Leonhard Euler: Perbedaan antara revisi
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{{Redirect|Euler}}
{{Infobox scientist
|name = Leonhard Euler
|image = Leonhard Euler 2.jpg
|image_size = 220px
|caption = Lukisan oleh Johann Georg Brucker (1756)
|
|birth_place = [[Basel]], [[Konfederasi Swiss Lama|Switzerland]]
|death_date = {{death date and age|1783|9|18|1707|4|15|df=y}}<br /><small><nowiki>[</nowiki>[[:en:Old Style and New Style dates|OS]]: 7 September 1783<nowiki>]</nowiki></small>
|
|residence = [[Kerajaan Prusia]], [[Kekaisaran Rusia]]<br /> [[Konfederasi Swiss Lama|Switzerland]]
|nationality = [[Konfederasi Swiss Lama|Swiss]]
|
|work_institutions = [[Akademi Ilmu Pengetahuan Rusia|Akademi Ilmu Pengetahuan Imperial Rusia]]<br />[[Akademi Sains Prusia|Akademi Berlin]]
|alma_mater = [[Universitas Basel]]
|doctoral_advisor = [[Johann Bernoulli]]
|doctoral_students = [[:en:Nicolas Fuss|Nicolas Fuss]]<br />[[:En:Johann Hennert|Johann Hennert]]<br />[[:en:Stepan Rumovsky|Stepan Rumovsky]]
|notable_students = [[Joseph Louis Lagrange]]
|known_for = [[Daftar hal-hal yang dinamai menurut Leonhard Euler|Lihat daftar]]
|prizes =
|religion = [[Calvinisme|Calvinis]]<ref>{{cite book|title=Scientists of Faith|url=https://archive.org/details/scientistsoffait00grav|author=Dan Graves|location=Grand Rapids, MI|year=1996|publisher=Kregel Resources|pages=[https://archive.org/details/scientistsoffait00grav/page/85 85]–86}}</ref><ref>{{cite book|title=Men of Mathematics, Vol. 1|author=E. T. Bell|location=London|year=1953|publisher=Penguin|page=155}}</ref>
|footnotes = Dia adalah ayah matematikawan <br />Menurut otoritas genealogi akademik dia dianggap setara dengan penasihat doktoral Joseph Louis Lagrange.<ref>{{MathGenealogy|id=17864}}</ref>
|signature = Euler's signature.svg
}}
'''Leonhard Euler''' ({{IPA-all|ˈɔɪleːʀ|Pengucapan Jerman Swiss:|LeonhardEulerByDrsDotChRadio.ogg}}, {{IPA-all|ˈɔʏlɐ|Standar Jerman:|De-Leonard_Euler.ogg}}, {{IPA-all|ˈɔɪlɹ̩|Inggris:}}<small>, mirip dengan 'oiler'</small>;<ref>Cara pengucapan {{IPA-all|ˈjuːlər|}} tidaklah benar. "Euler", [[Oxford English Dictionary]], edisi kedua, Oxford University Press, 1989 [http://www.merriam-webster.com/dictionary/Euler "Euler"], [[:en:Webster's Dictionary|Merriam–Webster's Online Dictionary]], 2009. [http://ahdictionary.com/word/search.html?q=Euler%2C+Leonhard&submit.x=40&submit.y=16 "Euler, Leonhard"], [[The American Heritage Dictionary of the English Language]], edisi keempat, Houghton Mifflin Company, Boston, 2000. {{cite book|title=Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections|url=https://archive.org/details/netspuzzlespostm00higg|author=Peter M. Higgins|year=2007|publisher=Oxford University Press|page=[https://archive.org/details/netspuzzlespostm00higg/page/n51 43]}}</ref>
15 April 1707{{spaced ndash}}18 September 1783) adalah seorang [[matematikawan]] dan [[fisikawan]] pionir dari [[Swiss]]. Dia membuat penemuan-penemuan penting dalam bidang yang beragam seperti [[kalkulus]] dan [[teori graf]]. Dia juga mengenalkan banyak notasi dan terminologi matematika modern, terutama untuk [[analisis matematika]], seperti konsep [[Fungsi (matematika)|fungsi matematika]].<ref name="function">{{harvnb|Dunham|1999|p=17}}</ref> Dia juga dikenal melalui karyanya dalam [[mekanika]], [[dinamika fluida]], [[optik]], dan [[astronomi]]. Euler menghabiskan masa dewasanya di [[St. Petersburg]], [[Kekaisaran Rusia|Rusia]], dan di [[Berlin]], [[Kerajaan Prusia|Prusia]]. Ia dianggap sebagai matematikawan unggulan abad ke-18, dan salah satu matematikawan terhebat yang pernah ada. Dia juga merupakan salah satu matematikawan paling produktif; hasil karyanya termuat dalam 60–80 jilid kuarto.<ref name="volumes">{{cite journal|last = Finkel|first = B.F.|year = 1897|title = Biography- Leonard Euler|journal = The American Mathematical Monthly| volume = 4| issue = 12<!--| page = 300 -->|jstor = 2968971|pages = 297–302}}</ref> Sebuah ungkapan dari [[Pierre-Simon Laplace]] memperlihatkan pengaruh Euler dalam matematika: "Baca Euler, baca Euler, dia adalah master dari kita semua."<ref name="Laplace">{{harvnb|Dunham|1999|p=xiii}} "Lisez Euler, lisez Euler, c'est notre maître à tous."</ref>
== Kehidupan ==
=== Tahun-tahun awal ===
[[Berkas:Euler-10 Swiss Franc banknote (front).jpg|jmpl|Uang kertas Swiss [[Swiss Franc|10 Franc]] lama untuk mengenang Euler]]
Euler lahir di [[Basel]], 15 April 1707. Ayahnya adalah Paul Euler, seorang [[pastor|pastur]] [[Calvinisme]]. Ibunya adalah Marguerite Brucker, anak dari seorang pastur. Dia memiliki dua adik perempuan Anna Maria dan Maria Magdalena. Segera setelah kelahiran Leonhard, keluarga Euler pindah dari Basel menuju Riehen, tempat dia menjalani masa kanak-kanaknya. Paul Euler merupakan teman dari salah seorang anggota keluarga Bernoulli—[[Johann Bernoulli]], yang dianggap sebagai matematikawan Eropa terkemuka, yang nantinya menjadi pengaruh penting terhadap Leonhard muda.
Pendidikan formal Euler berawal di Basel. Di sana dia tinggal bersama nenek dari pihak ibunya. Di usianya yang ketigabelas, dia mendaftar di [[Universitas Basel]], dan pada tahun 1723 pada usia 16 tahun, dia menerima gelar ‘’Master of Philosophy’’ dengan disertasi yang membandingkan filsafat dari [[René Descartes|Descartes]] dan [[Isaac Newton|Newton]]. Setelah kelulusannya, dia mengambil les Sabtu sore dari Johann Bernoulli, yang dengan cepat menemukan bakat luar biasa dari murid barunya itu dalam matematika.<ref name="childhood">{{cite book|last= James|first= Ioan|title= Remarkable Mathematicians: From Euler to von Neumann|publisher= Cambridge|year= 2002|page=2|isbn= 0-521-52094-0}}</ref> Dari sini, Euler mempelajari [[teologi]], [[bahasa Yunani]], dan [[bahasa Ibrani]] karena desakan ayahnya, agar ia menjadi seorang pastor, tetapi Bernoulli meyakinkan Paul Euler bahwa Leonhard telah ditakdirkan untuk menjadi seorang matematikawan hebat. Pada tahun 1726, Euler merampungkan disertasi tentang [[Kecepatan suara|perambatan suara]] dengan judul ''De Sono''.<ref>[http://www.17centurymaths.com/contents/euler/e002tr.pdf Euler's Dissertation De Sono: E002. Translated & Annotated by Ian Bruce]. (PDF) . 17centurymaths.com. Retrieved on 2011-09-14.</ref> Kemudian, dia berusaha mendapatkan posisi di Universitas Basel (yang akhirnya gagal).
