Mathematical Genius and Bible-believing Christian
Published: 14 December 2017 (GMT+10)
Leonhard Euler (pronounced oi-la), (1707–1783) was not only one of the greatest mathematicians and theoretical physicists of all time, but he was also the most prolific.1 He contributed to almost every area of pure and applied mathematics—especially calculus, number theory, notation, optics, and celestial, rational and fluid mechanics.
By applying mathematics to physical problems, he made many practical advances in cartography, chronology, ship-building, bridge-building, and ballistics. These multiple achievements place him in the company of Archimedes, Newton, and Gauss. Eminent Swiss science historian Emil Fellmann says that Euler was “not only by far the most productive mathematician in all of human history, but also one of the greatest scholars of all time.”2
Early life and a talent for mathematics
Leonhard was born on 15 April 1707 in Basel, a city on the Rhine River, Switzerland, the first child of Paul and Margaretha Euler.3 Paul had studied mathematics and theology in acquiring a Master of Arts degree from the University of Basel, and he then became the pastor of an Evangelical-Reformed Church in Riehen (near Basel) which stressed “the Christian inner life of rebirth, brotherly love, and living belief”.4 These were convictions which Leonhard accepted for himself and from which he never wavered.5
Young Leonhard was introduced to the elements of mathematics at home by his father. Then, after schooling at the Basel Gymnasium, in 1720 he enrolled at the University of Basel, where he studied theology, law, philosophy, Greek and Hebrew with a view to becoming a pastor. However, mathematics was taught there by Johann Bernoulli, a family friend, later one of the foremost mathematicians in Europe. He was so impressed with Leonhard’s exceptional talent and zeal for this subject that he not only gave him private Saturday-afternoon tutorials in mathematics, astronomy and physics,6 but also convinced Paul that his son was predestined for a career in mathematics rather than in theology.
Entering the Paris Prize competition
In 1727, Euler submitted a paper in Latin (E4)7 to the Paris Academy of Science’s annual Paris Prize competition, which was then the most distinguished scientific award in Europe.8 For ‘even’ years, a prize of 2,500 livres was offered for the best treatise on a theory to do with astronomy, matter, mechanics, optics, or physics. For ‘odd’ years, the prize was 2,000 livres for solving practical nautical problems such as determining longitude, improving navigation, or advancing ship construction.
The problem for that year was the best way to arrange a sailing ship’s masts—their number, placement, and height—to achieve maximum ship speed. Although Euler had yet to see any vessel more sea-going than a Rhine River freighter, he achieved an accessit (honourable mention), and over the years he won or shared this Prize 12 times and the accessit three times from his 15 submissions.9
Time well spent in Russia
From 1727 to 1741, Euler taught at the Imperial Academy of Sciences in St Petersburg,10 where he quickly mastered Russian. He became professor of physics in 1731, and two years later was appointed head of the mathematics department. His biographer, Ronald Calinger, writes that his research ranged widely over “algebra, arithmetic, astronomy, ballistics, conic sections, differential geometry, elasticity, infinite series, music theory, number theory and oscillations, while his main field was rational mechanics. Convinced of the unity of the mathematical sciences, Euler set about perfecting each branch.”11 Much of this was published in the St Petersburg Academy’s journal Commentarii academiae scientiarum Petropolitanae (replaced by Novi Comentarii from 1747).
He was also involved in such practical activities as designing fire engines, advising the Russian navy, writing textbooks for Russian schools, as well as two volumes on elementary arithmetic in German (E17, E35) for use in the St Petersburg scholastic gymnasia.12 He also had the task of helping prepare the first accurate large scale map of the complete Russian empire, for which accurate determinations of longitude and latitude were needed. This enterprise, comprising 20 sectional maps, was published as the Russian Atlas in 1745.
