1096:
429:
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53:
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794:, including probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; and the law of large numbers.
1388:(This is a problem of another kind: The question is, if some lender were to invest sum of money interest, let it accumulate, so that every moment were to receive proportional part of annual interest; how much would he be owed end of year?) Bernoulli constructs a power series to calculate the answer, and then writes:
456:
In 1684, Bernoulli married Judith
Stupanus; they had two children. During this decade, he also began a fertile research career. His travels allowed him to establish correspondence with many leading mathematicians and scientists of his era, which he maintained throughout his life. During this time, he
640:
is equivalent to solving a first-order nonlinear differential equation. The isochrone, or curve of constant descent, is the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. It had been studied by
Huygens
897:
One example is an account that starts with $ 1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $ 2.00; but if the interest is computed and added twice in the year, the $ 1 is multiplied by 1.5 twice, yielding $ 1.00×1.5 = $
521:
Jacob collaborated with his brother on various applications of calculus. However the atmosphere of collaboration between the two brothers turned into rivalry as Johann's own mathematical genius began to mature, with both of them attacking each other in print, and posing difficult mathematical
537:
Jacob
Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines.
775:, published in Basel in 1713, eight years after his death. The work was incomplete at the time of his death but it is still a work of the greatest significance in the theory of probability. The book also covers other related subjects, including a review of
1367:
Jacob
Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the
984:
spiral "may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self". Bernoulli died in 1705, but an
545:
and published his law of large numbers in probability theory. Jacob
Bernoulli published five treatises on infinite series between 1682 and 1704. The first two of these contained many results, such as the fundamental result that
797:
Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity.
1386:"Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?"
892:
518:. It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were among the first to try to understand and apply Leibniz's theories.
750:
was first conceived by Jacob
Bernoulli in 1694. In 1695, he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced.
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in a discussion of the exponential series. Inspired by
Huygens' work, Bernoulli also gives many examples on how much one would expect to win playing various games of chance. The term
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615:
574:
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905:) for more and smaller compounding intervals. Compounding weekly yields $ 2.692597..., while compounding daily yields $ 2.714567..., just two cents more. Using
1080:
2523:
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of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the
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from 1676 to 1682, learning about the latest discoveries in mathematics and the sciences under leading figures of the time. This included the work of
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2.25. Compounding quarterly yields $ 1.00×1.25 = $ 2.4414..., and compounding monthly yields $ 1.00×(1.0833...) = $ 2.613035....
232:
Solutionem tergemini problematis arithmetici, geometrici et astronomici... (Solution to a triple problem in arithmetics, geometry and astronomy...)
342:
1653:
487:, leading to his interest in infinitesimal geometry. Apart from these, it was between 1684 and 1689 that many of the results that were to make up
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on mathematical topics. The two brothers began to study the calculus as presented by
Leibniz in his 1684 paper on the differential calculus in "
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40 years earlier and was proved by Nicole Oresme in the 14th century already. Bernoulli could not find a closed form for
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succumbed in the year of grace 1705, the 16th of August, at the age of 50 years and 7 months, awaiting the resurrection.
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compounding, the account value will reach $ 2.7182818.... More generally, an account that starts at $ 1, and yields (1+
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in 1687 and
Leibniz in 1689. After finding the differential equation, Bernoulli then solved it by what we now call
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appears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called the
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in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother
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1480:< 3. (** The reference is to a problem which Jacob Bernoulli posed and which appears in the
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challenges to test each other's skills. By 1697, the relationship had completely broken down.
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645:. Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term
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and his two children have erected a monument to the husband and father they miss so much.
980:('Although changed, I rise again the same') engraved on his tombstone. He wrote that the
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779:, in particular the work of van Schooten, Leibniz, and Prestet, as well as the use of
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and entered the ministry. But contrary to the desires of his parents, he also studied
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which required him to find the value of the following expression (which is in fact
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diverges, which
Bernoulli believed were new but they had actually been proved by
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1070:(in Latin). Vol. 2. Genève: héritiers Cramer & frères Philibert. 1744.
