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Sophia Rare Books

An Investigation of the Laws of Thought

An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities

BOOLE, George First edition, first issue (see below). In this main work of Boole's he gave the first proper presentation of Boolean algebra - "Boole invented the first practical system of logic in algebraic form, which enabled more advances in logic to be made in the decades of the nineteenth century than in the twenty-two centuries preceding. Boole's work led to the creation of set theory and probability theory in mathematics, to the philosophical work of Peirce Russell, Whitehead, and Wittgenstein, and to computer technology via the master's thesis of Claude Shannon, who recognized that the true/false values in Boole's two-valued logic were analogous to the open and closed states of electric circuits." (Hook & Norman, Origins of Cyberspace, no. 224). ?OOC 224 (1st issue, re-backed); Erwin Tomasch B198 (2nd issue); Haskell Norman 266 (3rd issue). "Since Boole showed that logics can be reduced to very simple algebraic systems - known today as Boolean Algebras - it was possible for Babbage and his successors to design organs for a computer that could preform the necessary logical tasks. Thus our debt to this simple, quit man, George Boole, is extraordinarily great. . His remark about a 'special law to which the symbols of quantity are not subject' is very important: this law in effect is that x2=x for every x in his system. Now in numerical terms this equation or law has as its only solution 0 and 1. This is why the binary system plays so vital a role in modern computers: their logical parts in effect carrying out binary operations. In Boole's system 1 denotes the entire realm of discourse, the set of all objects being discussed, and 0 the empty set. There are two operations in this system which we may call + and &infty; or we may say or and and. It is most fortunate for us that all logics can be comprehended in so simple a system, since otherwise the automation of computation would probably not have occurred - or at least not when it did" (Goldstine: The Computer from Pascal to von Neumann, pp.37-38). This is the rare first issue. "The probable first issue of Boole's Laws of Thought, of which the Origins of Cyberspace copy is an example, has the errata leaf bound in the back, and a binding of black zigzag cloth with blindstamped border, panel, lozenge, and corner- and side-ornaments. The probable second issue has the errata leaf following the last numbered leaf of preliminaries, an additional printed 'Note' leaf following page 424 concerning a complex error, an eight-page Walton and Maberly publisher's catalogue, and a binding of black blindpaneled zigzag cloth without the central lozenge. Both issues have an integral title-leaf with imprint reading 'London: Walton and Maberly, Upper Gower-Street, and Ivy Lane, Paternoster Row. Cambridge: Macmillan and Co.' A later issue has been noted in a green pebble-cloth binding, with a cancel title-leaf and imprint reading 'London: Macmillan and Co.'"(Hook & Norman). 8vo (225 x 143 mm), uncut. Original publishers black blind-paneled cloth with gilt spine lettering (bound by Edmonds & Remnants, London), top of hinges with a very small tear, covers with some very light discoloration, in all a very fine copy of this fragile binding, entirely unrested. Pp [10], 424 [2:errata], text fresh and clean. Rare in such fine condition.
Interferenz-Erscheinungen bei Röntgenstrahlen." - "Eine quantitative Prüfung der Theorie für den Interferenz-Erscheinungen bei Röntgenstrahlen." Offprint (containing both papers) from theSitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften Mathematisch-physikalische Klasse(1912)

Interferenz-Erscheinungen bei Röntgenstrahlen.” – “Eine quantitative Prüfung der Theorie für den Interferenz-Erscheinungen bei Röntgenstrahlen.” Offprint (containing both papers) from theSitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften Mathematisch-physikalische Klasse(1912)

LAUE, Max von, Walter FRIEDRICH & Paul KNIPPING First edition, very rare offprint issue, of Laue's Nobel Prize-winning report of "one of the most beautiful discoveries in physics" (Einstein). X-rays had been in wide use since their discovery in 1895 but their exact nature as electromagnetic waves of short wavelength was first elucidated by Laue and his collaborators in the present papers. Laue (1879-1960) had moved in 1909 from Berlin (where he was Planck's assistant) to Ludwig Maximillians University in Münich, where he was Arnold Sommerfeld's Privatdozent. In the spring of 1912 he was asked by Sommerfeld's doctoral student Paul Ewald a question about the arrangement of atoms in a crystal. In attempting to answer this question "Laue had the crucial idea of sending X-rays through crystals. At this time scientists were very far from having proven the supposition that the radiation that Röntgen had discovered in 1895 actually consisted of very short electromagnetic waves. Similarly, the physical composition of crystals was in dispute, although it was frequently stated that a regular structure of atoms was the characteristic property of crystals. Laue argued that if these suppositions were correct, then the behavior of X-radiation upon penetrating a crystal should be approximately the same as that of light upon striking a diffraction grating" (DSB), an instrument used for measuring the wavelength of light, inapplicable to X-rays because their wavelength is too short. Sommerfeld was initially skeptical but Laue persisted, enlisting the help of Sommerfeld's experimental assistant Walter Friedrich (b. 1883) in his spare time as well as that of the doctoral student Paul Knipping. OnApril 12, 1912, Friedrich and Knipping succeeded in producing a regular pattern of dark spots on a photographic plate placed behind a copper sulphate crystal which had been bombarded with X-rays. Laue's second paper contains his complicated mathematical explanation of the phenomenon. "The awarding of the Nobel Prize in physics for 1914 to Laue indicated the significance of the discovery that Albert Einstein called 'one of the most beautiful in physics'. Subsequently it was possible to investigate X-radiation itself by means of wavelength determinations as well as to study the structure of the irradiated material. In the truest sense of the word scientists began to cast light on the structure of matter" (DSB).The following year the Prize was granted to the father and son team W. H. and W. L. Bragg for their exploration of crystal structure using X-rays.ABPC/RBH lists three other copies of this offprint (Christie's, 4 October 2002, lot 151, $5736; Sotheby's, 11 January 2001, lot 333, $10,200; Christie's 29 October 1998, lot 1161, $16,100). In 1912, "the nature of the X rays discovered by Röntgen in 1895 was not known. Röntgen himself conjectured that they might be longitudinal ether waves as opposed to the transverse ones, the electromagnetic waves found by Hertz. Since in Röntgen's original experiment the X-rays originated from the point where cathode rays, i.e., electrons, hit matter, Wiechert and also Stokes suggested already in 1896 that X-rays were emitted by electrons while the latter were decelerated. In Maxwell's theory an electric charge with a velocity, which is not constant, emits electromagnetic waves. In the Hertzian dipole antenna the charges oscillate to and fro. In a Röntgen tube electrons lose their velocity hitting a piece of matter. The fact that interference effects, characteristic of all waves, in particular, light and Hertzian waves, were not observed for X-rays, did not preclude that they were electromagnetic waves. Their wavelength might be too small for the detection of interference . "The nature of X-rays, homogeneous or heterogeneous, remained a mystery. They could be understood as electromagnetic waves of short wavelengths or as new neutral particles. The former standpoint was taken, for instance, by Barkla, the latter by William Henry Bragg. One of Bragg's arguments ran like this: X-rays, produced by electrons falling on matter, fly more or less in the same direction as the incident electrons. That is easily understood if one assumes them to be particles. The production of X-rays can then be seen as a collision process, just as one billiard ball hitting another. For some time the two scientists fought out the Barkla-Bragg controversy in the columns of Nature. Sommerfeld showed that, contrary to the expectations of Bragg and others, electromagnetic radiation is emitted mostly in forward direction if a fast electron suffers a sudden deceleration. The German term bremsstrahlung [breaking radiation] is still used commonly in the literature. "It was the work of Laue and the experiment done by Friedrich and Knipping on his suggestion that cleared up the nature of X-rays once and for all and that, moreover, beautifully demonstrated that crystals are composed of atoms arranged in a regular lattice. Laue had studied mathematics and physics in Strasbourg, Göttingen, Munich, and Berlin, where in 1903 he took his Ph.D. with a thesis under Planck. Feeling that he still had to continue his studies he went for another two years to Göttingen. In 1905 Planck offered him a position as his assistant. Laue worked with Planck on the latter's speciality, the entropy of radiation. In the autumn of 1905 Planck gave a talk in the Berlin Physics Colloquium on Einstein's first paper The Electrodynamics of Moving Bodies. Laue was deeply impressed. In 1906, when on a mountaineering trip in Switzerland, as one of the first (possibly the very first) visitor from abroad, he looked up Einstein in the patent office in Bern. In 1907 he published a paper in which he showed that classic experiment by Fizeau, who had measured the velocity of light in a moving liquid, was in accordance with Einstein's theory. Laue became a Privatdozent in Berlin and, also in that capacity, moved to Munich University in 1909. In 1910 he wrote the first book on the theory of relativity . "The t
Beschreibung eines Augen-Spiegels zur Untersuchung der Netzhaut im lebenden Auge

Beschreibung eines Augen-Spiegels zur Untersuchung der Netzhaut im lebenden Auge

HELMHOLTZ, Hermann von First edition, an exceptionally fine copy in original printed wrappers, of this famous work which describes Helmholtz's announcement of his invention of the ophthalmoscope, one of the most important clinical tools in medicine, which greatly improved the ability of ophthalmologists to diagnose eye disease and revolutionized visual science. The invention of the ophthalmoscope by Helmholtz has been called "the greatest event in the history of ophthalmology, which advanced it toward the goal of independence as a specialty" (Gorin). This invention was a by-product of Helmholtz's attempt to demonstrate to his physiology students that when the human eye is made to glow with reflected light, the light emitted from the pupil follows the same course it took in entering. Realizing that if the light could be brought to a focus the details of the retina would be made visible, he invented a device to accomplish this objective. "With this instrument it was possible for the first time to examine the interior of the living eye. Although crude attempts had been made earlier to see into the eye, it was Helmholtz's invention of a workable instrument in 1850 and the publication of his monograph in 1851 that laid the basis of scientific ophthalmology. Helmholtz's invention of the ophthalmoscope arose from an attempt to demonstrate for his class in Königsberg the nature of the glow of reflected light sometimes seen in the eyes of animals such as the cat. When the great ophthalmologist A. von Graefe first saw the fundus of the living eye, with its disc and blood-vessels, his face flushed with excitement, and he cried 'Helmholtz has unfolded to us a new world!' (Hagerstrom Library). The Augen-Spiegel was printed in a very small edition: there are no copies in the Becker, Osler or Cushing Collections. Rare on the market in such fine condition. "Helmholtz's invention had its roots in earlier attempts to see the back of the eye, though these were insufficient to permit proper inspection of the human fundus. In 1703, Jean Méry (1645-1722), who worked at the Hôtel Dieu, found that the luminosity of the cat's eye could be seen when the animal was held under water, showing that it was essentially an optical phenomenon. Philippe de la Hire, 6 years later, thought it was owing to abolition of corneal refraction under water that the incident light rays emerged divergent and were thus seen by the observer's eye. In the fourth essay in his Oeuvres (2 volumes, Leiden, 1717), Edmé Mariotte (1620-1684), who was both physicist and priest, observed that a dog's eye is luminous because its choroid is white; and the darker choroid in man and animals allowed no clear image. Richter provoked further interest when it was found that luminosity could still be present in a blind eye, and in 1792 Georg Joseph Beer had observed the luminosity of the fundus in aniridia. However, spontaneous luminosity in man remained unexplained. "Bénédict Prévost, Professor of Philosophy at Montaubon in France (1755-1819), repeated Mariotte's experiments, examining the eyes of a cat in the dark, and explained that the retina was invisible: 'It is not the light which proceeds from the eye to an object that enables the eye to perceive that object, but the light which arrives in the eye from it.' This was an important discovery that dispelled the accepted notions that light came from within the eye to permit animals to see in the dark. "In 1821, the Swedish naturalist Karl Asmund Rudolphi (1771-1832) shone a light into a decapitated cat's eyes and showed that the reflecting eye emitted light along the same line as the direction of the in-going rays. Twenty-seven years before Helmholtz's work, in 1823, Jan Evangelista Purkinje (1787-1869), Professor of Physiology at Breslau, had observed that under certain illumination human eyes could be made luminous: in 1825 Purkinje started to use lenses to examine the back of the eye. His crucial work, published in Latin, was unrecognized for many years: 'I examined the eye of a dog by using the spectacle lens of a myope and placing a candle behind the dog's back . I found the light as the source, which is reflected from the concavity of the spectacle lens into the interior of the eye. From there it is again reflected. I immediately repeated the experiment on a human eye and found the same phenomenon'. "The pupil too appeared black. The 'beautiful orange glow was reflected when light was thrown into it'. Unnoticed, this was rediscovered independently by William Cumming in England, who, in 1846, wrote 'On a luminous appearance of the human eye and its application to the detection of disease of the retina and posterior part of the eye.' He explained that the axis of illumination and observation had to be coincident to view the fundus. A year later, the Berlin physiologist Ernst Wilhelm Ritter von Brücke (1819-1892) made the same observation: 'A short time ago in the evening as I was standing between the chandelier and the door in the auditorium of this university, I saw a young man whose pupils were illuminated with a bright red light as he turned to close the door through which he had just passed . If one wishes to see this reflex in human eyes . Take the usual oil lamp with its cylindrical wick and the glass chimney, . and regulate the wick in such fashion that it burns with a short, intense flame. Then set the lamp close to you, but place the subject 8 to 10 feet away, . If [the subject] then looks with widely opened lids towards the darkness adjacent to the lamp, or if he slowly moves his eyes to and from, then the pupils will be illuminated with a reddish light, while the iris, in contrast, will appear slightly greenish.' Helmholtz was later generously to say: 'Brucke himself was but a hair's breadth away from the invention of the ophthalmoscope. He had only failed to ask himself what optical image was formed by the rays reflected from the luminous eye. Had it occurred to him, he was the man to answer it just as quick
Elements of Vector Analysis. [Offered with:] Autograph letter from Gibbs to John Monroe Van Vleck

Elements of Vector Analysis. [Offered with:] Autograph letter from Gibbs to John Monroe Van Vleck

