This book examines the ongoing (1655-1680) conflict between Thomas Hobbes and Oxford's Savilian Professor of Mathematics, John Wallis, over the squaring of the circle and related problems of Euclidian geometry. To the end of his long life, Hobbes asserted that he had succeeded in squaring the circle, and Wallis refuted him again and again. This "war" extended beyond mathematics, to, to each man's religious beliefs and politics.
What interested me most immediately was the cast: the people involved and their intellectual and personal relationships. Hobbes' first public involvement with squaring the circle, was his intervention in the dispute between the Danish mathematician Longoburg (Longomontanus) and John Pell. Interestingly, Hobbes demonstrated that Pell was correct in rejecting Longomontanus's claim to having squared the circle. Pell was author of an essay regarding establishment of a library of mathematical books and instruments. That essay was published as an appendix to his friend John Dury's The Reformed Librarian. Pell and Dury were part of the Hartlib circle, as was Theodor Haak, who reported Hobbes' role in the Pell/Longomontanus dispute to John Aubrey. After that we see Henry Oldenburg, secretary of the Royal Society, asking Hobbes to make up a list of mathematical texts. Presumably, the list was for Robert Boyle. In 1658 we find Thomas White and Kenelm Digby carrying communications between Fermat and Fenicle de Bessey in France and Lord Brouncker and Wallis in England. White, a Jesuit, had headed the English College at Douai; he had engaged Hobbes in debate over Galileo's Dialogues; and, at the end of a long life of intellectual distinction, he was caught up in the Popish Plot fabricated by Simon Oates. Etc.
It's worth describing the problem of squaring the circle, because it accounts for so much. Simply put: can a square be constructed that has the same area as a given circle, or vice versa? The key to the answer is in the word "constructed." By means of certain curves, such a square or circle can be defined. But those curves cannot be constructed using the classical instruments: ruler and compasses. Essentially, we're dealing with the transcendental character of pi, something that wasn't proved mathematically until the early 19th century.
Jesseph asks the question: why did Hobbes insist so long, to the end of his life, that he had squared the circle? Wallis refuted him again and again, but Hobbes hung on like a snapping turtle. Answering the question opens a window on the political, philosophical, social, and mathematical developments of the time. To summarize:
Hobbes's materialist philosophy brought him to support Isaac Barrow in Barrow's contention with Wallis over the primacy of geometry over arithmetic. These were the two components of classical mathematics, geometry dealing with continuous quantities and arithmetic with discrete. Barrow held that geometry was primary because it dealt with real physical reality, like distances and area, whereas numbers were abstractions: 1, 2, 3, etc. are concepts drawn from one egg, two brothers, three ships, and the like.
Hobbes' Erastianism--his contention in Leviathan that the king should be ruler over the religion of his subjects--placed him on the side of the Independents (congregationalists) against the Presbyterians in Protectorate religious politics. Hobbes, himself an atheist, felt that the Independents could be more easily be brought to heel by the King than the Presbyterians. Wallis was a Presbyterian, had even been Secretary of the Westminster Assembly.
Hobbes shared with the eduicational reformer John Webster and the Independent preacher William Dell a mistrust of the universities as temples to the scholasticism deriving from Aquinas. Wallis and his friend Seth Ward, Savilian Professor of Astronomy, were both actively involved in defending the universities from these attacks.
Of particular interest is the first of these issues. The debate between geometry and arithmetic came to the fore at this time because the distinction between the two was breaking down. Descartes (1637) and then Viete (1646) had defined analytical geometry. Napier (1620s?) had derived the logarithm from his work on the geometrical problem of the parallax. Cavalieri (1635) developed a "method of indivisibles": parallel slices of a geometric figure which, taken together, define the figure. Where this all is going, it seems to us in Whiggish hindsight, is toward notions of infinity, infinite series, and Newton's calculus.
I'll close by noting that Jesseph responds to Shapin and Schaffer, who, in their Leviathan and the Air-Pump, make "the success or failure of a scientific research program entirely independent of the truth or falsehood of the program's central claims." Jesseph calls their approach Wittgensteinian, but it was recognized when the book came out as post-modernist radicalism: science as pure social construct, without necessary basis in either the physical world or agreed canons of reason and method. The fact that Hobbes persisted so long in his claims is certainly less important than that after 1670 he was no longer taken seriously as a mathematician, except insofar as Wallis took the time and effort to refute him in the Transactions of the Royal Society.