Pada tahun 1727, dia mengikuti kompetisi ''Paris Academy Prize Problem'' (kompetisi memecahkan masalah), yang pada saat itu tantangannya adalah menemukan cara terbaik untuk menempatkan tiang kapal pada sebuah perahu. Dia mendapat juara kedua, kalah dari [[:en:Pierre Bouguer|Pierre Bouguer]]—yang sekarang dikenal sebagai "Bapa arsitekur angkatan laut." Euler kemudian memenangkan kompetisi tahunan yang didambakan ini dua belas kali sepanjang kariernya.<ref name="prizes">{{harvnb|Calinger|1996|p=156}}</ref>
===
Sebelumnya, kedua anak Johan Bernoulli, [[Daniel Bernoulli|Daniel]] dan [[:en:Nicolaus II Bernoulli|Nicolaus]], tengah bekerja di [[Akademi Ilmu Pengetahuan Rusia|Akademi Ilmu Pengetahuan Imperial Rusia]] di [[St Petersburg]]. Kemudian pada 10 Juli 1726, Nicolaus meninggal akibat [[apendisitis]] yang telah menjangkitinya selama satu tahun di Rusia, dan saat Daniel harus mengisi posisi saudaranya di divisi matematika/fisika, dia menyarankan bahwa salah satu bagian di bidang fisiologi yang kosong ditempati oleh temannya, Euler. Pada November 1726, Euler menerima tawaran itu dengan senang hati, tetapi dia menunda kepergiannya menuju St Petersburg karena dia telah mengajukan lamaran untuk menjadi dosen fisika di Universitas Basel, yang sayangnya tidak diberikan kepadanya.<ref name="stpetersburg">{{harvnb|Calinger|1996|p=125}}</ref>
[[Berkas:Euler-USSR-1957-stamp.jpg|jmpl|kiri|Perangko tahun 1957 dari [[Uni Soviet]] untuk memperingati ulang tahun Euler ke-250. Tertulis: Sudah 250 tahun sejak kelahiran seorang matematikawan hebat, akademikus Leonhard Euler.]]
Euler tiba di [[St Petersburg|ibu kota Rusia]] pada 17 Mei 1727. Dia naik jabatan dari posisi junior di departemen kesehatan ke salah satu posisi di departemen matematika di akademi tersebut. Dia menginap di rumah Daniel Bernoulli, orang yang selalu bekerja bersamanya dalam kolaborasi yang akrab. Euler menguasai [[bahasa Rusia]] dan hidup menetap di St Petersburg. Dia juga mengambil kerja sampingan sebagai pembantu medis di [[:en:Russian Navy|Angkatan Laut Rusia]].<ref name="medic">{{harvnb|Calinger|1996|p=127}}</ref>
Akademi di St. Petersburg itu, yang didirikan oleh raja [[Pyotr I dari Rusia|Peter I]], memiliki visi memajukan pendidikan di Rusia dan menghilangkan kesenjangan ilmiah dengan dunia barat. Hasilnya, akademi tersebut secara khusus menjadi perhatian para sarjana asing seperti Euler. Akademi tersebut memiliki sumber daya keuangan yang mencukupi dan sebuah perpustakaan yang lengkap yang meniru perpustakan pribadi Peter dan juga seperti perpustakaan peribadi milik kaum bangsawan lain. Hanya beberapa murid yang mendaftar di akademi tersebut untuk menjadi pengajar di fakultas yang ada, dan akademi tersebut menekankan terhadap pengadaan riset dan memberikan waktu dan kebebasan kepada fakultas-fakultasnya untuk mengikuti berbagai pertanyaan ilmiah.<ref name="prize">{{harvnb|Calinger|1996|p=124}}</ref>
[[Yekaterina I dari Rusia|Catherine I]], seorang donatur wanita dari akademi tersebut yang telah meneruskan kebijakan progressif mendiang suaminya, meninggal pada hari kedatangan Euler. Kemudian para bangsawan Rusia mendapat kekuasaan lebih atas naik tahtanya [[Pyotr II dari Rusia|Peter II]] yang baru berumur dua belas tahun. Para bangsawan pun menaruh curiga pada para ilmuwan asing yang mengajar di akademi tersebut, dan oleh karena itu mereka memotong pemberian. Hal ini memunculkan kesulitan lain bagi Euler dan koleganya.
Kondisi ini perlahan mulai membaik setelah kematian Peter II, dan Euler segera naik pangkat menjadi profesor fisika pada tahun 1731. Dua tahun kemudian, Daniel Bernoulli, yang sudah tidak betah dengan adanya pembatasan dan permusuhan yang ia dapati di St. Petersburg, pergi menuju Basel. Euler menggantikannya sebagai kepala departemen matematika.<ref name="promotion">{{harvnb|Calinger|1996|pp=128–9}}</ref>
Pada tanggal 7 Januari 1734, dia menikahi Katharina Gsell (1707-1773), putri dari [[:en:Georg Gsell|Georg Gsell]], seorang pelukis pada ''Academy Gymnasium''.<ref>{{Cite book|first1=I.R.|last1=Gekker|first2=A.A.|last2=Euler|chapter=Leonhard Euler's family and descendants|chapterurl=http://books.google.com/books?id=Ta9bz1wv79AC&pg=PA402|title={{harvnb|Bogoli︠u︡bov|Mikhaĭlov|I︠U︡shkevich|2007|page=402}}|ref={{harvid|Gekker|Euler|2007}}}}</ref> Pasangan muda ini membeli rumah dekat [[Sungai Neva]]. Dari ketigabelas anak mereka, hanya lima anak yang berhasil hidup melampaui masa kanak-kanak.<ref name="wife">{{cite web| url=http://www-history.mcs.st-and.ac.uk/~history/Extras/Euler_Fuss_Eulogy.html| title = Eulogy of Euler by Fuss| accessdate =30 August 2006| last = Fuss| first = Nicolas}}</ref>
=== Berlin ===
Karena kerusuhan terus menerus di Rusia, Euler meninggalkan St. Petersburg pada tanggal 19 Juni 1741 untuk menduduki jabatan pada ''[[Akademi Sains Prusia|Akademi Berlin]]'', yang ditawarkan kepadanya oleh [[Friedrich II dari Prusia]]. Ia tinggal 25 tahun di [[Berlin]], di mana ia menulis lebih dari 380 articles. Di Berlin, ia menerbitkan dua karya yang membuatnya sangat terkenal: ''[[:en:Introductio in analysin infinitorum|Introductio in analysin infinitorum]]'', suatu teks mengenai fungsi-fungsi matematika diterbitkan pada tahun 1748, dan ''[[:en:Institutiones calculi differentialis|Institutiones calculi differentialis]]'',<ref name=dartm/> diterbitkan pada tahun 1755 mengenai [[kalkulus diferensial]].<ref name="Friedrich"/> Pada tahun 1755, ia diangkat sebagai anggota orang asing (bukan orang Swedia) pada [[Akademi Ilmu Pengetahuan Kerajaan Swedia]]
Lebih lanjut, Euler diminta untuk menjadi tutor bagi [[:en:Friederike Charlotte of Brandenburg-Schwedt|Friederike Charlotte dari Brandenburg-Schwedt]], putri bangsawan [[:en:Anhalt-Dessau|Anhalt-Dessau]], yang adalah keponakan perempuan Frederick. Euler menulis lebih dari 200 surat kepadanya pada awal tahun 1760-an, yang kemudian dikumpulkan menjadi suatu volume terlaris berjudul ''[[:en:Letters to a German Princess|Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess]]''.<ref name='Digital Copy of "Letters to a German Princess"'/> Karya ini memuat penjelasan Euler mengenai berbagai topik fisika dan matematika, selain juga pandangan berharga mengenai kepribadian dan kepercayaan agama Euler. Buku ini menjadi lebih banyak dibaca daripada karya-karya matematikanya, dan diterbitkan di seluruh Eropa dan Amerika Serikat. Kepopuleran buku ini membuktikan kemampuan Euler untuk menyampaikan hal-hal ilmiah secara efektif bagi orang awam, suatu kemampuan yang jarang ditemukan pada ilmuwan-ilmuwan peneliti yang berdedikasi.<ref name="Friedrich"/>
Meskipun Euler banyak berkontribusi bagi prestasi Akademi Berlin, ia akhirnya tidak disukai oleh [[Friedrich II dari Prusia|Friedrich]] dan harus meninggalkan Berlin untuk kembali ke St. Petersburg.