GIBBS, Josiah Willard First edition, first issue (see below), of this extremely rare pamphlet which "marks the beginning of modern vector analysis" (Crowe, p. 150). "Nearly all branches of classical physics and many areas of modern physics are now presented in the language of vectors, and the benefits derived thereby are many. Vector analysis has likewise proved a valuable aid for many problems in engineering, astronomy and geometry" (ibid. p. v). The genesis of the present work was described in Gibbs's own words in an 1888 letter to Victor Schlegel: "My first acquaintance with quaternions was in reading Maxwell's E & M [i.e. Treatise on Electricity and Magnetism, 1873] where quaternion notations are considerably used. I became convinced that to master those subjects, it was necessary for me to commence by mastering those methods. At the same time I saw, that although the methods were called quaternionic the idea of the quaternion was quite foreign to the subject. In regard to the product of vectors, I saw that there were two important functions (or products) called the vector part & the scalar part of the product, but that the union of the two to form what was called the (whole) product did not advance the theory as an instrument of geom[etric] investigation. Again with respect to the operator as applied to a vector I saw that the vector part & the scalar part of the result represented important operations, but their union (generally to be separated afterwards) did not seem a valuable idea . I therefore began to work out ab initio, the algebra of the two kinds of multiplication, the three differential operations applied to a scalar, & the two operations to a vector . This I ultimately printed but never published, although I distributed a good many copies among such persons as I though might possibly take an interest in it" (ibid. pp. 152-3). This is the first issue; a second issue, with two additional chapters, was published in 1884. Both issues of Gibbs's pamphlet are extremely rare in commerce. We have been unable to locate any copy of this first issue in auction records, and only two of the second issue: the Horblit copy (Christie's, 16 February 1994) and the Richard Green copy (Christie's, 17 June 2008). "In the year 1844 two remarkable events occurred, the publication by [William Rowan] Hamilton of his discovery of quaternions, and the publication by [Hermann Günther] Grassmann of his 'Ausdehnungslehre.' With the advantage of hindsight we can see that Grassmann's was the greater contribution to mathematics, containing the germ of many of the concepts of modern algebra, and including vector analysis as a special case" (Dyson). "During the 1880's Gibbs seems to have concentrated on optics and particularly on Maxwell's electromagnetic theory of light . Gibbs's reading Maxwell's Treatise on Electricity and Magnetism led him to a study of quaternions, since Maxwell had used the quaternion notation to a limited extent in that work. Gibbs decided, however, that quaternions did not really provide the mathematical language appropriate for theoretical physics, and he worked out a simpler and more straightforward vector analysis" (DSB). From Schlegel's letter, we learn that "Gibbs commenced his search for a vector analysis 'with some knowledge of Hamilton's methods' and ended up with methods that were 'nearly those of Hamilton' . Gibbs also stated that he was not 'conscious that Grassmann exerted any particular influence on my V-A.' This is to be expected since Gibbs had begun searching for a new vector system 'long before my acquaintance with Grassmann.' When (1877 or later) Gibbs finally began to read Grassmann, he found a kindred spirit. Although Gibbs admitted he had never been able to read through either of Grassmann's books, he did recognize Grassmann's priority and warmly praised his ideas on numerous occasions" (Crowe, pp. 153-4). "In 1879 Gibbs gave a course in vector analysis with applications to electricity and magnetism, and in 1881 he arranged for the private printing of the first half of his Elements of Vector Analysis; the second half appeared in 1884. In an effort to make his system known, Gibbs sent out copies of this work to more than 130 scientists and mathematicians. Many of the leading scientists of the day received copies, for example, Michelson, Newcomb, J. J. Thomson, Rayleigh, FitzGerald, Stokes, Kelvin, Cayley, Tait, Sylvester, G. H. Darwin, Heaviside, Helmholtz, Clausius, Kirchhoff, Lorentz, Weber, Felix Klein, and Schlegel. Though the work was not given the advertisement that a regular publication would have had, such a selective distribution must have aided in making it known. "Some idea of the form of Gibbs' Elements of Vector Analysis may be obtained from Gibbs' introductory paragraph: 'The fundamental principles of the following analysis are such as are familiar under a slightly different form to students of quaternions. The manner in which the subject is developed is somewhat different from that followed in treatises on quaternions, since the object of the writer does not require any use of the conception of the quaternion, being simply to give a suitable notation for those relations between vectors, or between vectors and scalars, which seem most important, and which lend themselves most readily to analytical transformations, and to explain some of these transformations. As a precedent for such a departure from quaternionic usage, Clifford's Kinematic may be cited. In this connection, the name of Grassmann may also be mentioned, to whose system the following method attaches itself in some respects more closely than to that of Hamilton.' "Although Gibbs mentioned only Clifford and Grassmann in the introductory paragraph, the previously cited letter makes it clear that his chief debt was not to either Clifford or Grassmann but to the quaternionists. In the discussion of Gibbs' book this point will be illustrated; specifically it will be suggested that Gibbs was strongly influenced b
The Principles of Mr. Harrison's Time-Keeper

The Principles of Mr. Harrison’s Time-Keeper, with plates of the same. Published by Order of the Commissioners of Longitude

HARRISON, John, [and Nevil MASKELYNE.] First edition, one of the very few copies with the plates printed on India paper, of the "description of the famous solution to the centuries-old world-wide problem of finding the longitude" (Grolier/Horblit). "Harrison's chronometer not only supplied navigators with a perfect instrument for observing the true geographical position at any moment during their voyage, but also laid the foundation for the compilation of exact charts of the deep seas and the coastal waters of the world . There has possibly been no advance of comparable importance in aids to navigation until the introduction of radar" (PMM 208). In 1714 the Board of Longitude offered a reward of £20,000, a colossal sum at the time, to anyone who could find a reliable and accurate method for determining longitude at sea. In 1730 the clockmaker John Harrison (1693-1776) completed a manuscript describing some of his chronometrical inventions, including a chronometer "accurate enough to measure time at a steady rate over long periods, thus permitting the measurement of longitude by comparison of local solar time with an established standard time" (Norman). On the strength of his descriptions, Harrison obtained a loan from George Graham, a leading maker of clocks and watches, for the construction of his timekeeper. After numerous attempts, most of which either Harrison himself or his son William tested on ocean voyages, Harrison succeeded in constructing a chronometer 'H4' that was both accurate and convenient in size. Following successful tests on voyages to the West Indies in 1761 and 1764, Harrison felt that he had a right to the prize, but the Board of Longitude hedged, insisting on a demonstration and full written description of his invention. The demonstration took place on 22 August 1765, in the presence of the astronomer-royal Nevil Maskelyne and a six-member committee of experts appointed by the Board. The results were written up and published in this pamphlet by Maskelyne, along with Harrison's own description of his timekeeper. Officially the Board intended Principles to enable other clockmakers to construct H4, but it "was both incomplete of enough information to allow the duplication of the watch . and contain[ed] some accidentally-on-purpose errors. This was likely done as much to help maintain the hard-won knowhow of its inventor, as well as to protect any military advantage, given the importance of the H4 to maritime navigation" (Lake). Maskelyne's Preface explains the reason for this special issue of Principles: "for the sake of the curious, and particularly artists who may be desirous to construct other watches after the model of Mr. Harrison's, I have caused a few impressions of the plates to be taken off upon India paper; which, if it be made only a little damp, by being put for a few minutes between two wet sheets of paper, will receive the impression from the plates perfect, and will not shrink at all in the drying" (p. vii). Since this issue was also printed on oblong sheets, rather than the regular 4to, the plates remain unfolded, save the considerably larger 7th plate. Only one copy of this issue has sold at auction since the Frank S. Streeter copy (Christie's, 16 April, 2007, lot 254, $228,000), and according to Christie's no other copy had sold for 30 years previously. "In the early 1700s, European monarchies aspired to power by building world-spanning networks of colonies and commercial ventures. As a result, the merchant fleets and navies that connected and protected these assets were critically important. Eighteenth-century sailors led dangerous lives, not least because they seldom knew their exact location on the open ocean. Although navigators readily determined latitude, or north-south position, by estimating the height of certain stars at their zenith, they could not determine longitude. This failure caused shipwrecks that killed thousands of mariners and lost cargoes worth fortunes. Several countries offered immense financial rewards for a solution to the problem; Britain promised £20,000 (several million dollars in today's currency) for a way to establish longitude to within half a degree (30 nautical miles at the equator) after a journey from England to the West Indies. To judge proposed solutions, the crown established a Board of Longitude, made up of the Astronomer Royal, various admirals and mathematics professors, the Speaker of the House of Commons and 10 members of Parliament. "In effect, determining longitude depended on knowing the difference between local time and the time in Greenwich, site of the Royal Observatory. In principle, if a ship had a clock keeping Greenwich time, the navigator could measure the angle of the Sun to note local noon and compare it to the clock. If the clock read 2 p.m., his longitude was two hours, or 30 degrees, west of Greenwich. The problem lay in finding a clock reliable enough to keep time during the long voyages of that era. The best pendulum clocks of the day were accurate enough, but were useless on a heaving ship at sea. Alternately, a less reliable clock might be used if some means could be devised to correct it frequently. In practice this meant an astronomical method, the best of which became known as the method of lunar distances, in reference to the fact that the Moon's orbit causes it to continually change position in the sky. For example, a new moon, which appears close to the Sun, will have moved 180 degrees by the time it becomes a full moon two weeks later. The idea was for astronomers to provide tables of this angle between Moon and Sun (or Moon and selected stars in the night sky) as a function of Greenwich time. A measurement of this angle every few days would provide a correction to the mechanical clock. This scheme had two drawbacks: The first was that, at least initially, astronomers could not accurately predict the Moon's motion; the second was that the mathematical calculations required of the mariner were very complex-they took hou
Mémoires sur l'action mutuelle de deux courans électriques

Mémoires sur l’action mutuelle de deux courans électriques, sur celle qui existe entre un courant électrique et un aimant ou le globe terrestre, et celle de deux aimans l’un sur l’autre

AMPERE, Andre-Marie First edition, probable first issue, extremely rare and inscribed by Ampère, of this continually evolving collection of important memoirs on electrodynamics by Ampère and others. "Ampère had originally intended the collection to contain all the articles published on his theory of electrodynamics since 1820, but as he prepared copy new articles on the subject continued to appear, so that the fascicles, which apparently began publication in 1821, were in a constant state of revision, with at least five versions of the collection appearing between 1821 and 1823 under different titles" (Norman). The collection begins with 'Mémoires sur l'action mutuelle de deux courans électriques', Ampère's "first great memoir on electrodynamics" (DSB), representing his first response to the demonstration on 21 April 1820 by the Danish physicist Hans Christian Oersted (1777-1851) that electric currents create magnetic fields; this had been reported by François Arago (1786-1853) to an astonished Académie des Sciences on 4 September. In this article he "demonstrated for the first time that two parallel conductors, carrying currents traveling in the same direction, attract each other; conversely, if the currents are traveling in opposite directions, they repel each other" (Sparrow, Milestones, p. 33). This first paper is mostly phenomenological, but it is followed here by the important and much more mathematical sequel, 'Additions au mémoire précédent - note sur les expériences électro-magnétiques de MM. Oersted, Ampère, Arago et Biot,' in which Ampère gave the first quantitative expression for the force between current carrying conductors. Ampère attempted to explain his observations by postulating a new theory of magnetism - according to him, magnetic forces were the result of the motion of two electric fluids; permanent magnets contained these currents running in circles concentric to the axis of the magnet and in a plane perpendicular to this axis. By implication, the earth also contained currents which gave rise to its magnetism. Ampère's theory was attacked by the great Swedish chemist Jöns Jacob Berzelius in a letter to his French colleague Claude Louis Berthollet, to which Ampère replied in a letter to François Arago. These are the third and fourth items in this collection; the fifth and final part is the text of a lecture to the Académie on 2 April 1821 in which Ampère again stressed the identity of electricity and magnetism. The bibliographical complexity of this work is a direct result of Ampère's modus operandi: "His work was marked by flashes of insight, and it often happened that he would publish a paper in a journal one week, only to find the next week that he had thought of several new ideas that he felt ought to be incorporated into the paper. Since he could not change the original, he would add the revisions to the separately published reprints of the paper and even modify the revised versions later if he felt it necessary" (Norman). Only three other copies of this work listed by ABPC/RBH (all later issues). OCLC lists only one copy of this issue of the collection (University College, London), and we found no record of any earlier issue. Provenance: Société Philotechnique d'Ostende (presentation inscription from the author). The collection opens with the 'Premier Mémoire' [1] (numbering as in the list of contents, below), first published in Arago's Annales de Chimie et de Physique at the end of 1820 (Series 2, Tome 15, pp. 59-76 in the October issue & 170-218 in November, read 18 & 25 September). "There is some confusion over the precise nature of Ampère's first discovery. In the published memoir, "Mémoire sur 1'action mutuelle de deux courants électriques," he leaped immediately from the existence of electromagnetism to the idea that currents traveling in circles through helices would act like magnets. This may have been suggested to him by consideration of terrestrial magnetism, in which circular currents seemed obvious. Ampère immediately applied his theory to the magnetism of the earth, and the genesis of electrodynamics may, indeed, have been as Ampère stated it. On the other hand, there is an account of the meetings of the Académie des Sciences at which Ampère spoke of his discoveries and presented a somewhat different order of discovery. It would appear that Oersted's discovery suggested to Ampere that two current-carrying wires might affect one another. It was this discovery that he announced to the Académie on 25 September. Since the pattern of magnetic force around a current-carrying wire was circular, it was no great step for Ampère the geometer to visualize the resultant force if the wire were coiled into a helix. The mutual attraction and repulsion of two helices was also announced to the Académie on 25 September. What Ampère had done was to present a new theory of magnetism as electricity in motion . "Ampère's first great memoir on electrodynamics was almost completely phenomenological, in his sense of the term. In a series of classical and simple experiments, he provided the factual evidence for his contention that magnetism was electricity in motion. He concluded his memoir with nine points that bear repetition here, since they sum up his early work. Two electric currents attract one another when they move parallel to one another in the same direction; they repel one another when they move parallel but in opposite directions. It follows that when the metallic wires through which they pass can turn only in parallel planes, each of the two currents tends to swing the other into a position parallel to it and pointing in the same direction. These attractions and repulsions are absolutely different from the attractions and repulsions of ordinary [static] electricity. All the phenomena presented by the mutual action of an electric current and a magnet discovered by M. Oersted . are covered by the law of attraction and of repulsion of two electric currents that has just been enunciat
Ueber sehr schnelle electrische schwingungen. Offprint from Annalen der Physik

Ueber sehr schnelle electrische schwingungen. Offprint from Annalen der Physik, Bd. 31 (1887). With six other offprints documenting Hertz’s seminal work which demonstrated the existence of electromagnetic waves, which thereby provided the experimental proof of Maxwell’s theory and formed the foundation for wireless communication