On the surface, this book seems an unlikely candidate for my enthusiasm. It appears to be 500 pages of minutia. In fact, the author starts out by saying it is an expansion of a footnote to an early work on the relatively obscure Calvinist mathematician Wallis. How perfectly academic! If that doesn't put the book out of reach, look forward to reading about 1000 footnotes.

Regardless, I think this a great book. Once I got comfortable with the terms, I realized this might be construed as something much more interesting than the traditional 'discovery' of mathematical truths. I'm still not exactly sure how to characterize it, but I'm having fun thinking it a history of science fiction. After all, 'squaring the circle' is the 17th century equivalent to predicting the winner of the Kentucky Derby or tomorrow's change in the Dow Jones Industrials. I may be stretching a bit here, but there is clearly more on the plate than justifying 17th century mathematic revolutions with apples falling on the head of a reclining Newton.

Rather than placing the subject matter in purely mathematical terms, Jesseph considers his material in a wider context, one that makes room for Restoration style science fiction. Keep in mind that Newton's alchemy was an early form of teleportation and the monads of Leibnitz took advantage of an early warp drive. Despite overtly humiliating Hobbs for his mathematical errors, its clear that Jesseph finds Hobbs the ultimate winner. Hobbs suffers no more from his lapse of academic rigor than any contemporary science fiction author. And like Jules Verne, the spirit of his ideas has won if not his details. The political science advocated by Hobbs in Leviathan is hard to distinguish from contemporary standards. Hobbs advocated severely limiting the legal authority of church bishops, scientific materialism and the notion that good laws could produce a good society. Meanwhile, Wallis would be known as a dogmatic right-wing Christian fundamentalist. Further, most of us are convinced that science has 'solved' the problem of 'squaring the circle' which is all Hobbs was advocating, anyway.

And so, Jesseph does a subtle job of indicting my modern sensibility. Painting Hobbs to be the fool, he is actually pointing a finger at my happy secular humanism. Bravo!

So, why is squaring the circle so tricky?

In short, an attempt to find the circle which is exactly 1 square foot in area forces us to confront conflicting intuitions about how we prove the existence of 'real' objects. Try it out for yourself. It is something you can attempt with pencil and paper, or better... try it with a home computer. One of the great things about this book is that it lays out the mathematical issues clearly enough that anyone with high school algebra and maybe an ability to write an excel spreadsheet, can play the 17th century geometer and mathematician. I had a great time doing a 'quadrature' in Excel. If you are interested, I'll email you the spreadsheet (see users.htcomp.net/markmills).

In summary, I think Jessup's book fits into a broad, ongoing reappraisal of mathematical history. I can identify 3 trends, 'Squaring the circle belonging to the 3rd and most important of them:

1. Bringing non-western mathematics to western readers. The best of this is the ongoing research into ancient Chinese mathematics. See 'Chinese Mathematics: A Concise History', Li Yan, Du Shiran, John N. Crossley, Anthony W.-C. Lun, Shih-Jan Tu or 'Astronomy and Mathematics in Ancient China: The Zhou Bi Suan Jing', Christopher Cullen. In short, most of what the Europeans called 'new' math in the 1500s had been around for 500 years in China.

2. Finding a physiological basis for mathematic intuitions. Check out the cognitive research described by 'Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being', George Lakoff, Rafael E. Nunez, Rafael Nuñez. All you really need to read is the first chapter. The rest is rather speculative.

3. Reappraising the conventional myths about heroic ancient European mathematicians. Unless your ambition is a tenured job teaching the history of math, you will have a great time reading sensible inquiries into pre-modern western math. Take a look at 'Biographies of Scientific Objects , Lorraine Daston (Editor), or Pappus of Alexandria and the Mathematics of Late Antiquity, Serafina Cuomo.

One cannot help wondering why two scholars did engage in such a fiercely dispute for half of their intellectual life. John Wallis, Oxford's Savilian professor of geometry had no trouble in exposing time and again the mathematical short-commings in the preposterous claims of so ill equiped an amateur mathematician as Thomas Hobbes. But why was the battle so intense and why should it cover nearly a quarter of a century, producing hundreds of letters, books and publications? Even more bewildering is the fact that the subject of the debate, squaring the circle, was later proved to be impossible and that neither the claims of Hobbes nor the rebuttals of Wallis contributed anything significant to the field of mathematics. Fortunately, Douglas Jesseph provides adequate answers to these questions in this excellent work on 17th century history of mathematics. A study of these important controversies sheds light on the reform of mathematics in the 17th century and exposes the widely divergent philosophical conceptions of mathematics to which both antagonists adhered.

