Al-Battani: Pioneering Muslim Mathematician-Astronomer Extraordinaire (Part-2)


Aristotle, Ptolemy and the other Greek masters might have been great theorists of science, but the patient work of direct observation and studied inference based on obtainable evidence was left to stand apart as the hallmark of the early Muslim scientists inspired by the Qur’anic revelation: a revelation that compelled its believers to observe and ponder over the creation of the universe and the world all around – and, indeed, within – them. Not only does al-Battani’s Kitāb az-Zīj give us the results of his observations and analysis, but his inferences also allow data given for one era to be converted to another era, writes BIJU ABDUL QADIR in this concluding part of the series on the life and achievements of one of the greatest astronomers of all time: the ninth century Muslim scientist, Abū ʿAbdAllāh Muḥammad ibn Jābir ibn Sinān al-Raqqī al-Ḥarrānī al-Ṣābiʾ al-Battānī. 

Moving Past Ptolemy: Trigonometric Innovations over Greek Geometry

PtolemyThe strongest and most lasting influence of al-Battani’s investigations, perhaps, was his work with the ratios related to the sides and angles of right triangles – or to describe this discipline in one word: trigonometry. Al-Battani’s innovative use of the sine, tangent-cotangent functions, and his discovery and deployment of the secant and cosecant functions were, in fact, a vast improvement over the techniques employed by Ptolemy (c.127CE), the greatest of Greek astronomers. Some of al-Battani’s famous innovations in trigonometry led to equations such as the following:

Thus putting forward a number of important trigonometric functions, he wrote accurate tables for cotangents for angles from 0-900. He found the cotangents in his tables as represented by the following functions:

Cot x = Cos x/ Sin x

In his book, Mathematics History, Florian Cajori noted that:

“Al-Battani did not follow the Greek in many of their Geometry solutions; he instead solved problems by another way: [for e.g.,] m = Sin x/ Cos x. But,

Thus, the value x for the angle could be obtained.”

According to one Western source:

“Al-Battānī used al-Marwazi’s idea of tangents (‘shadows’) to develop equations for calculating tangents and cotangents, compiling tables of them. He also discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants, which he referred to as a ‘table of shadows’ (in reference to the shadow of a gnomon), for each degree from 1° to 90°.” []

However, staying true to the tradition of Arab astronomers of not belittling Ptolemy’s genius, al-Battani fought shy of openly critiquing the great Greek even when he was himself certain of Ptolemy’s several errors. Al-Battani sought, rather, to correct Ptolemy’s findings in a tacit manner without direct reference to any of the latter’s misjudgments. Carrying on the work of reform and updation in this honourable way, al-Battani soon compiled new tables of the sun and moon, correcting, in the process, some of Ptolemy’s results which were, for long, held as indisputable truths.

While Ptolemy had correctly surmised the direction of the Sun’s apogee, it was left to al-Battani to discover that this direction was actually changing. This change in direction is now known as occurring due to the changing direction of the eccentricity vector of the Earth’s orbit.

Based again on his observations, al-Battani also came to the conclusion that the longitude of the sun’s apogee had actually gone up by 16° 47′ since Ptolemy’s time. This, of course, led to the subsequent – and important – discovery of the motion of the solar apsides and of a slow variation in the equation of time. As mentioned earlier, al-Battani’s measurement of the length of the solar year was the most accurate up to that point in history; this was while Ptolemy’s own measurement of the same value had overstepped the correct value by 6 minutes and 26 seconds.

Staying true to his original streak in research, al-Battani preferred, even in the company of other astronomers of his time, to employ a uniform rate for precession in his tables, instead of adopting the theory of trepidation attributed to his learned contemporary, Thabit b. Qurra, and which Copernicus still held as viable centuries after both these Arab masters had passed away. In Encyclopedia Britannica, we read the following with regard to one in several of al-Battani’s propositions which have held their ground through the centuries:

“Al-Battani gave a rule for finding the elevation θ of the sun above the horizon in terms of the length s of the shadow cast by a vertical gnomon of height h. Al-Battani’s rule, s = h sin (900 – θ)/ sin θ, is equivalent to the formula s = h cot θ. Based on this rule, he constructed a ‘table of shadows’ – essentially a table of cotangents – for each degree from 10 to 900.” (


The Kitābaz-Zīj: Astronomy Encyclopedia of Generations

BattaniAl-Battani is known to have written several tracts on the subject of his astronomical studies, but few were as consequential and far-reaching in its legacy as his Magnum Opus, the Kitāb az-Zīj, or the Book of Astronomical Tables, sometimes also referred to as Az-Zīj as-Ṣābi’. It is known that some of his observations mentioned in the Kitāb az-Zīj were made in the year 880CE and, later on, in the year 900CE. Al-Battani put forth his first edition of this work before 900CE, but he then revised the same sometime after 901CE. This was so as to incorporate the results of his observation of two eclipses – a solar and a lunar eclipse – which he had the chance to see that same year while on a visit to Antioch in Syria.