HERTZ, Heinrich Rudolf First edition, extremely rare offprint, of the first of Hertz's papers on electromagnetic waves, accompanied by offprints of six further papers on the same subject, including 'Ueber elektrodynamische Wellen im Luftraume und deren Reflexion', in which Hertz first demonstrated the existence of electromagnetic waves propagating in air ('Hertzian waves'). "In his Treatise on Electricity and Magnetism (1873) [Maxwell] gave no theory of oscillatory circuits or of the connection between currents and electromagnetic waves. The possibility of producing electromagnetic waves in air was inherent in his theory, but it was by no means obvious and was nowhere spelled out. Hertz's proof of such waves was in part owing to his theoretical penetration into Maxwell's thought" (DSB). "Experimental proof by Hertz of the Faraday-Maxwell hypothesis that electrical waves can be projected through space was begun in 1887, eight years after Maxwell's death. The two main requirements were (a) a method of producing the waves, supposing that they existed, and (b) a method of detecting them once they were produced. Hertz found the first problem easy to solve. He used the oscillatory discharge of a condenser. Detection was much more difficult, because there then existed no means of detecting currents alternating at the high speed of these waves. Hertz in fact used an effect as old as the discovery of electricity itself - the electric spark. By inducing the waves to produce an electrical spark at a distance, with no apparent connection between the oscillator and the spark gap, and by moving the sparking apparatus so that the length of the spark varied, Hertz proved beyond question the passage of electric waves through space. The experiments were reported periodically from 1887 onward in Annalen der Physik und Chemie" (PMM). "This discovery [of electromagnetic waves] and its demonstration led directly to radio communication, television and radar" (Dibner). "In the early 1890's the young inventor Guglielmo Marconi read of Hertz's electric wave experiments in an Italian electrical journal and began considering the possibility of communication by wireless waves. Hertz's work initiated a technological development as momentous as it physical counterpart" (DSB). We can find no other copies of any of the papers [1]-[8] in auction records. The Smithsonian holds a copy of each of the offprints [1]-[7]; OCLC adds two other copies of [1], one in York (though not listed in their library catalogue), and one in Japan (not verified); two other copies of [2], one in Bern and one in Japan (not verified); and three other copies of [5], one in Bern and two in Yale. The offered papers are as follows: Uber sehr schnelle electrische schwingungen [On very rapid electrical oscillations]. Offprint from Annalen der Physik, Bd. 31 (1887), pp. 421-448 and one folding plate. Contemporary wrappers, upper wrapper renewed with matching paper with manuscript title label, lower wrapper with publisher's imprint. Uber Inductionserscheinungen, hervorgerufen durch die elektrischen Vorgänge in Isolatoren [On electromagnetic effects produced by electrical disturbances in insulators]. Offprint from Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 10 November, 1887, pp. 1-12. Original printed wrappers (creased vertically and horizontally for posting and with postmark and remains of stamp on rear cover). Ueber die Einwirkung einer geradlinigen electrischen Schwingung auf eine benachbarte Strombahn [The action of a rectilinear electric oscillation on a neighbouring circuit]. Offprint from Annalen der Physik, Bd. 34 (1888), pp. 155-170 and one folding plate. Original printed wrappers. Ueber elektrodynamische Wellen im Luftraume und deren Reflexion [On electrodynamic waves in air and their reflection]. Offprint from Annalen der Physik, Bd. 34 (1888), pp. 609-623. Original printed wrappers. Uber Strahlen elektrischer Kraft [On rays of electric force]. Offprint from Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 13 December, 1887, pp. 1-11. Original printed wrappers (creased vertically and horizontally for posting). Die Krafte electrischer Schwingungen behandelt nach der Maxwell'schen Theorie [The forces of electric oscillations treated according to Maxwell's theory]. Offprint from Annalen der Physik, Bd. 36 (1888), pp. 1-22. Original printed wrappers. Uber die Fortleitung electrischer Wellen durch Drähte [On the propagation of electric waves by means of wires]. Offprint from Annalen der Physik, Bd. 37 (1889), pp. 395-408 and one folding plate. Original printed wrappers, upper wrapper with English inscription by Hertz. Hertz's work on electric waves began with Helmholtz's proposal in 1879 of a prize problem connected with the behaviour of unclosed circuits in Maxwell's theory. "Central to Maxwell's theory was the assumption that changes in dielectric polarization yield electromagnetic effects in precisely the same manner as conduction current do. Helmholtz wanted an experimental test of the existence of these effects or, conversely, of the electromagnetic production of dielectric polarization. Although at the time Hertz declined to try the Berlin Academy problem because the oscillations of Leyden jars and open induction coils which he was familiar with did not seem capable of producing observable effects, he kept the problem constantly in mind; and in 1886 shortly after arriving in Karlsruhe, he found that the Riess or Knochenhauer induction coils he was using in lecture demonstrations he was using in lecture demonstrations were precisely the means he needed for undertaking Helmholtz' test of Maxwell's theory . "He produced electric waves with an unclosed circuit connected to an induction coil, and he detected them with a simple unclosed loop of wire. He regarded his detection device as his most original stroke, since no amount of theory could have predicted that it would work. Across the darkened K
Epitome cosmographica

Epitome cosmographica, o Compendiosa introduttione all’astronomia, geografia, & idrografia, per l’uso, dilucidatione, e fabbrica delle sfere, globi, planisferi, astrolabi, et tavole geografiche, e particolarmente degli stampati, e spegiati nelle publiche lettioni

CORONELLI, Vincenzo Maria First edition, rare when complete, of this sumptuously illustrated work, a uniquely valuable source for the documentation of several of the most elaborate large-scale globes and astronomical mechanisms, some now lost, constructed during the latter decades of the seventeenth century. Coronelli, a Franciscan monk, was the official cosmographer of the Venetian Republic, the greatest maker of terrestrial globes and maps during the last half of the seventeenth century, and the founder of the Accademia Cosmografica degli Argonauti, the first geographical society. The Epitome is particularly important for the information Coronelli includes on the highly decorative and massive globes he constructed for Louis XIV, one of which is illustrated in this work (they can be seen today in the Bibliothèque Nationale in Paris). The work contains four large foldout celestial maps in circular format engraved in a spectacular baroque style; they were based on the most recent astronomical observations and were copied into the eighteenth century. Also included are two large terrestrial maps - the western and eastern hemispheres. Of the 37 double-page plates, many illustrate globes, spheres, astronomical diagrams and instruments. As a leading cosmographer, Coronelli's career bears on the history of astronomy at many points. "In the Epitome he listed everything that seemed to him important in astronomy and geography, describing, without making any value judgements, the systems of Ptolemy (18 lines), Tycho Brahe (17 lines), Copernicus (132 lines), and Descartes (40 lines). Not only does he give, for many constellations, an account of their history or the origins of their names, and for all the positions of the stars, . every star is accorded a number . The appearances of comets since ancient times are all listed. The chapter 'Geography' contains summary descriptions of the continents, but also of the earthquakes and volcanic eruptions recorded since ancient times. The explorers of the more important regions are listed, together with lists of writings on astronomy and geography by authors back to the ancient Greeks" (Schmidt & Bridge, p. 100). Riccioli was his main source of technical information, for the earth as well as the heavens; other celestial information, including the only telescopic stars he included, was taken from Bayer and Hevelius. Coronelli was a long-standing friend of Edmond Halley and Robert Hooke, observing a lunar eclipse with them in London in 1696. ABPC/RBH list only two complete copies in last 40 years: Christie's, 21 March 2012, £6875; Sotheby's, 7 December 1989, £2640. Although reasonably well represented in institutional collections, it is unclear how many of those copies are complete (the copy in Cambridge University Library, for example, lacks one plate; and of the two British Library copies, one lacks two of the volvelles at p. 361 and the other lacks the engraved title). The Epitome is divided into three books. The first, comprising 35 chapters (pp. 1-208), begins with a discussion of spherical geometry, latitude and longitude, great circles, the tropics, winds, and climate. This is followed by a discussion of the stars and planets, their distances and number, the constellations, comets, and solar and lunar eclipses. The second book, comprising 17 chapters (pp. 209-324), is devoted to geography: the land and sea, the various regions of the earth (Europe, Asia, Africa, America, the poles), with tables of latitude and longitude of the major cities. There follow several chapters devoted to earthquakes and volcanic eruptions. The third book, divided into two parts, comprising 39 chapters, is devoted to the description of various globes, celestial and terrestrial, and how to use them, as well as information on the construction and use of popular instruments such as armillary spheres, planispheres, and astrolabes. Part one (pp. 325-342) describes and illustrates some of the most spectacular globes to be found across Europe (England, France, Germany). Chapter 1 describes the 'English Globe' produced in 1679 by the Earl of Castlemaine (1634-1705), in collaboration with Joseph Moxon (1627-91). It was an immobile globe whose sphere does not rotate but is fixed in place over a planisphere; this allowed complex calculations to be performed more easily than with a turning sphere. Chapter 2 is devoted to the great globe of Gottorp, Germany, constructed under the supervision of Adam Olearius (1599-1671), which had a map of the earth's surface on the outside and a map of star constellations with astrological and mythological symbols on the inside. Turned by water-power, it demonstrated the 'movement' of the heavens to those seated inside in candlelight - it was a predecessor of the modern planetarium. In chapter 3, Coronelli describes the 'Globus Pancosmus' of Erhardt Weigel (1625-99), which had a circumference of 9.5 metres with interior effects such as a breeze which could be made to blow from any desired quarter, various elements such as rain, hail and thunder and images of people of different nationalities. Chapter 4 is devoted to Christopher Treffler's 'Sphaericum Automatum', a self-moving celestial globe, now lost - it was for sale for the price of 8000 Talari in 1688 at the time Coronelli was passing through Augsburg, and his description of it appears to be the only surviving evidence that it existed. The fifth and final chapter in this part is devoted to the globes Coronelli constructed for Louis XIV, described below. In part two, in 34 chapters (pp. 343-406), Coronelli describes the construction of globes in general: how the globe gores are designed and printed to represent the earth on a flat plane, how the heavens can also be represented in the plane by means of planispheres, how the globes themselves are assembled by gluing the gores onto a large ball made of wood and papier-mâché and finished with plaster, and how the information on the globes is to be presented. Coronelli was celebrated
Autograph manuscript

Autograph manuscript, unsigned, entitled ‘Talk at Vancouver New Particles etc.’ Vancouver, Canada, 22 November 1975

FEYNMAN, Richard Phillips A detailed draft for a talk given to the Canadian Association of Physics Students, in Vancouver,on the contemporary state of subatomic particle physics, its current difficulties and possible future developments. Although a 'popular' talk, it is pitched at a high level, appropriate to postgraduate students in physics. Feynman manuscripts with scientific content are very rare on the market - this is one of a small collection of such manuscripts that was retained by Feynman's family until 2018 when it was consigned to auction. Widely regarded as the most brilliant, influential, and iconoclastic figure in theoretical physics in the post-World War II era, Feynman shared the Nobel Prize in Physics 1965 with Sin-Itiro Tomonaga and Julian Schwinger "for their fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles." Feynman refers briefly to a talk given to students in Vancouver in Surely You're Joking Mr. Feynman! (p. 343): 'In Canada they have a big association of physics students. They have meetings; they give papers, and so on. One time the Vancouver chapter wanted to have me come and talk to them. The girl in charge of it arranged with my secretary to fly all the way to Los Angeles without telling me. She just walked into my office. She was really cute, a beautiful blonde. (That helped; it's not supposed to, but it did.) And I was impressed that the students in Vancouver had financed the whole thing. They treated me so nicely in Vancouver that now I know the secret of how to really be entertained and give talks: Wait for the students to ask you.' In fact, Feynman gave more than one talk to the Vancouver students, the offered notes probably referring to the second such talk, for in a letter to Mariela Johansen of April 1975, he writes: 'I often remember vividly my most enjoyable trip. Many things bring it to mind - like the dark blue T-shirt in my drawer at home - or the interference picture in my office - or just now when my secretary asked me if I wanted to talk to students at a nearby university (USC 20 miles) or high school. My answer was that I will talk to students anytime they are near enough to home, or are at Vancouver, B.C.' (The Quotable Feynman (2015), p. 296). That these notes probably refer to the second Vancouver talk is confirmed by the first sentence on p. 1, in which Feynman writes: 'Difference from last time . last year .' The slides for this talk, which Feynman refers to at one point in these notes, are preserved in the Feynman archives at Caltech. By 1975, what is now called the 'Standard Model' of particle physics was close to being established. It provides a 'unified' description of three of the four forces through which subatomic particles interact - the electromagnetic, weak and strong forces; the fourth force, gravity, has still not been unified with the other three. In this manuscript, Feynman summarizes the current understanding of the Standard Model in 1975, and discusses several significant 'loose ends'. Some of these were resolved in the years following his talk, while others are still open today. The first two of the three forces to be 'unified' were the weak and electromagnetic. In 1957 Julian Schwinger postulated that three different bosons (particles with whole number spin, that obey Bose-Einstein statistics) must be involved in transmitting the weak force to take account of all the possible different ways the nucleons can interact in the nucleus. Two of these bosons were required to exchange positive and negative charges, now called the W+ and W- (weak) bosons; a third neutral boson, the Z0 (which Feynman calls the W0) was required for reactions in which no charge was transferred. In 1973, 'neutral weak currents' (i.e., interactions between particles that involve the exchange of W0 bosons) were observed at CERN, and the electroweak theory became widely accepted. However, the W+, W- and W0 bosons themselves were not observed experimentally until 1983. The theory of the strong force, called quantum chromodynamics (QCD), acquired its modern form in 1973-74. In 1964, Murray Gell-Mann and George Zweig (a student of Feynman) independently postulated that baryons (protons and neutrons) were composed of triplets of very small, strongly interacting, fundamental particles which Gell-Mann called 'quarks'. It was also predicted that mesons were similarly composed of these same fundamental particles but in the form of quark-antiquark pairs. The proposed quarks had very unusual properties in that their charge had fractional rather than integer values. At the time only three types (also known as flavours) of quarks were known: 'up, 'down' and 'strange' (u, d and s) with electric charges 2/3, -1/3, -1/3, respectively. The proton contains 2 up quarks and 1 down quark giving it a total charge of 1; the neutron contains 2 down quarks and 1 up quark giving it a total charge of 0; mesons could be composed of a variety of quark/antiquark pairs such as uu, dd, ud, du and others. Nobody has actually isolated or seen a single individual quark since they are permanently 'confined' within observable particles like the proton and neutron from which single quarks cannot escape due to the strong inter-quark (nuclear) force, which holds the particle together. In 1964 Oscar Greenberg pointed out that having two identical quarks in the hadron's triplet of quarks violated Pauli's exclusion principle, a basic rule of quantum physics which does not allow a particle to contain more than one quark in the same quantum state. To overcome this problem he suggested that quarks should have three new degrees of freedom. In 1965 Greenberg's idea was taken up by Moo-Young Han and Yoichiro Nambu who introduced the notion of a quantum 'colour charge' with three possible values, red, green or blue; colours can also be positive or negative. Analogous to the electromagnetic force, like-coloured charges repel each other and different-coloured cha
Erklarung der Perihelbewegung des Merkur aus der allgemeinen Relativitatstheorie

Erklarung der Perihelbewegung des Merkur aus der allgemeinen Relativitatstheorie, pp. 831-839 in Sitzungsberichte der Königlich preussischen Akademie der Wissenschaften, XLVII, 18 November 1915

EINSTEIN, Albert First edition, journal issue in original printed wrappers, of one of Einstein's most important papers, in which "he presents two of his greatest discoveries. Each of these changed his life" (Pais, p. 253). "In the fall of 1915, Einstein came to the painful realization that the 'Entwurf' field equations are untenable. Casting about for new field equations, he fortuitously found his way back to equations of broad covariance that he had reluctantly abandoned three years earlier. He had learned enough in the meantime to see that they were physically viable after all . and on November 4, 1915, presented the rediscovered old equations to the Berlin Academy. He returned a week later with an important modification, and two weeks after that with a further modification. In between these two appearances before his learned colleagues, he presented yet another paper showing that his new theory explains the anomalous advance of the perihelion of Mercury. Fortunately, this result was not affected by the final modification of the field equations presented the following week" (Janssen, pp. 59-60). "The first result [reported in the present paper] was that his theory [of general relativity] 'explains . quantitatively . the secular rotation of the orbit of Mercury, discovered by Le Verrier, . without the need of any special hypothesis.' This discovery was, I believe, by far the strongest emotional experience in Einstein's scientific life, perhaps in all his life. Nature had spoken to him. He had to be right. 'For a few days, I was beside myself with joyous excitement'. Later, he told Fokker that his discovery had given him palpitations of the heart. What he told de Haas is even more profoundly significant: when he saw that his calculations agreed with the unexplained astronomical observations, he had the feeling that something actually snapped in him" (Pais, p. 253). "Einstein devoted only half a page to his second discovery: the bending of light [by gravity] is twice as large as he had found earlier. 'A light ray passing the sun should suffer a deflection of 1".7 (instead of 0".85)'" (Pais, p. 255). The confirmation of this prediction four years later by Dyson and Eddington not only confirmed Einstein's theory, but also made Einstein world famous. "Einstein's discovery resolved a difficulty that was known for more than sixty years. Urbain Jean Joseph Le Verrier had been the first to find evidence for an anomaly in the orbit of Mercury and also the first to attempt to explain this effect. On September 12, 1859, he submitted to the Academy of Sciences in Paris the text of a letter to Herve Faye in which he recorded his findings. The perihelion of Mercury advances by thirty-eight seconds per century due to 'some as yet unknown action on which no light has been thrown . a grave difficulty, worthy of attention by astronomers.' The only way to explain the effect in terms of known bodies would be (he noted) to increase the mass of Venus by at least 10 per cent, an inadmissible modification. He strongly doubted that an intramercurial planet, as yet unobserved, might be the cause. A swarm of intramercurial asteroids was not ruled out, he believed. 'Here then, mon cher confrere, is a new complication which manifests itself in the neighborhood of the sun.' Perihelion precessions of Mercury and other bodies have been the subject of experimental study from 1850 up to the present. The value 43 seconds per century for Mercury, obtained in 1882 by Simon Newcomb, has not changed. The present best value is 43".11 + 0.45. The experimental number quoted by Einstein on November 18, 1915, was 45" ± 5. "In the late nineteenth and early twentieth centuries, attempts at a theoretical interpretation of the Mercury anomaly were numerous. Le Verrier's suggestions of an intramercurial planet or planetary ring were reconsidered. Other mechanisms examined were a Mercury moon (again as yet unseen), interplanetary dust, and a possible oblateness of the sun. Each idea had its proponents at one time or another. None was ever generally accepted. All of them had in common that Newton's 1/r2 law of gravitation was assumed to be strictly valid. There were also a number of proposals to explain the anomaly in terms of a deviation from this law . These attempts either failed or are uninteresting because they involve adjustable parameters. Whatever was tried, the anomaly remained puzzling. In his later years, Newcomb tended 'to prefer provisionally the hypothesis that the sun's gravitation is not exactly as the inverse square'. "Against this background, Einstein's joy in being able to give an explanation 'without any special hypothesis' becomes all the more understandable" (Pais, pp. 253-4). "Let us briefly recapitulate Einstein's progress in understanding the bending of light. 1907. The clerk at the patent office in Bern discovers the equivalence principle, realizes that this principle by itself implies some bending of light, but believes that the effect is too small to ever be observed. 1911. The professor at Prague finds that the effect can be detected for starlight grazing the sun during a total eclipse and finds that the amount of bending in that case is 0''.87. He does not yet know that space is curved and that, therefore, his answer is incorrect. He is still too close to Newton, who believed that space is flat and who could have himself computed the 0''.87 (now called the Newton value) from his law of gravitation and his corpuscular theory of light. 1912. The professor at Zürich discovers that space is curved. Several years pass before he understands that the curvature of space modifies the bending of light. 1915. The member of the Prussian Academy discovers that general relativity implies a bending of light by the sun equal to 1".74, the Einstein value, twice the Newton value. This factor of 2 sets the stage for a confrontation between Newton and Einstein . "An opportunity to observe an eclipse in Venezuela in 1916 had to be passed up beca
Antropologium de hominis dignitate