A first fundamental issue concerned Hobbes' materialistic foundation of mathematics where Wallis reasoned from the traditional account that mathematical facts do not depend on the structure of the material world. A second source of dispute was the completely different concept of ratios: Wallis defended that those ratios could only be applied to homogoneous quantities. Thirdly, the angle of contact between a circle and its tangent was a subject of wilful misunderstanding from the part of Wallis. Finally, the important 17th-century debate on infinitesimal small quantities was heavily criticised by Hobbes and although he did not develop an alternative, his objections to some of the obscurities of Wallis' arithmetic of infinities were well justified. These disputed foundations on the philosophy of mathematics were not the only sources of the irreconcilable conflict. Wallis and Hobbes also held opposing views on methodological issues such as the nature of demonstration and the centuries-old discussion on analytic and synthetic methods. For Hobbes all demonstration must arise from causes and as such he rejected techniques from algebra and analytic geometry in which one starts by assuming the truth of the proposition that is ultimately sought and deduces consequences from that assumption. In doing so he tossed aside the tools that might have helped him in his desperate attempt to make his mark as mathematician. Apart from diverging views on the fundaments of mathematics and methodological issues, religious and political positions play part in explaining the controversy. Wallis fitted neatly into the reformed tradition while Hobbes' religious opinions stood far apart. Hobbes was excluded from the newly established Royal Society for ideological and personal reasons and as such was deprived from a forum to respond to his critics. His political opinions and his vitriolic polemics at universities brought him into conflict with many important people and explain at least some of the vehemence with which the dispute was conducted. However, Jesseph refrains from pursueing sociological reductionist account and spends part of the last chapter convincingly demonstrating the inadequacy of a purely sociological explanation of the dispute. By uncovering the conceptual gulf dividing Hobbes and Wallis, Jesseph succeeds in demonstrating important differences in the philosophy of mathematics in the 17th century and explains why these two men engaged in such ferocious fight. Objectively, Hobbes miserably failed attempt at the solution of classic geometric problems makes him the loser of this dispute. But one can feel in this book also some sympathy for the consequent way in which Hobbes rigorously applied principles of his philosophy to mathematics and ultimately rejected classical geometry to avoid an even worse fate: the refutation of his own philosophy.

It's fun reading this collection of digressive informal anecdotes about famous (and some obscure) Englishmen. If you enjoyed "An Instance of the Fingerpost" (where some of thc characters appear) you'd like this. As a primary source for information it gets less reliable the further back it goes. Aubrey was born in 1626 so his accounts of Shakespeare and Elizathans are a generation removed, but he had met Harvey and Penn and had been through the Civil War and the rule of Cromwell.

Because its author never completed most of the entries for this biographical work, and never published it, what he did set down about his varied noble and ignoble subjects is uncensored, gossipy, perhaps unsubstantiated, and delightful. If you like browsing in Pepys' diary, or are fascinated by English life in the 17th century, this is the book to leave about for the occasional free moment.

Leo Strauss, in his first printed work, does a superb job of distilling Hobbes essential ideas on man. By showing us the educational background of Hobbes and the philosophers he read Strauss, shows how Hobbes comes to believe that "man acts out of fear of death". The preservation of life is the primary goal of man in the "Hobbesean" world. "Vanity is the force that makes men blind, fear is the force which makes men see".

Strauss points out that Hobbes started out as a classical political philosopher who was influenced by his readings of Aristotle and Plato. Not until Hobbes was forty years old and he discovered the works of Euclid did Hobbes move away from the "humanist" view to a more "scientific" approach to political philosophy. Euclid teaches Hobbes that politics must have a philosophy; it can't just be studied through history. Hobbes criticism of Aristotle and historism was; "that the historian takes the great as his standard, while the philosopher is concerned with the right and true". Hobbes is the first to see clearly between "right" and "law" the state is founded on the "right" "law" is a mere consequence. Strauss points out that, "Hobbes becomes the first philosopher to realize that politics can be raised to the rank of science".

This book is not an easy read for the casual reader but is essential for one to understand the concept of political philosophy.

