‎Mathematics‎

‎"SHEWHART, W. A.‎

‎Quality Control.‎

‎New York, American Telephone and Telegraph Company, 1927. 8vo. In the original printed blue wrappers. In ""The Bell System Technical Journal"", Volume VI, October, Number 4, 1927. Entire issue offered. Light miscolouring to spine. A fine fine and clean copy. Pp. 722-735. [Entire issue: 551-752 pp.].‎

‎"SHEWHART, W.A.‎

‎Economic Control of Quality of Manufactured Product.‎

‎New York, American Telephone and Telegraph Company, 1930. Original full fabrikoid, gilt lettering to spine. In ""The Bell System Technical Journal"", Vol. 9, 1930. (10),842 pp., textillustrations and plates. The entire volume offered. Shewhart's paper: pp.364-389, textfigs. Clean and fine.‎

‎"SIEBECK.‎

‎Ueber die graphische Darstellung imaginärer Functionen.‎

‎Berlin, G. Reimer, 1858. 4to. As extracted from ""Journal für die reine und angewandte Mathematik, 55. Band, 1858""., without backstrip. Fine and clean. [Siebeck:] Pp. 221-253" [Steiner)] Pp. 356-378.‎

‎"SLONIMSKI, CH. Z. [CHAIM ZELIG SLONIMSKI], [HAYYIM SELIG].‎

‎Allgemeine Bemerkungen über Rechenmaschinen, und Prospectus eines neu erfundenen Rechen-Instrument (+) Eine allgemeine Formel für die gesammte jüdische Kalenderberechnung. - [FIRST RUSSIAN CALCULATOR]‎

‎Berlin, G. Reimer, 1844. 4to. In ""Journal für die reine und angewandte Mathematik, 28 Band, 2. Heft, 1844"". In the original printed wrappers, without backstrip. Fine and clean. [Slonimskyi:] Pp. 179-183"" Pp. 184-190. [Entire issue: Pp. 97-192, (2) + 1 plate.].‎

‎First printing of Slonimsky's paper on the design of his calculator or multiplying device, probably the earliest Russian contribution to designing a calculator. ""In the middle of the last century, the mathematician Zinovy (Zelig) J. Slonimsky (1810-1904) invented a simple multiplication device to implement one of the theorems, which he had proven. The device could multiply every number of permitted length (i.e. the length permitted by the device-design) by every simple number. In other words, the device was something like a mechanical multiplication table. Later, the Slonimsky theorem was used for another simple multiplication device, namely for the calculating bars designed by Jofe. Both devices are interesting as examples of technical implementations of complicated mathematical ideas rather than as mechanical innovations."" (Trogemann, Computing in Russia, P. 29).‎

Bookseller reference : 45144

‎"SLONIMSKI, CH. Z. [CHAIM ZELIG SLONIMSKI], [HAYYIM SELIG].‎

‎Allgemeine Bemerkungen über Rechenmaschinen, und Prospectus eines neu erfundenen Rechen-Instrument (+) Eine allgemeine Formel für die gesammte jüdische Kalenderberechnung. - [FIRST RUSSIAN CALCULATOR]‎

‎Berlin, G. Reimer, 1844. 4to. In ""Journal für die reine und angewandte Mathematik, 28 Band, 2. Heft, 1844"". In the original printed wrappers, without backstrip. Fine and clean. [Slonimskyi:] Pp. 179-183" " Pp. 184-190. [Entire issue: Pp. 97-192, (2) + 1 plate.].‎

‎"SONIN, N. (NIKOLAY YAKOVLEVICH).‎

‎(Sur la fonction discontinue (x) et ses applications par N. Sonin). In Russian, but transcribed by the author in his own hand on the title-page.‎

‎Varsovie, 1889. Orig. printed wrappers. (2),78 pp. Inscribed to the Danish mathematician Niels E. Nørlund: ""A Monsieur N. Nörlund/ hommage de l'auteur"" together with the title translated into French by Sonin.‎

