# Sophia Rare Books

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## A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae.

HUBBLE, Edwin. First edition, a copy of the very rare offprint with outstanding provenance, of Hubble?s landmark paper, which ?made as great a change in man?s conception of the universe as the Copernican revolution 400 years before? (DSB). This paper ?is generally regarded as marking the discovery of the expansion of the universe? (Biographical Encyclopedia of Astronomers). It established what would later become known as Hubble?s Law: that galaxies recede from us in all directions and more distant ones recede more rapidly in proportion to their distance. ?? the repercussions were immense. The galaxies were not randomly dashing through the cosmos, but instead their speeds were mathematically related to their distances, and when scientists see such a relationship they search for a deeper significance. In this case, the significance was nothing less than the realization that at some point in history all the galaxies in the universe had been compacted into the same small region. This was the first observational evidence to hint at what we now call the Big Bang? (Simon Singh, Big Bang). Hubble?s ?result has come to be regarded as the outstanding discovery in twentieth-century astronomy? (DSB). Provenance: Herbert McLean Evans (1882-1971), anatomist, endocrinologist and bibliophile (bookplate); Important Scientific Books: The Richard Green Library, Christie?s, New York, 17 June 2008, lot 185.In the early 1920s most astronomers believed that the universe was static and unchanging on the large scale. Einstein himself had introduced his ?cosmological constant? in 1917 to allow solutions of the equations of general relativity corresponding to a static universe. Two such solutions were found: Einstein?s matter-filled universe and Willem de Sitter?s empty universe. The latter model attracted much interest because it predicted redshifts for very distant objects, something which had been observed as early as 1912 by Vesto Slipher. However, De Sitter?s model was conceived by astronomers to be no less static than Einstein?s. In 1922 Alexander Friedmann developed a model of an evolutionary universe, which could be expanding, and this was re-discovered by Georges Lemaître in 1927. But Lemaître went further: he established theoretically the proportional relationship between the rate of expansion and distance. Important as these theoretical developments were, it was only observational data that could establish which of the models, if any, corresponded to the actual universe.Edwin Powell Hubble (1889-1953) ?was born in 1889 in Missouri. As a young man, he was tall and athletic, known especially for his talent at boxing, basketball, and track. He earned an undergraduate degree in math and astronomy at the University of Chicago, and then studied law at Oxford on a Rhodes scholarship, following his father?s wishes. Hubble returned to the US and joined the Kentucky bar, but quickly decided law wasn?t for him. He taught high school Spanish for a year before heading back to the University of Chicago to earn his PhD in astronomy in 1917. After serving in the Army in World War I, he went to southern California to work at the Mt. Wilson observatory, home of the 100-inch Hooker telescope, the largest in the world at the time.?In the early 1920s many astronomers believed that objects then known as nebulae were nearby gas clouds in our own galaxy, and that the Milky Way was the entire universe, while others thought the nebulae were actually more distant ?island universes? separate from our own galaxy.?At Mt. Wilson, Hubble began measuring the distances to nebulae to try to resolve the issue, using a method based on an earlier discovery by Henrietta Leavitt. She had found that a type of star known as a Cepheid variable had a predictable relationship between its luminosity and its pulsation rate. Measuring the period of the star?s fluctuations in brightness would give its absolute brightness, and comparing that with the star?s apparent brightness would yield a measure of the-
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## Disquisitiones generales circa superficies curvas.

GAUSS, Carl Friedrich. First edition, the very rare separately-paginated offprint from Commentationes Societatis Regiae Scientiarum Göttingensis (Vol. VI, 1828, pp. 99-146), of this ?masterpiece of the mathematical literature? (Zeidler, p. 16). ? the crowning contribution of the period, and his last great breakthrough in a major new direction of mathematical research, was Disquisitiones generales circa superficies curvas (1828), which grew out of his geodesic meditations of three decades and was the seed of more than a century of work on differential geometry? (DSB). ?A decisive influence on the entire course of development of differential geometry was exerted by the publication of a remarkable paper of Gauss, ?Disquisitiones generales circa superficies curvas? (Göttingen, 1828), written in Latin, as was the custom in the seventeenth and eighteenth centuries. It was this paper, carefully polished and containing a wealth of new ideas, that gave this area of geometry more or less its present form and opened a large circle of new and important problems whose development provided work for geometers for many decades? (Kolmogorov & Yushkevitch, p. 7). Gauss?s Disquisitiones was, in particular, the basis for Riemann?s famous 1854 Habilitationsschrift ?Uber die Hypothesen welche die Geometrie zu Grunde liegen? (see below). ABPC/RBH list only four copies sold in the last 40 years (Gedeon, Honeyman, Norman, and Stanitz).?The surface theory of Gauss was strongly influenced by Gauss? work as a surveyor. Under great physical pains, Gauss worked from 1821 to 1825 as a land surveyor in the kingdom of Hannover ? In 1822 he submitted his prize memoir ?General solution of the problem of mapping parts of a given surface onto another surface in such a way that image and pre-image become similar in their smallest parts? to the Royal Society of Sciences in Copenhagen ? When writing his prize memoir, Gauss had apparently already worked on a more general surface theory, because he added the following Latin saying to his title page: Ab his via sterniture ad maiora (From here the path to something more important is prepared). The development of the general surface theory, however, was difficult, though the basic ideas were known to Gauss since 1816. On February 19, 1826, he wrote to Olbers: ?I hardly know any period in my life, where I earned so little real gain for truly exhausting work, as during this winter. I found many, many beautiful things, but my work on other things has been unsuccessful for months.? Finally on October 8, 1827 Gauss presented the general surface theory. The title of the paper was ?Disquisitiones generales circa superficies curvas? (Investigations about curved surfaces). The most important result of this masterpiece in the mathematical literature is the Theorema egregium? (Zeidler, p. 15).?Gauss made the parametric representation of a surface and the corresponding expression for its element of length [the ?metric?] into the foundations of the Disquisitiones. He was the first to formulate clearly and explicitly the concept of intrinsic geometry of a surface, and he proved that the curvature could be measured by a quantity (the Gaussian curvature) that belongs to intrinsic geometry, i.e., does not vary when the surface is deformed. He further developed the theory of geodesic lines [shortest paths on a surface], which also belong to intrinsic geometry ??Another useful innovation due to Gauss was the use of spherical mapping in geometry, which was usually applied in astronomy. Every oriented line is assigned the point on the unit sphere having radius vector parallel to the line. Thus a region of the surface is mapped to a region on the unit sphere using the normal. Relying on this mapping Gauss introduced the concept of a measure on the curvature K (the Gaussian curvature of the surface at the given point) as the ratio of the areas of the corresponding infinitesimal regions on the sphere and on the surface? (Kolmogorov & Yushkevitch, pp. 7-9).Gauss established the new and unexpected fact that the curvature K could be expressed entirely in terms of the metric of the surface, and therefore belongs to the intrinsic geometry of the surface. This, ?as Gauss pointed out, leads to his ?great theorem? (Theorema Egregium): If a curved surface is developed on any other surface, the measure of the curvature at each point remains invariant (ibid., p. 10). ?Gauss? Theorema egregium had an enormous impact on the development of modern differential geometry and modern physics culminating in the principle ?force equals curvature? This principle is basic for both Einstein?s theory of general relativity on gravitation and the Standard Model in elementary particle physics? (Zeidler, p. 16).?The concept of a geodesic, i.e., a shortest line, also belongs to the intrinsic geometry, since geodesic lines remain geodesics under deformation. For that reason Gauss, in studying problems of intrinsic geometry, found the equations of the geodesic lines in curvilinear coordinates and studied their behavior further. He introduced the notion of a geodesic circle, i.e., the geometric locus of the endpoints of geodesic radii of constant length emanating from a single point, and he showed that it was orthogonal to its radii? (Kolmogorov & Yushkevitch, p. 11).A second major result contained in the present paper, which perhaps had even greater ramifications in mathematics than the Theorema egregium, is a version of what is now known as the Gauss-Bonnet theorem. ?The remarkable formula found by Gauss for the sum of the angles of a geodesic triangle amounts to the statement that the excess over 180° of the sum of the angles of such a triangle in the case of a surface of positive curvature, or the deficiency in the case of a surface of negative curvature, equals the area of the spherical image of the triangle, called by Gauss the total curvature (curvature integra) of the triangle. This formula has a direct connection with Gauss? reflections and computatio-
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## BARDEEN, J. & BRATTAIN, W. H. The transistor, a semi-conductor triode, pp. 230-1 [AND] BRATTAIN, W. H. & BARDEEN, J. Nature of the forward current in Germanium point contacts, pp. 231-2 [AND] SHOCKLEY, W. & PEARSON, W. L. Modulation of conductance of thin films of semi-conductors by surface charges, pp. 232-3, in Physical Review Vol. 74, No. 2, July 15, 1948. [Offered with:] BARDEEN, J. & BRATTAIN, W. H. Physical principles involved in transistor action, pp. 1208-25 in Physical Review Vol. 75, No. 8, April 15, 1949. [Offered with:] SHOCKLEY, William, SPARKS, Morgan & TEAL, Gordon K. p-n junction transistors, pp. 151-162 in Physical Review Vol. 83, No. 1, July 1, 1951.

BARDEEN, J.; BRATTAIN, W. H.; SHOCKLEY, W.; PEARSON, W. L.; William, SPARKS, M.; TEAL, G. First edition, journal issues, documenting the invention of the transistor, ?which has been called ?the most important invention of the 20th Century.? Developed from semiconductor material, the transistor was the first device that could both amplify an electrical signal, as well as turn it on and off, allowing current to flow or to be blocked. It was small in size, generated very low heat, and was very dependable, making possible a breakthrough in the miniaturization of complex circuitry. The transistor heralded in the ?Information Age? and paved the way for the development of almost every electronic device, from radios to computers to space shuttles. For their monumental ?researches on semiconductors and their discovery of the transistor effect,? Bardeen, Shockley and Brattain were presented with the Nobel Prize in Physics in 1956 ?for their researches on semiconductors and their discovery of the transistor effect?.?The genesis of the transistor emanates, interestingly enough, from a marketing problem. In the early part of the 20th Century, AT&T was engrossed in expanding its telephone service across the continent in an effort to beat the competition. The company turned to its research and development arm, Bell Laboratories, to develop innovations to meet this need.?At the time, telephone technology was based on vacuum tubes, which were essentially modified light bulbs that controlled electron flow, allowing for current to be amplified. But vacuum tubes were not very reliable, and they consumed too much power and produced too much heat to be practical for AT&T?s needs. Furthermore, as scientists at Bell Labs discovered, transcontinental telephone communication required the use of ultrahigh frequency waves and the vacuum tubes were incapable of picking up rapid vibrations.?An all-star team of scientists was assembled at Bell Labs to develop a replacement for the vacuum tubes based on solid-state semiconductor materials. Shockley, who had received his Ph.D. in physics from the Massachusetts Institute of Technology in 1936 and joined Bell Labs the same year, was selected as the team leader. He recruited several scientists for the project, including Brattain and Bardeen.?Walter Brattain had been working for Bell Labs since 1929, the year he received his Ph.D. in physics from the University of Minnesota. His main research interest was on the surface properties of solids.John Bardeen was a theoretical physicist with an industrial engineering background. With a Ph.D. in physics from Princeton University, he was working as an assistant professor at the University of Minnesota when Shockley invited him to join the group.?The team commenced work on a new means of current amplification. In 1945, Shockley designed what he hoped would be the first semiconductor amplifier, an apparatus that consisted of ?a small cylinder coated thinly with silicon, mounted close to a small, metal plate? The device didn’t work, and Shockley assigned Bardeen and Brattain to find out why.?In 1947, during the so-called ?Miracle Month? of November 17 to December 23, Brattain and Bardeen performed experiments to determine what was preventing Shockley?s device from amplifying. They noticed that condensation kept forming on the silicon. Could this be the deterrent? Brattain submerged the experiment in water ?inadvertently creating the largest amplification thus far.? Bardeen was emboldened by this result, and suggested they modify the experiment to include a [gold] metal point that would be pushed into the silicon surrounded by distilled water. At last there was amplification, but disappointingly, at a trivial level.?But the scientists were galvanized by the meager result, and over the next few weeks, experimented with various materials and set ups. They replaced the silicon with germanium, which resulted in amplification 330 times larger than before. But it only functioned for low frequency currents, whereas phone lines, for example, would need to handle the many complicated frequencies of the human voice.?Next, they replaced the liquid with a layer of germanium dioxide. When some of the oxide layer accidentally washed away, Brattain continued the experiment shoving the gold point into the germanium and voila! Not only could he still achieve current amplification, but he could do so at all frequencies. The gold contact had put holes in the germanium and the punctures ?canceled out the effect of the electrons at the surface, the same way the water had.? Their invention was finally increasing the current at all frequencies.?Bardeen and Brattain had achieved two special results: the ability to get a large amplification at some frequencies, and a small amplification for all frequencies. Their goal now was to combine the two. The essential components of the device thus far were the germanium and two gold point contacts that were fractions of a millimeter apart. With this in mind, Brattain placed a gold ribbon around a plastic triangle, and cut it through one of the points. When the point of the triangle touched the germanium, electric current entered through one gold contact and increased as it rushed out the other. They had done it ? it was the first point-contact transistor. On December 23, Shockley, Bardeen and Brattain presented their ?little plastic triangle? to the Bell Labs VIPs and it became official: the super star team had invented the first working solid state amplifier.?Following the triumph of the transistor, the three amplifying architects went their separate ways. Shockley left Bell Labs in 1955 to become the Director of the ?Shockley Semi-Conductor Laboratory of Beckman Instruments, Inc. in Mountain View, Ca. His company was one of the first of its kind in Northern California and quickly attracted more semiconductor labs and related computer firms to the area. Soon the region had a new moniker: Silicon Valley.?Bardeen left Bell Labs in 1951 for a professorial appointment in electrical engineering and physics at t-
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## Atomic theory of liquid helium near absolute zero. Offprint from Physical Review, Vol. 91, No. 6, September 15, 1953.

FEYNMAN, Richard Phillips. First edition, extremely rare offprint, of the first of Feynman?s important papers which provided a quantum mechanical explanation of the superfluidity of liquid helium at temperatures below the ?lambda-point? of 2.18K. ?The dramatic announcement of superfluidity of liquid He4 in 1938 is one of the defining moments in modern physics? (Griffin, p. 1). In 1938, Pyotr Kapitza in Moscow and John Allen and Donald Misener in Cambridge (UK) discovered independently that, at sufficiently low temperatures, liquid He4 has zero viscosity. Phenomenological theories of this property of superfluidity were developed by Fritz London and Laszlo Tisza before World War II, and by Lev Landau in the 1940s. Successful as these theories were, they lacked an atomistic foundation. ?Between 1953 and 1958, Feynman published a seminal series of papers on the atomic theory of superfluid helium ? A significant part of Feynman?s central contribution was the demonstration that these phenomenological concepts arose directly from the fundamental quantum mechanics of interacting bosonic atoms with strong repulsive cores. One of his earliest helium papers [offered here] showed in detail how the symmetric character of the many-body wave function severely restricts the allowed class of low-lying excited states? (Selected Papers of Richard Feynman (2000), p. 313). 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.? No copies in auction records, or on OCLC.Helium was first liquefied by the Dutch physicist Heike Kamerlingh Onnes (1853?1926) in 1908, who is best known for his discovery three years later of superconductivity in mercury at very low temperatures. ?Kamerlingh Onnes also observed the transition to the related phenomenon of superfluidity in liquid helium in an experiment performed in 1908, without recognizing it. The nominal discovery of superfluidity came in 1937 when Pyotr Kapitza in the Soviet Union and John Allen and Donald Misener at Cambridge independently discovered it. Three papers were published, one after the other, in Nature on January 8, 1938. Appallingly, when Kapitsa was awarded the Nobel Prize in 1978, no mention was made of Allen?s simultaneous discovery, probably because of the dominance of Kapitsa?s group after the war? (Purrington, p. 337). (See Griffin for a detailed analysis of the relation between the work of Kapitsa and Allen & Misener.)?The most spectacular signature of the transition of liquid 4He into the superfluid phase is the sudden onset of the ability to flow without apparent friction through capillaries so small that any ordinary liquid (including 4He itself above the lambda transition) would be clamped by its viscosity; thus, a vessel that was ?helium-tight? in the so-called normal phase (i.e., above the lambda temperature) might suddenly spring leaks below it. Related phenomena observed in the superfluid phase include the ability to sustain persistent currents in a ring-shaped container; the phenomenon of film creep, in which the liquid flows without apparent friction up and over the side of a bucket containing it; and a thermal conductivity that is millions of times its value in the normal phase and greater than that of the best metallic conductors. Another property is less spectacular but is extremely significant for an understanding of the superfluid phase: if the liquid is cooled through the lambda transition in a bucket that is slowly rotating, then, as the temperature decreases toward absolute zero, the liquid appears gradually to come to rest with respect to the laboratory even though the bucket continues to rotate. This non-rotation effect is completely reversible; the apparent velocity of rotation depends only on the temperature and not on the history of the system? (Britannica). The papers of Kapitsa and Allen & Misener ?stimulated feverish activity in the period leading up to World War II, and in the 1950s developed into a major research area called ?quantum fluids.? The phenomenon of flow without any measurable viscosity suggested that liquid He4 below the transition temperature of 2.18K was some strange new phase of matter. Within a few weeks after the discovery, Fritz London [and Laszlo Tisza] suggested that this new phase might have some connection with the phenomenon of Bose-Einstein condensation (BEC). This was originally predicted by Einstein to occur in an ideal gas of atoms in a 1925 paper, but this had been largely discounted as wrong over the next decade? (Griffin, p. 1). Landau rejected the description of He4 below the lambda-point as an ideal Bose-Einstein gas, and proposed instead to derive the properties of the superfluid from a consistent quantum-mechanical approach to a fluid. His phenomenological ?two-fluid? model of superfluidity, published in 1941, led to his award of the Nobel Prize for Physics in 1962.Then, in the spring of 1953, ?Richard Feynman entered the scene. He set himself the task of providing a theoretical understanding of the problem of liquid helium on an atomic basis, which could only be done if one approached the problem from first principles. While he greatly admired Landau?s contributions to and successes in the field, Feynman pointed out several weaknesses in Landau?s theory. Notably, Landau?s quantum hydrodynamical approach treated Helium II [i.e., He4 below the lambda-point] as a continuous medium which right from the beginning sacrificed the atomic structure of the liquid and thus forestalled the possibility of calculating the various characteristics of the system, such as the various parameters, on an atomic basis ? By writing ?the exact partition function as an integral over trajectories, using the space-time approach to quantum mechanic-
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## Historiae coelestis libri duo: quorum prior exhibet catalogum stellarum fixarum Britannicum novum & locupletissimum, una cum earundem planetarumque omnium observationibus sextante, micrometro, &c. habitis; posterior transitus syderum per planum arcus meridionalis et distantias eorum a vertice complectitur. Observante Johanne Flamsteedio in Observatorio Regio Grenovicensi continua serie ab anno 1676 ad annum 1705 completum.

FLAMSTEED, John. The true first edition, extremely rare, of Flamsteed?s catalogue of fixed stars and sextant observations, the foundation of modern observational astronomy. Flamsteed?s catalogue was far more extensive and accurate than anything that had gone before. It was the first constructed with instruments using telescopic sights and micrometer eyepieces; Flamsteed was the first to study systematic errors in his instruments; he was the first to urge the fundamental importance of using clocks and taking meridian altitudes; and he insisted on having assistants to repeat the observations and the calculations. The catalogue contains about 3000 naked eye stars (Ptolemy and Tycho listed 1000, Hevelius 2000) with an accuracy of about 10 seconds of arc. However, Flamsteed, although appointed Astronomer Royal in 1675, by the turn of the eighteenth century had still not published any of his observations. Isaac Newton and Edmond Halley pressed him to do so; Flamsteed?s refusal led to one of the most famous, and bitterest, disputes in the history of astronomy, and to the present work being published against Flamsteed?s will. Flamsteed?s response, in 1716, was to destroy 300 of the 400 copies printed, so just a few years after publication no more than 100 copies survived. Flamsteed published his own, ?authorised?, version of his star catalogue in 1725. ABPC/RBH list three copies: 1. Sotheby?s, April 3, 1985, lot 287, £11,000; Bonham?s, November 26, 1975, lot 171, £5,400; previously sold: Sotheby?s, May 7, 1935, lot 98, £29 (Halley?s annotated copy, lacking the star catalogue). 3. Sotheby?s, May 7, 1935, lot 99, £10.10s (the present copy). OCLC lists 11 copies in the US.Provenance: Edward Henry Columbine (1763-1811), hydrographer and colonial governor (signature ?E. H. Columbine? on title); Radcliffe Observatory, Oxford (Sotheby?s Catalogue of the Valuable Library Removed From, The Radcliffe Observatory, Oxford, Tuesday, 7th May, 1935).?Born a somewhat sickly child at Denby, near Derby, Flamsteed?s condition seems to have worsened in 1660 by what sounds like an attack of rheumatic fever. He was taken away from school and devoted himself to the study of mathematics and astronomy. A visit to Ireland in 1665 to be touched by Vincent Greatrakes, a famous healer of the day as a seventh son of a seventh son, had no effect upon his health. Shortly afterwards, however, his work began to be noticed by a number of Fellows of the Royal Society. Amongst these was Sir Jonas Moore, who was considering building a private observatory for Flamsteed. It proved unnecessary, for in 1675 Flamsteed was appointed to be the first Astronomer Royal by Charles II. As the first holder of the post, Flamsteed was responsible for the building and organisation of the new observatory at Greenwich. He also found that on a salary of £100 a year he was expected to engage and pay his own staff, and to provide his own instruments. Although some instruments were donated by Moore and others, Flamsteed still found it necessary to spend £120 of his own money on a mural arc. Made and divided by Abraham Sharp it was ready for use in September 1689. As a result of this expenditure, all observations made after 1689 seemed to Flamsteed to be unarguably his own property, and his to do with as he willed.?He met Newton for the first time in Cambridge in 1674. The first substantial issue between them arose over the nature of the comet of 1680-1. Newton was convinced that two comets were present and in letters to Flamsteed argued so at length. Flamsteed, however, insisted only one comet was present, a position Newton finally accepted in September 1685. Relations remained cordial and in 1687 Flamsteed was one of the few scholars selected to receive a presentation copy of Principia. It contained, he noted, only ?very slight acknowledgements? to his Greenwich observations.?On 1 September 1694 Newton paid his first visit to Greenwich. He spoke with Flamsteed about the moon. Newton was keen to examine Flamsteed?s lunar data in order to correct and improve the lunar theory presented in Principia. Flamsteed offered to loan Newton 150 ?places of the moon? on two conditions: firstly, that Newton would not show the work to anyone else; secondly, and more unreasonably, Newton would have to agree not to reveal any results derived from Flamsteed?s observations to any other scholar. It was the beginning of an ill-tempered dispute which would last until Flamsteed?s death. His own version of the quarrel is contained in his History of his own Life and Labors published in Baily (An Account of the Revd John Flamsteed (1966), pp. 7-105). It is a most bitter document.?None of Newton?s proposals found favour with Flamsteed. The offer in November 1694 ?to gratify you to your satisfaction? brought the answer that he was not tempted with ?covetousness? and the lament that Newton could have ever thought so meanly of him. An offer in 1695 to pay Flamsteed?s scribe two guineas for his transcriptions brought an equally forthright rejection. It was enough, Newton was told, to offer ?verball acknowledgements?; a ?superfluity of monys?, he found, ?is always pernicious to my Servants it makes them run into company and wast their time Idly or worse? If Newton asked for ?your Observations only?, Flamsteed complained of being treated like a drudge; if, however, calculations were asked for as well, Flamsteed would respond that such work required all kinds of tedious analysis for which he had little time ??Over the period 1694-5 Newton received another 150 observations. They were, however, none too reliable, having been made with the help of a stellar catalogue constructed with the help of a sextant alone. By this time Flamsteed was beginning to resent Newton?s somewhat imperial tone. ?But I did not think myself obliged?, he complained, ?to employ my pains to serve a person that was so inconsiderate as to presume he had a right to that which was only a courtesy (Baily, p. 63). Consequently, he returne-
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## Über formal unentscheidbare Sätze der Principia Mathematica undver wandter Systeme I. Offprint from: Monatshefte für Mathematik und Physik 38, 1931. [Bound with:] Über die Vollständigkeit der Axiome des logischen Funktionenkalküls. Offprint from: Monatshefte für Mathematik und Physik 37, 1930. Leipzig: Akademische Verlagsgesellschaft, 1931, 1930. [Bound with:] VON WRIGHT, Georg Henrik. Typed letter signed Georg Henrik von Wright in Swedish on Academy of Finland letterhead.

GÖDEL, Kurt. First edition, extremely rare author?s presentation offprint, of Gödel?s famous incompleteness theorem, ?one of the major contributions to modern scientific thought? (Nagel & Newman). ?Every system of arithmetic contains arithmetical propositions, by which is meant propositions concerned solely with relations between whole numbers, which can neither be proved nor be disproved within the system. This epoch-making discovery by Kurt Gödel, a young Austrian mathematician, was announced by him to the Vienna Academy of Sciences in 1930 and was published, with a detailed proof, in a paper in the Monatshefte für Mathematik und Physik, Volume 38, pp. 173-198? (R. B. Braithwaite in Gödel/Meltzer, p. 1). ?This theorem is an important limiting result regarding the power of formal axiomatics, but has also been of immense importance in other areas, such as the theory of computability? (Zach, p. 917). Gödel ?obtained what may be the most important mathematical result of the 20th century: his famous incompleteness theorem, which states that within any axiomatic mathematical system there are propositions that cannot be proved or disproved on the basis of the axioms within that system; thus, such a system cannot be simultaneously complete and consistent. This proof established Gödel as one of the greatest logicians since Aristotle, and its repercussions continue to be felt and debated today? (Britannica). The offprint of Gödel?s incompleteness theorem is here accompanied by an author?s presentation offprint of his earlier completeness theorem for first-order logic. ?In his doctoral thesis, ?Über die Vollständigkeit des Logikkalküls? (?On the Completeness of the Calculus of Logic?), published in a slightly shortened form in 1930, Gödel proved one of the most important logical results of the century?indeed, of all time?namely, the completeness theorem, which established that classical first-order logic, or predicate calculus, is complete in the sense that all of the first-order logical truths can be proved in standard first-order proof systems. This, however, was nothing compared with what Gödel published in 1931?namely, the incompleteness theorem: ?Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I? (?On Formally Undecidable Propositions of Principia Mathematica and Related Systems?)? (Britannica). Gödel intended to write a second part to the 1931 paper, but this was never published. OCLC lists two copies of the 1931 offprint (both in Canada), and none of the 1930 offprint. ABPC/RBH list two copies of each offprint, the most recent being those sold at Christie?s, London, 19 November 2014, which realised £104,500 ($167,000) and £35,000 ($55,930), respectively.Provenance: The history of the present volume is explained in the accompanying letter from von Wright to von Plato, which reads, in translation:11 Oct. 2000Dear Jan,The two essays were in the estate of Eino Kaila. In all probability, he had them directly from the Author. I hope that you appreciate having them. I had them bound together and hand them now, on the day of your inaugural lecture as Swedish professor of philosophy, to you with my wishes for the best of luck.Your devoted,Georg Henrik von Wright.The Finnish philosopher Eino Kaila (1890-1958) worked in the early 1930s in Vienna and became associated to the Vienna Circle. He introduced its ideas to Finnish philosophical debate in Der Logistische Neupositivismus (1930, The new logical-positivism) and Inhimillinen tieto (1939, Human knowledge), an overview of the epistemological theory of logical empiricism. Kaila knew personally several members of the Circle and took part in its sessions, as did Gödel. After Kaila?s death, the offprints were acquired by Georg Henrik von Wright (1916-2003), the famous Finnish philosopher (partly of Scottish ancestry) who had studied under Kaila at the University of Helsinki. Von Wright was also a relative of Kaila: his mother was the cousin of Kaila?s wife Anna. Von Wright, who made major contributions to logic and the philosophy of science, and latterly in ethics and the humanities, was deeply influenced by Ludwig Wittgenstein, succeeded him as professor at Cambridge University from 1948-52, and was later executor of Wittgenstein?s estate. Von Wright was the first holder of the Swedish-language Chair of Philosophy at the University of Helsinki (he was a member of the Swedish-speaking minority in Finland), a post he held from 1946 until his retirement in 1961, when he was appointed to the 12-member Academy of Finland. He is one of the very few philosophers to whom a volume is dedicated in the Library of Living Philosophers series. Its current editor, Randall E. Auxier, has written: ?There is no Nobel Prize in philosophy, but being selected for inclusion in the Library of Living Philosophers is, along with the Gifford Lectures, perhaps the highest honor a philosopher can receive.?In the year 2000, von Wright had the two offprints bound together as the present volume, with the spine lettered ?Gödel: ZWEI AUFSATZE? (Gödel: Two Essays), and presented it to his successor as Swedish-language Chair of Philosophy at the University of Helsinki, Jan von Plato (b. 1951). Von Plato works on proof theory, and is the author of Structural Proof Theory (2001) and Proof Analysis: A Contribution to Hilbert’s Last Problem (2011).Following his graduation from the Gymnasium in Brno, Moravia, in 1924, Gödel (1906-78) went to Vienna to begin his studies at the University. Vienna was to be his home for the next fifteen years, and in 1929 he was also to become an Austrian citizen. Gödel?s principal teacher was the German mathematician Hans Hahn (1879-1934), who was interested in modern analysis and set-theoretic topology, as well as logic, the foundations of mathematics, and the philosophy of science. It was Hahn who introduced Gödel to the group of philosophers around Moritz Schlick (1882-1936); this group was later baptized as the ?Vienna Circle? and be-
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## Autograph report in Hookes hand, and signed by him, as surveyor of the City of London following the Great Fire, concerning a disagreement arising from the rebuilding of a structure on Ludgate Hill in the burnt district. Countersigned by Hookes fellow City Surveyor John Oliver. Dated 4 July 1670. [MATTED WITH:] HOLLAR, Wenceslaus. A Map or Groundplot of the Citty of London and the Suburbes thereof, that is to say all which is within the iurisdiction of the Lord Mayor or properlie calld’t London: by which is exactly demonstrated the present condition thereof since the last sad accident of fire. The blanke space signifeing the burnt part & where the houses are exprest, those places yet standig [sic]. London: Sold by John Overton at the White House in little Brittaine, next door to S. Bartholomew gate, 1666.

