Map Details      Question or Comment about this Map?
Number: 1344
Continent: Africa
Region: Continent
Place Names:
Year of Origin: 1753
Title: Tab: Geogr: Africae ad emendatiora qua ad huc prodierunt exempla jufsu Acad Reg: Scient: et eleg: Litt: Pruss: descripta
Language: Latin
Publish Origin: Berlin
Height: 30.9
Width: 39.1
Units: centimeters
Size Class.: Medium
Scale: 1 : 30,000,000
Color Type: No Color
Click for high-resolution zoomable image
Cartographer: Leonhard Euler
Engraver: Johann Ernst Gericke
Publisher: Prussian Royal Academy of Sciences and Literature
Other Contributors: N. Sauerbrey
Northernmost Latitude:
Southernmost Latitude:
Westernmost Longitude:
Easternmost Longitude:
Notes: Garwood/Voigt; Euler, Widely published Swiss intellectual in mathematics, physics, astronomy, and philosophy; this mep is from his 'Atlas Geographicus' map-38 for the Prussian Royal Academy of Sciences and Literature in Berlin. Leonhard Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history. He is also widely considered to be the most prolific, as his collected works fill 92 volumes, more than anyone else in the field. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin. He wrote extensive works on Mathematics, Calculus, Astronomy and Physics, ?Theoria motuum planetarum et cometarum. Continens methodum facilem ex aliquot observationibus orbitas cum planetarum tum cometarum determinandi. Una cum calculo, quo cometae, qui annis 1680 et 1681. Itemque ejus, qui nuper est visus, motus verus investigator?, In this work Euler gives "the solutions of the main problems of theoretical astronomy dealing with the structure, nature, motion and action of comets and planets. With regard to the theory of perturbed motion of celestial bodies, Euler formulated the perturbation theory in general terms so that it can be used to solve the mathematical problem of dynamic models and particular problems of theoretical astronomy . He gave an extensive mathematical treatment of the problem of improving approximations of orbits within the framework of the two-body problem and taking perturbations into account. In his Theoria motuum planetarum et cometarum published in 1744, Euler gave a complete mathematical treatment of the two-body problem consisting of a planet and the Sun." (Debnath, The Legacy of Leonhard Euler, p. 364). Other works in Mathematics and Calculus, ?Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici?, ?Institutiones Calculi Differentialis cum eius Usu in Analysi Finitorum ac Doctrina Serierum ? and the OPUSCULA ANALYTICA? Academia Imperialis Scientiarum, St. Petersburg, 1783. collection of 28 previously unpublished papers by Euler (listed below), of which the most important is 'Observationes circa divisionem quadratorum per numeros primos', which gives the first clear statement of the 'law of quadratic reciprocity,' called the 'golden theorem' by Gauss who gave its first proof in his Disquisitiones Arithmeticae (1801). The first volume was published in the year of Euler's death, most of the papers having been presented to the St. Petersburg Academy a decade or more earlier. The majority of the papers in these volumes deal with topics in number theory (3, 5, 7-11, 14, 19). Euler had studied questions related to quadratic reciprocity for decades, starting with a letter to Christian Goldbach dated 28 August, 1742. He gradually accumulated numerical evidence, but did not feel able to formulate the 'law' until he did so in (3). He continued to work on topics related to quadratic reciprocity - paper (5) is an example - but a proof eluded him, as it did Legendre later. Apart from the papers dealing with quadratic reciprocity, the most interesting of the number-theoretic papers is perhaps 'Nova subsidia pro resolutione formulae axx + 1 = yy,' which gives a new method of solving Pell's equation x2- dy2 = 1. Euler had developed a method of solution using continued fractions in 1765 ('De usu novi algorithmi in problemate Pelliano solvendo') but noted that for some values of d this method leads to very tedious calculations (he mentions the case d = 61, for which the smallest