[[Berkas:Leonhard Euler.jpg|jmpl|Lukisan portret Euler pada tahun 1753 karya [[:en:Emanuel Handmann|Emanuel Handmann]], menunjukkan masalah pada mata kanan Euler, kemungkinan [[:en:strabismus|strabismus]]. Mata kiri Euler yang saat itu tampak sehat kemudian terkena [[katarak]].<ref name="blind"/>]]
=== Penyakit mata ===
[[Penglihatan]] Euler terus memburuk sepanjang karier matematikanya. Pada tahun 1738, tiga tahun setelah hampir mati akibat demam tinggi, mata kanannya terkena infeksi dan hampir sama sekali buta. Selama tinggal di Jerman, ia hanya bisa melihat dengan mata kiri. Tidak berapa lama setelah tiba di St. Petersburg, mata kirinya terkena [[katarak]] pada tahun 1766. Hanya beberapa minggu setelahnya, ia menjadi buta total. Namun, produktivitasnya malah meningkat berkat ingatannya yang luar biasa. Euler berkata, "Sekarang perhatianku lebih sedikit gangguannya".<ref>[https://books.google.ca/books?id=KUYLhOVkaV4C&pg=PA17&lpg=PA17&dq=%22now+i+will+have+fewer+distractions%22&source=bl&ots=cELiNguUQ9&sig=vGQmYpZ7EUbtpOyh8CIb3uDCgh8&hl=en&sa=X&ved=0ahUKEwiX6c_V7KLSAhWH6YMKHabkCbYQ6AEILjAE#v=onepage&q=%22now%20i%20will%20have%20fewer%20distractions%22&f=false]</ref> Dalam keadaan buta, Euler dapat mengatakan isi ''[[Aeneid]]'' karya [[Virgil]] dari awal sampai akhir tanpa ragu-ragu, dan untuk setiap halaman edisi itu ia dapat menunjukkan baris pertama dan baris terakhir. Dengan bantuan seorang juru tulis, ia semakin banyak berkarya dalam berbagai bidang ilmu. Rata-rata ia menerbitkan satu artikel matematika setiap minggu pada tahun 1775.<ref name="volumes"/><!--The Eulers bore a double name, Euler-Schölpi, the latter of which derives from ''schelb'' and ''schief'', signifying squint-eyed, cross-eyed, or crooked. This suggests that the Eulers may have had a susceptibility to eye problems.<ref>{{cite book |last=Calinger |first=Ronald |date=2016 |title=Leonhard Euler mathematical genius in the Enlightenment |url=http://press.princeton.edu/titles/10531.html |location= |publisher=Princeton University Press |page=8 |isbn=9781400866632 }}</ref>
-->
=== Kembali ke Rusia dan kematian ===
<!--In 1760, with the [[Seven Years' War]] raging, Euler's farm in Charlottenburg was ransacked by advancing Russian troops. Upon learning of this event, [[Ivan Petrovich Saltykov|General Ivan Petrovich Saltykov]] paid compensation for the damage caused to Euler's estate, later [[Empress Elizabeth|Empress Elizabeth of Russia]] added a further payment of 4000 roubles – an exorbitant amount at the time.<ref>Gindikin, S.G., Гиндикин С. Г., МЦНМО, НМУ, 2001, с. 217.</ref> The political situation in Russia stabilized after [[Catherine II of Russia|Catherine the Great's]] accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant – a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. All of these requests were granted. He spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. -->
Pada tahun 1773, istri Euler, Katharina, meninggal dunia setelah menikah 40 tahun. Tiga tahun setelah kematian istrinya, Euler menikah dengan saudari-tiri Katharina, Salome Abigail Gsell (1723–1794).<ref name=gekker2/> Perkawinan ini berlanjut sampai kematiannya.
Pada tahun 1782 ia diangkat sebagai ''Foreign Honorary Member'' pada [[American Academy of Arts and Sciences]].<ref name=AAAS/>
Di St. Petersburg pada tanggal 18 September 1783, setelah makan siang dengan keluarganya, Euler berdiskusi mengenai planet [[Uranus]] yang baru ditemukan dan [[orbit]]-nya bersama dengan rekan [[akademikus]] [[:en:Anders Johan Lexell|Anders Johan Lexell]], ketika ia tiba-tiba pingsan akibat [[Hemorrhagia cerebral|pendarahan otak]]. Ia meninggal beberapa jam kemudian.<ref name="euler"/> [[:de:Jacob von Staehlin|Jacob von Staehlin-Storcksburg]] menulis suatu obituari singkat untuk [[Akademi Ilmu Pengetahuan Rusia]] dan [[matematikawan]] Rusia [[:en:Nicolas Fuss|Nicolas Fuss]], salah satu murid Euler, menulis eulogi yang lebih detail,<ref name=novaacta/> yang dibacakannya pada upacara pengenangan. Dalam eulogi bagi ''French Academy'', matematikawan dan filsuf Prancis, [[Marquis de Condorcet]], menulis:
{{quote|''il cessa de calculer et de vivre''—... ia berhenti berhitung dan hidup.<ref name=condorcet/>}}
Euler dimakamkan di samping Katharina pada [[:en:Smolensk Lutheran Cemetery|Smolensk Lutheran Cemetery]] di [[:en:Goloday Island|Pulau Goloday]]. Pada tahun 1785, [[Akademi Ilmu Pengetahuan Rusia]] membuat patung dada marmer Leonhard Euler pada suatu tugu dekat kantor Directur dan, pada tahun 1837, menempatkan batu nisan pada makam Euler. Untuk mengenang hari ulang tahun Euler ke-250, batu nisan itu dipindahkan pada tahun 1956, bersama dengan jenazahnya, ke nekropolis abad ke-18 pada [[:en:Alexander Nevsky Lavra|Biara Alexander Nevsky]].
[[Berkas:Euler Grave at Alexander Nevsky Monastry.jpg|jmpl|ka|250px|Makam Euler di [[:en:Alexander Nevsky Lavra|Biara Alexander Nevsky]]]]
== Kontribusi Terhadap Matematika dan Fisika ==
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{{E (mathematical constant)}}-->
Euler berkarya dalam hampir semua bidang [[matematika]], seperti [[geometri]], [[kalkulus]] [[:en:infinitesimal|infinitesimal]], [[trigonometri]], [[aljabar]], dan [[:en:number theory|teori bilangan]], selain juga [[Mekanika kontinuum|fisika kontinuum]], [[:en:lunar theory|teori lunar]] dan bidang-bidang [[fisika]] lainnya. Ia merupakan tokoh utama dalam sejarah matematika; jika dicetak, karya-karyanya, kebanyakan pada landasan ilmu, akan menjadi 60 sampai 80 volume [[:en:quarto (text)|quarto]].<ref name="volumes"/> Nama Euler juga terkait dengan [[Daftar hal-hal yang dinamai menurut Leonhard Euler|banyak topik]].
