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418:
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42:
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783:, 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.
1377:(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:
445:
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
629:
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
886:
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 = $
510:
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
526:
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.
764:, 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
1356:
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
973:
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
534:
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
786:
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.
1375:"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?"
881:
507:. 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.
739:
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.
772:
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
714:
604:
563:
1028:
1017:
894:) for more and smaller compounding intervals. Compounding weekly yields $ 2.692597..., while compounding daily yields $ 2.714567..., just two cents more. Using
751:
1687:
723:
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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398:
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
887:
2.25. Compounding quarterly yields $ 1.00×1.25 = $ 2.4414..., and compounding monthly yields $ 1.00×(1.0833...) = $ 2.613035....
221:
Solutionem tergemini problematis arithmetici, geometrici et astronomici... (Solution to a triple problem in arithmetics, geometry and astronomy...)
1084:
476:, 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
1718:
493:
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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1550:
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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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105:
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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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1469:< 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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634:. Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term
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8:
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1986:
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and his two children have erected a monument to the husband and father they miss so much.
969:('Although changed, I rise again the same') engraved on his tombstone. He wrote that the
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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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1059:(in Latin). Vol. 2. Genève: héritiers Cramer & frères Philibert. 1744.
1049:(in Latin). Vol. 1. Genève: héritiers Cramer & frères Philibert. 1744.
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member of the Royal Academies of Paris and Berlin; famous for his writings.
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1024:(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
1329:. 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
876:{\displaystyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}}
455:
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186:
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D. J. Struik (1986) A Source Book In Mathematics, 1200-1800, p. 320
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606:, but he did show that it converged to a finite limit less than 2.
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1554:. Vol. 2. New York: Charles Scribner's Sons. pp. 46–51.
338:, which he made numerous contributions to; along with his brother
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1301:(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
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352:. 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
625:, 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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410:. During this time he also produced an incorrect theory of
346:. He also discovered the fundamental mathematical constant
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Die Werke von Jakob Bernoulli: Bd. 2: Elementarmathematik
1033:(in Latin). Amstelaedami: apud Henricum Wetstenium. 1683.
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is also named after him jointly with his brother Johann.
427:(1682) wherein was published the critique of Bernoulli's
1022:(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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1379:" ... quæ nostra serie &c. major est. ... si
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1584:Landmark Writings in Western Mathematics 1640–1940
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1685:Correspondence Regarding the Art of Conjecturing"
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1519:(First trade paperback ed.). New York City:
1457:, the geometric series reduces to the series for
485:He was appointed professor of mathematics at the
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1215:. 3. Auflage. Mannheim/Wien/Zürich, Dudenverlag.
1040:, Basileae, impensis Thurnisiorum Fratrum, 1713.
985:Jacob Bernoulli, the incomparable mathematician.
825:
1271:Jan Hendrik Oort: Master of the Galactic System
994:Of a chronic illness, of sound mind to the end;
618:which came out of examining compound interest.
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978:was engraved rather than a logarithmic one.
530:By 1689, he had published important work on
356:, where he derived the first version of the
1415:( ... which our series is larger . ... if
1229:The MacTutor History of Mathematics archive
798:In 1683, Bernoulli discovered the constant
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382:. Following his father's wish, he studied
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2528:Members of the French Academy of Sciences
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334:and was an early proponent of Leibnizian
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1231:, School of Mathematics and Statistics,
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503:. They also studied the publications of
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1643:MacTutor History of Mathematics Archive
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791:Discovery of the mathematical constant
599:{\displaystyle \sum {\frac {1}{n^{2}}}}
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1683:Gottfried Leibniz and Jakob Bernoulli
906:in each interval, the limit for large
468:'s supplements of it. He also studied
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940:dollars with continuous compounding.
621:In May 1690, in a paper published in
438:from 1683. His doctoral dissertation
495:Nova Methodus pro Maximis et Minimis
342:, he was one of the founders of the
758:Bernoulli's most original work was
709:{\displaystyle y'=p(x)y+q(x)y^{n}.}
558:{\displaystyle \sum {\frac {1}{n}}}
332:Leibniz–Newton calculus controversy
97:(D.Th., 1676; Dr. phil. hab., 1684)
13:
1551:Dictionary of Scientific Biography
1541:
1322:Pfeiffer, Jeanne (November 2006).
1249:. Historisches Lexikon der Schweiz
1238:
981:Translation of Latin inscription:
835:
521:
14:
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2513:18th-century Swiss mathematicians
2493:17th-century Swiss mathematicians
1656:"Tractatus de Seriebus Infinitis"
1602:
1373:, Bernoulli poses the question:
1268:Kruit, Pieter C. van der (2019).
