### Struik: A Concise History of Mathematics: The Beginnings of Western Europe

#### by Carson Reynolds

Relative to the East, the Western portion of the Roman empire was less advanced. A slave-based agricultural economy was gradually replaced by tenant framers, commerce, and a money economy. The Catholic Church and its monasteries (and a few laymen) kept Greco-Roman Traditions alive. The philosopher Boetius’ *Instutio arithmetica* (a translation of Nicomachus) provided pythagorean number theory and was authoritative more than a thousand years, despite being poor in content. Following the disappearance of large scale economy and Charlemagne coronation as emperor of the Holy Roman Empire, Western society became ecclesiastical and feudal.

During the early centuries there was little appreciation of math. Most math associated with monasteries was concerned with the computation of Easter time. Alcuin and his *Problems for the Quickening of the Mind of the Young* influenced textbooks for many centuries. Pope Sylvester II, was also important in that he studied Arabic math in Spain.

The development of Western feudalism is notable because it was less reliant on slavery. The burghers had to rely on inventive genius to improve their lot. Conflict between feudal landlords and cities led to further technology developments. Partnerships between some feudal princes and cities led to the establishment of Europe’s national states.

These cities established commercial and cultural relationships with the Orient in peaceful and violent ways (think: crusades). When Toledo was retaken from the Moors, scholars (Plato of Tivoli, Gherardo of Cremona, Adelard of Bath, Robert of Chester) flocked there, and with the help of interpreters, translated Arabic math manuscripts, among them the Greek classics.

The first powerful commercial cities were in Italy (Genoa, Pisa, Venice, Milan, and Florence). The 12th and 13th centuries saw exploration (e.g. Marco Polo) and the beginnings of capitalism (banking). Leonardo of Pisa (AKA Fibonacci) wrote *Liber Abaci* about arithmetic and algebraic information he’d collected during travels to the East. *Practica Geometriae* (1220) did the same for geometry and trigonometry. He introduced the fibonacci series and the problem to which it is a solution (the growth of a rabbit population). He solved cubic equations that could not be solved using compass and ruler to six sexagesimal places. He also made use of Hindu-Arabic numbers, although not the first. The oldest recorded use is the *Codex Vigilanus* (976 CE). But introduction was slow, it was met with cultural resistance, not being really in wide use for accounting till the 14th century.

As trade extended so did interest in math, but usually the practical kind, as the use of computation by “reckon masters” attests. Some philosophers (St. Augustine in his *Civitas Dei* and St. Thomas Aquinas) made conjectures about mathematical matters like the nature of infinity: (e.g. “a continuum cannot consist of indivisibles”). Thomas Bradwardine, Archbishop of Canterbury, investigated star polynomials. Nicole Oresme, Bishop of Lisieux played with fractional powers in *De latitudinbus formarum* he had 2D graphs with a dependent and independent variable (which he called *latitudo* and *longitudo*). Decartes may have been influenced by its publication.

Northern academics and laypeople were both struck with *Rechenhaftigkeit*, a belief in the use of computation. Their work was aided when Greek scholars arrived following the fall of Constantinople. Regiomontanus was the greatest mathematician of the 15th Century. He translated many Greek works, and did much to separate trigonometry from astronomy. Nasir al-din had done the same, but Regiomontanus’ work influenced many. His trigonometric tables had sines as line segments, which were dependent on the circle’s radius. It was not till Euler’s systematic use of radius 1 that our modern notation arose.

Western work at this point had not exceeded that of Greece and the Arabic world. Scipio del Ferro’s development of a general solution to cubic equations changed this. Luca Pacioli’s *Summa de Arithmetica* (1494) remarks that cubic equations were unsolved at the time. Work at the University of Bologna (of which Copernicus was a student) found solutions to third degree equations. The solution was lost and rediscovered by Tartagila (“the stammerer”) (1535). He disclosed his solution to Hieronimo Cardano, who published them againts his will in 1545 in *Ars magna*. This text also contained the reduction of biquadratic equations to cubic equations and also considered negative roots, but found them “irreducible.” Raffel Bombelli’s *Algebra* addressed this by providing a consistent theory of imaginary complex numbers.

Astronomy remained an important domain of math studies, and was bolstered by the astronomical theories of Copernicus, Brahe, and Kepler. Trigonometric tables increased in accuracy (15 places by 1613). In 1593 Adriaen van Roomen publicly challenged someone to solve a 45th degree equation. Francois Viete found a solution by relating it to sin. He also reduced Cardon’s solution to cubic equations to a trigonometric one, discarding imaginary numbers. His main contribution, though was the use of letters to denote coefficients in polynomials. He also found pi to 9 decimals, soon to be surpassed by Ludolph van Coolen who found it to 35 places. Viete also had an expression for pi as the sum of an infinite product.

Engineers in France, England, and the Netherlands made additional discoveries. Simon Stevin in *La disme* (1585) introduced decimal fractions. The invention of logarithms was also a great improvement. John Napier in *Mirifici logarithmorum canonis descriptio* (1614) constructed two sequences related so that as one increased arithmetically, the other increased geometrically. His first attempts were clumsy, choosing base 7. Henry Briggs collaborated with Napier and after his death published 10 base logarithms at 14 places for 1…2×10^4 and 9*10^4…100,000. The whole range of 1…100,000 was published by Ezechiel Decker in 1627. It is interesting because the exponential, to which logarithms are usually related, didn’t occur till the 17th century.