Euler adalah satu-satunya matematikawan yang mempunyai ''dua'' bilangan dengan namanya: [[E (konstanta matematika)|Bilangan Euler]] yang terkenal dalam [[kalkulus]], ''e'', kira-kira setara dengan 2.71828, dan [[:en:Euler–Mascheroni constant|konstanta Euler–Mascheroni]] γ ([[gamma]]) yang kadang kala hanya disebut "Konstanta Euler" ("''Euler's constant''"), kira-kira setara dengan 0.57721. Sampai sekarang belum diketahui apakah γ adalah suatu [[bilangan rasional]] atau [[bilangan irasional|irasional]].<ref name=derbysh/>
<!-- The biography could use more correlation with his mathematical activities. When was his most prolific period and discoveries, and how did they fit in with his general life? -->
=== Notasi matematika ===
Euler memperkenalkan dan mempopulerkan sejumlah konvensi notasi matematika melalui buku-buku teksnya yang berjumlah sangat banyak dan tersebar luas. Terutama, ia memperkenalkan konsep [[Fungsi (matematika)|fungsi matematika]]<ref name="function"/> dan yang pertama menuliskan ''f''(''x'') untuk menandai suatu fungsi ''f'' yang diterapkan pada argumen ''x''. Ia juga memperkenalkan notasi modern untuk [[Fungsi trigonometrik|fungsi-fungsi trigonometri]], huruf {{math|''e''}} untuk dasar [[logaritma natural]] (sekarang dikenal sebagai [[E (konstanta matematika)|Bilangan Euler atau ''Euler's number'']]), huruf Yunani [[Sigma|Σ]] untuk penjumlahan dan huruf {{math|''i''}} untuk menandai [[Bilangan imajiner|unit imajiner]].<ref name=Boyer/> Penggunaan huruf Yunani ''[[Pi|π]]'' untuk menandai [[Pi|rasio keliling suatu lingkaran dengan diameternya]] juga dipopulerkan oleh Euler, meskipun berasal dari matematikawan [[Wales]], [[:en:William Jones (mathematician)|William Jones]].<ref name="pi">{{cite web| url = http://www.stephenwolfram.com/publications/mathematical-notation-past-future/| title = Mathematical Notation: Past and Future| accessdate=23 September 2014| last = Wolfram| first = Stephen}}</ref>
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===Analisis ===
The development of [[infinitesimal calculus]] was at the forefront of 18th-century mathematical research, and the [[Bernoulli family|Bernoullis]]—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of [[mathematical rigor|mathematical rigour]]<ref name = "Basel"/> (in particular his reliance on the principle of the [[generality of algebra]]), his ideas led to many great advances.
Euler is well known in [[Mathematical analysis|analysis]] for his frequent use and development of [[power series]], the expression of functions as sums of infinitely many terms, such as
:<math>e^x = \sum_{n=0}^\infty {x^n \over n!} = \lim_{n \to \infty} \left(\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\right).</math>
Notably, Euler directly proved the power series expansions for {{math|''e''}} and the [[inverse tangent]] function. (Indirect proof via the inverse power series technique was given by [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]] between 1670 and 1680.) His daring use of power series enabled him to solve the famous [[Basel problem]] in 1735 (he provided a more elaborate argument in 1741):<ref name="Basel"/>
:<math>\sum_{n=1}^\infty {1 \over n^2} = \lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}.</math>
[[Image:Euler's formula.svg|thumb|A geometric interpretation of [[Euler's formula]]]]
Euler introduced the use of the [[exponential function]] and [[logarithms]] in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and [[complex number]]s, thus greatly expanding the scope of mathematical applications of logarithms.<ref name=Boyer/> He also defined the exponential function for complex numbers, and discovered its relation to the [[trigonometric function]]s. For any [[real number]] [[φ|{{math|φ}}]] (taken to be radians), [[Euler's formula]] states that the [[Exponential function#On the complex plane|complex exponential]] function satisfies
:<math>e^{i\varphi} = \cos \varphi + i\sin \varphi.</math>
A special case of the above formula is known as [[Euler's identity]],
:<math>e^{i \pi} +1 = 0 </math>
called "the most remarkable formula in mathematics" by [[Richard P. Feynman]], for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, {{math|''e''}}, {{math|''i''}} and {{pi}}.<ref name="Feynman"/> In 1988, readers of the ''[[Mathematical Intelligencer]]'' voted it "the Most Beautiful Mathematical Formula Ever".<ref name=MathInt/> In total, Euler was responsible for three of the top five formulae in that poll.<ref name=MathInt/>
[[De Moivre's formula]] is a direct consequence of [[Euler's formula]].
In addition, Euler elaborated the theory of higher [[transcendental function]]s by introducing the [[gamma function]] and introduced a new method for solving [[quartic equation]]s. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern [[complex analysis]]. He also invented the [[calculus of variations]] including its best-known result, the [[Euler–Lagrange equation]].
Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, [[analytic number theory]]. In breaking ground for this new field, Euler created the theory of [[Generalized hypergeometric series|hypergeometric series]], [[q-series]], [[hyperbolic functions|hyperbolic trigonometric functions]] and the analytic theory of [[generalized continued fraction|continued fractions]]. For example, he proved the [[infinitude of primes]] using the divergence of the [[harmonic series (mathematics)|harmonic series]], and he used analytic methods to gain some understanding of the way [[prime numbers]] are distributed. Euler's work in this area led to the development of the [[prime number theorem]].<ref name="analysis"/>
===Number theory===
Euler's interest in number theory can be traced to the influence of [[Christian Goldbach]], his friend in the St. Petersburg Academy. A lot of Euler's early work on number theory was based on the works of [[Pierre de Fermat]]. Euler developed some of Fermat's ideas and disproved some of his conjectures.
Euler linked the nature of prime distribution with ideas in analysis. He proved that [[Proof that the sum of the reciprocals of the primes diverges|the sum of the reciprocals of the primes diverges]]. In doing so, he discovered the connection between the [[Riemann zeta function]] and the prime numbers; this is known as the [[Proof of the Euler product formula for the Riemann zeta function|Euler product formula for the Riemann zeta function]].
Euler proved [[Newton's identities]], [[Fermat's little theorem]], [[Fermat's theorem on sums of two squares]], and he made distinct contributions to [[Lagrange's four-square theorem]]. He also invented the [[totient function]] φ(''n''), the number of positive integers less than or equal to the integer ''n'' that are [[coprime]] to ''n''. Using properties of this function, he generalized Fermat's little theorem to what is now known as [[Euler's theorem]]. He contributed significantly to the theory of [[perfect number]]s, which had fascinated mathematicians since [[Euclid]]. He proved that the relationship shown between perfect numbers and [[Mersenne prime]]s earlier proved by Euclid was one-to-one, a result otherwise known as the [[Euclid–Euler theorem]]. Euler also conjectured the law of [[quadratic reciprocity]]. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of [[Carl Friedrich Gauss]].<ref name="numbertheory"/>
By 1772 Euler had proved that 2<sup>31</sup> − 1 = [[2147483647|2,147,483,647]] is a Mersenne prime. It may have remained the [[largest known prime]] until 1867.<ref name=caldwell/>
=== Teori Graf ===
[[Image:Konigsberg bridges.png|frame|right|Map of [[Königsberg]] in Euler's time showing the actual layout of the [[Seven Bridges of Königsberg|seven bridges]], highlighting the river Pregel and the bridges.]]
In 1735, Euler presented a solution to the problem known as the [[Seven Bridges of Königsberg]].<ref name="bridge"/> The city of [[Königsberg]], [[Kingdom of Prussia|Prussia]] was set on the [[Pregel]] River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no [[Eulerian path|Eulerian circuit]]. This solution is considered to be the first theorem of [[graph theory]], specifically of [[planar graph]] theory.<ref name="bridge"/>
Euler also discovered the [[Planar graph#Euler's formula|formula]] <math>V - E + F = 2</math> relating the number of vertices, edges and faces of a [[Convex polytope|convex polyhedron]],<ref name=cromw/> and hence of a [[planar graph]]. The constant in this formula is now known as the [[Euler characteristic]] for the graph (or other mathematical object), and is related to the [[genus (mathematics)|genus]] of the object.<ref name=gibbons/> The study and generalization of this formula, specifically by [[Augustin-Louis Cauchy|Cauchy]]<ref name="Cauchy"/> and [[Simon Antoine Jean L'Huilier|L'Huilier]],<ref name="Lhuillier"/> is at the origin of [[topology]].