HOOKE, Robert. A very rare document related to the Great Fire of London written and signed by the great polymath Robert Hooke (1635-1703), with an equally rare separately-issued map showing the destruction caused by the fire. Starting at a bakery on Pudding Lane sometime after midnight on September 2, 1666, The Great Fire of London consumed over 13,000 houses, as well as numerous churches (including St. Paul?s cathedral) and other buildings. Charles II sought to rebuild as soon as possible to limit unrest and possible rebellion and called for plans from Robert Hooke, John Evelyn, Christopher Wren, and others. Hooke was appointed Surveyor of the City of London and, with Wren, was the chief architect for its rebuilding. As Surveyor Hooke was the arbiter of disputes erupting out of the staking-out process whereby party walls had been altered or streets widened. The present document is a report on such a dispute, between William Sanders (or Saunders) Draper and John Rowly Skinner over the rebuilding of their shop and residence on Ludgate Hill within the burnt district. Autograph documents by Hooke are extremely rare, with only two examples on the market in the last quarter century: Hooke?s manuscript notebook recording proceedings of the Royal Society (sold by private treaty to the Royal Society by Bonham’s in 2006 for a reported £1,000,000) and a signed document being a King’s Warrant for a patent for Hooke’s watches with springs (sold by Bloomsbury Auctions for £23,100 in 1991). The present autograph document is accompanied by an important map of London following the fire, published in December 1666, and described by John Evelyn as ?the most accurate hitherto extant? (see Letterbooks, epistle CCLXXXI). ?Hollar was to be employed in the preparation of surveys for rebuilding the city and was in close touch with the cartographic elite of his day, the quality of his work is apparent? (Glanville). The present map is an example of the first state, with Overton’s ?White horse in little Brittaine? address. We find no examples of this map appearing on the market, and only three institutional holdings (British Library, Harvard and the Bibliothèque Nationale).?In the early morning of Sunday, September 2, 1666, embers in the oven of Thomas Farriner?s bakery set fire to the wharves along the Thames. Despite the dry summer beforehand, the city administration reacted without much concern; Lord Mayor Thomas Bludworth, London?s chief official, infamously quipped that ?a woman might piss it out.? As if in a Greek tragedy, hubris in the face of a mightier power became the city?s downfall. Whipped up by the wind and enabled by a lack of adequate firebreaks, the fire spread rapidly, engulfing the city for three more days. Forced onto a boat on the Thames, diarist Samuel Pepys watched the flames from nearly the same view as the creators of the city?s maps and prints. Instead of an idyllic medieval town, Pepys saw ?one entire arch of fire from this to the other side of the bridge, and in a bow up the hill.? Only when the winds died down on Wednesday the 5th did the blaze subside, revealing the extent of the devastation. Evelyn?s diary entry from the 10th reads in full: ?I went again to the ruines, for it was now no longer a Citty.? Indeed, while only eight people perished in the flames, London was left fundamentally changed. Over four-fifths of the walled city lay in ashes, with at least 13,000 houses and hundreds of shops, halls, and churches destroyed. Hundreds of thousands of people wandered without shelter, displaced from their now charred homes. Beyond the human cost, London?s former cityscape, upon which the city had long been mapped and conceived, lay ruined. The conflagration ?obliterated at a stroke virtually every trace of a medieval city that had been six centuries in the making,? observed historian Neil Hanson. Whether tragedy or opportunity, the Great Fire burnt down one London and left open the possibility of creating another. Evelyn did not exaggerate in concluding, ?London was, but is no more.??Still staggering from the scale of the losses, King Charles II and the city government acted swiftly but without a coherent plan. Five days after the fire, the Court of Common Council forbid property owners from immediate reconstruction. Charles himself then issued a proclamation on the matter three days later. On the surface, he promised an idealistic vision of ?a much more beautiful city? that would become ?the most convenient and noble for the advancement of trade of any city in Europe.? He prohibited hasty and unplanned rebuilding, authorizing the removal of any unapproved construction. Nonetheless, Charles denied that ?any particular person?s right and interest [would] be sacrificed to the public benefit or convenience.? As such, his grand ideas, like widening the main streets and building a city wharf, lacked any specific locational detail. Instead, he pledged a comprehensive survey of the destroyed properties before any plan was finalized and promised ?a plot or model ? for the whole building through those ruined places.? Regardless of the specifics, Charles recognized the necessity of cartography and surveys in order to realize his vision. Mapping would no longer be a years-long pursuit for travel guides and artists. Charles needed a map?a new kind of map?and he needed it fast.?The king?s plan required two elements: a detailed survey of land ownership and a map of which areas had been burnt down. For the latter, Charles turned to the man most experienced at depicting London: Wenceslaus Hollar. Within days, Hollar?s request to map the fire?s results received an enthusiastic response from a government desperate to use cartography to reshape the city. On September 10, Hollar and associate Francis Sandford were tasked ?to take an exact plan and survey of the city, as it now stands after the calamity of the late fire.? They set to work immediately, surveying the damage and creating a map at an unprecedented speed ??Ho-
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## Disquisitiones Arithmeticae.

GAUSS, Carl Friedrich. First edition, rare, of Gauss? masterpiece, ?a book that begins a new epoch in mathematics ? Gauss ranks, together with Archimedes and Newton, as one of the greatest geniuses in the history of mathematics? (PMM). ?Published when Gauss was just twenty-four, Disquisitiones arithmeticae revolutionized number theory. In this book Gauss standardized the notation; he systemized the existing theory and extended it; and he classified the problems to be studied and the known methods of attack and introduced new methods ? The Disquisitiones not only began the modern theory of numbers but determined the direction of work in the subject up to the present time. The typesetters of this work were unable to understand Gauss? new and difficult mathematics, creating numerous elaborate mistakes which Gauss was unable to correct in proof. After the book was printed Gauss insisted that, in addition to an unusually lengthy four-page errata, the worst mistakes be corrected by cancel leaves to be inserted in copies before sale ? Gauss?s highly technical work was printed in a small edition, and the difficulty of understanding it was hardly alleviated by the sloppy typesetting? (Norman). ?In the late eighteenth century [number theory] consisted of a large collection of isolated results. In his Disquisitiones Gauss summarized previous work in a systematic way, solved some of the most difficult outstanding questions, and formulated concepts and questions that set the pattern of research for a century and still have significant today. He introduced congruence of integers with respect to a modulus (a ? b (mod c) if c divides a – b), the first significant algebraic example of the now ubiquitous concept of equivalence relation. He proved the law of quadratic reciprocity, developed the theory of composition of quadratic forms, and completely analyzed the cyclotomic equation. The Disquisitiones almost instantly won Gauss recognition by mathematicians as their prince? (DSB).?The awe that [Disquisitiones arithmeticae] inspired in mathematicians was displayed to the cultured public of the Moniteur universel ou Gazette nationale as early as March 21, 1807, when Louis Poinsot, who would succeed Joseph-Louis Lagrange at the Academy of Sciences six years later, contributed a full page article about the French translation of the Disquisitiones arithmeticae: ?The doctrine of numbers, in spite of [the works of previous mathematicians] has remained, so to speak, immobile, as if it were to stay for ever the touchstone of their powers and the measure of their intellectual penetration. This is why a treatise as profound and as novel as his Arithmetical Investigations heralds M. Gauss as one of the best mathematical minds in Europe.??A long string of declarations left by readers of the book, from Niels Henrik Abel to Hermann Minkowski, from Augustin-Louis Cauchy to Henry Smith, bears witness to the profit they derived from it. During the XIXth century, its fame grew to almost mythical dimensions. In 1891, Edouard Lucas referred to the Disquisitiones Arithmeticae as an ?imperishable monument [which] unveils the vast expanse and stunning depth of the human mind,? and in his Berlin lecture course on the concept of number, Leopold Kronecker called it ?the Book of all Books? ? Gauss?s book is now seen as having created number theory as a systematic discipline in its own right, with the book, as well as the new discipline, represented as a landmark of German culture ??Gauss began to investigate arithmetical questions, at least empirically, as early as 1792, and to prepare a number-theoretical treatise in 1796 (i.e., at age 19 and, if we understand his mathematical diary correctly, soon after he had proved both the constructibility of the 17-gon by ruler and compass and the quadratic reciprocity law). An early version of the treatise was completed a year later. In November 1797, Gauss started rewriting the early version into the more mature text which he would give to the printer bit by bit. Printing started in April 1798, but proceeded very slowly for technical reasons on the part of the printer. Gauss resented this very much, as his letters show; he was looking for a permanent position from 1798. But he did use the delays to add new text, in particular to sec. 5 on quadratic forms, which had roughly doubled in length by the time the book finally appeared in the summer of 1801.?The 665 pages and 355 articles of the main text are divided unevenly into seven sections. The first and smallest one (7 pp., 12 arts.) establishes a new notion and notation which, despite its elementary nature, modified the practice of number theory:?If the number a measures the difference of the numbers b, c, then b and c are said to be congruent according to a; if not, incongruent; this a we call the modulus. Each of the numbers b, c are called a residue of the other in the first case, a nonresidue in the second.? The corresponding notation b ? c (mod a) is introduced in art. 2. The remainder of sec. 1 contains basic observations on convenient sets of residues modulo a and on the compatibility of congruences with the arithmetic operations ??Section 2 (33 pp., 32 arts.) opens with several theorems on integers including the unique prime factorization of integers (in art. 16), and then treats linear congruences in arts. 29?37, including the Euclidean algorithm and what we call the Chinese remainder theorem. At the end of sec. 2, Gauss added a few results for future reference which had not figured in the 1797 manuscript, among them: (i) properties of the number ?(A) of prime residues modulo A (arts. 38?39); (ii) in art. 42, a proof that the product of two polynomials with leading coefficient 1 and with rational coefficients that are not all integers cannot have all its coefficients integers; and (iii) in arts. 43 and 44, a proof of Lagrange?s result that a polynomial congruence modulo a prime cannot have more zeros than its degree.?Section 3 (51 pp., 49 arts.) is entitl-
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## The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning.

BOOLE, George. First edition, very rare in commerce, of Boole?s first book, the birth of modern symbolic logic and the first presentation of ?Boolean algebra? ?Boole?s work also contains what Bertrand Russell called the greatest discovery of the nineteenth century: the nature of pure mathematics? (OOC). ?Self-taught mathematician George Boole (1815?1864) published a pamphlet in 1847 ? The Mathematical Analysis of Logic ? that launched him into history as one of the nineteenth century?s most original thinkers? (Introduction to the CUP reprint). ?The Mathematical Analysis of Logic marks the beginning of symbolic logic in the modern sense. Boole showed that classical logic was actually a branch of mathematics which gave rise to a hitherto unconsidered type of algebra. Boole?s book however went considerably further. It threw a great deal of light on the nature of pure mathematics; it opened up possibilities of an extension of the subject into totally new and unexpected areas ? classical mathematics had concentrated on the notions of shape and number and even when symbols were employed, they were generally interpreted in terms of number. Boole had now introduced the notion of interpreting symbols as classes or sets of objects, a concept breathtaking in its scope because it meant that the study of all well-defined sets of objects now came under the realm of mathematics ? By enlarging the horizons of mathematics so enormously, Boole unwittingly (but perhaps subconsciously, wittingly) highlighted a topic that has come to influence virtually every aspect of present-day life ? the storage and processing of information, which in turn has led to the development of computer science. Not only is Boole?s algebra the ?correct? and most economical tool for handling information, but the electronic machines which now do the work actually operate according to principles determined by that self-same algebra. Boole has been called the ?Father of Symbolic Logic? and the ?Founder of Pure Mathematics?, but he is just as deserving of the title, ?Father of Computer Science?? (MacHale, p. 82). ABPC/RBH list only two copies since Honeyman: the OOC copy, Christie?s 2005, $10,800, and Bonham?s 2013, £25,000 = $38,000. Both of these copies were in modern bindings (the OOC copy with the original front wrapper bound in).?Boole?s contribution to logics made possible the works of subsequent logicians including Turing and Von Neumann ? Even Babbage depended a great deal on Boole?s ideas for his understanding of what mathematical operations really are ? 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 perform the necessary logical tasks. Thus our debt to this simple, quiet 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 ×; 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, pp. 37-38).?Early in the spring of 1847, Boole’s long-dormant interest in the connections between mathematics and logic was dramatically reawakened. At this time, a furious controversy was raging between the supporters of de Morgan and those of Sir William Hamilton, the Scottish philosopher and metaphysician (not to be confused with Sir William Rowan Hamilton). Hamilton was a logician who distrusted mathematics, but he was an innovator in logic and had, about this time, introduced the notion of ‘quantification of the predicate’ which was to lead to a widening of the scope of logic. Classical logic had concentrated on the ‘four forms’ of statement ? all A are B, no A are B, some A are B, some A are not B. In Hamilton’s approach, the predicate, or second term B, is quantified by considering statements of the type: all A are all B, any A is not some B, and so on. De Morgan too was at this time working on a more mathematical theory of logic which included a notion equivalent to quantification of the predicate. Hamilton at once accused de Morgan of plagiarism, despite the fact that the notion in question was not original to either of them nor, as it transpired, of any great significance per se in the development of logic. Hamilton’s charges were unjust, even absurd, but controversy raged for many years and attracted a great deal of attention ??From his neutral position, Boole was able to judge the merits and defects of the approaches of both Hamilton and de Morgan. Though Hamilton disliked mathematics and even poured scorn on the subject, yet his approach seemed to suggest that logic should concentrate on ‘equations’ connecting ‘collections of objects or classes’. De Morgan’s approach, on the other hand, seemed to concentrate on a purely symbolic representation of logical processes, yet his notation was cumbersome and unwieldy. Why not, thought Boole, synthesise the two approaches by representing each class of objects by a single symbol and allow relations between classes to be expressed by algebraic equations between the symbols? This devastatingly simple but ingenious notion intrigued and excited him and he set to work at once on a book expounding a new mathematical theory of logic? (MacHale, p. 79-80).?The priority dispute triggered Boole to write his first book; but its content was much influe-
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## The High-Frequency Spectra of the Elements, I-II.

MOSELEY, Henry Gwyn Jeffreys. First edition, an exceptionally fine set of both parts of this landmark work, journal issue, in the original printed wrappers. "In 1913 and 1914, respectively, Moseley (1887-1915) published two papers which, once and for all, established a firm connection of the Periodic Table, which was based on empirical chemistry, to the physical structure of atoms" (Brandt, p. 97). "Moseley, working under Rutherford at Manchester, used the method of X-ray spectroscopy devised by the Braggs to calculate variations in the wavelength of the rays emitted by each element. These he was able to arrange in a series according to the nuclear charge of each element . These figures Moseley called atomic numbers. He pointed out that they also represented a corresponding increase in extra-nuclear electrons and that it is the number and arrangement of these electrons rather than the atomic weight that determines the properties of an element. It was now possible to base the periodic table on a firm foundation, and to state with confidence that the number of elements up to uranium is limited to 92" (PMM). On the basis of his results, Moseley also predicted the existence of four new elements, later discovered and named hafnium, rhenium, technetium and promethium.?PMM 407; Evans 62; Norman 1599."Before 1913 the order of the elements in the periodic system was universally taken to be given by the atomic weight. Although this caused some anomalies, such as that related to the reversed atomic weights of tellurium (Te = 127.6) and iodine (I = 126.9), the convention or dogma of atomic weight being the defining property of a chemical element was rarely questioned According to Charles Galton Darwin, who at the time was a lecturer at Manchester University, the 1913 scattering experiments of Geiger and Marsden convinced Rutherford and his group that the nuclear charge was the defining quantity of a chemical element. The idea certainly was in the air, but it took until November 1913 before it was explicitly formulated, and then from the unlikely source of a Dutch amateur physicist. Trained as a lawyer, Antonius van den Broek had since 1907 published articles on radioactivity and the periodic system In a short communication to Nature dated November 10 he disconnected the ordinal number from the atomic weight and instead identified it with the nuclear charge N (or Z, as it subsequently became symbolized). This hypothesis, he said, holds good for Mendeleevs table but the nuclear charge is not equal to half the atomic weight. Van den Broeks suggestion was quickly adopted by Soddy, Bohr, and Rutherford and his group In an address of 1934 celebrating the centenary of Mendeleevs birth, Rutherford credited Bohr for first having recognized the significance of an ordinal number for the chemical elements: The idea that the nuclear charge of an element might be given by its ordinal or atomic number was first suggested and used by Bohr in developing his theory of spectra. By a strange oversight, Bohr himself gave the credit of this suggestion to van den Broek, who later discussed the applicability of this conception to the elements in general "Besides the successes from the spectra of hydrogen and helium, the strongest experimental support for Bohrs theory came from X-ray spectroscopy, a branch of science that did not yet exist when Bohr completed his trilogy The existence of monochromatic X-rays characteristic of the element emitting the rays had been known since 1906, when the phenomenon was discovered by Charles Glover Barkla, a physicist at the University of Liverpool. Although Barkla could not determine the wavelengths of the characteristic rays he could study and classify them by means of their penetrating power. He soon found that there were two kinds of rays, which he named K and L radiation and where the first had a greater penetrating power than the latter. What was missing, among other things, was a method of determining the wavelength of the radiation, but such a method was provided after William Henry Bragg and his son William Lawrence Bragg in 1912 invented the X-ray spectrometer based on the reflection of X-rays on crystals. "In Manchester, Henry Gwyn Moseley, who was Bohrs junior by two years, set out to employ the method of the Braggs to measure and understand the wavelengths of the characteristic radiation. He had earlier collaborated with Darwin on X-ray diffraction, but from the summer of 1913 he pursued the new research programme alone. Bohr knew Moseley, but it was only in July 1913 that he had a long discussion with him and told him about his new atomic theory. The two physicists evidently had shared interests, such as the periodic system and its relation to the atomic number. Moseleys research programme was to a large extent motivated by the possibility of confirming by means of X-ray spectroscopy van den Broeks hypothesis or the van den Broek-Bohr hypothesis of the atomic number. My work was undertaken for the express purpose of testing Broeks hypothesis, which Bohr has incorporated as a fundamental part of his theory of atomic structure, he wrote. Moseley constructed a new kind of X-ray tube where the targets could be easily interchanged and moved in position opposite to the cathode, to give out their characteristic rays. To determine the wavelengths he developed a photographic method. Having surmounted the inevitable experimental difficulties, in October 1913 he was ready to collect data, starting with the K lines from calcium to zinc" (Kragh, pp. 32-3 & 104). "In a very short time, Moseley produced the first of his two famous papers in which he showed the spectra of K radiation of ten different substances Moseley arranged the spectra, one below the other in a step-like fashion, in such a way that a given wavelength was in the same position for all spectra. It then became clear by simple inspection of this step ladder that the spectrum of K radiation of each element contains two str-
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## Die formale Grundlage der allgemeinen Relativitätstheorie. Author?s presentation offprint from Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, XLI, 19 November 1914.

EINSTEIN, Albert. First edition of this extremely rare offprint, a remarkable association copy inscribed by the theoretical physicist Gunnar Nordström, often designated by modern writers as ?The Einstein of Finland? Einstein had an extended correspondence with Nordström on the subject of Nordström?s own competing theory of gravitation, which at the time was considered a serious competitor to Einstein?s, and which he completed in the same year as the present paper. A few years later Nordström also assisted Einstein in his work on gravitational waves. The present paper was the crucial step between Einstein?s Entwurf theory of 1913 and the final form of general relativity which Einstein completed in November 1915: it develops the mathematical techniques necessary for the final formulation, namely the ?absolute differential calculus? of Tullio Levi-Civita, as well as the expression of the field equations in terms of a variational principle, which later proved to be of great importance. This author?s presentation offprint, with ?Überreicht vom Verfasser? printed on upper wrapper, must not to be confused with the much more common trade offprint which lacks this printed statement (see below). We have located only one copy of this author?s presentation offprint at auction, in the collection belonging to Einstein?s son Hans Albert sold at Christie?s in 2006 (there was no copy in Einstein?s own collection of his offprints sold by Christie?s in 2008).Provenance: Gunnar Nordström (1881-1923) (?G. Nordström? written in pencil on upper wrapper in Nordström?s hand). Mathematical annotations in pencil to margin of p. 1077 (in Nordström?s hand?). Later inscription in Russian on upper wrapper.?In summer 1914, Einstein felt that the new theory should be presented in a comprehensive review. He also felt that a mathematical derivation of the field equations that would determine them uniquely was still missing. Both tasks are addressed in a long paper, presented in October 1914 to the Prussian Academy for publication in its Sitzungsberichte. It is entitled ?The formal foundation of the general theory of relativity?; here, for the first time, Einstein gave the new theory of relativity the epithet ?general? in lieu of the more cautious ‘generalized’ that he had used for the Entwurf? (Landmark Writings in Western Mathematics 1640-1940). ?In the year that he was called to Berlin, on October 29, 1914, Einstein was able to present his work ?Die formale Grundlage der allgemeinen Relativitätstheorie? ? The ?formal foundation? of the general theory of relativity was the tensor calculus. Without the tensor calculus, the general theory of relativity could not have been formulated ? By October 1914, Einstein was finally able to present his results in mathematical form, and indeed in a manner that became the basis of his general theory of relativity of 1916. He introduced general covariants, contravariants, and also?what was new?mixed tensors, in order to represent the individual arithmetic operations, above all, the various types of multiplication. Thus the mathematical calculus necessary for the general theory of relativity was at the ready in 1914? (Reich). ?The principal novelty [in the present paper] lies in the mathematical formulation of the theory. Drawing on earlier work with [Marcel] Grossman, Einstein formulated his gravitational field equations using a variation principle? (Calaprice, 47).The first important stage in the development of Einstein?s theory of gravitation was accomplished, with his friend and classmate the mathematician Marcel Grossmann, in their 1913 work Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation. ?In this book, Einstein and Grossman investigated curved space and curved time as they relate to a theory of gravity. They presented virtually all the elements of the general theory of relativity with the exception of one striking omission: gravitational field equations that were not generally covariant. Einstein soon reconciled himself to this lack of general covariance through the ?hole argument,? which sought to establish that generally covariant gravitational field equations would be physically uninteresting? (Calaprice 40). Einstein?s ?hole argument?, he believed, implied that general covariance was incompatible with the requirement that the distribution of mass-energy should determine the gravitational field uniquely. He believed, therefore, that the field equations should only be valid in certain coordinate systems, which he called ?adapted?, and that only coordinate transformations from one adapted system to another adapted system should be allowed ? he called these ?justified coordinate transformations?.?Einstein?s move to Berlin in April 1914 marked the end of his collaboration with Grossmann. Fortunately, by this time Einstein no longer seems to have needed Grossmann?s mathematical guidance. By October 1914, he had completed a lengthy summary article [offered here] on his new theory, whose form and detailed nature suggest that Einstein felt his theory had reached its final form. The article contained a review of the methods of tensor calculus used in the theory and, flexing his newfound mathematical muscles, Einstein could even promise to give new and simpler derivations of the basic laws of the ?absolute differential calculus? Of great importance was the fact that Einstein had taken the new mathematical techniques of his last paper with Grossmann, generalized them and found in them a quite new derivation of the field equations? (Norton, p. 293).This new derivation made use, for the first time in Einstein?s work on the theory of gravity, of an action principle (or variational principle). Einstein worked initially with an action that was an arbitrary function of the metric tensor and its first derivatives, and then showed that with a particular choice of the action he could recover the Entwurf field equations. He further believed that he had found a simple general cov-
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## The art of dialling performed geometrically by scale and compasses: arithmetically, by the canons of sines and tangents: instrumentally, by a trigonal instrument, accommodated with lines for that purpose; The geometrical part whereof is performed by projecting of the sphere in plano, upon the plain it self, whereby not only the making, but the reason also of dials is discovered.

LEYBOURN, William. First edition, first issue, extremely rare, of the first of Leybourn?s books on the subject. This issue has ?dialling:? in line 2 of the title and lacks imprint date; the more common (but still rare) second issue has ?Dialling,? and imprint date 1669. This, the Kenney copy, is one of perhaps only two known examples of the first issue, ESTC listing only the BL copy. ?In 1669 Leybourn authored The Art of Dialling, a book on the use of sundials and astrolabes in determining the position of vessels at sea. The contemporary expansion of the Royal Navy and Merchant Marines created a significant demand for such manuals, and The Art of Dialling was well written, easy to understand and cheaply produced? (Wikipedia).?The design of sundials represented a steady source of income for independent mathematicians such as William Leybourn. It is quite straightforward to design a standard garden sundial on the horizontal plane with the gnomon angled toward the north celestial pole. It is another matter to design a dial on a wall of a building that is situated at an odd angle to the cardinal compass points. Leybourn was well-known for his books on dialling that treat such arcane subjects as how to create a dial on a ceiling, the sun being reflected from a mirror attached to the windowsill. In this, the first of his several volumes on the subject, he does not consider these very unusual situations but introduces the reader to such fundamental concepts as how to measure the two required angles (reclination and declination) of the surface on which the dial is to be mounted. After a short tutorial on astronomy and the movement of the sun, he covers three different methods of designing dials: geometrically by ruler and compass, arithmetically by means of a table of sines and tangents, and finally by the use of a very simple instrument of hisown design. The work concludes with a table of the sun?s declination for each day of the year and a table of the latitudes for each of the major places in Britain. The frontispiece, a portrait of the author, is the same as that later used in his 1672 book Panorganon? (Tomash). Leybourn (1626-1716) enjoyed a fine reputation in his day, both as a fellow (described by John Gadbury as of ?a facetious, pleasant and cheerful disposition?), and as a mathematician, ranked by William Derham with William Oughtred and Jonas Moore, and some of his textbooks had a life of over a century. ?A notable example of the mathematical career that flourished largely away from elite court circles and the universities is that of William Leybourn ? Leybourn might with justification be thought one of the most significant London mathematicians of his day. His long life and career paralleled [Christopher] Wren?s closely, though at a distinct social remove? (Edwards, p. 100). No other copies of this first issue located in auction records.?Born in 1626, Leybourn worked originally with his brother Robert as a printer, based at Monkswell Street, Cripplegate. Whilst the Leybourn brothers produced many books for writers involved in technical experimentation and reform, William Leybourn increasingly gave up his printing work to write his own books on mathematics; to teach private pupils whom he boarded at his home in Southall; and to work as a mathematical practitioner, taking part in private and public projects such as the great fire survey, and the surveying of estates forfeited in the Civil War.?Leybourn?s practical work and his writing were clearly highly symbiotic in the maintenance of his career. Leybourn?s books made evident his knowledge and competence in mathematics. They also directly advertised his services. The Line of Proportion, for instance, first published in 1667, offered a straightforward, accessible guide to the use of ?Gunter?s line?, a logarithmic series designed to help artisans with little mathematics compute areas and volumes mechanically. An unfussy duodecimo volume, it was first published in the immediate aftermath of the great fire and dedicated to the City grandees who oversaw the surveying of the ruins. As well as advertising Leybourn?s employment in this survey, later editions of this simple, practical book were used to tout for further business with a wider clientele. One edition (1673) carries the following notice:?If any Gentleman, or other Person, desire to be instructed in any of the Sciences Mathematical, as Arithmetick, Geometry, Astronomy, the Use of the Globes, Trigonometry, Navigation, Surveying of Land, Dialling, or the like; the Author will be ready to attend them at times appointed.Also, If any Person would have his Land, or any Ground for Building Surveyed, or any Edifice of Building Measured, either for the Carpenters, Bricklayers, Plaisterers, Glaziers, Joyners or Masons work, he is ready to perform the same either for Master Builder or workman: ?You may hear of him where these Books are to be sold.??Self-marketing such as this is utterly characteristic of Leybourn?s long and fertile publishing career. Leybourn published his first mathematical work, a treatise on surveying, in 1650. By 1682 he was able to fund production of a folio edition of his book on Dialling by subscription, expanding a text first published in quarto form in 1669 [offered here], and furnishing it with his portrait. An insider to the seventeenth-century print trade, Leybourn joined other early modern writers beginning to capitalize on the social currency of their knowledge; owning and selling shares in it. Having made a name for himself, Leybourn effectively franchised his name. He contributed prefaces to other writers? works, and published in many instances to promote instruments made by his associates: an early example of tie-in marketing. He was also cannily sensitive to the different markets for his books. Some, such as The Line of Proportion, are simply and cheaply produced, with the accent on practical use. Cursus Mathematicus, on the other hand, [a] relatively lavish folio production pub-
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## The sequence of the human genome.

VENTER, J. Craig, et al. First edition, journal issue in the original printed wrappers, signed by Craig Venter, of the first published announcement of Celera Genomics? sequencing of the human genome. The problem of finding the order of the building blocks of the nucleic acids that make up the entire genetic material of a human was first proposed in 1985, but it was not until 1990 that the Human Genome Project (HGP) was officially initiated in the United States under the direction of the National Institutes of Health (NIH) and the U.S. Department of Energy with a 15-year, $3 billion plan for sequencing the entire human genome composed of 2.9 billion base pairs. Other countries such as Japan, Germany, the United Kingdom, France, and China also contributed to the global sequencing effort. Venter was a scientist at the NIH during the early 1990s when the project was initiated. In 1998 his company Celera announced its intention to build a unique genome sequencing facility, to determine the sequence of the human genome over a 3-year period. The Celera approach to genome sequencing was very different from the map-based public efforts. They proposed to use ?shotgun sequencing? (sequencing of DNA that has been randomly fragmented into pieces) of the genome, subsequently putting it together. This approach was widely criticized but was shown to be successful after Celera sequenced the genome of the fruit fly Drosophila melanogaster in 2000 using this method. The Celera effort was able to proceed at a much more rapid rate, and about 10% of the cost, of the HGP because it relied upon data made available by the publicly funded project. Venter announced in April 2000 that his group had finished sequencing the human genome during testimony before Congress on the future of the HGP, a full three years before that project had been expected to be complete. Venter?s article ?The Sequence of the Human Genome? was published in Science ten months later. The publicly funded HGP reported their findings one day earlier in Nature, thus preventing Celera from patenting the genetic information. Venter was listed on Time magazine?s 2007 and 2008 ?Time 100? list of the most influential people in the world, and in 2008 he received the National Medal of Science from President Obama. We are not aware of any other copy of this historic article signed by Venter having appeared on the market. When the HGP was begun in 1990, it was far too expensive to sequence the complete human genome. The National Institutes of Health therefore adopted a ?shortcut?, which was to look just at sites on the genome where many people have a variant DNA unit. The genome was broken into smaller pieces, approximately 150,000 base pairs in length. These pieces were then ligated into a type of vector known as ?bacterial artificial chromosomes?, which are derived from bacterial chromosomes which have been genetically engineered. The vectors containing the genes can be inserted into bacteria where they are copied by the bacterial DNA replication machinery. Each of these pieces was then sequenced separately as a small ?shotgun? project and then assembled. The larger, 150,000 base pairs go together to create chromosomes. This is known as the ?hierarchical shotgun? approach, because the genome is first broken into relatively large chunks, which are then mapped to chromosomes before being selected for sequencing. Celera used a technique called ?whole genome shotgun sequencing,? employing pairwise end sequencing, which had been used to sequence bacterial genomes of up to six million base pairs in length, but not for anything nearly as large as the three billion base pair human genome. Celera initially announced that it would seek patent protection on ?only 200?300? genes, but later amended this to seeking ?intellectual property protection? on ?fully-characterized important structures? amounting to 100?300 targets. The firm eventually filed preliminary (?place-holder?) patent applications on 6,500 whole or partial genes. Celera also promised to publish their findings in accordance with the terms of the 1996 ?Bermuda Statement?, by releasing new data annually (the HGP released its new data daily), although, unlike the publicly funded project, they would not permit free redistribution or scientific use of the data. The publicly funded competitors were compelled to release the first draft of the human genome before Celera for this reason. Special issues of Nature (which published the publicly funded project?s scientific paper) and Science (which published Celera’s paper) described the methods used to produce the draft sequence and offered analysis of the sequence. These drafts covered about 83% of the genome (90% of the euchromatic regions with 150,000 gaps and the order and orientation of many segments not yet established). In February 2001, at the time of the joint publications, press releases announced that the project had been completed by both groups. Improved drafts were announced in 2003 and 2005, filling in approximately 92% of the sequence.In the publicly funded HGP, researchers collected blood (female) or sperm (male) samples from a large number of donors. Only a few of many collected samples were processed as DNA resources. Thus the donor identities were protected so neither donors nor scientists could know whose DNA was sequenced. In the Celera project, DNA from five different individuals was used for sequencing. Venter later acknowledged (in a public letter to Science) that his DNA was one of 21 samples in the pool, five of which were selected for use. ?The work on interpretation and analysis of genome data is still in its initial stages. It is anticipated that detailed knowledge of the human genome will provide new avenues for advances in medicine and biotechnology. Clear practical results of the project emerged even before the work was finished. For example, a number of companies, such as Myriad Genetics, started offering easy ways to administer genetic tests that can s-
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## A Method for the Calculation of the Zeta-Function, pp. 180-197 in Proceedings of the London Mathematical Society, Series 2, Vol. 48, No. 3, December 15, 1943. London: C. F. Hodgson and Son, 1943. [Offered with:] Some calculations of the Riemann zeta-function, pp. 99-117 in ibid., Series 3, Vol. 3, No. 9, March 1953.