‎"STEICHEN, E.‎

‎Exposé de diverses remarques et reflexions sur les moments et d'autres sujets de statique.‎

‎Berlin, Georg Reimer, 1852. 4to. In the original wrappers, without backstrip. In ""Journal für die reine und angewandte Mathematik"", 44 band, 3. heft. Entire issue offered. Fine and clean. Pp. 181-219. [Entire issue: Pp. 181-276 + 2 folded plates].‎

‎First printing.‎

Bookseller reference : 47447

‎"STEICHEN, E.‎

‎Mémoire de mécanique, relatif au mouvement de rotation et au mouvement naissant des corps solides.‎

‎Berlin, G. Reimer, 1852. 4to. In the original printed wrappers, without back strip. In ""Journal für die reine und angewandte Mathematik"", 43. band, Heft 3, 1852. Entire third heft offered. Fine and clean. Pp. 185-244. [Entire volume: Pp. 185-282 + 3 plates.].‎

‎First printing of Steichen's paper the rotational movement of solids.‎

Bookseller reference : 47536

‎"STEINER, J.‎

‎Sur le maximum et le minimum des figures dans le plan, sur la sphére et dans l'espace en général. Second mémoire.‎

‎(Berlin, G. Reimer, 1842) 4to. As extracted from ""Journal für die reine und angewandte Mathematik, 34 Band, Heft, 1842"". Without wrappers and backstrip. Fine and clean. Pp.189-250.‎

‎First printing of Swiss mathematician Steiner's ""outstandingly clear and brilliant exposition"" on the sphere in general (Klein, Development of Mathematics in the 19th Century P 118).""Students and contemporaries wrote of the brilliance of Steiner's geometric research and of the fiery temperament he displayed in leading others into the new territory he had discovered. Having set himself the task of reforming geometry, Steiner sought to discover simple principles from which many seemingly unrelated theorems in the subject could be deduced in a natural way."" (DSB)‎

Bookseller reference : 47252

‎"STEICHEN, E.‎

‎Exposé de diverses remarques et reflexions sur les moments et d'autres sujets de statique.‎

‎Berlin, Georg Reimer, 1852. 4to. In the original wrappers, without backstrip. In ""Journal für die reine und angewandte Mathematik"", 44 band, 3. heft. Entire issue offered. Fine and clean. Pp. 181-219. [Entire issue: Pp. 181-276 + 2 folded plates].‎

‎"STEINER, J.‎

‎Sur le maximum et le minimum des figures dans le plan, sur la sphére et dans l'espace en général. Second mémoire.‎

‎(Berlin, G. Reimer, 1842) 4to. As extracted from ""Journal für die reine und angewandte Mathematik, 34 Band, Heft, 1842"". Without wrappers and backstrip. Fine and clean. Pp.189-250.‎

‎"STIELTJES, (THOMAS). - THE ""STIELTJES INTEGRAL"" INTRODUCED.‎

‎Recherches sur les fractions continues.‎

‎(Paris, Gauthier-Villars), 1894. 4to. No wrappers. In: ""Comptes Rendus Hebdomadaires des Séances de L'Academie des Sciences"", Tome 118, No 25. Pp. (1371-) 1442. (Entire issue offered). Le Chatelier's paper: pp. 1401-1411.‎

‎"SYLVESTER, J.J. (JAMES JOSEPH). - THE PROOF OF NEWTON'S RULE.‎

‎Algebraic Researches, containing a disquisition on Newton's Rule for the Discovery of Imaginary Roots, and an allied Rule applicable to a particular class of Equations, together with a complete invariantive determination of the character of the Roots ...‎

‎(London, Taylor and Francis, 1865). 4to. No wrappers as extracted from ""Philosophical Transactions"", Vol. 154 - Part III, pp. 579-666 and 2 lithographed plates. Some faint brownspots to theplates. Otherwise clean and fine.‎

