ياكوب برنولي

لمعلومات عن other family members named Jacob، انظر Bernoulli family.

ياكوب برنولي
Jacob Bernoulli
ياكوب برنولي
وُلِدَ	(1654-12-27)27 ديسمبر 1654 بازل، سويسرا
توفي	16 أغسطس 1705(1705-08-16) (aged 50) بازل، سويسرا
القومية	سويسري
التعليم	University of Basel (D.Th., 1676; Dr. phil. hab., 1684)
عـُرِف بـ	معادلة برنولي التفاضلية عدد برنولي (صيغة برنولي عديدات حدود برنولي Bernoulli map) محاولة برنولي (عملية برنولي Bernoulli scheme Bernoulli operator نموذج برنولي الخفي Bernoulli sampling توزيع برنولي Bernoulli random variable مبرهنة برنولي الذهبية) Bernoulli's inequality Lemniscate of Bernoulli
السيرة العلمية
المجالات	عالم رياضيات
الهيئات	جامعة بازل
الأطروحات	Primi et Secundi Adami Collatio (1676) Solutionem tergemini problematis arithmetici, geometrici et astronomici... (Solution to a triple problem in arithmetics, geometry and astronomy...) (1684)
المشرف على الدكتوراه	Peter Werenfels (1676 thesis advisor)
مشرفون أكاديميون آخرون	گوتفريد لايبنتس (بالمراسلة)
طلاب الدكتوراه	يوهان برنولي ياكوب هرمان نيكولاوس الأول برنولي
طلاب بارزون آخرون	Johann Bernoulli
ملاحظات
شقيق يوهان برنولي.

لأشخاص آخرين في العائلة باسم ياكوب، انظر عائلة برنولي.

ياكوب برنولي (أحيانا جيمس ، أو جاك) (و. 27 ديسمبر 1654، بازل - 16 أغسطس 1705)، كان واحدا عدة علماء رياضيات بارزين في عائلة برنولي. He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibnizian calculus, to which he made numerous contributions. A member of the Bernoulli family, he, along with his brother Johann, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant e. However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi.

تحقيقا لرغبة والده، درس ياكوب علم اللاهوت والتحق بالكهنوت. لكنه خلافا لرغبة والديه درس أيضا الرياضيات والفلك. تجول في اوروبا في الفترة من 1676 - 1682 ، واطلع على أحدث الاكتشافات في الرياضيات والعلوم. وشمل ذلك أعمال روبرت بويل، وروبرت هوك.

وكان له خمس بنات وولدين.

 السيرة

Jacob Bernoulli was born in Basel in the Swiss Confederation; the son and grandson of Protestant spice merchants on his father's side. His mother was born into a family engaged in banking and city governing.

Following his father's wish, he studied theology and entered the ministry. But contrary to the desires of his parents, he also studied mathematics and astronomy. He traveled throughout Europe 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 Johannes Hudde, Robert Boyle, and Robert Hooke. During this time he also produced an incorrect theory of comets.

ملف:Acta Eruditorum - X astronomia, 1682 – BEIC 13349171.jpg

Image from Acta Eruditorum (1682) wherein was published the critique of Bernoulli's Conamen novi systematis cometarum

Bernoulli returned to Switzerland, and began teaching mechanics at the University of Basel from 1683. His doctoral dissertation Solutionem tergemini problematis was submitted in 1684. It appeared in print in 1687.

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 studied the new discoveries in mathematics, including Christiaan Huygens's De ratiociniis in aleae ludo, Descartes' La Géométrie and Frans van Schooten's supplements of it. He also studied Isaac Barrow and John Wallis, 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 Ars Conjectandi were discovered.

People believe he was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on the differential calculus in "Nova Methodus pro Maximis et Minimis" published in Acta Eruditorum. They also studied the publications of von Tschirnhaus. 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.

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 challenges to test each other's skills. By 1697, the relationship had completely broken down.

The lunar crater Bernoulli is also named after him jointly with his brother Johann.

 الأعمال الهامة

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.

By 1689, he had published important work on infinite series 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 $\sum {\frac {1}{n}}$ diverges, which Bernoulli believed were new but had actually been proved by Pietro Mengoli 40 years earlier and was proved by Nicole Oresme in the 14th century already. Bernoulli could not find a closed form for $\sum {\frac {1}{n^{2}}}$, but he did show that it converged to a finite limit less than 2. Euler was the first to find the limit of this series in 1737. Bernoulli also studied the exponential series which came out of examining compound interest.

In May 1690, in a paper published in Acta Eruditorum, Jacob Bernoulli showed that the problem of determining the isochrone 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 in 1687 and Leibniz in 1689. After finding the differential equation, Bernoulli then solved it by what we now call separation of variables. Jacob Bernoulli's paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called the Bernoulli differential equation,

$y'=p(x)y+q(x)y^{n}.$

Jacob Bernoulli also discovered a general method to determine evolutes 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 parabola, the logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli 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.

ملف:Bernoulli - Ars conjectandi, 1713 - 058.tif

Ars conjectandi, 1713 (Milano, Fondazione Mansutti).

Bernoulli's most original work was Ars Conjectandi, 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 combinatorics, in particular the work of van Schooten, Leibniz, and Prestet, as well as the use of Bernoulli numbers 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 Bernoulli trial resulted from this work.