===Applied mathematics===
Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the [[Bernoulli numbers]], [[Fourier series]], [[Euler number]]s, the constants [[E (mathematical constant)|{{math|e}}]] and [[pi|{{pi}}]], continued fractions and integrals. He integrated [[Gottfried Leibniz|Leibniz]]'s [[differential calculus]] with Newton's [[Method of Fluxions]], and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the [[numerical approximation]] of integrals, inventing what are now known as the [[Euler approximations]]. The most notable of these approximations are [[Euler's method]] and the [[Euler–Maclaurin formula]]. He also facilitated the use of [[differential equations]], in particular introducing the [[Euler–Mascheroni constant]]:
:<math>\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right).</math>
One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the ''Tentamen novae theoriae musicae,'' hoping to eventually incorporate [[musical theory]] as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.<ref name="music"/>
===Physics and astronomy===
{{Classical mechanics|cTopic=Scientists}}
Euler helped develop the [[Euler–Bernoulli beam equation]], which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in [[classical mechanics]], Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the [[solar parallax|parallax]] of the sun. His calculations also contributed to the development of accurate [[History of longitude|longitude tables]].<ref name=yousch/>
In addition, Euler made important contributions in [[optics]]. He disagreed with Newton's [[corpuscular theory of light]] in the ''[[Opticks]]'', which was then the prevailing theory. His 1740s papers on optics helped ensure that the [[wave theory of light]] proposed by [[Christiaan Huygens]] would become the dominant mode of thought, at least until the development of the [[wave-particle duality|quantum theory of light]].<ref name="optics"/>
In 1757 he published an important set of equations for [[inviscid flow]], that are now known as the [[Euler equations (fluid dynamics)|Euler equations]].<ref name=euler2/> In differential form, the equations are:
:<math>
\begin{align}
&{\partial\rho\over\partial t}+
\nabla\cdot(\rho\bold u)=0\\[1.2ex]
&{\partial(\rho{\bold u})\over\partial t}+
\nabla\cdot(\bold u\otimes(\rho \bold u))+\nabla p=\bold{0}\\[1.2ex]
&{\partial E\over\partial t}+
\nabla\cdot(\bold u(E+p))=0,
\end{align}
</math>
di mana
* ''ρ'' is the fluid [[mass density]],
* '''''u''''' is the fluid [[velocity]] [[Vector (geometric)|vector]], with components ''u'', ''v'', and ''w'',
* ''E'' = ''ρ e'' + ½ ''ρ'' (''u''<sup>2</sup> + ''v''<sup>2</sup> + ''w''<sup>2</sup>) is the total energy per unit [[volume]], with ''e'' being the [[internal energy]] per unit mass for the fluid,
* ''p'' is the [[pressure]],
* ''⊗'' denotes the [[tensor product]], and
* '''0''' being the [[zero vector]].
Euler is also well known in structural engineering for his formula giving the critical [[buckling]] load of an ideal strut, which depends only on its length and flexural stiffness:<ref name="SIAM"/>
:<math>F=\frac{\pi^2 EI}{(KL)^2}</math>
di mana
* ''F'' = maximum or critical [[force]] (vertical load on column),
* ''E'' = [[modulus of elasticity]],
* ''I'' = [[area moment of inertia]],
* ''L'' = unsupported length of column,
* ''K'' = column effective length factor, whose value depends on the conditions of end support of the column, as follows.
::For both ends pinned (hinged, free to rotate), ''K'' = 1.0.
::For both ends fixed, ''K'' = 0.50.
::For one end fixed and the other end pinned, ''K'' = 0.699…
::For one end fixed and the other end free to move laterally, ''K'' = 2.0.
* ''K L'' is the effective length of the column.
-->
=== Logika ===
Euler juga dikenang dengan hasil karyanya berupa [[Kurva|kurva tertutup]] untuk menggambarkan pemikiran [[silogisme]] (1768). Diagram ini telah dikenal dengan nama [[:en:Euler diagram|diagram Euler]].<ref name="logic">{{cite journal |last=Baron |first=M. E. |title=A Note on The Historical Development of Logic Diagrams |journal=The Mathematical Gazette |volume=LIII |issue=383 |pages=113–125 |date=May 1969 |jstor=3614533}}</ref>
<!--
An Euler diagram is a [[diagram]]matic means of representing [[Set (mathematics)|sets]] and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict [[Set (mathematics)|sets]]. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the [[element (mathematics)|elements]] of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships ([[intersection (set theory)|intersection]], [[subset]] and [[Disjoint sets|disjointness]]). Curves whose interior zones do not intersect represent [[disjoint sets]]. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the [[intersection (set theory)|intersection]] of the sets). A curve that is contained completely within the interior zone of another represents a [[subset]] of it. Euler diagrams were incorporated as part of instruction in [[set theory]] as part of the [[new math]] movement in the 1960s. Since then, they have also been adopted by other curriculum fields such as reading.<ref name=quest/>
===Music===
Even when dealing with music, Euler’s approach is mainly mathematical. His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that did not leave him throughout his life.<ref>Peter Pesic, ''Music and the Making of Modern Science'', p. 133.</ref>
A first point of Euler’s musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2<sup>m</sup>A, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2<sup>m</sup> (where "m is an indefinite number, small or large, so long as the sounds are perceptible"<ref>Leonhard Euler, ''Tentamen novae theoriae musicae'', St Petersburg, 1739, p. 115</ref>), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2<sup>m</sup>.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2<sup>m</sup>.5, major third + minor sixth (C–E–C); the fourth is 2<sup>m</sup>.3<sup>2</sup>, two fourths and a tone (C–F–B{{music|b}}–C); the fifth is 2<sup>m</sup>.3.5 (C–E–G–B–C); etc. Genres 12 (2<sup>m</sup>.3<sup>3</sup>.5), 13 (2<sup>m</sup>.3<sup>2</sup>.5<sup>2</sup>) and 14 (2<sup>m</sup>.3.5<sup>3</sup>) are corrected versions of the diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2<sup>m</sup>.3<sup>3</sup>.5<sup>2</sup>) is the "diatonico-chromatic", "used generally in all compositions",<ref>Eric Emery, ''Temps et musique'', Lausanne, L’Âge d’homme, 2000, pp. 344–45.</ref> and which turns out to be identical with the system described by Johann Mattheson.<ref>Johannes Mattheson, ''Grosse General-Baß-Schule'', Hamburg, 1731, Vol. I, p. 104-106, mentioned by Euler; and ''Exemplarische Organisten-Probe'', Hamburg, 1719, p. 57-59.</ref> Euler later envisaged the possibility of describing genres including the prime number 7.<ref>Wilfrid Perret, ''Some Questions of Musical Theory'', Cambridge, 1926, p. 60-62; "What is an Euler-Fokker genus?", http://www.huygens-fokker.org/microtonality/efg.html, retrieved 12-6-2015.</ref>
Euler devised a specific graph, the ''Speculum musicum'',<ref>Leonhard Euler,''Tentamen novae theoriae musicae'', St Petersburg, 1739, p. 147; ''De harmoniae veris principiis'', St Petersburg, 1774, p. 350.</ref> to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, reminding his interest for the Seven Bridges of Königsberg (see [[Leonhard Euler#Graph theory|above]]). The device knew a renewed interest as the [[Tonnetz]] in neo-Riemannian theory (see also [[Lattice (music)]]).<ref>Edward Gollin, "Combinatorial and Transformational Aspects of Euler’s ''Speculum Musicum''", ''Mathematics and Computation in Music'', T. Klouche and Th. Noll eds, Springer, 2009, pp. 406–411.</ref>
