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Thomas Harriot's Doctrine of Triangular Numbers: the 'Magisteria Magna'

Janet Beery and Jacqueline Stedall, editors
European Mathematical Society
Publication Date: 
Number of Pages: 
Heritage of European Mathematics
[Reviewed by
Fernando Q. Gouvêa
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In the early 1600s, Thomas Hariott wrote a treatise entitled “De numeris triangularibus et inde de progressionibus arithmeticis Magisteria Magna,”  which translates to something like “The Great Doctrine of triangular numbers and, through them, of arithmetic progressions.” The text was circulated to friends and other interested people in manuscript form. Even after Harriot’s death (in 1621) it remained unpublished, prompting Sir Charles Cavendish, in 1651, to express the hope that it would be published soon. Well, thanks to Janet Beery and the indefatigable Jacqueline Stedall, that has finally happened.

As Beery and Stedall explain in their useful introduction, this treatise is contained in 38 of the about 4,000 pages of Harriot manuscripts currently held in the British Library. Stedall has published the part of this material containing Harriot’s work on polynomial equations in The Greate Invention of Algebra. These pages, which don't quite add up to a “treatise,” were published, after Harriot’s death and with many additions and alterations, as Artis Analyticae Praxis, which has also been translated recently. The “Magisteria Magna” seems much more like a “finished product”: Harriot even gave it a title page. 

As in Stedall’s earlier edition, the manuscript pages are reproduced photographically. On facing pages, there is a description of the contents and some much needed discussion of what is going on. Most pages contain nothing but calculations and formulas, with no explanation. (Might explanations have been given orally?) One can certainly see the reason for Cavendish’s comment that “I doute I vnderstand it not all”!

Harriot studies triangular numbers and their generalizations, mostly by means of tables of finite differences, both numerical and symbolic. Using these, he obtains formulas for these numbers. He then proceeds to more general interpolation problems. The editors point out that these results were not surpassed until Newton, who obtained his results on polynomial interpolation in the 1660s.

In addition to describing the mathematical contents, the editors discuss the reception of this material by 17th century English mathematicians. The “Magisteria Magna” seems to have been copied out and shared by several mathematical practitioners of the time, who did not, however, say much about this mathematics in print. Such informal networks were a fairly common way of communicating mathematics in the 17th century, but of course this makes it very hard to track developments. Thus, historians of the period and those interested in the history of algebra and of analysis will find much here that they want to read and ponder.

There are far too few books that, like this one, reproduce, translate, and make accessible important mathematical works. Thomas Harriot’s Doctrine of Triangular Numbers is part of a series entitled Heritage of European Mathematics, published by EMS and distributed in the U.S. by the American Mathematical Society. I hope that this means that we will see more volumes of this kind in the future!

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. 


Thomas Harriot and the `Magisteria magna': A short chronology

Thomas Harriot's `Magisteria magna' and constant difference interpolation in the seventeenth century


De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna