This volume assembles ten studies of the life and work of Thomas Harriot (1560-1621). These are based on lectures that have been given annually at Oriel College, Oxford since 1990, by such authorities as Hugh Trevor Roper, David Quinn and John D. North. The contributions to Thomas Harriot. An Elizabethan man of science shed new light on all the main aspects of Harriot's life and stand as an important contribution to the re-evaluation of one of (...) the most gifted and intriguing figures in early modern British science. (shrink)

Thomas Harriot was the finest English mathematician before Isaac Newton, but his work on the coinage of his country is almost unknown, unlike Newton’s. In the early 1600s Harriot studied several aspects of the gold and silver coins of his time. He investigated the ratio between the values of gold and silver, using data derived from the official weights of the coins; he used hydrostatic weighing to determine the composition of the coins; and he studied the methods (...) used to calculate how metals should be combined in order to produce the required standard of purity. This work required not only arithmetical skill, but also great insight into the relationships between the various quantities, and here his ‘greate invention of algebra’ played its part. The gold and silver coins were critically important in the national economy at that time, because they had intrinsic value and represented the country’s store of wealth. In his lifetime Harriot was supported by the patronage of Henry Percy, the Earl of Northumberland, who was a member of the Privy Council in 1604 when great changes to the English coinage were being made. After Harriot’s death his papers remained under the control of one of his executors, Sir Thomas Aylesbury, who became Master of the Tower Mint in 1635. These facts suggest that his work on the coins was considered to be important for practical reasons, not just academic ones. In this article the interplay between the scholarly work and its application to matters of national significance will be analysed in depth. The conclusions also throw some light on the long-standing mystery surrounding the fate of Harriot’s papers. (shrink)

From the early eighteenth century onward, primacy for the invention of binary numeration and arithmetic was almost universally credited to the German polymath Gottfried Wilhelm Leibniz (1646–1716). Then, in 1922, Frank Vigor Morley (1899–1980) noted that an unpublished manuscript of the English mathematician, astronomer, and alchemist Thomas Harriot (1560–1621) contained the numbers 1 to 8 in binary. Morley’s only comment was that this foray into binary was “certainly prior to the usual dates given for binary numeration”. Almost thirty (...) years later, John William Shirley (1908–1988) published reproductions of two of Harriot’s undated manuscript pages, which, he claimed, showed that Harriot had invented binary numeration “nearly a century before Leibniz’s time”. But while Shirley correctly asserted that Harriot had invented binary numeration, he made no attempt to explain how or when Harriot had done so. Curiously, few since Shirley’s time have attempted to answer these questions, despite their obvious importance. After all, Harriot was, as far as we know, the first to invent binary. Accordingly, answering the how and when questions about Harriot’s invention of binary is the aim of this short paper. (shrink)

Some time in the late 1590s, the Welsh amateur mathematician John Bulkeley wrote to Thomas Harriot asking his opinion about the properties of a truly gargantuan (but totally imaginary) plano-spherical convex lens, 48 feet in diameter. While Bulkeley’s original letter is lost, Harriot devoted several pages to the optical properties of “Mr Bulkeley his Glasse” in his optical papers (now in British Library MS Add. 6789), paying particular attention to the place of its burning point. Harriot’s (...) calculational methods in these papers are almost unique in Harriot’s optical remains, in that he uses both the sine law of refraction and interpolation from Witelo’s refraction tables in order to analyze the passage of light through the glass. For this and other reasons, it is very likely that Harriot wrote his papers on Bulkeley’s glass very shortly after his discovery of the law and while still working closely with Witelo’s great Optics; the papers represent, perhaps, his very first application of the law. His and Bulkeley’s interest in this giant glass conform to a long English tradition of curiosity about the optical and burning properties of large glasses, which grew more intense in late sixteenth-century England. In particular, Thomas Digges’s bold and widely known assertions about his father’s glasses that could see things several miles distant and could burn objects a half-mile or further away may have attracted Harriot and Bulkeley’s skeptical attention; for Harriot’s analysis of the burning distance and the intensity of Bulkeley’s fantastic lens, it shows that Digges’s claims could never have been true about any real lens (and this, I propose, was what Bulkeley had asked about in his original letter to Harriot). There was also a deeper, mathematical relevance to the problem that may have caught Harriot’s attention. His most recent source on refraction—Giambattista della Porta’s De refractione of 1593—identified a mathematical flaw in Witelo’s cursory suggestion about the optics of a lens (the only place that lenses appear, however fleetingly, in the writings of the thirteenth-century Perspectivist authors). In his early notes on optics, in a copy of Witelo’s optics, Harriot highlighted Witelo’s remarks on the lens and della Porta’s criticism (which he found unsatisfactory). The most significant problem with Witelo’s theorem would disappear as the radius of curvature of the lens approached infinity. Bulkeley’s gigantic glass, then, may have provided Harriot an opportunity to test out Witelo’s claims about a plano-spherical glass, at a time when he was still intensely concerned with the problems and methods of the Perspectivist school. (shrink)

Recent influential criticisms attack the reputation of Thomas Harriot by citing the contents of his ethnographic and economic survey, A briefe and true report of the new found land of Virginia, first published in 1588. This interpretation makes Harriot, together with Shakespeare and others, agents of a colonialist project. But profound differences are indicated in the comparison of the relatively unbiased depiction and analysis by Harriot and his artist collaborator John White with the interpretations of America (...) and Americans by some of their contemporaries. A briefe and true report conveys, and John White's drawings support, insight into particular non-European features of the cultural and economic lives of the Algonquian peoples. On the contrary, Captain Arthur Barlowe's Discourse and the illustrations by Theodore de Bry for the 1600 edition of Harriot's book instance romanticizing, stereotyping, and unanalytic condescension very unlike Harriot's and White's. These differences prove the faultiness of the recent assessment. (shrink)

Summary This paper investigates the fate of Thomas Harriot's algebra after his death in 1621 and, in particular, the largely unsuccessful efforts of seventeenth-century mathematicians to promote it. The little known surviving manuscripts of Nathaniel Torporley have been used to elucidate the roles of Torporley and Walter Warner in the preparation of the Praxis, and a partial translation of Torporley's important critique of the Praxis is offered here for the first time. The known whereabouts of Harriot's mathematical (...) papers, both originals and copies, during the seventeenth century and later are summarised. John Wallis's controversial 1685 account of Harriot's algebra is examined in detail and it is argued that John Pell's influence on Wallis was far more significant than has previously been realised. The paper ends with a reassessment of Harriot's underrated and important contribution to the development of modern algebra. (shrink)

'The Greate Invention of Algebra' casts new light on the work of Thomas Harriot, an innovative thinker and practitioner in several branches of the mathematical sciences, including navigation, astronomy, optics, geometry, and algebra. Although on his death Harriot left behind over four thousand manuscript sheets, much of his work remains unpublished. This book focuses on one hundred and forty of Harriot's manuscript pages, those concerned with the structure and solution of equations. The original material has been (...) carefully ordered, translated, and annotated to provide the first complete edition of his work on this subject, and an extended introduction provides the reader with a lucid background to the work and explains its contents. Illustrations from the manuscripts provide fascinating reference material. The appendix discusses correlations between Harriot's manuscripts and the texts of his contemporaries Viète, Warner, and Torporley. The clear and concise exposition makes this an excellent reference volume for historians of mathematics and those interested in the history of science. This is an important new resource for understanding the development of algebra in seventeenth-century England. (shrink)

This is a book review-article. No abstract is required by me.

Torporley's ‘Congestor analyticus’, completed in 1627 in the library of the Earl of Northumberland at Petworth, was seen by Rigaud in the 1830s among the mathematical manuscript collection of the Earl of Macclesfield. Torporley's additional copy of the introductory part, preserved at Sion College, has been used for the present report. Torporley's prime objective was the presentation of some of Harriot's work. His first example concerns a classical problem in number theory. The complete solution, by an inductive process based (...) on tables of differences, is given in a few folios of the British Library collection of Harriot manuscripts. Torporley's transcription and commentary contrast forcibly today with the immediate illuminating impact of Harriot's original. (shrink)

In a manuscript summary, probably written in 1619, of his otherwise unknown earlier work on the oblique impact of elastic spheres, Thomas Harriot gives a largely ‘correct’ theory for their subsequent motion. He derives various consequences from his theory, but gives little indication of the observations or the first principles on which it may have been based. The text of the summary, and of some related fragments, is given.

The article investigates the role of symbolic means of knowledge representation in concept development using the historical example of medieval diagrams of change employed in early modern work on the motion of fall. The parallel cases of Galileo Galilei, Thomas Harriot, and René Descartes and Isaac Beeckman are discussed. It is argued that the similarities concerning the achievements as well as the shortcomings of their respective work on the motion of fall can to a large extent be attributed (...) to their shared use of means of knowledge representation handed down from antiquity and the Middle Ages. While the interpretation of medieval diagrams was unproblematic in the scholastic context from which they arose, in the early modern context, which was characterized by the confluence of natural philosophy and practical mathematics, it became ambiguous. It was the early modern mathematicians’ work within this contradictory framework that brought about a new conceptualization of motion which, in particular, eventually led to an infinitesimal concept of velocity. In this process, the diagrams themselves remained largely unchanged and thus functioned as a catalyst for concept development. (shrink)

Gottfried Wilhelm Leibniz is the self-proclaimed inventor of the binary system and is considered as such by most historians of mathematics and/or mathematicians. Really though, we owe the groundwork of today’s computing not to Leibniz but to the Englishman Thomas Harriot and the Spaniard Juan Caramuel de Lobkowitz, whom Leibniz plagiarized. This plagiarism has been identified on the basis of several facts: Caramuel’s work on the binary system is earlier than Leibniz’s, Leibniz was acquainted—both directly and indirectly—with Caramuel’s (...) work and Leibniz had a natural tendency to plagiarize scientific works. (shrink)

Giordano Bruno’s visit to Elizabethan England in the 1580s left its imprint on many fields of contemporary culture, ranging from the newly-developing science, the philosophy of knowledge and language, to the extraordinary flowering of Elizabethan poetry and drama. This book explores Bruno's influence on English figures as different as the ninth Earl of Northumberland, Thomas Harriot, Christopher Marlowe and William Shakespeare. Originally published in 1989, it is of interest to students and teachers of history of ideas, cultural history, (...) European drama and renaissance England. Bruno's work had particular power and emphasis in the modern world due to his response to the cultural crisis which had developed - his impulse towards a new ‘faculty of knowing’ had a disruptive effect on existing orthodoxies – religious, scientific, philosophical, and political. (shrink)

Critics of climate collapse and colonization in the Americas rightly identify the origin of these twin crises in early modern political theologies. They seek to combat these crises with new political theologies of nature that pay greater reverence to “native” peoples’ ecological knowledge. But in doing so, these critics subtly, perhaps unwittingly, recall elements of the colonial power they criticize. I explain why this is the case, examining Bartolomé de Las Casas’s use of naturales in his critiques of Spanish Conquest, (...) and Thomas Harriot’s use of naturall inhabitants in his writing on English colonization to describe “native” Americans. Both authors aimed to promote politico-theological reverence for “native” peoples and their relationships with “nature.” This set into motion a productive form of power operating in modern political theologies. This power works by legitimizing the European-Christian presence in the Americas through their ability to recognize, respect, and protect “native” relationships with “nature.”. (shrink)

ABSTRACT At the beginning of the 13th century Fibonacci described the rules for making mixtures of all kinds, using the Hindu-Arabic system of arithmetic. His work was repeated in the early printed books of arithmetic, many of which contained chapters on ‘alligation', as the subject became known. But the rules were expressed in words, so the subject often appeared difficult, and occasionally mysterious. Some clarity began to appear when Thomas Harriot introduced a modern form of algebraic notation around (...) 1600, and he was almost certainly the first to express the basic rule of alligation in algebraic terms. Thus a link was forged with the work on Diophantine problems that occupied mathematicians like John Pell and John Kersey in the 17th century. Joseph Fourier's work on mechanics led him to suggest a procedure for handling linear inequalities based on a combination of logic and algebra; he also introduced the idea of describing the set of feasible solutions geometrically. In 1898, inspired by Fourier’s work, Gyula Farkas proved a fundamental theorem about systems of linear inequalities. This topic eventually found many applications, and it became known as Linear Programming. The theorem of Farkas also plays a significant role in Game Theory. (shrink)

Between the appearance of Copernicus’ De Revolutionibus in 1543 and the works of Kepler and Galileo that appeared in 1609–10, there were probably no more than a dozen converts to physical heliocentrism. Following Westman we take this list to include Rheticus, Maestlin, Rothmann, Kepler, Bruno, Galileo, Digges, Harriot, de Zúńiga, and Stevin, but we include Gemma Frisius and William Gilbert, and omit Thomas Harriot. In this paper we discuss the reasons this tiny group of true Copernicans give (...) for believing that Copernicus, not Ptolemy or Tycho Brahe, was correct. We conclude that the early followers of Copernicus can be divided into two main groups, designated mathematicians and physicists. Two main arguments appear in the works of the former: first, the relationship between the velocity of a planet and its distance from the center of the world; second, the explanation of retrogradations. Early Copernicans differed on the reality of celestial spheres, and several attempted to reconcile heliocentrism with scripture. The mathematicians continued to accept a traditional definition of the scope of astronomy and the methods of science, both issues challenged by Kepler and Galileo. Hence the work of Copernicus set in motion a train of events that led to a decisive epistemological shift, but did not itself represent such a shift. The real revolution is the replacement of the methods and goals of Ptolemaic astronomy and Aristotelian physics with Copernicanism in its modern form, which incorporates the conceptual structure of Kepler and the astronomical evidence of Galileo. (shrink)

