Thomas Harriot
Thomas Harriot was one of those rare minds recounted in history who seems to presage the most modern insights in times long before we would expect their having come into being known. The sheer excellence of his work in what prefigures the Calculus so often attributed between Newton and Leibniz should need little introduction, but here we are nonetheless.
The so-called Stirling Numbers were approached by Harriot in his 1618 work on Triangle Numbers accessible in English at IBSN: 9783037190593.
Torporley, Warner, Pell, Collins, and Mercator, all explored Harriot’s methods before or independently of Newton’s rediscovery of them in 1665. The mathematics of his day passed by way of friends instead of publication. Methods would resurface in later generations after these unwritten doctrines would propagate. What the texts show us often are only a fragment of the excited chatter of the day.
Work by an Alexandrian of the Second or Third Century CE known as Diophantus focused on Arithemetic and something known as Polygonal numbers. In the case of Polygonal Numbers for 3 or Triangle Numbers, we may imagine arranging pebbles in a triangle. This work surfaced in Arabic and some books of Diophantus we retain only from Arabic. The work on Polygonal numbers appears mostly lost. Thomas Harriot arrived at results related to these lapsed or misplaced doctrines.
The earliest introduction to figurate numbers to circulate widely in western Europe was ‘De institutione arithmeticae’ of Boethius (c. 500 ad). Largely based on an earlier work, the ‘Arithmetike isagoge’ of Nicomachus (c. 100 ad), Boethius’ text preserved some elementary Euclidean number theory, and in the later sections introduced figurate numbers: linear, triangular, square, pentagonal, hexagonal, and heptagonal, together with various kinds of pyramidal numbers, and even spherical numbers. 7 Boethius showed how each kind of number can be generated (beautiful coloured diagrams accompany many manuscript copies of his text), and he also gave examples of a few simple relationships, for example, that the sum of two consecutive triangular numbers is a square number. In fact he claimed through this and similar examples that the simple triangular numbers are the basis of all others. Harriot twice mentioned Boethius as a source of information on triangular numbers, 8 and it is possible that he had access to a manuscript copy of ‘De institutione’, but he also probably knew the detailed commentary on it published by Jacques Lefevre
— 9783037190593 pg 5
Citation Chain
9783037190593: Thomas Harriots doctrine of triangular numbers : the Magisteria magna European Mathematical Society, Heritage of European mathematics, 2009 Janet Beery; Jacqueline A Stedall; European Mathematical Society https://f005.backblazeb2.com/file/wwwmdf/Thomas_Harriot_Triangle_Numbers-9783037190593.pdf