a multitude of changes, eventually developed into the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 used today. The development of numerals and place-valued number systems are studied in the article Indian numerals.

The Vedic religion with its sacrificial rites began to wane and other religions began to replace it. One of these was Jainism, a religion and philosophy which was founded in India around the 6^th century BC. Although the period after the decline of the Vedic religion up to the time of Aryabhata I around 500 AD used to be considered as a dark period in Indian mathematics, recently it has been recognised as a time when many mathematical ideas were considered. In fact Aryabhata is now thought of as summarising the mathematical developments of the Jaina as well as beginning the next phase.

The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. More surprisingly the Jaina developed a theory of the infinite containing different levels of infinity, a primitive understanding of indices, and some notion of logarithms to base 2. One of the difficult problems facing historians of mathematics is deciding on the date of the Bakhshali manuscript. If this is a work which is indeed from 400 AD, or at any rate a copy of a work which was originally written at this time, then our understanding of the achievements of Jaina mathematics will be greatly enhanced. While there is so much uncertainty over the date, a topic discussed fully in our article on the Bakhshali manuscript, then we should avoid rewriting the history of the Jaina period in the light of the mathematics contained in this remarkable document.

You can see a separate article about Jaina mathematics.

If the Vedic religion gave rise to a study of mathematics for constructing sacrificial altars, then it was Jaina cosmology which led to ideas of the infinite in Jaina mathematics. Later mathematical advances were often driven by the study of astronomy. Well perhaps it would be more accurate to say that astrology formed the driving force since it was that "science" which required accurate information about the planets and other heavenly bodies and so encouraged the development of mathematics. Religion too played a major role in astronomical investigations in India for accurate calendars had to be prepared to allow religious observances to occur at the correct times. Mathematics then was still an applied science in India for many centuries with mathematicians developing methods to solve practical problems.

Yavanesvara, in the second century AD, played an important role in popularising astrology when he translated a Greek astrology text dating from 120 BC. If he had made a literal translation it is doubtful whether it would have been of interest to more than a few academically minded people. He popularised the text, however, by resetting the whole work into Indian culture using Hindu images with the Indian caste system integrated into his text.

By about 500 AD the classical era of Indian mathematics began with the work of Aryabhata. His work was both a summary of Jaina mathematics and the beginning of new era for astronomy and mathematics. His ideas of astronomy were truly remarkable. He replaced the two demons Rahu, the Dhruva Rahu which causes the phases of the Moon and the Parva Rahu which causes an eclipse by covering the Moon or Sun or their light, with a modern theory of eclipses. He introduced trigonometry in order to make his astronomical calculations, based on the Greek epicycle theory, and he solved with integer solutions indeterminate equations which arose in astronomical theories.

Aryabhata headed a research centre for mathematics and astronomy at Kusumapura in the northeast of the Indian subcontinent. There a school studying his ideas grew up there but more than that, Aryabhata set the agenda for mathematical and astronomical research in India for many centuries to come. Another mathematical and astronomical centre was at Ujjain, also in the north of the Indian subcontinent, which grew up around the same time as Kusumapura. The most important of the mathematicians at this second centre was Varahamihira who also made important contributions to astronomy and trigonometry.

The main ideas of Jaina mathematics, particularly those relating to its cosmology with its passion for large finite numbers and infinite numbers, continued to flourish with scholars such as Yativrsabha. He was a contemporary of Varahamihira and of the slightly older Aryabhata. We should also note that the two schools at Kusumapura and Ujjain were involved in the continuing developments of the numerals and of place-valued number systems. The next figure of major importance at the Ujjain school was Brahmagupta near the beginning of the seventh century AD and he would make one of the most major contributions to the development of the numbers systems with his remarkable contributions on negative numbers and zero. It is a sobering thought that eight hundred years later European mathematics would be struggling to cope without the use of negative numbers and of zero.

These were certainly not Brahmagupta's only contributions to mathematics. Far from it for he made other major contributions in to the understanding of integer solutions to indeterminate equations and to interpolation formulas invented to aid the computation of sine tables.