Euler further used the principle of the "exponent" to propose a derivation of the ''gradus suavitatis'' (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered just intonation, i.e. 1 and the prime numbers 3 and 5 only.<ref>Mark Lindley and Ronald Turner-Smith, ''Mathematical Models of Musical Scales'', Bonn, Verlag für systematische Musikwissenschaft, 1993, pp. 234–239. See also Catherine Nolan, "Music Theory and Mathematics", ''The Cambridge History of Western Music Theory'', Th. Christensen ed., New York, CUP, 2002, pp. 278–79.</ref> Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form
:''ds'' = {{SIGMA}} ''(k<sub>i</sub>p<sub>i</sub> – k<sub>i</sub>)'' + 1
where ''p<sub>i</sub>'' are prime numbers and ''k<sub>i</sub>'' their exponents.<ref>Patrice Bailhache, "La Musique traduite en Mathématiques: Leonhard Euler", http://patrice.bailhache.free.fr/thmusique/euler.html, retrieved 12-6-2015.</ref>
-->
== Filsafat dan Kepercayaan ==
Euler dan temannya [[Daniel Bernoulli]] bertolak belakang dengan [[:en:monadism|monadisme]] [[Gottfried Leibniz|Leibniz]] dan filosofi [[Christian Wolff]]. Euler bersikeras bahwa pengetahuan didirikan atas dasar hukum kuantitatif yang tepat, hal yang tidak dapat dijelaskan oleh monadisme dan ilmu pengetahuan Wolffian. Kecenderungan religius Euler mungkin juga menjadi alasan ketidaksukaannya terhadap doktrin; dia bertindak lebih jauh dan menyebut ideologi Wolff sebagai "kafir dan ateis".<ref name="wolff">{{harvnb|Calinger|1996|pp=153–4}}</ref>
Keyakinan agama Euler bisa dilihat dari suratnya kepada seorang Putri Jerman dan karyanya sebelumnya, ''Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister'' (''Mempertahankan Wahyu Ilahi terhadap Keberatan Para [[Pemikiran bebas|Pemikir Bebas]]''). Karya-karya inilah yang menunjukkan bahwa Euler adalah seorang penganut [[Kristen]] taat yang percaya akan ilham [[Alkitab|Injil]]; ''Rettung'' semula adalah argumen untuk [[Ilham Alkitab|ilham kitab suci Ilahi]].<ref name="teologi"/>
Ada satu legenda yang terkenal,<ref name="diderot">{{cite journal| last = Brown | first = B.H.| year = 1942| month = May| title = The Euler-Diderot Anecdote| journal =The American Mathematical Monthly| volume = 49| issue = 5| pages = 302–303| doi = 10.2307/2303096| jstor = 2303096}}; {{cite journal| last = Gillings | first = R. J.| year = 1954| month = February| title = The So-Called Euler-Diderot Incident| journal =The American Mathematical Monthly| volume = 61| issue = 2| pages = 77–80| doi = 10.2307/2307789| jstor = 2307789}}</ref> terinspirasi dari argumen-argumen antara Euler dengan para filsuf sekuler yang terjadi selama masa tugas kedua Euler di Akademi St. Petersburg. Filsuf Prancis [[Denis Diderot]] berkunjung ke Rusia atas undangan [[Yekaterina II dari Rusia|Katerina Yang Agung]]. Namun, sang [[Tsarina]] telah diperingatkan bahwa paham ateisme yang dibawa filsuf tersebut telah mempengaruhi anggota sidangnya, hingga Euler diminta untuk menghadapi pria Prancis tersebut. Diderot kemudian diberitahu bahwa seorang matematikawan terpelajar telah memproduksi bukti mengenai [[Filsafat ketuhanan|keberadaan Tuhan]]: Diderot bersedia untuk menyaksikan bukti tersebut, yang dipresentasikan dalam suatu sidang. Euler muncul, maju mendekati Diderot, dan dengan nada berkeyakinan sempurna, ia mengumumkan perkataan ''[[:en:Non sequitur (literary device)|non-sequitur]]'' ini: "Tuan, <math>\frac{a+b^n}{n}=x</math>, jadi: Allah ada — jawablah!" Diderot, yang menurut ceritanya menganggap matematika itu omong kosong, tidak bisa menjawab apa-apa, sementara suara gemuruh tawa akan meledak di persidangan. Karena merasa malu, Diderot minta izin meninggalkan Rusia, dan izin ini dengan senang hati diberikan oleh sang Ratu. Meskipun anekdot ini menarik, dicurigai tidak pernah benar-benar terjadi karena Diderot sendiri pernah melakukan riset dalam matematika<ref>{{cite web|last=Marty|first=Jacques |title=Quelques aspects des travaux de Diderot en Mathematiques Mixtes. |url= http://www.persee.fr/web/revues/home/prescript/article/rde_0769-0886_1988_num_4_1_954}}</ref> Cerita ini pertama kalinya dituturkan oleh [[Dieudonné Thiébault]]<ref name="brown">{{cite journal | journal = [[American Mathematical Monthly]] | volume=49 | issue=5 | last1 = Brown | first1 = B.H. | title = The Euler-Diderot Anecdote | pages = 302–303 |date=May 1942 | doi=10.2307/2303096}}</ref> dan dikembangkan oleh [[Augustus De Morgan]].<ref name="Struik">{{cite book|title = A Concise History of Mathematics|url = https://archive.org/details/concisehistoryof0000stru_m6j1|edition = 3rd revised|last1 = Struik|first1 = Dirk J.|publisher = [[Dover Books]]|year = 1967|page = [https://archive.org/details/concisehistoryof0000stru_m6j1/page/129 129]|authorlink = Dirk Jan Struik|isbn = 0486602559 }}</ref><ref name="gillings">{{cite journal | journal = [[American Mathematical Monthly]] | volume=61 | issue=2 | last1 = Gillings | first1 = R.J. | title = The So-Called Euler-Diderot Anecdote | pages = 77–80 |date=Feb 1954 | doi=10.2307/2307789}}</ref>
== Penghormatan ==
Euler muncul pada seri keenam uang kertas Swiss pecahan 10-[[Franc Swiss|franc]] dan pada beberapa [[prangko]] Swiss, Jerman, dan Rusia. [[Asteroid]] [[Euler 2002]] mengambil namanya sebagai penghormatan atasnya. Dia juga dikenang oleh [[Gereja Lutheran]] melalui [[Kalender Santo (Lutheran)|Kalender Santo]] mereka pada 24 Mei—dia merupakan penganut Kristen taat (dan yakin akan [[ineransi Alkitab]]) yang telah menulis [[apologetik]] dan menentang tegas ateis terkemuka pada masanya.<ref name="teologi">{{cite journal| last = Euler| first = Leonhard | editor = Orell-Fussli| year = 1960| title = Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister| journal = Leonhardi Euleri Opera Omnia (series 3)| volume = 12 }}</ref>
== Referensi ==
{{Reflist|colwidth=30em|refs=
<!--<ref name=mathg>{{MathGenealogy|id=38586}}</ref>
<ref name=pronun>The pronunciation {{IPAc-en|ˈ|juː|l|ər}} is incorrect. "Euler", [[Oxford English Dictionary]], second edition, Oxford University Press, 1989 [http://www.merriam-webster.com/dictionary/Euler "Euler"], [[Webster's Dictionary|Merriam–Webster's Online Dictionary]], 2009. [http://ahdictionary.com/word/search.html?q=Euler%2C+Leonhard&submit.x=40&submit.y=16 "Euler, Leonhard"], [[The American Heritage Dictionary of the English Language]], fifth edition, Houghton Mifflin Company, Boston, 2011. {{cite book|title=Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections|url=https://archive.org/details/netspuzzlespostm00higg|author=Peter M. Higgins|year=2007|publisher=Oxford University Press|page=[https://archive.org/details/netspuzzlespostm00higg/page/n51 43]}}</ref>
<ref name="function">{{harvnb|Dunham|1999|p=17}}</ref>
<ref name=eulerarch>{{cite web|url=http://eulerarchive.maa.org/pages/E033.html |author=Saint Petersburg |title=Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae |publisher= |date=1739 |accessdate= }}</ref>
<ref name="volumes">{{cite journal |last = Finkel |first = B. F. |year = 1897 |title = Biography—Leonard Euler |journal = The American Mathematical Monthly |volume = 4 | issue = 12 |jstor = 2968971|pages = 297–302|doi=10.2307/2968971 }}</ref>