TURING, Alan. First edition, journal issues in the original printed wrappers, of Turing?s ground-breaking work outlining a method to decide the most famous open problem in mathematics, the so-called Riemann hypothesis. This is a conjecture about the location of the zeros of the ?Riemann zeta function? ? it asserts that, apart from some ?trivial? zeros, they all lie on a certain ?critical line.? If true, this would have enormous implications for the study of prime numbers. Turing had worked on the zeta function since 1939 and in ?A Method for the Calculation of the Zeta-Function? he outlined a method of calculating the zeros using a mechanical computer. ?The Turing archive contains a sketch of a proposal, in 1939, to build an analog computer that would calculate approximate values for the Riemann zeta-function on the critical line. His ingenious method was published in 1943 [as the present work]? (Downey, p. 11). Although he received a grant to build his zeta-function machine, the outbreak of World War II, and Turing?s role in it as cryptanalyst, postponed the work, and the machine was never constructed. After the War, Turing returned to the Riemann hypothesis and developed a new procedure, now known as ?Turing?s method?, for checking the Riemann Hypothesis (described in Section 4 of the 1953 paper). He then used the Manchester Mark I digital computer to implement this method. ?Of Turing?s two published papers [both offered here] on the Riemann zeta function, the second is the more significant. In that paper, Turing reports on the first calculation of zeros of [the zeta function] ever done with the aid of an electronic digital computer. It was in developing the theoretical underpinnings for this work that Turing?s method first came into existence? (Hedjhal & Odlyzko, p. 265). ?Some calculations of the Riemann zeta-function? was Turing?s last published mathematical paper. ?This work was one of the first announcing a new chapter in which experimental mathematics performed with computers would play an important role? (Mezzadri and Snaith, Recent Perspectives in Random Matrix and Number Theory). Rare on the market in unrestored original printed wrappers ? we know of only one copy of the first paper at auction, in the Weinreb Computer Collection (Bloomsbury Book Auctions, 28 October 1999), and no other copy of the second.The Riemann zeta function is defined as the sum of an infinite series?(s) = 1/1s + 1/2s + 1/3s + 1/4s + ? This actually makes sense when s is any complex number (except s = 1, when the sum is infinite). It is known that ?(s) = 0 when s = ?2, ?4, ?6, ? ? these are called the ?trivial zeros? The Riemann hypothesis (RH) is the assertion that all the non-trivial zeros are complex numbers of the form s = ½ + t?-1, where t is a real number ? these complex numbers form a line in the complex plane, called the ?critical line? The RH, first put forward by Bernhard Riemann in 1859, is known to be true for the first 1013 non-trivial zeros, but remains unproven. ?The RH is widely regarded as the most famous unsolved problem in mathematics. It was one of the 23 famous problems selected by [David] Hilbert in 1900 as among the most important in mathematics, and it is one of the seven Millennium Problems selected by the Clay Mathematics Institute in 2000 as the most important for the 21st century? (Hedjhal & Odlyzko, p. 266).?The first computations of zeros of the zeta function were performed by Riemann, and likely played an important role in his posing of the RH as a result likely to be true. His computations were carried out by hand, using an advanced method that is known today as the Riemann-Siegel formula. Both the method and Riemann?s computations that utilized it remained unknown to the world-at-large until the early 1930s, when they were found in Riemann?s unpublished papers by C. L. Siegel ? In the mid-1930s, after Siegel?s publication of the Riemann-Siegel formula, [the Oxford mathematician] E. C. Titchmarsh obtained a grant for a larger computation. With the assistance of L. J. Comrie, tabulating machines, some ?computers? (as the mostly female operators of such machinery were called in those days), and the recently published algorithm, Titchmarsh established that the 1041 nontrivial zero with 0 t ibid., p. 268).?Turing encountered the Riemann zeta function as a student, and developed a life-long fascination with it. Though his research in this area was not a major thrust of his career, he did make a number of pioneering contributions? (ibid., p. 266). ?Apparently he had decided that the Riemann hypothesis was probably false, if only because such great efforts had failed to prove it. Its falsity would mean that the zeta-function did take the value zero at some point which was off the special line, in which case this point could be located by brute force, just by calculating enough values of the zeta-function ? There were two aspects to the problem. Riemann?s zeta-function was defined as the sum of an infinite number of terms, and although this sum could be re-expressed in many different ways, any attempt to evaluate it would in some way involve making an approximation. It was for the mathematician to find a good approximation, and to prove that it was good: that the error involved was sufficiently small. Such work did not involve computation with numbers, but required highly technical work with the calculus of complex numbers. Titchmarsh had employed a certain approximation which ? rather romantically ? had been exhumed from Riemann?s own papers at Göttingen where it had lain for seventy years. But for extending the calculation to thousands of new zeroes a fresh approximation was required; and this Alan set out to find and to justify.?The second problem, quite different, was the ?dull and elementary? one of actually doing the computation, with numbers substituted into the approximate formula, and worked out for thousands of different entries. It so happened that the formula was rather like those w-
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## Ondes et Mouvements. Collection de Physique Mathématique; [Offered with:] La Mecanique Ondulatoire.

BROGLIE, Louis Victor Pierre Raymond De. First edition of these expanded presentations of the ideas in de Broglie?s epoch-making doctoral thesis on the quantum theory, which, Einstein said, ?lifted a corner of the great veil? (Isaacson, Einstein: His Life and Universe, p. 327). In that work he developed the startling and revolutionary idea that material particles such as electrons have a wave as well as a corpuscular nature, analogous to the dual behavior of light demonstrated by Einstein and others in the first two decades of the twentieth century. ?He made the leap in his September 10, 1923, paper [?Ondes et quanta,? Comptes Rendus, t. 177]: E = h? should hold not only for photons but also for electrons, to which he assigns a ?fictitious associated wave?? (Pais, Subtle is the Lord, p. 436). ?Louis de Broglie achieved a worldwide reputation for his discovery of the wave theory of matter, for which he received the Nobel Prize for physics in 1929. His work was extended into a full-fledged wave mechanics by Erwin Schrödinger and thus contributed to the creation of quantum mechanics? (DSB). De Broglie was awarded the 1929 Nobel Prize in physics ?for his discovery of the wave nature of electrons.? De Broglie?s book Ondes et mouvements (1926) was selected by Carter and Muir for the Printing and the Mind of Man exhibition and catalogue (1967).In his 1905 paper ?Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt? (?On a Heuristic Viewpoint Concerning the Production and Transformation of Light?), ?Einstein postulated that light is composed of individual quanta (later called photons) that, in addition to wavelike behavior, demonstrate certain properties unique to particles. In a single stroke he thus revolutionized the theory of light and provided an explanation for, among other phenomena, the emission of electrons from some solids when struck by light, called the photoelectric effect? (Britannica). ?The central idea of de Broglie?s work ? was that the formula E = h? by which Einstein had related the frequency ? of light to the energy E of light quanta should not only apply to light but also to material particles. For a particle at rest with mass m he concluded, since its energy is E = mc2, that it performs an internal oscillation with frequency ? = mc2/h. He considered the motion of a particle, carefully taking into account the effects of the special theory of relativity, and was able to construct a wave which was always in phase with the internal oscillation of the particle ? he gave an application of his theory by showing that he could naturally explain the discrete electron orbits in Bohr?s model of the hydrogen atom. Each stable orbit should be closed in the sense that the same phase should be assumed by the matter wave after completion of an orbit? (Brandt, The Harvest of a Century, Chapter 32, p. 133).In a second Comptes Rendus note, De Broglie also predicted on the basis of his theory that ?a stream of electrons traversing a very small aperture will show the phenomenon of diffraction.? This was experimentally observed by C. J. Davisson and L. H. Germer in 1927, work for which they received the Nobel Prize in Physics in 1937. ?Thus the duality of both light and matter had been established, and physicists had to come to terms with fundamental particles which defied simple theories and demanded two sets of ‘complementary’ descriptions, each applicable under certain circumstances, but incompatible with one another? (PMM 417). Finally, in a third note, De Broglie derived from his theory ?a result of Planck on the kinetic theory of gases, by making the assumption that ?the state of a gas will be stable only if the waves corresponding to all the atoms form a system of stationary waves.? He also showed that ?the interplay between the propagation of a particle and of its associated matter wave could be expressed in more formal terms as an identity between the fundamental variational principles of Pierre de Fermat (rays), and Pierre Louis Maupertuis (particles)? (DSB).When de Broglie first published his theory of matter waves he was practically unknown in scientific circles, although his elder brother Maurice had already done important experimental work on X-rays and his illustrious family was famous in France. His ideas became widely known only with the publication of his doctoral thesis Recherches sur la théorie des quanta in the summer of 1924, which is an elaboration of the content of the three Comptes Rendus notes. Einstein?s support for de Broglie?s ideas brought them to the attention of the principal actors in the development of quantum theory, notably Schrödinger, whose wave mechanics was an extension and completion of de Broglie?s work. An account of Schrödinger?s development of de Broglie?s ideas into what became wave mechanics is given in the 1928 work offered here.?In his communications of 1923, and later in his 1924 PhD thesis, de Broglie did not want to commit himself to any physical interpretation of the waves. He granted physical relevance only to these wave features which could be directly related to the particle motion, namely their phase, while eluding any questions pertaining to their amplitude and proper dynamics. They were, as he dubbed them, ?fictitious.? However, in the months following his PhD, de Broglie started to explore the consequences of his wave-particle model for the problem of the interaction of light with matter. He also considered the possibilities of more physically interpreting his particle-associated waves. Willing to acknowledge the reality of the particles, he tried to conceive them as embodied by the singularities of the waves. However, he had then to cope with the Schrödinger view, where only continuous matter waves were considered. He first attempted to save his dualism by conceiving Schrödinger?s equation as actually admitting pairs of solutions characterized by a common phase. He thought of each pair as consisting of a singular solution, with the sing-
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## Essay d’Analyse sur les Jeux de Hazard.

MONTMORT, Pierre Rémond de]. First edition, first issue, very rare, of the first separately published textbook of probability. This issue has significant textual differences from what is usually referred to as the first edition. ?In 1708 [Montmort] published his work on Chances, where with the courage of Columbus he revealed a new world to mathematicians? (Todhunter, p. 78). ?The Essay (1708) is the first published comprehensive text on probability theory, and it represents a considerable advance compared with the treatises of Huygens (1657) and Pascal (1665). Montmort continues in a masterly way the work of Pascal on combinatorics and its application to the solution of problems on games of chance. He also makes effective use of the methods of recursion and analysis to solve much more difficult problems than those discussed by Huygens. Finally, he uses the method of infinite series, as indicated by Bernoulli (1690)? (Hald, p. 290). ?Montmort?s book on probability, Essay d?analyse sur les jeux de hazard, which came out in 1708, made his reputation among scientists? (DSB). Based on the problems set forth by Huygens in his De Ratiociniis in Ludo Aleae (1657) (published as an appendix to Frans van Schooten?s Exercitationum mathematicarum), the Essay spawned Abraham de Moivre?s two important works De Mensura Sortis (1711) and Doctrine of Chances (1718). ABPC/RBH record the sale of just three other copies of the first edition (Christie?s 1981, Hartung 1987 and the Tomash copy). As Sotheby?s correctly noted in the Tomash library sale catalogue (18 September 2018, lot 434), ?This book was first issued in 1708 without illustrations and an uncorrected text,? and indeed the three large folding tables found in the regular issue are not present in this first issue, which also has a shorter list of errata than the regular issue. The existence of two textually different issues of this work, both published in 1708, has not, as far as we are aware, been noted in the academic literature.The modern theory of probability is generally agreed to have begun with the correspondence between Pierre de Fermat and Blaise Pascal in 1654 on the solution of the ?Problem of points?; this was published in Fermat?s Varia Opera (1679). Pascal included his solution as the third section of the second part of his 36-page Traité du triangle arithmétique (1665), which was essentially a treatise on pure mathematics. ?Huygens heard about Pascal?s and Fermat?s ideas [on games of chance] but had to work out the details for himself. His treatise De ratiociniis in ludo aleae ? essentially followed Pascal?s method of expectation. ? At the end of his treatise, Huygens listed five problems about fair odds in games of chance, some of which had already been solved by Pascal and Fermat. These problems, together with similar questions inspired by other card and dice games popular at the time, set an agenda for research that continued for nearly a century. The most important landmarks of this work are Bernoulli?s Ars conjectandi (1713), Montmort?s Essay d’analyse sur les jeux de hazard (editions in 1708 and 1711 [i.e., 1713]) and De Moivre?s Doctrine of Chances (editions in 1718, 1738, and 1756). These authors investigated many of the problems still studied under the heading of discrete probability, including gamblers ruin, duration of play, handicaps, coincidences and runs. In order to solve these problems, they improved Pascal and Fermat?s combinatorial reasoning, summed infinite series, developed the method of inclusion and exclusion, and developed methods for solving the linear difference equations that arise in using Pascal?s method of expectations.? (Glenn Schafer in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (1994), Grattan-Guiness (ed.), p. 1296).?It is not clear why Montmort undertook a systematic exposition of the theory of games of chance. Gaming was a common pastime among the lesser nobility whom he frequented, but it had not been treated mathematically since Christiaan Huygens? monograph of 1657. Although there had been isolated publications about individual games, and occasional attempts to come to grips with annuities, Jakob I Bernoulli?s major work on probability, the Ars conjectandi, had not yet been published. Bernoulli?s work was nearly complete at his death in 1705; two obituary notices give brief accounts of it. Montmort set out to follow what he took to be Bernoulli?s plan ?[Montmort] continued along the lines laid down by Huygens and made analyses of fashionable games of chance in order to solve problems in combinations and the summation of series? (DSB).?In this first edition of the Essai d’Analyse Montmort begins by finding the chances involved in various games of cards. He discusses such simple games as Pharaoh, Bassette, Lansquenet and Treize, and then, not so fully or successfully, Ombre and Picquet. The work is easy to read in that he prefaces each section with the rules of the game discussed, so that what he is trying to do can be explicitly understood. Possibly he found it necessary to do this because different versions of the games were in vogue, but this does not always occur to other writers. Having set down the rules, he solves simple cases in a method somewhat reminiscent of Huygens, and then takes a plunge into a general solution which appears to be correct but is not always demonstrably so. The Problèmes divers sur le jeu du treize are interesting indeed in that he gives the matching distribution and its exponential limit. Treize has survived today as the children’s game of Snap.?The players draw first of all as to who shall be the Bank. Let us suppose that this is Pierre, and the number of players whatever one likes. Pierre having a complete pack of 52 shuffled cards, turns them up one after the other. Naming and pronouncing one when he turns the first card, two when he turns the second, three when he turns the third, and so on until the thirteenth which is the King. Now if in all this procee-
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## Ars conjectandi, Opus posthumum. Accedit tractatus de seriebus infinitis, et epistola Gallicè scripta de ludo pilae reticularis.

BERNOULLI, Jakob. First edition, a truly exceptional copy, uncut in original boards. It is hard to imagine a finer copy. ?Jakob 1 Bernoulli?s posthumous treatise, edited by his nephew [Nicholas I Bernoulli], (the title literally means ?the art of[dice] throwing?) was the first significant book on probability theory: it set forth the fundamental principles of the calculus of probabilities and contained the first suggestion that the theory could extend beyond the boundaries of mathematics to apply to civic, moral and economic affairs. The work is divided into four parts, the first a commentary on Huygens?s De ratiociniis in ludo aleae (1657), the second a treatise on permutations (a term Bernoulli invented) and combinations, containing the Bernoulli numbers,and the third an application of the theory of combinations to various games of chance. The fourth and most important part contains Bernoulli?s philosophical thoughts on probability: probability as a measurable degree of certainty, necessity and chance, moral versus mathematical expectation, a priori and a posteriori probability, etc. It also contains his attempt to prove what is still called Bernoulli?s Theorem: that if the number of trials is made large enough, then the probability that the result will lie between certain limits will be as great as desired? (Norman). This was the first statement of the law of large numbers.?In the first Part (pp. 2-71) Jakob Bernoulli complemented his reprint of Huygens?s tract by extensive annotations which contained important modifications and generalisations. Bernoulli?s additions to Huygens?s tract are about four times as long as the original text. The central concept in Huygens?s tract is expectation. The expectation of a player A engaged in a game of chance in a certain situation is identified by Huygens with his share of the stakes if the game is not played or not continued in a ?just? game. For the determination of expectation Huygens had given three propositions which constitute the ?theory? of his calculus of games of chance. Huygens?s central proposition III maintains:?If the number of cases I have for gaining a is p, and if the number of cases I have for gaining b is q, then assuming that all cases can happen equally easily, my expectation is worth (pa + qb)/(p + q).??Bernoulli not only gives a new proof for this proposition but also generalizes it in several ways ??Huygens?s propositions IV to VII treat the problem of points, also called the problem of the division of stakes, for two players; propositions VIII and IX treat three and more players. Bernoulli returns to these problems in Part II of the Ars Conjectandi. In his annotations to Huygens?s proposition IV he generalised Huygens?s concept of expectation ? This is the only instance in the annotations and commentaries to Huygens?s tract where Bernoulli uses the word ?probabilitas?, or probability as understood in everyday life. Later in Part IV of the Ars Conjectandi Bernoulli replaced Huygens?s main concept, expectation, by the concept of probability for which he introduced the classical measure of favourable to all possible cases. The remaining propositions X to XIV of Huygens?s tract deal with dicing problems of the kind: What are the odds to throw a given number of points with two or three dice? or: With how many throws of a die can one undertake it to throw a six or a double six? ? The meaning of Huygens?s result of proposition X, that the expectation of a player who contends to throw a six with four throws of a die is greater than that of his adversary, is explained by Bernoulli in a way which relates to the law of large numbers proved in Part IV of the Ars Conjectandi ??In the second Part (pp. 72-137) Bernoulli deals with combinatorial analysis, based on contributions of van Schooten, Leibniz, Wallis, and Jean Prestet ? [It] consists of nine chapters dealing with permutations, the number of combinations of all classes, the number of combinations of a particular class, figurate numbers and their properties (especially the multiplicative property), sums of powers of integers, the hypergeometric distribution, the problem of points for two players with equal chances to win a single game, combinations with repetitions and with restricted repetitions, and variations with repetitions and with restricted repetitions.?Evidently Bernoulli did not know Blaise Pascal?s Triangle arithmétique, published posthumously in 1665, though Leibniz had alluded to it in his last letter to him in 1705. Not only does Bernoulli not mention Pascal in the list of authors that he had consulted concerning combinatorial analysis, except for Pascal?s letter to Fermat of 24 July 1654; it would also be difficult to explain why he repeated results already published by Pascal in the Triangle arithmétique, such as the multiplicative property for binomial coefficients for which Bernoulli claims the first proof for himself. His arrangement differs completely from that of Pascal, whose proof for the multiplicative property of the binomial coefficients has been judged to be clearer than Bernoulli?s. It is fair to add that in the Ars Conjectandi, which Bernoulli left as an unpublished manuscript, he was much more honest concerning the achievements of his predecessors than Pascal in the Triangle arithmétique. It is also true that Bernoulli was concerned with combinatorial analysis in the Ars Conjectandi first of all because it constituted for him a most useful and indispensable universal instrument for dealing numerically with conjectures, since ?every conjecture is founded upon combinations of the effective causes? (p. 73) ??In the third Part (pp. 138-209) Bernoulli gives 24 problems concerning the determination of the modified Huygenian concept of expectation in various games. Here he uses extensively conditional expectations without, however, distinguishing them from unconditional expectations. All the games are games of chance with dice and cards including games en vogue at the French-
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## Mirifici logarithmorum canonis constructio; et eorum ad naturales ipsorum numeros habitudines; una cum appendice, de aliâ eâque præstantiore logarithmorum specie contenda. Quibus accessere propositiones ad triangula sphærica faciliore calculo resolvenda: Unà cum annotationibus aliquot doctissimi D. Henrici Briggii, in eas & memoratam appendicem. Edinburgh: Andrew Hart, 1619. [Bound with:] GREGORY, James. Vera circuli et hyperbolae quadratura, in propria sua proportionis specie, inventa, & demonstrata. Padua: Giacomo Cadorino,[1667].

NAPIER, John. First edition, extremely rare, of this complement to Napier?s epoch-making Mirifici logorithmorum canonis descriptio (1614) ? while the Descriptio gave the first ever table of logarithms, it was in the Constructio that Napier explained the method of their construction. It is here bound with the first edition of James Gregory?s first mathematical work, highly important in the pre-history of calculus, and if anything even rarer than the Napier. ?Probably no work has ever influenced science as a whole, and mathematics in particular, so profoundly as this modest little book [the Descriptio]. It opened the way for the abolition, once and for all, of the infinitely laborious, nay, nightmarish, processes of long division and multiplication, of finding the power and the root of numbers? (Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times (1958), p. 402). ?The ?Mirifici logorithmorum canonis constructio? is the most important of all of Napier?s works, presenting as it does in a most clear and simple way the original conception of logarithms. It is, however, so rare as to be very little known, many writers on the subject never having seen a copy? (Macdonald, p. xvii). ?Historically, it is important to note that in the Constructio the decimal notation is used with ease and power practically for the first time? (Henderson, p. 253). The second work in this volume is by the brilliant Scottish mathematician James Gregory. ?Of British mathematicians of the seventeenth century, Gregory was excelled only by Newton? (Gjertsen, p. 245). The Quadratura ?contained an astonishing number of novel and fundamental concepts, precisely formulated: concepts such as convergence, functionality, algebraic and transcendental functions, classes of transcendency, the process of iteration, the inherent likeness between circular and hyperbolic functions and the existence of functions invariant over an infinite sequence of values of their arguments. Incidentally, he calculated ? to thirteen places and was the first to give the number 2.3025850929940456, or loge10, for the zone of the hyperbola. This battery of ideas was directed with the sole aim of proving the transcendence of ? and e, an investigation that was finally completed by Lindemann at the close of the nineteenth century? (Turnbull, p. 5). ?Although [his] proof was defective and in consequence rapidly incurred a storm of criticism it is to Gregory?s credit that he was the first to formulate a proposition of this class? (Baron, p. 231). ?Indeed, by his speculations Gregory opens a new realm of mathematics ? It is surprising that he quotes three important problems solved today: the squaring of the circle, the impossibility of solving the general algebraic equation, and the impossibility of reducing the pure equation of the nth degree [xn ? 1 = 0] to quadratic equations? (Turnbull, p. 495). No other copies of either the Constructio or the Quadratura on ABPC/RBH in the last half-century.Provenance: Earls of Macclesfield (South Library bookplate on front paste-down and blind-stamp on title of Constructio), (Sotheby?s, 4 November 2004, lot 885, £7,800). Erwin Tomash (book label on front paste-down).The basic idea of what logarithms were to achieve is straightforward: to replace the wearisome task of multiplying two numbers by the simpler task of adding together two other numbers. To each number there was to be associated another, which Napier called at first an ?artificial number? and later a ?logarithm? (a term which he coined from Greek words meaning something like ?ratio-number?), with the property that from the sum of two such logarithms the result of multiplying the two original numbers could be recovered. An idea of this kind was known to the Greeks: take an arithmetic progression (in which there is a constant difference between successive terms) and a geometric progression (in which there is a constant ratio between successive terms); writing one progression next to the other, one sees that adding any two terms of the arithmetic progression corresponds to multiplying the corresponding terms of the geometric progression. In the Constructio, Napier uses this idea but expresses it in kinematical terms. Whiteside suggests that Napier may have derived the idea of using motion in his construction from the writings of William Heytesbury and Nicole Oresme.Napier (1550-1617) imagines two points, P and L, each moving along its own straight line. P starts at a point P0 and moves towards a fixed point Z in such a way that its speed is proportional to the distance PZ still to go, while L starts at the same time at L0 and travels at constant speed ? if the constant speed of L is 1 we can think of L0L as the time taken by P to travel from P0. Napier defines the time L0L to be the logarithm of the distance PZ, which we will denote by NapLog(PZ). If the distance PZ = x, and the factor of proportionality r, so that the speed at P is x/r, it is easy for us to show (using calculus) thatNapLog(x) = r ln(r/x),where ?ln? is our natural logarithm with base e. This means that, apart from the factor r, Napier?s logarithms are the same as our logarithms, but with base 1/e. Note that NapLog(r) = 0, so that r is the distance P0Z. For us the logarithm of 1 is zero, but Napier chooses r = 107 (see below).How did Napier calculate his logarithms? If Q is another point and QZ = y, where y is less than x, the moving point is slowing down as it travels from P to Q so the time taken to travel the distance PQ = x ? y is greater than the time it would have taken if the point had travelled at the speed x/r it had at P and less than the time it would have taken if it had travelled at the speed y/r it had at Q. The actual time taken is NapLog(y) ? NapLog(x), so(r/x)(x ? y) y) ? NapLog(x) r/y)(x ?y).If the difference between x and y is very small, the two end values are almost equal so we can take the middle term to be their average:NapLog(y) ? NapLog(x) is nearly equal to ½(r/x-
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## Appendix. Scientiam Spatii Absolute Veram exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) independentem: adjecta ad casum falsitatis, quadratura circuli geometrica. [in:]BÓLYAI, Farkas. Tentamen Juventutem Studiosam in Elementa Matheseos Purae. Tomus primus [-secundus].

BOLYAI, János. First edition of ?the most extraordinary two dozen pages in the history of thought? (Halsted) and one of the few absolute rarities among the classics of science. This work contains the independent foundation (along with the work of Lobachevsky) of non-Euclidean geometry. I have located some 23 other copies worldwide, all of them exhibiting variations in issue or completeness (the present copy represents the most complete state of the text for both volumes).Lobachevsky and János Bolyai had independently created non-Euclidean systems by challenging the ?parallel postulate? of Euclid. János Bolyai?s work was conceived in 1823, when he wrote to his father ?I have now resolved to publish a work on the theory of parallels . I have created a new universe from nothing? It was published as an appendix to his father?s mathematical treatise, the Tentamen, 1832?3. Lobachevsky’s work appeared in a Kazan academic periodical between 1829?1830, and in fuller form as Geometrische Untersuchungen, Berlin 1840. Whereas Lobachevsky initially had only demonstrated the possibility of a geometry in which Euclid?s fifth postulate (or 11th axiom) was untrue, János developed a geometry completely independent of the fifth postulate and applicable to varieties of curved space. However, the epochal significance of the work of these two was to remain largely unappreciated until the beginning of the twentieth century when it provided the mathematical basis for the Theory of Relativity.Provenance: stamp of the publisher, press of the Reformed College, Maros Vásárhely (now the Romanian city of Târgu Mures) on free flyleaf and title of first volume. János began working on his new geometry early in the 1820s. His father tried to discourage him from attempting to prove or refute Euclid?s fifth postulate:?You should not tempt the parallels in this way. I know this way until its end ? I also have measured this bottomless night, I have lost in it every light, every joy of my life . You should shy away from it as if from lewd intercourse, it can deprive you of all your leisure, your health, your peace of mind and your entire happiness. This infinite darkness might perhaps absorb a thousand giant Newtonian towers, it will never be light on earth, and the miserable human race will never have something absolutely pure, not even geometry? (quoted from DSB). Farkas Bolyai, however diffidently he felt about his son?s researches, did send the manuscript of the Appendix to Gauss: the first letter went unanswered, and a second letter only elicited the reply that Gauss could not praise it, because he himself had reached the same conclusions some 30 years earlier although he had not published his discovery! This assertion so discouraged János that it effectively terminated his career in creative mathematics, but his father did publish his paper as an appendix to his own textbook. Published in this form, in a small edition (the two lists of subscribers give some 79 names accounting for 156 copies), by an obscure Hungarian college publisher, in a small town in Transylvania, the work was guaranteed immediate oblivion. It remained a forgotten masterpiece until, 35 years later, Riemann?s paper on the hypotheses of geometry reawakened interest in this field, with profound consequences for the mathematical description of real space.In the rediscovery of János?s masterpiece, the father?s work was largely neglected. Farkas Bolyai was a close friend of Gauss and regarded by the latter as the only man who fully understood Gauss?s metaphysics of mathematics. ?He can be taken as a precursor of Gottlob Frege, Pasch, and Georg Cantor; but, as with many pioneers, he did not enjoy the credit that accrued to those that followed him? (DSB). He had worked on the parallel postulate and the possibilities of a non-Euclidean geometry from his earliest days as a mathematician in Göttingen, and had corresponded with Gauss on the subject, even sending him a manuscript entitled Theoria parallelarum, but it was his son János who was to achieve the breakthrough. The Appendix appears at the end of volume one, and is separately paginated [ii] 26 [2, errata], and with one plate in the volume specifically pertaining to the Appendix. There are further substantial references to the Appendix in the main body of Farkas?s text, primarily in the section ?Generalis conspectus geometriae? (Vol I, pp 442?502) and an important supplement to the Appendix in the second volume (pp 380?383). Apart from the Appendix, hardly any two copies of the Tentamen agree in collation, and the great variation amongst them, including cancel leaves and gatherings, indicates that the publishing history of this work was confused, and remains confusing.This copy is unusual in a further respect; the first volume is larger than any others known. The Norman copy was also uncut but measured only 219 x 137 mm, whereas its second volume was 225 x 143 mm, closer to the measurements of our first volume.Bolyai illustrates his textbook with 14 folding plates, five of which are augmented with numerous small flaps. These plates contain as many as 10 slips, often concealed one behind the other; plate 10 also displays a single volvelle, which has gone unrecorded in most bibliographies to date; although not described in the printed or on-line catalogue entries, it is present in most copies. One point of bibliographic confusion has been clarified: the Horblit/Grolier Catalogue (based on the Smithsonian copy) lists an overslip on plate 6 that is not recorded in any other copy. Upon investigation, it appears that an integral part of the plate (the lower portion of the diagram labelled T.144) was inadvertently detached during rebinding and subsequently reattached on a stub, leading to the conclusion that this was a required flap.Currently 24 copies of the Tentamen are known to exist, including the present copy, and one (Berlin) that was lost in WWII. Of these 24, one comprises Janos Bolyai?s Appendix only. A further three-
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## Mémoires Mathématiques, Contenant ce en quoy sest exercé le très-illustre, très-excellent Prince et Seigneur Maurice Prince dOrange, Conte de Nassau translate en François par Jean Tuning.