1060:(in Latin). Vol. 1. Genève: héritiers Cramer & frères Philibert. 1744.
17:
52:
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member of the Royal Academies of Paris and Berlin; famous for his writings.
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1035:(title roughly translates as "A new hypothesis for the system of comets".)
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The Golden Ratio: The Story of Phi, the World's Most Astonishing Number
1340:. Journal Électronique d'Histoire des Probabilités et de la Statistique
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Bernoulli returned to Switzerland, and began teaching mechanics at the
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27 December 1654] – 16 August 1705) was one of the many prominent
887:{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}
406:
214:
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D. J. Struik (1986) A Source Book In Mathematics, 1200-1800, p. 320
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1565:. Vol. 2. New York: Charles Scribner's Sons. pp. 46–51.
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1312:(in Italian). Springer Science & Business Media. p. 92.
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as the number of compounding intervals, with interest of 100% /
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In the last part of the book, Bernoulli sketches many areas of
363:. However, his most important contribution was in the field of
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Jacob Bernoulli also discovered a general method to determine
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Bernoulli noticed that this sequence approaches a limit (the
636:, Jacob Bernoulli showed that the problem of determining the
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Professor at the University of Basel For more than 18 years;
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Hoffman, J.E. (1970–1980). "Bernoulli, Jakob (Jacques) I".
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421:. During this time he also produced an incorrect theory of
357:. He also discovered the fundamental mathematical constant
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Die Werke von Jakob Bernoulli: Bd. 2: Elementarmathematik
1044:(in Latin). Amstelaedami: apud Henricum Wetstenium. 1683.
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is also named after him jointly with his brother Johann.
438:(1682) wherein was published the critique of Bernoulli's
1033:(in Latin). Amstelaedami: apud Henr. Wetstenium. 1682.
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studied the new discoveries in mathematics, including
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was submitted in 1684. It appeared in print in 1687.
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1595:Landmark Writings in Western Mathematics 1640–1940
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1696:Correspondence Regarding the Art of Conjecturing"
1530:(First trade paperback ed.). New York City:
1468:, the geometric series reduces to the series for
496:He was appointed professor of mathematics at the
1226:. 3. Auflage. Mannheim/Wien/Zürich, Dudenverlag.
1051:, Basileae, impensis Thurnisiorum Fratrum, 1713.
996:Jacob Bernoulli, the incomparable mathematician.
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1282:Jan Hendrik Oort: Master of the Galactic System
1005:Of a chronic illness, of sound mind to the end;
629:which came out of examining compound interest.
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8:
989:was engraved rather than a logarithmic one.
541:By 1689, he had published important work on
367:, where he derived the first version of the
1426:( ... which our series is larger . ... if
1240:The MacTutor History of Mathematics archive
809:In 1683, Bernoulli discovered the constant
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393:. Following his father's wish, he studied
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345:and was an early proponent of Leibnizian
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514:. They also studied the publications of
1654:MacTutor History of Mathematics Archive
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802:Discovery of the mathematical constant
610:{\displaystyle \sum {\frac {1}{n^{2}}}}
1694:Gottfried Leibniz and Jakob Bernoulli
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951:dollars with continuous compounding.
632:In May 1690, in a paper published in
449:from 1683. His doctoral dissertation
7:
506:Nova Methodus pro Maximis et Minimis
353:, he was one of the founders of the
769:Bernoulli's most original work was
720:{\displaystyle y'=p(x)y+q(x)y^{n}.}
569:{\displaystyle \sum {\frac {1}{n}}}
343:Leibniz–Newton calculus controversy
108:(D.Th., 1676; Dr. phil. hab., 1684)
1562:Dictionary of Scientific Biography
1333:Pfeiffer, Jeanne (November 2006).