Antropologium de hominis dignitate, natura et proprietatibus, de elementis, partibus et membris humani corporis

HUNDT, Magnus An outstanding copy, in untouched contemporary binding from the collection of Jean Blondelet, of one of the earliest works with anatomical illustrations, "includes the first illustrations of the viscera in a printed book" (GM). "Hundt's best-known work, Antropologia de hominis dignitate, natura et proprietatibus de elementis, published in 1501, is one of the three or four earliest printed books to include anatomic illustrations. At one time, Hundt's work was looked upon as the oldest printed book with original anatomic illustrations, but that is no longer believed to be the case. His Antorpologia included five full-page woodcuts, including two identical reproductions of the human head, which appeared on the back of the title page as well as later in the book. The woodcuts are crude and schematic and not done from nature, and although one of the woodcuts pictures the entire body and lists the various external parts, there is no attempt to equate the anatomical term with the actual representation. There is also a full-page woodcut of a hand with chiromantic markings, and of the internal organs of the throat and abdomen. Smaller woodcuts, including plates of the stomach, intestines, and cranium, are inserted throughout the text. The work gives a clear idea of anatomy prior to the work of Berengario da Carpi, and can be regarded as typifying late-fifteenth-century concepts. Hundt held that the stars exert more influence on the human body than on other composites of elements, and his book includes generalizations about human physiognomy and chiromancy as well as anatomy. He subscribed to the notion of the seven-celled uterus, which he apparently derived from Galen" (DSB). This is a very rare book on the market: APPC/RBH lists just the Norman copy, Christie's 1998 $85,000 modern binding; Swann Galleries 1979 $8,600 modern binding; Sotheby's 1974 $6,000 disbound. Provenance: Rear paste-down with the marking of Blondelet, and with his preferred custom morocco box by Duval. Numerous contemporary annotations throughout. "Jean Blondelet was probably the greatest, but least known, French collector of rare medical and scientific books in the 20th century" (Jeremy Norman). "Hundt's best-known work, Antropologia de hominis dignitate, natura et proprietatibus de elementis, published in 1501, is one of the three or four earliest printed books to include anatomic illustrations. At one time, Hundt's work was looked upon as the oldest printed book with original anatomic illustrations, but that is no longer believed to be the case. His Antropologia included five full-page woodcuts, including two identical reproductions of the human head, which appeared on the back of the title page as well as later in the book. The woodcuts are crude and schematic and not done from nature, and although one of the woodcuts pictures the entire body and lists the various external parts, there is no attempt to equate the anatomical term with the actual representation. There is also a full-page woodcut of a hand with chiromantic markings, and of the internal organs of the throat and abdomen. Smaller woodcuts, including plates of the stomach, intestines, and cranium, are inserted throughout the text. The work gives a clear idea of anatomy prior to the work of Berengario da Carpi, and can be regarded as typifying late-fifteenth-century concepts. Hundt held that the stars exert more influence on the human body than on other composites of elements, and his book includes generalizations about human physiognomy and chiromancy as well as anatomy. He subscribed to the notion of the seven-celled uterus, which he apparently derived from Galen" (DSB). "The Antropologium . contains four large and several small woodcuts, which are accepted among the earliest of anatomical illustrations that are a little more than schematic representation. His work contains illustrations of the internal organs but without images of bones or muscles and this work seems to be the most comprehensive representation of all the internal parts up to that time. One of those illustrations shows that trachea on the right side of the neck, passing downward to the lungs; on the left side the oesophagus is represented. In the thorax are seen the lungs and the heart. The pericardium has been opened and the stomach and intestines are figured crudely. In addition, a figure of the uterus depicting the anatomy of the uterus with seven cells (Figura matricis) is noted. These illustrations also give a clear idea of pre-Berengarian anatomy and seem to be the aggregate of the views entertained in the fifteenth century as to the position and shape of the anatomic parts" (Gurunluoglu et al, 'The history and illustration of anatomy in the Middle Ages,' Journal of Medical Biography 21 (2013), 219-229). The representation of the head was reproduced in several later works. "It is believed that Hundt's scheme had its origin in the 1493 edition of Albertus Magnus' Philosophia naturalis. A similar representation can be found in the Margarita philosophica (1503) written by Gregor Reisch (ca. 1470-1523) . The figure was subsequently reproduced in a number of medical texts published in the sixteenth century, such as Giovanni Battista Porta's (ca. 1535-1615) De humani Physiognomia, published in Padua in 1593 . In the later sixteenth century, a notable figure is Otto Casmann (d. 1607), also known as Otto Casmannus, a physician-theologian from Stade, near Hamburg in Lower Saxony, who published a number of texts such as Psycholgia anthropologica (1594) and Somatologia (1598), which appear to be elaborations of the "anthropology" of Hundt" (History of Physical Anthropology, Vol. 1, F. Spencer (ed.) (1997), p. 425). "Hundt's Antropologium discussed anatomy and physiology in their premodern forms as well as the religious and philosophical aspects of humans. Thomas Bendyshe (1865) called the Antropolgium "a purely anatomical work", and Joseph Barnard Davis (1868) added that it was "ornamented with rud
Novelle Arithmetique binaire'. [With:] ?Explication de l'Arithmetique binaire

Novelle Arithmetique binaire’. [With:] ?Explication de l’Arithmetique binaire, qui se sert des seuls caracteres 0 & 1; avec des remarques sur son utilité, & sue ce qu’elle donne le sens des anciens figues Chinoises de Fohy’. Pages 58-63 (Histoires) and 85-9 (Mémoires) in Histoire de l’Académie Royale des Sciences Année MDCCIII. Avec les Mémoires de Mathématiques & de Physique, pour la même Année

LEIBNIZ, Gottfried Wilhelm First edition, first issue, of Leibniz's invention of binary arithmetic, the foundation of the electronic computer industry. This is the second of Leibniz's great trilogy of works on mathematics and computation, following Nova methodus pro maximis et minimis (1684), his independent invention of calculus, and preceding Brevis descriptio machinae arithmeticae (1710), his (decimal) mechanical calculating machine. "A dated manuscript by Gottfried Wilhelm Leibniz, preserved in the Niedersachsische Landesbibliothek, Hannover, 'includes a brief discussion of the possibility of designing a mechanical binary calculator which would use moving balls to represent binary digits.' Though Leibniz thought of the application of binary arithmetic to computing in 1679, the machine he outlined was never built, and he published nothing on the subject until [the offered work]" (Norman). Leibniz viewed binary arithmetic less as a computational tool than as a means of discovering mathematical, philosophical and even theological truths. It was a candidate for the characteristica generalis, his long sought-for alphabet of human thought. With base 2 numeration Leibniz witnessed a confluence of several intellectual strands in his world view, including theological and mystical ideas of order, harmony and creation. ABPC/RBH list only one copy of this first issue (Zisska & Schauer, May 4, 2011, lot 461, ?5,616). The copy of the extracted leaves sold at the Hans Merkle sale (Reiss, Auktion 85, October 15, 2002, lot 696) realized ?6500. "In the domain of mathematics, Leibniz regarded binary notation as intrinsically superior to decimal notation. Over and above this advantage, however, he believed that it contained the key to resolving both the problem of conceptual primitives and the problem of adequate characters. If it could be established, as Leibniz speculated from about 1679 onwards, that the only truly primitive concepts were those of God and Nothingness (or Being and Privation), then the symbols 1 and 0 would form the basis for an adequate characteristic, whose simplest signs would stand in an immediate relation to the two conceptual primitives" (Jolley, pp. 236-237). "About this time [1679] Leibniz also outlined a design for a calculating machine to operate the four rules in binary arithmetic, though he recognised that the development of such a machine would not be easy. Owing to the great number of wheels needed, the problems related to friction and smooth movement already encountered with the ordinary calculating machine would be more serious, while the greatest difficulty would be the mechanical conversion of ordinary numbers into binary and the binary answers into ordinary numbers. Perhaps it was on account of these seemingly insuperable obstacles that Leibniz failed to mention the binary calculating machine in his correspondence. Concerning the 'binary progression' itself, he remarked to Tschirnhaus in 1682 that he anticipated from its use discoveries in number theory that other progressions could not reveal" (Aiton, p. 104). ". in April [1697] he [Leibniz] edited a collection of letters and essays by members of the Jesuit mission in China, entitled Novissima Sinica . One of the copies of the Novissima Sinica that Leibniz sent to to Verjus [Antoine Verjus, the leader of the mission] came into the hands of Joachim Bouvet, a member of the Mission who had just returned home to Paris on leave. Bouvet wrote to Leibniz on 18 October 1697 expressing his commendation of the Novissima Sinica and giving him more recent news from China . In the years that followed , the correspondence with Bouvet proved to be of great importance in relation to the dissemination of Leibniz's binary arithmetic" (ibid., pp. 213-4). "In his reply of 2 December 1697 to Bouvet's first letter, Leibniz described the nature of his own researches, in which he had shown by mathematics that the Cartesians did not have the true laws of nature. To arrive at these, he explained, it was necessary to suppose in nature not only matter but also force, and the forms or entelechies of the ancients were nothing other than forces. Bouvet, in his letter of 28 February 1698, written before his return to Peking, expressed the view that the ancient Chinese philosophy did not differ from that of Leibniz, for it supposed in nature only matter and movement, which was the same as form, or what Leibniz called force. The ancient Chinese philosophy, he added, was embodied in the hexagrams of the I ching, of which he had found the true meaning. In his view they represented in a very simple and natural manner the principles of all the sciences, or rather a complete system of a perfect metaphysics, of which the Chinese had lost the knowledge a long time before Confucius. It is in the 'Great appendix' of the I ching that the words 'yin' and 'yang' make their first appearance in philosophical terms, used to describe the fundamental forces of the universe, symbolising the broken and full lines of the trigrams and hexagrams" (ibid., p. 245). "Early in 1700 Leibniz was elected a foreign member of the reconstituted Royal Academy of Sciences in Paris. This brought him into correspondence with Fontenelle . In return for his election to the Academy, he contributed papers on the binary system of arithmetic [offered here]" (ibid., p. 218). "During his visit to Berlin in the summer of 1700, Leibniz evidently sought the collaboration of the Court mathematician Philippe Naudé in further researches on the binary system. For on his return from the conversations on Church reunion with Buchhaim in Vienna, he received a letter from Naudé containing tables of series of numbers in binary notation, including the natural numbers up to 1023. Thanking Naudé for the pains he had taken to compile these tables, Leibniz explained his intention to investigate the periods in the columns of the various series of numbers. For it was remarkable that series - such as the natural numbers, triples, squares,
A Method of Reaching Extreme Altitudes [with:] Liquid Propellant Rocket Development

A Method of Reaching Extreme Altitudes [with:] Liquid Propellant Rocket Development

GODDARD, Robert Hutchins First edition, presentation copies in the original printed wrappers, inscribed by Goddard, and with a highly important association, of the most influential early works on rocketry, which provided the foundation for the modern space age. "Having explored the mathematical practicality of rocketry since 1906 and the experimental workability of reaction engines in laboratory vacuum tests since 1912, Goddard began to accumulate ideas for probing beyond the earth's stratosphere. His first two patents in 1914, for a liquid-fuel gun rocket and a multistage step rocket, led to some modest recognition and financial support from the Smithsonian Institution . The publication in 1919 of his seminal paper 'A Method of Reaching Extreme Altitudes' gave Goddard distorted publicity because he had suggested that jet propulsion could be used to attain escape velocity and that this theory could be proved by crashing a flash-powder missile on the moon. Sensitive to criticism of his moon-rocket idea, he worked quietly and steadily toward the perfection of his rocket technology and techniques . Among Goddard's successful innovations were fuel-injections systems, regenerative cooling of combustion chambers, gyroscopic stabilization and control, instrumented payloads and recovery systems, guidance vanes in the exhaust plume, gimbaled and clustered engines, and aluminium fuel and oxidizer pumps" (DSB). The 1919 paper described work on rockets that were fed with a continuous stream of solid charges, but this method eventually proved unfeasible, and in 1922 Goddard went back to an earlier idea of his, proposed independently by Oberth in Germany and also noted by Tsiolkovsky in Russia: a liquid-fuel rocket. Goddard successfully launched the first such rocket on March 16, 1926, ushering in an era of space flight and innovation. The results of Goddard's research into liquid-fuel rockets are presented in the 1936 paper offered here. "Years after his death, at the dawn of the Space Age, Goddard came to be recognized as one of the founding fathers of modern rocketry, along with Robert Esnault-Pelterie, Konstantin Tsiolkovsky, and Hermann Oberth. He not only recognized the potential of rockets for atmospheric research, ballistic missiles and space travel but was the first to scientifically study, design and construct the rockets needed to implement those ideas. NASA's Goddard Space Flight Center was named in Goddard's honor in 1959" (Wikipedia). We have located only one presentation copy of the first work in auction records (Christie's, 13 December 2006, lot 145, £4200), a copy rebound in modern wrappers and with later institutional stamps; and one of the second (RR Auction, 2014, $4773). Provenance: "With the author's compliments" written in ink in Goddard's hand at lower right corner of front wrapper of first work, "With the author's compliments // R. H. Goddard" written in ink in Goddard's hand at upper right corner of front wrapper of second work, presented to; Clarence Hickman (signed by him in ink at top of front wrapper and on first page of text of first work). Goddard (1882-1945) became interested in space when he read H. G. Wells' science fiction classic The War of the Worlds at 16 years old. He received his B.S. degree in physics from Worcester Polytechnic in 1908, and after serving there for a year as an instructor, he began his graduate studies at Clark University in the fall of 1909. Goddard received his M.A. degree in physics from Clark in 1910, and then stayed on to complete his Ph.D. in physics in 1911. After another year at Clark as an honorary fellow in physics, in 1912 he accepted a research fellowship at Princeton University. By this time he had in his spare time developed the mathematics which allowed him to calculate the position and velocity of a rocket in vertical flight, given the weight of the rocket and weight of the propellant and the velocity (with respect to the rocket frame) of the exhaust gases. In effect he had independently developed the Tsiolkovsky rocket equation published a decade earlier in Russia. In early 1913, Goddard became seriously ill with tuberculosis and had to leave his position at Princeton. He then returned to Worcester, where he began a prolonged process of recovery at home. It was during this period of recuperation that Goddard began to produce some of his most important work. By the fall of 1914, Goddard's health had improved, and he accepted a part-time position as an instructor and research fellow at Clark University. His position at Clark allowed him to further his rocketry research, but by 1916 the cost of Goddard's rocket research had become too great for his modest teaching salary to bear. He began to solicit potential sponsors for financial assistance, beginning with the Smithsonian Institution. The Smithsonian was interested and asked Goddard to elaborate upon his initial inquiry. Goddard responded with a detailed manuscript he had already prepared, entitled A Method of Reaching Extreme Altitudes. Two years later, Goddard arranged for the Smithsonian to publish this manuscript, updated with footnotes. "In late 1919, the Smithsonian published Goddard's groundbreaking work, A Method of Reaching Extreme Altitudes. The report describes Goddard's mathematical theories of rocket flight, his experiments with solid-fuel rockets, and the possibilities he saw of exploring Earth's atmosphere and beyond. Along with Konstantin Tsiolkovsky's earlier work, The Exploration of Cosmic Space by Means of Reaction Devices, which was not widely disseminated outside Russia, Goddard's report is regarded as one of the pioneering works of the science of rocketry, and 1750 copies were distributed worldwide. Goddard also sent a copy to individuals who requested one, until his personal supply was exhausted. Smithsonian aerospace historian Frank Winter said that this paper was 'one of the key catalysts behind the international rocket movement of the 1920s and 30s.' "Goddard described extensiv
Sur l'Homme et le Développement de ses Facultés

