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umber nor a magnitude if from the greater there be a magnitude greater that its half and from that which is left a magnitude greater than its half and if this process be repeated continually there will be left some magnitude which is less than the lesser magnitude set out 3 source 4 either the axiom of archimedes or the proof of the method of exhaustion or both are sometimes credited to hippocrates of chios which given his work on squaring the circle and his demonstration of the squaring of the lune is not unlikely but there is no actually proof of this euclid brings another five examples of proof using the method of exhaustion in book xii of the elements proposition 5 the volumes of two tetrahedra of the same height are proportional to the areas of their triangular bases proposition 10 the volume of a cone is a third of the volume of the corresponding cylinder which has the same base and height proposition 11 the volume of a cone or cylinder of the same height is proportional to the area of the base proposition 12 the volume of a cone or cylinder that is similar to another is proportional to the cube of the ratio of the diameters of the bases proposition 18 the volume of a sphere is proportional to the cube of its diameter all five are thought to be by eudoxus you won t however find the name method of exhaustion there as it was first coined by the jesuit mathematician grégoire de saint vincent 1584 1667 in his opus geometricum quadraturae circuli et sectionum coni decem libris comprehensum jan van meurs jacob van meurs antwerp 1647 the method having been revived by nicolo tartaglia 1499 1557 in his edition of archimedes in 1543 opera archimdedes syracusani philosophi et mathematici ingeniosissimi archimedes having extensively used the method as we will see in the next episode the method of exhaustion is regarded as an early predecessor to integral calculus and although there are very strong similarities and the basic concept is common to both there are significant differences the method of exhaustion is entirely geometrical and even goes so far as to avoid numbers completely being based on magnitudes which are non numerical integral calculus is arithmetical and embraces the entire set of real numbers which of course includes the incommensurable numbers that caused eudoxus to turn to magnitudes secondly the method of exhaustion avoiding infinity does not have a limit concept one can reiterate the process as often as one will increasing the accuracy of the approximation but one can never take it to the conclusion the limit concept is totally central to integral calculus and would as will see much later in this series cause several generations of mathematicians a lot of problems 1 euclid the thirteen books of the elements translated with introduction and commentary by sir thomas heath volume 2 dover 1956 p 114 2 euclid the thirteen books of the elements translated with introduction and commentary by sir thomas heath volume 2 dover 1956 pp 371 374 3 carl b boyer the history of calculus and its conceptual development the concepts of the calculus 4 euclid the thirteen books of the elements translated with introduction and commentary by sir thomas heath volume 2 dover 1956 p 12 8 comments filed under history of mathematics july 23 2026 7 50 am reading the mind of god i make no secret of the fact that i think johannes kepler was a far more significant scientist at the beginning of the seventeenth century than galileo galilei and that he contributed far more to the development of modern science in that century second only in this aspect to newton despite this the literature about kepler is but a fraction of that about galileo or newton the title of this post is the title of a recent volume on kepler s life and work which helps to redress the balance the full title of the book is reading the mind of god johannes kepler and the reform of astronomy it was edited by a e l davis j v field and t j mahoney and published in 2024 by springer and the royal astronomical society whose librarian sian prosser who knowing my interest in kepler was kind enough to send me a copy the book has a long and interesting genesis in 2009 during the international year of astronomy which was declared to celebrate the four hundredth anniversary of galileo s telescopic astronomical discoveries and the publication of kepler s astronomia nova at the general assembly of the international astronomical union in rio de janeiro a session was held to celebrate kepler s life and work following the session the participants set up the johannes kepler working group jkwg who decided to publish a multi authored conspectus of research on kepler aimed at astronomers and undergraduate students who would normally not read the specialist history of science publications this book is that conspectus in what follows i will make brief comments about each of the articles it contains the book opens with introduction kepler s place in the history of science which gives brief sketches of those areas to which he made significant contribution starting with astronomy and optic followed by a brief discussion of his various contributions to mathematics music and astrology reform covers kepler s attitude to research the introduction closes with a brief sketch of each of the following chapters followed by a brief note on the calendar reform the first paper is kepler religion and natural philosophy a theological biography by charlotte methuen methuen s paper explains the title of the book this is fundamental to understanding kepler and the motivation behind his entire scientific endeavours he was deeply religious his whole life and he came to believe that his god was not just a mathematician but he was mathematics and his creation was a totally logical mathematical structure kepler believed he was paying homage to his god by revealing the structure of that creation this is an excellent brief introduction to kepler s theological beliefs i did however miss one thing there is a widespread myth that kepler was studying theology in tübingen and he had to break off his theology studies to take up his first teaching post in graz i think that max casper in the first modern biography of kepler created this myth in an earlier paper methuen showed that kepler took theology courses to prepare him for his future posting as either a school teacher or a village pastor but that he was never registered as a theology student methuen does not include this information in this paper the second paper is kepler s cosmology by j v field which ties into the first as i have just explained kepler s cosmology was driven by his theological beliefs field gives a good analysis of the mysterium cosmographicum kepler s first book from 1596 which launched his career and his attempt to analyse the mathematical cosmos of his mathematical god and its sequel harmonice mundi a wonderful synthesis of mathematics music astrology and astronomy already planned in 1596 but first finished and published twenty two years later in 1618 field emphasises the importance of these two text in understanding kepler s entire work proceeding chronologically through kepler s life we now get introduced to tycho brahe in measuring the heavens how tycho brahe revolutionised observational astronomy by t j mahoney this s a very good forty five page synopsis of tycho s life and work which closes with kepler coming to prague to work with tycho tycho of course delivered the data with out of which