The way that the contributions of these mathematicians were prompted by a study of methods in spherical astronomy is described in [25]:-

The Hindu astronomers did not possess a general method for solving problems in spherical astronomy, unlike the Greeks who systematically followed the method of Ptolemy, based on the well-known theorem of Menelaus. But, by means of suitable constructions within the armillary sphere, they were able to reduce many of their problems to comparison of similar right-angled plane triangles. In addition to this device, they sometimes also used the theory of quadratic equations, or applied the method of successive approximations. ... Of the methods taught by Aryabhata and demonstrated by his scholiast Bhaskara I, some are based on comparison of similar right-angled plane triangles, and others are derived from inference. Brahmagupta is probably the earliest astronomer to have employed the theory of quadratic equations and the method of successive approximations to solving problems in spherical astronomy.

Before continuing to describe the developments through the classical period we should explain the mechanisms which allowed mathematics to flourish in India during these centuries. The educational system in India at this time did not allow talented people with ability to receive training in mathematics or astronomy. Rather the whole educational system was family based. There were a number of families who carried the traditions of astrology, astronomy and mathematics forward by educating each new generation of the family in the skills which had been developed. We should also note that astronomy and mathematics developed on their own, separate for the development of other areas of knowledge.

Now a "mathematical family" would have a library which contained the writing of the previous generations. These writings would most likely be commentaries on earlier works such as the Aryabhatiya of Aryabhata. Many of the commentaries would be commentaries on commentaries on commentaries etc. Mathematicians often wrote commentaries on their own work. They would not be aiming to provide texts to be used in educating people outside the family, nor would they be looking for innovative ideas in astronomy. Again religion was the key, for astronomy was considered to be of divine origin and each family would remain faithful to the revelations of the subject as presented by their gods. To seek fundamental changes would be unthinkable for in asking others to accept such changes would be essentially asking them to change religious belief. Nor do these men appear to have made astronomical observations in any systematic way. Some of the texts do claim that the computed data presented in them is in better agreement with observation than that of their predecessors but, despite this, there does not seem to have been a major observational programme set up. Paramesvara in the late fourteenth century appears to be one of the first Indian mathematicians to make systematic observations over many years.

Mathematics however was in a different position. It was only a tool used for making astronomical calculations. If one could produce innovative mathematical ideas then one could exhibit the truths of astronomy more easily. The mathematics therefore had to lead to the same answers as had been reached before but it was certainly good if it could achieve these more easily or with greater clarity. This meant that despite mathematics only being used as a computational tool for astronomy, the brilliant Indian scholars were encouraged by their culture to put their genius into advances in this topic.

A contemporary of Brahmagupta who headed the research centre at Ujjain was Bhaskara I who led the Asmaka school. This school would have the study of the works of Aryabhata as their main concern and certainly Bhaskara was commentator on the mathematics of Aryabhata. More than 100 years after Bhaskara lived the astronomer Lalla, another commentator on Aryabhata.

The ninth century saw mathematical progress with scholars such as Govindasvami, Mahavira, Prthudakasvami, Sankara, and Sridhara. Some of these such as Govindasvami and Sankara were commentators on the text of Bhaskara I while Mahavira was famed for his updating of Brahmagupta's book. This period saw developments in sine tables, solving equations, algebraic notation, quadratics, indeterminate equations, and improvements to the number systems. The agenda was still basically that set by Aryabhata and the topics being developed those in his work.

The main mathematicians of the tenth century in India were Aryabhata II and Vijayanandi, both adding to the understanding of sine tables and trigonometry to support their astronomical calculations. In the eleventh century Sripati and Brahmadeva were major figures but perhaps the most outstanding of all was Bhaskara II in the twelfth century. He worked on algebra, number systems, and astronomy. He wrote beautiful texts illustrated with mathematical problems, some of which we present in his biography, and he provided the best summary of the mathematics and astronomy of the classical period.

Bhaskara II may be considered the high point of Indian mathematics but at one time this was all that was known [26]:-

For a long time Western scholars thought that Indians had not done any original work till the time of Bhaskara II. This is far from the truth. Nor has the growth of Indian mathematics stopped with Bhaskara II. Quite a few results of Indian mathematicians have been rediscovered by Europeans. For instance, the development of number theory, the theory of indeterminates infinite series expressions for sine, cosine and tangent, computational mathematics, etc.

