Before the Rabbits

IThe line you can fill

In the old Indian metres counted by duration (mātrā-vṛtta, "moraic verse"), every syllable is one of two lengths. A short syllable — laghu, "light" — lasts one beat. A long syllable — guru, "heavy" — lasts two. A line of a fixed duration can be filled many ways, and the poets asked the obvious question: how many?

So fill a line. Choose how many beats (morae) long it is, and the loom below lays out every rhythm that fills it exactly — short (•, one beat) and long (▬, two). Tap any row to hear it: a short tick, a long held twice as long. Then read the count.

One beat: just • — one way. Two beats: •• or ▬ — two ways. Three: three ways. Four: five. Five: eight. The counts are 1, 2, 3, 5, 8, 13, 21, 34, 55… — each the sum of the two before it. These are the Fibonacci numbers, arriving by a door that has nothing to do with rabbits.

IIWhy it is always the sum of the two before

The poets did not count by brute force forever. They found the reason — and the reason is a single, clean observation you can watch the loom make.

Look at any rhythm that fills n beats. Its last syllable is either short or long, and that is the whole proof:

• If it ends in a short (one beat), strip that syllable off and what remains is some rhythm filling n−1 beats — any of them, each exactly once.
• If it ends in a long (two beats), strip it off and what remains fills n−2 beats — again, any of them, exactly once.

Every rhythm falls into exactly one of those two buckets, and nothing is left out. So the number filling n beats is the number filling n−1 plus the number filling n−2. Press “split by last syllable” on the loom and watch the rhythms sort themselves into those two groups — the upper group is a perfect copy of the shorter line, the lower group a copy of the line shorter still. The recurrence is not asserted here; the loom builds the bijection in front of you.

This is exactly the rule the prosodist Virahāṅka set down, in his Prākrit treatise on metre, the Vṛttajātisamuccaya, somewhere between the sixth and eighth centuries — the first surviving text to state it explicitly:

“The variations of two earlier metres being mixed, the number is obtained. That is a direction for knowing the number of variations of the next mātrā-vṛtta.” Virahāṅka, Vṛttajātisamuccaya — trans. H. D. Velankar, via P. Singh (1985)

Virahāṅka's own book is lost. We have his rule because a later commentator, Gopāla (writing before 1135), quoted it — and went further, working the numbers out one by one: “…three happens, five happens, eight is obtained, thirteen, twenty-one… the process should be followed in all mātrā-vṛttas.” He is the first author known to write the sequence down as a sequence.

IIIThe triangle underneath

There is an older Indian instrument still, and the Fibonacci numbers are hiding along its diagonals — which is exactly why the attribution is so easy to get wrong.

Centuries before Virahāṅka, Pingala — author of the Chandaḥśāstra, the founding text of Sanskrit prosody, of uncertain date (anywhere from the last few centuries BCE to perhaps the early centuries CE) — worked on the other kind of metre, counted by number of syllables rather than duration. His tradition tabulated how many k-of-n patterns exist: how many lines of n syllables carry exactly k heavy ones. That table is the array later named the meru-prastāra, the “staircase of Mount Meru” — and it is, cell for cell, Pascal's triangle, six hundred years before Pascal.

Now put the two ideas together. A moraic line of n beats with exactly k long syllables has n−2k shorts, so n−k syllables in all, and the only freedom is which of them are the longs: that count is the binomial coefficient C(n−k, k) — a single entry of the triangle. Add up the entries for every possible number of longs and you have all the rhythms of n beats. The strip below is Pascal's triangle; drag the line length and watch the cells that sum to your rhythm-count light up. They lie along a shallow diagonal — and the diagonals of Pingala's triangle are the Fibonacci numbers.

So Pingala's combinatorics already contained the Fibonacci numbers — as a diagonal sum — without anyone yet reading them off it. This is the honest seam in the whole story: it is tempting, and common online, to credit Pingala himself with “discovering Fibonacci.” He did not state the additive rule. The one phrase sometimes offered as proof — his cryptic miśrau ca, “and the two are mixed” — is given its Fibonacci reading by a commentator a thousand years later, not by Pingala. The triangle is his; the diagonal sum is later.

The triangular array that does tie the binomials to the Fibonacci numbers directly even has a name — mātrā-meru — but it appears later still, in the Prākṛta Paiṅgala of about the fourteenth century, and it denotes the array, not the linear sequence. Naming it for Pingala is a folklore that has hardened into footnotes.

IVThe rabbits came late

Set the dates side by side and the familiar story inverts. Tap a name to see what they actually did.

The clearest medieval statement of the rule is Hemacandra's, a Jain polymath writing his Chandonuśāsana around 1150:

“The sum of the last and the last-but-one numbers is the number of the mātrā-vṛtta coming next.” … “Statement — 1, 2, 3, 5, 8, 13, 21, 34, and in this way, afterwards.” Hemacandra, Chandonuśāsana, c. 1150 — trans. H. D. Velankar, via P. Singh (1985)

Half a century after Hemacandra, in 1202, Leonardo of Pisa — Fibonacci — published the Liber Abaci and, to show off the new Hindu-Arabic numerals, posed a puzzle: a pair of rabbits breeds a new pair each month from its second month on, none die; how many pairs in a year? The monthly tally runs 1, 1, 2, 3, 5, 8, 13… The same numbers. He never called them his own — the name “Fibonacci sequence” was coined by Édouard Lucas in the 1870s, six centuries later again.

The numbers were not waiting to be born in a counting-house. They had already been heard, as the ways a line of verse can keep its time.

The honest shape of the correction is precise, and it cuts both ways. The recurrence is documented in India centuries before Fibonacci — that much is solid, and Fibonacci himself did nothing wrong by re-finding it; there is no evidence he knew of the Indian work, and the consensus is independent discovery in an unrelated domain. But the popular over-correction — “Pingala discovered the Fibonacci numbers in 200 BCE” — is its own distortion. The chain that actually holds is later and more interesting: implicit in the combinatorics of Pingala's triangle; first stated as a rule by Virahāṅka; first enumerated by Gopāla; stated most clearly by Hemacandra. Four hands, four centuries, all before the rabbits.

⚖The honest edges

“First,” here, means first in a surviving text to state the rule explicitly. Singh notes earlier implicit traces — in Bharata's dramaturgy as read by the commentator Abhinavagupta, and in the Viṣṇudharmottara Purāṇa — so even “Virahāṅka was first” is a claim about the record we still have, not about who first had the thought. The sequence is sometimes called the Virahāṅka–Fibonacci sequence for exactly this reason; in the OEIS it is A000045, whose notes carry the same history.

✓Show the check

Every count on this page is recomputed from scratch by one offline verifier, by generating the actual rhythms and tallying them — none is quoted.

The verifier enumerates every short/long rhythm of each length, checks the count equals the Fibonacci number, and — rather than asserting the recurrence — partitions the rhythms by their final syllable and proves that stripping it is a clean bijection onto the shorter lines. It then re-derives the same numbers two more independent ways (the closed-form Binet formula, and Hemacandra's purely additive array) and confirms the binomial-diagonal identity that ties the triangle to the sequence. The numbers it writes to artifact.json are the numbers this page draws.

drift guard: checking the page's numbers against the verified artifact…