1019:Conamen novi systematis cometarum
429:Conamen novi systematis cometarum
1608:
1365:), in the year (anno) 1685.**),
1363:Ephemerides Eruditorum Gallicanæ
1213:Duden — Das Aussprachewörterbuch
1189:Longman Pronunciation Dictionary
1115:
1083:
1068:
614:in 1737. Bernoulli also studied
440:Solutionem tergemini problematis
314:in French; 6 January 1655 [
212:Primi et Secundi Adami Collatio
1503:
1480:
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1038:Ars conjectandi, opus posthumum
952:Jacob Bernoulli's tombstone in
640:Bernoulli differential equation
106:Bernoulli differential equation
19:For other family members named
16:Swiss mathematician (1655–1705)
1671:"Bernoulli, Jakob (1654–1705)"
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2503:18th-century writers in Latin
1627:Mathematics Genealogy Project
1587:. Elsevier. pp. 88–104.
1487:J J O'Connor; E F Robertson.
1245:Nagel, Fritz (11 June 2004).
1170:
804:by studying a question about
2498:18th-century apocalypticists
2488:17th-century apocalypticists
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452:De ratiociniis in aleae ludo
374:Jacob Bernoulli was born in
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248:Other academic advisors
7:
2538:Scientists from Basel-Stadt
273:Other notable students
10:
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1423:, will be owed more than
254:(epistolary correspondent)
158:Bernoulli's Golden Theorem
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2508:18th-century male writers
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1473:of 1685 at the bottom of
1295:Bernoulli, Jakob (2006).
1274:. Springer. p. 639.
1225:Jacob (Jacques) Bernoulli
1191:(3rd ed.). Longman.
1160:[ˈjaːkɔbbɛrˈnʊli]
776:resulted from this work.
394:. He traveled throughout
328:Gottfried Wilhelm Leibniz
291:
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252:Gottfried Wilhelm Leibniz
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154:Bernoulli random variable
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2523:Burials at Basel Münster
1667:Weisstein, Eric Wolfgang
1648:University of St Andrews
1233:University of St Andrews
1098:
1010:
781:mathematical probability
612:the limit of this series
1491:. St Andrews University
737:lemniscate of Bernoulli
632:separation of variables
166:Lemniscate of Bernoulli
2109:Nicolaus IV Bernoulli
1579:Grattan-Guinness, Ivor
1569:Schneider, I. (2005).
1156:Swiss Standard German:
1003:his wife for 20 years,
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877:
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710:
616:the exponential series
610:was the first to find
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559:
431:
344:calculus of variations
162:Bernoulli's inequality
150:Bernoulli distribution
142:Hidden Bernoulli model
2543:Probability theorists
1969:Nicolaus II Bernoulli
1387:, debebitur plu quam
1211:Mangold, Max (1990).
1076:De gravitate aetheris
1030:De gravitate aetheris
951:
878:
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380:Old Swiss Confederacy
243:(1676 thesis advisor)
118:Bernoulli polynomials
2548:Swiss mathematicians
2445:Hans Benno Bernoulli
2102:Daniel II Bernoulli
2094:Johann III Bernoulli
1960:Nicolaus I Bernoulli
1634:Robertson, Edmund F.
1617:at Wikimedia Commons
966:Eadem mutata resurgo
821:
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358:law of large numbers
267:Nicolaus I Bernoulli
2436:Elisabeth Bernoulli
1987:Johann II Bernoulli
1849:Nicolaus Bernoulli
1759:Nicolaus Bernoulli
1654:Bernoulli, Jacobi.
1632:O'Connor, John J.;
1523:. pp. 116–17.
1471:Journal des Sçavans
1359:Journal des Savants
959:Bernoulli wanted a
912:is the number that
752:Fondazione Mansutti
487:University of Basel
436:University of Basel
197:University of Basel
114:Bernoulli's formula
94:University of Basel
2117:Jakob II Bernoulli
1690:2016-04-06 at the
1409:& minus quam 3
1247:"Bernoulli, Jacob"
976:Archimedean spiral
961:logarithmic spiral
957:
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729:logarithmic spiral
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466:Frans van Schooten
448:Christiaan Huygens
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146:Bernoulli sampling
138:Bernoulli operator
84:Basel, Switzerland
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2464:
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1638:"Jacob Bernoulli"
1613:Media related to
1594:978-0-08-045744-4
1571:"Jakob Bernoulli
1561:978-0-684-10114-9
1324:"Jacob Bernoulli"
1308:978-3-7643-1891-8
1281:978-3-030-17801-7
1198:978-1-4058-8118-0
932:compound interest
892:force of interest
860:
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806:compound interest
770:Bernoulli numbers
735:around 1692. The
594:
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514:The lunar crater
482:were discovered.