Following Bhaskara II there was over 200 years before any other major contributions to mathematics were made on the Indian subcontinent. In fact for a long time it was thought that Bhaskara II represented the end of mathematical developments in the Indian subcontinent until modern times. However in the second half of the fourteenth century Mahendra Suri wrote the first Indian treatise on the astrolabe and Narayana wrote an important commentary on Bhaskara II, making important contributions to algebra and magic squares. The most remarkable contribution from this period, however, was by Madhava who invented Taylor series and rigorous mathematical analysis in some inspired contributions. Madhava was from Kerala and his work there inspired a school of followers such as Nilakantha and Jyesthadeva.

Some of the remarkable discoveries of the Kerala mathematicians are described in [26]. These include: a formula for the ecliptic; the Newton-Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier's formula for the circumradius of a cyclic quadrilateral. Of particular interest is the approximation to the value of π which was the first to be made using a series. Madhava's result which gave a series for π, translated into the language of modern mathematics, reads

π R = 4R - 4R/3 + 4R/5 - ...

This formula, as well as several others referred to above, were rediscovered by European mathematicians several centuries later. Madhava also gave other formulae for π, one of which leads to the approximation 3.14159265359.

The first person in modern times to realise that the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus by nearly 300 years was Charles Whish in 1835. Whish's publication in the Transactions of the Royal Asiatic Society of Great Britain and Ireland was essentially unnoticed by historians of mathematics. Only 100 years later in the 1940s did historians of mathematics look in detail at the works of Kerala's mathematicians and find that the remarkable claims made by Whish were essentially true. See for example [15]. Indeed the Kerala mathematicians had, as Whish wrote:-

... laid the foundation for a complete system of fluxions ...

and these works:-

... abound with fluxional forms and series to be found in no work of foreign countries.

There were other major advances in Kerala at around this time. Citrabhanu was a sixteenth century mathematicians from Kerala who gave integer solutions to twenty-one types of systems of two algebraic equations. These types are all the possible pairs of equations of the following seven forms:

x + y = a, x - y = b, xy = c, x² + y² = d, x² - y² = e, x³ + y³ = f, and x³ - y³ = g.

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. See [12] for more details.

Now we have presented the latter part of the history of Indian mathematics in an unlikely way. That there would be essentially no progress between the contributions of Bhaskara II and the innovations of Madhava, who was far more innovative than any other Indian mathematician producing a totally new perspective on mathematics, seems unlikely. Much more likely is that we are unaware of the contributions made over this 200 year period which must have provided the foundations on which Madhava built his theories.

Our understanding of the contributions of Indian mathematicians has changed markedly over the last few decades. Much more work needs to be done to further our understanding of the contributions of mathematicians whose work has sadly been lost, or perhaps even worse, been ignored. Indeed work is now being undertaken and we should soon have a better understanding of this important part of the history of mathematics.

The Arabic numeral system

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 12^th 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 [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° 21' 14", would have been written as "mj ka yd" in this base 60 version of the "abjad" letters for calculating.

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°. There is a reason for this change which came about due to the way that scribes wrote, for they wrote on a scroll which they wound from right to left across their bodies as they sat cross-legged. The scribes therefore, instead of writing from right to left (the standard way that Arabic was written) wrote in lines from top to bottom. The script was rotated when the scroll was read and the characters when then in the correct orientation.

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° so that they would appear correctly when the scroll was rotated to be read.

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.

Babylonian numerals

The Babylonian civilisation in Mesopotamia replaced the Sumerian civilisation and the Akkadian civilisation. We give a little historical background to these events in our article Babylonian mathematics. Certainly in terms of their number system the Babylonians inherited ideas from the Sumerians and from the Akkadians. From the number systems of these earlier peoples came the base of 60, that is the sexagesimal system. Yet neither the Sumerian nor the Akkadian system was a positional system and this advance by the Babylonians was undoubtedly their greatest achievement in terms of developing the number system. Some would argue that it was their biggest achievement in mathematics.

Often when told that the Babylonian number system was base 60 people's first reaction is: what a lot of special number symbols they must have had to learn. Now of course this comment is based on knowledge of our own decimal system which is a positional system with nine special symbols and a zero symbol to denote an empty place. However, rather than have to learn 10 symbols as we do to use our decimal numbers, the Babylonians only had to learn two symbols to produce their base 60 positional system.