‎First printing of a main paper of the founder (together with Cayley) of the theory of algebraic invariants, in which Sylvester investigates the roots of quintic equations and the proof of Newton's rule.""Another problem of great importence investigated in two long memoirs of 1853 and 1864 (the paper offered) concerns the nature of the roots of quintic equation. Sylvester took the functions of the coefficients that serve to decide the reality of the roots, and treated them as coordinates of a point in n-dimensional space. A point is or is not ""facultative"" according to whether there correspons, or fail to correspond, an equation with real coefficients. The character of the roots depends on the bounding surface or surfaces of the facultative regions, and on a single surface depending on the ""discriminant.""(DSB XIII, p. 219).‎

Bookseller reference : 42883

‎"SYLVESTER, J.J. (JAMES JOSEPH). - THE PROOF OF NEWTON'S RULE.‎

‎Algebraic Researches, containing a disquisition on Newton's Rule for the Discovery of Imaginary Roots, and an allied Rule applicable to a particular class of Equations, together with a complete invariantive determination of the character of the Roots ...‎

‎(London, Taylor and Francis, 1865). 4to. No wrappers as extracted from ""Philosophical Transactions"", Vol. 154 - Part III, pp. 579-666 and 2 lithographed plates. Some faint brownspots to theplates. Otherwise clean and fine.‎

‎"TARSKI, ALFRED.‎

‎Pojecie Prawdy w Jezykach Nauk Dedukcyjnych (Polish). [The Concept of Truth in Formalized Languages]. - [FOUNDING MODERN LOGICAL SEMANTICS]‎

‎Warszawa, 1933. Small 4to. Orig. printed wrappers, sunned at the edges, but otherwise near mint condition, also internally. An excellent copy. VII, (1), 116, (1, - errata) pp.‎

‎The exceedingly scarce first printing of Tarski's most important and influential work, ""The Concept of Truth in Formalized Languages"", which founded modern logical semantics.The work appeared in an extremely small number, in Polish, and many copies of the article have later been destroyed, thus, the work is of the utmost scarcity. In this seminal article the Polish-American logician and mathematician Alfred Tarski devotes himself to ""the definition of truth"". ""Its task is to construct -with reference to a given language- a materially adequate and formally correct definition of the term ""true sentence""."" (Introduction, English translation, 1956). With this work the face of logic was changed forever. The ""Concept of Truth"" constitutes a landmark event in 20th century analytic philosophy, and it ranks as one of the most important contributions to symbolic logic, semantics and philosophy of language. In this work Tarski develops the semantic theory of truth for formal languages and determines the fact that no language can contain its own truth predicate. Tarski thus concluded that the semantic theory could not be applied to any natural language. -This was later used by e.g. Davidson to construct his truth-conditional semantics, and the problems solved by Tarski are some of the same that Russell and Whitehead struggled to solve in their ""Principia Mathematica"".Tarski (1901-1983) has contributed seminally to the fields of mathematics and logic in a number of ways, and together with Frege, Russell and Gödel, he now ranks as one of the most important contributors to the field of modern logic. At the time of Franz Brentano (1838-1917), one of the philosophers of the greatest significance for contemporary philosophy and in many ways a forerunner of present-day empiricism, it was very unusual for a metaphysician to acknowledge that philosophical investigation must go hand in hand with an analysis of language. Linguistic analysis has thus been almost totally limited to the pure empiricists of philosophy, who reject all forms of metaphysics. Meanwhile, ontologists and metaphysicians have been satisfied with the ordinary language and asked no questions about its possible limitations, merely dismissing the logical faults and adding the odd neologisms. Today, however, especially within the English speaking tradition, linguistic analysis has reached a degree unheard of at the time of Brentano, and it is now generally accepted that certain logical and epistemological problems can be solved only by forsaking ordinary language and substituting it for artificially constructed language systems that follow certain principles. Thus, difficulties that appeared within earlier philosophical doctrines are meant to disappear if the theory can be formulated more precisely, and one of the most important examples is the ""adequacy theory of truth"". Tarski shows that the concept of truth of the adequacy theory can be introduced in a perfectly exact way within the formalized language systems that are equipped with precise rules of interpretation, and thus he rids us of the usual misgivings against the concept of truth. And thus he has developed one of the most important theories of modern logic.""Tarski's investigations are of singular philosophical significance for another reason as well. Within the framework of semantics, which he founded and which Carnap later developed further, it becomes possible for the first time to introduce the notion of an analytic judgment (or an analytic statement) in a form that is both sufficiently general and of the utmost precision. This notion also plays an exceptionally important role in Brentano's philosophy, especially in his studies in formal logic."" (Stegmüller, Main Currents... p. 56). When constructing a semantical system, a vocabulary of the desired object language must be determined as the first. Then formulation rules must be specified, before the rules of interpretation are laid down, and finally the rules of application are supplied. The most important rules here are the rules of truth, and the concept of truth is one of the most important semantical concepts at all, for without them no understanding of the sentences within the system would be ensured. And, of course, the truth definitions must satisfy a condition of adequacy. ""...This form of an adequacy condition that must be satisfied by every semantical truth concept goes back to the Polish logician, Stanislaw Lesniewski. But it was the logician Alfred Tarski who above all made use of this notion, and who first studied in detail the possibilities of introducing a formally exact and materially adequate concept of truth into the precise languages of science. Carnap's accounts of semantical systems rest largely on the prior works of Tarski."" (Stegmüller, p. 311). Tarski also pointed out that it is necessary for all semantical concepts, and especially for the concept of truth, to strictly separate object language and metalanguage. Otherwise we would put ourselves in the unlucky position of being able to prove both a statement and its negation at the same time. In the English translation from 1956 of Tarski's works, ""Logic, Semantics, Metamathematics"", the bibliographical information about this article is erroneous.‎