In the last part of the book, Bernoulli sketches many areas of mathematical probability, including probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori and 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.

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.

أكثر ما اشتهر به ياكوب برنولي هو كتابه فن الحدس Ars Conjectandi، الذي نُشر بعد ثمان سنوات من وفاته في 1713 من قِبل ابن أخيه نيكولاوس. ففي هذا العمل، وصف النتائج المعروفة في نظرية الاحتمالات وفي الترقيم، وكثيراً ما يُعطي براهين بديلة لنتائج معروفة. كما ضم هذا العمل تطبيق نظرية الاحتمالات في ألعاب الفرص وتقديمة مبرهنة عُرفت بإسم قانون الأعداد الكبيرة. المصطلحات محاولة برنولي وأعداد برنولي نتجت من هذا العمل. الفوهة القمرية برنولي مسماة على اسمه بالتشارك مع شقيقة يوهان.

 اكتشاف الثابت الرياضي e

اكتشف برنولي الثابت e بدراسة السؤال حول الفائدة المركبة التي تطلبت منه أن يجد قيمة التعبير التالي (الذي هو في الواقع e):

$e=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$

أحد الأمثلة على ذلك هو حساب مصرفي يبدأ بمبلغ $1.00 دولار ويدفع 100% فائدة في السنة. لو الفائدة أضيفت مرة واحدة، في نهاية العام، فإن القيمة تصبح $2.00 دولار؛ ولكن إذا حـُسِبت الفائدة وأضيفت مرتين في السنة، فإن مبلغ $1 دولار يـُضرّب في 1.5 مرتين، فينتج عنه $1.00×1.5² = $2.25. وإذا أضفنا الفائدة فصلياً (كل ربع سنة)، فإن المبلغ بنهاية السنة يكون $1.00×1.25⁴ = $2.4414...، وبتحصيل الفائدة شهرياً يصبح المبلغ $1.00×(1.0833...)¹² = $2.613035....

لاحظ برنولي أن هذا التسلسل يقترب من نهاية (قوة الفائدة) كلما ازدادت عدد وقصرت مدد الفترات. فالاضافة الأسبوعية للفائدة تنتج $2.692597...، بينما الاضافة اليومية للفائدة تنتج $2.714567...، أي بزيادة سنتين فقط. باستخدام n كعدد فترات التركيب، بفائدة 100%/n في كل فترة، فإن النهاية لقيمة n كبيرة تكون هي الرقم الذي أصبح معروفاً باسم e؛ فبتركيب متصل، ستصل قيمة الحساب المصرفي إلى $2.7182818.... وبصيغة عمومية، فإن الحساب المصرفي الذي يبدأ بمبلغ $1 دولار، وينمو بفائدة (1+R) دولاراً في فائدة بسيطة، سينتج e^R دولار بالتضاعف المركب.

ضريح برنولي

 Tombstone

Jacob Bernoulli's tombstone in Basel Münster

Bernoulli wanted a logarithmic spiral and the motto Eadem mutata resurgo ('Although changed, I rise again the same') engraved on his tombstone. He wrote that the self-similar 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 Archimedean spiral was engraved rather than a logarithmic one.

Translation of Latin inscription:

Jacob Bernoulli, the incomparable mathematician.

Professor at the University of Basel For more than 18 years;

member of the Royal Academies of Paris and Berlin; famous for his writings.

Of a chronic illness, of sound mind to the end;

succumbed in the year of grace 1705, the 16th of August, at the age of 50 years and 7 months, awaiting the resurrection.

Judith Stupanus,

his wife for 20 years,

and his two children have erected a monument to the husband and father they miss so much.

 Works

• Conamen novi systematis cometarum (in اللاتينية). Amstelaedami: apud Henr. Wetstenium. 1682. (title roughly translates as "A new hypothesis for the system of comets".)
• De gravitate aetheris (in اللاتينية). Amstelaedami: apud Henricum Wetstenium. 1683.
• Ars conjectandi, opus posthumum, Basileae, impensis Thurnisiorum Fratrum, 1713.
• Opera (in اللاتينية). Vol. 1. Genève: héritiers Cramer & frères Philibert. 1744.
  • Opera (in اللاتينية). Vol. 2. Genève: héritiers Cramer & frères Philibert. 1744.

• Bernoulli - De gravitate aetheris, 1683 - 1216514.jpg

  De gravitate aetheris, 1683

• Bernoulli, Jakob – Opera, vol 1, 1744 – BEIC 12199963.jpg

  Opera, vol 1, 1744

 قراءات إضافية

• Hoffman, J.E. (1970–80). "Bernoulli, Jakob (Jacques) I". Dictionary of Scientific Biography. Vol. 2. New York: Charles Scribner's Sons. pp. 46–51. ISBN 0684101149. {{cite encyclopedia}}: Cite has empty unknown parameter: |coauthors= (help)CS1 maint: date format (link)
• Schneider, I., 2005, "Ars conjectandi" in Grattan-Guiness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 88-104.
• Livio, Mario, 2002, The golden ratio: the story of Phi, the extraordinary number of nature, art, and beauty. London.

 وصلات خارجية

• ياكوب برنولي at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "ياكوب برنولي", MacTutor History of Mathematics archive 
• Jakob Bernoulli: Tractatus de Seriebus Infinitis (pdf)
• Weisstein, Eric W., Bernoulli, Jakob (1654-1705) at ScienceWorld.