What the sequence is and what it was called
The Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13, 21, and so on, with each term the sum of the two before it. It is named after Leonardo of Pisa, known as Fibonacci, who set a puzzle about breeding rabbits in his book of 1202. But the name is a later habit: the French mathematician Edouard Lucas is usually credited with popularising it in the nineteenth century. Leonardo did not claim to have invented the numbers, and sources in India had already described them.
The rabbit problem in 1202
Leonardo’s book, the Liber Abaci, is best known in Europe for teaching Hindu-Arabic arithmetic to merchants. Among its many practical problems is one about a pair of rabbits. Each pair becomes fertile after a month and then produces one new pair every month. How many pairs are there after a year?
Following the rules gives 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 pairs. The problem is idealised and not a model of real rabbits, and Leonardo treated it as an exercise, listing the monthly totals and noting that each one is the sum of the two previous. The puzzle is the first appearance of the sequence in Europe that we can point to, and it is what attached his name to it.
The Indian prosody problem
The same numbers appear in Sanskrit writing on metre, the study of poetic rhythm. Classical Sanskrit poetry builds lines from syllables classed as short (laghu), counted as one beat, and long (guru), counted as two. A natural question for a poet or a teacher of metre is how many different patterns fill a given number of beats.
Counting by hand shows the answer:
- one beat: 1 pattern (short)
- two beats: 2 patterns (short-short, or one long)
- three beats: 3 patterns (short-short-short, short-long, long-short)
- four beats: 5 patterns
- five beats: 8 patterns
Each total is the sum of the previous two, for a simple reason. A pattern of n beats either ends with a short syllable, leaving n minus 1 beats to fill, or ends with a long one, leaving n minus 2. Add those counts and you get the next term. That reasoning is the whole sequence, and it needs no rabbits.
Pingala, Virahanka and later commentators
The oldest text in this tradition is the Chandahshastra, attributed to Pingala. Its date is uncertain; scholars usually place it somewhere in the last few centuries BCE, but the dating is debated. The text lays out rules for enumerating metres, and later commentators have read its methods as containing the key to the sequence, though the surviving verses are terse and readers interpret them differently.
A clearer statement comes from Virahanka, usually dated to roughly the seventh or eighth century CE, though the date is not fixed. He is credited with a rule that the number of variations of a metre is the sum of the two preceding ones. Later writers repeated and extended it. Gopala, around the early twelfth century, and Hemachandra, a Jain scholar of the mid-twelfth century, both described the same rule, with Hemachandra including it in his own work on prosody. That is why the sequence is sometimes called the Virahanka or Hemachandra numbers in discussions of Indian mathematics.
Did the idea travel to Europe?
It is tempting to draw a line from the Sanskrit texts to Leonardo, who learned arithmetic in North Africa and travelled in the Mediterranean world. The Liber Abaci certainly draws on Arabic-language mathematics, which in turn absorbed Indian methods such as the Brahmi numerals that became our digits. But no surviving document shows the prosody rule reaching Pisa. It is safer to say that the same simple recursion was found more than once, and that the earliest clear record is Indian.
What the sequence is not
Popular accounts credit the sequence with explaining everything from seashells to stock markets. The honest picture is narrower. The numbers do appear in the spiral counts of some plants, though by no means all. The ratio of one term to the next settles toward about 1.618, which is the golden ratio, and that mathematical fact is real. Stronger claims about art and architecture are a separate and more doubtful matter, and are taken up in the history of that ratio.
A pattern found for different reasons
What makes the story attractive is not a rivalry over credit but the way one small rule appears wherever someone counts arrangements in a line. A poet counting syllable patterns and a merchant’s son counting rabbit pairs reach the same list because both are applying one addition rule to a pair of earlier totals. Seen that way, the sequence is less an invention than a fact about counting, and each culture that met it described it in its own terms. The Indian prosodists simply got there first on the written record we have.