While the influence of Ptolemy and of other Greco-Syriac sources is discernible at several places in this compendium of astronomical data, the book shows minimal, if any, Indian or Persian dependence. Publishing many of his observations and inferences in the Kitāb az-Zīj, al-Battani further enhanced the value of the work by extensively supporting it with tables. A curious interest in astrology is peculiar to al-Battani’s study inasmuch as his subjecting Ptolemy’s Tetrabiblos to rigorous scrutiny was combined with his own study on the signs of the Zodiac in the same work. Indeed, a whole chapter in the Kitāb az-Zīj is entitled: ‘On Ascensions of the Signs of the Zodiac.’ It is possible that the position of astrology, in the ancient world, as a relevant science, did not go without its effect on al-Battani, for there are quite a few of his shorter essays which sought to address different aspects of astrology. Moreover, seven chapters of the Kitāb az-Zīj are dedicated to a discussion of problems in astrology, rather than astronomy.

In all, Kitāb az-Zīj consists of 57 chapters, headed by a description of the celestial sphere which al-Battani divided into degrees. There is then an introduction to the requisite mathematical instruments like arithmetical operations on sexagesimal fractions as also trigonometric functions. Another chapter – the fourth – provides data from al-Battani’s own observations. No less than 21 chapters which follow on this fourth chapter are dedicated to the discussion of numerous problems in astronomy; to a certain extent, at least, al-Battani is seen here as having followed through with the content of Ptolemy’s Almagest.

The next five chapters of Kitāb az-Zīj offer a discussion of the motion of the sun, moon and five planets with some reference to Ptolemy’s theory. However, to al-Battani, the theory was after all just that – theory – and had little meaning unless, and until, supported by the practical aspects of direct observation and experience. This, in fact, represented the classic Arab-Muslim stance on all scientific matters: the preference for empirical observation over abstract, or even plausible, theorization. Aristotle, Ptolemy and the other Greek masters might have been great theorists of science, but the patient work of direct observation and studied inference based on obtainable evidence was left to stand apart as the hallmark of Muslim scientists inspired by the Qur’anic revelation: a revelation that compelled its believers to observe and ponder over the creation of the universe and the world all around – and, indeed, within – them.

Not only does al-Battani’s Kitāb az-Zīj give us the results of his observations and analysis, but his inferences also allow data given for one era to be converted to another era. In fact, he then puts forward sixteen chapters which explain how his tables are to be read and understood. The next seven chapters address problems in astrology, while the second last chapter – the 56th – is, in its entirety, dedicated to a discussion on the construction of a single instrument: the sundial. Moving further ahead on instruments, al-Battani ends his book with a final chapter which explains the construction of several other types of astronomical instruments.

In many ways, the Kitābaz-Zīj was a landmark accomplishment, not only for al-Battani personally, but also for the history of astronomy, mathematics and even instrumentation. Cataloging, within the pages of the Zij, the position of no less than 489 stars among other important findings, al-Battani also recorded for posterity, his refined value for the length of the solar year and of the seasons, the precession of the equinoxes as well as his accurately observed value for the inclination of the ecliptic. The Zijalso showed how, unlike Ptolemy, who worked with geometrical methods, al-Battani had employed trigonometrical techniques which ultimately proved an undeniable advance for astronomical and mathematical research. One among several important formulas for right-angled triangles, which was developed by al-Battani – and recorded in the Zij– is the following:

bSinθ = a Sin (90-θ)

The Kitāb az-Zīj Survives – Through a Fortunate Irony of History

In time, theZij came to be rendered into Latin twice in the twelfth century: once by Plato of Tivoli and then by Robert Retinensis, the latter translator probably being the same Robert of Chester who was the first scholar to produce a Latin translation of the Qur’an. However, only one of these two valuable translations survived: the one done by Plato of Tivoli in 1116CE, which came out under the title, De Motu Stellarum(‘On the Motion of the Stars’), with later annotations by Regiomontanus. There is, fortunately, one other translation of the Zijwhich has outlived the vicissitude of the ages: the Spanish version commissioned by King Alfonso X of Spain in the thirteenth century.