STEVIN, Simon. Very rare first edition in French of this collection of works, which was published almost simultaneously in Dutch, French and Latin. They deal, among other topics, with geometry, trigonometry, perspective, and double-entry book-keeping ? Stevin was one of the first authors to compose a treatise on governmental accounting. The Appendice Algébraique, which Sarton called ?one of Stevin?s most important publications,? is the first published general method of solving algebraic equations; it uses what is now called the ?intermediate value theorem,? a remarkable anticipation, as it was not rigorously formulated by mathematicians until the nineteenth century. All the works appearing in this volume were first published in this collection (with one exception, where the version here is the earliest extant ? see below). Stevin (1548-1620) was perhaps the most original scientist of the second half of the 16th century (the major works of Galileo did not appear until the 17th century). ?He was involved in geometry, algebra, arithmetic (pioneering a system of decimals), dynamics and statics, almost all branches of engineering and the theory of music? (Kemp, p. 113). ?Stevin unconditionally supported [the Copernican system], several years before Galileo and at a time when few other scientists could bring themselves to do likewise? (DSB XIII: 48). In 1593 Prince Maurice of Nassau (1567-1625) appointed Stevin quartermaster-general of the Dutch armies, a post he held until his death. From 1600 Stevin organized the mathematical teaching at the engineering school attached to Leiden University. ?The Prince used to carry manuscripts of [Stevin?s lectures] with him in his campaigns. Fearing that he might lose them, he finally decided to have them published, not only in the original Dutch text [Wisconstighe Gedachtenissen] ? but also in a Latin translation by Willebrord Snel [Hypomnemata mathematica] ? and in a French translation by Jean Tuning [offered here]? (Sarton, p. 245). The Dutch and Latin editions were published in five parts, of which the fourth consisted principally of reprints of his works on statics that had appeared separately in 1586. This fourth part was not translated into French because, we are told at the beginning of the fifth part, of the printer?s impatience ? he was tired of keeping the sheets already printed and suggested that additional materials could be published later when the author had prepared them. The printer?s impatience also accounts for the fact that several works that are announced on the title pages of the individual volumes did not in fact appear in the Dutch, French or Latin editions. The only other complete copy of this French edition listed by ABPC/RBH is the De Vitry copy, in a nineteenth-century binding (Sotheby?s, April 11, 2002, lot 779, £15,200 = $21,935). OCLC lists Columbia, Harvard and UCLA only in US.Provenance: L. Cundier, early inscription on title-pages, i.e., Louis Cundier (c. 1615- 1681), French geometer, surveyor and engraver. He was professor of mathematics at Aix, and was responsible for a Carte géographique de Provence, published about 1640. Contemporary marginal annotation on R6v of final part.The first part of the work, entitled Cosmographie (1608), is a treatise on the trigonometrical techniques used in the observation of the heavens, together with extensive tables of sines, tangents and secants. ?The first to use the term trigonometry seems to have been Pitiscus, whose book Trigonometria made its first appearance in 1595, but in 1608, when Stevin?s book appeared, the term had not yet been generally accepted. The book consists of four parts, the first dealing with the construction of goniometrical tables, the second with plane triangles, and the remaining two parts with spherical trigonometry ? It is mainly of interest to those who wish to see what trigonometry was like in the sixteenth century, long before Euler, in 1748, introduced the present notation. It also has some distinction as the first complete text on trigonometry written in Dutch; and one of the first ? if not the first ? written in any vernacular? (Works, IIb, p. 751).Part II, De la Practique de Géométrie (1605) [in Dutch, De Meetdaet], ?is primarily a textbook for the instruction of those who, like Prince Maurice, wanted to learn some of the more practical aspects of geometry. The course was not one for beginners, knowledge of Euclid?s Elements being a prerequisite, while the reader was also supposed to know something about the measurement of angles and Stevin?s own calculus of decimal fractions ? Parts of the contents were taken from the Problemata Geometrica, the book which Stevin published in 1583, but to which he, curiously enough, never refers. Other parts show the influence of Archimedes and of contemporary writers such as Del Monte and Van Ceulen. Although in accordance with the title strong emphasis is laid on the practical applications of geometry, many theoretical problems are discussed. For Stevin theory and application always went hand in hand.?The Meetdaet appeared in 1605, but it was drafted more than twenty years before. Already in the Problemata Geometrica Stevin refers to a text on geometry, ?which we hope shortly to publish? and in which the subject was to be treated by a method parallel to that used in arithmetic. At that time Stevin?s L’Arithmétique was either finished or well advanced. We get the impression that in this period, 1583-85, Stevin decided to publish his full text on arithmetic, but of his text on geometry only those parts which he considered novel. The general outline of the two texts was laid out at the same time, and in close parallel. When at last the Meetdaet appeared, it had undergone many changes, resulting partly or wholly from lengthy discussions with the Prince of Orange. The underlying idea, however, remained the same.?In the introduction to the Meetdaet Stevin explains what he means by this parallelism of arithmetic and geometry. In arithmeti-
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## A Mathematical Theory of Communication. Offprint from Bell System Technical Journal, Vol. 27 (July and October).

SHANNON, Claude Elwood. First edition, the rare offprint, of ?the most famous work in the history of communication theory? (Origins of Cyberspace). ?Probably no single work in this century has more profoundly altered man’s understanding of communication than C. E. Shannon?s article, ?A mathematical theory of communication?, first published in 1948? (Slepian). ?Th[is] paper gave rise to ?information theory?, which includes metaphorical applications in very different disciplines, ranging from biology to linguistics via thermodynamics or quantum physics on the one hand, and a technical discipline of mathematical essence, based on crucial concepts like that of channel capacity, on the other? (DSB). ?A half century ago, Claude Shannon published his epic paper ?A Mathematical Theory of Communication.? This paper [has] had an immense impact on technological progress, and so on life as we now know it ? One measure of the greatness of the [paper] is that Shannon?s major precept that all communication is essentially digital is now commonplace among the modern digitalia, even to the point where many wonder why Shannon needed to state such an obvious axiom? (Blahut & Hajek). ?In 1948 Shannon published his most important paper, entitled ?A mathematical theory of communication? This seminal work transformed the understanding of the process of electronic communication by providing it with a mathematics, a general set of theorems rather misleadingly called information theory. The information content of a message, as he defined it, has nothing to do with its inherent meaning, but simply with the number of binary digits that it takes to transmit it. Thus, information, hitherto thought of as a relatively vague and abstract idea, was analogous to physical energy and could be treated like a measurable physical quantity. His definition was both self-consistent and unique in relation to intuitive axioms. To quantify the deficit in the information content in a message he characterized it by a number, the entropy, adopting a term from thermodynamics. Building on this theoretical foundation, Shannon was able to show that any given communications channel has a maximum capacity for transmitting information. The maximum, which can be approached but never attained, has become known as the Shannon limit. So wide were its repercussions that the theory was described as one of humanity?s proudest and rarest creations, a general scientific theory that could profoundly and rapidly alter humanity?s view of the world. Few other works of the twentieth century have had a greater impact; he altered most profoundly all aspects of communication theory and practice? (Biographical Memoirs of Fellows of the Royal Society, Vol. 5, 2009). Remarkably, Shannon was initially not planning to publish the paper, and did so only at the urging of colleagues at Bell Laboratories.?Relying on his experience in Bell Laboratories, where he had become acquainted with the work of other telecommunication engineers such as Harry Nyquist and Ralph Hartley, Shannon published in two issues of the Bell System Technical Journal his paper ?A Mathematical Theory of Communication.? The general approach was pragmatic; he wanted to study ?the savings due to statistical structure of the original message? (p. 379), and for that purpose, he had to neglect the semantic aspects of information, as Hartley did for ?intelligence? twenty years before. For Shannon, the communication process was stochastic in nature, and the great impact of his work, which accounts for the applications in other fields, was due to the schematic diagram of a general communication system that he proposed. An ?information source? outputs a ?message,? which is encoded by a ?transmitter? into the transmitted ?signal.? The received signal is the sum of the transmitted signal and unavoidable ?noise.? It is recovered as a decoded message, which is delivered to the ?destination.? The received signal, which is the sum between the signal and the ?noise,? is decoded in the ?receiver? that gives the message to destination. His theory showed that choosing a good combination of transmitter and receiver makes it possible to send the message with arbitrarily high accuracy and reliability, provided the information rate does not exceed a fundamental limit, named the ?channel capacity.? The proof of this result was, however, nonconstructive, leaving open the problem of designing codes and decoding means that were able to approach this limit.?The paper was presented as an ensemble of twenty-three theorems that were mostly rigorously proven (but not always, hence the work of A. I. Khinchin and later A.N. Kolmogorov, who based a new probability theory on the information concept). Shannon?s paper was divided into four parts, differentiating between discrete or continuous sources of information and the presence or absence of noise. In the simplest case (discrete source without noise), Shannon presented the [entropy] formula he had already defined in his mathematical theory of cryptography, which in fact can be reduced to a logarithmic mean. He defined the bit, the contraction of ?binary digit? (as suggested by John W. Tukey, his colleague at Bell Labs) as the unit for information. Concepts such as ?redundancy,? ?equivocation,? or channel ?capacity,? which existed as common notions, were defined as scientific concepts. Shannon stated a fundamental source-coding theorem, showing that the mean length of a message has a lower limit proportional to the entropy of the source. When noise is introduced, the channel-coding theorem stated that when the entropy of the source is less than the capacity of the channel, a code exists that allows one to transmit a message ?so that the output of the source can be transmitted over the channel with an arbitrarily small frequency of errors.? This programmatic part of Shannon?s work explains the success and impact it had in telecommunications engineering. The turbo codes (error correction codes) achieved a l-
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## A precious sammelband containing eight extremely rare works, all in first edition, notably: HOOD, The making and use of the geometricall instrument, called a sector . London: J. Windet and sold by S. Shorter, [1598] & MOHR, Euclides Danicus . Amsterdam: Jacob van Velsen for the author, 1672.

HOOD, Thomas; MOHR, Georg; BEDWELL, William; STURM, Johannes. One of the most remarkable sammelbands from the Macclesfield library, containing the extremely rare first edition of the first published work on the ?sector?, also called the ?geometrical compass? by Galileo who developed it independently in the late 1590s as an instrument for military engineering (although he did not publish an account of it until 1606). ?Hood?s sector was the first mechanical calculating device of general practical use to be published since the abacus of remote antiquity? (Stillman Drake, p. 17). ?Although credit for the sector is often given to Galileo, it is clear that the instrument was well known and used in England before Galileo published his work on it? (Tomash & Williams, p. 1416). ?The sector was one of the most familiar of mathematical instruments between the 17th and 19th centuries. It was however devised just before 1600 and was first published in 1598 by the English mathematical practitioner Thomas Hood. An independent version developed by Galileo Galilei in the 1590s was published early in the 17th century [1606], and many other designs subsequently followed? (mhc.ox.ac.uk). OCLC lists no copies in North America, but we have located one (Folger), though it lacks the plates (present in this copy); ABPC/RBH list three copies (including Horblit and Kenney), all lacking the plates. Also included in this volume are three geometrical works by the Danish mathematician Georg Mohr that are so rare that they were thought to be lost until a copy of one of them, Euclides Danicus, was discovered in 1928. This work proves the ?Mohr-Mascheroni? theorem, according to which all geometrical constructions that can be carried out with ruler and compasses can, in fact, be carried out using compasses alone ? it was proved independently by Lorenzo Mascheroni (1750-1800) 125 years after Mohr in his Geometria del Compasso. OCLC lists only one copy of Euclides Danicus in North America (Harry Ransom Center, University of Texas); only one other copy has appeared at auction. The present volume includes two further works attributed to Mohr, as rare as Euclides Danicus, also on geometrical constructions (no copies on OCLC). They are accompanied by three rare works by William Bedwell (1561-1632), two on architectural measuring instruments, the ?carpenter?s rule? and the ?trigon,? the third being the earliest published work on Tottenham, where Bedwell resided (now part of London but then a village to the north of the City). The final work in this volume, by Johannes Sturm (1507-89), is a contribution to the controversy which raged in the late sixteenth and early seventeenth century between Clavius, van Roomen, Viète, and Scaliger over the squaring of the circle. HOOD, Thomas. The making and use of the geometricall instrument, called a sector. Whereby many necessarie geometricall conclusions concerning the proportionall description, and division of lines, and figures, the drawing of a plot of ground, the translating of it from one quantitie to another, and the casting of it up geometrically, the measuring of heights, lengths and breadths may be mechanically performed with great expedition, ease, and delight to all those, which commonly follow the practise of the mathematicall arts. London: Printed by Iohn Windet, and are to solde at the great North dore of Paules Church by Samuel Shorter, [1598].First edition of the first published work on the sector. ?The sector, also known as the proportional, geometric, or military compass, was an analog calculating instrument used widely from the late sixteenth century until modern times ? Requirements for extensive arithmetic calculation grew rapidly during the Renaissance and in the early years of the scientific and industrial revolutions. It soon became apparent to practitioners that calculation by hand, particularly the multiplication and division of large numbers, was both laborious and error-prone. It was small wonder that talented mathematicians and scientists sought to develop methods and mechanisms that would lessen the burden of computation while increasing accuracy? (Tomash & Williams, p. 1456).Hood?s sector consisted of ?a pair of flat rules hinged stiffly at one end and bearing identical scales engraved on the two arms, different on the two faces ? It had three scales and was fitted with removable sights and a graduated quadrant, plumb line, and accessory graduated arm ? Hood?s principal scale was one of equal linear divisions from pivot to end of either arm. On the other face he provided a scale which gave the side of various regular polygons inscribed in a circle of diameter equal to the separation of the ends, and another which gave the side of a square having an area which was an integral multiple of the area of a unit square. The enormous value of Hood?s sector for speedy mechanical approximation to a wide variety of commonest practical mathematical problems is obvious, and he explained these at great length in his book? (Drake, pp. 17-18). ?We know little of Thomas Hood (1556-1620) other than that he was the first mathematical lecturer for the City of London and gave public lectures there on topics such as the sector and other instruments. We do no know where or how Hood might have first learned of the instrument, but we presume that he learned of it through contacts with the military. In 1598, he published The making and use of the geometricall instrument, called a sector. With this book title he seems to have coined the English word sector (at least as it applies to a mathematical instrument). The book is well organized and contains useful diagrams, examples and exercises. It is obviously not a work created in haste, and this fact leads one to the conclusion that Hood must have been familiar with the sector for some time prior to 1598. Further, the book notifies the reader that Hood?s sectors were available for sale around 1594-1611 by the instrument maker Charles Whitwell, who engraved the illustrations for Hood?s book. Indeed, a-
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## Did The Atlantic Close And Then Re-Open? Offprint from: Nature, Vol. 211, No. 5050, August 13, 1966.

TUZO WILSON, John. First edition, very rare offprint, of this landmark paper elucidating the history and mechanism of continental drift by ?one of the most imaginative Earth scientists of his generation? (DSB). ?In 1966, J. Tuzo Wilson published ?Did the Atlantic Close and then Re-Open?? in the journal Nature. The Canadian author introduced to the mainstream the idea that continents and oceans are in continuous motion over our planet?s surface. Known as plate tectonics, the theory describes the large-scale motion of the outer layer of the Earth. It explains tectonic activity (things like earthquakes and the building of mountain ranges) at the edges of continental landmasses (for instance, the San Andreas Fault in California and the Andes in South America)? (Heron). Alfred Wegener (1880-1930) had already suggested in the early 1900s that continents move around the surface of the earth, specifically that there had been a super-continent (Pangaea) where now there is a great ocean (the Atlantic). In the present paper, Wilson explained the geological evidence that North America and Europe were once separated across an ocean before the Atlantic Ocean. This ocean closed in stages as the continents that used to be separated by the ocean converged by subduction and eventually collided in a mountain-building event. The combined continent was then sliced apart and the continents drawn away from each other once more as the modern Atlantic Ocean opened. The paper combined the nascent ideas of divergent and convergent plate boundaries into a conceptual model that matched observations of geological features around the world. The tectonic cycle he described now goes by ?the Wilson Cycle? or the ?Supercontinent Cycle? and still governs how we think of the evolution of tectonic plates through time. ?Wilson?s great idea was a crucial step forward. It reopened the whole question of ?what happened before Pangea?? By suggesting that his ?proto-Atlantic? had opened within an earlier supercontinent (just as the Atlantic did within Pangea) he also linked his process to a grander cycle leading from one supercontinent Earth to another? (Nield). As was often the case for offprints from Nature (e.g., the famous Watson/Crick DNA offprint), this offprint is printed in a smaller format than the journal issue, with the text reset. No copies in auction records or on OCLC.In the early twentieth century the prevailing wisdom regarding how mountain belts were formed and why the sea is deep was that the Earth started out as a molten ball and gradually cooled. When it cooled, heavier metals such as iron sank down and formed the core, while lighter metals such as aluminium stayed up in the crust. The cooling also caused contraction and the pressure produced by contraction caused some parts of the crust to buckle upwards, forming mountains, while other parts of the crust buckled downwards, creating ocean basins. ?Originally a devotee of the contracting-Earth hypothesis, [Tuzo Wilson] became a convert to [continental] drift as he was entering his fifties (by which time he had been Professor of Geophysics at Toronto for a decade). Swiftly recanting his former views, Tuzo saw the way the Earth?s mountain belts were often superimposed upon one another, and set about explaining it in terms of plate tectonics. In a classic paper published in Nature in 1966 and titled ?Did the Atlantic close and then reopen?? he addressed the coincidence of the modern Atlantic with two mountain ranges called the Caledonides in Europe and the Appalachians in the USA. It was the very first time the new plate tectonics had been extended back to the pre-Pangean Earth. ?These two mountain ranges are really one and the same ? except that they are now separated by the Atlantic Ocean, which cut the range in two at a low angle when it opened between them. At one time the two belts had been joined, end-to-end, Caledonides in the north, Appalachians in the south; and the collision that had created them was one event among many that built the supercontinent Pangea. Indeed, the matching of the now separated halves of this once-mighty chain provided Wegener with one of his key ?proofs? ? part of his geological matching of opposing Atlantic shores ??Wegener did not speculate about how his Pangea had come together. But as the new plate tectonics emerged from studies of the ocean floor and began to revitalize drift theory, the time was ripe to see the break-up of Pangea as part of a bigger process. Professor Kevin Burke of the University of Houston, Texas, recalls that on 12 April 1968 in Philadelphia, at a meeting titled ?Gonwanaland Revisited? at the Philadelphia Academy of Sciences, Wilson told his audience how a map of the world showed you oceans opening in some places and closing in others. Burke recalls: ?He therefore suggested that, because the ocean basins make up the largest areas on the Earth?s surface, it would be appropriate to interpret Earth history in terms of the life cycles of the opening and closing of the ocean basins ? In effect he said: for times before the present oceans existed, we cannot do plate tectonics. Instead we must consider the life cycles of the ocean basins.? This key insight had by then already provided Wilson with the answer to an abiding puzzle in the rocks from either side of the modern Atlantic.?Nothing pleased Tuzo more than a grand, overarching framework that made sense of those awkward facts that get thrown aside because they don?t fit ? ideas that philosopher William James dubbed the ?unclassified residuum.? Geologists had been aware since 1889 that within the rocks forming the Caledonian and Appalachian mountains ? that is, rocks dating from the early Cambrian to about the middle Ordovician (from 542 to 470 million years ago) ? were fossils that fell into two clearly different groups or ?assemblages.? This was especially true for fossils of those animals that in life never travelled far, but lived fixed to, or grubbing around in, the seabed. B-
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## Numerical Inverting of Matrices of High Order, pp. 1021-1099 in Bulletin of the American Mathematical Society, Vol. 53, No. 11, November, 1947. [Offered with:] Numerical Inverting of Matrices of High Order II, pp. 188-202 in Proceedings of the American Mathematical Society, Vol. 2, No. 2, April, 1951.

VON NEUMANN, John. & GOLDSTINE, Herman H. First edition, journal issues in the original printed wrappers, of two of von Neumann?s major papers. ?The 1947 paper by John von Neumann and Herman Goldstine, ?Numerical Inverting of Matrices of High Order? (Bulletin of the AMS, Nov. 1947), is considered as the birth certificate of numerical analysis. Since its publication, the evolution of this domain has been enormous? (Bultheel & Cools). ?Just when modern computers were being invented (those digital, electronic, and programmable), John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did concerning the accuracy of mechanized Gaussian elimination ? but their paper was about more than that. Von Neumann and Goldstine described what they surmised would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them? (Grcar, p. 607). ?In sum, von Neumann?s paper contains much that is unappreciated or at least unattributed to him. The contents are so familiar, it is easy to forget von Neumann is not repeating what everyone knows. He anticipated many of the developments in the field he originated, and his theorems on the accuracy of Gaussian elimination have not been encompassed in half a century. The paper is among von Neumann’s many firsts in computer science. It is the first paper in modern numerical analysis, and the most recent by a person of von Neumann?s genius? (Vuik). Von Neumann & Goldstine?s 1947 paper is here accompanied by its sequel (the 1947 paper comprises Chapters I-VII, the sequel Chapters VIII-IX), in which the authors reassess the error estimates proved in the first part from a probabilistic point of view. The only other copy of either paper listed on ABPC/RBH is the OOC copy of part I (both journal issue and offprint).?Before computers, numerical analysis consisted of stopgap measures for the physical problems that could not be analytically reduced. The resulting hand computations were increasingly aided by mechanical tools which are comparatively well documented, but little was written about numerical algorithms because computing was not considered an archival contribution. ?The state of numerical mathematics stayed pretty much the same as Gauss left it until World War II? [Goldstine, The Computer from Pascal to Von Neumann (1972), p. 287]. ?Some astronomers and statisticians did computing as part of their research, but few other scientists were numerically oriented. Among mathematicians, numerical analysis had a poor reputation and attracted few specialists? [Aspray, John von Neumann and the Origins of Modern Computing (1999), pp. 49?50]. ?As a branch of mathematics, it probably ranked the lowest, even below statistics, in terms of what most university mathematicians found interesting? [Hodges, Alan Turing: the Enigma (1983), p. 316].?In this environment John von Neumann and Herman Goldstine wrote the first modern paper on numerical analysis, ?Numerical Inverting of Matrices of High Order?, and they audaciously published the paper in the journal of record for the American Mathematical Society. The inversion paper was part of von Neumann?s efforts to create a mathematical discipline around the new computing machines. Gaussian elimination was chosen to focus the paper, but matrices were not its only subject. The paper was the first to distinguish between the stability of a mathematical problem and of its numerical approximation, to explain the significance in this context of the ?Courant criterium? (later CFL condition), to point out the advantages of computerized mixed precision arithmetic, to use a matrix decomposition to prove the accuracy of a calculation, to describe a ?figure of merit? for calculations that became the matrix condition number, and to explain the concept of inverse, or backward, error. The inversion paper thus marked the first appearance in print of many basic concepts in numerical analysis.?The inversion paper may not be the source from which most people learn of von Neumann?s ideas, because he disseminated his work on computing almost exclusively outside refereed journals. Such communication occurred in meetings with the many researchers who visited him at Princeton and with the staff of the numerous industrial and government laboratories whom he advised, in the extemporaneous lectures that he gave during his almost continual travels around the country, and through his many research reports which were widely circulated, although they remained unpublished. As von Neumann?s only archival publication about computers, the inversion paper offers an integrated summary of his ideas about a rapidly developing field at a time when the field had no publication venues of its own.?The inversion paper was a seminal work whose ideas became so fully accepted that today they may appear to lack novelty or to have originated with later authors who elaborated on them more fully. It is possible to trace many provenances to the paper by noting the sequence of events, similarities of presentation, and the context of von Neumann?s activities? (Grcar, pp. 609-610).We are fortunate to have an account of the genesis and content of these two important papers in Goldstine?s own words. In the years immediately following the end of World War II, Von Neumann, Goldstine and others instituted the ?electronic computer project? at the Institute for Advanced Study at Princeton, NJ. One of the first topics discussed ?was the solution of large systems of linear equations, since they arise almost everywhere in numerical work. V. Bargmann and D. Montgomery collaborated-
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## Larismethique nouellement composee par maistre Estienne de La Roche dict Villefra[n]che natif de Lyo[n] sus le Rosne diuisee en deux parties dont la p[re]miere tracte des p[ro]prietes p[er]fectio[n]s et regles de la dicte scie[n]ce: come le no[m]bre entire, le no[m]bre rout, le regle de troys, la regle dune faulse position, de deux faluses position[n]s, dapposition et remotio[n], de la regle la chose, et de la qua[n]tite des p[ro]gressio[n]s et p[ro]portio[n]s. La seco[n]de tracte de la practique dicelle applicquee en fait des mo[n]oyes, en toutes marcha[n]dises comme drapperie, espicerie, mercerie et en toutes aultres marcha[n]dises qui se vendent a mesure au pois ou au nombre, en co[m]paignies et en tro[n]ques, es changes et merites, en fin dor et dargent et en lavaluer diceux. En arge[n]t le rey et en fin darge[n]t doze. Es deneraulx allyages et effaiz, tant de lot que de large[n]t. Et en geometrie aplicquee aux ars mecha[n]ique come aux masons charpe[n]tiers et a tous aultres bes

DE LA ROCHE, Estienne. First edition, extremely rare, of the first published work on algebra in French. This is a fine copy in a beautiful contemporary binding. Born in Lyon, then the principal commercial centre of France, La Roche was a student of Nicolas Chuquet and published for the first time in the present work large sections from Chuquet?s Le Triparty en la Science des Nombres, the most original mathematical work of the fifteenth century. Chuquet?s work, of which a single manuscript survives (BNF Fonds français 1346), remained unpublished until 1881. La Roche?s work thus printed for the first time several important innovations in arithmetic and algebra introduced by Chuquet: the use of exponents to denote powers of a number, often credited to Descartes who introduced them in his Géométrie more than a century later; the use of the ?second unknown? (see below) in the solution of systems of linear equations, which was an important step towards the invention of symbolic algebra by Viète; the use of negative numbers in the solution of equations; and the introduction of our terms ?million?, ?billion? and ?trillion? for powers of 106. La Roche also includes Chuquet?s ?règle des nombres moyens,? according to which a fraction could be found between any two given fractions by taking the sum of their numerators and dividing by the sum of their denominators; this rule could be used to find the solution of any problem soluble in rational numbers, once an upper and a lower bound for the solution had been found. La Roche intended his work to serve the mercantile class, and his account of commercial arithmetic goes considerably beyond Chuquet. ?The second, and greater, part of La Roche?s work has, apart from some geometrical calculations at the end, a commercial character. The author states that as a basis he used ?the flower of several masters, experts in the art? of arithmetic, such as Luca Pacioli, supplemented by his own knowledge of business practice ? [La Roche?s work] presented an outstanding view of contemporary methods of computation and their applications in trade? (DSB). OCLC lists copies at Columbia and Harvard only in North America. ABPC/RBH list only the Macclesfield copy (rebound in the 19th century) since Honeyman (Sotheby?s, April 14, 2005, lot 1204, £19,200 = $36,409). The present copy was offered by Librairie Thomas-Scheler in 1996 (Catalogue Nouvelle Série No. 15, n. 296, 120,000F). ?We do not know much about de la Roche (c. 1470-1530). Tax registers from Lyon reveal that his father lived in the Rue Neuve in the 1480s and that Estienne owned more than one property in Villefranche, from which he derived his nickname. De la Roche is described as a ?master of argorisme? as he taught merchant arithmetic for 25 years at Lyon. He owned the manuscript of the Triparty after the death of Chuquet (1488). It is therefore considered that de la Roche was on friendly terms with Chuquet and possibly learned mathematics from him.?The importance of the Larismethique has been seriously underestimated. There are several reasons for this. Probably the most important one is Aristide Marre?s misrepresentation of the Larismethique as a grave case of plagiarism. Marre discovered that the printed work by Estienne de la Roche, contained large fragments that were literally copied from Chuquet?s manuscript (Marre, Le Triparty ? par Maistre Nicolas Chuquet (1881), introduction)? (Heeffer, pp. 1-2). But Barbara Moss argues that ?the charge of plagiarism against Estienne de la Roche is largely an anachronism ? Before the spread of printing, academic knowledge had been disseminated through the copying of manuscripts, and Chuquet, like many of his contemporaries, must have written down for reference a large number of examples from the work of others, with or without a note of their source ? De la Roche?s use of citations and sources is similar to that in a number of printed arithmetics of that period. Following the usual commendation of mathematics for its ?great utility and necessity?, he continues: ?I have collected and amassed the flowers of several masters expert in this art, such as master Nicolas Chuquet, Parisian, Philippe Frescobaldi, Florentine, and Brother Luke of Borgo [Pacioli], with some small addition of what I have been able to invent and test out in my time in its practice? [first (unnumbered) page]? (Moss, pp. 117-9).Moreover, ?giving a transcription of the problem text only, Marre withholds that for many of the solutions to Chuquet?s problems de la Roche uses different methods and an improved symbolism. In general, the Larismethique is a much better structured text than the Triparty and one intended for a specific audience. Chuquet was a bachelor in medicine educated in Paris within the scholarly tradition and well acquainted with Boethius and Euclid. On the other hand, de la Roche was a reckoning master operating within the abbaco tradition. It becomes clear from the structure of the book that de la Roche had his own didactic program in mind. He produced a book for teaching and learning arithmetic and geometry which met the needs of the mercantile class. He rearranges Chuquet?s manuscript using Pacioli?s Summa as a model. He even adopts Pacioli?s classification in books, distinctions and chapters. He moves problems from Chuquet?s Appendice to relevant sections within the new structure. He adds introductory explanations to each section of the book, such as for the second unknown, discussed below. With the judgment of an experienced teacher, he omits sections and problems from the Triparty which are of less use to merchants and craftsmen and adds others which were not treated by Chuquet such as problems on exchange and barter? (Heeffer, p. 2).?As far as the first book of the Triparty is concerned, de la Roche is reasonably faithful to his teacher. He does include an extra chapter, on the connotations of the numbers 1 to 12, which he took from Pacioli, and Pacioli from St Augustine?s Civita Dei. He also prefers some-
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## De vi percussionis liber. Bologna: Giacopo Monti, 1667. [Bound with:] [Drop-title:] Risposta . alle considerazioni fatte sopra alcuni luoghi del suo libro della forza della percossa del R. P. F. Stefano de gl Angeli . allillustrissim. Messina [after 29 February, 1668].