I read somewhere that Strauss carried this book in a water-tight container when he crossed the channel to England so that, even if the ship went down, his work would survive. However that may be, it is the rare opportunity to see Strauss genuinely struggle with a problem. The prevailing opinion, I am told, is that Hobbes' science, or the discovery of Galileo's analytic-synthetic method, was the origin of Hobbes political philosophy (the analysis of the prevailing order (state of nature), the synthesis or construction of a new order (Leviathan)). Strauss makes the convincing argument that not the scientific method, but instead Aristotelian humanism (in particular, the anthropology of the Rhetoric) was the "source" for Hobbes' Staatslehre. Central to this is a critique of aristocratism, and the aristocratic valorization of courage. Not courage but cowardice and the fear of death is the ruling passion of the Hobbesian bourgeois (if Bloom learned anything from Strauss, it was that). In particular, Hobbes borrowed from the Rhetoric the treatment of anger, in which the most asocial human passion is the desire for revenge (and justice). Strauss later repudiated (in Natural Right) the argument against Hobbes scientism, but the point was made.

this book includes one of the most helpful accounts of the connection between the 'social contract'and the 'second discourse'in rousseau's philosophy that i've ever read. hoffman is clear and to the point, but manages, at the same time, to bring up original and insightful points about the rousseauian project. i highly recommend it.

When I read British philosophy as an undergraduate, I skimmed over Hobbes and focused primarily on Locke, Berkeley, and Hume. It was not until recently that I realized the importance of Hobbes's political thought. Therefore, I decided to read Hobbes's "Leviathan." Having previously discovered the outstanding little books in the "Past Masters" series published by the Oxford University Press, I first looked to see if the series included a title on Hobbes, and I found Tuck's book, which I read before reading "Leviathan." Tuck's "Hobbes" provided me with a good foundation for reading "Leviathan," and Tuck greatly increased my appreciation for Hobbes. Tuck is particularly careful to describe not only Hobbes's political philosophy; he also provides an introduction to Hobbes's thought regarding religion, science, ethics, and philosophical method. By gaining an overall picture of Hobbes's thought, I came to appreciate Tuck's claim that "Hobbes created English-language philosophy." I recommend this book to anyone approaching Hobbes for the first time.

One, if not the first, in a series of biographies of European philosophers by Cambridge University Press, this volume more than holds its own and is bound to becomne the standard text on the life of Thomas Hobbes.

Deftly written and extremely well researched, this is a volume not only for the scholar of English philosophy or history, but for the well-read layman as well. Martinich presents his subject chronologically, as any good biography should, with brief stopovers for analysis of each Hobbes text both philosophically and within the historical context against which it was written. Martinich is most unusual in that he does not take his own words as the last ones on the subject; there are pages on his disagreements with other writers on interpretations of both the life and thought of Hobbes, which makes this volume both unusual and valuable to any understanding of its subject.

Pricey, but strongly recommended, especially if one has any of the other volumes in the Cambridge series. If possible, wait for the paperback . . . but not too long, for there is much about Hobbes one will miss.

this accounts a principle line of arguementation in the christianization of hobbes.

This is one of the best books on Hobbes's political theory. Hampton is a clear author, and she does a fine job explaining both Hobbes's argument favoring a strong sovereign and where his argument falls short. A particular strength of this book is that she gives Hobbes a sympathetic reading but pulls no punches. The aim of the book is to take his most valuable ideas and to rework them to create a new and stronger social-contract theory.

Deborah Baumgold understands what so many Hobbes commentators have forgotten. Namely, that Hobbes wrote about politics and the conditions necessary to prevent elite competition wherein the many are sacrificed to the ambitiousness of the few. Examining, as no one else before her has, how Hobbes treats the state as constituted out of the many different roles that ordinary subjects and rulers are called upon to fill, Baumgold persuades us that Leviathan is, indeed, meant to protect the innocent by chastening the proud. Baumgold's Hobbes is political through and through, a point unfortunately obscured by those studies that would convert Hobbes into a methodological individualist or, even worse, a theorist of rational choice. The discussion of subjects and soldiers alone makes the book a treat.

The Rhetoric of Leviathan is a deep study about the systematic construction of Thomas Hobbes's political philosophy. The author leaves of an unusual analysis of the theory of history of young Hobbes and sustains, differently of classics like Leo Strauss, that effective rupture doesn't exist in uncoiling of the historical thought for the scientific-philosophical thought. Both base on the causal idea. Johnston notices Hobbes as the ideologist that indeed was. It demonstrates that the objective of its political theory was to do the common, ignorant and irrational man, to accept the idea of the state of nature, rational indicator of the need of an organized State. He sees the rhetoric of the political theory of Hobbes as a technique of cultural transformation that objectified to impede the sedition and the civil war in the England of seventy century. A fantastic and essential book.