Marriage and family
In 1734, now financially established, Euler married Katharina Gsell, daughter of the Swiss-born artist Georg Gsell, who was court painter, keeper of the imperial art gallery, and teacher at the art academy in St Petersburg. The couple had 13 children, of whom sadly only five survived early childhood, and only three outlived their parents. Calinger writes:
“Every evening after dinner Euler gathered his children, servants, and students lodging at his house, for a domestic devotion; there were biblical readings and sometimes explanations and discussions. Before bed, he also often read passages from the Bible or scriptural stories for the children.”13
Solving the Basel problem
In 1735, Euler achieved immediate fame by solving a numerical puzzle that had baffled the world’s greatest mathematicians ever since it was posed in 1644. The problem is named after the home city of Euler and the Bernoulli family (who failed to solve the problem). It asks for the precise summation, with proof thereof, of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:
Euler showed that the sum of this series is π2/6 as n approaches infinity. His proof, De summis serierum reciprocarum (E41), (On the sums of series of reciprocals), in English, for ‘mathematically-literate’ readers, is given in wikipedia.org/wiki/Basel_problem.14
Resolving the ‘Seven Bridges of Königsberg’ problem
The old city of Königsberg in East Prussia (now Kaliningrad, Russia) formerly had four land masses connected by seven bridges across the Pregel River, as per the diagram. Citizens occupied their spare time and no doubt improved their health (if not their equanimity) by trying to access these four areas in a single walk by crossing each bridge only once.
Although no one had been able to do it, it seems that no one had the courage (or perhaps the erudition) to say categorically that it couldn’t be done. Enter Euler! In 1735, he resolved the problem with a huge helping of logic, and a speck of arithmetic. He pointed out that:
The choice of route inside each area is irrelevant.
Two different bridges must be used to enter and leave each area.
So each area must be served by an even number of bridges.
However, one area has five bridges and three areas each have three bridges.
Therefore the walk as specified is impossible. Q.E.D.!15
Euler’s comprehensive solution to the above problem, called Solutio problematic ad geometriam situs pertinentis (E53), was published in the Academy’s above-mentioned journal Comentarii in 1736, and comprised 21 numbered paragraphs.16 In this, he used a diagram with the capital letters A, B, C, and D to represent the land areas, and the lower case letters a, b, c, d, e, f, and g to represent the bridges. This was a new concept for that era, and is regarded as laying the foundations for modern graph theory, where a graph is a collection of vertices (≡ land areas) and edges (≡ bridges).17 And it prefigured the concept of topology.18
In 1736, the St Petersburg Academy published his mathematical analysis of Newton’s dynamics titled Mechanica, sive motus scientia analytice exposita (E15 Vol. 1, and E16 Vol. 2) (Analysis of the science of motion). This work used the then new differential and integral calculus and represents the first treatise on what is now called analytic or rational mechanics.
In 1738, he experienced a dangerously high fever, and strong infection led to an abscess in his right eye; a subsequent cataract led to loss of vision in that eye. This was to have serious consequences later in his life.
In Berlin, Euler vs Bible skeptics
In 1741, following turmoil in Russia after the death of Czarina Anna,19 Euler accepted a post at the Berlin (Royal Prussian) Academy of Sciences. This was at the request of King Frederick II of Prussia (Frederick the Great), who wanted the Academy in Berlin to be comparable to those in Paris, London, and St Petersburg. For this he needed the superstars of science and philosophy, such as Euler, who was widely renowned for having won the Paris Prize for the three previous years, 1738, 1739, and 1740.20
Euler lived during the period misnamed ‘the Enlightenment’, when skeptic philosophers such as Voltaire (1694–1778), Hume (1711–1776), Kant (1724–1804), and their fellow ‘freethinkers’, mocked the biblical concept of God, denied the Christian faith, and declared that humanity could be improved solely through rational changes. Through all of this, Euler steadfastly maintained his Christian faith, and in 1746 wrote an impassioned response to the skepticism of his day entitled Defence of the Revelation [i.e. the Bible] Against the Objections of Freethinkers. This was a pamphlet consisting of 53 numbered paragraphs, originally written in German (E92) and printed in Berlin, and later translated into French.21
In this, he began by asserting that happiness involves understanding the truth, because God is truth and the world is the product of His omnipotence and wisdom (Defence paragraphs #1&2). A perfect knowledge of God and His works would be infinite (#3). God is the source of all truth and is the ultimate good (#4). God has written natural law in the hearts of men and requires that men’s actions conform to this law (#5). Since this law originates from God Himself, disobedience to it is rebellion against Almighty God, and this brings divine judgement (#6).
Concerning the Bible, Euler argued that it presents the unique and true source of all our duties in a way that cannot be attributed to the talents of its authors, and so we regard the Bible as having come from God (#26). A multitude of Christians not only saw Christ after He rose from the dead, but they also communicated with Him, so this was not the product of their imagination (#34). Thus the resurrection of Jesus Christ is an incontestable fact, solely the work of God, and hence we can believe all the promises of the Gospel both for this life and the one to come (#36).