Now although the Babylonian system was a positional base 60 system, it had some vestiges of a base 10 system within it. This is because the 59 numbers, which go into one of the places of the system, were built from a 'unit' symbol and a 'ten' symbol.

Here are the 59 symbols built from these two symbols

Now given a positional system one needs a convention concerning which end of the number represents the units. For example the decimal 12345 represents

1 10⁴ + 2 10³ + 3 10² + 4 10 + 5.

If one thinks about it this is perhaps illogical for we read from left to right so when we read the first digit we do not know its value until we have read the complete number to find out how many powers of 10 are associated with this first place. The Babylonian sexagesimal positional system places numbers with the same convention, so the right most position is for the units up to 59, the position one to the left is for 60 n where 1 ≤ n ≤ 59, etc. Now we adopt a notation where we separate the numerals by commas so, for example, 1,57,46,40 represents the sexagesimal number

1 60³ + 57 60² + 46 60 + 40

which, in decimal notation is 424000.

Here is 1,57,46,40 in Babylonian numerals

Now there is a potential problem with the system. Since two is represented by two characters each representing one unit, and 61 is represented by the one character for a unit in the first place and a second identical character for a unit in the second place then the Babylonian sexagesimal numbers 1,1 and 2 have essentially the same representation. However, this was not really a problem since the spacing of the characters allowed one to tell the difference. In the symbol for 2 the two characters representing the unit touch each other and become a single symbol. In the number 1,1 there is a space between them.

A much more serious problem was the fact that there was no zero to put into an empty position. The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation and now there was no way that spacing could help. The context made it clear, and in fact despite this appearing very unsatisfactory, it could not have been found so by the Babylonians. How do we know this? Well if they had really found that the system presented them with real ambiguities they would have solved the problem - there is little doubt that they had the skills to come up with a solution had the system been unworkable. Perhaps we should mention here that later Babylonian civilisations did invent a symbol to indicate an empty place so the lack of a zero could not have been totally satisfactory to them.

An empty place in the middle of a number likewise gave them problems. Although not a very serious comment, perhaps it is worth remarking that if we assume that all our decimal digits are equally likely in a number then there is a one in ten chance of an empty place while for the Babylonians with their sexagesimal system there was a one in sixty chance. Returning to empty places in the middle of numbers we can look at actual examples where this happens.

Here is an example from a cuneiform tablet (actually AO 17264 in the Louvre collection in Paris) in which the calculation to square 147 is carried out. In sexagesimal 147 = 2,27 and squaring gives the number 21609 = 6,0,9.

Here is the Babylonian example of 2,27 squared

Perhaps the scribe left a little more space than usual between the 6 and the 9 than he would have done had he been representing 6,9.

Now if the empty space caused a problem with integers then there was an even bigger problem with Babylonian sexagesimal fractions. The Babylonians used a system of sexagesimal fractions similar to our decimal fractions. For example if we write 0.125 then this is ¹/₁₀ + ²/₁₀₀ + ⁵/₁₀₀₀ = ¹/₈. Of course a fraction of the form a/b, in its lowest form, can be represented as a finite decimal fraction if and only if b has no prime divisors other than 2 or 5. So ¹/₃ has no finite decimal fraction. Similarly the Babylonian sexagesimal fraction 0;7,30 represented ⁷/₆₀ + ³⁰/₃₆₀₀ which again written in our notation is ¹/₈.

Since 60 is divisible by the primes 2, 3 and 5 then a number of the form a/b, in its lowest form, can be represented as a finite decimal fraction if and only if b has no prime divisors other than 2, 3 or 5. More fractions can therefore be represented as finite sexagesimal fractions than can as finite decimal fractions. Some historians think that this observation has a direct bearing on why the Babylonians developed the sexagesimal system, rather than the decimal system, but this seems a little unlikely. If this were the case why not have 30 as a base? We discuss this problem in some detail below.

