In 1593, the German astronomer Christophorus Clavius ??made a small mark on a table showing triangular proportions in his treatise Astrolabium. The Jesuit had just left a mark that was going to change the history of mathematics forever.
Clavius ??has since been known as the first to use the decimal point, the symbol that indicates the separation between the whole part and the fractional part of a number. Still, Dr. Glen Van Brummelen of Trinity Western University has just discovered that it was actually Giovanni Bianchini, an Italian court astrologer, who used this sign many decades earlier.
Van Brummelen explains in an article published in the journal Historia Mathematica that Clavius’ treatise on the astrolabe was “a curious place to introduce such a significant new idea, and the fact that he never took advantage of it in his later writings remains unexplained.”
Throughout history, different civilizations, such as the Arab and Asian, had already invented the decimal fractional mark, although these previous forms of notation did not persist until today. Clavius ??(and from now on also Bianchini), on the other hand, played a decisive role in establishing the modern decimal point and leading it towards the dominant system it is today.
The Trinity Western historian specializing in mathematics in ancient cultures, sometimes described as the world’s only historian of trigonometry, was analyzing astronomical tables from the late Middle Ages and early Renaissance when he noticed an unexpected use of a dot decimal in a text from 1440.
“Mathematics does not develop on its own, as people often think, but in response to situations that arose in other scientific contexts or even in commerce or finance,” says Van Brummelen. “This dating reveals the richness of medieval scientific activity, dispelling the popular notion that this was a time of intellectual stagnation,” he adds.
Bianchini, in addition to being a professor of mathematics and astronomy at the University of Ferrara and an astrologer at the court of the Marquis Leonello d’Este (1407-1450), was also hired by members of the same court to help them manage their businesses and commercial activities. .
One hundred and fifty years before Clavius, a young Giovanni Bianchini, barely 30 years old, wrote several astronomical works that made extensive and increasingly refined use of decimal points. Van Brummelen argues that the German Jesuit simply copied the Italian’s mathematical innovation.
Bianchini was a Venetian merchant who had probably trained in arithmetic and algebra. At some point in the 1430s, the House of Este hired him as administrator of the estate. It was during this stage that he wrote his astronomical treatises, probably because his role involved astrological predictions for his noble employers.
He wrote five important works between 1440 and 1460: Astronomical tables, sobre los movimientos de los planetas; Tables of the first mobile A, on spherical astronomy and mathematical astrology; Flores Almagesti, his masterpiece on mathematical astronomy. Tables of the first mobile B, a rewritten version of the Tables of the first mobile A; y Tables of eclipses, sobre la prediction de eclipses.
At that time in Europe, astronomy was performed using sexagesimal arithmetic, a system that uses 60 as a base in the same way that our modern decimal system is based on units of 10. Sexagesimal arithmetic is still applied today to the measurement of time and of the amplitude of the angles.
Another of Bianchini’s contributions that has changed the history of mathematics, and that we can still see today, is his invention of what became the tangent function in trigonometry, now a standard button on our calculators and a common tool of science, engineering and technology.
It was during an analysis of trigonometric functions in Tabulae primi mobilis B that the Italian astrologer introduced the decimal point. When describing the use of interpolation to find the sine and cosine of an arc, he casually dropped the number 10.4 and explained how to multiply it by 8, giving the answer 83.
The product of 10.4 and 8 is actually 83.2, but Bianchini’s work shows that he is using the decimal point and understanding it, the same way we use and understand it today. Later, more signs appear in other tables of the same work.
This is exactly how, almost two centuries later, the decimal point appears in the Astrolabium, used in exactly the same way to perform the same function. And, most tellingly, the German astronomer never used it again, as far as is known.
“Clavius ??probably had access to Bianchini’s table (or someone who had borrowed it) and copied the structure of that table in his own work,” explains historian Glen Van Brummelen in his article.
“He recognized the usefulness of decimal fraction notation in the context of interpolation within a numerical table; but it was not his idea, and he himself had probably not invested significant effort in exploring its use. The result was that the origin of decimal fractional notation and the decimal point in Europe remained hidden until now,” concludes the researcher.