Concerning the freethinkers, Euler said that they cannot put forward anything against the arguments on which the divinity of Scripture firmly rests (#37). Yes the Bible does contain things the freethinkers disagree with; if it didn’t, this would be harmful to the Bible (#38). With regard to apparent contradictions in the Bible, there is no science, including mathematics, against which similar or even stronger criticisms re contradictions cannot be made. Yet no one dismisses the certainty of mathematics (#39–41). The objections of freethinkers have long been thoroughly refuted, but because they are not motivated by a desire for the truth, they reject refutations and continually repeat weak and absurd objections (#46). They do not believe that the world had a beginning or will have an end because this would acknowledge the direct action of God (#47).
In this period of 25 years in Berlin, Euler wrote some 380 articles applying mathematics to a host of subjects; of these, 96 were published by the St Petersburg Academy, with whom he maintained a good relationship. He also wrote the two books for which he would become most renowned. These were his two-volume Introductio in analysin infinitorum (E101) in 1748,22 a text on functions and probably the most influential mathematical textbook in modern history, and the Institutiones calculi differentialis (E212) in 1755,23 on differential calculus.
His third landmark book from this period was his two-volume Scientia navalis (E110 Vol.1, and E111 Vol. 2), both in 1749. These dealt with ship design to achieve maximum stability, handling, and speed—features that often work against one another in practise. Soon after its publication he became concerned that it was too technical for navigators and began an elementary abridgment, which in 1773 became his last major work.
The aforementioned Johann Bernoulli, 40 years Euler’s senior, was widely hailed as the undisputed ‘Princeps Mathematicorum’ (prince of mathematics) after the death of Leibniz (1716), and Newton’s withdrawal from the field in advanced age.24 Yet he recognized his one-time pupil’s brilliance early on. His letters to Euler show an early respect, addressing him as “highly educated and brilliant” when Euler was only 21. The salutations in following letters rapidly escalated to frank awe; in 1745 Bernoulli, known as definitely not given to flattery, addressed his letter to the 38-year-old Euler: “To the incomparable Leonhard Euler, the prince among the mathematicians”.25
Letters to a German princess: pioneering science popularization
In 1759, Euler was asked by his close friend Friedrich Heinrich to tutor the latter’s 14-year-old daughter, Friederike Charlotte Leopoldine Louise, who was a second cousin of King Frederick II and became known as the Princess of Prussia. To do this, over the next two years Euler wrote her 234 letters in lucid and cogent layman’s terms, but without equations or formulas, in French, the language of the Prussian court. In these he addressed her as Votre Altesse, Your Highness. These Letters of Euler on Different Subjects in Physics and Philosophy Addressed to a German Princess were originally published in French by the St Petersburg Academy (E343 Vol. 1, and E344 Vol. 2 in 1768, and E417 Vol. 3 in 1772).
A second French edition of three volumes was published in Paris in 1787–1789, after Euler’s death. This was edited by Voltaire’s disciple, Nicolas de Condorcet, who took exception to Euler’s references to God and to Adam and Eve (which showed Euler believed in Genesis), and purposely eliminated some of them. The English editor, Henry Hunter, purposely restored most of these in his English translation of the Letters.
Calinger describes the Letters as “The most exhaustive and authoritative science popularization written during the 18th century.”26 They indeed constituted a unique encyclopedia. Subjects included gravity, tides, the sun, moon and planets of our solar system, Newton’s laws of motion, the nature of sound, light, electricity and magnetism, the atmosphere, heat and cold, the trajectory of cannonballs, and much, much more. He explained scientific devices such as thermometers, telescopes, and microscopes, plus vision and the structure of the eye, etc. And he also taught her logic, for which he used syllogistic diagrams (see box).