Sur l’Homme et le Développement de ses Facultés, ou Essai de Physique Sociale

QUETELET, Lambert Adolphe Jacques First edition of Quetelet's principal work in which he presented his conception of the homme moyen ("average man") as the central value about which measurements of a human trait are grouped according to the normal distribution (this was the first time the normal distribution had been used other than as an error law). "With Quetelet's work of 1835 a new era in statistics began. It presented a new technique of statistics, or, rather, the first technique at all. The material was thoughtfully elaborated, arranged according to certain pre-established principles, and made comparable. There were not very many statistical figures in the book, but each figure reported made sense. For every number, Quetelet tried to find the determining influences, its natural causes, and the perturbations caused by man. The work gave a description of the average man as both a static and dynamic phenomenon. This work was a tremendous achievement, but Quetelet had aimed at a much higher goal: social physics, as the subtitle of the work said; the same title under which, since 1825, Comte had taught what he later called sociology. Terms and analogies borrowed from mechanics played a great part in Quetelet's theoretical explanation. To find the laws that govern the social body, one has to do what one does in physics: to observe a large number of cases and then take averages. Quetelet's average man became a slogan in nineteenth-century discussions on social science" (DSB). "A rare, three-part review in the Athenaeum concluded by remarking: 'We consider the appearance of these volumes as forming an epoch in the literary history of civilization'" (Stigler, p. 170). This work occasionally appears on the market, but we have not been able to locate any copy in original printed wrappers sold at auction. "It was in writings published in the 1830s that Quetelet (1796-1874) established the theoretical foundations of his work in moral statistics or, to use the modern term, sociology. First there was the idea that social phenomena in general are extremely regular and that the empirical regularities can be discovered through the application of statistical techniques. Furthermore, these regularities have causes: Quetelet considered his averages to be "of the order of physical facts," thus establishing the link between physical laws and social laws. But rather than attach a theological interpretation to these regularities-as Sussmilch and others had done a century earlier, finding in them evidence of a divine order-Quetelet attributed them to social conditions at different times and in different places. This conclusion had two consequences: It gave rise to a large number of ethical problems, casting doubt on man's free will and thus, for example, on individual responsibility for crime; and in practical terms it provided a basis for arguing that meliorative legislation can alter social conditions so as to lower crime rates or rates of suicide. "On the methodological side, two key principles were set forth very early in Quetelet's work. The first states that 'Causes are proportional to the effects produced by them'. This is easy to accept when it comes to man's physical characteristics; it is the assumption that allows us to conclude, for example, that one man is 'twice as strong' as another (the cause) simply because we observe that he can lift an object that is twice as heavy (the effect). Quetelet proposed that a scientific study of man's moral and intellectual qualities is possible only if this principle can be applied to them as well. The second key principle advanced by Quetelet is that large numbers are necessary in order to reach any reliable conclusions-an idea that can be traced to the influence of Laplace, Fourier, and Poisson . "Quetelet was greatly concerned that the methods he adopted for studying man in all his aspects be as 'scientific' as those used in any of the physical sciences. His solution to this problem was to develop a methodology that would allow full application of the theory of probabilities. For in striking contrast to his contemporary Auguste Comte, Quetelet believed that the use of mathematics is not only the sine qua non of any exact science but the measure of its worth . "The two memoirs which form the basis for all of Quetelet's subsequent investigations of social phenomena appeared in 1831. By then he had decided that he wanted to isolate, from the general pool of statistical data, a special set dealing with human beings. He first published a memoir entitled Recherches sur la loi de la croissance de I'homme, which utilized a large number of measurements of people's physical dimensions. A few months later he published statistics on crime, under the title Recherches sur le penchant au crime aux differens ages. While the emphasis in these publications is on what we would call the life cycle, both of them also include many multivariate tabulations, such as differences in the age-specific crime rates for men and women separately, for various countries, and for different social groups . In 1833 Quetelet published a third memoir giving developmental data on weight, Recherches sur le poids de Vhomme aux differens ages" (DSB). "Quetelet made two important advances toward the statistical analysis of social data: the first of these was formulating the concept of the average man, the second the fitting of distributions. Quetelet's first awakening to the variety of relationships latent in society may have come with his investigation of population data, but his interests soon spread. From 1827 to 1835 he examined scores of potentially meaningful relationships through the compilation of tables and the preparation of graphical displays. He examined birth and death rates by month and city, by temperature, and by time of day. He calculated the month of conception from the birth month and tried to relate it to marriage statistics. He investigated mortality by age, by profession, by locality, by season, in prisons, and
Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance

Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance

CARNOT, Nicolas Leonard Sadi First edition, very rare, and a fine copy in original state, of Carnot's only published work, which led directly to the first and second laws of thermodynamics. "Carnot, one of the most original thinkers among physical scientists, applied himself to the analysis of the cyclical operation of [heat] engines" (Dibner). "Carnot's treatise on the motive power of heat, which contains the first (albeit imperfect) statement of the second law of thermodynamics, was written to address a practical engineering problem that had occupied French physicists since 1815 - namely, how heat could be used most economically in the production of motive power . Carnot's originality lay in his recognition that the motive power of a heat engine was independent of the nature of the substance generating it - that it was a function, instead, of the transfer of heat from a warmer to a colder body. He also introduced the fundamental thermodynamic concept of completeness of cycle, in which the engine and working substance return to their original conditions. Carnot's achievement was largely ignored by his contemporaries, and the Réflexions remained forgotten until rediscovered by William Thomson (Lord Kelvin) in the 1840s; Kelvin, one of the founders of modern thermodynamics, said of Carnot's work that 'nothing in the whole range of natural philosophy is more remarkable than the establishment of general laws by such a process of reasoning' (quoted in Fox, p. 1). The first edition of Réflexions was published in an edition of six hundred copies (see Fox, p. 23, illustrating the printer's bill)" (Norman). The Réflexions is now regarded as one of the great rarities of 19th-century science, and copies such as ours in completely original state are extremely difficult to find. "After a concise review of the industrial, political, and economic importance of the steam engine, Carnot [in the Réflexions] raised two problems that he felt prevented further development of both the utility and the theory of steam engines. Does there exist an assignable limit to the motive power of heat, and hence to the improvement of steam engines? Are there agents preferable to steam in producing this motive power? As Carnot conceived it, the Réflexions was nothing more, nor less, than a 'deliberate examination' of these questions. Both were timely problems and, although French engineers had investigated them for a decade, no generally accepted solutions had been reached. In the absence of a clear concept of efficiency, proposed steam-engine designs were judged largely on practicality, safety, and fuel economy . The usual approach to these problems was either an empirical study of the fuel input and the work output of individual engines or the application of the mathematical theory of gases to the abstract operations of a specific type of engine. In his choice of problems Carnot was firmly in this engineering tradition; his method of attacking them, however, was radically new and is the essence of his contribution to the science of heat. "Previous work on steam engines, as Carnot saw it, had failed for want of a sufficiently general theory, applicable to all imaginable heat engines and based on established principles. As the foundations for his study Carnot carefully set out three premises. The first was the impossibility of perpetual motion, a principle that had long been assumed in mechanics and had recently played an important role in the work of Lazare Carnot [Sadi's father]. As his second premise Carnot used the caloric theory of heat, which, in spite of some opposition, was the most accepted and most developed theory of heat available. In the Réflexions, heat (calorique) was always treated as a weightless fluid that could neither be created nor be destroyed in any process. As an element in Carnot's demonstrations this assumption asserted that the quantity of heat absorbed or released by a body in any process depends only on the initial and final states of the body. The final premise was that motive power can be produced whenever there exists a temperature difference. The production of motive power was due 'not to an actual consumption of caloric, but to its transportation from a warm body to a cold body.' Making the analogy with a waterwheel, Carnot observed that this motive power must depend on both the amount of caloric employed and the size of the temperature interval through which it falls. In his concept of reversibility Carnot also implicitly assumed the converse of this premise, that the expenditure of motive power will return caloric from the cold body to the warm body. "The analysis of heat engines began . with an abstract, three-stage steam-engine cycle. The incompleteness of this cycle proved troublesome, and Carnot pushed the abstraction one step further, producing the ideal heat engine and the cycle that now bear his name. The 'Carnot engine' consisted simply of a cylinder and piston, a working substance that he assumed to be a perfect gas, and two heat reservoirs maintained at different temperatures. The new cycle incorporated the isothermal and adiabatic expansions and the isothermal compression of the steam engine, but Carnot added a final adiabatic compression in which motive power was consumed to heat the gas to its original, boiler temperature. In describing the engine's properties, Carnot introduced two fundamental thermodynamic concepts, completeness and reversibility. At the end of each cycle the engine and the working substance returned to their original conditions. This complete cycle not only provided an unambiguous definition of the input and output of the engine, but also rendered superfluous the detailed examination of each stage of the cycle. With each cycle the engine transferred a certain quantity of caloric from the high-temperature reservoir to the low-temperature reservoir and thereby produced a certain amount of motive power. Since each stage of the cycle could be reversed, the entire engine was reversible.
An Act for Providing a Publick Reward for such Person or Persons as shall Discover the Longitude at Sea

An Act for Providing a Publick Reward for such Person or Persons as shall Discover the Longitude at Sea

LONGITUDE ACT] First edition, and a fine copy, of the Act of 1714 establishing a reward for the discovery of a method of determining longitude at sea. "Anearly example ofameansadopted by a government for encouraging scientific discoveryand progress" (Grolier/Horblit). John Harrison (1693-1776), among others, was so encouraged and eventually solved the longitude problem with the invention of his chronometer H4. "Harrison's chronometer not only supplied navigators with a perfect instrument for observing the true geographical position at any moment during their voyage, but also laid the foundation for the compilation of exact charts of the deep seas and the coastal waters of me world . There has possibly been no advance of comparable importance in aids to navigation until the introduction of radar" (PMM 208). "The Act of 1714 constituted 24 Commissioners either by name or office; if five or more thought a longitude proposal promising, they could direct the Commissioners of the Navy to have their Treasurer issue up to £2000 in total to conduct trials. After experiments were made, the Commissioners of the Longitude or 'the major part of them' were to determine whether the tested proposal was 'Practicable, and to what Degree of Exactness'. The Act set up a three-tiered reward system for methods which were deemed successful" (Baker). This and the ensuing longitude acts passed between 1714 and 1828 set a precedent for government funding and, within fifty years, also gave rise to a unique standing body, the Board of Longitude, that encouraged and helped to define British science and technology at large. Although it is its conflict with John Harrison, who claimed the prize, which now most characterizes the Board of Longitude, and Parliament's longitude legislation, in the public mind, the Board involved itself in wide-ranging scientific, technological, and maritime activities - such as the annual publication of the Nautical Almanac, the improvement of diverse technologies, the establishment of observatories abroad, and voyages including those of Captain Cook and of Arctic exploration. The act was issued both separately, as here, and as part of the collected acts of Parliament for the 12th year of Queen Anne's reign. "The latitude at sea could easily be determined from the altitude of celestial bodies, but the early sailors had no way to measure the longitude, other than by estimating the number of miles sailed east or west, which was often little more than inspired guesswork. The lack of method was increasingly felt in the seventeenth century and was, in fact, the main reason for the founding of the Royal Observatory at Greenwich in 1675. "Two particular incidents accelerated the founding of the Board [of Longitude]. In 1707 a squadron under Sir Clowdisley Shovel ran aground off [the Isles of] Scilly with the loss of some 2,000 lives: Britain's worst maritime disaster. Then in 1713 the mathematicians William Whiston and Humphrey Ditton suggested a scheme for determining longitude by anchoring ships along the main sea-lanes and firing a shell timed to explode at a height of over a mile. The time between the flash and the corresponding sound would give the distance to any ship within range. "Although completely impractical it received widespread publicity and encouraged a petition to Parliament by several sea-captains and London merchants which suggested that Parliament should offer a prize for finding a solution. The government took this seriously and a Parliamentary committee was set up to report on the problem. Among others this included Isaac Newton, Edmond Halley and Samuel Clarke, and the committee recommended that a reward should be offered for finding longitude at sea. "A Bill was presented in June 1714 'for Providing a Publick Reward for such Person or Persons as shall Discover the Longitude at Sea.' It received the Royal Assent by Queen Anne on July 20, 1714, only 12 days before she died. "The Act offered rewards of up to £20,000 for discovering longitude at sea to within certain limits of accuracy: £10,000 if accurate to one degree of the great circle (60 nautical miles), £15,000 if to 2/3° (40 nautical miles) and £20,000 if to 1/2° (30 nautical miles). This sum was unprecedented by the standard of the day. By comparison, the Astronomer Royal's salary was originally only £100, rising to £300 under [Nevil] Maskelyne. It is estimated that £20,000 in 1714 is the equivalent of at least £1 million in the currency of the 1980s. "Half the reward was to be paid if the method extended to 80 nautical miles from shore, the place of greatest danger, the other half if successful over a longer distance, such as the voyage to the West Indies. The method had to be 'practicable and useful at sea,' a vague term which was to be the subject of much controversy later. "Small sums could be advanced towards schemes that seemed promising for further experimentation. If a method did not reach the listed limits of accuracy but was still considered to be useful, a smaller reward could be offered. These rewards were open to people of all nationalities and many applicants came from overseas, especially from France and Spain, the other two leading seafaring nations of the day. The reward also stimulated mathematicians and astronomers the world over to work on the problem. "The Longitude Act appointed a group of Commissioners who came to be known as the Board of Longitude. Their function was to consider the suitability of schemes and pass on their opinions to the government. They obviously had to be selected so that they were competent in dealing with such scientific and technical matters. They comprised admirals, the Master of Trinity House, the President of the Royal Society, the Astronomer Royal, professors at Oxford and Cambridge, and ten members of Parliament [the Astronomer Royal had the final say as to the suitability of schemes and instruments] . "The basic theory behind discovering longitude is simplicity itself. As each 15
Thèse présentée à la faculté des sciences de Paris. Sur les vitesses rélatives de la lumière dans l'air et dans l'eau