<ref name="Laplace">{{harvnb|Dunham|1999|p=xiii}} "Lisez Euler, lisez Euler, c'est notre maître à tous."</ref>
<ref name=libri>The quote appeared in Gugliemo Libri's review of a recently published collection of correspondence among eighteenth-century mathematicians: Gugliemo Libri (January 1846), Book review: "Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIe siècle, … " (Mathematical and physical correspondence of some famous geometers of the eighteenth century, … ), ''Journal des Savants'', [http://gallica.bnf.fr/ark:/12148/bpt6k57253t/f52.image.langEN page 51.] From page 51: ''" … nous rappellerions que Laplace lui même, … ne cessait de répéter aux jeunes mathématiciens ces paroles mémorables que nous avons entendues de sa propre bouche : 'Lisez Euler, lisez Euler, c'est notre maître à tous.' "'' ( … we would recall that Laplace himself, … never ceased to repeat to young mathematicians these memorable words that we heard from his own mouth: 'Read Euler, read Euler, he is our master in everything.)</ref>
<ref name="childhood">{{cite book |last= James |first= Ioan |title= Remarkable Mathematicians: From Euler to von Neumann |publisher= Cambridge |year= 2002|page=2 |isbn= 0-521-52094-0}}</ref>
<ref name=17cent>{{cite web|url=http://www.17centurymaths.com/contents/euler/e002tr.pdf |author=Ian Bruce |title=Euler's Dissertation De Sono : E002. Translated & Annotated |publisher=17centurymaths.com |date= |accessdate=14 September 2011 }}</ref>
<ref name="prize">{{harvnb|Calinger|1996|p=156}}</ref>
<ref name="stpetersburg">{{harvnb|Calinger|1996|p=125}}</ref>
<ref name="medic">{{harvnb|Calinger|1996|p=127}}</ref>
<ref name="promotion">{{harvnb|Calinger|1996|pp=128–9}}</ref>
<ref name=gekker>{{Cite book | first1=I. R. | last1=Gekker | first2=A. A. | last2=Euler | chapter=Leonhard Euler's family and descendants |chapterurl=https://books.google.com/books?id=Ta9bz1wv79AC&pg=PA402 |title={{harvnb|Bogolyubov|Mikhaĭlov|Yushkevich|2007|page=402}} |ref={{harvid|Gekker|Euler|2007}}}}</ref>
-->
<ref name="wife">{{cite web| url=http://www-history.mcs.st-and.ac.uk/~history/Extras/Euler_Fuss_Eulogy.html| title = Eulogy of Euler by Fuss| accessdate =30 August 2006| last = Fuss| first = Nicolas}}</ref>
<ref name=dartm>{{cite web| title = E212 – Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum|publisher=Dartmouth|url=http://www.math.dartmouth.edu/~euler/pages/E212.html}}</ref>
<ref name="Friedrich">{{harvnb|Dunham|1999|pp=xxiv–xxv}}</ref>
<ref name='Digital Copy of "Letters to a German Princess"'>{{cite web|last=Euler|first=Leonhard|title=Letters to a German Princess on Diverse Subjects of Natural Philosophy|url=https://archive.org/details/letterseulertoa00eulegoog|publisher=Internet Archive, Digitzed by Google|accessdate=15 April 2013}}</ref>
<!--
<ref name=fredlett>{{cite book | title=Letters of Voltaire and Frederick the Great, Letter H 7434, 25 January 1778 | author=Frederick II of Prussia | author-link=Frederick II of Prussia | publisher=Brentano's | location=New York | year=1927 | others=[[Richard Aldington]] }}</ref>
-->
<ref name="blind">{{harvnb|Calinger|1996|pp=154–5}}</ref>
<ref name=gekker2>{{harvnb|Gekker|Euler|2007|p=[https://books.google.com/books?id=Ta9bz1wv79AC&pg=PA405 405]}}</ref>
<ref name=AAAS>{{cite web|title=Book of Members, 1780–2010: Chapter E|url=http://www.amacad.org/publications/BookofMembers/ChapterE.pdf|publisher=American Academy of Arts and Sciences|accessdate=28 July 2014}}</ref>
<ref name="euler">{{cite book|title=Leonhard Euler|author=A. Ya. Yakovlev|year=1983|publisher=Prosvesheniye|location=M.}}</ref>
<ref name=novaacta>{{cite journal| year = 1783| title = Eloge de M. Leonhard Euler. Par M. Fuss| journal = Nova Acta Academiae Scientiarum Imperialis Petropolitanae | volume = 1| pages = 159–212 |url=https://www.biodiversitylibrary.org/item/38629#page/177/mode/1up}}</ref>
<ref name=condorcet>{{cite web| url = http://www.math.dartmouth.edu/~euler/historica/condorcet.html| title = Eulogy of Euler – Condorcet| accessdate =30 August 2006| author =Marquis de Condorcet}}</ref>
<ref name=derbysh>{{cite book|last=Derbyshire|first=John|title=[[Prime Obsession]]: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics|year=2003|publisher=Joseph Henry Press|location=Washington, D.C.|page=[https://archive.org/details/primeobsessionbe00derb_046/page/n438 422]}}</ref>
<!--<ref name="Basel">{{cite book| last = Wanner| first = Gerhard|author2=Hairer, Ernst | title = Analysis by its history| edition = 1st|date=March 2005| publisher = Springer| page = 63}}</ref>
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<ref name=Boyer>{{cite book|title = A History of Mathematics|url = https://archive.org/details/historymathemati00boye_328|last= Boyer|first=Carl B.|author2=Merzbach, Uta C.|author2-link= Uta Merzbach |publisher= [[John Wiley & Sons]]|isbn= 0-471-54397-7|pages = [https://archive.org/details/historymathemati00boye_328/page/n458 439]–445|year = 1991}}</ref>
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<ref name="Feynman">
{{cite book |last= Feynman|first= Richard|title= The Feynman Lectures on Physics: Volume I|page=10 |chapter= Chapter 22: Algebra |date=June 1970}}</ref>
<ref name=MathInt>{{cite journal | last= Wells | first= David | year= 1990 | title = Are these the most beautiful? | journal = Mathematical Intelligencer | volume = 12 | issue = 3 | pages= 37–41 | doi= 10.1007/BF03024015 }}<br />{{cite journal | last= Wells | first= David | year= 1988 | title = Which is the most beautiful? | journal = Mathematical Intelligencer | volume = 10 | issue = 4 | pages= 30–31 | doi= 10.1007/BF03023741 }}</ref>
<ref name="analysis">{{harvnb|Dunham|1999|loc=Ch. 3, Ch. 4}}</ref>
<ref name="numbertheory">{{harvnb|Dunham|1999|loc=Ch. 1, Ch. 4}}</ref>
<ref name=caldwell>Caldwell, Chris. [http://primes.utm.edu/notes/by_year.html ''The largest known prime by year'']</ref>
<ref name="bridge">{{cite journal| last = Alexanderson| first = Gerald|authorlink=Gerald L. Alexanderson|date=July 2006| title = Euler and Königsberg's bridges: a historical view| journal = Bulletin of the American Mathematical Society| doi = 10.1090/S0273-0979-06-01130-X| volume = 43| page = 567| issue = 4}}</ref>
<ref name=cromw>{{cite book |first=Peter R. |last=Cromwell |title=Polyhedra |url=https://books.google.com/books?id=OJowej1QWpoC&pg=PA189 |year=1999 |publisher=Cambridge University Press |isbn=978-0-521-66405-9 |pages=189–190}}</ref>
<ref name=gibbons>{{cite book |first=Alan |last=Gibbons |title=Algorithmic Graph Theory |url=https://books.google.com/books?id=Be6t04pgggwC&pg=PA72 |year=1985 |publisher=Cambridge University Press |isbn=978-0-521-28881-1 |page=72}}</ref>
<ref name="Cauchy">{{cite journal|author=Cauchy, A. L.|year=1813|title=Recherche sur les polyèdres—premier mémoire|journal=[[Journal de l'École Polytechnique]]|volume= 9 (Cahier 16)|pages=66–86}}</ref>
<ref name="Lhuillier">{{cite journal|author=L'Huillier, S.-A.-J.|title=Mémoire sur la polyèdrométrie|journal=Annales de Mathématiques|volume=3|year=1861|pages=169–189}}</ref>
<ref name="music">{{harvnb|Calinger|1996|pp=144–5}}</ref>
<ref name=yousch>{{cite book |url= |title=Dictionary of Scientific Biography |first=A P |last=Youschkevitch |year=1970–1990 |publisher=New York |isbn= |page= }}</ref>
<ref name="optics">{{cite journal
| author = Home, R. W.