1260:. Historisches Lexikon der Schweiz
992:Translation of Latin inscription:
846:
25:
2524:18th-century Swiss mathematicians
2504:17th-century Swiss mathematicians
1667:"Tractatus de Seriebus Infinitis"
1384:, Bernoulli poses the question:
1279:Kruit, Pieter C. van der (2019).
1030:Conamen novi systematis cometarum
440:Conamen novi systematis cometarum
1619:
1376:), in the year (anno) 1685.**),
1374:Ephemerides Eruditorum Gallicanæ
1224:Duden — Das Aussprachewörterbuch
1200:Longman Pronunciation Dictionary
1126:
1094:
1079:
625:in 1737. Bernoulli also studied
451:Solutionem tergemini problematis
325:in French; 6 January 1655 [
223:Primi et Secundi Adami Collatio
1049:Ars conjectandi, opus posthumum
963:Jacob Bernoulli's tombstone in
651:Bernoulli differential equation
117:Bernoulli differential equation
30:For other family members named
27:Swiss mathematician (1655–1705)
1682:"Bernoulli, Jakob (1654–1705)"
843:
701:
695:
683:
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1:
2514:18th-century writers in Latin
1638:Mathematics Genealogy Project
1598:. Elsevier. pp. 88–104.
1498:J J O'Connor; E F Robertson.
1256:Nagel, Fritz (11 June 2004).
815:by studying a question about
2509:18th-century apocalypticists
2499:17th-century apocalypticists
463:De ratiociniis in aleae ludo
385:Jacob Bernoulli was born in
259:Other academic advisors
2549:Scientists from Basel-Stadt
284:Other notable students
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1434:, will be owed more than
265:(epistolary correspondent)
169:Bernoulli's Golden Theorem
29:
2519:18th-century male writers
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1484:of 1685 at the bottom of
1306:Bernoulli, Jakob (2006).
1285:. Springer. p. 639.
1236:Jacob (Jacques) Bernoulli
1202:(3rd ed.). Longman.
1171:[ˈjaːkɔbbɛrˈnʊli]
787:resulted from this work.
405:. He traveled throughout
339:Gottfried Wilhelm Leibniz
302:
293:
263:Gottfried Wilhelm Leibniz
182:
165:Bernoulli random variable
50:
2534:Burials at Basel Münster
1678:Weisstein, Eric Wolfgang
1659:University of St Andrews
1244:University of St Andrews
792:mathematical probability
623:the limit of this series
1502:. St Andrews University
748:lemniscate of Bernoulli
643:separation of variables
177:Lemniscate of Bernoulli
2120:Nicolaus IV Bernoulli
1590:Grattan-Guinness, Ivor
1580:Schneider, I. (2005).
1167:Swiss Standard German:
1014:his wife for 20 years,
967:
888:
766:
721:
627:the exponential series
621:was the first to find
611:
570:
442:
355:calculus of variations
173:Bernoulli's inequality
161:Bernoulli distribution
153:Hidden Bernoulli model
2554:Probability theorists
1980:Nicolaus II Bernoulli
1398:, debebitur plu quam
1222:Mangold, Max (1990).
1087:De gravitate aetheris
1041:De gravitate aetheris
962:
889:
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391:Old Swiss Confederacy
254:(1676 thesis advisor)
129:Bernoulli polynomials
2559:Swiss mathematicians
2456:Hans Benno Bernoulli
2113:Daniel II Bernoulli
2105:Johann III Bernoulli
1971:Nicolaus I Bernoulli
1645:Robertson, Edmund F.
1628:at Wikimedia Commons
977:Eadem mutata resurgo
832:
660:
584:
550:
369:law of large numbers
278:Nicolaus I Bernoulli
2447:Elisabeth Bernoulli
1998:Johann II Bernoulli
1860:Nicolaus Bernoulli
1770:Nicolaus Bernoulli
1665:Bernoulli, Jacobi.
1643:O'Connor, John J.;
1534:. pp. 116–17.