Now we have already suggested the notation that we will use to denote a sexagesimal number with fractional part. To illustrate 10,12,5;1,52,30 represents the number

10 60² + 12 60 + 5 + ¹/₆₀ + ⁵²/_60² + ³⁰/_60³

which in our notation is 36725 ¹/₃₂. This is fine but we have introduced the notation of the semicolon to show where the integer part ends and the fractional part begins. It is the "sexagesimal point" and plays an analogous role to a decimal point. However, the Babylonians has no notation to indicate where the integer part ended and the fractional part began. Hence there was a great deal of ambiguity introduced and "the context makes it clear" philosophy now seems pretty stretched. If I write 10,12,5,1,52,30 without having a notation for the "sexagesimal point" then it could mean any of:

0;10,12, 5, 1,52,30

  10;12, 5, 1,52,30

  10,12; 5, 1,52,30

  10,12, 5; 1,52,30

  10,12, 5, 1;52,30

  10,12, 5, 1,52;30

  10,12, 5, 1,52,30

in addition, of course, to 10, 12, 5, 1, 52, 30, 0 or 0 ; 0, 10, 12, 5, 1, 52, 30 etc.

Finally we should look at the question of why the Babylonians had a number system with a base of 60. The easy answer is that they inherited the base of 60 from the Sumerians but that is no answer at all. It only leads us to ask why the Sumerians used base 60. The first comment would be that we do not have to go back further for we can be fairly certain that the sexagesimal system originated with the Sumerians. The second point to make is that modern mathematicians were not the first to ask such questions. Theon of Alexandria tried to answer this question in the fourth century AD and many historians of mathematics have offered an opinion since then without any coming up with a really convincing answer.

Theon's answer was that 60 was the smallest number divisible by 1, 2, 3, 4, and 5 so the number of divisors was maximised. Although this is true it appears too scholarly a reason. A base of 12 would seem a more likely candidate if this were the reason, yet no major civilisation seems to have come up with that base. On the other hand many measures do involve 12, for example it occurs frequently in weights, money and length subdivisions. For example in old British measures there were twelve inches in a foot, twelve pennies in a shilling etc.

Neugebauer proposed a theory based on the weights and measures that the Sumerians used. His idea basically is that a decimal counting system was modified to base 60 to allow for dividing weights and measures into thirds. Certainly we know that the system of weights and measures of the Sumerians do use ¹/₃ and ²/₃ as basic fractions. However although Neugebauer may be correct, the counter argument would be that the system of weights and measures was a consequence of the number system rather than visa versa.

Several theories have been based on astronomical events. The suggestion that 60 is the product of the number of months in the year (moons per year) with the number of planets (Mercury, Venus, Mars, Jupiter, Saturn) again seems far fetched as a reason for base 60. That the year was thought to have 360 days was suggested as a reason for the number base of 60 by the historian of mathematics Moritz Cantor. Again the idea is not that convincing since the Sumerians certainly knew that the year was longer than 360 days. Another hypothesis concerns the fact that the sun moves through its diameter 720 times during a day and, with 12 Sumerian hours in a day, one can come up with 60.

Some theories are based on geometry. For example one theory is that an equilateral triangle was considered the fundamental geometrical building block by the Sumerians. Now an angle of an equilateral triangle is 60° so if this were divided into 10, an angle of 6° would become the basic angular unit. Now there are sixty of these basic units in a circle so again we have the proposed reason for choosing 60 as a base. Notice this argument almost contradicts itself since it assumes 10 as the basic unit for division!

I [EFR] feel that all of these reasons are really not worth considering seriously. Perhaps I've set up my own argument a little, but the phrase "choosing 60 as a base" which I just used is highly significant. I just do not believe that anyone ever chose a number base for any civilisation. Can you imagine the Sumerians setting set up a committee to decide on their number base - no things just did not happen in that way. The reason has to involve the way that counting arose in the Sumerian civilisation, just as 10 became a base in other civilisations who began counting on their fingers, and twenty became a base for those who counted on both their fingers and toes.

Here is one way that it could have happened. One can count up to 60 using your two hands. On your left hand there are three parts on each of four fingers (excluding the thumb). The parts are divided from each other by the joints in the fingers. Now one can count up to 60 by pointing at one of the twelve parts of the fingers of the left hand with one of the five fingers of the right hand. This gives a way of finger counting up to 60 rather than to 10. Anyone convinced?

A variant of this proposal has been made by others. Perhaps the most widely accepted theory proposes that the Sumerian civilisation must have come about through the joining of two peoples, one of whom had base 12 for their counting and the other having base 5. Although 5 is nothing like as common as 10 as a number base among ancient peoples, it is not uncommon and is clearly used by people who counted on the fingers of one hand and then started again. This theory then supposes that as the two peoples mixed and the two systems of counting were used by different members of the society trading with each other then base 60 would arise naturally as the system everyone understood.

