Middle Ages to Early Modern — Diophantus, Arabic Mathematics, Fibonacci, and Seki Takakazu

Filling the “European Gap”

After Archimedes in the previous article, European mathematics enters almost 1500 years of stagnation. But mathematics continued to develop elsewhere:

• 3rd-century Alexandria — the algebra of Diophantus
• 9th-century Baghdad — Islamic preservation and extension
• 12th-century Italy — Fibonacci’s reintroduction to the West
• 17th-century Japan — the independent development of Seki Takakazu

This article traces those 1500 years of “non-European mathematical history,” and how they became the bridge to the modern era.

Diophantus — “The Father of Algebra”

Diophantus of Alexandria (Διόφαντος, c. AD 200–284) was a late-antiquity Greek mathematician working in Alexandria. His Arithmetica (Ἀριθμητικά), in 13 books (10 extant), is a classic of the theory of integer-solution equations, and its author is remembered as the “father of algebra.”

Biographical information is essentially nonexistent, but the Greek Anthology preserves a puzzle epitaph:

God gave him 1/6 of his life for boyhood, and adorned his youth for 1/12 more, and after 1/7 lit the wedding torch. Five years after marriage he was granted a son, who died at half his father’s age; four years later, the father followed.

As an equation: x/6 + x/12 + x/7 + 5 + x/2 + 4 = x. Solution: x = 84. He died at 84.

Toward Symbolic Algebra

Diophantus introduced, unusually for ancient Greek mathematics, an algebraic notation with abbreviations:

• ς — the unknown (modern x)
• Δ^Υ — x²
• Κ^Υ — x³
• ↑ — subtraction / negative
• M — constant term

This is a critical step from rhetorical algebra (equations in words) toward syncopated algebra (equations with abbreviations), and a distant precursor to the modern symbolism of Viète and Descartes.

Diophantine Equations

Polynomial equations for which one seeks integer (or rational) solutions. x² + y² = z² (Pythagorean triples), x³ + y³ = z³ (Fermat’s Last Theorem for n = 3), and so on — a central object of modern number theory.

The Arithmetica and Fermat’s Marginal Note

The Arithmetica reached Europe in 1621 through Bachet de Méziriac’s Greek-Latin edition, which is the version Fermat read. The problem “divide a square into two squares” (Book II, Problem 8) is the very margin where Fermat’s future Last Theorem would appear as a scribble.

A margin in a book more than a thousand years old gave birth to a central problem of modern mathematics.

Arabic Mathematics — The House of Wisdom in Baghdad

In the 8th–9th centuries, Greek, Indian, and Persian mathematics were synthesized under the Abbasid caliphs at Baghdad’s House of Wisdom (Bayt al-Hikma). The central figure was al-Khwārizmī (c. 780–850).

Two Words Made from One Name

• Algorithm — from the Latinization of al-Khwārizmī’s name
• Algebra — from the title of his book [al-Kitāb al-Jabr wa al-Muqābala] (c. 830)

Two central terms in modern mathematics and computing come from the name and one book of a single ninth-century Arab mathematician. No other person is quite so deeply embedded in the vocabulary of mathematics.

Arabic Mathematical Contributions

• Adoption and transmission of Indian numerals and zero (India → Arabia → the West)
• General solution of quadratic equations
• Development of trigonometry (al-Tūsī, al-Kāshī)
• Precursors of symbolic algebra

Major figures:

• al-Khwārizmī (9th c.) — the founder of algebra and algorithm
• al-Karajī (11th c.) — polynomial algebra
• Omar Khayyām (11–12th c.) — geometric solutions of cubic equations; poet of the Rubaiyat
• al-Tūsī (13th c.) — trigonometry

Fibonacci — Liber Abaci and Zero’s Arrival in the West

Leonardo Pisano, commonly called Fibonacci (c. 1170–1250), was an Italian mathematician from Pisa. His book Liber Abaci (1202) was the decisive vehicle by which Hindu-Arabic numerals and decimal place-value notation were established in Europe.

Life

• Born c. 1170 in Pisa; his father Guglielmo was a Pisan merchant and customs officer
• As a boy in Bugia (modern Bejaia, Algeria), his father’s posting, he studied Islamic mathematics
• Traveled the Mediterranean (Egypt, Syria, Greece, Sicily, Provence) absorbing mathematics
• 1202, at about 32 — the Liber Abaci completed
• 1220s — gained fame at the court of Emperor Frederick II
• 1240 — a Pisan document grants him an annual pension of 20 lire in recognition

“Fibonacci” is a nickname coined by the 19th-century historian Guglielmo Libri, from filius Bonacci, “son of Bonacci.” He called himself “Leonardo Pisano” or “Leonardo Bigollo” (“Leonardo the Traveler”).

Opening of the Liber Abaci

“The nine figures of the Arabs are these: 9, 8, 7, 6, 5, 4, 3, 2, 1. With these nine figures and the sign 0 which the Arabs call zephirum, any number can be written.”

This is the substantial introduction of Hindu-Arabic numerals (especially zero) into the West. Multiplying “MMXXIII × CXLVII” in Roman numerals is essentially impossible; in Hindu-Arabic numerals it is trivial by written arithmetic. Fibonacci’s book was a computational revolution for merchants and craftsmen.