kepler created his revolutionary reform of astronomy that reform is the subject of the next paper kepler s discovery of the planetary orbit the goldilocks solution by a e l davis i should note that davis was one of the editors of this volume but he unfortunately died before the book was published one should not speak ill of the dead but this is one of the chapters that i have mixed feelings about although davis explanation of kepler s twisted path to the first two of his laws is technically first class i have the feeling that for the readers that this book is aimed at it is too technical and for the non expert hard to follow we stay with the astronomia nova for the next very short paper the translation of the title of kepler s astronomia nova by andrew gregory the full title of kepler s master piece is astronomia nova αιτιολογητος seu physica coelestis tradita commentariis de motibus stellae martis ex observationibus g v tychonis brahe and it is the greek αιτιολογητος in the middle of that title that concerns gregory apparently what kepler intended to say with this word here is more than somewhat ambiguous fascinating and new to me but is it really necessary in a general volume on kepler s life and work we turn from astronomy to astrology with kepler and the reform of astrology by sheila j rabin this is a very good short summary on the status of astrology and the role that it played during the renaissance which leads into the role that astrology played in the life and work of kepler from astrology we proceed to optics an area of science where kepler played a very major role with a chapter on kepler s work on optics by w h donahue donahue is probably the expert on kepler s optics and the paper is as good as one would expect if i have a complaint it is that it could have been longer j v field returns this time with a paper kepler and galileo her coverage of the early part of their somewhat strange relationship is good and the final part on the comet debate post 1618 is good as far as it goes however i have two serious objections firstly field treats the debate on the status of comets su or superlunar as if there were two equal parties debating like everybody else she completely ignores the fact that clavius the leading catholic ptolemaic astronomer had already in the sixteenth century clearly accepted that the supernova of 1572 and the great comet of 1577 were supralunar he continued to confirm the supralunar status of the supernova of 1604 and the various post 1577 comets this meant that the official catholic stance on comets was that they were supralunar secondly she like everybody else emphasises that galileo s il saggiatore contains his empirical observation and mathematics based modern methodology however in the debate triggered by the great comet of 1618 both of the jesuit astronomers johann baptist cysat and orazio grassi followed that methodology empirical observations using the telescope the instrument that launched galileo s fame which were carefully mathematically analysed producing the correct scientific result galileo s theory was an unsubstantiated piece of waffle dreamt up on the spur of the moment rather than admit that grassi was right kepler of course as field correctly states supported tycho s analysis that comets were supralunar whereas galileo was simply wrong the next paper the long life of the rudophine tables once again by j v field is quite simply excellent the next paper is johnes kepler the kepler spacecraft and transits by jay m pasachoff takes off in a different direction it starts with the role that kepler s rudolphine tables played in the first observations of transits of the inner planets venus and mercury then moves into a brief synopsis of the history of transit observations culminating with the most modern observations using space telescopes and satellites also covering nasa s kepler mission to search for exoplanets we now turn to kepler s contributions to mathematics by eberhard knobloch knobloch is an excellent historian of mathematics and his analysis of kepler s various mathematical endeavours is first class however i miss something and that is context kepler was not a pure mathematician who did mathematics for its own sake but his mathematics was always developed to fulfil a particular purpose in his scientific activities knobloch doesn t mention kepler s reasons for his contributions to a particular area at all for example knobloch delivers an excellent analysis of the mathematics in his new stereometry and how it functions as a precursor to infinitesimal mathematics and the calculus but never mentions why kepler devoted his only mathematics book to the seemingly obscure topic of determining the volumes of wine barrels most modern readers would not be aware that gauging as this determination is known was one of the major branches of practical mathematics in the sixteenth and seventeenth centuries wine was transported over long distances and in the process often crossed many borders and at each border the trader had to pay customs duty it order to calculate the duty due the customs officials had to determine or gauge the volume of the barrel hence the discipline of gauging kepler became interested in the topic when he himself transported a shipment of wine across several borders we now turn to kepler s posthumously published book that he originally wrote as a student and then expanded throughout his life kepler s dream and lunar astronomical phenomena by jarosław wloarczyk which gives a good explanation of how kepler intended it to be an illustration of a heliocentric system the final paper in the book is once again w h donahue this time on translating kepler donahue has published translations of both kepler s pars optica and the astronomia nova as someone who has worked as a translator of scientific texts i found donahue s paper really excellent in particular he gives a detailed analysis as to why there were so few translations of kepler s works before the twentieth century a truly eye opening analysis the book closes with a three and a half page chronology of kepler life and work followed by a fairly extensive glossary there is also a very good index each paper has footnotes which largely refer to the extensive bibliographies at the end of each paper there are black and while illustration at the appropriate places throughout the text this book is an important and useful addition to the available kepler literature and definitely becomes the renaissance mathematicus seal of approval the only negative point is that the original price of 96 29 places it outside the budget of most undergraduates however the hardback is currently discounted and available for 23 99 with free shipping direct from springer 14 comments filed under book reviews july 15 2026 12 56 pm take it to the limit one more time a history of calculus i i in the previous episode i presented what i have always taken to be euclid s proof that there is no greatest prime number an elegant reduction ad absurdum proof over the decades i must have seen it present as such in at least a dozen different sources including as an example of a reductio ad absurdum proof an illustration of the idea i have never thought to question this i presented it thus in the elements euclid fl 300 bce also avoids real infinity presenting only potential infinity people often refer to his famous proof that there are infinite prime numbers but he doesn t prove that rather euclid in a w...
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