Thèse présentée à la faculté des sciences de Paris. Sur les vitesses rélatives de la lumière dans l’air et dans l’eau

FOUCAULT, Jean Bernard Léon First edition, rare, and an exceptionally fine copy, of Foucault's doctoral thesis on the speed of light, in which he provides a convincing proof for the wave theory of light. In the 1840s Foucault undertook a series of optical experiments using an apparatus of rotating mirrors to determine the velocity of light. Originally developed by Charles Wheatstone to measure the velocity of electricity, the rotating mirror apparatus had been proposed as an instrument for the measurement of light in 1838 by Dominique-François Arago who failed in his own attempts to carry out the experiment. Foucault's initial work was carried out in conjunction with the physicist Armand Hippolyte Louis Fizeau (1819-1896); but a personal dispute broke up their partnership in 1847 and the two collaborators became rivals, working separately on the same problem using the same technique. Both reached the same conclusion, but while Fizeau was the first to obtain, in 1849, a precision measurement of the velocity of light, Foucault pre-empted him in announcing, on 30 April 1850, that light travels faster in air than in water, a decisive argument in favour of the wave theory of light, which by the mid-nineteenth century had become generally accepted. In his thesis Foucault gives a detailed account of his experiment, illustrating his apparatus; it was not until 1862 that he was able to determine a numerical value for the speed of light, of about 298,000 kilometers per second, a figure significantly smaller, and more accurate, than Fizeau's. Foucault is today best known for the pendulum experiments demonstrating the rotation of the earth which he performed in 1851. Perhaps he considered these experiments to be unsuitable as a thesis topic as the result (the rotation of the earth) was well known to everyone, whereas the results of his air-and-water experiments, though expected by most scientists, were new. ABPC/RBH list five copies in the last 40 years: Christie's 2008, $17,395; Christie's, Paris 2004, ?9000; Christie's 2004, $8,812; Christie's 1999, $10,350; Christie's 1998, $7,475. "The early-to-mid 1800s were a period of intense debate on the particle-versus-wave nature of light. Although the observation of the Arago spot in 1819 may seem to have settled the matter definitively in favor of Fresnel's wave theory of light, various concerns continued to appear to be addressed more satisfactorily by Newton's corpuscular theory . "In 1834, Charles Wheatstone developed a method of using a rapidly rotating mirror to study transient phenomena, and applied this method to measure the velocity of electricity in a wire and the duration of an electric spark ['An Account of Some Experiments to Measure the Velocity of Electricity and the Duration of Electric Light,' Philosophical Transactions of the Royal Society of London, vol. 124, pp. 583-591]. He communicated to François Arago the idea that his method could be adapted to a study of the speed of light. Arago expanded upon Wheatstone's concept in an 1838 publication ['Sur un système d'expériences à l'aide duquel la théorie de l'émission et celle des ondes seront soumises à des épreuves décisives,' Comptes rendus hebdomadaires des séances de l'Académie des sciences, vol. 7, pp. 954-960], emphasizing the possibility that a test of the relative speed of light in air versus water could be used to distinguish between the particle and wave theories of light" (Wikipedia). "A comparison of this velocity in air and in water would be a clear experimental test between the wave and particle theories of light, since the former required light to travel faster in air; the latter, in water" (DSB). "In 1845, Arago suggested to Fizeau and Foucault that they attempt to measure the speed of light. Sometime in 1849, however, it appears that the two had a falling out, and they parted ways pursuing separate means of performing this experiment. In 1848-49, Fizeau used, not a rotating mirror, but a toothed wheel apparatus to perform an absolute measurement of the speed of light in air . "In 1850 and in 1862, Léon Foucault made improved determinations of the speed of light substituting a rotating mirror for Fizeau's toothed wheel. The apparatus involves light from a slit S reflecting off a rotating mirror R, forming an image of the slit on the distant stationary mirror M, which is then reflected back to reform an image of the original slit. If mirror R is stationary, then the slit image will reform at S regardless of the mirror's tilt. The situation is different, however, if R is in rapid rotation. As the rotating mirror R will have moved slightly in the time it takes for the light to bounce from R to M and back, the light will be deflected away from the original source by a small angle. "Guided by similar motivations as his former partner, Foucault in 1850 was more interested in settling the particle-versus-wave debate than in determining an accurate absolute value for the speed of light. Foucault measured the differential speed of light through air versus water by inserting a tube filled with water between the rotating mirror and the distant mirror. His experimental results, announced shortly before Fizeau announced his results on the same topic, were viewed as 'driving the last nail in the coffin' of Newton's corpuscle theory of light when it showed that light travels more slowly through water than through air. Newton had explained refraction as a pull of the medium upon the light, implying an increased speed of light in the medium. The corpuscular theory of light went into abeyance, completely overshadowed by wave theory. This state of affairs lasted until 1905, when Einstein presented heuristic arguments that under various circumstances, such as when considering the photoelectric effect, light exhibits behaviors indicative of a particle nature. "In contrast to his 1850 measurement, Foucault's 1862 measurement was aimed at obtaining an accurate absolute value for the speed of light, since his concern was to
Ueber die Bestimmung des Inhaltes eines Polyëders'

Ueber die Bestimmung des Inhaltes eines Polyëders’, pp. 31-68 in Berichte der Königlichen Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physisiche Classe 17 (1865). [With:] ?Theorie der elementaren Verwandtschaft’, pp. 18-57 in ibid. 15 (1863)

MÖBIUS, August Ferdinand First edition, journal issues in the original printed wrappers, of two of the most important papers in the early history of topology, including the introduction (and illustration) in the 1865 paper of the famous 'Möbius band' (or 'Möbius strip'). "August Möbius was one of the nineteenth century's most influential mathematicians and astronomers" (Fauvel et al.). "Möbius first described the 'Möbius band' in a paper presented to the Paris Academy in 1861 as an entry to a competition on the theme "Improve in some important point the geometric theory of polyhedra." Möbius' paper, written in bad French and containing many new ideas, was not understood by the jury, and like the other papers submitted to the competition, was not awarded the prize. The contents of the paper were published by Möbius in his articles 'Theorie der elementaren Verwandtschaft' (1863) and 'Ueber die Bestimmung des Inhaltes eines Polyëders' (1865)" (Kolmogorov & Yushkevich, p. 101). From an examination of Möbius's notebooks it is known that he discovered the Möbius strip in 1858; it was discovered independently in the same year by Johann Listing (who had coined the term 'topology' in 1847). On his discovery of a one-sided surface, Ian Stewart writes (Fauvel et al, p. 159): "It was typical that Möbius should notice a simple fact that anyone could have seen in the previous two thousand years - and typical that nobody did". Norman Biggs (ibid., p. 112) speculates that both Listing and Möbius may have been influenced in their discovery by the great Carl Friedrich Gauss (1777-1855). Gauss, Listing and Möbius all worked for many years at Göttingen; Möbius studied under Gauss and Listing freely acknowledges that he was trying to develop the topological ideas of Gauss, who himself never published anything on the subject. The Möbius band is a surface obtained by taking a strip of paper, giving one of the two ends a half twist, and then gluing together the two ends. Unlike the cylinder, obtained by joining the ends of the strip without a twist, the Möbius band has only one side. This is sometimes illustrated by saying that an ant walking around a Möbius band will return to its starting position but will be on the opposite side. This is famously illustrated in M. C. Escher's woodcut 'Möbius Strip II (Red Ants).' A further surprising property of the Möbius band is that, whereas a cylinder has two boundary curves, the Möbius band has only one: if you start at any point on the boundary and move along you will pass through every point on the boundary before returning back to your starting place. The 1865 paper contains the first published illustration of a Möbius band. In his 1865 paper, "Möbius pointed out that 'having only one side,' while intuitively clear, is difficult to make precise, and proposed a related property that could be defined in complete rigour. This property was 'orientability'. A surface is 'orientable' if you can cover it with a network of triangles, with arrows circulating round each triangle, so that whenever two triangles have a common edge the arrows point in opposite directions. If you draw a network on a plane, for example, this is what happens. On a Möbius band, no such network exists" (Stewart). By gluing the circular edge of a disc to the single edge of his band, Möbius constructed the first example of a non-orientable 'closed' surface (i.e., one without boundary curves); this is now called a 'projective plane' and is the simplest of an infinite number of such surfaces. Möbius also showed that there is no well-defined concept of volume ('Inhalt') for a non-orientable closed surface. Möbius's 1863 paper gave a classification of closed orientable surfaces. Such surfaces had been studied by Bernhard Riemann as part of his theory of complex functions (they are now called 'Riemann surfaces' in that context), and he had grasped intuitively that they are classified by the number of 'holes' (zero for a sphere, one for the surface of a doughnut, etc.), a concept that had been introduced by Simon L'Huilier in 1813, building on work of Leonhard Euler in 1752. Möbius gave the first rigorous proof of this important result. For this it was first necessary to agree when two such surfaces are to be considered 'equivalent'. Möbius introduced the idea of elementaren Verwandtschaften ('elementary relationships') between two surfaces, in which 'each point of one corresponds to a point of the other, in such a way that two infinitely neighbouring points always correspond to two infinitely neighbouring points' (such transformations are now called 'homeomorphisms'). Möbius showed that if two closed orientable surfaces have the same number of holes, there is an 'elementary relationship' from one to the other (i.e., they are 'equivalent'). His proof was remarkably modern. "He classified singular points of a 'height' function into 'elliptic' and 'hyperbolic' points and developed what from a 20th century point of view reads as a geometric presentation of the Morse theory of differentiable closed orientable surfaces" (James, p. 37). In his 1865 paper, Möbius showed that the classification theorem no longer holds for closed surfaces that are not necessarily orientable. August Ferdinand Möbius (November 17, 1790 - September 26, 1868) was born in Schulpforta, Saxony-Inhalt, and was descended on his mother's side from religious reformer Martin Luther. He studied mathematics under Carl Friedrich Gauss and Johann Pfaff. His early work was devoted to astronomy, culminating in his Die Elemente der Mechanik des Himmels (1843), which gives a through treatment of celestial mechanics without the use of higher mathematics. His best-known book, Der barycentrysche Calcul (1827) is celebrated for the introduction of homogeneous coordinates into projective geometry. Many mathematical concepts are named after him, including the Möbius transformations, important in projective geometry, and the Möbius function and the Möbius inversion formula in number th
Thirteen papers by Gödel on the logical foundations of mathematics

Thirteen papers by Gödel on the logical foundations of mathematics, together with von Neumann’s historic paper on general economic equilibrium, all first editions, in Ergebnisse eines mathematischen Kolloquiums, unter Mitwirkung von Kurt Gödel und Georg Nöbeling. Herausgegeben von Karl Menger. Heft 1-8

GÖDEL, Kurt, VON NEUMANN, John (& others) First editions, and a fine complete set in the original printed wrappers, of all eight issues of these proceedings to which Gödel contributed thirteen important papers and remarks on the foundations of logic and mathematics. The last three issues are particularly rare, and are important for containing several seminal papers on mathematical economics, notably von Neumann's "A model of general economic equilibrium" in Heft 8, which "E. Roy Weintraub, current President of the History of Economics Society, described as 'the greatest paper in mathematical economics that was ever written'" (Cabral, p. 126). "In stark contrast to the short eight years of its existence, the colloquium that met in Vienna from 1928 to 1936 had a long lasting influence on economic theory" (Debreu - winner of the 1983 Nobel Prize in Economics). The most important of the Gödel papers are perhaps 'Über Vollständigkeit und Widerspruchsfreiheit' ('On completeness and consistency') in Heft 3 and 'Zur intuitionistischen Arithmetik und Zahlentheorie' ('On intuitionist arithmetic and number theory') in Heft 4. Based on the lecture at the Colloquium required for his Habilitation, in the first paper Gödel presented a different approach to his epochal incompleteness theorem, published just a few months earlier in Monatshefte für Mathematik: instead of Russell's theory of types, in the present version he used Peano's axioms for the natural numbers; this soon became the standard approach. In the second paper, Gödel proved that intuitionist mathematics is no more certain, or more consistent, than ordinary mathematics. "By invitation, in October 1929 Gödel began attending Menger's mathematics colloquium, which was modelled on the Vienna Circle. There in May 1930 he presented his dissertation results, which he had discussed with Alfred Tarski three months earlier, during the latter's visit to Vienna. From 1932 to 1936 he published numerous short articles in the proceedings of that colloquium (including his only collaborative work) and was co-editor of seven of its volumes. Gödel attended the colloquium quite regularly and participated actively in many discussions, confining his comments to brief remarks that were always stated with the greatest precision" (DSB XVII: 350). Von Neumann also attended the colloquium in the early years. Although subsets of this collection occasionally appear on the market, complete sets of all eight issues are very rare. Working under Hans Hahn, Karl Menger (1902-85) received his PhD from the University of Vienna in 1924 and accepted a professorship there three years later. "During the academic year 1928/29, several students asked Menger to direct a Mathematical Colloquium, somewhat analogous to the philosophically motivated Vienna Circle . This Colloquium, which met on alternate Tuesdays during semester time, had a flexible agenda including lectures by members or invited guests, reports on recent publications and discussion of unsolved problems. Menger kept a record of these meetings, which he published, regularly in November of the following year, under the title 'Ergebnisse eines mathematischen Kolloquiums' . "Gödel had entered the university in 1924, and Menger first met him as the youngest and most silent member of the Vienna Circle. In 1928, Gödel started working on Hilbert's program for the foundation of mathematics, and in 1929 he succeeded in solving the first of four problems of Hilbert, proving in his PhD thesis (under Hans Hahn) that first order logic is complete: Any valid formula could be derived from the axioms. "At that time Menger, who was greatly impressed by the Warsaw mathematicians, had invited Alfred Tarski to deliver three lectures at the Colloquium. Gödel, who had asked Menger to arrange a meeting with Tarski, soon took a hand in running the Colloquium and editing its Ergebnisse. "Menger was visiting the USA [in 1931] when Gödel discovered the incompleteness theorem and used it to refute the remaining three of Hilbert's conjectures. He learned by letter that Gödel had lectured in the Colloquium 'On Completeness and Consistency'. This was the lecture required for Gödel's habilitation. The paper required for the same procedure, 'On the undecidability of certain propositions in the Principia Mathematica,' had been published in Hahn's 'Monatshefte'. In his Colloquium lecture, Gödel presented a simpler approach. Instead of Russell's theory of types, he used Peano's axioms for the natural numbers. This soon became the standard approach . "Menger was particularly fond of Gödel's results on intuitionism. These vindicated his own tolerance principle. Specifically, Gödel proved that intuitionist mathematics is no more certain, or more consistent, than ordinary mathematics ('Zur intuitionistischen Arithmetik und Zahlentheorie', Heft 4) . Menger brought Oswald Veblen to the Colloquium when Gödel lectured on this result. Veblen, who had been primed by John von Neumann, was tremendously impressed by the talk and invited Gödel to the Institute for Advanced Study during its first full year of operation: A signal honour that proved a blessing in Gödel's later years" (Karl Sigmund in Selecta Mathematica, pp. 14). The Gödel papers contained in these five volumes are as follows, with summaries based on the Annotated Bibliography of Gödel by John Dawson: 'Ein Spezialfall des Entscheidungsproblems der theoretischen Logik,' Heft 2, pp. 27-28. This undated contribution was not presented to a regular meeting of the colloquium, but appeared among the Gesammelte Mitteilungen for 1929/30. In the context of the first-order predicate calculus without equality, Gödel describes an effective procedure for deciding whether or not a certain formula is satisfiable; the procedure is related to the method used in [his dissertation Die Vollstandigkeit der Axiome des logischen Funktionenkalküls] to establish the completeness theorem. 'Über Vollständigkeit und Widerspruchsfreiheit,' Heft 3, pp. 12-13. Closely related to
Über die Erhaltung der Kraft