BORELLI, Giovanni Alfonso. First edition of the first published book on the laws of percussion, and containing important hitherto unpublished material from the lectures of Galileo and Torricelli. This copy is bound with Borelli?s very rare Risposta, intended as a supplement to De vi percussionis (it was issued without a separate title-page), which contains his reply to criticisms by Stefano degli Angeli of Borelli?s views on the motion of bodies in free fall under gravity. ?In this, Borelli?s first book on mechanics, he quotes Galileo?s youthful work on percussion, the fourth Dialogo, and lectures by Torricelli. As well as the detailed discussion of impact, the book deals with the dynamics of falling bodies, vibration, gravity, fluid mechanics, magnetism, and pendular motion ? he gives the name resilience for the first time to a number of problems now classed under this name? (Roberts & Trent). This is ?the earliest book on the laws of percussion, which undoubtedly influenced John Wallis who, in 1668, published his discovery of the laws governing the percussion of non-elastic bodies, and Christiaan Huygens, who deals with the percussion of elastic bodies in his treatise De motu corporum ex percussione, published in 1669? (Zeitlinger I, 174). Thanks to the Risposta, Borelli ?can be credited to be the first person to have examined in quantitative detail the deflection that falling bodies undergo due to the earth?s diurnal rotation? (Theo Gerkema, On Borelli?s analysis concerning the deflection of falling bodies, 2009). Borelli regarded De vi percussionis, together with his De motionibus naturalibus (1670), as necessary preparation for his masterpiece, De motum animalium (1680-81), on which he had worked since the early 1660s. Although De vi percussionis is found without great difficulty on the market, this is the first copy we have seen that is complete with its supplement, the Risposta. OCLC records five copies of the Risposta in US (Burndy; Hagley Library, Columbia; Cornell; New York Academy of Medicine; and Wisconsin); no copies in auction records.Provenance: Bookplate of G[iovanni]. B[attista]. Tomaselli (1650-1730) on front paste-down; faded contemporary ownership inscription on title.?In May 1665, Cardinal Michelangelo Ricci, Roman correspondent and adviser to the Tuscan Court, wrote to Borelli?s patron, Leopoldo de Medici, encouraging Borelli to apply himself to the composition of a treatise on motion. According to Ricci, motion was a particularly important topic since so many contemporaries, famed for their contributions to mathematics and philosophy, had dedicated so much time to the topic and had explained so many of nature?s secrets. Borelli?s initial response was that he was instead concentrating on a treatise on anatomy within which he would insert some words regarding collision of moving bodies. At some point in this discussion, seemingly prompted by an insistence from Ricci, Borelli decided to publish On the Force of Percussion independently from his main project. The intention of the book on colliding bodies was to establish crucial propositions concerning motion as a means of introducing issues related to human and animal movements.?The main problem in question, as Aristotle had put it, was to explain why a heavy axe, as an example, has virtually no effect when rested on a piece of wood but has a much greater impact when it is made to fall from a significant height. Aristotelians believed that the increased force is a result simply of the velocity of the movable; the velocity supposedly artificially increases the weight of the object. For Italian natural philosophers in the seventeenth century, the first point of reference in response to this Aristotelian position is the work carried out by Galileo concerned with motion and mechanics including percussion. In his Mechanics (c. 1590), Galileo claimed that to study percussion, one must consider ?that which has been seen to happen in all other mechanical operations, which is that the force, the resistance, and the space through which the motion is made respectively follow that proportion and obey those laws by which a resistance equal to the force will be moved by this force through an equal space and with equal velocity to that of the mover.? That was to say that it is not only the weight of the body in motion that determines the force of percussion but the distance it travels and its velocity before impact that is required to overcome the resistance of the body being impacted upon. Galileo elaborated on his argument in Discourse Concerning Two New Sciences (1638), where he presented several experiments in which the force of percussion was tested and measured by relying on the proportions of opposing forces (including distances and velocities) rather than simply differences in weight.?In On the Force of Percussion, Borelli agrees with the Galilean proposition that the energies of colliding bodies are not measurable through weight alone. To prove his point, he begins with a series of propositions explaining how a body must be first moved by an impeller in order to acquire a ?motive virtue? or ?impetus? Upon colliding with another body at rest, that impetus is transmitted to the stationary body, overcomes its resistance proportional to the mass and velocity of the first body and itself sets in motion. Borelli puts it succinctly: ?Despite the horror of some Aristotelians for the migration of the motive virtue, it seems certain that part of the virtue or impetus which was concentrated in the impelling body is distributed and expanded in the struck body.? The ?distribution and expansion? of impetus does not mean that the struck body acquires the same speed as the first, ?impelling? body, only that the motive virtue is preserved and shared between the two bodies?the reactions of these bodies to the collision is proportional to their respective masses. In sum, the impact of colliding bodies occurs in only a moment, but the result of that instant of time?the c-
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## Arithmetica Universalis; sive de Compositione et Resolutione Arithmetica Liber. Ciu accessit Helleiana Aequationum Radices Arithmetice Inveniendi Methodus .

NEWTON, Sir Isaac. First edition of Newton?s treatise on algebra, or ?universal arithmetic,? his ?most often read and republished mathematical work? (Whiteside). ?Included are ?Newton?s identities? providing expressions for the sums of the ith powers of the roots of any polynomial equation, for any integer i [pp. 251-2], plus a rule providing an upper bound for the positive roots of a polynomial, and a generalization, to imaginary roots, of René Descartes? Rule of Signs [pp. 242-5]? (Parkinson, p. 138). About this last rule for determining the number of imaginary roots of a polynomial (which Newton offered without proof), Gjertsen (p. 35) notes: ?Some idea of its originality ? can be gathered from the fact that it was not until 1865 that the rule was derived in a rigorous manner by James Sylvester.?Provenance: Jesuit College at Ghent (ink inscription ?Bibliotheca Collegii Gandavensis Soc[ietatis] Jesu.? and shelfmark on title); extensive marginal annotations by a well-informed contemporary reader. This reader was possibly the English Jesuit Christopher Maier (1697-1767). Born in Durham, England, Maier entered the Society of Jesus in 1715. He taught at Liège, where he became interested in astronomy. In 1750, Maire was commissioned by Pope Benedict XIV to measure two degrees of the meridian from Rome to Rimini with fellow Jesuit Roger Boscovich, with a view to mapping the Papal States; in turn, they proved that the earth is an oblate spheroid, as Newton had proposed in Principia, publishing their results in Litteraria Expeditione (1755). Maier spent his final years at the English Jesuit College in Ghent.?In fulfillment of his obligations as Lucasian Professor, Newton first lectured on algebra in 1672 and seems to have continued until 1683. Although the manuscript of the lectures in [Cambridge University Library] carries marginal dates from October 1673 to 1683, it should not be assumed that the lectures were ever delivered. There are no contemporary accounts of them and, apart from Cotes who made a transcript of them in 1702, they seem to have been totally ignored. Whiteside (Papers V, p. 5) believes that they were composed ?over a period of but a few months? during the winter of 1683-4? (Gjertsen, pp. 33-4). The course of lectures stemmed from a project on which Newton had embarked in the autumn of 1669, thanks to the enthusiasm of John Collins: the revision of Mercator?s Latin translation of Gerard Kinckhuysen?s Dutch textbook on algebra, Algebra ofte stel-konst (1661). Newton composed a manuscript, ?Observations on Kinckhuysen?, in 1670 (see Whiteside, Papers II) and used it in the preparation of his lectures. He took the opportunity not only to extend Cartesian algebraic methods, but also to restore the geometrical analysis of the ancients, giving his lectures on algebra a strongly geometric flavor.?When Newton resigned his Lucasian professorship to his deputy William Whiston in December 1701, it was natural that the latter should wish to familiarize himself with the deposited lectures of his predecessor? (Whiteside, Papers V, p. 8). Whiston later claimed (in his Memoirs, London: 1749) that Newton gave him his reluctant permission to publish the lectures. Whiston arranged with the London stationer to underwrite the expense of printing the deposited manuscript and then subsequently, between September 1705 and the following June, corrected both specimen and proof sheets as they emerged from the University Press. The completed editio princeps finally appeared in May 1707, priced at 4s. 6d., without Newton?s name on the title page, although references inside the work made no attempt to hide the author?s identity. It included an appended tract by Halley on ?A new, accurate and easy method for finding the roots of any equations generally, without prior reduction? (pp. 327-343). Publication of the work had been delayed by Newton, who complained that the titles and headings were not his and that it contained numerous mistakes. Yet when he prepared a second edition in 1722 the changes he introduced were ?primarily reorderings of his own manuscript, not corrections of Whiston?s additions? (Westfall, p. 649). In reality, Newton?s misgivings probably derived more from his reluctance to place before the public a relatively immature and poorly organized work, and one that did not take into account the developments in the subject that had taken place in the quarter century since the manuscript was composed.For a book that was to become Newton?s most often republished mathematical work, the Arithmetica initially made little impact in Britain, and was not even graced by a review in the Philosophical Transactions. On the Continent the reception accorded the lectures was more positive. ?Leibniz, unhesitatingly divining their author beneath the cloak of anonymity, gave them a long review in the Acta Eruditorum of Leipzig in 1708. Written thirty years before, he noted, and now deservingly printed by William Whiston, he assured the reader that ?you will find in this little book certain particularities that you will seek in vain in great tomes on analysis.? His close associate, Johann Bernoulli, despite some adverse remarks paid Newton the compliment in 1728 of basing his own course on the elements of algebra upon Newton?s text. Perhaps partly in consequence of Newton?s recent death, in Britain too the book began about this time to arouse greater interest than when it was first issued in 1707? (Hall, p. 174).Despite the impressive contributions of the work to the theory of equations, mentioned earlier, it is difficult to pigeonhole the work as being either algebraic or geometric. From one point of view, the Arithmetica can be seen as a fulfillment of the programme outlined by Descartes in the Géométrie because it teaches how geometrical problems (and also arithmetical and mechanical ones) can be translated into the language of algebra. Paradoxically, however, Newton criticized Descartes, maintaining that, at least in some case-
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## The Doctrine of Chances: or, A Method of Calculating the Probability of Events in Play.

MOIVRE, Abraham de. First edition, and an unusually fine copy without any restoration, of this classic on the theory of probability, the first original work on the subject in English. ?De Moivre?s book on chances is considered the foundation for the field of probability and statistics? (Tomash). ?De Moivre?s masterpiece is The Doctrine of Chances? (DSB). ?His work on the theory of probability surpasses anything done by any other mathematician except P. S. Laplace. His principal contributions are his investigations respecting the Duration of Play, his Theory of Recurring Series, and his extension of the value of Daniel Bernoulli?s theorem by the aid of Stirling?s theorem? (Cajori, A History of Mathematics, p. 230). ?He was among the intimate friends of Newton, to whom this book is dedicated. It is the second book devoted entirely to the theory of probability and a classic on the subject? (Babson 181). De Moivre?s interest in probability was raised by Pierre-Rémond de Montmort?s Essay d?analyse sur les jeux de hazard (1708), the first separately-published work on probability. ?The [Doctrine] is in part the result of a competition between De Moivre on the one hand and Montmort together with Nikolaus Bernoulli on the other. De Moivre claimed that his representation of the solutions of the then current problems tended to be more general than those of Montmort, which Montmort resented very much. This situation led to some arguments between the two men, which finally were resolved by Montmort?s premature death in 1719 ? De Moivre had developed algebraic and analytical tools for the theory of probability like a ?new algebra? for the solution of the problem of coincidences which somewhat foreshadowed Boolean algebra, and also the method of generating functions or the theory of recurrent series for the solution of difference equations. Differently from Montmort, De Moivre offered in [Doctrine] an introduction that contains the main concepts like probability, conditional probability, expectation, dependent and independent events, the multiplication rule, and the binomial distribution? (Schneider, p. 106).Provenance: Nathaniel Cholmley (1721-91), British Member of Parliament from 1756 to 1774 (bookplate on front paste-down). Erwin Tomash (book label on front paste-down).The modern theory of probability is generally agreed to have begun with the correspondence between Pierre de Fermat and Blaise Pascal in 1654 on the solution of the ?Problem of points? Pascal included his solution as the third section of the second part of his 36-page Traité du triangle arithmétique (1665), which was essentially a treatise on pure mathematics. ?Huygens heard about Pascal?s and Fermat?s ideas [on games of chance] but had to work out the details for himself. His treatise De ratiociniis in ludo aleae ? essentially followed Pascal?s method of expectation. ? At the end of his treatise, Huygens listed five problems about fair odds in games of chance, some of which had already been solved by Pascal and Fermat. These problems, together with similar questions inspired by other card and dice games popular at the time, set an agenda for research that continued for nearly a century. The most important landmarks of this work are [Jakob] Bernoulli?s Ars conjectandi (1713), Montmort?s Essay d’analyse sur les jeux de hazard (editions in 1708 and 1711 [i.e., 1713]) and De Moivre?s Doctrine of Chances (editions in 1718, 1738, and 1756). These authors investigated many of the problems still studied under the heading of discrete probability, including gamblers ruin, duration of play, handicaps, coincidences and runs. In order to solve these problems, they improved Pascal and Fermat?s combinatorial reasoning, summed infinite series, developed the method of inclusion and exclusion, and developed methods for solving the linear difference equations that arise in using Pascal?s method of expectations.? (Glenn Schafer in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (1994), Grattan-Guiness (ed.), p. 1296).?De Moivre?s earliest book on probability, the first edition of the Doctrine of Chances, was an expansion of a long (fifty-two pages) memoir he had published in Latin in the Philosophical Transactions of the Royal Society in 1711 under the title ?De mensura sortis? (literally, ?On the measurement of lots?). De Moivre tells us that in 1711 he had read only Huygens? 1657 tract De Ratiociniis in Ludo Aleae and an anonymous English 1692 tract based on Huygens? work (now known to have been written by John Arbuthnot). By 1718 he had encountered both Montmort?s Essay d?analyse sur les jeux de hazard (2nd ed., 1713) and Bernoulli?s Ars Conjectandi (1713), although the latter had no pronounced effect on De Moivre at that early date? (Stigler, p. 71).The Doctrine consists of an introduction with definitions and elementary theorems, followed by a series of numbered problems. ?De Moivre begins with the classical measure of probability, ?a fraction whereof the numerator be the number of chances whereby an event may happen, and the denominator the number of all the chances whereby it may either happen or fail? He gives the summation rule for probabilities of disjunct events explicitly only for the case of the happening and the not happening of an event. Expectation is still on the level of Huygens defined as the product of an expected sum of money and the probability of obtaining it, the expectation of several sums is determined by the sum of the expectations of the singular sums. He defines independent and dependent events and gives the multiplication rule for both. But whereas today the criterion for independence of two events is the validity of the multiplication rule in the [Doctrine], the multiplication rule follows from the independence of the events, which seems to be a self-evident concept for De Moivre ??With these tools ?those who are acquainted with Arithmetical Operations? (as De Moivre remarked in the preface) could tackle man-
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## Essay d’Analyse sur les Jeux de Hazard.

MONTMORT, Pierre Rémond de]. A fine copy, of the first separately published textbook of probability. "In 1708 [Montmort] published his work on Chances, where with the courage of Columbus he revealed a new world to mathematicians" (Todhunter, p. 78). "The Essay (1708) is the first published comprehensive text on probability theory, and it represents a considerable advance compared with the treatises of Huygens (1657) and Pascal (1665). Montmort continues in a masterly way the work of Pascal on combinatorics and its application to the solution of problems on games of chance. He also makes effective use of the methods of recursion and analysis to solve much more difficult problems than those discussed by Huygens. Finally, he uses the method of infinite series, as indicated by Bernoulli (1690)" (Hald, p. 290). "Montmorts book on probability, Essay danalyse sur les jeux de hazard, which came out in 1708, made his reputation among scientists" (DSB). Based on the problems set forth by Huygens in his De Ratiociniis in Ludo Aleae (1657) (published as an appendix to Frans van Schootens Exercitationum mathematicarum), the Essay spawned Abraham de Moivres two important works De Mensura Sortis (1711) and Doctrine of Chances (1718), as well as Jacob I Bernoullis celebrated Ars Conjectandi (1713). ABPC/RBH list just two copies of this first edition (Christies 1981 and Hartung 1987).The modern theory of probability is generally agreed to have begun with the correspondence between Pierre de Fermat and Blaise Pascal in 1654 on the solution of the Problem of points; this was published in Fermats Varia Opera (1679). Pascal included his solution as the third section of the second part of his 36-page Traité du triangle arithmétique (1665), which was essentially a treatise on pure mathematics. "Huygens heard about Pascals and Fermats ideas [on games of chance] but had to work out the details for himself. His treatise De ratiociniis in ludo aleae essentially followed Pascals method of expectation. At the end of his treatise, Huygens listed five problems about fair odds in games of chance, some of which had already been solved by Pascal and Fermat. These problems, together with similar questions inspired by other card and dice games popular at the time, set an agenda for research that continued for nearly a century. The most important landmarks of this work are Bernoullis Ars conjectandi (1713), Montmorts Essay d’analyse sur les jeux de hazard (editions in 1708 and 1711 [i.e., 1713]) and De Moivres Doctrine of Chances (editions in 1718, 1738, and 1756). These authors investigated many of the problems still studied under the heading of discrete probability, including gamblers ruin, duration of play, handicaps, coincidences and runs. In order to solve these problems, they improved Pascal and Fermats combinatorial reasoning, summed infinite series, developed the method of inclusion and exclusion, and developed methods for solving the linear difference equations that arise in using Pascals method of expectations." (Glenn Schafer in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (1994), Grattan-Guiness (ed.), p. 1296). "It is not clear why Montmort undertook a systematic exposition of the theory of games of chance. Gaming was a common pastime among the lesser nobility whom he frequented, but it had not been treated mathematically since Christiaan Huygens monograph of 1657. Although there had been isolated publications about individual games, and occasional attempts to come to grips with annuities, Jakob I Bernoullis major work on probability, the Ars conjectandi, had not yet been published. Bernoullis work was nearly complete at his death in 1705; two obituary notices give brief accounts of it. Montmort set out to follow what he took to be Bernoullis plan "[Montmort] continued along the lines laid down by Huygens and made analyses of fashionable games of chance in order to solve problems in combinations and the summation of series. For example, he drew upon the game that he calls "treize," in which the thirteen cards of one suit are shuffled and then drawn one after the other. The player who is drawing wins the round if and only if a card is drawn in its own place, that is, if the nth card to be drawn is itself the card n. In the generalized game, the pack consists of m cards marked in serial order "The greatest value of Montmorts book lay perhaps not in its solutions but in its systematic setting out of problems about games, which are shown to have important mathematical properties worthy of further work. The book aroused Nikolaus I Bernoullis interest in particular and the 1713 edition includes the mathematical correspondence of the two men. This correspondence in turn provided an incentive for Nikolaus to publish the Ars conjectandi of his uncle Jakob I Bernoulli "The work of De Moivre is, to say the least, a continuation of the inquiries of Montmort. Montmort put the case more stronglyhe accused De Moivre of stealing his ideas without acknowledgment. De Moivres De mensura sortis appeared in 1711 and Montmort attacked it scathingly in the 1713 edition of his own Essay. Montmorts friends tried to soothe him, and largely succeeded. He tried to correspond with De Moivre, but the latter seldom replied. In 1717 Montmort told Brook Taylor that two years earlier he had sent ten theorems to De Moivre; he implied that De Moivre could be expected to publish them" (DSB). "The value of Montmorts work resides partly in his scholarship. He was well-versed in the work of chance of his predecessors (Pascal, Fermat, Huygens), met Newton on one of a number of visits to England, corresponded with Leibniz, but remained on good terms with both sides during the strife between their followers. The summation of finite series is an element of Montmorts mathematical interests which enters into his probability work and distinguishes it from the earlier purely combinatorial problems arising out of enu-
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## De Sedibus, et Causis Morborum per anatomen Indagatis libri quinque. Dissectiones, et Animadversiones, nunc primum editas, complectuntur propemodum innumeras, medicis, chirurgis, anatomicis profuturas.

MORGAGNI, Giovanni Battista. First edition, first issue, an exceptional copy, completely untouched in the original printers interim-boards, of "one of the most important [works] in the history of medicine" (Garrison & Morton)."After Antonio Benivieni [1443-1502], Giovanni Battista Morgagni is considered the founder of pathological anatomy. His De sedibus [the offered work], regarded as one of the most important books in the history of medicine, established a new era in medical research." (Haskell F. Norman). "Morgagnis contribution to the understanding of disease may well rank with the contributions of Vesalius in anatomy and Harvey in physiology." (Heirs of Hippocrates). "On the basis of direct examination and records of some 700 post mortem dissections, he advanced the procedure of basing diagnosis, prognosis and treatment on a detailed and comprehensive knowledge of the anatomical conditions of common diseases. In the above volumes, some of the cases are given with a precision and details hardly surpassed in medical history. His proposal was a shift of emphasis from the traditional nature of a diseace to its anatomaical seat. It combined the approach of anatomist and pathologist, making their special knowledge available to the diagnostician." (Dibner). "Morgagnis most important work, , is his De sedibus et causis morborum per anatomen indagatis of 1761. This book grew out of a concept of Malpighi, which Morgagni then developed into a major work. The concept may be stated simply as the notion that the organism can be considered as a mechanical complex. Life therefore represents the sum of the harmonious operation of organic machines, of which many of the most delicate and minute are discernible, hidden within the recesses of the organs, only through microscopic examination."Like inorganic machines, organic machines are subject to deterioration and breakdowns that impair their operation. Such failures occur at the most minute levels, but, given the limits of technique and instrumentation, it is possible to investigate them only at the macroscopic level, by examining organic lesions on the dissecting table. These breakdowns give rise to functional impairments that produce disharmony in the economy of the organism; their clinical manifestations are proportional to their location and nature. "This thesis is implicit in the very title De sedibus et causis morborum per anatomen indagatis. In this book Morgagni reasons that a breakdown at some point of the mechanical complex of the organism must be both the seat and cause of a disease or, rather, of its clinical manifestations, which may be conceived of as functional impairments and investigated anatomically. Morgagnis conception of etiology also takes into account what he called external causes, including environmental and psychological factors, among them the occupational ones suggested to Morgagni by Ramazzini. "The parallels that exist between anatomical lesion and clinical symptom served Morgagni as the basis for his historiae anatomico-medicae, the case studies from which he constructed the De sedibus. There had, to be sure, been earlier collections of case histories, in particular Théophile Bonets Sepulchretum (1679), but Bonets work was, as René Laënnec wrote of it, an undigested and incoherent compilation, while the special merit of Morgagnis work lies in its synthesis of case materials with the insights provided by his own anatomical investigations. In his book Morgagni made careful evaluations of anatomic medical histories drawn exhaustively from the existing literature. In addition, he describes a great number of previously unpublished cases, including both those that he had himself observed in sixty years of anatomical investigation and those collected by his immediate predecessors, especially Valsalva, whose posthumous papers Morgagni meticulously edited and commented upon. The case histories collected in the De sedibus therefore represent the work of an entire school of anatomists, beginning with Malpighi, then extending through his pupils Valsalva and Albertini to Morgagni himself. "Morgagni may thus be considered to be the founder of pathological anatomy. This work was, in turn, developed by Baillie, who classified organic lesions as types (1793); Auenbrugger and Laënnec, who recognized organic lesions in the living subject (1761 and 1819, respectively); Bichat, who found the pathological site to be in the tissue, rather than the organ (1800); and Virchow, who traced the pathology from the tissue to the cell (1858)." (Luigi Belloni in DSB)."The first issue of De sedibus had the title page of Volume I printed in red and black. In a second issue, also of 1761, the title page was printed entirely in black, and there exists at least one copy of the first edition with imprint, Venice: Ex Typographia Remondiniana, 1762. The work was reprinted in Naples in 1762 and at Padua in 1765, as well as 1769, a German translation in 1771-76, a French translation in 1821-24, and an Italian version in 1823-29." (Haskell Norman in Grolier/Medicine). Evans 98; PMM 206; Dibner 125; Grolier/Medicine 46; Heirs of Hippocrates 792; Norman 1547; Garrison-Morton 2276; Waller 6672; Osler 1178; Lilly Library, Notable Medical Books, p. 125. Two volumes, large folio: 390 x 245 mm, both volumes entirely ntouched in their original state from the printer; uncut and un-pressed in carta rustica, internally fine and clean with only occasional light spotting, an exceptional set. Engraved frontispiece portrait by Jean Renard after Giovanni Volpato, pp. [i-viii] ix-xcvi, [1-2] 3-298 [2]; [1-2] 3-452. Highly rare in such fine condition.-
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## De Centro Gravitatis Solidorum Libri Tres. Rome: B[artolomeo] Bonfadino, 1604. [Bound with]: Quadratura Parabolae per simplex falsum. Et altera quàm secunda Archimedis expeditior ad Martium Columnam. Rome: L[epido] Facio, 1606.

VALERIO, Luca. First editions of these two rare and fundamental works on determining centres of gravity ? these are fascinating copies with numerous contemporary annotations in an untouched contemporary binding. In these books Valerio transformed the Archimedean ?method of exhaustion? into a series of important theorems ?so that for a whole class of convex curves and solids it was no longer necessary to establish results by special methods? (Baron, p. 101). Valerio was thus able to determine for the first time the centres of gravity of a number of ?Archimedean solids,? notably the hemispheroid and hyperboloid ?which is undoubtedly one of the most difficult results in classical-Renaissance mathematics? (Napolitani & Saito, p. 108). These works were also important for initiating the transition from classical Greek geometry to modern mathematics (see below) ? for example, according to Divizia, De CentroGravitatis anticipated the concept of limit and the integral calculus. Valerio strongly influenced Galileo, through his correspondence and these two books, and he was singled out for praise in the Discorsi, where he is described as ?our greatest geometer, the New Archimedes of our age? ?Valerio introduced important changes and novelties into the mathematics of his time ? especially at the methodological level. He opened up a new road which was to be followed by several others: notably, Cavalieri, with his theory of indivisibles, and, above all, Descartes with his geometry of curves. Mathematics, as a result, was totally transformed, and its language and methods have since undergone a revolution? (ibid., p. 74). ?Valerio was the first to break with the Greek model of the mathematical object: he introduces, and uses, classes of figures defined by one or more properties, to prove theorems that are more general and productive (in particular Theorem II-32); he invented [sic] the method of exhaustion, codifying in a series of general theorems, the classical technique to establish quantitative relationships between geometric figures ? The way to these methods had been opened by Valerio, and Cavalieri proceeded further ahead: but then little remained to be done. The introduction of ars analytica into geometry was to break this wall; it was not so long before Descartes opened a practically infinite new world for mathematicians, by identifying the curves with their equations. It is difficult, however, to imagine Cavalieri and Descartes without Valerio, to imagine their radical innovations without the methodological novelties of De centro gravitatis solidorum ? the impact of Valerio?s work provoked a crisis in the rigidity of the classical paradigm, and helped to create the conceptual context that was to lead to the birth of modern mathematics? (ibid., pp. 120-3). OCLC lists six copies of De Centro Gravitatis and four of Quadratura Parabolae in US.Provenance: Signature on title ?Ignatis Braccii? (?), possibly Ignatius Braccius, author of Phoenicis effigies in numismatis, et gemma, quae in Museo Gualdino asseruantur (1637). Full page of manuscript remarks in a contemporary hand on front free endpaper discussing the work of Archimedes, Galileo and Valerio on centres of gravity, and numerous marginal annotations in the same hand throughout.?The rediscovery of Greek mathematics, especially of Greek geometry, in the sixteenth century lies at the basis of many of the conceptual revolutions of the following century ? The period of rediscovery may be considered to have come to an end in 1575, the year in which both Maurolico and Commandino died. This date marks the beginning of the assimilation phase, an activity which soon became increasingly important. The assimilation of Archimedes involved the completion and the revision of his work; it implied a methodological reflection on the whole subject of geometry of measure and its relationship to mechanics ??Luca Valerio was one of the most important figures in this process of assimilation of ancient geometry. He was born in Naples in 1553, and entered the Collegio Romano of the Jesuits at the age of 17. There he attended the academy of mathematics directed by Christopher Clavius; when he left the Society in 1580, he was already an established mathematician ? However, it was only after several years that he succeeded in finding a position worthy of his talents: in 1600 he obtained the chair of mathematics at the University ?La Sapienza? in Rome, thanks to the protection of Pope Clement VIII Aldobrandini. And thus, at last, he was able to devote himself to research. His work, Three Books on the Center of Gravity of Solids (De Centro Gravitatis Solidorum libri tres) was published in Rome in 1604, and won him a lasting reputation, which continued long after his death in 1618. Valerio?s reflections on the classics regularly follow the same pattern: he takes a demonstrative technique already available (which can often easily be identified with a technique used by Archimedes) for a particular case, and then he transforms the specific properties of that particular case which makes the technique valid, into the definition of a particular class of figures (so that that same technique is applicable to all the figures of this newly defined class). This was a completely original approach, which led to new vistas of mathematical research ??The objects of classical Greek geometry were always particular objects, given by a more or less axiomatized construction procedure ? And the aim of geometry was to study these objects, and determine their properties ? An immediate consequence of this notion of mathematical objects is that for the Greeks, general objects could not exist. Nothing like our ?curves? existed in classical geometry. Various curved lines existed, of course ? the circle, the conic sections, the conchoid, the cissoid, the spiral, the quadratrix ? but no single conceptual operational category existed that included all of them ? If no general object existed, then general methods could-
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## Nebulae.

SLIPHER, Vesto Melvin. First edition, extremely rare separately-paginated offprint, and a remarkable association copy, signed by Edwin Hubble. In this profoundly significant paper, Slipher reports his finding that, of 25 spiral ?nebulae? examined, 21 are receding from us at high velocities, thus anticipating Hubble in his two most important discoveries: that the universe is expanding, and that the nebulae are separate galaxies outside our own, implying that the universe is vastly larger than our own Milky Way galaxy. Slipher?s ?systematic observations of the extraordinary radial velocities of spiral galaxies provided the first evidence supporting the expanding-universe theory? (Britannica). ?For the eminent cosmologist Fred Hoyle (1915-2001), Slipher?s pioneering observations of redshifts [i.e. velocities of recession] should have led to his being credited with the discovery of the expanding universe? (Kragh & Smith, p. 143). When in 1929 Hubble published the law now named after him stating that galaxies were receding from us at velocities proportional to their distance away, it was Slipher?s velocity measurements he was using, although he neglected to give Slipher any credit at the time. Slipher also notes in this paper that his measurements imply that we are not at rest with respect to the nebulae on average. He deduces a mean velocity of 700 km/s and makes a tremendous intellectual leap: ?For us to have such motion and the stars not show it means that our whole stellar system moves and carries us with it. It has for a long time been suggested that the spiral nebulae are stellar systems seen at great distances ? This theory, it seems to me, gains favor in the present observations? (p. 7). This was eight years before Hubble measured the distance of the Andromeda nebula and settled definitively the ?island universe? question of whether the nebulae were objects inside or outside the Milky Way. ?Vesto Melvin Slipher, a pioneer in the field of astronomical spectroscopy during his long career at Lowell Observatory at Flagstaff, Arizona, probably made more fundamental discoveries than any other observational astronomer of the twentieth century? (Hoyt, p. 411). OCLC lists Observatoire de Paris only; no copies in auction records.Provenance: Edwin Powell Hubble (1889-1953) (signature on last page of text).In 1901, Slipher (1875-1969) took a position working at the new Lowell Observatory with Percival Lowell (1855-1916). Gradually, Slipher became highly skilled with the observatory?s spectrograph, learning to take spectra of planetary atmospheres. In 1909 Lowell asked Slipher to turn his attention to the long-standing problem of the spiral nebulae, which Lowell believed were solar systems in the process of formation within the Milky Way. ?For three centuries astronomers had observed and speculated about these numerous, but faint, diffuse objects, yet almost nothing about them was then known. Some believed that they were vast aggregations of stars beyond the Milky Way, ?island universes? as suggested by philosopher Immanuel Kant in 1755. Others felt they might be embryonic planetary systems in early stages of evolution and thus analogs of the primordial solar system ? [Lowell] thought that if such objects were indeed incipient solar systems, they might show spectrographic similarities to the solar system itself. Early in 1909 he asked Slipher to observe what were then only classified as ?green? and ?white? nebulae, the latter group containing the enigmatic spirals, and to compare the spectra of their ?outer parts? with his spectra of the giant outer planets? (Hoyt, p. 421). It took Slipher four years of difficult and demanding work before he obtained significant results: he detected a blue-shift in the wavelength of light from the Andromeda nebula due to the Doppler effect, indicating that the nebula was moving at high speed toward us. ?On February 3, 1913 he wrote Lowell that the Great Nebula in Andromeda was approaching the earth at the then unheard-of speed of 300 km/sec, the value, incidentally, that is accepted today. ?It looks as if you had made a great discovery,? Lowell replied. ?Try some other spiral nebulae for confirmation.??Slipher now turned his attention to a spindle-shaped, edge-on spiral in Virgo, designated NGC4594, and by April his spectrograms showed that its spectral lines were shifted far toward the red, indicating that it was receding from the earth at about 1000 km/sec, an astounding velocity at that time. ?This nebula is leaving the solar system [sic],? he pointed out to Lowell, ?hence it seems safe to conclude that motion in the line of sight is the real cause of these great displacements in their nebular spectra? ? Slipher continued these observations through the next year. In August 1914, at the American Astronomical Society?s seventeenth meeting at Evanston, Illinois, he could announce radial velocities for fifteen spirals. ?In the great majority of cases,? he reported, ?the nebula is receding; the largest velocities are all positive ? The striking preponderance of the positive sign indicates a general fleeing from us or the Milky Way?? (Hoyt, pp. 424-5).Always very cautious about the interpretation of his observations, Slipher was reluctant to challenge in print Lowell?s view of the nature of the spiral nebulae, and did not do so until 1917 [in the present paper], by which time he had measured the velocities of 10 more spirals. Other astronomers immediately realised the significance of Slipher?s data. ?The brilliant Danish astronomer Ejnar Hertzsprung (1873-1967) sent to Slipher: ?harty [sic] congratulations on your beautiful discovery of the great radial velocity of some spiral nebulae. It seems to me, that with this discovery the great question, if the spirals belong to the system of the Milky Way or not, is answered with great certainty to the end, that they do not.? To Hertzsprung and others, the speeds of the spirals seemed altogether too great for them to be gravitationally bound to th-
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## Memoir on a Trans-Neptunian Planet.

