Varahamihira of Ujjain Set Five Older Astronomical Systems Side by Side in the Pancha-siddhantika, a Sixth-Century Handbook That Is Now a Main Source for Astronomy in India Before 500 CE

The Pancha-siddhantika is a sixth-century Sanskrit handbook by the astronomer Varahamihira that sets out five older systems of astronomy: the Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas. A siddhanta is a full treatise on mathematical astronomy. The Pancha-siddhantika is a karana, a compact manual of quick rules for finding where the Sun, Moon and planets stand on a given date and when eclipses fall. G. Thibaut, its first modern editor, describes manuals of this type as giving concise rules for speedy calculation instead of setting out the full theory.

Five systems in one handbook

MacTutor, the history-of-mathematics archive at the University of St Andrews, quotes the scholar Shukla calling the book “one of the most important sources for the history of Hindu astronomy before the time of” Aryabhata I. Aryabhata finished his own Aryabhatiya in 499, by his own statement, and that book has an article of its own on this site.

Pingree, who wrote the Varahamihira entry in the Dictionary of Scientific Biography, gives the reason. Very few other sources show how astronomy was done in India before the year 500, and Varahamihira’s book records several traditions that cannot be studied anywhere else. Thibaut, who edited and translated it with Sudhakara Dvivedi at Benares in 1889, wrote that Varahamihira seemed to be the only Indian writer on astronomy, as far as anyone could then judge, who thought it worth setting out all the more important schools current in his day.

When Thibaut wrote, only one of the five had come down as a separate text, the Surya Siddhanta, and he distinguished that surviving version from the older Surya that Varahamihira summarises. Alberuni, whose account of India dates to about 1030, reported that he had been unable to obtain any of the books on his list except Pulisa’s and Brahmagupta’s.

Ujjain, and a date that is really an epoch

Varahamihira called himself a native of Avanti, the western Malwa region around Ujjain, and lived at a village named Kapitthaka, writes Pingree, who identifies it probably with the ruins at Kayatha about twelve miles from the city. MacTutor treats Kapitthaka as the most likely guess for his birthplace and describes Ujjain as an important centre of mathematics from about the year 400. His father, Adityadasa, belonged to the Maga Brahmanas, a community descended from Iranian Zoroastrians who had settled in northern India and been absorbed into Hinduism, Pingree adds.

The book’s own calculations start from the Saka year 427, which is 505 CE. Thibaut explains the practice: writers of such manuals chose a starting year close to the date of writing, because that keeps the arithmetic short and the errors small. Pingree says Varahamihira’s date rests on that epoch and on Brahmagupta’s citing him as an authority in 628.

What the 505 stands for is argued. MacTutor gives his life as 505 to 587 and dates the Pancha-siddhantika to 575. A commentator named Amaraja, as quoted by Bhau Daji, puts Varahamihira’s death in Saka 509, which is 587, according to Thibaut. Kern had taken 505 as the birth year. Thibaut set that idea aside but saw the death date as an argument against finishing the book in 505. V. Subrahmanya Sastri, in his 1946 translation of the Brihat Samhita, judged 505 to be the birth year. All of them place him in the sixth century, and they differ on the year.

A further suggestion, which Pingree reports without endorsing, links Varahamihira to the court of the Aulikara kings at Dasapura, now Mandasor, and to Yasodharman, who was reigning in 532. Popular tradition also lists Varahamihira among the “nine gems” at the court of a king Vikramaditya, a list that another article on this site shows first appears centuries later.

What the five systems taught

Varahamihira’s own ranking, as Thibaut reports it, puts the Surya first, the Romaka and Paulisa next as about equally correct, and the Paitamaha and Vasistha well below them. Space follows rank. The Surya and Paulisa get long treatment, the Romaka a shorter one, and the Paitamaha is squeezed into the twelfth chapter, five stanzas long. In the first chapter Varahamihira explains that he wanted the book to concentrate on calculating solar eclipses.

Pingree sorts the five by origin. In his account the Paitamaha is Vedanga-style astronomy of the first century, the oldest layer. Thibaut found it running on the five-year cycle known from the Vedanga Jyotisha, with 366-day years, sixty solar months and sixty-two lunar months to a cycle, and no sign of Greek influence. Pingree counts the Vasistha and Paulisa among the Indian versions of Greek and Babylonian methods for the Sun, Moon and planets, and calls the Romaka and Lata’s Surya essentially Hellenistic.

