Chapter 3: Medieval Astronomy & Navigation
1000 – 1500 CE
The Maragha Revolution
At the Maragha Observatory in Persia (founded 1259), Nasir al-Din al-Tusi and his colleagues developed the Tusi couple — a mathematical device that converts circular motion into linear motion using two nested circles.
The Tusi couple: if a small circle of radius \(r\) rolls inside a large circle of radius \(2r\), a point on the small circle traces a straight line:
\( x(t) = 2r\cos t, \quad y(t) = 0 \)
This device appeared almost identically in Copernicus' De Revolutionibus (1543). Historians now believe Copernicus had access to the Maragha models through Byzantine intermediaries — a remarkable case of mathematical ideas crossing civilizations to trigger a scientific revolution.
Indian Mathematics & Trigonometry
Indian mathematicians made transformative contributions. Aryabhata (476–550 CE) introduced the sine function (as ardha-jya, “half-chord”), computed \(\pi \approx 3.1416\), and proposed that Earth rotates on its axis.
Brahmagupta (598–668) formulated rules for zero and negative numbers, and solved the “Pell equation” \(x^2 - Ny^2 = 1\) centuries before European mathematicians. Madhava of Sangamagrama (c. 1350–1425) discovered the power series expansions:
\( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \)
\( \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \)
These series — often attributed to Taylor, Leibniz, or Gregory — were discovered by the Kerala school 250 years earlier. They represent the earliest known use of calculus-like methods for computing transcendental functions.
Chinese Mathematics & Astronomy
Chinese mathematicians independently developed sophisticated methods. The Nine Chapters on the Mathematical Art (c. 200 BCE–200 CE) contains systems of linear equations solved by Gaussian elimination — 1,800 years before Gauss.
Zu Chongzhi (429–500) computed \(\pi\) as \(\frac{355}{113} = 3.1415929...\), accurate to 7 decimal places — a record that stood for a thousand years. Chinese astronomers made the most detailed records of supernovae, comets, and eclipses in the pre-telescopic world, data still used by modern astronomers.
Navigation & Computational Demand
The Age of Exploration (15th century) created enormous demand for mathematical tools. Navigators needed to determine latitude from star positions, requiring accurate trigonometric tables and spherical geometry.
Regiomontanus (1436–1476) wrote De Triangulis, the first European treatise on trigonometry as an independent mathematical discipline. His tables of sines and tangents enabled both navigation and the astronomical observations that Copernicus would use to overthrow the geocentric model.
Bridge to Physics: Computation Drives Theory
The practical need for accurate navigation tables drove the development of computational mathematics. This pattern repeats throughout history: WWII drove early computing, particle physics drove the development of the World Wide Web, and today AI is driving new mathematics.
On the Eve of Revolution
By 1500, the mathematical tools were in place for a revolution: algebra from the Islamic world, trigonometry refined by Indian, Islamic, and European scholars, computational techniques driven by navigation, and the Ptolemaic astronomical models reaching their mathematical limits.
Copernicus, armed with these tools (and possibly the Tusi couple), was about to rearrange the cosmos. The next chapter — the invention of calculus — would give physics its most powerful mathematical language.