Bookseller reference : 35832

‎"TCHEBYCHEFF, P. [PAFNUTY LVOVICH CHEBYSHEV]‎

‎Sur la représentation des valeurs limites des intégrales par des résidus intégraux.‎

‎Berlin, Stockholm, Paris, Beijer, 1886. 4to. With the original wrappers in ""Acta Mathematica, 9:1. Band]. No backstrip. Fine and clean. Pp. 35-56. [Entire issue: 104 pp. ].‎

‎First translation of Tchebycheff's paper on Moments - a quantitative measure of the shape of a set of points. Tchebycheff is known for his work in the field of probability, statistics and number theory.‎

Bookseller reference : 46202

‎"TCHEBYCHEFF, P. [CHEBYSHEV] [TCHEBYCHEV].‎

‎Sur deux théorèmes relatifs aux probabilités. - [THE FOUNDATION FOR THE APPLICATION OF PROBABILITY THEORY TO STATISTICS]‎

‎Berlin, Stockholm, Paris, Almqvist & Wiksell, 1890. 4to. As extracted from ""Acta Mathematica"", Vol, 14, 1890. No backstrip. A fine and clean copy. Pp. 305-15.‎

‎"TETENS, JOHANN NICOLAI.‎

‎Formula Polynomiorum. En almindelig Formel for Coefficienterne udi Polynomier.‎

‎(Kiøbenhavn, Johan Rudolph Thiele, 1799). 4to. Without wrappers. In: ""Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter."" Vol. V. Pp. 253-286.‎

‎An importent mathematical memoir by the author of ""Einleitung zur Berechnung der Leibrenten und Anwartschaften"", a landmark work in actuarial science.‎

Bookseller reference : 45698

‎"TETENS, JOHANN NICOLAI.‎

‎Formula Polynomiorum. En almindelig Formel for Coefficienterne udi Polynomier.‎

‎(Kiøbenhavn, Johan Rudolph Thiele, 1799). 4to. Without wrappers. In: ""Nye Samling af det Kongelige Danske Videnskabernes Selskabs Skrifter."" Vol. V. Pp. 253-286.‎

