De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna

Last updated

De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna is a 38-page mathematical treatise written in the early 17th century by Thomas Harriot, lost for many years, and finally published in facsimile form in 2009 in the book Thomas Harriot's Doctrine of Triangular Numbers: the "Magisteria Magna". Harriot's work dates from before the invention of calculus, and uses finite differences to accomplish many of the tasks that would later be made more easy by calculus.

Contents

De numeris triangularibus

Thomas Harriot wrote De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna in the early 1600s, and showed it to his friends. [1] By 1618 it was complete, [2] but in 1621 Harriot died before publishing it. Some of its material was published posthumously, in 1631, as Artis analyticae praxis, but the rest languished in the British Library among many other pages of Harriot's works, [1] and became forgotten until its rediscovery in the late 1700s. [2] It was finally published in its entirety, as part of the 2009 book Thomas Harriot’s Doctrine of Triangular Numbers: the "Magisteria Magna". [1]

The title can be translated as "The Great Doctrine of triangular numbers and, through them, of arithmetic progressions". [1] Harriot's work concerns finite differences, and their uses in interpolation for calculating mathematical tables for navigation. [2] Harriot forms the triangular numbers through the inverse process to finite differencing, partial summation, starting from a sequence of constant value one. Repeating this process produces the higher-order binomial coefficients, which in this way can be thought of as generalized triangular numbers, and which give the first part of Harriot's title. [3]

Harriot's results were only improved 50 years later by Isaac Newton, and prefigure Newton's use of Newton polynomials for interpolation. [1] [4] As reviewer Matthias Schemmel writes, this work "shows what was possible in dealing with functional relations before the advent of the calculus". [3]

The work was written as a 38-page manuscript in Latin, and Harriot wrote it up as if for publication, with a title page. However, much of its content consists of calculations and formulas with very little explanatory text, [1] [4] leading at least some of Harriot's contemporaries such as Sir Charles Cavendish to complain of the difficulty of understanding it. [1]

Thomas Harriot’s Doctrine

The monograph Thomas Harriot’s Doctrine of Triangular Numbers: the "Magisteria Magna", edited by Janet Beery and Jackie Stedall, was published in 2009 by the European Mathematical Society in their newly created Heritage of European Mathematics series. Its subject is De numeris triangularibus, and the third of its three sections consists of a facsimile reproduction of Harriot's manuscript, with each page facing a page of commentary by the editors, [1] [2] including translations of its Latin passages. [3] The earlier parts of Beery and Stedall's book survey the material in Harriot's work, the context for this work, the chronology of its loss and recovery, and the effect of this work on the 17th-century mathematicians who read it. [2] [4]

Although reviewer Matthias Schemmel suggests that the 2009 monograph is primarily aimed at historians of mathematics, who "will welcome this book as providing new insights into the development of mathematics", he suggests that it may also be of interest to other mathematicians and could perk their interest in the history of mathematics. [3]

Related Research Articles

Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.

<span class="mw-page-title-main">Discrete mathematics</span> Study of discrete mathematical structures

Discrete mathematics is the study of mathematical structures that can be considered "discrete" rather than "continuous". Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics".

<span class="mw-page-title-main">Isaac Newton</span> English mathematician and physicist (1642–1727)

Sir Isaac Newton was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author who was described in his time as a natural philosopher. He was a key figure in the Scientific Revolution and the Enlightenment that followed. His pioneering book Philosophiæ Naturalis Principia Mathematica, first published in 1687, consolidated many previous results and established classical mechanics. Newton also made seminal contributions to optics, and shares credit with German mathematician Gottfried Wilhelm Leibniz for developing infinitesimal calculus, though notably he developed calculus well before Leibniz. He is considered one of the greatest and most influential scientists in history.

<span class="mw-page-title-main">Nonstandard analysis</span> Calculus using a logically rigorous notion of infinitesimal numbers

The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.

A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.

In mathematics, the term linear function refers to two distinct but related notions:

<span class="mw-page-title-main">Infinitesimal</span> Extremely small quantity in calculus; thing so small that there is no way to measure it

In mathematics, an infinitesimal number is a quantity that is closer to 0 than any standard real number, but that is not 0. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence.

In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method.

The Analyst is a book by George Berkeley. It was first published in 1734, first by J. Tonson (London), then by S. Fuller (Dublin). The "infidel mathematician" is believed to have been Edmond Halley, though others have speculated Sir Isaac Newton was intended.

Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Infinitesimal calculus was developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other. An argument over priority led to the Leibniz–Newton calculus controversy which continued until the death of Leibniz in 1716. The development of calculus and its uses within the sciences have continued to the present day.

<span class="mw-page-title-main">Thomas Harriot</span> English astronomer, mathematician, ethnographer and translator

Thomas Harriot, also spelled Harriott, Hariot or Heriot, was an English astronomer, mathematician, ethnographer and translator to whom the theory of refraction is attributed. Thomas Harriot was also recognized for his contributions in navigational techniques, working closely with John White to create advanced maps for navigation. While Harriot worked extensively on numerous papers on the subjects of astronomy, mathematics and navigation, he remains obscure because he published little of it, namely only The Briefe and True Report of the New Found Land of Virginia (1588). This book includes descriptions of English settlements and financial issues in Virginia at the time. He is sometimes credited with the introduction of the potato to the British Isles. Harriot invented binary notation and arithmetic several decades before Gottfried Wilhelm Leibniz, but this remained unknown until the 1920's. He was also the first person to make a drawing of the Moon through a telescope, on 5 August 1609, about four months before Galileo Galilei.

Johan Frederik Steffensen was a Danish mathematician, statistician, and actuary who did research in the fields of calculus of finite differences and interpolation. He was professor of actuarial science at the University of Copenhagen from 1923 to 1943. Steffensen's inequality and Steffensen's method are named after him. He was an Invited Speaker at the 1912 International Congress of Mathematicians (ICM) in Cambridge, England and at the 1924 ICM in Toronto.

... His more important works included the theory of statistics (1923), interpolation (1925), insurance mathematics (1934) and the calculation of interest (1936). ... He was President of the Danish Actuarial Society in 1922-24 and 1930-33, and of the Danish Mathematical Society in 1930-36 ...

<span class="mw-page-title-main">Timeline of calculus and mathematical analysis</span> Summary of advancements in Calculus

A timeline of calculus and mathematical analysis.

<span class="mw-page-title-main">Infinity</span> Mathematical concept

Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .

Brahmagupta's interpolation formula is a second-order polynomial interpolation formula developed by the Indian mathematician and astronomer Brahmagupta in the early 7th century CE. The Sanskrit couplet describing the formula can be found in the supplementary part of Khandakadyaka a work of Brahmagupta completed in 665 CE. The same couplet appears in Brahmagupta's earlier Dhyana-graha-adhikara, which was probably written "near the beginning of the second quarter of the 7th century CE, if not earlier." Brahmagupta was one of the first to describe and use an interpolation formula using second-order differences.

<span class="mw-page-title-main">Continuous or discrete variable</span> Types of quantitative variables in mathematics

In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by measuring or counting, respectively. If it can take on two particular real values such that it can also take on all real values between them, the variable is continuous in that interval. If it can take on a value such that there is a non-infinitesimal gap on each side of it containing no values that the variable can take on, then it is discrete around that value. In some contexts a variable can be discrete in some ranges of the number line and continuous in others.

<span class="mw-page-title-main">Jackie Stedall</span> British mathematics historian

Jacqueline Anne "Jackie" Stedall was a British mathematics historian. She wrote nine books, and appeared on radio on BBC Radio 4's In Our Time programme.

Janet Lynn Beery is an American mathematician and historian of mathematics who serves as a professor of mathematics and computer science at the University of Redlands. She also served as the editor-in-chief of mathematics history journal Convergence from 2009 to 2019, and has authored a book on the mathematics of Thomas Harriot.

Muriel Seltman was British left-wing activist, mathematics educator, historian of mathematics, and author of books on mathematics, religion, politics, and philosophy.

References

  1. 1 2 3 4 5 6 7 8 Gouvêa, Fernando Q. (March 2009), "Review of Thomas Harriot's Doctrine of Triangular Numbers", MAA Reviews, Mathematical Association of America
  2. 1 2 3 4 5 mbec (May 2011), "Review of Thomas Harriot's Doctrine of Triangular Numbers", EMS Reviews, European Mathematical Society
  3. 1 2 3 4 Schemmel, Matthias (September 2010), "Before calculus (review of Thomas Harriot's Doctrine of Triangular Numbers)", Notes and Records of the Royal Society of London, 64 (3): 303–304, doi:10.1098/rsnr.2010.0016, JSTOR   20753908, S2CID   202575019
  4. 1 2 3 Shea, William R. (2010), "Review of Thomas Harriot's Doctrine of Triangular Numbers", Mathematical Reviews , MR   2516550
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.