LOWELL, Percival. LOWELL, Percival. ?Memoir on a trans-Neptunian planet.? Offprint from Memoirs of the Lowell Observatory, Vol. 1, No. 1. Lynn, Mass.: T. P. Nichols & Son, 1915.First edition, the most desirable possible presentation/association copy, of Lowell?s prediction of the existence of a planet beyond the orbit of Neptune, which initiated the search that culminated in 1930 with the discovery of Pluto at Lowell Observatory by Clyde W. Tombaugh (1906-97): this copy is signed both by Lowell and Tombaugh. ?Clyde Tombaugh?s February 18, 1930, discovery of Pluto concluded a three-stage search at Lowell Observatory in Flagstaff, Arizona, spanning 25 years. What started with Percival Lowell?s musings about a theoretical ninth planet led to mathematical and photographic studies that survived Lowell?s death and set the stage for Tombaugh?s eureka moment ? The first phase of Percival Lowell?s hunt for Planet X ran from 1905 to 1909 ? Lowell and his small team of ?computers? ? led by head computer Elizabeth Williams ? carried out a series of calculations based on the observed perturbations of Uranus, pinpointing likely locations for the new planet ? [In 1910] Lowell redoubled his mathematical efforts by incorporating the latest technology ?With new calculations and improved equipment, Lowell estimated locations of Planet X and published his findings in the 1915 Lowell Observatory publication, Memoir on a Trans-Neptunian Planet. Unfortunately, Lowell died the following year, before he had a chance to complete a photographic search in the targeted area of the sky. Eleven years after Lowell?s death, the final phase of the Observatory?s search for Planet X commenced ? Abbott Lawrence Lowell, Percival’s younger brother and president of Harvard University, gave $10,000 for the construction of a new telescope, a 13-inch photographic instrument usually referred to as an astrograph. To operate the telescope, director V. M. Slipher hired a young man from Kansas of ?the self-made type,? Clyde Tombaugh. Tombaugh arrived in Flagstaff in 1929 and soon took over the systematic search for Planet X, examining the area of sky Lowell had indicated in Memoir on a Trans-Neptunian Planet. The 13-inch telescope was ideal for the search, and Tombaugh had the patience and attention to detail necessary for the work. On February 18, 1930, he discovered what would soon be named Pluto, completing the search begun by Percival Lowell 25 years earlier? (Schindler). ?More than any other astronomer of his generation, Percival Lowell (1855-1916) had a profound influence on the general public. His thesis that the planet Mars was the abode of intelligent life continued to excite the public mind decades after his death ? Lowell?s name is also forever linked with Pluto; although he did not live to see that distant planet, there is no doubt that his inspiration advanced the date of its detection by many years? (DSB).Provenance: Percival Lowell (signed book label on half-title); Clyde W. Tombaugh (signature on half-title).A member of the distinguished Lowell family of Massachusetts, Percival Lowell devoted his early years to literature and travel. Beginning in the winter of 1893?94, inspired by Giovanni Schiaparelli?s discovery of ?canals? on Mars, Lowell decided to dedicate himself to the study of astronomy, founding the observatory at Flagstaff, AZ, which bears his name. In 1904, Lowell received the Prix Jules Janssen, the highest award of the Société Astronomique de France, the French astronomical society. For the last 23 years of his life astronomy, Lowell Observatory, and his and others? work at his observatory, were the focal points of his life.?As early as 1902, Lowell was lecturing and writing about his conviction that a planet existed beyond the orbit of Neptune. He had studied the orbits of a number of comets and had noticed gaps and groupings which he felt were significant. In 1905, the first of his two extended searches for ?Planet X? began. This phase lasted until 1909, and was essentially carried out by Lowell, a trio of graduate assistants and William T. Carrigan, a ?computer? from the US Naval Observatory?s Nautical Almanac Office in Washington DC. Lowell employed Carrigan to perform a number of calculations on the orbits of Uranus and Neptune. Lowell was interested in ?residuals,? differences between the calculated positions of the planets and the positions which were actually observed.?Lowell?s second search spanned the years 1910-1915. This search was based upon calculations using the formulae of celestial mechanics. In essence, it was similar to the predictions of Adams and Le Verrier [for the orbit of Neptune], but using residuals which were much smaller. The calculations, thus, were much more difficult and involved. This search culminated in the publication entitled ?Memoir on a Trans-Neptunian Planet,? in Memoirs of the Lowell Observatory, vol. 1, 1915? (Bakich, p. 299).?Lowell himself recognized the difficulty of the problem he was trying to solve. In his 1915 paper ?Memoir on a Trans-Neptunian Planet,? he pointed out: ?We cannot use Neptune as a finger-post to a trans-Neptunian as Uranus was used for Neptune because we do not possess observations of Neptune far enough back.? (In fact, Galileo had unknowingly observed Neptune, on at least two and possibly as many as four occasions, from late December 1612 into early 1613. These observations might have helped Lowell enormously; however, Galileo?s observations of Neptune were not recognized until 1979, in work by Stillman Drake and Charles Kowal.) Lowell further noted that in 1845, when Adams and Le Verrier used the residuals in the orbit of Uranus to predict the existence of Neptune, the difference between the predicted and observed positions of Uranus sometimes amounted to 133 seconds of arc. In contrast, after accounting for the gravitational influence of Neptune on Uranus, the remaining residuals in the orbit of Uranus never exceeded 4.5 seconds of arc.?Lowell used the tried-
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## Co-relations and their measurement, chiefly from anthropometric data.

GALTON, Francis. First edition, journal issue in original printed wrappers, of Galton?s invention of the statistical concept of correlation, one of the most ?fundamental and ubiquitous? ideas in statistics (Stigler, p. 73). ?Like all major scientific discoveries, correlation did not appear in a vacuum. It was a concluding step in a 20-year research project? (ibid.). The ?correlation coefficient? measures the strength of the linear relationship between two observed phenomena. It ranges in value from ?1 to +1; the closer the correlation coefficient is to 1, the stronger the relationship. It is positive if the increase in the value of one variable may be followed by an increase in the value of the other; if it is negative the increase in the value of one variable may be followed by the decrease in the value of the other. ?If one individual can be credited as the founder of the field of behavioural and educational statistics, that individual is Francis Galton . He is responsible for the terms correlation (from co-relation), he discovered the phenomenon of regression to the mean, and he is responsible for the choice of r (for reversion or regression) to represent the correlation coefficient? (Clauser, p. 440). ?The major components of what we take to be correlation were in place by 1886 ? [notably] a rather full development of the ideas of regression. Galton summarized all this work in his book Natural Inheritance, published in 1889 ? But if correlation was not far away, it was still not there, and the word does not appear in Natural Inheritance? (Stigler, p. 75).Galton wrote an account of his discovery of correlation in a paper published in 1890 (?Kinship and correlation,? North American Review, vol. 150, pp. 419-431). ?The story is told in his 1890 article of how, late in 1888, after Galton had parted with the final revision of the page proofs of Natural Inheritance, he was simultaneously pursuing two superficially unrelated investigations. One was a question in anthropology: If a single thigh bone is recovered from an ancient grave, what does its length tell the anthropologist about the total height or stature of the individual to whom it had belonged? The other was a question in forensic science: What, for the purposes of criminal identification, could be said about the relationship between measurements taken of different parts of the same person (the lengths of different limbs surely did not constitute independent bits of data for purposes of identification)? Galton recognized these problems were identical, and he set to work on them with a data set he had on measures made on 348 adult males ? In his 1890 article Galton described how, while plotting these data, it suddenly came to him that the problem was the same as that he had considered in studying heredity, ?that not only were the two new problems identical in principle with the old one of kinship which I had already solved, but that all three of them were no more than special cases of a much more general problem ? namely that of Correlation ??There is a breathless quality to part of this narrative: ?Fearing that this idea, which had become so evident to myself, would strike many others as soon as Natural Inheritance was published, and that I should be justly reproached for having overlooked it, I made all haste to prepare a paper for the Royal Society with the title of Correlation. It was read some time before that the book was published, and it even made its appearance in print a few days the earlier of the two.? Actually the title of the article was ?Co-relations and their measurements, chiefly from anthropometric data.? The spelling ?correlation? was common at the time (and used by Galton in subsequent writings) ??To Galton, correlation meant what we might today call intraclass correlation ? two variables are correlated because they share a common set of influences. He described the effect of correlation on the dispersion of differences (the difference in heights of two random Englishmen is said to have median 2.4 inches, the difference in heights of two brothers had median 1.4 inches). Galton seems to have only conceived of correlation as a positive relationship; negative correlations play no role in his discussion.?Galton gave three examples to illustrate the concept of correlation, examples where he could make concrete the common factors behind the relationship. The first of these, on kinship, seems hazy and unsatisfactory to modern eyes, but perhaps that is because we view the problem through a clarifying lens, Mendelian genetics, that was not available to Galton. The other two examples are superb ? the trip time for two clerks travelling home taking the same bus over part of the journey, and the stock portfolios of two investors who hold some shares in the same commercial ventures.?Galton was able to use his examples to underscore the fact that correlation did not in any way depend upon the choice of origin. At first glance he might seem to have faltered on the question of dependence on the scaling of measurements; because of the difference of scales, he tells us, ?There is relation between stature and length of finger, but no real correlation.? But he quickly recovers and explains that a simple multiplication (to measure the quantities in units of ?probable error,? where this is a term that denotes a median deviation for a symmetric distribution) will turn the relationship into correlation, and that he will henceforth tacitly assume that has been done.?He also tells us that the concepts only apply to variables that have at least a ?quasi-normal? (approximately normal) distribution. Here, as elsewhere in his writings, he is enchanted by this ?singularly beautiful law,? and we might even accuse him of over- enthusiasm. ?Now, when a series of measures are submitted to a competent statistician, it is a very simple matter for him to discover whether they vary normally or not.? But in the end he is cautious, and his insistence upon a check of dis-
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## Saggi di natvrali esperienze fatte nell’Accademia del Cimento sotto la protezione del Serenissimo principe Leopoldo di Toscana, e descritte dal segretario di essa Accademia.

MAGALOTTI, Lorenzo]. First edition, first issue, a magnificent copy bound in contemporary Italian red morocco gilt, probably for presentation, of the only publication of the Accademia del Cimento, the first organisation founded for the sole purpose of making scientific experiments (the more common second issue has the title page reset and dated 1667). The Accademia sought to extend the work of Galileo by performing experiments which would demonstrate the folly of continuing opposition to the new science. Its patrons were Prince Leopoldo of Tuscany and his elder brother, Ferdinando II, Grand Duke of Tuscany, both amateur scientists. Leopoldo, who ?set the experimental agenda? (Biagioli, Galileo Courtier (1993), p. 358) provided the group with the first well-equipped physical sciences laboratory in Europe, established in the Palazzo Pitti in Florence, and devised some of the methods and instruments used in it. Although active for only a decade, the Accademia exerted a lasting influence on the development of experimental scientific methods and devices during the following century. Led by the polymath Giovanni Alfonso Borelli (1608-1679) and Vincenzio Viviani (1622-1703), pupil of Galileo and Torricelli, its members included the naturalist Francesco Redi, the anatomist and geologist Niels Stensen, the astronomer Gian Domenico Cassini, and the linguist and writer Lorenzo Magalotti, who became the secretary and who edited the Saggi. The collection was published anonymously, an expression of the Accademia?s (or Leopoldo?s) policy of submerging the members? individual identities and presenting itself as a group. The essays focus strictly on the identification and description of physical phenomena, and avoid stating any potentially controversial theories or conclusions. Although the Saggi describe only a portion of the experiments carried out by the Academy, omitting some interesting investigations, including observations of comets, they include several scientific novelties: ?[The Saggi] contains the description of the first true thermometer (as opposed to Galileo’s thermo-baroscope), which had its tube for the first time sealed and a graduated scale attached to it . It describes the first true hygrometer, and an improved barometer, giving also classical experiments on air pressure, experiments on the velocity of sound, radiant heat, phosphorescence, and the first experiments showing the expansion of water in freezing. Of great importance were the experiments on the compressibility of water, discovering at the same time the porosity of silver, and the pendulum experiments, when the deviation of the pendulum from its true plane of oscillation was first discovered, which was afterwards used by Foucault to prove the earth’s rotation? (Zeitlinger, Bibliotheca chemico-mathematica (1921), 22505). The 800 copies printed (including both issues) were not sold by booksellers: they were all distributed as gifts to Leopoldo?s friends and correspondents. Only two other copies bound in contemporary red morocco are recorded on ABPC/RBH, both in bindings nearly identical to that of the present copy: the Norman copy (second issue) and a copy of the first issue sold at Sotheby?s, London, on 27 September 1988 (lot 152). The Accademia del Cimento was founded in 1657 by the Grand Duke of Tuscany Ferdinando II de? Medici (1610-70) and by his brother Prince Leopoldo (1617-75). Founded to follow Galileo?s agenda, the Accademia?s activity gradually diverged from his mathematical approach to nature, making way for an experimental view of science. Linked to this intense experimental programme was the emergence of the crucial role of instruments and laboratories, two aspects of scientific research which had not previously been given the same significance. Viviani, who had been Galileo?s secretary during the last years of his life, and Borelli were the true leaders of the experimental activities, though the role of Ferdinando and Leopoldo in the promotion and coordination of the experiments to be performed should not be underestimated.?The choice of members for the Accademia, which was decided by the Medici, reflected the scientific priorities the academy intended to pursue. With the presence of Francesco Redi, medicine and natural history flanked physics, astronomy and mathematics. Though not officially members of the Accademia, a considerable number of naturalists, both Italian and foreign, participated in the academy?s activities in various roles. Among them were the doctor and geologist Niels Stensen, summoned to take part in several activities, and the future president of The Royal Society, Robert Southwell, who had been a protégé of Viviani, from whom he learned the methods and organization which governed the academy. Other scientists, such as Huygens and Thévenot, had established intense epistolary exchanges with the Florentine scientists and sovereigns, the contents of which are directly traceable to the scientific research performed in the rooms of the Palazzo Pitti.?Whatever the object of investigation, the distinctive element of the sessions was public experimentation. Naturally, for this kind of approach to be successful it was necessary to dispose of a rich laboratory ? During the 10-year run of the academy?s activity, thousands of astronomical, physical, chemical and meteorological instruments were either commissioned or constructed in the bosom of the Cimento. If we consider that Prince Leopoldo?s collection alone contained 1282 glass instruments, ? we can only imagine the enormous wealth of the laboratory in the Palazzo Pitti ? the production and construction of scientific instruments and the upkeep of the laboratory required a separate budget which expended very considerable resources. The costs of acquiring such instruments ? had necessarily led science to become a collegial undertaking ? supported by a solid and prestigious financial institution which would guarantee its function and development. In the case of the Accademia d-
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## In duos Archimedis Aequeponderantium libros paraphrasis scholijs illustrata. Pesaro: apud Hieronymum Concordiam, 1588 [colophon, 1587]. [Bound with:] SCALETTI, Carlo Cesare. Scuola mecanico-speculativo-pratica in cui si esamina la proporzione, che hà la potenza alla resistenza del corpo grave, e la causa per la quale la suddetta potenza si estenda a maggiorattività mediante la machina; opera utile alluso civile, e militare necessaria ad ogni matematico, ingegniero, architetto, machinista, e bombardiere. Bologna: Costantino Pisarri, 1711.

MONTE, Guidobaldo, Marchese del. First edition, the beautiful Macclesfield copy, of Guidobaldo?s Paraphrasis, the important companion to his Mechanicorum liber (1577), regarded as the greatest work on statics since the Greeks, which employed the mathematically rigorous proofs of Archimedes to the investigation of problems in mechanics. ?This work complements the Mechanicorum liber, and together they represent the greatest opus of 16th century mechanics? (Roberts & Trent, p. 13). They had a profound and lasting impact on the methodology adopted by contemporary century Italian scientists, most notably Galileo: ?Guidobaldo was Galileo?s patron and friend and was possibly the greatest single influence on the mechanics of Galileo? (DSB). The Mechanicorum liber had reduced the study of simple machines to the law of the lever, according to which bodies balance on the ends of a lever when their distances from the fulcrum are inversely proportional to their weights. Guidobaldo?s 1588 work is a paraphrase of Archimedes? ?On the equilibrium of plane figures,? which included a geometrical proof of the law of the lever. This is Archimedes? most important surviving work in mechanics, but much of it ?is undoubtedly not authentic, consisting as it does of inept later additions or reworkings? (DSB). Guidobaldo?s purpose in the Paraphrasis is to explicate and correct the Archimedean text, thereby providing a secure geometrical foundation for the law of the lever, and hence for the whole of the science of statics. ?The Law of the Lever was among the first laws of nature to be formulated in quantitative terms. It dates back at least to Archimedes? On the Equilibrium of Planes and possibly even to Aristotle?s Mechanics. Shortly after its first formulation, scholars like Archimedes and Euclid were already seeking to prove it by means of deduction from general axioms and postulates. The Law of the Lever thus comprises the very core of rational mechanics? (Schlaudt, p. 93). Like Archimedes, Guidobaldo also considers the application of the law of the lever to the determination of centres of gravity: of plane figures bounded by straight lines in the first book, and in the second book segments of conic sections (treated by approximating these curved figures by inscribed polygons). Monte?s work is here bound with the rare first edition of Scaletti?s treatise on civil engineering, with chapters on statics, mechanics, on the practical construction of fountains, buildings, bridges, etc. (BL only on COPAC; only three other copies located in auction records). ?Guido Ubaldo, Marquis del Monte, was born at Pesaro on 11 January 1545. He entered the University of Padua in 1564, where one of his companions in study was the poet Torquato Tasso. On his return from the university, he continued his studies in mathematics under Federico Commandino at Urbino? (Drake, p. 44). ?Writing to a friend in Paris in 1633, Galileo declared that ?at the age of twenty-one, after studying geometry for two years he worked out a number of propositions about the center of gravity of solids.? Galileo had become acquainted with Commandino?s Liber de centro gravitatis solidorum that had been published in 1565 and had opened, or rather reopened, a new field of research but suffered from what Galileo called ?some imperfections.? These he sought to set right by following the example of ?that very great mathematician,? Guidobaldo del Monte, to whom he sent his demonstrations? (Shea, pp. 97-98). ?In 1588 [Guido Ubaldo] received from Galileo some theorems on centers of gravity with a request for his opinion. In this way a correspondence was opened which continued until his death in 1607. Guido Ubaldo was favourably impressed with Galileo?s talents, and sent to him a copy of his second important contribution to mechanics, a paraphrase of and commentary on the work of Archimedes on plane equilibrium [the offered work] ? In appraising probable influences on Galileo, one should remember that, before Galileo wrote anything on motion, he had received this book from his most valued patron, [and] that it was a book on Archimedes (whom Galileo admired above all other writers)? (Drake, pp. 45-46).?Guido Ubaldo?s two chief works on theoretical mechanics make clear his devotion to the idea of mathematical rigor of treatment and his repugnance for medieval writings on the science of weights and for Tartaglia?s adherence to that tradition. The Mechanics contains numerous criticisms of these writers, and in the Paraphrasis of 1588 Guido Ubaldo wrote: ?And however much Jordanus Nemorarius (whose followers include Niccolo Tartaglia and others) struggled in his book De ponderibus to prove this same proportion of the general lever by many means, yet not any of the proofs were worthy to be called demonstrations, and were scarcely to be credited. For he put things together which in no way command conviction and perhaps do not even persuade anyone by probability, when in mathematical demonstrations the most precise reasons are required. And on that account it never seemed to me that this Jordanus should even be reckoned among writers on mechanics?? (ibid.).?With the Paraphrasis, Guidobaldo attended to restore the integrity of the Equilibrium of Planes, Archimedes?s principal work of mechanics. The corrupted text presented Guidobaldo with problems of essentially three kinds: minor technical problems, like missing argumentative steps in the demonstrations; completely inconclusive demonstrations that requested a massive intervention in the text with lemmata or auxiliary propositions; and, most seriously, obscurities regarding the key notions of Archimedean mechanics. Approaching this challenge, Guidobaldo adopted a quite ?philological? modus operandi: firstly, to establish a correct text, he had recourse to the Greek version of the editio princeps (1544), which appeared to him less corrupt than the existing Latin translations. In the course of the Paraphrasis, he indicated Greek passages which did not seem to ma-
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## Rerum arithmeticarum libri sex, quoru[m] primi duo adiecta habent scholia, maximi (ut conjectura est) Planudis. Item liber de numeris polygonis seu multiangulis. Opus incomparabile, veræ arithmeticæ logisticæ perfectionem continens, paucis adhuc visum. A Guil[lelmo] Xylandro Augustano incredibili labore Latinè redditum, & commentariis explanatum, inq[ue] lucem editum. Ad illustriss. principe[m] Ludovicum Vuirtembergensem.

DIOPHANTUS of Alexandria. First edition, very rare, of the first systematic treatise on algebra (Smith, Rara Arithmetica, p. 348), which ?inspired the rebirth of number theory? (Britannica), translated from the Greek by Wilhelm Holtzmann [Xylander] ? the original Greek text was not published until 1621. ?The appearance of [this] translation had an immediate and enormous influence on the development and shaping of Algebra?? (Heath, Diophantus of Alexandria (1910), p. 26). ?The work marks the high point of Alexandrian Greek algebra: Diophantus introduced symbolism into algebra, dealt with powers as high as six (in contrast to classical Greek mathematicians, who did not consider powers higher than three), and delved extensively into the solution of indeterminate equations, founding the branch of algebra now known as Diophantine analysis? (Norman). ?Xylander was an enthusiast for Diophantus, and his preface and notes are often delightful reading. Unfortunately the book is now very rare? (Heath, History of Greek Mathematics (1921), Vol. 2, p. 454). ?The Arithmetica is essentially a logistical work, but with the difference that Diophantus? problems are purely numerical with the single exception of problem V, 30. In his solutions Diophantus showed himself a master in the field of indeterminate analysis, and apart from Pappus he was the only great mathematician during the decline of Hellenism? (DSB). ?The Arithmetica is a collection of problems and even though the solutions presented by Diophantus are always quite specific, his solutions do tend to suggest general methods. As a result, Diophantus has often been called the father of algebra, in part because of these methods, but also because of the systematic use of notation and terminology that he introduced in this work. For example, even though he did not have the notation we now use for exponents, he nonetheless had his own effective symbolic way of representing polynomials. But the spirit of the Arithmetica has far more in common with modern number theory than with today?s practice of algebra? (Watkins, Number Theory: A Historical Approach (2013), pp. 91-2). Xylander?s translation also includes a fragment of the only other work of Diophantus that has come down to us, his treatise on ?polygonal numbers? (or ?figurate numbers?) ? the numbers of dots that can be arranged in the shape of a regular polygon. It was famously in his copy of the 1621 reprint of Xylander?s translation that Pierre de Fermat made his marginal annotations, including his statement of ?Fermat?s last theorem.? ABPC/RBH list just two other copies since Honeyman, both in modern bindings. OCLC lists 8 copies in North America.?The Arithmetica begins with an introduction addressed to Dionysius?arguably St. Dionysius of Alexandria. After some generalities about numbers, Diophantus explains his symbolism?he uses symbols for the unknown (corresponding to our x) and its powers, positive or negative, as well as for some arithmetic operations?most of these symbols are clearly scribal abbreviations. This is the first and only occurrence of algebraic symbolism before the 15th century. After teaching multiplication of the powers of the unknown, Diophantus explains the multiplication of positive and negative terms and then how to reduce an equation to one with only positive terms (the standard form preferred in antiquity). With these preliminaries out of the way, Diophantus proceeds to the problems. Indeed, the Arithmetica is essentially a collection of problems with solutions, about 260 in the part still extant. The introduction also states that the work is divided into 13 books. Six of these books were known in Europe in the late 15th century, transmitted in Greek by Byzantine scholars and numbered from I to VI; four other books were discovered in 1968 in a 9th-century Arabic translation by translation by Qusta ibn Luqa. However, the Arabic text lacks mathematical symbolism, and it appears to be based on a later Greek commentary?perhaps that of Hypatia (c. 370?415)?that diluted Diophantus?s exposition. We now know that the numbering of the Greek books must be modified: Arithmetica thus consists of Books I to III in Greek, Books IV to VII in Arabic, and, presumably, Books VIII to X in Greek (the former Greek Books IV to VI). Further renumbering is unlikely; it is fairly certain that the Byzantines only knew the six books they transmitted and the Arabs no more than Books I to VII in the commented version.?The problems of Book I are not characteristic, being mostly simple problems used to illustrate algebraic reckoning. The distinctive features of Diophantus?s problems appear in the later books: they are indeterminate (having more than one solution), are of the second degree or are reducible to the second degree (the highest power on variable terms is 2, i.e., x2), and end with the determination of a positive rational value for the unknown that will make a given algebraic expression a numerical square or sometimes a cube. (Throughout his book Diophantus uses ?number? to refer to what are now called positive, rational numbers; thus, a square number is the square of some positive, rational number.) Books II and III also teach general methods. In three problems of Book II it is explained how to represent: (1) any given square number as a sum of the squares of two rational numbers; (2) any given non-square number, which is the sum of two known squares, as a sum of two other squares; and (3) any given rational number as the difference of two squares. While the first and third problems are stated generally, the assumed knowledge of one solution in the second problem suggests that not every rational number is the sum of two squares. Diophantus later gives the condition for an integer: the given number must not contain any prime factor of the form 4n + 3 raised to an odd power, where n is a non-negative integer. Such examples motivated the rebirth of number theory. Although Diophantus is typically satisfied to obtain one s-
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## Tabulae Rudolphinae, quibus astronomicae scientiae, temporum longinquitate collapsae Resauratio continetur a Phnice illo Astronomorum Tychone, ex illustri & generosa Braheorum in regno Daniae familia oriundo equite, primum animo concepta et destinata a anno Christi MDLXIV: exinde observationibus siderum accuratissimis, post annum praecipue MDLXXII, quo sidus in Cassiopeiae constellatione novum effulsit, serio affectata; variisque operibus, cum mechanicis, tum librariis, impenso patrimonio amplissimo, accedentibus etiam subsidiis Friderici II. Daniae Regis, regali magnificentia dignis, tracta per annos XXV, potissimum in insula freti Sundici Huenna, & arce Uraniburgo, in hos usus a fundamentis extructa: tandem traducta in Germaniam, in que aulam et nomen Rudolphi Imp. anno M D IIC. Tabulas ipsas, jam et nuncupatas, et affectas, sed morte authoris sui anno MDCI desertas, jussu et stipendiis fretus trium Imppp. Rudolphi, Matthiæ, Ferdinandi, annitentibus haeredibus Braheanis; ex fundamen

KEPLER, Johannes. First edition of this great scientific classic, ?the chief vehicle for the recognition of his astronomical accomplishments? (DSB), with the magnificent and very rare world map (see below). ?The tables are extraordinarily important, for they document in a unique way Kepler?s great contributions to astronomy? (Gingerich). This was Kepler?s last lifetime publication and his crowning achievement, ?the foundation of all planetary calculations for over a century? (Sparrow). Kepler himself called the tables ?my chief astronomical work? (Gingerich). ?In 1601 Kepler was charged by the dying Tycho Brahe to complete his proposed Rudolphine tables of planetary motion, to be based upon Tycho?s great storehouse of observations. When the tables finally appeared twenty-six years later, Kepler excused the long delay in his preface, in which he cited not only salary and wartime difficulties, but also ?the novelty of my discoveries and the unexpected transfer of the whole of astronomy from fictitious circles to natural causes, which were most profound to investigate, difficult to explain, and difficult to calculate, since mine was the first attempt? (Gesammelte Werke 10, pp. 42-43; quoted in DSB). Kepler?s work was shaped not only by his Copernican bias and his discovery of the laws of planetary motion, but by the ?happy calamity?, in 1618, of his initiation into logarithms, which enabled him to make the complex calculations necessary for determining planetary orbits. Kepler was thus able to take into account the relative heliocentric positions of the earth and planets, calculating these positions separately and combining them to produce the geocentric position; this yielded far more accurate positions than in previous tables, which had erred by as much as five degrees. This improvement constituted a strong endorsement of the Copernican system, and insured the tables? dominance in the field of astronomy throughout the seventeenth century? (Norman). ?The tables? accuracy was strikingly demonstrated four years later. On 7 November 1631, when Kepler himself had been dead a year, the French astronomer Pierre Gassendi became the first observer in history to see Mercury crossing the face of the Sun, in fulfilment of a prediction by Kepler. Kepler?s tables were in error by only one-third of the solar diameter, whereas even the Copernican tables they had replaced were in error by thirty times that amount? (Hoskins, The Cambridge Concise History of Astronomy, pp. 110-111). ABPC/RBH record seven other copies with the map in the last half-century: Christie?s, 25 June 2009, lot 62, ?103,000 = $133,076; Christie?s, 17 June 2008, lot 209, $134,500 (Richard Green copy); Sotheby?s, 15 November 2007, lot 105, £38,400 = $78,978; Christie?s, 16 April 2007, lot 297, $120,000 (Honeyman-Streeter copy); Christie?s, 25 June 2004, lot 102, ?100,000 = $112,963; Christie?s, 2 July 1994, lot 67, £22,000 = $34,019; Sotheby?s, 30 June 1983, lot 418, £4180 = $6476; Sotheby?s, 12 May 1980, lot 1802, £4800 = $11,059 (Honeyman copy). Provenance: contemporary astronomical annotations to text in several leaves, including some emendations and additions to the text; small ex libris of Swedish historian Brje Israelson (18811931) on inner margin of blank verso of last leaf.Johannes Kepler (1571-1630) came from a very modest family in the small German town of Weil der Stadt and was one of the beneficiaries of the ducal scholarship; it made possible his attendance at the Lutheran Stift, or seminary, at the University of Tübingen where he began his studies in 1589. At Tübingen, the professor of mathematics was Michael Maestlin (1550-1631), one of the most talented astronomers in Germany, and a Copernican (though a cautious one). Maestlin lent Kepler his own heavily annotated copy of De revolutionibus, and so while still a student, Kepler made it his mission to demonstrate rigorously Copernicus? theory.In 1594 Kepler moved to Graz in Austria to take up a position as provincial mathematician and as a teacher at the Lutheran school there. Just over a year after arriving in Graz, Kepler discovered what he thought was the key to the universe: ?The earth?s orbit is the measure of all things; circumscribe around it a dodecahedron, and the circle containing this will be Mars; circumscribe around Mars a tetrahedron, and the circle containing this will be Jupiter; circumscribe around Jupiter a cube, and the circle containing this will be Saturn. Now inscribe within the earth an icosahedron, and the circle contained in it will be Venus; inscribe within Venus an octahedron, and the circle contained in it will be Mercury. You now have the reason for the number of planets.? This remarkable idea was published in Mysterium cosmographicum (1596), ?the first unabashedly Copernican treatise since De revolutionibus? (DSB).?Kepler sent copies of his remarkable book to various scholars, including the most famous astronomer of the day, Tycho Brahe (1546-1601). Although unwilling to accept all these strange arguments, the Danish astronomer immediately recognized the author?s genius, and invited Kepler to visit him. However, the long journey was out of the question for the impecunious young man. Thus, wrote Kepler, ?I ascribe it to Divine Providence that Tycho came to Bohemia.? Tycho, fearing the loss of royal support by the King of Denmark, had in the meantime resolved to join the court of Rudolph II in Prague. Emperor Rudolph was a moody, eccentric man whose twin loves were the occult and his collection of curiosities. He was more than willing to support a distinguished astronomer whose accurate planetary positions could make horoscopes more accurate. Tycho arrived at Prague in 1599, and Kepler, forced out of Graz by religious controversy, joined him there the following January. To Kepler was assigned the analysis of the observations of Mars. The encounter between the young German theoretician and the famous Danish observer turned the course of astronomy ? within two y-
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## Méchanique analitique. [Bound with:] Théorie des fonctions analytiques.

