This very important mathematical point is much older than we thought

Some inventions are so ubiquitous that it’s easy to forget that anyone invented them in the first place. Take the comma for example. There was a time when if we wanted to write a number between zero and one, we could pretty much just use a fraction. At some point, though, all that changed – and it appears that point may have been about a century and a half earlier than we previously thought.

“The earliest known appearance of the comma was in the interpolation column of a sine table in Christopher Clavius’ Astrolabe (1593),” writes Glen Van Brummelen, professor of mathematical sciences at Trinity Western University and historian of mathematics and astronomy, in a new article that examines the history of the minute symbol.

“But this is a curious place to introduce such an important new idea,” he argues, “and the fact that Clavius ​​never abused it in his later writings remains unexplained.”

It turns out that there is a simple solution to these riddles: Clavius ​​​​wasn’t the one who came up with the comma at all. “We trace Clavius’ use of decimal fraction numbering and the comma back to the work of Giovanni Bianchini (1440),” explains Van Brummelen, “whose decimal system was a distinctive feature of his calculations in spherical astronomy and metrology.”

So who was this mysterious Bianchini, who gave us such a fundamental part of our interpretation of the world? Don’t worry if you don’t remember him from your math books: he wasn’t actually a mathematician at all, but a Venetian merchant and administrator of the locally powerful d’Este family.

Nevertheless, he apparently had some interest in the subject, as evidenced by a short article on geometry that he seems to have written sometime in the 1640s. In this text he used an instrument called a biffa Unpleasant “[invent] an equivalent of the metric system,” Van Brummelen writes:

… let the line of each foot (pedis) be divided into ten equal parts, bounded by lines of less length than the lines bounding the feet; these divisions are called detached. And also loosening are divided into ten parts and also indicated with smaller lines or dots; these divisions are called minuta. And also the minuta are divided, if that is possible, into ten parts, which are in congruent intervals; these divisions are called secunda… And note that these divisions are always bounded in tens by tens, so that multiplication and division can be made thereby, according to the doctrine I will teach below, will work more easily.

If that doesn’t look too groundbreaking, don’t worry: it’s not. As Van Brummelen notes, Bianchini was far from the first person to ever use decimal expansion, period. “In China, the early emergence of decimal fractions led to a continuing tradition from the Middle Ages,” he notes; “[the] Mid-10th century Damascene scholar Abū al-Hasan al-Uqlīdisī […] usage[d] a short vertical notch to indicate the units in a series of decimal digits,” and numerous other scientists around the world have independently devised equivalent notation hacks and shortcuts at various points in history.

But what makes Bianchini’s treatise so special is the specific notation he chose: a small dot, which separates the whole units from the fractional part.

“The first time Bianchini refers to a length that requires more than one unit of measurement, he names each unit as follows: ‘sitque ipsa distanceia pedes .0. loosen .7. minute .4. etc second .6.'”, writes van Brummelen.

“But when he focuses on multiplication, division and extracting roots, metrology disappears,” he continues. Bianchini further abbreviates the representation, for example by ‘.746.’ to write, which he notes can easily be read as 746 second. At one point he squares the distance by 92 feet9 not0 minute9 second. He writes this quantity as ‘.92909.’”

use of the comma from the Compositio instrumenti

The smoking gun. From Bianchini’s Compositio instrumenti.

Van Brummelen, 2024, taken from the Biblioteca Estense di Modena (CC BY-NC-ND 4.0)

It’s a clever breakthrough, but it would be little more than a footnote in the history books if it hadn’t been spotted by a pair of highly influential mathematical astronomers: Clavius ​​himself and Johannes Müller von Königsberg – better known as Regiomontanus.

Regiomontanus “learned from Bianchini, adopted a number of the latter’s innovations, and in some respects expanded the paradigms that Bianchini had established,” van Brummelen writes. table, and the fact that he never used it again is easily explained.”

Clavius ​​”had access to Bianchini’s Sine table,” Van Brummelen concludes, “and he copied the structure of that table in his own work.”

The article was published in the journal Historia Mathematica.

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