zbMATH — the first resource for mathematics

Thomas Harriot’s doctrine of triangular numbers: The “Magisteria Magna”. (English) Zbl 1168.01001
Heritage of European Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-059-3/hbk). v, 135 p. (2009).
Bei Interpolation denkt man wohl an Isaac Newton. Hier geht es jedoch um eine Vorgeschichte, die zumindest für Interessierte am Kontinent neu ist: “By about 1614, Thomas Harriot (1560–1621) had developed finite difference interpolation methods to aid the construction of navigational tables. In 1618 (or slightly later) he composed a treatise entitled ‘De numeris triangularibus et inde de progressionibus arithmeticis, Magisteria magna’, in which he derived symbolic interpolation formulae and showed how to use them.” (Preface); zugänglich in Add MS 6782, pp. 107–144, British Library, “…which has survived amongst the several hundred pages of mathematical work left behind …” (p. 3).
In Thomas Harriot’s ‘Magisteria magna’ and constant difference interpolation in the seventeenth century wird gefolgert: “One has to penetrate only a little way to recognize that there is some very interesting mathematics here, beautifully and concisely written, but it is not immediately easy to see what Harriot was aiming at as he moved apparently so effortlessly from one page to the next, and only in the later stages does the purpose of the treatise become clear. Harriot wrote for himself and a few close friends, who would have already known the context and reason for the work […] this introductory essay offers an overview of the structure and aims of the ‘Magisteria’ and of its influence on Harriot’s contemporaries and successors.” (p. 3), wobei (p. 3f.):
1) Harriot’s relevante Formulierungen bewegen sich nahe moderner algebraischer Schreibweise.
2) Die hierbei verwendeten Methoden “were the subject of discussion and research amongst English mathematicians […] from 1610 to 1670”; so arbeiteten hieran – später freilich ebenfalls unbeachtet – Nathaniel Torporley, Walter Warner, John Pell, John Collins und Nicolaus Mercator, “before or independently of Newton’s rediscovery of them in 1665. […] A topic of enduring interest in early seventeenth-century English mathematical circles has therefore until now passed largely unnoticed.” 3) Diese neuen Ideen wurden innerhalb kleiner Gruppen “through verbal explanation and discussion” ausgetauscht, von Generation zu Generation ungedruckt tradiert: “In studying Harriot’s ‘Magisteria’, therefore, we uncover not only a fascinating and little known piece of seventeenth-century mathematics but also a network of connections and communications between friends and acquaintances, extending in time for over half a century.”
Das aufwendig gestaltete und auch deshalb nicht gerade billige Buch ist deutlich gegliedert. Passend werden jeweils historische Fakten eingeflochten: In Triangular numbers wird die Vorgeschichte bei Boetius, Jacques Lefevre d’Etaples, Francisco Maurolico, Michael Stifel und Girolamo Cardano, auch mit Blick auf gedankliche Vorgänger, angesprochen: “Thus Harriot knew both from existing writings and from his own investigations that the general triangular numbers appear both in binomial expansions and in calculations of combinations. In his ‘Magisteria’ he was to find yet another use for them.” (p. 7) – In Harriot’s difference method werden gemäß “Harriot’s theory was concerned with any sequence of numbers that has constant first, second, third, or higher differences.” (p. 7) die Schemata einzelner “difference tables” aufgebaut und im Vergleich mit dem Original diskutiert; die ohne jede erläuternde Bemerkung verfaßten Magisteria sind, ausführlich kommentiert, geschlossen im Anhang abgebildet.