LAGRANGE, Joseph Louis de. An exceptional volume, in a fine contemporary binding, containing the first edition of Lagrange?s masterpiece, the Méchanique, ?one of the outstanding landmarks in the history of both mathematics and mechanics? (Sarton, p. 470) and ?perhaps the most beautiful mathematical treatise in existence, together with the corrected second printing of the Théorie, containing Lagrange?s formulation of calculus in terms of infinite series, which provided the basis for Augustin-Louis Cauchy?s development of complex function theory in the first decades of the next century. The Méchanique contains the discovery of the general equations of motion, the first epochal contribution to theoretical dynamics after Newton?s Principia? (Evans). ?Lagrange?s masterpiece, the Méchanique Analitique (Paris, 1788), laid the foundations of modern mechanics, and occupies a place in the history of the subject second only to that of Newton?s Principia? (Wolf). ?The year 1797 ? saw the appearance of the famous work of Lagrange, Théorie des fonctions analytiques ? This book developed with care and completeness the characteristic definition and method in terms of ?fonctions dérivées,? based upon Taylor?s series, which Lagrange had proposed in 1772 ? Lagrange?s Théorie des fonctions was only one, but by far the most important, of many attempts made about this time to furnish the calculus with a basis which would logically modify or supplant those given in terms of limits and infinitesimals? (Cajori).?With the appearance of the Mécanique Analytique in 1788, Lagrange proposed to reduce the theory of mechanics and the art of solving problems in that field to general formulas, the mere development of which would yield all the equations necessary for the solution of every problem . [it] united and presented from a single point of view the various principles of mechanics, demonstrated their connection and mutual dependence, and made it possible to judge their validity and scope. It is divided into two parts, statics and dynamics, each of which treats solid bodies and fluids separately. There are no diagrams. The methods presented require only analytic operations, subordinated to a regular and uniform development. Each of the four sections begins with a historical account which is a model of the kind.? (DSB).?In [Méchanique Analitique] he lays down the law of virtual work, and from that one fundamental principle, by the aid of the calculus of variations, deduces the whole of mechanics, both of solids and fluids. The object of the book is to show that the subject is implicitly included in a single principle, and to give general formulae from which any particular result can be obtained. The method of generalized co-ordinates by which he obtained this result is perhaps the most brilliant result of his analysis. Instead of following the motion of each individual part of a material system, as D?Alembert and Euler had done, he showed that, if we determine its configuration by a sufficient number of variables whose number is the same as that of the degrees of freedom possessed by the system, then the kinetic and potential energies of the system can be expressed in terms of those variables, and the differential equations of motion thence deduced by simple differentiation ? Amongst other minor theorems here given I may mention the proposition that the kinetic energy imparted by the given impulses to a material system under given constraints is a maximum, and the principle of least action. All the analysis is so elegant that Sir William Rowan Hamilton said the work could only be described as a ?scientific poem?? (Rouse Ball, A Short Account of the History of Mathematics).The Méchanique analitique ?was certainly regarded as the most important unification of rational mechanics at the turn of the 18th century and as its ?crowning? (Dugas). This achievement of unification and the abstract-formal nature of the work, physically reflected in immediate applications, earned the extravagant praise of Ernst Mach: ?Lagrange [?] strove to dispose of all necessary considerations once and for all, including as many as possible in one formula. Every case that arises can be dealt with according to a very simple, symmetric and clearly arranged scheme [?] Lagrangian mechanics is a magnificent achievement in respect of the economy of thought? (Mach, Die Mechanik in ihrer Entwicklung, 1933, 445)? (Pulte, p. 220).?Lagrange produced the Méchanique analitique during his time in Berlin. He referred as early as 1756 and 1759 to an almost complete textbook of mechanics, now lost; a later draft first saw the light of day in 1764. But it was not until the end of 1782 that Lagrange seems to have put the textbook into an essentially complete form, and the publication of the book was delayed a further six years? (Pulte, p. 209).?By 1790 a critical attitude had developed both within mathematics and within general scientific culture. As early as 1734 Bishop George Berkeley in his work The Analyst had called attention to what he perceived as logical weaknesses in the reasonings of the calculus arising from the employment of infinitely small quantities. Although his critique was somewhat lacking in mathematical cogency, it at least stimulated writers in Britain and the Continent to explain more carefully the basic rules of the calculus. In the 1780s a growing interest in the foundations of analysis was reflected in the decisions of the academies of Berlin and Saint Petersburg to devote prize competitions to the metaphysics of the calculus and the nature of the infinite. In philosophy Immanuel Kant?s Kritik der reinen Vernunft (1787) set forth a penetrating study of mathematical knowledge and initiated a new critical movement in the philosophy of science ??The full title of the Théorie explains its purpose: ?Theory of analytical functions containing the principles of the differential calculus disengaged from all consideration of infinitesimals, vanishing limits or fluxions and re-
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## Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundation of Thermodynamics.

GIBBS, Josiah Willard. First edition, inscribed presentation copy to the great French mathematician and mathematical physicist Henri Poincaré. ?Of Gibbs [Einstein] wrote in 1918: ?[His] book is ? a masterpiece, even though it is hard to read and the main points are found between the lines?? (Pais, Subtle is the Lord (1983), p. 73). This book was ?a major advance in statistical mechanics, the branch of science in which a purely mechanical view of natural phenomena is replaced by one combining mechanics with probability? (Norman). Gibbs? book was ?a triumph of the rigorous axiomatic method, which placed him beside Clausius, Maxwell, and Boltzmann as one of the principal founders of statistical mechanics? (Mehra, p. 1786). ?Albert Einstein ? who independently developed his own version of statistical mechanics from 1902 to 1904, having no knowledge of Gibbs? work ? remarked in 1910 ?Had I been familiar with Gibbs? book at that time, I would not have published those papers at all, but would have limited myself to the discussion of just a few points?? (Inaba, p. 102). ?Gibbs’ book on statistical mechanics became an instant classic and has remained so for almost a century? (Mehra). In this book, Gibbs formulated statistical mechanics in terms of ?ensembles? of systems, which were collections of large numbers of copies of the system of interest, all identical except for their physical properties (volume, temperature, etc.). ?In most of his elegant Principles in Statistical Mechanics of 1902, [Gibbs] described the underlying mechanical system in a formal manner, by generalised coordinates subject to Hamilton?s equations ? He introduced and systematically studied the three fundamental ensembles of statistical mechanics: the micro-canonical, the canonical, and the grand-canonical ensemble (in which the number of molecules may vary). He examined the relations between these three ensembles and their analogies with thermodynamic systems, including fluctuation formulas? (Buchwald & Fox, p. 784). ?A year before his death, Einstein paid Gibbs the highest compliment. When asked who were the greatest men, the most powerful thinkers he had known, he replied ?Lorentz?, and added ?I never met Willard Gibbs; perhaps, had I done so, I might have placed him beside Lorentz?? (Pais, p. 73). According to Emilio Segré (From Falling Bodies to Radio Waves (1984), p. 250), ?even Jules-Henri Poincaré found [Elementary Principles] difficult to digest? ? the present copy is presumably the one Poincaré puzzled over. Although reasonably well represented in institutional collections, this is a very rare book on the market. ABPC/RBH lists only one other copy in the last 35 years (and that copy lacked the dust-jacket). Provenance: Jules-Henri Poincaré (1854-1912), presentation inscription on front free endpaper: ?M. J.-H. Poincaré with the respects of the author? Poincaré was ?one of the greatest mathematicians and mathematical physicists at the end of 19th century. He made a series of profound innovations in geometry, the theory of differential equations, electromagnetism, topology, and the philosophy of mathematics? (Britannica). Although Poincaré did not work directly on statistical mechanics, his work on the three-body problem in celestial mechanics had an important impact upon it. In 1890, he proved his ?recurrence theorem?, according to which mechanical systems governed by Hamilton?s equations will, after a sufficiently long time, return to a state very close to the initial state. This theorem created serious difficulties for any mechanical explanation of the laws of thermodynamics, as it apparently contradicts the Second Law, which says that large dynamical systems evolve irreversibly towards states with higher entropy, so that if one starts with a low-entropy state, the system will never return to it. ?Josiah Willard Gibbs was born in 1839: his father was at that time a professor of sacred literature at Yale University. Gibbs graduated from Yale in 1858, after he had compiled a distinguished record as a student. His training in mathematics was good, mainly because of the presence of H. A. Newton on the faculty. Immediately after graduation he enrolled for advanced work in engineering and attained in 1863 the first doctorate in engineering given in the United States. After remaining at Yale as tutor until 1866, Gibbs journeyed to Europe for three years of study divided between Paris, Berlin, and Heidelberg. Not a great deal of information is preserved concerning his areas of concentration during these years, but it is clear that his main interests were theoretical science and mathematics rather than applied science. It is known that at this time he became acquainted with Möbius? work in geometry, but probably not with the systems of Grassmann or Hamilton. Gibbs returned to New Haven in 1869 and two years later was made professor of mathematical physics at Yale, a position he held until his death [in 1903].?His main scientific interests in his first year of teaching after his return seem to have been mechanics and optics. His interest in thermodynamics increased at this time, and his research in this area led to the publication of three papers, the last being his now classic ?On the Equilibrium of Heterogeneous Substances,? published in 1876 and 1878 in volume III of the Transactions of the Connecticut Academy. This work of over three hundred pages was of immense importance. When scientists finally realized its scope and significance, they praised it as one of the greatest contributions of the century? (Crowe, p. 151).?During the academic year 1889?1890 Gibbs announced ?A short course on the a priori Deduction of Thermodynamic Principles from the Theory of Probabilities,? a subject on which he lectured repeatedly during the 1890s? (DSB). ?Lord Rayleigh, writing on 5 June 1892 about an optical problem to Josiah Willard Gibbs in New Haven, Connecticut, concluded his letter as follows:?Have you ever thought of bringing out a new edition of, or a-
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## Considérations sur les Résultats dun Allègement indéfini des Moteurs.

ESNAULT-PELTERIE, Robert. First edition, exceptionally rare separately-paginated offprint, inscribed by Esnault-Pelterie, of the published version of his lecture, delivered in 1912 in both St. Petersburg and Paris, which first demonstrated theoretically that space travel was possible; it marks the beginning of theoretical astronautics. ?The lecture contains all the theoretical bases of self-propulsion, destroying the myth that rockets need atmospheric support and giving the real equation of motion. Anticipated is the use of auxiliary propulsion for guidance and complete maneuverability of rockets. Also contained are calculations of the escape velocity, the phases of a round-trip voyage to the Moon, and the times, velocities, and durations, of trips to the Moon, Mars, and Venus, as well as thermal problems related notably to the surface facing the sun . . . This 1912 lecture is the first purely scientific study marking the birth of astronautics. While Tsiolkovsky had the prescience and talent to first suggest, in 1903, rocket propulsion to space, REP [as Esnault-Pelterie liked to be called] was the first to develop the equations of the problem and to establish the mathematical theory of interplanetary flight. REP is thus the founder of theoretical astronautics? (Blosset, p. 9). As far as we can determine, no other copy of this offprint has appeared on the market; OCLC lists Bibliothèque Nationale only.Provenance: Boldly inscribed and signed by Esnault-Pelterie on front wrapper: ?En souvenir affectueux des premiers balbutiements ? que dis-je ? des premiers tressaillements d?un art aviatique en gestation!? (?In affectionate memory of the first faltering steps ? how do I put it? the first quickening of an aviation art in gestation!?).Born eleven months before American rocketry pioneer Robert Goddard, Esnault-Pelterie (1881-1957), French inventor and engineer, made substantial contributions to both aeronautics and astronautics, the design and flying of planes and the design of rockets. REP graduated in engineering at the Sorbonne, and was the fourth man in France to obtain a pilot?s license. He made a series of remarkable contributions to the development of airplanes: he built his first powered aircraft in 1907, the REP 1, which used ailerons rather than wing-warping to steer, had internally stressed wings instead of the external wire struts of the Wright brothers, and utilized a fuselage with an aluminium frame rather than wood. A serious crash in 1908 ended Esnault-Pelterie?s flying career, and although he persisted in building REP aircraft through the First World War, after his crash he turned to thinking about the possibility of space travel. ??When flying became a fact,? he wrote later, ?having once been only a dream, it was apparent to me, as one who remembered the time when there were even no automobiles, that it would develop rapidly, and I wondered what the next stage might be. Once the atmosphere had been conquered, there remained nothing more but to strike out into the empty space of the universe?? (French & Burgess, p. 54).?In the public mind in the beginning of the 20th century, rocketry and space exploration belonged more to the realm of science fiction than to the field of ?serious? research pursuit ? It was at this time that isolated visionaries and thinkers, including amateurs, sketched the sinews of the spaceflight concept. Technical details of rocketry and space travel had precarious credibility. Consequently, usual all-knowing intellectuals of the day dismissed them as ridiculous. The word astronautics did not yet exist. Actually, science fiction literature had used the word astronaut by that time, but astronautics was unknown as a term of science and engineering ? Esnault-Pelterie?s credibility of an accomplished engineer and fame as an aviation pioneer helped him to gain acceptance by mainstream scientific audiences? (Gruntman, pp. 1-2). By 1912 France had become the leading nation in aviation progress, and Esnault-Pelterie was a star, surpassed only by Bleriot and the Wright brothers in terms of notoriety and fame.The results of Esnault-Pelterie?s deliberations were first presented in his lecture, delivered on 14 February 1912 in St. Petersburg as a guest of the Imperial All-Russian Aerial Club, and then again in Paris on 15 November 1912 to the French Physical Society. His lecture celebrated ??the Rocket? as the sole machine capable of realizing fiction?s dream of ?traveling from planet to planet? A series of impressive mathematical formulas and planetary trajectories confirmed the propellant forces necessary to accelerate the rocket beyond Earth?s gravity: an escape velocity of just over eleven kilometres per second, what he called the ?critical velocity of liberation?, enough to take human beings ?to infinity? His conclusion: only atomic power (radium) would suffice to provide the propulsive force for such extreme velocities. Esnault-Pelterie also considered the physiological effects of spaceflight, artificial atmospheres and zero-gravity, and heat and energy sources from the sun. None of these challenges were insurmountable. His itinerary was bold: the Moon, Mars and on to Venus. Humanity was destined, in his concluding words, to become a new ?Halley?s comet?, to reach its fantastic velocities into interplanetary space? (Smith, p. 82) ?Esnault-Pelterie emphasized the importance of addressing physical foundations of spaceflight:?Numerous authors made a man traveling from star to star a subject for fiction ? No one has ever thought to seek the physical requirements and the orders of magnitude of the relevant phenomena necessary for realization of this idea ? This is the only aim of the present study? [the present offprint, p. 3].?In his lecture, Esnault-Pelterie discussed the acceleration of a rocket and derived the rocket equation. He considered the energetic properties of guncotton, hydrogen-oxygen mixture, and radium as propellants. Then he provided estimates of the required velocity increm-
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## Analysis per quantitatum series, fluxiones, ac differentias: cum enumeration linearum tertii ordinis.

NEWTON, Sir Isaac] (William Jones, ed.). First edition of the third of Newton?s great works on physics and mathematics, following Principia (1687) and Opticks (1704), and certainly the rarest of the three. This is a very fine copy in untouched contemporary English calf. This work contains ?De Analysi per Aequationes Numero Terminorum Infinitas,? written in 1669 and published here for the first time, containing Newton?s theory of infinite series; ?Methodis differentialis,? a treatise on interpolation written in 1676 and published here for the first time, the basis of the calculus of finite differences; two treatises, ?De quadratura curvarum? and ?Enumeratio linearum tertii ordinis,? first published in the Opticks but written in 1693 and 1695; the ?Epistola prior? and ?Epistola posterior,? first published in vol. III of John Wallis? Opera (1699), a letter from Newton to Collins, written November 8, 1676, and one to Wallis dated August 27, 1692. Newton described De analysi ?to Oldenburg as ?a compendium of the method of these [infinite] series, in which I let it be known that, from straight lines given, the areas and lengths of all the curves and the surfaces and volumes of all the solids [formed] could be determined, and conversely with these [taken as] given the straight lines could be determined, and I illustrated the method there outlined by several series.? Despite the use of the words ?method of series? rather than ?method of fluxions? (in the letter quoted Newton made no open reference of ?fluxions? at all), it is obvious from the inversion (lines to areas, areas to lines) that differentiation and integration, that is, the method of fluxions, is in question? (Hall, pp. 16-17). ?Modern workers, Duncan Fraser noted in 1927, had only just struggled up to the level reached by Newton in 1676 [in ?Methodus differentialis?]. Whiteside was equally impressed by the work, claiming that ?During the years 1675-76 ? Newton laid down the ? modern elementary theory of interpolation by finite differences but ? diffidently kept back his insights and discoveries therein for nearly forty years more? (Papers, IV, pp. 7-8)? (Gjertsen, p. 357). Newton?s decision to allow the publication of the 1669 tract at this time was heavily influenced by the on-going priority dispute with Leibniz over the invention of calculus. ?The work [Analysis per quantitatum series ?] also contained a Preface drafted, no doubt, with Newton?s assistance. It contained no mention of Leibniz. It did, however, contain the claim that Newton had ?Deduced the quadrature of the circle, hyperbola, and certain other curves by means of infinite series ? and that he did so in 1665; then he devised a method of finding the same series by division and extraction of roots, which he made general the following year?? (ibid., p. 18). For good measure, the tract was included in its entirety the following year in Commercium epistolicum, the official report on the priority dispute, largely drafted by Newton himself. ?De analysi, the work which established Newton?s reputation outside the walls of Trinity College, was first heard of in a letter from Barrow to Collins dated 20 June 1669. ?A friend of mine,? Barrow wrote, ?brought me the other day some papers, wherein he hath sett downe methods of calculating the dimension of magnitudes like that of Mr Mercator concerning the Hyperbola; but very Generall; as also of resolving equations? (Correspondence, I, p. 13). The manuscript, with Newton?s permission, was sent to Collins on June 31. The author?s name was revealed to Collins on 20 August, when Barrow wrote that the author was ?Mr Newton, a fellow of our College, and very young ? but of an extraordinary genius and proficiency in these things? (ibid., pp. 14-15).?Not only was Collins the first outside Cambridge to see important work of Newton; he had also, although inadvertently, provoked the work. In the early months of 1669 he had sent Barrow a copy of Mercator?s Logarithmotechnia (1668), a work which contained the series for log(1 + x). Barrow was aware that Newton had worked out for himself a general method for infinite series some two years before. Mercator?s book warned Barrow, and through him Newton, that others were working along similar lines. Newton?s reaction was to write, probably in a few summer days of 1669, his treatise De analysi which showed, by its generality, how far ahead he was of all other rivals.?Collins, like Barrow, had no difficulty in recognising the originality and power of Newton?s technique and, consequently, brought up the question of publication. An appendix to Barrow?s forthcoming optical lectures seemed a suitable place. Newton revealed, however, for the first time, his ability to frustrate even such skilled and persistent suitors as Collins. Immediate publication was rejected out of hand; thereafter Newton deployed a variety of excuses: a need to revise the work, a desire to add further material, the pressures of other business and, as a last resort when demands became too pressing, he simply failed to reply. As a result De analysi remained, with a good deal more of Newton?s early mathematical work, unpublished for half a century.?Newton?s reluctance to publish did not prevent Collins from copying and distributing the work. One copy was found by Jones in 1709 and is now to be seen in the Royal Society. Another copy was sent to John Wallis, at some point passed to David Gregory, and is at present in the Gregory papers at St. Andrew?s. Others who heard from Collins of Newton?s work were James Gregory, de Sluse and, above all, Leibniz. In October 1676 Leibniz visited London, saw Collins, and was allowed to read De analysi. He took thirteen printed pages of notes, an event construed by Newton as undoubted evidence of Leibniz?s reliance upon the discoveries of others in his mathematical development.?The work, Newton began, would present a general method ?for measuring the quantity of curves by an infinite series of terms.? To this end, three rules were formulated?-
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## Astronomiae Instauratae Progymnasmata, quorum hunc cassel pars prima de restitutione motuum solis et lunae, stellarumque inerrantium tractat et praeterea de admiranda nova stella anno 1572 exorta luculenter agit. Frankfurt: Gottfried Tampach, 1610. [With:] De Mundi Aetherei recentioribus Phaenomenis. Liber Secundus. Excudi primun coeptius Uraniburgi Daniae, ast Pragae Bohemiae absolutus. Frankfurt: Gottfried Tampach, 1610 (Colophon: Pragae Bohemorum, Absolvebatur Typis Schumanianis, Anno Domini 1603). [With:] Epistolarum Astronomicarum Libro. Quorum primus hic Illustriss. et Laudatiss. Principis Gulielmi Hassiae Landtgravij ac ipsius Mathematici Literas, unaque Responsa ad singulas complectitur Frankfurt: Gottfried Tampach, 1610 (Colophon: Uraniburgi, Ex officina Typographica Authoris, Anno Domini, 1596).

BRAHE, Tycho. An extraordinary sammelband, uniting the three most important works of the great Danish astronomer, all in first edition, Frankfurt issues, in two volumes uniformly bound in contemporary vellum. ?Tycho Brahe?s contributions to astronomy were enormous ? He revolutionized astronomical instrumentation. He also changed observational practice profoundly. Whereas earlier astronomers had been content to observe the positions of planets and the Moon at certain important points of their orbits, Tycho and his cast of assistants observed these bodies throughout their orbits. Without these complete series of observations of unprecedented accuracy, Kepler could not have discovered that planets move in elliptical orbits ? Tycho?s observations of the new star [now recognized to have been a supernova] of 1572 and comet of 1577, and his publications on these phenomena, were instrumental in establishing the fact that these bodies were above the Moon and that therefore the heavens were not immutable as Aristotle had argued and philosophers still believed ? Further, if comets were in the heavens, they moved through the heavens. Up to now it had been believed that planets were carried on material spheres that fit tightly around each other. Tycho?s observations showed that this arrangement was impossible because comets moved through these spheres? (Galileo Project). ?Astronomiae instauratae progymnasmata was produced in 1602 by the author?s own press at Uraniborg, and only a small number were printed for dedication purposes. It contains important investigations on the new star of 1572 which Brahe had discovered in Cassiopeia. This discovery led to far-reaching consequences in the history of astronomy as this work became the foundation on which Kepler, and later Newton, built their astronomical systems? (Sparrow). De mundi aetherei contains Brahe?s observations of the great comet of 1577, the brightest of the century, and, most importantly, includes the first account of his geoheliocentric theory of the universe, according to which the inferior and superior planets of Mercury, Venus, Mars, Jupiter and Saturn revolved around the Sun, but the Sun and the Moon orbited the Earth. ?There can be little doubt that Tycho regarded it [the geoheliocentric system] as his most significant achievement, and in the short term it surely was. As a geometrical equivalent of the Copernican system, it was capable of representing every aspect of the astronomical phenomena without demanding allegiance to a moving Earth, for which there would be no proof until much later? (Thoren, p. 8). The Astronomiae and De mundi were intended to form the first two parts of a trilogy, together with a work on the comets of 1582 and 1585, but this was never completed. The Epistolarum contains correspondence between Brahe and the Landgrave Wilhelm IV of Hesse-Cassel and his astronomer Christopher Rothmann, mostly concerning astronomical observations and the construction of astronomical instruments. ?This correspondence covered all aspects of contemporary astronomy: instruments and methods of observing, the Copernican system (which Rothmann supported against Tycho?s system), comets, and auroras? (DSB, under Rothmann). Brahe?s description of Uraniborg contained here is one of the earliest descriptions of an astronomical observatory and its instruments. These three works were originally produced on Tycho?s private press at Uraniborg, and were then reissued with minor textual changes at Prague and Frankfurt. The Astronomiae is present here in its second issue, the other two works in their third issue; the first issues of all three works are exceptionally rare as the few copies printed were intended for presentation only. ?Tyge (Latinized as Tycho) Brahe was born on 14 December 1546 in Skane, then in Denmark, now in Sweden ? He attended the universities of Copenhagen and Leipzig, and then traveled through the German region, studying further at the universities of Wittenberg, Rostock, and Basel. During this period his interest in alchemy and astronomy was aroused, and he bought several astronomical instruments ? In 1572 Tycho observed the new star in Cassiopeia and published a brief tract about it the following year. In 1574 he gave a course of lectures on astronomy at the University of Copenhagen. He was now convinced that the improvement of astronomy hinged on accurate observations. After another tour of Germany, where he visited astronomers, Tycho accepted an offer from the King Frederick II to fund an observatory. He was given the little island of Hven in the Sont near Copenhagen, and there he built his observatory, Uraniburg, which became the finest observatory in Europe. Tycho designed and built new instruments, calibrated them, and instituted nightly observations. He also ran his own printing press. The observatory was visited by many scholars, and Tycho trained a generation of young astronomers there in the art of observing. After a falling out with King Christian IV, Tycho packed up his instruments and books in 1597 and left Denmark. After traveling several years, he settled in Prague in 1599 as the Imperial Mathematician at the court of Emperor Rudolph II. He died there in 1601? (Galileo Project). In 1600, Tycho met Kepler and asked him to be his assistant. This placed Kepler in a position not only to publish some of Brahe?s works after his death, but also, after many trials and tribulations, to acquire Tycho?s actual observations, from which he would deduce his laws of planetary motion.Although Tycho intended the Astronomiae to form the first work of his trilogy, ?the corrected star places which were necessary for the reduction of the observations of 1572-73 involved researches on the motion of the sun, on refraction, precession, etc., the volume gradually assumed greater proportions than was originally contemplated, and was never quite finished in Tycho?s lifetime? (Dreyer, pp. 162-3). The first to be completed, in 1588, was De mundi, Tycho?s-
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## Gliding Experiments. Offprint from: Journal of the Western Society of Engineers, Vol. 2, No. 5, October, 1897 (read 20 October).

CHANUTE, Octave Alexander. First edition, extremely rare offprint, of this seminal work in the history of aviation, and a remarkable association copy linking three great early American pioneers of aviation and technology. In this work, Chanute presented his findings from nearly 2,000 flights made at Dune Park, Indiana, on the shores of Lake Michigan, in the summer of the previous year. These test-flights, led to his introduction of the Pratt-trussed biplane configuration, the essential shape of the early airplane, which was later adopted by the Wright brothers, and which he described in the present article. ?It was a pivotal moment in the history of aircraft design, ranking with Henson and Stringfellow?s postulation of the Aerial Steam Carriage over half a century earlier and Cayley?s even more distant derivation of the modern airplane shape. It ushered in an era of strong, light, straightforward, uncomplicated (and easily analysed) rectangular structures that quickly superseded the convoluted curves and framing of older attempts such as those by Adler and Lilienthal. Chanute, in short, had taken both glider experimentation and structural design to a new level, contributions of seminal importance? (Hallion, p. 177). ?The Chanute glider, designed by Chanute but also incorporating the ideas of his young employee Herring with regard to automatic stability, was the most influential of all flying machines built before the Wright brothers began designing aircraft ? Wilbur Wright, whom Chanute befriended, understood the importance of the 1896 biplane glider. ?The double-deck machine,? Wright remarked, ?represented a very great structural advance, as it was the first in which the principles of the modern truss bridge were fully applied to flying machine construction.? Chanute?s rigid, lightweight structure provided the most basic model for all externally based biplanes. It was nothing less than the first modern aircraft structure? (Britannica). ?The Wright brothers began their long association with Chanute in 1899, when they started serious work on their airplane. The Wrights corresponded with Chanute regularly, carefully detailing their thoughts to him. He served as their mentor, encouraging their efforts and offering advice. In 1901 he visited the brothers and encouraged them in their gliding experiments. Chanute also witnessed the early Wright flights, including the 1902 glider and the 1904 and 1905 powered flyer. He published the Wright brothers? writings in America and abroad, which did much to stimulate interest in aviation? (Octave Chanute and His Photos of the Wright Experiments at the Kill Devil Hills, www.memory.local.gov). OCLC locates eight known copies, including those at the Library of Congress and the Smithsonian. No copies on ABPC/RBH.Provenance: James Means (1853-1920), industrialist and aviation pioneer, inscribed on front wrapper to: Francis Blake Jr. (1850-1913), engineer and inventor who partnered with Alexander Graham Bell in the invention of the telephone (see below).?Born in 1832, the son of a history professor at the Collège de France, [Chanute] emigrated with his parents to the United States at age six, where his father assumed the vice presidency of Jefferson College in Louisiana. Though he considered himself American, young Chanute grew up in an intellectually oriented household so European in outlook that a pronounced Gallic accent would forever tinge his English. Gifted in mathematics, he chose a career in engineering while still a teenager, subsequently joining a railway survey crew and learning engineering first-hand. He eventually rose to the very top of his profession, earning a fortune while working with a series of railroad companies, as a noted bridge builder, and as the architect and chief engineer of the Union Stock Yards in Chicago and Kansas City ? But Chanute harboured a dark secret, something he feared that if learned could hurt his reputation: he had accumulated a wealth of material on early flying attempts, paths taken, and configurations chosen. He was in fact a closet aerophile, had been since taking a trip to France with his wife and children in 1875 that had exposed him to the work of Pénaud, Wenham and others, and in a few more years, he hoped, he might be able to come out in the open ??Only when in his late 50s, so distinguished, accomplished, and professionally secure as to no longer fear ridicule for advocating and discussing aviation, did he now devote his full attention to flight. And he did so with the characteristic energy and enthusiasm he had brought to his career as a practicing engineer. He became a virtual one-man aeronautical information clearinghouse and also bankrolled a number of individuals studying aviation ? Upon Chanute?s death in 1910, Wilbur Wright would state: ?No one was too humble to receive a share of his time. In patience and goodness of heart he has rarely been surpassed. Few men were more universally respected and loved.??In 1891 he wrote the first of a series of articles for American Engineer and Railroad Journal, which he would pull together and publish as a book three years later. He also sponsored professional meetings, most notably a four-day international Conference on Aerial Navigation held in Chicago on August 1-4, 1894, that drew together most of the major names in American aviation, and some international figures as well ??Chanute clearly enunciated his own thoughts on flight in his address at the opening of the conference. Flight to this point, he said, ?has hitherto been associated with failure,? its advocates viewed ?as eccentric ? to speak plainly, as ?cranks?? But the record of ballooning and airship development, and now increasingly that of winged aviation, held great promise, even if the precise commercial and military promise of such craft could not yet be clearly seen. Most significant, Chanute emphasized the importance of seeking integrated solutions to the problems of flight. ?It is a mistake,? he wrote, ?to suppose that th-
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## De revolutionibus orbium coelestium, libri VI: in quibus stellarum et fixarum et erraticarum motus, ex veteribus atq[ue] recentibus observationibus, restituit hic autor : praeterea tabulas expeditas luculentasq[ue] addidit, ex quibus eosdem motus ad quodvis tempus mathematum studiosus facillime calculare poterit. Item, De libris revolutionum Nicolai Copernici narratio prima, per Georgium Ioachimum Rheticum ad Ioan. Schonerum scripta.