1482:Journal des Sçavans
1370:Journal des Savants
970:Bernoulli wanted a
923:is the number that
763:Fondazione Mansutti
498:University of Basel
447:University of Basel
208:University of Basel
125:Bernoulli's formula
105:University of Basel
2128:Jakob II Bernoulli
1701:2016-04-06 at the
1420:& minus quam 3
1258:"Bernoulli, Jacob"
987:Archimedean spiral
972:logarithmic spiral
968:
884:
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740:logarithmic spiral
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607:
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477:Frans van Schooten
459:Christiaan Huygens
443:
157:Bernoulli sampling
149:Bernoulli operator
95:Basel, Switzerland
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2475:
2471:
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1649:"Jacob Bernoulli"
1624:Media related to
1605:978-0-08-045744-4
1582:"Jakob Bernoulli
1572:978-0-684-10114-9
1335:"Jacob Bernoulli"
1319:978-3-7643-1891-8
1292:978-3-030-17801-7
1209:978-1-4058-8118-0
943:compound interest
903:force of interest
871:
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817:compound interest
781:Bernoulli numbers
746:around 1692. The
605:
564:
525:The lunar crater
493:were discovered.
312:
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270:Doctoral students
184:Scientific career
141:Bernoulli process
121:Bernoulli numbers
16:(Redirected from
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2544:Number theorists
2529:Bernoulli family
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1989:Daniel Bernoulli
1872:Johann Bernoulli
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1739:Bernoulli family
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337:. He sided with
335:Bernoulli family
306:Johann Bernoulli
288:Johann Bernoulli
246:Doctoral advisor
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145:Bernoulli scheme
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2427:Maria Bernoulli
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1852:Jacob Bernoulli
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1703:Wayback Machine
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1634:Jacob Bernoulli
1626:Jakob Bernoulli
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1584:Ars conjectandi
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1553:Further reading
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1456:and less than 3
1447:
1444:
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1437:
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1411:
1408:
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1404:
1401:
1399:
1380:, pp. 219–23.
1378:Acta eruditorum
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906:
856:
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851:
830:
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820:
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785:Bernoulli trial
772:Ars Conjectandi
759:Ars conjectandi
704:
663:
658:
657:
634:Acta Eruditorum
595:
582:
581:
548:
547:
543:infinite series
535:
533:Important works
516:von Tschirnhaus
511:Acta Eruditorum
508:" published in
490:Ars Conjectandi
435:Acta Eruditorum
383:
374:Ars Conjectandi
358:
317:(also known as
315:Jacob Bernoulli
308:
276:
253:
251:Peter Werenfels
240:
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46:
45:Jacob Bernoulli
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18:James Bernoulli
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1614:External links
1612:
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1571:
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1547:
1540:
1532:Broadway Books
1513:
1500:"The number e"
1490:
1476:, so 2.5 <
1360:
1351:
1325:
1318:
1298:
1291:
1271:
1248:
1228:
1215:
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1196:Wells, John C.
