When you write the number 2,026, you are using a system invented in India, carried west through the Islamic world, and named in Europe after the people who passed it on. Mathematics and astronomy are among the clearest cases where scholars writing in Arabic did not merely keep older knowledge but changed it. You do not need to solve equations to follow the story; the ideas can be told in words, much as the original authors told them.

Al-Khwarizmi and the birth of algebra

Muhammad ibn Musa al-Khwarizmi (about 780 to about 850), whose name points to the region of Khwarazm in Central Asia, worked in Baghdad under al-Ma'mun. Around the 820s he wrote a short book on calculation by "completion and balancing." The word algebra comes from al-jabr in its title, meaning restoring or completing: moving a subtracted quantity to the other side of an equation so that both sides are whole. Al-Khwarizmi wrote for practical users. Much of the book deals with dividing inheritances under Islamic law, which assigns fixed fractions to different relatives, along with measuring land and settling trade accounts.

Two features of his algebra stand out. First, he wrote everything in words, with no symbols at all; a problem might read "a square and ten roots equal thirty-nine," where we would write x² + 10x = 39. Second, he sorted problems into standard types and gave a rule for each, backed by a geometric picture: he drew the square and the "roots" as areas and completed the figure to find the answer. This mix of Greek-style geometric proof with Babylonian and Indian calculation rules made algebra a subject in its own right rather than a bag of tricks.

Al-Khwarizmi also wrote a book explaining Indian numerals and the place-value system, including a sign for zero. That book survives only in Latin, and its opening words, "Dixit Algorizmi," "so said al-Khwarizmi," gave us another word: algorithm comes from the Latin form of his name. What Europeans later called "Arabic numerals" were, as Arab authors themselves said, Indian in origin.

Later mathematicians

Others built on this work. The Persian poet and mathematician Omar Khayyam (1048 to 1131) classified cubic equations and solved them geometrically by intersecting curves; he also helped reform the Persian calendar. Mathematicians developed trigonometry far beyond its Greek and Indian forms, working out general rules for functions such as sine and tangent, largely because astronomers needed them.

Astronomy for prayer, calendar and curiosity

Religious practice created steady demand for astronomy. Muslims pray toward Mecca at times set by the sun's position, and the months of the Islamic calendar begin with the sighting of the new crescent moon. Finding the qibla, the direction of Mecca, from any city is a problem in spherical geometry, and astronomers produced tables giving it for hundreds of places. The astrolabe, a hand-held brass instrument inherited from the Greeks, was refined into a versatile computing device that could tell time, find the height of stars and solve such problems.

Astronomers also measured. Under al-Ma'mun, teams measured a degree of latitude on the plain of Sinjar to estimate the size of the Earth. The scholar al-Biruni (973 to about 1050) devised a way of finding the Earth's radius from the angle at which the horizon dips when seen from a mountaintop.

Doubting Ptolemy

The ruling model of the heavens was Ptolemy's, from the Greek work that Arabic readers called the Almagest. Astronomers respected it but noticed a problem: to fit the observations, Ptolemy had used a device called the equant, which made planets move at uneven speeds around the centers of their circles and seemed to break his own principle of uniform circular motion. In the thirteenth century Nasir al-Din al-Tusi devised new geometric models to avoid the equant, including a pairing of two circles now called the Tusi couple, and led a team of astronomers at the Maragha observatory in Iran. In fourteenth-century Damascus, Ibn al-Shatir built models that are mathematically very close to some used by Copernicus over a century later. Whether Copernicus learned of them, perhaps through Byzantine manuscripts, is still debated: the similarities are striking, but no direct link has been proven.

Worked example: an inheritance problem

A man dies leaving a widow, two sons and a daughter. Islamic law gives a widow one eighth when there are children, and divides the rest among the children with each son taking twice a daughter's share. How would a judge divide it? After the widow's eighth, seven eighths remain. Count shares: two sons at two shares each and one daughter at one share make five shares, so each share is seven fortieths of the estate. The widow gets five fortieths, each son fourteen fortieths and the daughter seven fortieths. Jurists defended the larger male share on the ground that men were obliged to support their families; many people today see it as unequal, and the debate continues within Muslim societies. The example shows why algebra grew from daily needs: a society that cared about getting each heir's share right needed reliable methods to compute it.