‎"THOMÉ, LUDVIG G.‎

‎13 papers on ""Linearen Differentialgleichungen"" und ""Differentialgleichungen mitt algebraischen Koeffizienten"" etc.‎

‎Berlin, G. Reimer, 1887-1901. 4to. All 13 offprints from ""Journal für die reine und angewandte Mathematik. Hrsg. von A.L. Crelle"", Bde. 100-123. Some of the offprints with traces after having been folded.‎

‎All first printing.‎

Bookseller reference : 41691

‎"THOMÉ, L. W.‎

‎Zur Theorie der algebraischen Funktionen mit Bezugnahme auf die Theorie der linearen Differentialgleichungen (+) Über eine Anwendung der Theorie der linearen Differentialgleichungen in der Variationsrechnung (+) Über simultane lineare Differentialglei... - [COLLECTION OF OFFPRINTS SIGNED BY THE AUTHOR]‎

‎Berlin, Georg Reimer, 1888 - 1901. Small folio. Collection of 13 offprint all in the original printed wrappers from ""Journal für reine und angewandte Mathematik"", A. L. Crelle. All with the printed presentation statement on top of frontwrappers ""Überreicht vom Verfasser"" [i.e. ""Given by the author""] and the author's signature: ""Thomé"". All have been bended vertically, otherwise fine and clean.‎

‎A collection of offprints by the influential German mathematician L. W. Thomé, all signed by the author. L.W. Thomé worked closely with Bernhard Riemann on continued fractions.‎

Bookseller reference : 48884

‎"THOMÉ, L. W.‎

‎Zur Theorie der algebraischen Funktionen mit Bezugnahme auf die Theorie der linearen Differentialgleichungen (+) Über eine Anwendung der Theorie der linearen Differentialgleichungen in der Variationsrechnung (+) Über simultane lineare Differentialglei... - [COLLECTION OF OFFPRINTS SIGNED BY THE AUTHOR]‎

‎Berlin, Georg Reimer, 1888 - 1901. Small folio. Collection of 13 offprint all in the original printed wrappers from ""Journal für reine und angewandte Mathematik"", A. L. Crelle. All with the printed presentation statement on top of frontwrappers ""Überreicht vom Verfasser"" [i.e. ""Given by the author""] and the author's signature: ""Thomé"". All have been bended vertically, otherwise fine and clean.‎

‎"TINSEAU (D'AMONDANS, CHARLES de).‎

‎Sur quelques Propriètés des Solides renfermés par des Surfaces composées de Lignes droites.‎

‎(Paris, Moutard, Panckoucke, 1780). 4to. Extract from ""Mémoires fe Mathematique et de Physique, Présentés à l'Academie des Sciences par divers Savans"", Tome IX. Pp. 625-642 and 2 folded engraved plates.‎

‎First appearance of an importent papwer in the history of analytic geometry.""Two of the three memoirs that constitute Tinseau’s oeuvre deal with topics in the theory of surfaces and curves of double curvature: planes tangent to a surface, contact curves of circumscribed cones or cylinders, various surfaces attached to a space curve, the determination of the osculatory plane at a point of a space curve, problems of quadrature and cubature involving ruled surfaces, the study of the properties of certain special ruled surfaces (particularly conoids), and various results in the analytic geometry of space. In these two papers the equation of the tangent plane at a point of a surface was first worked out in detail (the equation had been known since Parent), methods of descriptive geometry were used in determining the perpendicular common to two straight lines in space, and the Pythagorean theorem was generalized to space (the square of a plane area is equal to the sum of the squares of the projections of this area on mutually perpendicular planes). (DSB). Although Tinseau published very little, his papers are of great interest as additions to Monge’s earliest works. Indeed, Tinseau appears to have been Monge’s first disciple.‎

