Dicuil (9th century) on triangular and square numbers

Abstract

Dicuil was a ninth-century Irish monk who taught at the Carolingian school of Louis the Pious. He wrote a Computus or astronomical treatise in Latin in about 814–16, which contains a chapter on triangular and square numbers. Dicuil describes two methods for calculating triangular numbers: the simple method of summing the natural numbers, and the more complex method of multiplication, equivalent to the formula n(n + 1)/2. He also states that a square number is equal to twice a triangular number minus the generating number, equivalent to n² = 2[n(n + 1)/2] – n. The multiplication formula for triangular numbers was first explicitly described in about the third century AD by the Greek authors Diophantus and Iamblichus. It was also known as a solution to other mathematical problems as early as 300 BC. It reappeared in the West in the sixteenth century. Dicuil thus fills a gap in our medieval knowledge.

Acknowledgements

Helen Ross is grateful to Meg Bateman for introducing her to Dicuil; Gavin J.S. Ross for suggesting Diophantus as a source; David J Murray for suggesting Wertheimer’s experiments; Ellen Beard, Donald Smith and Roger Webster for many ideas; several anonymous authors on the internet, and one anonymous referee, for useful leads.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Helen Elizabeth Ross http://orcid.org/0000-0003-0800-5474