Über die Erhaltung der Kraft, eine physikalische Abhandlung, vorgetragen in der Sitzung der physikalischen Gesellschaft zu Berlin am 23sten Juli 1847

HELMHOLTZ, Hermann First edition, rare, of "the first comprehensive statement of the first law of thermodynamics: that all modes of energy, heat, light, electricity, and all chemical phenomena, are capable of transformation from one to the other but are indestructible and cannot be created" (PMM). "On the basis of this short paper, written when he was only twenty-six, Helmholtz is ranked as one of the founders, along with Joule and Mayer, of the principle of conservation of energy. The paper sets forth the philosophical and physical basis of the energy conservation principle: Helmholtz maintained that the scientific world view was based on two abstractions, matter and force, and since the only possible relationship that can exist among the ultimate particles of matter is a spatial one, then ultimate forces must be moving forces radically directed. This can be inferred from the impossibility of producing work continually from nothing. Helmholtz analyzed different forms of energy and different types of force and motion, grouping them into two categories, active (kinetic) and tension (potential). He also gave mathematical expression to the energy of motion, providing an experimental measure for research on all forces, including those of muscle physiology and chemistry" (Norman). "Intended expressly for 'physicists,' Helmholtz's 1847 paper must be counted as one of the most impressive first publications in the history of physics. Helmholtz had the highest regard for the principle he developed there, speaking of it fifteen years later, for example, as the most important scientific advance of the century because it encompassed all laws of physics and chemistry. On the occasion of Helmholtz's hundredth birthday, in 1921, his former student Wilhelm Wien could write that the significance of the principle was still growing" (Jungnickel & McCormach, p. 161). ABPC/RBH record 10 copies in the last fifty years, the most recent being a copy in a modern binding, without wrappers, sold at PBA Galleries in 2015, which made $27,000. "Benjamin Thompson, Count Rumford, the American-born scientist largely responsible for the foundation of the Royal Institution and the founder of the Royal Society's Rumford Medal, was the first to challenge successfully the accepted theory that heat was the manifestation of an imponderable fluid called 'caloric'. He declared, and gave experimental proof before the Royal Society in I798, that heat was a mode of motion. Rumford was, in fact, conspicuous in his day for what was considered his old-fashioned theory of heat. He harked back to the seventeenth-century views of Bacon, Locke and Newton in opposition to the fashionable modern theory of caloric, which, indeed, worked very well, especially in chemistry. "Sadi Carnot, in 1824, approached very close to the principle of the conservation of energy and his brother found among his papers an almost explicit statement of it, although Carnot had actually used the caloric theory in his researches. J. R. Mayer, in Liebig's Annalen, 1842, demonstrated its application in physiological processes, but his paper made little impression until it was reprinted as a polemic in 1867. J. P. Joule made a manuscript translation of Mayer's thesis for his own use, and, in a series of papers in the Philosophical Magazine, 1840-3, provided experimental proof of the mechanical equivalent of heat for physical phenomena" (PMM). Helmholtz worked out the principle of the conservation of force soon after completing his education in Berlin. While a student at the gymnasium in nearby Potsdam where his father taught classical languages, he had decided that he wanted to study physics. Since his father could afford this plan only if he studied physics within a medical education, in 1838 he entered the Friedrich-Wilhelms-Institut in Berlin. This state medical-surgical institution trained army physicians by providing them with a free medical education at Berlin University. Helmholtz wrote his dissertation at the university on the physiology of nerves under Johannes Muller and received his M.D. in 1842. In 1843 he published his first independent investigation in Muller's Archiv and that year took up his duties as army surgeon at Potsdam. There at the army post, he set up a small physical-physiological laboratory. He also kept up the scientific associations he had formed at Berlin University. In Berlin, Helmholtz was drawn into the circle of Muller's students, befriending especially du Bois-Reymond and Brucke, who were united in their desire to eliminate from physiology the concept of life force, in their eyes an unscientific concept left over from nature philosophy. They wanted to see how far physics and chemistry could go in explaining life processes, which brought them into contact with physicists in Berlin, above all with Magnus. Because a state examination for physicians required Helmholtz to spend a half year in Berlin, in the winter of 1845-46 he worked regularly in Magnus's private laboratory. Du Bois-Reymond, who had participated in Magnus's physical colloquium, introduced Helmholtz to the newly formed Berlin Physical Society, which was soon to provide the first audience for his work on the conservation of force. He regularly attended the meetings of the society, and for the first volume, as for later volumes, of the Fortschritte der Physik, he reported on researches in physiological heat. "Helmholtz's work on the conservation of force required a sound knowledge of mathematical physics, which he had acquired in his early years in Berlin. He had read extensively in the literature; in 1841, for example, after his first medical examinations were over, he was left with some free time, which he devoted to the study of mathematics and the advanced parts of mechanics. On his own, or with a friend, he studied the writings of Laplace, Biot, Poisson, Jacobi, and others. He attended no lectures in mathematical physics or in mathematics at Berlin; in these subjects he was largely self-t
Forces in molecules. Offprint from: Physical Review

Forces in molecules. Offprint from: Physical Review, Second Series, Vol. 56, No. 4, August 15, 1939

FEYNMAN, Richard Phillips First edition, extremely rare offprint, inscribed by Feynman, of Feynman's senior undergraduate thesis at MIT, a fundamental discovery "that has played an important role in theoretical chemistry and condensed matter physics" (Selected Papers, p. 1), published when he was just twenty-one. This is a remarkable paper, documenting the first steps in original research of one of the most brilliant minds of the twentieth century. "Feynman was one of the most creative and influential physicists of the twentieth century. A veteran of the Manhattan Project of World War II and a 1965 Nobel laureate in physics, he made lasting contributions across many domains, from electrodynamics and quantum theory to nuclear and particle physics, solid-state physics, and gravitation" (DSB). Feynman showed that "the force on an atom's nucleus is no more or less than the electrical force from the surrounding field of charged electrons - the electrostatic force. Once the distribution of charge has been calculated quantum mechanically, then from that point forward quantum mechanics disappears from the picture. The problem becomes classical; the nuclei can be treated as static points of mass and charge. Feynman's approach applies to all chemical bonds. If two nuclei act as though strongly attracted to each other, as the hydrogen nuclei do when they bond to form a water molecule, it is because the nuclei are each drawn toward the electrical charge concentrated quantum mechanically between them" (Gleick, Genius: The Life and Science of Richard Feynman). His discovery, now known as Feynman's theorem or the Feynman-Hellmann theorem, has endured as an efficient approach to the calculation of forces in molecules. "The importance of the forces on the atomic nuclei for molecular geometry, the theory of chemical binding, and for crystal structure is evident" (Selected Papers, p. 1). ABPC/RBH lists no copy of any offprint of any of Feynman's papers in Physical Review (where he published almost all of his most important work). Not on OCLC. Provenance: Signed 'R. P. Feynman' in pencil in top margin of first page. This offprint was signed by Feynman and given by him to Robert Kinsel Smith (1920-99), a classmate and personal friend of Feynman's at Princeton University, where both Feynman and Kinsel Smith studied for their PhDs (a letter from Kinsel Smith's son testifying to this provenance accompanies the offprint). Born in Far Rockaway in the Queens section of New York City, Feynman (1918-88) entered the Massachusetts Institute of Technology (MIT) in 1935 to begin his undergraduate studies. Although he originally majored in mathematics, he later switched to electrical engineering, as he considered mathematics to be too abstract. Noticing that he 'had gone too far,' he then switched to physics, which he claimed was 'somewhere in between.' To complete their bachelor's degree, all physics majors at MIT were then (as now) required to write a 'senior thesis'. "Thirteen physics majors completed senior theses in 1939. The world of accumulated knowledge was still small enough that MIT could expect a thesis to represent original and possibly publishable work. The thesis should begin the scientist's normal career and meanwhile supply missing blocks in the wall of organized knowledge, by analyzing such minutiae as the spectra of singly ionized gadolinium or hydrated manganese chloride crystals . Seniors could devise new laboratory instruments or investigate crystals that produced electrical currents when squeezed. Feynman's thesis began as a circumscribed problem like these. It ended as a fundamental discovery about the forces acting within the molecules of any substance" (Gleick). Feynman recounted his work on the thesis in an interview with Charles Weiner in March, 1966. "I went to Slater [the renowned solid-state theorist John Clarke Slater (1900-76)], and he gave me a problem, which was . why does quartz have such a small coefficient of expansion? He thought that maybe the possibility was that the quartz crystal has moveable - see it's silicone dioxide, SiO2 so I think there are oxygen's clinging to silicones, and in the motion the oxygen can swing back and forth, and it's a bent angle, turning back and forth, like the bores on the governor of an old steam engine, and when it turns - when this is oscillating, it's the same idea - it pulls the heads of the steam engine together, the ends, because the bore goes out by centrifugal force. And so the bent bottom will be shortened - I mean, it will be pulled together by the motion - and this will compensate the ordinary effects which tend to make something expand, so that the expansion will be much less than usual. Can I work out any details or estimates or something to show that in fact that's the reason that quartz doesn't expand? All right, that was the problem. I was very interested in it. The first thing I did was, I looked up the forms, crystobalite A, crystobalite B, crystal forms, and so on, to get the idea of the bonds and the angles and so on. I got in the crystal business. Then I realized I'd have to figure out how a change in forces will change the dimensions of the crystal. So then I got involved . with the connection between the forces between the atoms, and the forces - all together. For example, if a crystal is compressed, what is the compressent strength? Supposing I assume certain spring constants between all the atoms and I want to know what the elastic constants of the whole crystal are. I realized that what I had to do there was an infinite bridge truss problem, like the guys in applied engineering with bridges with a lot of members. I had an infinite number of members. But, because of the periodicity, I had an advantage that I could work out. Then I gradually developed the theory of the connection between the elastic bonds . So I worked that out, and then discovered by fooling around that I could get it for a principle of energy minimum . But anyway, in the meantime I'd f
Modular elliptic curves and Fermat's Last Theorem. [with] Ring-theoretic properties of certain Hecke algebras

Modular elliptic curves and Fermat’s Last Theorem. [with] Ring-theoretic properties of certain Hecke algebras

WILES, Andrew. & Richard Taylor First edition, journal issue, of his proof of Fermat's Last Theorem, which was perhaps the most celebrated open problem in mathematics. In a marginal note in the section of his copy of Diophantus' Arithmetica (1621) dealing with Pythagorean triples (positive whole numbers x, y, z satisfying x2 + y2 = z2 - of which an infinite number exist), Fermat stated that the equation xn + yn = zn, where n is any whole number greater than 2, has no solution in which x, y, z are positive whole numbers. Fermat followed this assertion with what is probably the most tantalising comment in the history of mathematics: 'I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.' Fermat believed he could prove his theorem, but he never committed his proof to paper. After his death, mathematicians across Europe, from the enthusiastic amateur to the brilliant professional, tried to rediscover the proof of what became known as Fermat's Last Theorem, but for more than 350 years none succeeded, nor could anyone disprove the theorem by finding numbers x, y, z which did satisfy Fermat's equation. When the great German mathematician David Hilbert was asked why he never attempted a proof of Fermat's Last Theorem, he replied, "Before beginning I should have to put in three years of intensive study, and I haven't that much time to squander on a probable failure." Soon after the Second World War computers helped to prove the theorem for all values of n up to five hundred, then one thousand, and then ten thousand. In the 1980's Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000 and more recently mathematicians could claim that Fermat's Last Theorem was true for all values of n up to four million. But no general proof was found until 1995. "Between 1954 and 1986 a chain of events of occurred which brought Fermat's Last Theorem back into the mainstream. The incident which began everything happened in post-war Japan, when Yutaka Taniyama and Goro Shimura, two young academics, decided to collaborate on the study of elliptic curves and modular forms. These entities are from opposite ends of the mathematical spectrum, and had previously been studied in isolation. "Elliptic curves, which have been studied since the time of Diophantus, concern cubic equations of the form: y2 = (x + a).(x + b).(x + c), where a, b and c can be any whole number, except zero. The challenge is to identify and quantify the whole solutions to the equations, the solutions differing according to the values of a, b, and c. "Modular forms are a much more modern mathematical entity, born in the nineteenth century. They are functions, not so different to functions such as sine and cosine, but modular forms are exceptional because they exhibit a high degree of symmetry. For example, the sine function is slightly symmetrical because 2p can be added to any number, x, and yet the result of the function remains unchanged, i.e., sin x = sin (x + 2p). However, for modular forms the number x can be transformed in an infinite number of ways and yet the outcome of the function remains unchanged, hence they are said to be extraordinarily symmetric . "Despite belonging to a completely different area of the mathematics, Shimura and Taniyama began to suspect that the elliptic curves might be related to modular forms in a fundamental way. It seemed that the solutions for any one of the infinite number of elliptic curves could be derived from one of the infinite number of modular forms. Each elliptic curve seemed to be a modular form in disguise. This apparent unification became known as the Shimura-Taniyama conjecture, reflecting the fact that mathematicians were confident that it was true, but as yet were unable to prove it. The conjecture was considered important because if it were true problems about elliptic curves, which hitherto had been insoluble, could potentially be solved by using techniques developed for modular forms, and vice versa . "Even though the Shimura-Taniyama conjecture could not be proved, as the decades passed it gradually became increasingly influential, and by the 1970s mathematicians would begin papers by assuming the Shimura-Taniyama conjecture and then derive some new result. In due course many major results came to rely on the conjecture being proved, but these results could themselves only be classified as conjectures, because they were conditional on the proof of the Shimura-Taniyama conjecture. Despite its pivotal role, few believed it would be proved this century. "Then, in 1986, Kenneth A Ribet of the University of California at Berkeley, building on the work of Gerhard Frey of the University of Saarlands, made an astonishing breakthrough. He was unable to prove the Shimura-Taniyama conjecture, but he was able to link it with Fermat's Last Theorem. The link occurred by contemplating the unthinkable - what would happen if Fermat's Last Theorem was not true? This would mean that there existed a set of solutions to Fermat's equation, and therefore this hypothetical combination of numbers could be used as the basis for constructing a hypothetical elliptic curve. Ribet demonstrated that this elliptic curve could not possibly be related to a modular form, and as such it would defy the Shimura-Taniyama conjecture. Running the argument backwards, if somebody could prove the Shimura-Taniyama conjecture then every elliptic curve must be related to a modular form, hence any solution to Fermat's equation is forbidden to exist, and hence Fermat's Theorem must be true. If somebody could prove the Shimura-Taniyama conjecture, then this would immediately imply the proof of Fermat's Last Theorem. By proving one of the most important conjectures of the twentieth century, mathematicians could solve a riddle from the seventeenth century. "The Shimura-Taniyama conjecture had remained unproven since the 1950s and so there was little optimism that it was a realistic route to a proof of Fermat'
Recherches sur le Spectre Solaire. [With:] Spectre normal du soleil. Atlas de six Planches