solution is x = 1766319049 and y = 226153980). The new method, which generates solutions of Pell's equation from solutions of closely related equations, is much more efficient, although it applies only to certain values of d. It has been repeatedly rediscovered over the centuries, and has recently been generalized to apply to any value of d. Another group of papers deals with finite and infinite series (1, 2, 20, 24, 25). Euler had famously solved the problem of exactly summing the zeta function series ? (n) = 1/1n + 1/2n + 1/3n + 1/4n + . in the case n = 2, which had defeated the Bernoullis, and in paper (25) he calculates ? (2n) up to n = 17. In 'De summa seriei ex numeris primis formatae 1/3 - 1/5 + 1/7 + 1/11 - 1/13 . ubi numeri primi formae 4n - 1 habent signum positivum, formae autem 4n + 1 signum negativum,' Euler notes that the sum of the reciprocals of the primes diverges, as does the harmonic series 1 + (1/2) + (1/3) + (1/4) + . His derivations start with the 'Leibniz' series, 1 - (1/3) + (1/5) - (1/7) + (1/9) - . = ?/4. In 'De eximio usu methodi interpolationum in serierum doctrina' Euler presents his discovery of the 'Lagrange interpolation formula,' a fundamental technique in numerical analysis. This was published by Lagrange in 1795, and earlier by Waring in 1779 ('Problems concerning interpolations,' Phil. Trans., Vol. 69, pp. 59-67), but Euler's paper was presented to the Academy on May 18, 1772. The last two papers discuss topics in probability, a subject not normally associated with Euler. 'Solutio quaestionis ad calculum probabilitatis pertinentis' treats a problem in annuities: How much should be paid by a couple, so that a certain sum of money can be paid to the heir after the death of the other? In 'Solutio quarundam quaestionum difficiliorum in calculo probabilium,' Euler studies the 'Genoese lottery,' a game of chance similar to today's lotteries in which numbered balls are placed in a large wheel, five or six are drawn at random, and players attempt to guess the numbers. Euler became interested in such lotteries after he was asked by King Frederick II for his analysis of a proposal for a state lottery involving the drawing of five numbers from 1 to 90. Other papers deal with definite integrals (15-17), infinite products (12, 13), and several with one of Euler's favourite topics, continued fractions (4, 21-23). The works included in these volumes are as follows: 1. De seriebus, in quibus producta ex minis terminis contiguis datam constituunt progressionem 2. Varia artificia in serierum indolem inquirendi 3. Observationes circa divisionem quadratorum per numeros primos 4. Observationes analyticae 5. Disquitio accuratior circa residua ex divisione quadratorum altiorumque potestatum per numeros primos relicta 6. De eximio usu methodi interpolationum in serierum doctrina 7. De criteriis aequationis fxx + gyy = hxx, utrum ea resolutionem admittat necne? 8. De quibusdam eximiis proprietatibus circa divisores potestatum occurrentibus 9. Proposita quacunque protressione ab unitate incipiente, quaeritur quot eius terminos ad minimum addi oporteat, ut omnes numeri producantur 10. Nova subsidia pro resolutione formulae axx + 1 = yy 11. Miscellanea analytica 12. Variae observationes circa angulos in progressione geometrica progredientes 13. Quomodo sinus et cosinus angulorum multiplorum per producta exprimi queant 14. Considerationes super theoremate Fermatiano de resolutione numerorum in numeros polygonales 15. Observation in aliquot theoremata illustrissimi de la Grange 16. Investigatio formulae integralis ? (xm-1 dx)/(1+xk)n casu, quo post intagrationem statuitur x = ? 17. Investigatio valoris integralis ? (xm-1 dx)/(1-2xkcos?+x2k) a termino x = 0 ad x = ? extensi 18. Theoremata quaedam analytica, quorum demonstratio adhuc desideratur 19. De relatione inter ternas pluresve quantitates instituenda 20. De resolutione fractionum transcendentium in infinitas fractiones simplices 21. De transformatione serierum in fractiones continuas, ubi simul haec theoria non mediocriter amplificatur 22. Methodus inveniendi formulas integrales, quae certis casibus datam inter se teneant rationem, ubi sumul methodus traditur fractiones continuas summandi 23. Summatio fractionis continuae cuius indices progressionem arithmeticam constituunt dum numeratores omnes sunt unitates ubi simul resolutio aequationis Riccatianae per huiusmodi fractiones docetur 24. De summa seriei ex numeris ; the mathematics of calculus, ? Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici ? 1744; on ship building, ?COMPLEAT THEORY OF CONSTRUCTION AND PROPERITIES OF VESSELS, with practical conclusions for the management of ships, made easy to navigators?. translated from 'Th?rie Complette de la Construction et de la Manoeuvre des Vaisseaux' of the celebrated Leonard[sic.] Euler, by Henry Watson, Esq, first published in French in 1773, and previously in Latin in 1749, both editions in St. Petersburg; and also works on telescope contruction; But his most famous work was ?LETTRES A UNE PRINCESSE d ALLEMAGNE sur divers sujets de physique & de phiBlosophie?,. Petersburg: Imperial Academy of Sciences, 1768. letters to the Princess Friederike Charlotte Leopoldine Louise of Brandenburg-Schwedt, to whom Euler had given lessons during his stay in Berlin. The work "had an immense success and profoundly influenced contemporary philosophy" (PMM 196, note). "Another multi-volume masterwork by Euler, the 'Lettres' are a principal document of the Enlightenment. Their passion for learning reflects that period's faith in education, including support of female learning. In the mid-nineteenth century, some scholars mistakenly believed that the title referred not to an actual person but to a technique in writing. The 'Lettres,' with their insightful explanation of the sciences and his core religious and spiritual positions, offer probably the best rounded view of Euler's character" (Calinger, Leonhard Euler: Mathematical Genius in the Enlightenment, pp. 467-8). Despite being written for the lay reader, these letters broke much new ground: letters 101-108 anticipate the invention of 'Venn diagrams' in the 1880s; letter 60 speculates on the existence of what are now called exoplanets orbiting the fixed stars (and whether they might support life), and letters 155-169 describe six methods of determining the longitude of a ship at sea. "The most exhaustive and authoritative science popularization written during the eighteenth century, the 'Lettres' critically examined in greater depth than did other works the complex and changing major Enlightenment natural philosophies, the Cartesian, Newtonian, Leibnizian, and Wolffian, and presented each in terms understandable to the educated European reading public. As one of the few leading men of science, well grounded in all four schools of thought and with a sure command of them, Euler could comment adeptly on each. In neither method nor content was Euler a Cartesian, as has been sometimes thought; instead he brought together consistent ideas in the sciences from different natural philosophers and new thought in mathematics and physics introduced by some members of the Bernoulli family, and he combined these with his original insights. This was the Eulerian system. Alexandre Koyr?included the 'Lettres' among the prominent Newtonian popularizations - Henry Pemberton's 'View of Sir Isaac Newton's Philosophy', Voltaire's 'Philosophical Letters' and 'Elements of Newton's Philosophy', Count Francesco Algarotti's 'Newtonianism for the Ladies', Colin Maclaurin's 'Account of Sir Isaac Newton s Philosophical Discoveries', and Pierre-Simon Laplace's subsequent 'System of the World' - but Euler's work was far greater" (Calinger). From the final years of Euler's life to the present, the 'Lettres' have met with phenomenal success. He probably prepared in Russia the first German translation, while between 1768 and 1774 Euler's student Stepan Rumovskij translated the 'Lettres' from French into Russian, the first of five Russian editions leading up to 1808. By 1800 they had gone through thirty editions and were translated into seven other languages: Danish, Dutch, English, German, Italian, Spanish, and Swedish. Enestrom 3
Last updated: Jun 30, 2020