COPERNICUS, Nicolaus. Second edition of the most important scientific publication of the sixteenth century and a ?landmark in human thought? (PMM). De revolutionibus was the first work to propose a comprehensive heliocentric theory of the cosmos, according to which the sun stood still and the earth revolved around it. It thereby inaugurated one of the greatest ever paradigm shifts in the history of human thought. ?Renaissance mathematicians, following Ptolemy, believed that the moon, sun and five planets were carried by complex systems of epicycles and deferents about the central earth, the fixed pivot of the whole system. In Copernicus?s day it was well known that conventional astronomy did not work accurately ? Copernicus, stimulated by the free entertainment of various new ideas among the ancients, determined to abandon the fixity of the earth ? With the sun placed at the center, and the earth daily spinning on its axis and circling the sun in common with other planets, the whole system of the heavens became clear, simple and harmonious. The revolutionary nature of his theory is evident in his famous diagram illustrating the concentric orbits of the planets [C1v]? (PMM). The text of the second edition of De revolutionibus follows the 1543 first edition almost exactly, including Andreas Osiander?s notorious unsigned preface, in which he attempted to placate potential critics of the work by stating that ?these hypotheses need not be true nor even probable? ? all that was necessary was that they should allow astronomers to correctly calculate the motions of the heavenly bodies. Petri added, at the end of the ?Index capitulorum?, a five-line recommendation by ?our leading mathematician? Erasmus Reinhold, extracted from his Tabulae Prutenicae, stating that ?all posterity will gratefully remember the name of Copernicus, by whose labor and study the doctrine of celestial motions was again restored from near collapse? (translation from Gingerich, Eye of Heaven, p. 221). This second edition is the first to contain the Narratio prima of Copernicus?s disciple George Joachim Rheticus, which summarises and champions the Copernican heliocentric hypothesis, and records Rheticus?s indefatigable efforts to persuade Copernicus to publish. The first edition of the Narratio, published at Gdansk in 1540, is virtually unobtainable, and the second edition of 1541 is hardly more procurable. According to Gingerich (Census, p. XIV), about 500-600 copies of this second edition of De revolutionibus were printed, perhaps slightly more than the first edition.?The first speculations about the possibility of the Sun being the center of the cosmos and the Earth being one of the planets going around it go back to the third century BCE. In his Sand-Reckoner, Archimedes (d. 212 BCE), discusses how to express very large numbers. As an example he chooses the question as to how many grains of sand there are in the cosmos. And in order to make the problem more difficult, he chooses not the geocentric cosmos generally accepted at the time, but the heliocentric cosmos proposed by Aristarchus of Samos (ca. 310-230 BCE), which would have to be many times larger because of the lack of observable stellar parallax. We know, therefore, that already in Hellenistic times thinkers were at least toying with this notion, and because of its mention in Archimedes?s book Aristarchus’s speculation was well-known in Europe beginning in the High Middle Ages but not seriously entertained until Copernicus.?European learning was based on the Greek sources that had been passed down, and cosmological and astronomical thought were based on Aristotle and Ptolemy. Aristotle?s cosmology of a central Earth surrounded by concentric spherical shells carrying the planets and fixed stars was the basis of European thought from the 12th century CE onward. Technical astronomy, also geocentric, was based on the constructions of eccentric circles and epicycles codified in Ptolemy?s Almagest (2d. century CE).?In the fifteenth century, the reform of European astronomy was begun by the astronomer/humanist Georg Peurbach (1423-1461) and his student Johannes Regiomontanus (1436-1476). Their efforts were concentrated on ridding astronomical texts, especially Ptolemy?s, from errors by going back to the original Greek texts and providing deeper insight into the thoughts of the original authors. With their new textbook and a guide to the Almagest, Peurbach and Regiomontanus raised the level of theoretical astronomy in Europe.?Several problems were facing astronomers at the beginning of the sixteenth century. First, the tables (by means of which astronomical events such as eclipses and conjunctions were predicted) were deemed not to be sufficiently accurate. Second, Portuguese and Spanish expeditions to the Far East and America sailed out of sight of land for weeks on end, and only astronomical methods could help them in finding their locations on the high seas. Third, the calendar, instituted by Julius Caesar in 44 BCE was no longer accurate. The equinox, which at the time of the Council of Nicea (325 CE) had fallen on the 21st, had now slipped to the 11th. Since the date of Easter (the celebration of the defining event in Christianity) was determined with reference to the equinox, and since most of the other religious holidays through the year were counted forward or backward from Easter, the slippage of the calendar with regard to celestial events was a very serious problem. For the solution to all three problems, Europeans looked to the astronomers? (Galileo Project).Nicolaus Copernicus was born on 19 February 1473 in Thorn (modern day Torun) in Poland. His father was a merchant and local official. When Copernicus was 10 his father died, and his uncle, a priest, ensured that Copernicus received a good education. In 1491, he went to Krakow Academy, now the Jagiellonian University, and in 1496 travelled to Italy to study law. While a student at the University of Bologna he stayed with a mat-
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## Telescopium: sive Ars perficiendi novum illud Galilaei visorium instrumentum ad sydera in tres partes divisa.

SIRTORI, Girolamo. First edition, extremely rare, of ?the first book about the telescope, its invention and use? (Zinner). Written by the Milanese scholar Girolamo Sirtori in 1612, only four years after the telescope was invented, it contained a complete set of instructions and diagrams for building a refracting telescope, and gave in the second part the first detailed account of Galileo?s telescope. Dedicated to the Grand Duke Cosimo II de? Medici, Telescopium is also one of the most important sources for the history of the invention and first uses of the telescope. Sirtori records the arrival of a telescope in Milan in May 1609, brought by ?a Frenchman? who was an associate of its inventor, Hans Lipperhey, and presented to the Count of Fuentes; this was about a month before Galileo first learned of the invention (according to Siderius Nuncius). Sirtori claims to have seen and handled Lipperhey?s very first telescope, but also suggests that the device was known to others before the Dutchman. He also provides an account of a famous meeting organised by Federico Cesi, founder of the Accademia dei Lincei, dated by Rosen to 14 April 1611, at which a select group of natural philosophers and mathematicians, including Sirtori himself, met Galileo and experienced the performance of his telescope in person ? this was ?the public unveiling of the term telescope? (Rosen, p. 31). Galileo received a copy of Sirtori?s work in 1633 from the hands of Cassiano dal Pozzo, his fellow Lincean and great collector and begetter of the ?Museo Cartaceo? (Cassiano Dal Pozzo to Galileo [in Rome], Rome, 18 June 1633, Opere XV, p. 158); this is presumably the copy which belonged to François Arago, cited by both Brunet and Riccardi, and having notes in Galileo?s hand. ABPC/RBH list only three other complete copies, two of them in Macclesfield sammelbands, the other an ex-library copy with removed stamps. The Honeyman copy, a presentation copy from the author to Tommaso Mingoni (Menghin), Leibartzt of Rudolph II, and a friend of Kepler, lacked one of the plates. OCLC lists Chicago and Yale only in North America; VD17 gives only 5 locations. ?Almost immediately after its invention, the telescope evolved from a mere optical toy into a ?scientific instrument,? an instrument of a new type which at the time was called ?philosophical?: the manipulation of such instruments allowed scholars to attain natural philosophical truth. In this way, the telescope paved the way for other scientific instruments which also emerged in the course of the seventeenth century, such as the air pump, the barometer, and the microscope. The emergence of the telescope was an important episode in the history of science and technology not only because it marks the invention of a new device, or because it changed man?s image of the universe, but also because it helped change the ways in which natural philosophy was practiced and what counted as ?science? ??The search for the inventor of the telescope has a long tradition which began almost immediately after the invention of the instrument. In Telescopium, the earliest book on the telescope, published in 1618, but composed in 1612, Girolamo Sirtori already doubts whether Lipperhey, the first demonstrator of the instrument, was also the inventor of the device? (Van Helden, Origins, p. 2). Indeed, Sirtori writes (pp. 24-26, translation from Van Helden, Invention, pp. 50-51):?In the year 1609 [sic, for 1608] there appeared a genius or some other man, as yet unknown, of the race of Hollanders, who, in Middelburg in Zeeland, visited Johannes Lippersein [i.e., Lipperhey], a man distinguished from others by his remarkable appearance, and a spectacle maker. There was no other [spectacle-maker] in that city, and he ordered many lenses to be made, concave as well as convex. On the agreed day he returned, eager for the finished work, and as soon as he had them before him, raising two of them up, namely a concave and a convex one, he put the one and the other before his eye and slowly moved them to and fro, either to test the gathering point or the workmanship, and after that he left, having paid the maker. The artisan, by no means devoid of ingenuity, and curious about the novelty began to do the same and to imitate the customer, and quickly his wit suggested that these lenses should be joined together in a tube. And as soon as he had completed one, he rushed to the court of Prince Maurits and showed him the invention. The prince had one [or, had been acquainted with one] before, and lest it should be suspected that [the device] was of military value, and very necessary, had kept it a secret. But now that he found by chance that it had become known he disguised [his prior knowledge], rewarding the industry and good intentions of the artisan. Thence the novelty of so great a thing was spread through the whole world, and many other telescopes were made. But none of those turned out better or more apt than the first one (which I have seen and handled).?Sirtori goes on to describe his first encounter with a telescope.?In the month of May [1609] a Frenchman rushed into Milan and offered such a telescope to the Count of Fuentes. He said that he was an associate of the inventor from Holland. When the Count had given it to a silversmith to put it in a silver tube, it came into my hands. I handled it and examined it, and made similar ones, in which I observed that many inconveniences occurred because of the glass. I therefore went to Venice in order to obtain a supply [of lenses] from the artisans there and, being still unskilled in the art, I delivered a finished lens to someone so that he could make similar ones. I squandered some money uselessly and lost the lens, having learned nothing more than that the business is to be perfected by chance and by the laborious selection of lenses. As it happened when I acquired one, I imprudently ascended the tower of St. Mark, in order to try it out at a distance. Someone, having decried the novel-
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## A Treatise on Electricity and Magnetism.

MAXWELL, James Clerk. First edition, first issue, and a very fine copy in a contemporary prize binding, of Maxwell?s presentation of his theory of electromagnetism, advancing ideas that would become essential for modern physics, including the landmark ?hypothesis that light and electricity are the same in their ultimate nature? (Grolier/Horblit). ?This treatise did for electromagnetism what Newton?s Principia had done from classical mechanics. It not only provided the mathematical tools for the investigation and representation of the whole electromagnetic theory, but it altered the very framework of both theoretical and experimental physics. It was this work that finally displaced action-at-a-distance physics and substituted the physics of the field? (Historical Encyclopedia of Natural and Mathematical Sciences, p. 2539). ?From a long view of the history of mankind ? seen from, say, ten thousand years from now ? there can be little doubt that the most significant event of the 19th century will be judged as Maxwell?s discovery of the laws of electrodynamics? (R. P. Feynman, in The Feynman Lectures on Physics II (1964), p. 1-6). ?[Maxwell] may well be judged the greatest theoretical physicist of the 19th century . Einstein?s work on relativity was founded directly upon Maxwell?s electromagnetic theory; it was this that led him to equate Faraday with Galileo and Maxwell with Newton? (PMM). ?Einstein summed up Maxwell?s achievement in 1931 on the occasion of the centenary of Maxwell?s birth: ?We may say that, before Maxwell, Physical Reality, in so far as it was to represent the process of nature, was thought of as consisting in material particles, whose variations consist only in movements governed by [ordinary] differential equations. Since Maxwell?s time, Physical Reality has been thought of as represented by continuous fields, governed by partial differential equations, and not capable of any mechanical interpretation. This change in the conception of Reality is the most profound and the most fruitful that physics has experienced since the time of Newton?? (Longair).Provenance: Stanley Butter (presentation inscription in Latin on front free endpaper from Exeter College, Oxford, dated Michaelmas [autumn] term 1877). ?Maxwell?s great paper of 1865 established his dynamical theory of the electromagnetic field. The origins of the paper lay in his earlier papers of 1856, in which he began the mathematical elaboration of Faraday?s researches into electromagnetism, and of 1861?1862, in which the displacement current was introduced. These earlier works were based upon mechanical analogies. In the paper of 1865, the focus shifts to the role of the fields themselves as a description of electromagnetic phenomena. The somewhat artificial mechanical models by which he had arrived at his field equations a few years earlier were stripped away. Maxwell?s introduction of the concept of fields to explain physical phenomena provided the essential link between the mechanical world of Newtonian physics and the theory of fields, as elaborated by Einstein and others, which lies at the heart of twentieth and twenty-first century physics? (Longair).The 1865 paper ?provided a new theoretical framework for the subject, based on experiment and a few general dynamical principles, from which the propagation of electromagnetic waves through space followed without any special assumptions ? In the Treatise Maxwell extended the dynamical formalism by a more thoroughgoing application of Lagrange?s equations than he had attempted in 1865. His doing so coincided with a general movement among British and European mathematicians about then toward wider use of the methods of analytical dynamics in physical problems ? Using arguments extraordinarily modern in flavor about the symmetry and vector structure of the terms, he expressed the Lagrangian for an electromagnetic system in its most general form. [George] Green and others had developed similar arguments in studying the dynamics of the luminiferous ether, but the use Maxwell made of Lagrangian techniques was new to the point of being almost a new approach to physical theory?though many years were to pass before other physicists fully exploited the ground he had broken ??In 1865, and again in the Treatise, Maxwell?s next step after completing the dynamical analogy was to develop a group of eight equations describing the electromagnetic field ? The principle they embody is that electromagnetic processes are transmitted by the separate and independent action of each charge (or magnetized body) on the surrounding space rather than by direct action at a distance. Formulas for the forces between moving charged bodies may indeed be derived from Maxwell?s equations, but the action is not along the line joining them and can be reconciled with dynamical principles only by taking into account the exchange of momentum with the field? (DSB)."Maxwell once remarked that the aim of his Treatise was not to expound the final view of his electromagnetic theory, which he had developed in a series of five major papers between 1855 and 1868; rather it was to educate himself by presenting a view of the stage he had reached in his thinking. Accordingly, the work is loosely organized on historical and experimental, rather than systematically deductive, lines. It extended Maxwell?s ideas beyond the scope of his earlier work in many directions, producing a highly fecund (if somewhat confusing) demonstration of the special importance of electricity to physics as a whole. He began the investigation of moving frames of reference, which in Einstein?s hands were to revolutionize physics; gave proofs of the existence of electromagnetic waves that paved the way for Hertz?s discovery of radio waves; worked out connections between electrical and optical qualities of bodies that would lead to modern solid-state physics; and applied Tait?s quaternion formulae to the field equations, out of which Heaviside and Gibbs would develop vec-
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## Planiglobium coeleste ac terrestre: argentorati quondam [Planiglobium celeste, Hoc est globus coelestis nove forma ac norma in planum projectus, omnes orbis coelestis lineas, circulos, gradus, partes, stellas, sidera &c. in planis tabulis aeri incises artificiose exhibens. Adjecta succincta tum fabricate tum usus explicatione, omnium Problematum, quae vulgatis hactenus globis, Planisphaeris, Astrolabiis expediri solita sunt, stellas coeli quascunq; cognoscere & denominare possit.] [Planiglobium terrestre. Sive globus terrestris, novo modo, ac method in plano descriptus; omnes Orbis Terrestris lineas, circulos, Regiones, Regna, Provincias, Promontoria, Portus, Insulas, Maria: omnes deniq; Terrae Marisq; tractus & anfractus in tabulis aeri insculptis, accurate monstrans. Annexa simul perspicua tum structurae, tum usue explanation: ubi plurima as usum globorum terrestrium plenius intelligendum hinc inde inseruntur.]

HABRECHT II, Isaac & STURM, Johann Christoph. First edition of one of the most beautiful instrument books published in the seventeenth century and certainly one of the rarest, particularly with the full complement of plates. This work is an enlargement, by his student Sturm, of Habrecht?s famous treatise on the making of celestial and terrestrial globes, published in 1628/29. Much influenced by Blaeu and Hondius, Habrecht published a pair of printed celestial and terrestrial globes at Strasbourg in 1621. The first edition of Planiglobium included two large planispheres (although they are lacking from almost all copies), these being polar stereographic celestial charts of the northern and southern constellations. In the present edition Sturm augmented the text, reprinted these planispheres from the same plates (one of them is still dated 1628), and added to them a further 12 plates, including two handsome polar projections of the world, and ten engravings showing the various parts of his celestial and terrestrial globes. ?The plates, superbly executed by Jacob van der Heyden, were probably intended to be mounted and assembled to form several instruments, each with a revolving plate measuring 27cm in diameter and a movable pointer. Each was to be supported on an approximately 12cm base? (W.P. Watson, Cat. 18). Regarding the two planispheres, Warner writes (TheSky Explored, p. 104): ?Habrecht derived the bulk of the information for this globe from Plancius. The origin of Rhombus ? a constellation near the south pole that as Reticulum survives today ? is unclear. It may perhaps derive from the quadrilateral arrangement of stars seen by Vespucci around the Antarctic pole. In any case, Rhombus as such seems to have made its first appearance on Habrecht?s globe.? Habrecht added to his celestial globe several cometary paths, an innovation that was followed by many Central European globe makers. Despite being an obvious Americanum (see for example pp. 220, 237, 249, and America pictured on one of the maps), this work is not in Sabin, JCB, Palmer and other standard bibliographies. The Honeyman copy (Sotheby?s, November 6, 1979, £1100 = $2346) is the only other complete copy on ABPC/RBH since 1950 and it had significant defects (some plates torn or repaired, title page defective and repaired, O2 torn, one leaf with marginal repair, browned). The Macclesfield copy, sold in 2004 for £3600, lacked the title page. A set of the unfolded plate sheets (without the text) sold at Christie?s in 2010 for £10,000. OCLC lists Brown, Harvard (both lacking the plates) and Chicago only in North America.?European discovery of the New World helped to establish the status of the terrestrial globe as equal to its celestial counterpart, for it was ideally suited to whet the imagination of those who remained at home but were eager to learn of the new and hitherto unknown lands and people about whom so much speculation had long existed. This interest stimulated a real boom in the cartographic industry. Map and globe makers set out to produce different versions of the world by adjusting and correcting the existing Ptolemaic picture. Through this process of continual adjustment old and rare terrestrial globes have become valuable artefacts, on which the history of world exploration is recorded both visually, by the tracks of the various epoch-making circumnavigations, and verbally, by lengthy legends inscribed on the globes ??During the fifteenth century, when the first western terrestrial globes emerged in the wake of the Latin translation of Ptolemy?s Geography, the Earth was firmly believed to be immobile in the centre of the universe. Thus, as far as models of the Earth are concerned, it would have sufficed to make a terrestrial sphere with a fixed mounting. However, the dominant construction in early globe making was more complex. It consisted of a mobile sphere mounted in a stand with a number of accessories ? a movable meridian ring, a fixed horizon ring, and an hour circle with pointer. These accessories served to demonstrate the time-dependent phenomena of the world around us in terms of the then generally accepted Ptolemaic concept of the First Mover and the annual motion of the Sun around the Earth.?In the common pairing of the terrestrial and celestial globes the diurnal motion of the First Mover is realized by the rotation of both spheres around the poles of the world. In use, these spheres always have to be turned from east to west in accordance with the Ptolemaic world system. For a proper understanding of the common Ptolemaic globe, it is particularly important to realize that the mobility of a terrestrial example has nothing whatsoever to do with the motion of the Earth. Neither is it a matter of simple viewing convenience. When the sphere of a terrestrial globe is turned, it is the daily motion of the Sun that is reproduced ? Thus, in the terrestrial globe the motion of the First Mover is imparted to the ?sphere of the Sun? and in the celestial globe to the ?sphere of the fixed stars? With this construction the whole series of phenomena which mattered in daily life as well as in education, such as the rising and setting of the Sun (with the terrestrial globe) and of the stars (with the celestial globe), could be demonstrated. The meridian ring serves to rectify the globe for the latitude of a place. The hour circle with pointer on top of the meridian ring can be set to local time, as measured through the diurnal motion of the Sun.?The annual motion of the Sun around the Earth is realized, only indirectly, by two design features of these globes. First, the ecliptic is drawn on both the terrestrial and the celestial sphere. Second, the position of the Sun in the zodiac throughout the year is displayed graphically on the horizon ring. Having established the Sun?s position at a particular time of the year, it can then be located on the ecliptic drawn on the sphere and its motion for that season demonstrated.?Thus, the common terrestrial globe is not simp-
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## Autograph letter signed, with important scientific content concerning the vis viva controversy, from Basel, dated 27 July 1728, to Gabriel Cramer, Professor of Mathematics, presently in London.

BERNOULLI, Johann. An important autograph letter from Johann Bernoulli (1667-1748), then one of the elder scientific statesmen of Europe, and still one of its greatest mathematicians, to his gifted student Gabriel Cramer (1704-52), who despite his youth had been appointed to the chair of mathematics at Geneva the previous year, but was now traveling through Europe and England making the acquaintance of the leading mathematicians of the day. The letter concerns the problem of vis viva (?forces vives?, ?living force?), one of the most controversial topics of the day. This was the question of whether it is (to use modern terminology) momentum (?quantity of motion?, mass x velocity) or kinetic energy (?living force?, mass x velocity2) which is the true measure of the ?force? between colliding bodies in motion. As with so many other issues, this controversy pitted the supporters of Leibniz against those of Newton. Bernoulli had recently published a major contribution to the dispute, Discours sur les Loix de la Communication du Mouvement (1727), supporting the Leibnizian position, in which he presented an analysis of vis viva in terms of balls moved by releasing compressed springs. This was attacked by the young English Newtonian Benjamin Robins (1707-51) in May 1728 in an article in The Present State of the Republick of Letters, in which he gave a detailed discussion of the impact of elastic bodies. This article won Robins many admirers in England. In the present letter, Bernoulli refutes Robins? article, and writes that an experiment proposed by another English Newtonian, James Jurin (1684-1750), and carried out by the London instrument maker George Graham (1673-1751), involving dropping a lead weight onto an elastic plate, ?prouve rien contre la théorie des forces vives? Bernoulli also responds to a misunderstanding by Cramer of a point in his Discours which Cramer had raised in an earlier letter. In a postscript, Bernoulli conveys the compliments of his nephew, the mathematician Nicolas Bernoulli (1687-1759). Letters by Bernoulli are rare on the market, particularly those with significant scientific content.Transcription:Monsieur Robert CailleMarchand Banquier, pour faire tenir à Monsieur Cramer, Professeur en mathematique presant à LondresMonsieur,Ce mot de lettre n?est que pour vous don[n]er avis que je vous ecrivis jeudi passé une reponse à la votre du 22. Juin, que j?ai adressée à Mr. de Mairan à Paris. Elle contient quelques reflexions generales sur la piece de Mr. Robins, et une reponse à l?objection tirée de l?experience avec la plaque de cuivre par laquelle étant en oscillation on laisse tomber un poids de plomb. J?ai fait voir que cette experience ne prouve rien contre la theorie des forces vives, et qu?elle est semblable à celle qu?on feroit avec deux corps sans ressort dont l?un en mouvement choqueroit directement l?autre en repos, auquel si le choquant étoit egal, ne lui com[m]uniqueroit que la moitié de la vitesse et iroit avec lui après le choc de compagnie, en sorte que la moitié de la force vive paroitra étre perdue. Je vous avois aussi ecrit une tres grande lettre datée du 23. Mai en reponse à vos deux precedentes du 10. Mars et 15 Avril, mais de laquelle vous ne faites pas mention dans votre derniere du 22 Juin, auquel temps vous prairés [pourriez] déjà avoir reçû la mienne; ce silence me mettant en peine, je vous prie de m?en tirer au plutot pour savoir si en fin elle vous a été rendue : vous y aurés trouvé bien des choses pour la confirmation de la theorie des forces vives et une ample solution à votre difficulté, qui consistoit à me demander, d?où vient que c?est l?increment de la vitesse, et non pas celui de la force vive, qui dans un temps infiniment petit est proportionel à ce temps et à la pression: c?est-à-dire, pourquoi il faut faire du = pdx/u, et non pas df = pdx/u ? Je finis en vous temoignant que je suis toujours avec la plus parfait consideration Monsieur votre tres humble et tres obeissant serviteurJ BernoulliBale, ce 27. Juillet 1728 S. Mon neveu vous fait ses compliments ; il y a quelques semaines qu?il vous a ecrit une lettre sous l?adresse de Mr. Caille : dont je me sers aussi toujours en vous ecrivant. Translation:Mr Robert CailleMerchant Banker, for the attention of Mr. Cramer, Professor in mathematics present in LondonSir,This brief letter is only meant to give notice that I wrote you last Thursday a reply to your letter of 22 June which I addressed to Monsieur de Mairan in Paris. It contains some general reflections on Mr. Robins’ essay, and a response to the objection derived from the experiment with an oscillating copper plate on which a lead weight is dropped. I have shown that this experiment proves nothing against the theory of the live force, and that it is similar to that which would be made with two inelastic bodies, one of which in motion would directly collide with the other at rest. If the colliding body was equal to the one at rest, it would convey to it only half the speed and go with it after the collision, so that half the force will appear to be lost. I also wrote you a very long letter dated 23 May in reply to your two previous ones of 10 March and 15 April, which you do not mention in your last letter of 22 June, though at that time you may have already received mine; this silence is distressing and I beg you to put an end to it and let me know if eventually it was delivered to you: you will have found [in it] many things confirming the theory of live forces and an ample solution to your difficulty, which consisted in asking me, how come that it is the increment of speed, and not that of the live force, which in an infinitely small time interval is proportional to this interval and to pressure; that is to say, why should we do du = pdx/u, not df = pdx/u?I am ending this letter with the renewed assurance that I remain, with the most perfect consideration Your very humble and very obedient servantJ BernoulliBasel, 27 July 1728P.S. My nephew sends you h-
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## Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus.

LEIBNIZ, Gottfried Wilhelm. First edition of Leibniz?s invention of the differential calculus. ?Leibniz was an almost universal genius whose place in the history of mathematics depends on his being an independent inventor of the infinitesimal calculus and on his contributions to combinatorial analysis which foreshadowed the development of modern mathematical analysis ? The Acta Eruditorum was established in imitation of the French Journal des Sçavans in Berlin in 1682 and Leibniz was a frequent contributor. Another German mathematician (E.W. Tschirnhausen), having published in it his paper on quadratures, based on researches that Leibniz had communicated to him, Leibniz at last decided in 1684 to present to the world the more abstruse parts of his own work on the calculus. His epoch-making papers give rules of calculation without proof for rates of variation of functions and for drawing tangents to curves ? The infinitesimal calculus originated in the 17th century with the researches of Kepler, Cavalieri, Torrecelli, Fermat and Barrow, but the two independent inventors of the subject, as we understand it today, were Newton and Leibniz ? Although both Newton and Leibniz developed similar ideas, Leibniz devised a superior symbolism and his notation is now an essential feature in all presentation of the subject ? With the calculus a new era began in mathematics, and the development of mathematical physics since the 17th century would not have been possible without the aid of this powerful technique? (PMM). Leibniz ?applied his new method to the solution of the cubic parabola and the inverse methods of tangents and many problems left unsolved by Descartes? (Dibner). ?Although Newton had probably discovered the calculus earlier than Leibniz, Leibniz was the first to publish his method, which employed a notation superior to that used by Newton. The priority dispute between Newton and Leibniz over the calculus is one of the most famous controversies in the history of science; it led to a breach between English and Continental mathematics that was not healed until the nineteenth century? (Norman).?The invention of the Leibnizian infinitesimal calculus dates from the years between 1672 and 1676, when Gottfried Wilhelm Leibniz (1646?1716) resided in Paris on a diplomatic mission. In February 1667 he received the doctor?s degree by the Faculty of Jurisprudence of the University of Altdorf and from 1668 was in the service of the Court of the chancellor Johann Philipp von Schönborn in Mainz. At that time his mathematical knowledge was very deficient, despite the fact that he had published in 1666 the essay De arte combinatoria. It was Christiaan Huygens (1629?1695), the great Dutch mathematician working at the Paris Academy of Sciences, who introduced him to the higher mathematics. He recognised Leibniz?s versatile genius when conversing with him on the properties of numbers propounded to him to determine the sum of the infinite series of reciprocal triangular numbers. Leibniz found that the terms can be written as differences and hence the sum to be 2, which agreed with Huygens?s finding. This success motivated Leibniz to find the sums of a number of arithmetical series of the same kind, and increased his enthusiasm for mathematics. Under Huygens?s influence he studied Blaise Pascal?s Lettres de A. Dettonville, René Descartes?s Geometria, Grégoire de Saint-Vincent?s Opus geometricum and works by James Gregory, René Sluse, Galileo Galilei and John Wallis.?In Leibniz?s recollections of the origin of his differential calculus he relates that reflecting on the arithmetical triangle of Pascal he formed his own harmonic triangle in which each number sequence is the sum-series of the series following it and the difference-series of the series that precedes it. These results make him aware that the forming of difference-series and of sum-series are mutually inverse operations. This idea was then transposed into geometry and applied to the study of curves by considering the sequences of ordinates, abscissas, or of other variables, and supposing the differences between the terms of these sequences infinitely small. The sum of the ordinates yields the area of the curve, for which, signifying Bonaventura Cavalieri?s ?omnes lineae?, he used the sign ???, the firstletter of the word ?summa? The difference of two successive ordinates, symbolized by ?d?, served to find the slope of the tangent. Going back over his creation of the calculus Leibniz wrote to Wallis in 1697: ?The consideration of differences and sums in number sequences had given me my first insight, when I realized that differences correspond to tangents and sums to quadratures?.?The Paris mathematical manuscripts of Leibniz ? show Leibniz working out these ideas to develop an infinitesimal calculus of differences and sums of ordinates by which tangents and areas could be determined and in which the two operations are mutually inverse. The reading of Blaise Pascal?s Traité des sinus du quart de circle gave birth to the decisive idea of the characteristic triangle, similar to the triangles formed by ordinate, tangent and sub-tangent or ordinate, normal and sub-normal. Its importance and versatility in tangent and quadrature problems is underlined by Leibniz in many occasions, as well as the special transformation of quadrature which he called the transmutation theorem by which he deduced simply many old results in the field of geometrical quadratures. The solution of the ?inverse-tangent problems?, which Descartes himself said he could not master, provided an ever stronger stimulus to Leibniz to look for a new general method with optimal signs and symbols to make calculations simple and automatic.?The first public presentation of differential calculus appeared in October 1684 in the new journal Acta Eruditorum, established in Leipzig, in only six and an half pages, written in a disorganised manner with numerous typographical errors. In the title, ?A new method for maxima and mini-
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