Bookseller reference : 44932

‎"TINSEAU (D'AMONDANS, CHARLES de). - GENERALIZATION OF THE PYTHAGOREAN THEOREM.‎

‎Solution de quelques Problêmes relatifs à la Théorie des Surfaces courbes, & des Courbes à double courbure. (Présenté en 1774).‎

‎(Paris, Moutard, Panckoucke, 1780). 4to. Extract from ""Mémoires fe Mathematique et de Physique, Présentés à l'Academie des Sciences par divers Savans"", Tome IX. Pp. 593-624 and 2 folded engraved plates. Clean and fine.‎

‎"TORRES, M.L. (LEONARDO TORRES Y QUEVEDO).‎

‎Sur la constuction des machines algébriques. Extrait de la Revue de Mécanique, nos de septembre-octobre 1901. - [THE BIRTH OF COMPUTING - TORRES QUEVEDO'S CALCULATING MACHINES]‎

‎Paris, 1901. Small folio. Original printed wrappers. The fragile wrappers have been expertly restored along the edges. A stain to back wrapper, a closed horizontal tear, and spine re-enforced. A bit of soiling and discolouration. With a library-label to top of front wrapper, from the Bibliothèque des professeurs St-Stanislas, Mons, of which there is a book-plate to half-title and a stamp to title-page. Wrappers loose. Internally some brownspotting and finely restored marginal chipping, far from affecting text. 32 pp. Richly illustrated.‎