Fully abandoning Roman numerals took centuries — the Florentine bankers’ guild banned Arabic numerals in 1299 (fearing forgery).

The Rabbits and the Fibonacci Sequence

Chapter 12 of the Liber Abaci poses:

“A pair of rabbits produces a new pair every month; each newborn pair also produces from its second month. How many pairs are there after a year?”

The sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 (recurrence a(n+2) = a(n+1) + a(n)).

Consecutive ratios converge to the golden ratio φ = (1+√5)/2 ≈ 1.618. The sequence appears in phyllotaxis, pine cones, and sunflower spirals.

Similar sequences were studied earlier in India by Pingala (2nd century BC), Virahanka (6th century), and Hemachandra (12th century) in the theory of Sanskrit prosody. “Fibonacci sequence” is a Eurocentric name.

Seki Takakazu — The “Mathematical Sage” of Closed-Country Japan

Seki Takakazu (関孝和, c. 1640–1708) was an early-Edo Japanese mathematician who raised the domestic tradition of wasan to a world-class level. He is called the Mathematical Sage (sansei).

Independent Development under Sakoku

Seki lived under the closed-country system. Western mathematics — Descartes, Fermat, Newton, Leibniz — had not entered Japan. Starting from Chinese mathematics (the Suanxue Qimeng, Yang Hui suanfa, and other Song–Yuan works), he built a mathematics that decisively went beyond it.

This is a rare case of world-class independent invention. In an environment completely cut off from Europe, mathematics of comparable level was born.

Major Results, Compared with Europe

Result	Seki	Europe
Determinants	1683 (Kaifukudai-no-hō)	Leibniz 1693 (letter to L’Hôpital)
Bernoulli numbers	Katsuyō sanpō	Jacob Bernoulli 1713 (Ars Conjectandi)
π to 11 decimal places, 131072-gon		Parallel to Europe
Newton-like iterative approximation		Newton c. 1669

Seki thus arrived 10 to 30 years before Europe or in parallel.

Tenzan-jutsu — Symbolic Algebra with Multiple Unknowns

Seki systematized a symbolic algebra using variable characters, extending Chinese tianyuan (one unknown) to multiple unknowns:

• Tengen — first unknown
• Jingen — second
• Chigen — third
• Butsu-gen — fourth

Wasan could now handle serious multivariate algebra.

Main Works

• Hatsubi Sanpō (1674) — solutions to the “successor problems” left by Sawaguchi Kazuyuki; the debut of tenzan-jutsu
• Kaifukudai-no-hō — the discovery of determinants
• Katsuyō Sanpō (posthumous, 1712) — collected by his disciple Araki Muraide; π and Bernoulli numbers
• Taisei Sankei — a summation by Seki, Takebe Katahiro, and Takebe Katakira

Unique Wasan Culture — Idai and Sangaku

Idai — Problem-Driven Transmission

Edo-period wasan had a distinctive “idai” culture. A book ended by posing unresolved problems for the next mathematician to solve. Seki’s Hatsubi Sanpō was itself a response to fifteen idai of Sawaguchi. A problem-driven system of knowledge transmission.

Sangaku — Mathematics as Votive Offering

Wasan practitioners and enthusiasts wrote solved problems on wooden tablets and dedicated them to shrines and temples — sangaku.

• Commoners, samurai, and townspeople all participated
• Thousands of sangaku were dedicated across Japan throughout the Edo period
• Several hundred survive today
• A culture of popular participation in mathematics unmatched anywhere

“Mathematics as an offering to a shrine” — a distinctively Japanese conception. The opposite of the West’s academy-centered approach.

Takebe Katahiro — Seki’s Disciple and Enri

Takebe Katahiro (建部賢弘, 1664–1739) was Seki’s leading disciple. He developed enri — the theory of π, circular area, and arc length — and independently obtained infinite series expansions parallel to Newton, Leibniz, and Gregory.

Through Takebe, Seki’s work was passed down as the Seki school and remained the mainstream of Japanese mathematics until the Meiji Restoration.

Meiji and the End of Wasan

After the Meiji Restoration (1868), Western mathematics was introduced and wasan declined. The 1872 education law unified school mathematics on the Western model. Most wasan practitioners switched careers; some became teachers of Western mathematics.

But the legacy of wasan was not forgotten:

• Research by Fujiwara Matsusaburō, Hirayama Akira, Satō Ken’ichi, Ueno Ken’ichi, Takase Masahito, and others established the world-class quality of wasan
• 2008 — the Complete Works of Seki Takakazu (Iwanami Shoten)
• Cultural protection of sangaku

Reintegrated into modern history of mathematics as “world-class mathematics discovered independently.”

From This Article to the Next

We have traced 1500 years — Diophantus, Arabic mathematics, Fibonacci, Seki Takakazu. The next article enters the 17th-century Scientific Revolution. Descartes (1596–1650) unifies algebra and geometry through coordinate geometry, and Fermat (1607–1665) writes symbols in the margins of the Arithmetica — the first half of the “Scientific Revolution” in which mathematics leaps into modernity.