In several letters, Euler gave the princess his thoughts about God, prayer, eternal life, evil and sin, divine justice, the usefulness of adversity, and the conversion of sinners. In explaining the marvels of the eye to her in Letter 41, he wrote: “Though we are very far short of a perfect knowledge of the subject, the little we do know of it is more than sufficient to convince us of the power and wisdom of the Creator. We discover in the structure of the eye perfections which the most exalted genius could never have imagined.”27
In Letter 43, p. 174, in the second French Edition, the rationalist editor, Condorcet, deliberately left out Euler’s final long paragraph about God as the Creator of the eye. Hunter restored it as a footnote in his English version. Here is what Condorcet tried to prevent people from reading:
“But the eye which the Creator has formed is subject to not one of all the imperfections under which the imaginary construction of the freethinker labours. In this we discover the true reason why infinite wisdom has employed several transparent substances in the formation of the eye: it is thereby secured against all the defects which characterize every work of man. What a noble subject of contemplation! How pertinent the question of the Psalmist! He who formed the eye, shall he not see? And he who planted the ear, shall He not hear?28 The eye alone being a master-piece that far transcends the human understanding, what an exalted idea must we form of Him, who has bestowed this wonderful gift, and that in the highest perfection, not only on man, but on the brute creation, nay, on the vilest of insects!—E. E.”29
In Letter 110 Euler wrote: “The Holy Scriptures … inform us, that he who meditates only the destruction of his neighbour, suffering himself to be hurried [carried] away by a spirit of hatred, is as criminal in the sight of God, as the actual murderer; and that he who indulges a covetous desire of another’s property is, in his estimation, as much a thief as he who really steals.”30 And in Letter 113, he wrote: “Real happiness is to be found only in God himself; all other delights are but an empty shade, and are capable of yielding only momentary satisfaction.”31
Princess Friederike must surely have been the world’s most erudite teenager! She encouraged the publication of the letters, thereby making science as taught by Euler accessible to a wide range of readers. By 1800, they had gone through thirty editions and had been translated into Danish, Dutch, English, German, Italian, Russian, Spanish, and Swedish.32
Back to St Petersburg
In July 1766, Euler returned to St Petersburg to take up an offer sponsored by Czarina Catherine II (The Great) that involved himself in administration of the Academy there, employment for his three sons, and a future widow’s pension for his wife. Catherine aimed to restore the eminence of the St Petersburg Imperial Academy that had deteriorated since Euler’s departure in 1741, by re-employing ‘the foremost mathematician in Europe’.
In August/September 1766, Euler suffered a serious fever and this combined with a cataract in his one good (left) eye led to near blindness. Undeterred, he diligently pursued his duties and had scientific papers and correspondence read to him by assistants, to whom he dictated responses to be transcribed and forwarded by them. It was also a huge help that his son, Johann, was by then secretary of the Academy. He had this cataract removed in 1771, which briefly restored some sight, until another infection left him nearly completely blind. Calinger writes:
“Even in the onset of near blindness, his astonishing memory, rich imagination, steady willpower, insatiable curiosity, and disciplinary intuition continued to serve him well, and his addiction to research and delight in solving the most difficult problems, led him confidently to proclaim, ‘One more distraction removed’ in relation to his sight.”33
His prodigious memory is illustrated by the fact that he could recite the Aeneid by Virgil from memory, indicating which line was first and last on every page.34
None of this diminished his extraordinary literary productivity, in fact it now increased! “Alone, as lead author, or in the supervision of printing, he presided over a stream of articles and books numbering 415 … . Of these more than 150 did not get published until after his death.”35
In this period, he produced another ‘best seller’: the 500-page, 2-volume Vollständige Anleitung zur Algebra (E387, E388), (Complete Guide to Algebra), originally published in German. Translations quickly appeared in Russian and major European languages. Mathematician Walter Gautschi (1927–) writes: “It is indeed a delight to witness in this work Euler’s magnificent didactic skill, to watch him progress in ever so small steps from the basic principles of arithmetic to algebraic (up to quartic) equations, and finally to the beautiful art of Diophantine analysis.”36
In April 1773, he completed his last major book, his simplified navigation treatise, Théorie complete de la construction et de la manoeuvre des vaisseaux, mise à la portée des ceux, qui s’appliquant à la navigation (E426), (Complete theory of the construction and maneuver of ships brought into the reach of everyone involved in navigation).37 It covered all aspects of ship movement necessary for mariners, in language that sailors, navigators, mast-makers, and ship-builders could easily understand.
In November 1773, Katharina, Euler’s wife of almost 40 years died, aged sixty-six. Three years later he married her half-sister, Salome Abigail Gsell (1723–1794). She managed the household well, and was devoted to caring for her husband until his death seven years later. On the morning of 18 September 1783, he had been discussing with a colleague the newly discovered planet Uranus and the calculations for determining its orbit. That afternoon, while playing with one of his grandsons, he suffered a stroke and lost consciousness, which he never regained.