Harriot zog bereits allgemeine Zahlen, verständliche Bezeichnungen und Formeln heran, folglich ist sein schrittweises Vorgehen hilfreich für den Überblick; er verwendete zudem spezielle Symbole für die Grundrechenarten und führte kombinatorische Begriffe ein: “Such algebraic expressions were in themselves a remarkable achievement, the first ever examples of \(n\)th-term formulae.” (p. 10) Vermittels dieser Hilfsmittel gelang ihm der bereits großenteils allgemeine Aufbau seiner “difference tables” – siehe entsprechende Umrechnungsbeziehungen (pp. 10 und 11): “Suppose we are given a table of numbers […], where the third difference is assumed constant, and the first entries in each column are denoted by A, B, C, D.” (p. 10) –, und er beendete seine wohl zwischen 1611 und 1618 (pp. 13; 11) entstandenen Magisteria “with some worked examples in which they were put to use” (p. 11). – The origins and ‘many uses’ of Harriot’s method: “The interpolation of trigonometric tables was clearly one possible use of Harriot’s difference method. […] Harriot also worked for many years on the calculation of meridional parts.” (p. 15). Zu seinen Methoden: “…by 1614 he had derived algebraic formulae that allowed him to apply the method very efficiently to the interpolation of numerical tables.” (p. 17) “Another of Harriot’s uses of the difference method appears also to have been related to his experimentation with polynomials […] This is his discovery of formulae for sums of squares, or cubes, or higher powers.” (p. 19) “ Harriot used the same method to derive formulae for sums of consecutive numbers, squares, and fourth powers, and clearly he could have extended it to higher powers if he had wanted to.” (p. 20)
In The influence of the ‘Magisteria’ wird das Wirken damaliger Fachleute herausgestellt und das Fortwirken von Thomas Harriot’s “interpolation method, though never published, […] by some of his closest friends and was to remain in use amongst English mathematicians until the 1660s.” (p. 20) einschließlich vieler persönlicher Details aufgezeigt: Torporley, “who shared and understood Harriot’s mathematical interests better than anyone” (p. 20); “…Torporley nevertheless reproduced Harriot’s formulae.” (p. 23)
Zur Interpolation: “Briggs’ answers are identical to those that would be obtained using Harriot’s formulae, but his method is a little different, at first sight less theoretical and rather more intuitive.” (p. 26) “We simply do not know how or when Briggs devised his various interpolation methods […] All we can say with certainty is that Harriot was using constant difference interpolation methods at least ten years before an example appeared in print in Briggs’ Arithmetica logarithmica in 1624.” (p. 29) – “It fell to Warner to edit Harriot’s work on equations, and in 1631 he produced the Artis analyticae praxis, the only publication of any of Harriot’s mathematics until modern times.” (p. 29) – “Though he never displayed any great skill in mathematics, Sir Charles Cavendish took a keen interest in it from the 1620s onwards. He knew about the publication of Harriot’s manuscripts […]” (p. 37). – Nachdem offensichtliche Querverbindungen zu Pell und Mercator aufgezeigt wurden, gelangt sowohl die autarke Methode Newtons zur Sprache – “The integer columns […] contain binomial coefficients for the expansion of \((1+x)^n\), for \(n\) a positive integer, and Newton assumed that the interpolated entries gave him the corresponding coefficients when n is a fraction.” (p. 50) –, als auch: “By the end of 1670 James Gregory too had arrived at a form of what is now called the Newton–Gregory interpolation formula, but unlike Newton who left plenty of manuscript evidence, Gregory left little in the way of explanation.” (p. 51), wobei “…Gregory in 1670 did not yet know of Newton’s work, and worked out the formulae for himself, as Harriot had done half a century earlier.” (p. 51) – Der Kommentar endet: “.. we can say without any doubt that by 1611 or soon afterwards Harriot had worked out the essential ideas of an interpolation theory that remained in use for half a century or more, and not until the arrival of Newton and Gregory was it finally surpassed.” (p. 52)
Zu dieser beachtenswerten, allerdings nicht leicht lesbaren Darstellung wurden Original- und reichlich Sekundärliteratur herangezogen.

01-02 Research exposition (monographs, survey articles) pertaining to history and biography
01A40 History of mathematics in the 15th and 16th centuries, Renaissance
Biographic References:
Harriot, Thomas
Full Text: DOI