Recherches sur le Spectre Solaire. [With:] Spectre normal du soleil. Atlas de six Planches

ÅNGSTRÖM, Anders Jonas First edition, a very fine copy in unrestored original printed wrappers, and rare thus, of one of the founding works of spectroscopy in which Ångström demonstrated the presence of hydrogen and a number of other elements in the sun; the atlas contains his great map of the solar spectrum. Since the plates of the atlas were simply laid in to the printed wrappers (and not sewn), the wrappers were often lost or damaged; it is rare to find these wrappers present and in fine condition as in our copy. In his Nobel Prize lecture, Arthur Schawlow (who shared the 1981 prize for his contributions to the development of laser spectroscopy) wrote: "Fraunhofer had charted the dark lines in the spectrum of the Sun, and had measured their wavelengths. But it was Ångström who first identified some of these lines as corresponding to bright lines emitted by particular substances . Most importantly, he showed the red line of hydrogen." "After 1861 Ångström intensively studied the spectrum of the sun, noticing the presence of hydrogen in the solar atmosphere and confirming the probable existence there of a number of other elements. In 1868 he published the monumental Recherches sur le Spectre Solaire, which contained an atlas of the solar spectrum with measurements of the wavelengths of approximately a thousand lines determined by the use of diffraction gratings. Ångström expressed his results in units of one ten-millionth of a millimetre - a unit of length that has been named the Ångström unit in his honor. In order to have a precise basis for the new science of spectroscopy, accepted standards were needed . In 1861 Kirchhoff made a map of the solar spectrum and labeled lines with the corresponding scale readings of his own prismatic instrument. These rapidly became the almost universally accepted manner of designating spectral lines, but they were inconvenient because each observer had to correlate his own readings with those of the arbitrary Kirchhoff scale. Ångström's wavelength measurements provided a more precise and convenient reference and, after 1868, became a competing authoritative standard" (DSB). Spectroscopic studies were crucial to Max Planck's explanation of blackbody radiation, Albert Einstein's explanation of the photoelectric effect, and Niels Bohr's explanation of atomic structure. Spectra are used to detect, identify and quantify information about the chemical composition of substances in the laboratory, as well as in astronomy where they enable the determination of the chemical composition and physical properties of celestial objects. "By the time that Ångström began his studies on spectral analysis at Uppsala, a fair amount of information was known, more experimentally than theoretically, about the solar spectrum. Optics had become a subject of intensive study during the first half of the nineteenth century, but there was little interest in identifying the cause of the lines on the spectra and in appreciating their structural implications . no significant progress had been made in the eighteenth century since Newton's classic investigations on sunlight and his experiments with prisms and reflecting telescopes . Newton failed to note that the light from the sun is not perfectly homogeneous; instead it was [William Hyde] Wollaston (1766-1828) who first discovered this effect by observing the rays of sunlight admitted through a narrow slit in a window blind. Wollaston's initial observation of seven dark lines, followed by [Joseph von] Fraunhofer's work, which included a greater number of lines in the solar spectrum, lies at the root of all subsequent works, including that of Ångström. The studies of these pioneers had shown that whereas sunlight differed from ordinary white light in having a spectrum of dark lines, colored light differed from the same white light in having a spectrum in which bright lines could be seen. Ångström familiarized himself with all of this work ." (Reif-Acherman). "By the time he was appointed regular professor of physics, in 1858, Ångström had already published one of his two most famous contributions to the new scientific field of spectroscopy. The paper Optical Researches was published in Swedish in 1853 and in English and German two tears later. In it Ångström presented, in an unsystematic fashion, a number of experimental results concerning the absorption of light from electrical sparks in gases. He also made theoretical interpretations indicating, among other things, that gases absorb light of the same wavelengths that they emit when heated, and suggesting, somewhat obliquely, that the Fraunhofer lines could be explained in this way. "During the priority disputes that followed Gustav Kirchhoff's publication of the law of absorption and the explanation of the Fraunhofer lines [and his map of the solar spectrum] around 1860, Ångström and his collaborator at Uppsala University, Robert Thalén (1827-1905), vigorously defended the Swede's priority. Their claims were to some extent recognized also in Britain when the Royal Society elected Ångström foreign member in 1870 and awarded him the Rumford Medal two years later. These honors were also given in recognition of Ångström's other important spectroscopic work, an atlas of the solar spectrum published in 1868" (Biographical Encyclopedia of Astronomers). "In 1868, Ångström published his most important work, 'Recherches sur le Spectre Solaire', in Uppsala. The essay, a compendium of all of his experiments, received considerable international attention and became the standard of spectroscopy for at least a quarter of a century . Because of its considerable greater dispersive power [i.e., that of Ångström's spectrometer], the information included in Ångström's map surpassed the information found in Kirchhoff's map, and the number of visible bands rose accordingly . Several dark bands on Kirchhoff's map resolved themselves into arrays of tightly packed lines . The measurements give the places of and map the solar lines
Cursus Mathematici Practici Volumen Primum [all published]: continens Illustr. & Generosi Dn. Dn. Johannis Neperi Baronis Merchistonii &c. Scoti. Trigonometriam Logarithmicam

Cursus Mathematici Practici Volumen Primum [all published]: continens Illustr. & Generosi Dn. Dn. Johannis Neperi Baronis Merchistonii &c. Scoti. Trigonometriam Logarithmicam

URSINUS, Benjamin First edition, exceptionally rare, of the book that introduced logarithms to Continental Europe; in particular, it was through this work that Johannes Kepler, in 'a happy calamity,' as he called it, became aware of Napier's epoch-making work, a discovery that enabled him to complete his great Rudolphine Tables (1627), "the foundation of all planetary calculations for over a century" (Sparrow). "The earliest publication of Napier's logarithms on the Continent was in 1618, when Benjamin Ursinus included an excerpt from the canon, shortened by two places, in his Cursus mathematici practici. Through this work Kepler became aware of the importance of Napier's discovery and expressed his enthusiasm in a letter to Napier dated 28 July 1619, printed in the dedication of his Ephemerides (1620)" (DSB, under Napier). Ursinus assisted Kepler with the computations for the Rudolphine Tables, and Kepler presented and inscribed a copy to Ursinus (Honeyman 1800 - this copy is now held by the Adler Planetarium in Chicago); in the inscription, Kepler calls Ursinus and Tycho Brahe the scientific fathers of the tables. "The [Rudolphine] Tables was far more accurate than its predecessors - its margin of error staying within 10 seconds compared to up to 5 degrees with earlier tables. Instead of providing a sequence of planetary positions for specified days (which Kepler did in his Ephemerides), the Rudolphine Tables were set up to allow calculations of planetary positions for any time in the past or future. The finding of the longitude of a given planet at a given time was based on Kepler's equation and he exploited logarithms for this tabulation. The precise geocentric positions had to be worked out from combining the heliocentric positions of the planets and the earth that were calculated separately. Logarithmic tabulations were used again to facilitate calculation" ( Ursinus was for several years Kepler's assistant: in Prague, he made observations with Kepler of the newly discovered satellites of Jupiter, published in his Narratio (1611), and later, after Kepler had moved to Linz, lived there in Kepler's house for a year (1613/1614). A second issue of the Cursus Mathematici Practici was published in 1619 (same place and publisher). OCLC lists four copies (British Library, Chicago, Columbia, Göttingen); KVK adds no further copies outside Germany. As far as we can determine ours is the only copy of the first issue to have appeared in commerce; the Macclesfield copy of the second issue (Sotheby's, October 26, 2005, lot 2027), in an 18th century binding, realised £9600 ($16942). Provenance: Patrick Hume, 1st Earl of Marchmont (engraved armorial bookplate on front paste-down).Sir Patrick Hume (1641-1724) was a Scottish Presbyterian statesman and a supporter of William of Orange. He began his long political career in opposition during the reigns of Charles II and James VII and II. Because of his involvement in the 1685 anti-Catholic rebellion, Hume spent several years in exile in the Netherlands. He returned after the revolution of 1688 when he accompanied the Protestant William of Orange to Britain. His forfeited estates were returned to him and in 1696 he was appointed Lord Chancellor. Created Earl of Marchmont in 1697, he opposed the claims of the Jacobites and voted for Parliamentary union between Scotland and England. Benjamin Ursinus (originally Benjamin Behr, Latinized Ursinus), was born on July 15, 1587 in Sprottau in Silesia (now in Poland). Ursinus was a private tutor in Prague and then high school teacher at the Gymnasium of the Unity of the Bohemian brothers in Sobieslau and in Beuthen. From 1615 he taught at the Elector of Brandenburg's Gymnasium in Joachimsthal near Berlin, a school for gifted boys founded in 1607. From 1630 he was mathematics professor at the University of Frankfurt an der Oder, where he died in 1633 or 1634. We do not know exactly when Ursinus first came into contact with Kepler (some sources suggest that Ursinus was Kepler's student), but certainly by 1610 Ursinus was acting as Kepler's assistant. Following the publication of Sidereus nuncius (1610), Kepler, then Imperial Court Astronomer to Rudolph II, was keen to test Galileo's observations. At the end of August, "the Elector of Cologne passed through Prague and lent Kepler the very instrument earlier sent to him by Galileo. Consequently, in just over one week (from August 30 to September 8), Kepler was able to observe what he now called for the first time the 'satellites' of Jupiter, and he was careful to do so with the testimony of various named and carefully described witnesses. Presumably, these were the kind of testimonials that Kepler had expected from Galileo. The first was Benjamin Ursinus, 'a diligent student of astronomy who, from the start, because he loves this art and has decided to practice philosophizing in it, never dreams of ruining the credit necessary to a future astronomer by false witness.' But there was more to Ursinus's reliability than concern for his future reputation. Kepler explained: 'We adopted the following method: with a piece of chalk and out of sight of each other, each of us drew on a wall what he had been able to observe; afterward, each of us went at the same time to see the other's picture to see if it was in agreement. This [method] is also to be understood for the following [observations]' . From August 30 to September 5, Benjamin Ursinus was Kepler's principal co-witness" (Westman, p. 480). Kepler acknowledged Ursinus's assistance in the preface of his Narratio De Observatis a se quatuor Iouis satellitibus erronibus (1611). "In 1611 the political situation in Prague took an abrupt turn, ending Kepler's exhilarating atmosphere of intellectual freedom. The gathering storm of the Counter-Reformation reached the capital, and brought about the abdication of Rudolph II. As warfare and bloodshed surged around him, Kepler sought refuge in Linz, wh
Della Nuova Geometria Libri XV

Della Nuova Geometria Libri XV

PATRIZI, Francesco First edition, very rare, of Patrizi's important work on the concept of 'space', which is "one of the most significant and important documents for the history of mathematical epistemology in the Renaissance, and might indeed almost be considered the turning point and dividing line between ancient and modern geometry" (Prins, p. 56). "Patrizi's importance in the history of science rests primarily on his highly original views concerning the nature of space, which have striking similarities to those later developed by Henry More and Isaac Newton. His position was first set out in De rerum naturae libri II priores, alter de spacio physico, alter de spacio mathematico (Ferrara, 1587) [although this was probably published after the Nuova geometria - see below]. Rejecting the Aristotelian doctrines of horror vacui and of determinate 'place,' Patrizi argued that the physical existence of a void is possible and that space is a necessary precondition of all that exists in it. Space, for Patrizi, was 'merely the simple capacity (aptitudo) for receiving bodies, and nothing else.' It was no longer a category, as it was for Aristotle, but an indeterminate receptacle of infinite extent. His distinction between 'mathematical' and 'physical' space points the way toward later philosophical and scientific theories. The primacy of space (spazio) in Patrizi's system is also seen in his Della nuova geometria (Ferrara, 1587), the essence of which was later incorporated into the Nova de universis philosophia. In it Patrizi attempted to found a system of geometry in which space was a fundamental, undefined concept that entered into the basic definitions (point, line, angle) of the system. The full impact of Patrizi's works on later thought has yet to be evaluated" (DSB). "Patrizi's works seem to have been widely known throughout Europe and directly influenced some of the Cambridge Platonists, notably Joseph Glanville and Henry More. Henry More can be seen as a link between Patrizi and Sir Isaac Newton. Patrizi's long arguments for an isotropic, unchanging, immobile and infinite space, his vehement denunciation of the Aristotelian concept, and his establishment of space as a new philosophical term can finally be said to have taken root when Newton was able to discuss absolute space after writing: I do not define space as being well known to all" (John Christopher Henry, Francesco Patrizi and the concept of space, doctoral thesis, University of Leeds, 1977, pp. 167-168). OCLC records eight US locations, at Chicago, Columbia, Wisconsin, Burndy, Michigan, Illinois, Temple, and Cornell, and a copy at the Fisher Library, Toronto. ABPC/RBH list just three other copies: Swann, 21 October 2014, lot 346, $20,000; Sotheby's, 4 November 2004 (Macclesfield, part of a sammelband), £6250; Sotheby's, 11 April 2002 (de Vitry), £2585. "Patrizi's new geometry primarily presents a comprehensive axiomatic deductive system based on his own intuitive definitions of geometric concepts. As a religious mathematical realist, Patrizi holds that through the realization of these God given intuitions, or innate ideas, the mathematical structure of the universe can be obtained. First of all, he develops his geometric system in order to replace that of Euclid . Euclid's Elements had disregarded the concepts of continuous quantity and infinity, proceeding from basic assumptions, axioms, and postulates to develop a strictly deductive geometric system. Even though Euclid's axioms and postulates are faultless, Patrizi declares that the underlying philosophical foundation of Euclidean theory is not a suitable foundation for a science of geometry. He argues that although Euclid had defined all basic geometric concepts, including the point, line, surface and solid, he failed to formulate a proper philosophical system for defining all the other geometric concepts. In the history of thought, this was already often noted as a major logical shortcoming of Euclid's geometry. There is indeed an unbridgeable gap between Euclid's first seven definitions (of concepts such as point, line, and surface) in the first book of his Elements and the remaining definitions of the treatise. The former definitions concern only geometric concepts, which must be understood, in accordance with Aristotle, as abstractions from real things in the world. In line with this aversion to anything Aristotelian, Patrizi holds that such a system of definitions does not meet all formal logical requirements for a scientific geometry. In addition, he argues, as a mathematical realist against Aristotelian scholars, that the mathematics of the natural world is not something merely abstracted from bodies by the human mind, but that bodies themselves are constituted of mathematical space. Therefore, mathematics should precede natural philosophy. "In his Nuova geometria Patrizi defines points, lines, angles, surfaces, and solids as the subject matter of the discipline of geometry. He then goes on to argue that every space must have a minimum, maximum and mean. At this point in his explanation, he comes to the conclusion that each space must be one-dimensional (length), two-dimensional (length and breadth), or three-dimensional (length, breadth and depth) . Patrizi conceives of space as the first and most important undefined geometric concept which is lacking in Euclid's Elements. Points, lines, angles, surfaces, and solids must all be related to space as their founding concept . Patrizi holds that space, rather than the structure of the World-Soul, must be the prime subject of mathematics" (Prins, pp. 259-260). "Aristotle's philosophy of mathematics, which was the widest and most elaborate to be developed in the Classical Age and, what is more, the most influential for the following centuries, held the proper object of geometry to consist in magnitude, that is, in continuous quantity; in other words, it considered all the shapes and figures which elementary geometry studies (triangles, circles