| year = 1988
| title = Leonhard Euler's 'Anti-Newtonian' Theory of Light
| journal = Annals of Science
| volume = 45
| issue = 5
| pages = 521–533 | doi = 10.1080/00033798800200371
}}</ref>
<ref name=euler2>{{cite journal|last=Euler|first=Leonhard|title=Principes généraux de l'état d'équilibre d'un fluide|trans-title=General principles of the state of equilibrium of a fluid|journal=Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires|year=1757|volume=11|pages=217–273|url=http://arxiv-web3.library.cornell.edu/pdf/0802.2383.pdf}}</ref>
<ref name="SIAM">{{cite journal | url=http://www.cs.purdue.edu/homes/wxg/EulerLect.pdf | title=Leonhard Euler: His Life, the Man, and His Work | last=Gautschi | first=Walter | journal=SIAM Review | year=2008 | volume=50 | issue=1 | pages=3–33 | doi=10.1137/070702710|bibcode = 2008SIAMR..50....3G }}</ref>
<ref name="logic">{{cite journal |last=Baron |first=M. E. |title=A Note on The Historical Development of Logic Diagrams |journal=The Mathematical Gazette |volume=LIII |issue=383 |pages=113–125 |date=May 1969 |jstor=3614533}}</ref>
<ref name=quest>{{cite web|url=http://www.readingquest.org/strat/venn.html|author=|title=Strategies for Reading Comprehension Venn Diagrams|publisher=|date=|accessdate=|deadurl=yes|archiveurl=https://web.archive.org/web/20090429093334/http://readingquest.org/strat/venn.html|archivedate=29 April 2009|df=dmy-all}}</ref>
<ref name="wolff">{{harvnb|Calinger|1996|pp=153–4}}</ref>
<ref name="diderot">{{cite journal| last = Brown | first = B. H.|date=May 1942| title = The Euler–Diderot Anecdote| journal =The American Mathematical Monthly| volume = 49| issue = 5| pages = 302–303| doi = 10.2307/2303096| jstor = 2303096}}; {{cite journal| last = Gillings | first = R. J.|date=February 1954| title = The So-Called Euler–Diderot Incident| journal =The American Mathematical Monthly| volume = 61| issue = 2| pages = 77–80| doi = 10.2307/2307789| jstor = 2307789}}</ref>
<ref name=persee>{{cite web|last=Marty|first=Jacques|title=Quelques aspects des travaux de Diderot en Mathematiques Mixtes.|url=http://www.persee.fr/web/revues/home/prescript/article/rde_0769-0886_1988_num_4_1_954}}</ref>
<ref name="brown">{{cite journal | journal = [[American Mathematical Monthly]] | volume=49 | issue=5 | last1 = Brown | first1 = B.H. | title = The Euler–Diderot Anecdote | pages = 302–303 |date=May 1942 | doi=10.2307/2303096}}</ref>
<ref name="Struik">{{cite book | title = A Concise History of Mathematics | url = https://archive.org/details/concisehistoryof0000stru_m6j1 | edition = 3rd revised | last1 = Struik | first1 = Dirk J. | publisher = [[Dover Books]] | year = 1967 | page = [https://archive.org/details/concisehistoryof0000stru_m6j1/page/129 129] | authorlink = Dirk Jan Struik | isbn = 0486602559 }}</ref>
<ref name="gillings">{{cite journal | journal = [[American Mathematical Monthly]] | volume=61 | issue=2 | last1 = Gillings | first1 = R.J. | title = The So-Called Euler-Diderot Anecdote | pages = 77–80 |date=Feb 1954 | doi=10.2307/2307789}}</ref>
<ref name="theology">{{cite journal| last = Euler| first = Leonhard | editor = Orell-Fussli| year = 1960| title = Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister| journal = Leonhardi Euleri Opera Omnia (series 3)| volume = 12 }}</ref>
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<!--<ref name=indep>{{cite news|last=Williams|first=Rob|title=Google Doodle celebrates Leonhard Euler – Swiss mathematician considered one of the greatest of all time |url=http://www.independent.co.uk/life-style/gadgets-and-tech/news/google-doodle-celebrates-leonhard-euler--swiss-mathematician-considered-one-of-the-greatest-of-all-time-8573041.html|location=London|work=The Independent|date=15 April 2013}}</ref>-->
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<ref name=dartm2>[http://math.dartmouth.edu/~euler/pages/E065.html E65 — Methodus... entry at Euler Archives]. Math.dartmouth.edu. Retrieved on 14 September 2011.</ref>
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}}
== Pranala luar ==
{{wikisource|Author:Leonhard Euler}}
* [http://www.leonhardeuler.com/ LeonhardEuler.com]
* [http://www.britannica.com/eb/article-9033216/Leonhard-Euler Artikel Encyclopædia Britannica]
* {{MathGenealogy|id=38586}}
* [http://www.maa.org/news/howeulerdidit.html Bagaimana Euler melakukannya] berisi kolom-kolom yang menjelaskan bagaimana Euler menuntaskan beragam masalah
* [http://www.eulerarchive.org/ Arsip Euler]
* [http://portail.mathdoc.fr/cgi-bin/oetoc?id=OE_EULER_1_2 Leonhard Euler – Œuvres complètes] Gallica-Math
* [http://www.leonhard-euler.ch/ Komite Euler Akademi Ilmu Pengetahuan Swiss] {{Webarchive|url=https://web.archive.org/web/20110520092329/http://www.leonhard-euler.ch/ |date=2011-05-20 }}
* [http://www-history.mcs.st-andrews.ac.uk/References/Euler.html Referensi untuk Leonhard Euler]
* [http://www.euler-2007.ch/en/index.htm Tiga ratus tahun Euler 2007]
* [http://www.eulersociety.org/ Perkumpulan Euler]
* [http://www.pdmi.ras.ru/EIMI/2007/AG/ Kongres Leonhard Euler 2007]—St. Petersburg, Rusia
* [http://www.projecteuler.net Proyek Euler]
* [http://www.math.dartmouth.edu/~euler/historica/family-tree.html Pohon keluarga Euler]
* [http://friedrich.uni-trier.de/oeuvres/20/219/ Korespondensi Euler dengan Frederick Yang Agung, Raja Prusia]
* [http://www.gresham.ac.uk/event.asp?PageId=45&EventId=518 "Euler – 300th anniversary lecture"] {{Webarchive|url=https://web.archive.org/web/20081227054759/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=518 |date=2008-12-27 }}, persembahan dari Robin Wilson pada [[Perguruan Tinggi Gresham]], 9 Mei 2007 (bisa mengunduh file video atau audio)
* [http://euler413.narod.ru/ Dugaan Kuartic Euler]
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