I have heard the same theory proposed but with the two peoples who mixed to produce the Sumerians having 10 and 6 as their number bases. This version has the advantage that there is a natural unit for 10 in the Babylonian system which one could argue was a remnant of the earlier decimal system. One of the nicest things about these theories is that it may be possible to find written evidence of the two mixing systems and thereby give what would essentially amount to a proof of the conjecture. Do not think of history as a dead subject. On the contrary our views are constantly changing as the latest research brings new evidence and new interpretations to light.

Matrices and determinants

The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. However it was not until near the end of the 17^th Century that the ideas reappeared and development really got underway.

It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around 300 BC contains the following problem:-

There are two fields whose total area is 1800 square yards. One produces grain at the rate of ²/₃ of a bushel per square yard while the other produces grain at the rate of ¹/₂ a bushel per square yard. If the total yield is 1100 bushels, what is the size of each field.

The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. Indeed it is fair to say that the text Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. First a problem is set up which is similar to the Babylonian example given above:-

There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures. Two of the first, three of the second and one of the third make 34 measures. And one of the first, two of the second and three of the third make 26 measures. How many measures of corn are contained of one bundle of each type?

Now the author does something quite remarkable. He sets up the coefficients of the system of three linear equations in three unknowns as a table on a 'counting board'.

           1   2   3

           2   3   2

           3   1   1

          26  34  39

Our late 20^th Century methods would have us write the linear equations as the rows of the matrix rather than the columns but of course the method is identical. Most remarkably the author, writing in 200 BC, instructs the reader to multiply the middle column by 3 and subtract the right column as many times as possible, the same is then done subtracting the right column as many times as possible from 3 times the first column. This gives

           0   0   3

           4   5   2

           8   1   1

          39  24  39

Next the left most column is multiplied by 5 and then the middle column is subtracted as many times as possible. This gives

           0   0   3

           0   5   2

          36   1   1

          99  24  39

from which the solution can be found for the third type of corn, then for the second, then the first by back substitution. This method, now known as Gaussian elimination, would not become well known until the early 19^th Century.

Cardan, in Ars Magna (1545), gives a rule for solving a system of two linear equations which he calls regula de modo and which [7] calls mother of rules ! This rule gives what essentially is Cramer's rule for solving a 2 2 system although Cardan does not make the final step. Cardan therefore does not reach the definition of a determinant but, with the advantage of hindsight, we can see that his method does lead to the definition.

Many standard results of elementary matrix theory first appeared long before matrices were the object of mathematical investigation. For example de Witt in Elements of curves, published as a part of the commentaries on the 1660 Latin version of Descartes' Géométrie , showed how a transformation of the axes reduces a given equation for a conic to canonical form. This amounts to diagonalising a symmetric matrix but de Witt never thought in these terms.

The idea of a determinant appeared in Japan and Europe at almost exactly the same time although Seki in Japan certainly published first. In 1683 Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables in exactly the way the Chinese methods described above were constructed. Without having any word which corresponds to 'determinant' Seki still introduced determinants and gave general methods for calculating them based on examples. Using his 'determinants' Seki was able to find determinants of 2 2, 3 3, 4 4 and 5 5 matrices and applied them to solving equations but not systems of linear equations.

Rather remarkably the first appearance of a determinant in Europe appeared in exactly the same year 1683. In that year Leibniz wrote to de l'Hôpital. He explained that the system of equations

       10 + 11x + 12y = 0

       20 + 21x + 22y = 0

       30 + 31x + 32y = 0

had a solution because

10.21.32 + 11.22.30 + 12.20.31 = 10.22.31 + 11.20.32 + 12.21.30

which is exactly the condition that the coefficient matrix has determinant 0. Notice that here Leibniz is not using numerical coefficients but

two characters, the first marking in which equation it occurs, the second marking which letter it belongs to.

Hence 21 denotes what we might write as a₂₁.