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974:and the motto
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578:Pietro Mengoli
563:
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411:Johannes Hudde
382:
379:
331:mathematicians
321:in English or
310:
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114:
113:Known for
110:
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98:
97:
94:
92:(aged 50)
88:16 August 1705
86:
82:
81:
72:
70:6 January 1655
61:
57:
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47:
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26:
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1541:0-7679-0816-3
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1153:
1119:
1116:
1109:
1104:, vol 1, 1744
1103:
1097:
1092:
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1004:
1001:
998:
995:
994:
993:
990:
988:
983:
979:
978:
973:
966:
965:Basel Münster
961:
954:
952:
949:
945:, will yield
944:
941:) dollars at
936:
931:
926:
921:
915:
909:
904:
899:
879:
874:
868:
865:
860:
857:
853:
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828:
827:
826:
823:
818:
813:
805:
801:
799:
795:
793:
788:
786:
782:
778:
777:combinatorics
774:
773:
764:
760:
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749:
745:
741:
737:
733:
714:
709:
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366:
361:
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348:
344:
340:
336:
333:in the Swiss
332:
328:
324:
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316:
307:
301:
296:
292:
289:
286:
282:
279:
275:
274:Jacob Hermann
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257:
252:
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247:
243:
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185:
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142:
138:
134:
133:Bernoulli map
130:
126:
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118:
115:
111:
106:
103:
99:
87:
83:
79:
75:
62:
58:
54:
49:
42:
37:
33:
19:
1851:
1687:ScienceWorld
1685:
1652:
1594:
1583:
1560:
1526:
1522:Livio, Mario
1516:
1504:. Retrieved
1493:
1481:
1477:
1473:
1469:
1465:
1461:
1457:
1453:
1431:
1427:
1423:
1419:
1397:
1393:
1389:
1385:
1377:
1373:
1369:
1363:
1354:
1342:. Retrieved
1328:
1308:
1301:
1281:
1274:
1262:. Retrieved
1251:
1231:
1223:
1218:
1199:
1190:
1118:
1101:
1086:
1066:
1056:
1048:
1040:
1029:
991:
982:self-similar
975:
969:
947:
934:
929:
927:later named
919:
913:
907:
900:
896:
821:
808:
803:
796:
789:
770:
768:
758:
729:
633:
631:
540:
536:
524:
520:
509:
495:
488:
481:Isaac Barrow
472:La Géométrie
470:
462:
455:
450:
444:
439:
433:
419:Robert Hooke
415:Robert Boyle
384:
372:
371:in his work
322:
318:
314:
313:
231:
222:
204:Institutions
183:
90:(1705-08-16)
31:
2494:1705 deaths
2489:1655 births
2459:(1876–1959)
2450:(1873–1935)
2437:(1868–1963)
2430: [
2421:(1877–1962)
2326:(1837–1909)
2311:(1824–1913)
2210:(1793–1876)
2131:(1759–1789)
2122:(1754–1841)
2115:(1751–1834)
2108:(1744–1807)
2001:(1710–1790)
1992:(1700–1782)
1983:(1695–1726)
1974:(1687–1759)
1875:(1667–1748)
1862:(1662–1716)
1855:(1654–1705)
1772:(1623–1708)
744:epicycloids
485:John Wallis
432:Image from
399:mathematics
365:probability
341:during the
304:Brother of
194:Mathematics
78:Switzerland
2483:Categories
1506:2 November
1182:References
935:continuous
66:1655-01-06
2208:Nicolaus
1524:(2003) .
1486:page 314.
1382:On p. 222
1124:English:
955:Tombstone
847:∞
844:→
638:isochrone
588:∑
554:∑
527:Bernoulli
467:Descartes
403:astronomy
381:Biography
198:mechanics
101:Education
2324:Theodor
1699:Archived
1436:2
1400:2
1198:(2008).
736:parabola
732:evolutes
668:′
647:integral
395:theology
347:calculus
1680:(ed.).
1636:at the
1592:(ed.).
1586:(1713)"
1460:.) If
1451:
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933:; with
389:in the
323:Jacques
2309:Fritz
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1344:20 May
1316:
1289:
1264:20 May
1206:
1089:, 1683
738:, the
423:comets
417:, and
407:Europe
351:Johann
236:(1684)
234:
227:(1676)
225:
215:Theses
190:Fields
34:, see
2434:]
1670:(PDF)
1588:. In
1338:(PDF)
1246:, UK.
1110:Notes
1102:Opera
1067:Opera
1057:Opera
1022:Works
925:Euler
619:Euler
387:Basel
319:James
298:Notes
74:Basel
32:Jacob
1741:tree
1600:ISBN
1567:ISBN
1536:ISBN
1508:2016
1346:2016
1314:ISBN
1287:ISBN
1266:2016
1204:ISBN
1162:-lee
1158:bur-
742:and
483:and
475:and
401:and
327:O.S.
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