301:
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259:Doctoral students
173:Scientific career
130:Bernoulli process
110:Bernoulli numbers
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2533:Number theorists
2518:Bernoulli family
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1978:Daniel Bernoulli
1861:Johann Bernoulli
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491:Johann Bernoulli
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326:. He sided with
324:Bernoulli family
295:Johann Bernoulli
277:Johann Bernoulli
235:Doctoral advisor
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134:Bernoulli scheme
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1692:Wayback Machine
1658:
1623:Jacob Bernoulli
1615:Jakob Bernoulli
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1573:Ars conjectandi
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1542:Further reading
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532:infinite series
524:
522:Important works
505:von Tschirnhaus
500:Acta Eruditorum
497:" published in
479:Ars Conjectandi
424:Acta Eruditorum
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363:Ars Conjectandi
347:
306:(also known as
304:Jacob Bernoulli
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1521:Broadway Books
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1349:
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567:Pietro Mengoli
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400:Johannes Hudde
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310:in English or
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81:(aged 50)
77:16 August 1705
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59:6 January 1655
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954:Basel Münster
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941:
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934:, will yield
933:
930:) dollars at
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766:combinatorics
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1676:ScienceWorld
1674:
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1515:
1511:Livio, Mario
1505:
1493:. Retrieved
1482:
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1331:. Retrieved
1317:
1297:
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1263:
1251:. Retrieved
1240:
1220:
1212:
1207:
1188:
1179:
1107:
1090:
1075:
1055:
1045:
1037:
1029:
1018:
980:
971:self-similar
964:
958:
936:
923:
918:
916:later named
908:
902:
896:
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885:
810:
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792:
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759:
757:
747:
718:
622:
620:
529:
525:
513:
509:
498:
484:
477:
470:Isaac Barrow
461:La Géométrie
459:
451:
444:
439:
433:
428:
422:
408:Robert Hooke
404:Robert Boyle
373:
361:
360:in his work
311:
307:
303:
302:
220:
211:
193:Institutions
172:
79:(1705-08-16)
20:
2483:1705 deaths
2478:1655 births
2448:(1876–1959)
2439:(1873–1935)
2426:(1868–1963)
2419: [
2410:(1877–1962)
2315:(1837–1909)
2300:(1824–1913)
2199:(1793–1876)
2120:(1759–1789)
2111:(1754–1841)
2104:(1751–1834)
2097:(1744–1807)
1990:(1710–1790)
1981:(1700–1782)
1972:(1695–1726)
1963:(1687–1759)
1864:(1667–1748)
1851:(1662–1716)
1844:(1654–1705)
1761:(1623–1708)
733:epicycloids
474:John Wallis
421:Image from
388:mathematics
354:probability
330:during the
293:Brother of
183:Mathematics
67:Switzerland
2472:Categories
1495:2 November
1171:References
924:continuous
55:1655-01-06
2197:Nicolaus
1513:(2003) .
1475:page 314.
1371:On p. 222
1113:English:
944:Tombstone
836:∞
833:→
627:isochrone
577:∑
543:∑
516:Bernoulli
456:Descartes
392:astronomy
370:Biography
187:mechanics
90:Education
2313:Theodor
1688:Archived
1425:2
1389:2
1187:(2008).
725:parabola
721:evolutes
657:′
636:integral
384:theology
336:calculus
1669:(ed.).
1625:at the
1581:(ed.).
1575:(1713)"
1449:.) If
1440:
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922:; with
378:in the
312:Jacques
2298:Fritz
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1333:20 May
1305:
1278:
1253:20 May
1195:
1078:, 1683
727:, the
412:comets
406:, and
396:Europe
340:Johann
225:(1684)
223:
216:(1676)
214:
204:Theses
179:Fields
23:, see
2423:]
1659:(PDF)
1577:. In
1327:(PDF)
1235:, UK.
1099:Notes
1091:Opera
1056:Opera
1046:Opera
1011:Works
914:Euler
608:Euler
376:Basel
308:James
287:Notes
63:Basel
21:Jacob
1730:tree
1589:ISBN
1556:ISBN
1525:ISBN
1497:2016
1335:2016
1303:ISBN
1276:ISBN
1255:2016
1193:ISBN
1151:-lee
1147:bur-
731:and
472:and
464:and
390:and
316:O.S.
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