Leibniz was convinced that good mathematical notation was the key to progress so he experimented with different notation for coefficient systems. His unpublished manuscripts contain more than 50 different ways of writing coefficient systems which he worked on during a period of 50 years beginning in 1678. Only two publications (1700 and 1710) contain results on coefficient systems and these use the same notation as in his letter to de l'Hôpital mentioned above.

Leibniz used the word 'resultant' for certain combinatorial sums of terms of a determinant. He proved various results on resultants including what is essentially Cramer's rule. He also knew that a determinant could be expanded using any column - what is now called the Laplace expansion. As well as studying coefficient systems of equations which led him to determinants, Leibniz also studied coefficient systems of quadratic forms which led naturally towards matrix theory.

In the 1730's Maclaurin wrote Treatise of algebra although it was not published until 1748, two years after his death. It contains the first published results on determinants proving Cramer's rule for 2 2 and 3 3 systems and indicating how the 4 4 case would work. Cramer gave the general rule for n n systems in a paper Introduction to the analysis of algebraic curves (1750). It arose out of a desire to find the equation of a plane curve passing through a number of given points. The rule appears in an Appendix to the paper but no proof is given:-

One finds the value of each unknown by forming n fractions of which the common denominator has as many terms as there are permutations of n things.

Cramer does go on to explain precisely how one calculates these terms as products of certain coefficients in the equations and how one determines the sign. He also says how the n numerators of the fractions can be found by replacing certain coefficients in this calculation by constant terms of the system.

Work on determinants now began to appear regularly. In 1764 Bezout gave methods of calculating determinants as did Vandermonde in 1771. In 1772 Laplace claimed that the methods introduced by Cramer and Bezout were impractical and, in a paper where he studied the orbits of the inner planets, he discussed the solution of systems of linear equations without actually calculating it, by using determinants. Rather surprisingly Laplace used the word 'resultant' for what we now call the determinant: surprising since it is the same word as used by Leibniz yet Laplace must have been unaware of Leibniz's work. Laplace gave the expansion of a determinant which is now named after him.

Lagrange, in a paper of 1773, studied identities for 3 3 functional determinants. However this comment is made with hindsight since Lagrange himself saw no connection between his work and that of Laplace and Vandermonde. This 1773 paper on mechanics, however, contains what we now think of as the volume interpretation of a determinant for the first time. Lagrange showed that the tetrahedron formed by O(0,0,0) and the three points M(x,y,z), M'(x',y',z'), M"(x",y",z") has volume

¹/₆ [z(x'y" - y'x") + z'(yx" - xy") + z"(xy' - yx')].

The term 'determinant' was first introduced by Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms. He used the term because the determinant determines the properties of the quadratic form. However the concept is not the same as that of our determinant. In the same work Gauss lays out the coefficients of his quadratic forms in rectangular arrays. He describes matrix multiplication (which he thinks of as composition so he has not yet reached the concept of matrix algebra) and the inverse of a matrix in the particular context of the arrays of coefficients of quadratic forms.

Gaussian elimination, which first appeared in the text Nine Chapters on the Mathematical Art written in 200 BC, was used by Gauss in his work which studied the orbit of the asteroid Pallas. Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix.

It was Cauchy in 1812 who used 'determinant' in its modern sense. Cauchy's work is the most complete of the early works on determinants. He reproved the earlier results and gave new results of his own on minors and adjoints. In the 1812 paper the multiplication theorem for determinants is proved for the first time although, at the same meeting of the Institut de France, Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by Cauchy.

In 1826 Cauchy, in the context of quadratic forms in n variables, used the term 'tableau' for the matrix of coefficients. He found the eigenvalues and gave results on diagonalisation of a matrix in the context of converting a form to the sum of squares. Cauchy also introduced the idea of similar matrices (but not the term) and showed that if two matrices are similar they have the same characteristic equation. He also, again in the context of quadratic forms, proved that every real symmetric matrix is diagonalisable.

Jacques Sturm gave a generalisation of the eigenvalue problem in the context of solving systems of ordinary differential equations. In fact the concept of an eigenvalue appeared 80 years earlier, again in work on systems of linear differential equations, by D'Alembert studying the motion of a string with masses attached to it at various points.