His memory honoured
Calinger writes: “The scientific world recognised that it had been deprived of one of its great colleagues: the four major royal science academies in London, Paris, Berlin, and St Petersburg, along with societies in Basel, Lisbon, Munich, Stockholm, and Turin, all of which Euler belonged to, announced their profound loss.”38
After Euler’s death, the St Petersburg Academy had enough unpublished writings of his to continue issuing these for nearly 50 more years. Since 1911, the Swiss Academy of Sciences has been publishing Euler’s Opera omnia (Collected Works). So far (2017), some 72 quarto volumes have been printed, in three series—on mathematics, mechanics plus astronomy, and miscellany, respectively—covering approximately 35,000 pages. The 4th series, comprising 10 volumes of Euler’s 3,300 extant letters to 275 correspondents in French, Latin, German, Russian, and (a few) in English, is currently in production.
He has been featured on numerous Swiss, Russian, German, and other countries’ postage stamps, and the Swiss 10-franc banknote. And to honour Euler, astronomers have named a crater on the moon and an asteroid after him.
Gentle and modest, he was the greatest mathematician of his time—by the end of his life all the mathematicians of Europe regarded him as their teacher—which continued into the 19th and 20th centuries. For example, the eminent French mathematician (and atheist) Pierre Simon Laplace (1749–1827) said: “Read Euler; read Euler; he is the master of us all!”39 And he was a totally committed, God-honouring, Bible-believing Christian.
Euler’s influence in today’s mathematics
Much mathematical terminology and notation used today was created, popularized or standardized by Euler, including:
f(x) to denote functions.
x, y, and z as unknowns.
a, b, c for the sides of a triangle.
A, B, C for the opposite angles.
R and r for the circumradius and inradius of a triangle.
The abbreviations sin, cos, tan, cot, sec, csc, for their longer counterparts.
The extensive use of π (although he did not originate this term).
∑ for summation.
Δ for finite difference.
i for the imaginary unit √-1
e for the base of natural logarithms, e ≈ 2.71828.
Euler’s formula for the critical load of a column: Pcr = π2EI/(KL)2.
The formula eiπ = –1 attributed to Euler and called the Euler identity (modernized as eiπ + 1 = 0),40 which mathematicians eulogize as ‘one of the most beautiful formulas in all of mathematics’40 because (for them) it says so much, i.e. the five most important mathematical constants all appearing in one formulation.
Euler’s polyhedral formula also known as the Euler characteristic: V – E + F = 2 (where V = number of vertices, E = number of edges, and F = number of faces of a simple (i.e. without holes) three-dimensional convex polyhedron).
For many more see en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler.
One of the lesser known inventions commonly attributed to Euler is his use of syllogistic diagrams to teach Princess Frederike logic. He did this first in his Letter 103, p. 398, and applied it to various hypothetical situations through to Letter 108 inclusive. Since 1960, Euler diagrams (as they are now called) have been used, along with Venn diagrams (a special case of overlapping Euler diagrams conceived by John Venn c. 1880) to illustrate the relationship of different elements of a set. The significance is not the size or shape of the curves, but how they overlap. A curve that is contained completely within another represents a subset of it.
References and notes
- Euler wrote 80% of his works in Latin, the rest mostly in French, with some in German. He was also fluent in Russian. In 1913, Swedish Mathematician Gustav Eneström enumerated 866 distinct works—books, journal articles, and some letters he deemed to be especially important. Each of these was assigned a number from E1 to E866, which is now referred to as the ‘Eneström number’. Most scholars today use these numbers to identify Euler's writings quickly. For a list of these see eulerarchive.maa.org. We use these numbers in this article. Translations into English are proceeding, with 194 completed as of 23 June 2017, see eulerarchive.maa.org/translations. Return to text.
- Fellmann, Emil A., Leonhard Euler (in German), Rowohlt Taschenbuch Verlag, Reinbek bei Hamburg, Germany, p. 9. Fellmann (1927–) whose academic background is in mathematics, physics and philosophy, has been a visiting professor/lecturer at a number of European universities, including the University of Bonn. Return to text.
- This article is mostly based on Ronald S. Calinger’s monumental biography, Leonard Euler: Mathematical Genius in the Enlightenment, Princeton University Press, USA, 2016. Some details (such as name spellings, some dates) differ among writers, so we have used those of Prof. Calinger in this article. Return to text.