Greek methods in a Sanskrit handbook

The Romaka shows the link most plainly. Its cycle runs 2,850 years, which is 150 times the 19-year period that the Athenian astronomer Meton, around 430 BCE, showed to hold 235 lunar months. Worked through, Thibaut found, the year length the Romaka implies is the tropical year of Hipparchus, the Greek astronomer of the second century BCE. MacTutor adds that the system drew on Greek epicycle theory for the Sun and Moon, in which each body moves on a small circle that itself rides on a larger one.

Place names carry the link as well. According to Thibaut, the Romaka’s rule for counting days from the epoch is set for the meridian of Yavanapura, “city of the Yavanas”, the Indian name for the Greeks. The rules for the positions of the Sun, Moon and planets are set for the meridian of Ujjain, and Varahamihira gives the difference in longitude between the two places after the Paulisa. The earlier Roman-era sea trade with India is covered in the Muziris article.

Scholars read this differently. In Thibaut’s reading the book mentions facts and points of doctrine suggesting that Indian astronomy depended on the science of Alexandria. MacTutor records that some scholars trace the underlying theories to Babylon, while others argue that Indian astronomers refined Babylonian models through their own observation, and it says much more research is needed. A well-known verse in the Brihat Samhita has been drawn into the argument. It says that the Yavanas are foreigners (mlechchas), yet this science is well established among them, so even they deserve honour like sages. The World History Encyclopedia article renders it as a statement that the science originated with them. Sastri’s 1946 introduction quotes Swami Vivekananda reading the verse as praise meant to encourage foreign students of Indian learning, and offers it against the view that Indian sciences are echoes of Greek ones.

The chapters Varahamihira wrote himself

In Thibaut’s judgement several chapters are Varahamihira’s own treatment. Chapter thirteen gives a popular account of the earth’s roundness and of how the sky looks different at different latitudes. Chapter fourteen shows how results can be had by observation and by simple instruments as well as by calculation. Chapter fifteen refers to other astronomers, and elsewhere the book cites Aryabhata’s views, Thibaut notes. Alberuni recorded that Varahamihira held that everything perceived by the senses bears witness to the earth’s globular shape.

Rules for calculating a table of sines sit in the fourth chapter, a topic that other works usually place elsewhere. MacTutor credits Varahamihira with sine values more accurate than those of Aryabhata, whose work it elsewhere describes as giving “tables of half chords which now really are sine tables”. It also lists three identities that he found, written in modern notation as sin x = cos(90 degrees minus x), sin squared x plus cos squared x = 1, and half of (1 minus cos 2x) = sin squared x. Better tables mattered, MacTutor explains, because astronomy and astrology both depended on them, and that pushed the development of interpolation, the method of estimating values between table entries.

A perfume recipe that is also a magic square

Varahamihira’s other main work, the Brihat Samhita or “great compilation”, is a different kind of book, written at about the same time as the Pancha-siddhantika, Pingree says. Sastri’s edition notes that the text describes itself as 100 chapters and about 3,900 verses of standard length, and prints 107, some of which he thinks are later additions. Its contents run from rainfall, earthquakes and meteors to house-building, the search for underground water, the care of trees, the design of temples and the grading of gems. To Pingree it is one of the most complete surviving Sanskrit treatises on divination and a very valuable record of Indian geography and society.

Chapter 77 of Sastri’s edition deals with perfumes. By its count, choosing any four of sixteen substances gives 1,820 selections, and each can be blended in 96 variations. A table built by repeated addition leads to the 1,820, and MacTutor, citing a 1992 paper by Gupta, reads that table as Pascal’s triangle. A diagram of sixteen cells gives the parts of each ingredient. Sastri translates the rule as giving 18 proportions for each compound however four cells are taken, along a row, a column or a diagonal. That fits the pandiagonal magic square of order four that MacTutor says Hayashi examined, a grid in which even the wrapped-around diagonals share the same total.

How later scholars rated the book

Brahmagupta cited Varahamihira as an authority in 628, and the commentator Utpala wrote on the Brihat Samhita in 966. Alberuni translated Varahamihira’s shorter book of nativity horoscopes into Arabic. He was less impressed by the Pancha-siddhantika, writing that its title means only that the number of siddhantas is five, and that it is neither a distillation of them nor clearly better than they are.

Varahamihira was not original in his writings, Pingree states plainly, and relied on earlier authorities in astrology, divination and astronomy. For the Pancha-siddhantika that is the point of the book, since it reports what the older systems said. The modern edition, with translation and commentary, was published by O. Neugebauer and D. Pingree in two parts at Copenhagen in 1970 and 1971.