It should be stressed that neither Cauchy nor Jacques Sturm realised the generality of the ideas they were introducing and saw them only in the specific contexts in which they were working. Jacobi from around 1830 and then Kronecker and Weierstrass in the 1850's and 1860's also looked at matrix results but again in a special context, this time the notion of a linear transformation. Jacobi published three treatises on determinants in 1841. These were important in that for the first time the definition of the determinant was made in an algorithmic way and the entries in the determinant were not specified so his results applied equally well to cases were the entries were numbers or to where they were functions. These three papers by Jacobi made the idea of a determinant widely known.

Cayley, also writing in 1841, published the first English contribution to the theory of determinants. In this paper he used two vertical lines on either side of the array to denote the determinant, a notation which has now become standard.

Eisenstein in 1844 denoted linear substitutions by a single letter and showed how to add and multiply them like ordinary numbers except for the lack of commutativity. It is fair to say that Eisenstein was the first to think of linear substitutions as forming an algebra as can be seen in this quote from his 1844 paper:-

An algorithm for calculation can be based on this, it consists of applying the usual rules for the operations of multiplication, division, and exponentiation to symbolic equations between linear systems, correct symbolic equations are always obtained, the sole consideration being that the order of the factors may not be altered.

The first to use the term 'matrix' was Sylvester in 1850. Sylvester defined a matrix to be an oblong arrangement of terms and saw it as something which led to various determinants from square arrays contained within it. After leaving America and returning to England in 1851, Sylvester became a lawyer and met Cayley, a fellow lawyer who shared his interest in mathematics. Cayley quickly saw the significance of the matrix concept and by 1853 Cayley had published a note giving, for the first time, the inverse of a matrix.

Cayley in 1858 published Memoir on the theory of matrices which is remarkable for containing the first abstract definition of a matrix. He shows that the coefficient arrays studied earlier for quadratic forms and for linear transformations are special cases of his general concept. Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses. He gave an explicit construction of the inverse of a matrix in terms of the determinant of the matrix. Cayley also proved that, in the case of 2 2 matrices, that a matrix satisfies its own characteristic equation. He stated that he had checked the result for 3 3 matrices, indicating its proof, but says:-

I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree.

That a matrix satisfies its own characteristic equation is called the Cayley-Hamilton theorem so its reasonable to ask what it has to do with Hamilton. In fact he also proved a special case of the theorem, the 4 4 case, in the course of his investigations into quaternions.

In 1870 the Jordan canonical form appeared in Treatise on substitutions and algebraic equations by Jordan. It appears in the context of a canonical form for linear substitutions over the finite field of order a prime.

Frobenius, in 1878, wrote an important work on matrices On linear substitutions and bilinear forms although he seemed unaware of Cayley's work. Frobenius in this paper deals with coefficients of forms and does not use the term matrix. However he proved important results on canonical matrices as representatives of equivalence classes of matrices. He cites Kronecker and Weierstrass as having considered special cases of his results in 1874 and 1868 respectively. Frobenius also proved the general result that a matrix satisfies its characteristic equation. This 1878 paper by Frobenius also contains the definition of the rank of a matrix which he used in his work on canonical forms and the definition of orthogonal matrices.

The nullity of a square matrix was defined by Sylvester in 1884. He defined the nullity of A, n(A), to be the largest i such that every minor of A of order n-i+1 is zero. Sylvester was interested in invariants of matrices, that is properties which are not changed by certain transformations. Sylvester proved that

max{n(A), n(B)} ≤ n(AB) ≤ n(A) + n(B).

In 1896 Frobenius became aware of Cayley's 1858 Memoir on the theory of matrices and after this started to use the term matrix. Despite the fact that Cayley only proved the Cayley-Hamilton theorem for 2 2 and 3 3 matrices, Frobenius generously attributed the result to Cayley despite the fact that Frobenius had been the first to prove the general theorem.

An axiomatic definition of a determinant was used by Weierstrass in his lectures and, after his death, it was published in 1903 in the note On determinant theory. In the same year Kronecker's lectures on determinants were also published, again after his death. With these two publications the modern theory of determinants was in place but matrix theory took slightly longer to become a fully accepted theory. An important early text which brought matrices into their proper place within mathematics was Introduction to higher algebra by Bôcher in 1907. Turnbull and Aitken wrote influential texts in the 1930's and Mirsky's An introduction to linear algebra in 1955 saw matrix theory reach its present major role in as one of the most important undergraduate mathematics topic.