- Ref. 3, p. 11. Return to text.
- Ref. 3, p. 12. Return to text.
- Ref. 3, p. 23. Return to text.
- Title: Meditationes super problemate nautico, quod illustrissima regia Parisiensis Academia scientarum proposuit (Thoughts on a nautical problem, proposed by the illustrious Royal Academy of Sciences in Paris). A summary in English is available at eulerarchive.maa.org/translations-(E4). Return to text.
- The Paris Academy of Science (Academie royale des sciences de Paris) was created by Louis XIV’s chief minister Jean-Baptiste Colbert in 1666, and became a model for many other European academies, including those in St Petersburg and Berlin. Return to text.
- For a list of these, see eulerarchive.maa.org/publications/ParisPrizePieces.html Return to text.
- Planned by Czar Peter I of Russia, it was opened as Academia Scientiarum Imperialis Petropolitanae in 1725 by Peter’s widow and benefactress of the Academy, Czarina Catherine I of Russia. Its purpose was to improve education in Russia and close the scientific gap with Western Europe. Return to text.
- Ref. 3, p. 92. Return to text.
- Thiele, Rüdiger, The Mathematics and Science of Leonhard Euler, Ch. 5 in Mathematics and the Historian’s Craft, Springer, New York, 2005, p. 98. Return to text.
- Ref. 3, p. 188. Return to text.
- He later published a more rigorous proof, and contributions from other mathematicians are also available in this reference. Return to text.
- Q.E.D. = Quod Erat Demonstrandum (what was to be demonstrated). Return to text.
- A simplified version is available in the Mathematical Association of America article: maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem. Return to text.
- In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once. An Eulerian circuit (or Eulerian cycle) is an Eulerian trail that begins and ends on the same vertex. So a graph of the Königsberg bridges is not Eulerian. Return to text.
- Topology is the study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures. Return to text.
- Catherine I died in 1727, with her death being made known on the day Euler arrived (ref. 3, p. 65). She was succeeded by Peter II who in turn was succeeded by Anna Ivanovna, grand-niece of Peter I. Return to text.
- Ref. 3, p. 113; Euler won outright in 1739, while 1738 and 1740 were shared wins. Return to text.
- German title: Rettung der göttlichen Offenbahrung gegen die Einwürfe der Freigeister.French title: Défense de la Révélation contre les objections des esprits-forts. Return to text.
- Translated as Introduction to Analysis of the Infinite by John D. Blanton in 1988–89, (Springer-Verlag, Berlin). Return to text.
- Chapters 1–9 translated as Foundations of Differential Calculus by John D. Blanton in 2000, (Springer-Verlag, New York). Return to text.
- Ref. 2, p. 23. Return to text.
- Ref. 2, p. 24. Return to text.
- Ref. 3, p. 465. Return to text.
- Letters of Euler to a German Princess, trans. by Henry Hunter, Vol. 1, No. 41, p. 165. Note: all quotations and hence page numbers are from Hunter’s 1802 English Edition). Return to text.
- Psalm 94:9. Return to text.
- Ref. 27, No. 43, p. 174. “E. E.” stands for “English Editor” (Letters, Preface, pp. xxi–xxii), i.e. this was Henry Hunter’s way of identifying himself as the person with the authority to restore Condorcet’s deletion(s). Return to text.
- Ref. 27, No. 110, p. 432. Return to text.
- Ref. 27, No. 113, p. 441. Return to text.
- Ref. 3, p. 461. Return to text.
- Ref. 3, p. 454. Return to text.
- The Aeneid is a Latin epic poem, written by Virgil between 29 and 19 BC, that tells the legendary story of Aeneas, a Trojan who travelled to Italy, where he became the ancestor of the Romans. It comprises 9,896 lines in dactylic hexameter. Return to text.
- Ref. 3, p. 456. Return to text.
- Gautschi, W., Leonhard Euler: His life, the Man, and His Works, p. 26. Return to text.
- Published in French by the St Petersburg Academy in 1773, and an augmented second edition in 1776. Return to text.
- Ref. 3, p. 532. Return to text.
- Journal des Savants, January 1846, p. 51. Return to text.
- Sandifier, E., “e, π, and i: Why is ‘Euler’ in the Euler identity?”, How Euler Did It, August 2007. Return to text.