‎Exceedingly scarce original offprint of Torres y Quevedo's publication of his seminal algebraic machines, constituting a milestone in computing history. In this thoroughly illustrated publication, Torres y Quevedo explains the construction and operation of the first accurate calculating machines, following his explanation of them before the French Academy of Sciences earlier the same year. ""At the end of the nineteenth century, several analog machines had been proposed for solving algebraic equations. These machines -based not only on kinematics principles but also on dynamic or hydrostatic balances, electric or electromagnetic devices, etc.- had one important drawback: lack of accuracy. Leonardo Torres was the first to beat the challenge of designing and implementing a machine able to compute the roots of algebraic equations that, in the case of polynomials of degree eight, attained a precision down to 1/1000. The key element of Torres' machine was the endless spindle, an analog mechanical device designed to compute log(a +b) from log(a) and log(b). This short account gives a detailed description of this mechanism."" (Federico Thomas: A Short Account on Leonardo Torres' Endless Spindle). Leonardo Torres y Quevedo (1852 -1936), civil engineer and mathematician, was a one of the most important Spanish inventors of all time. Although little known outside of Spain and France in his life-time, he was an inventor of the utmost importance and would later be recognized as one of the most significant inventors of the beginning of the twentieth century. Apart from pioneering accurate automated calculation machines, he was also a pioneer in the development of the radio control, the inventor of a chess automaton (the first decision-making automaton), an innovative designer of the three-lobed non-rigid Astra-Torres airship as well as the Whirlpool Aero Car located in Niagara Falls, and with his Telekine, he created wireless remote-control operation principles.""Leonardo Torres was probably the most eminent Spanish engineer in the first half of the 20th century. He had a great influence, truncated by the Spanish Civil War, in the development of automatic control in Spain. As Randell points out in, we can only speculate on what might happened if Torres' writings had become better known to the English-speaking world. For instance, he qualifies Torres' paper Essays on Automatics as ""a fascinating work which well repays reading even today."" The paper contains what Randell believes to be the first proposal of the idea of floating-point arithmetic. It seems clear that Torres' contributions deserve much wider appreciation outside Spain"". (Federico Thomas: A Short Account on Leonardo Torres' Endless Spindle). ""Because of Torres Quevedo's work on analog machines, the Paris Academy of Sciences welcomed his ""Calculating Machines"" report in 1900. As a result, he was elected to the Academy of Sciences in Madrid in 1901, where he delivered a lecture on algebraic machines.The endless spindle was a critical component of the 8-term equation-solving computing machine. It created sums to answer the algorithm's central equation. The mechanism was the first of its sort, and it was the most intriguing and unique of Torres Quevedo's inventions.Torres Quevedo's calculating machine was both theoretically and practically groundbreaking. The endless spindle is a mechanical device that evaluates an expression's logarithm as the sum of numerous logarithms, overcoming the difficulty of reaching enough precision with an automatic set of parts."" (The History of the Computer) The algebraic machine of Torres was an analog computing device, featuring a mechanism based on cone-shaped pulley with a helical groove around it. The machine was used for the resolution of equations like: + Ax = B or + Ax = B. His publications marked the beginning of a new era in mathematical theory, and his contributions resulted in the creation of cutting-edge new machinery. ""Torres Quevedo's writings ushered in a new era in mathematical theory based on new concepts. His contributions led to the development of innovative new machines.He built a series of analog calculating machines to complement his theoretical work, all of which were mechanical."" (Torres Quevedo Museum - where another copy of the present offprint is displayed) ""Leonardo Torres (1852-1936), usually known as Leonardo Torres y Quevedo in Spanish-speaking countries, was a Spanish engineer and mathematician. He was president of the Academy of Sciences of Madrid, a member of the French Academy of Sciences, and famous-mainly in Spain and France- as a prolific and successful inventor... Some of the earliest Torres' inventions took the form of mechanical analog devices, considered of great originality, aiming at implementing what he called the Algebraic Machine. This machine had the ability of computing the values of arbitrary polynomial functions in one variable. Since it was an analogue machine, the variable could attain any value, not only a preestablished set of discrete values, contrarily to what happened, for example, with the celebrated Babbage's Difference Engine, an engine that used the method of finite differences to generate successive values of polynomial functions. In Torres' Algebraic Machine, all quantities were represented by means of angular displacements in logarithmic scale. Then, adding a counter to keep track of the number of turns, it was possible to compactly represent very large variations for all quantities. When the wheel representing the variable spun round, the final result was obtained as the angular displacement of another wheel that accumulated the addition of all involved monomials. When this result was zero, a root of the function was found. Using proper modifications, it was even possible to obtain the complex roots. The use of logarithms had two main advantages. Firstly, assuming that all absolute errors for the angular displacements were constant, all relative errors for the represented quantities were also kept constant. Secondly, the computation of monomials was greatly simplified. Actually, the logarithm of a monomial of the form aixi is an expression, ai+ilogx, that can be easily calculated using a differential transmission. Nevertheless, the use of logarithms had a number of disadvantages. It was necessary to introduce some transformations to ensure that all quantities were positive and, what was much more challenging, it was necessary to design a sophisticated mechanical device able to compute log(a +b) from log(a) and log(b) that would permit to accumulate the sum of all involved monomials. This device was the endless spindle. In 1893, Leonardo Torres presented, before the Academy of Exact, Physical and Natural Sciences, a memoir on his Algebraic Machine. In his time, this was considered an extraordinary success for Spanish scientific production. In 1895 the machine was presented at a congress in Bordeaux. Later on, in 1900, he would present his calculating machine before the Paris Academy of Sciences."" The present offprint, which richly illustrates, demonstrates, and presents his seminal calculating machines, was published after his presentation at the Academy of Sciences in Paris, and as he says in the preface to the present offprint ""the theoretical principles of these machines have very recently been exposed in a mémoire entitled ""Machines à calculer"" [See Tomash & Williams: T 47, printed 1902], presented by me at the l'Académie des Sciences, which should appear also in the ""Recueil des Savants étrangers."" (own translation from p. (5) ). This publication is of the utmost scarcity, and we can locate merely two copies of this offprint: at the Torres y Quevedo-museum in Spain, and at the Hungarian Academy of Sciences.‎

Bookseller reference : 60101