Francis Bacon could imagine the delights of the promised land of science, but he stands today as a propagandist rather than a practitioner. He was no Copernicus or Vesalius or Galileo; he was not even a Gabriel Harvey or a Thomas Harriot. As James Stephens observes in this study he was "never able to provide examples of the interpretation of nature" adequate to validate his claims. A great deal of his thought was taken up not with new experiments (...) but with the problem of how to express the new scientific philosophy in a way that would be intellectually rigorous and at the same time appealing to at least the better educated members of a general audience. Not unlike some modern analytic philosophers, he felt that language is one of the prime causes of human error. The use of language had been systematized for the ancients in Aristotle’s Rhetoric, with later contribution by Isocrates, Cicero, Quintilian, and Seneca, among others. If men’s minds were corrupted by false rhetoric, Bacon’s solution was to offer a true rhetoric. (shrink)

Thomas Sattig develops a novel philosophical picture of ordinary objects such as persons, tables, and trees. He carves a middle way between classical mereology and Aristotelian hylomorphism, and argues that objects lead double lives. They are compounds of matter and form, and each object's matter and form have different qualitative profiles.

The paper is about the interpretation of opaque verbs like “seek”, “owe”, and “resemble” which allow for unspecific readings of their (indefinite) objects. It is shown that the following two observations create a problem for semantic analysis: (a) The opaque position is upward monotone: “John seeks a unicorn” implies “John seeks an animal”, given that “unicorn” is more specific than “animal”. (b) Indefinite objects of opaque verbs allow for higher-order, or “underspecific”, readings: “Jones is looking for something Smith is looking (...) for” can express that there is something unspecific that both Jones and Smith are looking for. Given (a) and (b), it would seem that the following inference is hard to escape, if the premisses are construed unspecifically and the conclusion is taken on its under- specific reading: Jones is looking for a sweater. Smith is looking for a pen. Smith is looking for something Jones is looking for. (shrink)

Musing for Puncta special issue on "Critically Sick: New Phenomenologies Of Illness, Madness, And Disability.".

In the European Enlightenments it was often argued that moral conduct rather than adherence to certain theological doctrines was the true measure of religious belief. Thomas Ahnert argues that this characteristically “enlightened” emphasis on conduct in religion was less reliant on arguments from reason alone than is commonly believed. In fact, the champions of the Scottish Enlightenment were deeply skeptical of the power of unassisted natural reason in achieving “enlightened” virtue and piety. They advocated a practical program of “moral (...) culture,” in which revealed religion was of central importance. Ahnert traces this to theological controversies going back as far as the Reformation concerning the key question of early modern theology, the conditions of salvation. His findings present a new point of departure for all scholars interested in the intersection of religion and Enlightenment. (shrink)

In his Treatise on the Virtues, Aquinas discusses the character and function of habit; the essence, subject, cause, and meaning of virtue; and the separate intellectual, moral, cardinal, and theological virtues. His work constitutes one of the most thorough and incisive accounts of virtue in the history of Christian philosophy. John Oesterle's accurate and elegant translation makes this enduring work readily accessible to the modern reader.

This paper is about the semantic analysis of referentially opaque verbs like seek and owe that give rise to nonspecific readings. It is argued that Montague's categorization (based on earlier work by Quine) of opaque verbs as properties of quantifiers runs into two serious difficulties: the first problem is that it does not work with opaque verbs like resemble that resist any lexical decomposition of the seek ap try to find kind; the second one is that it wrongly predicts de (...) dicto (i.e. narrow scope) readings due to quantified noun phrases in the object positions of such verbs. It is shown that both difficulties can be overcome by an analysis of opaque verbs as operating on properties. This is a strongly modified version of a paper entitled lsquoDo We Bear Attitudes towards Quantifiers?rsquo that I have presented at conferences in Gosen (Gesellschaft für Sprachwissenschaft), Ithaca (SALT I), and Konstanz (Lexikon). I owe a special debt to Hans Kamp and Arnim von Stechow for shaping my views on the subject of this paper during the past ten years or so. Comments from and discussions with the following friends and colleagues have also led to considerable improvements: Heinrich Beck, Steve Berman, David Dowty, Veerle van Geenhoven, Fritz Hamm, Irene Heim, Wolfgang Klein, Angelika Kratzer, Michael Morreau, Barbara Partee, Mats Rooth, Roger Schwarzschild, Wolfgang Sternefeld, Emil Weydert, Henk Zeevat, and three referees. (shrink)