It is not without reason that the middle ages in Europe have often been called the ‘Dark Age.’ A period of religious diktats and persecution, several generations of Europeans were witness to this time of brutal repression by the Church which forbade any advance in science that put a question mark on Christian dogma in whatever form. Galileo Galilee and Nicolas Copernicus were themselves famous victims of this throttling of scientific endeavour by the Church authorities who often worked hand-in-glove with the State in matters of religious concern.

However, it is among the fortunate ironies of history, that even in this time of repression, when precious few books were authorized for publication in printed form, al-Battani’s Kitāb az-Zīj actually made it to the list of approved works. In 1537CE, the translation of the Zij by Plato of Tivoli was published from Nuremberg, Germany. Another edition of the same work then made its appearance in Bologna, Italy, in 1645CE. The Zij was thus fast becoming popular among the astronomers and mathematicians of central and northern Europe. Not only among Christian scientists, but even among astronomers of the Jewish community in Spain, was the Zij warmly received and to great acclaim, so much so that even as late as 1749CE, al-Battani’s observations of eclipses were still being referred to, and studied, by astronomers.

Closer to the modern period, it was during the years between 1899-1907CE that C. A. Nallino, the European historian of Islam, produced a large, three-volume, Arabic edition of the Kitāb az-Zīj. With the arrival of this new and substantial edition of the Zijat the turn of the last century, the book – and its author – has held the sustained attention of modern historians of science. Al-Battani is, today, well-respected within informed western academic circles as having been a keen follower – and reformer – of Ptolemaic theories of astronomy and as a vital connecting link between the astronomical sciences of the world of antiquity and those of the modern world.

The original manuscript of the Kitāb az-Zīj is, of late, maintained at the Vatican in Rome. Another valuable manuscript on astronomical chronology – also written by al-Battani – is known to be in the possession of the Escorial library.

Death and Legacy

As an astronomer, it is known that al-Battani remained active well into his 60th year (c.918CE). Another decade later, we find him again in 929CE, aged 71, but, nonetheless, accompanying a group of people from Raqqah on a journey to Baghdad. The apparent reason for the journey: to protest unfair taxes slapped on the populace by the government of the time. It has also been recorded that the aging, but determined, astronomer survived the onward journey, managing even to state his case before the authorities at Baghdad. Unfortunately, however, he did not survive the journey back to Raqqah. Al-Battani passed away at Qasr al-Jiss near what is today Samarrah in Iraq.

Nearly six decades after his death, al-Battani’s work was chronicled in the Fenrist compiled by the book-seller, Ibn Nadim. The Fehrist contained a useful list of Arabic literature (and the respective authors) extant then in the tenth century. About al-Battani, the Fehrist makes mention that he made his astronomical observations during the period between 877CE and 918CE, and that his star catalogue was from the year 880CE. Ibn an-Nadim was quite categorical in his praise for al-Battani whom he calls “one of the famous observers and a leader in geometry, theoretical and practical astronomy, and astrology.” Ibn an-Nadim further noted that:

“[al-Battani] composed a work on astronomy, with tables, containing his own observations of the sun and moon and a more accurate description of their motions than that given in Ptolemy’s Almagest. In it, moreover, he gives the motions of the five planets, with the improved observations he succeeded in making, as well as other necessary astronomical calculations. Some of his observations mentioned in his book of tables were made in the year 880 and later on in the year 900. Nobody is known in Islam who reached similar perfection in observing the stars and scrutinizing their motions. Apart from this, he took great interest in astrology, which led him to write on this subject too. Of his compositions in this field, I mention his commentary on Ptolemy’s Tetrabiblos.”

To be sure, Ibn an-Nadim’s comments on al-Battani’s life and achievements were not the exaggerations of a devout Muslim; they were, rather, an assertion of what was already known about this remarkable scientist throughout the Arab world of that time.

However, it was – and still is – the non-Arab, non-Muslim, Western world which, despite drawing on his pioneering scholarship centuries later for its own rebirth from its Dark Ages, remains generally and, perhaps, deliberately oblivious to the great legacy and civilizational contributions that al-Battani and other pioneering Muslim scientists, time and again, offered the world of men.

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