In Celebration of Leonhard Euler's 300th Birthday
15 April 2007.
In honor of Euler's 300th birthday a display of some of his works which are in the library of the United States Military Academy was arranged with the kind help of the librarians in special collections. Because of the fragility of the books, the display could not stay up for long. Below is the text that accompanied the books.
The Elements of Algebra was written when Euler was going blind. The book was written in German (1770), but first published in Russia (176869); This is an English translation of 1822. It was truly a best seller. It begins with the most elementary facts in algebra such as the arithmetic of signed numbers because Euler was training an amanuensis, a former tailor of modest ability who understood the work completely when finished, thus showing Euler's great ability as a teacher. It progresses in an orderly and precise way through all of elementary algebra and ends with a sophisticated discussion of Diophantine equations, i.e., equations where the only solutions considered are positive integers. A later English edition was reprinted in 1984 and is available for your reading pleasure. Euler's father – a minister who had studied mathematics with Jakob Bernoulli – taught his son algebra using an edition of Christoff Rudolff 's Coss (1525), the first German book entirely devoted to algebra.
Leonhard Euler, who was born 300 years ago on 15 April 1707, published a three volume work on the integral calculus: Institutionum calculi integralis (1768 1770). The second volume is open to the section where he discusses linear homogeneous differential equations with constant coefficients. The historical importance of this class of equations is that it forced Euler to conceive of trigonometric functions. Trigonometry is ancient, but sine and cosine did not become functions until 1739 when Euler needed functions to solve this class of differential equations. Look for the sines and cosines in examples II and III.
The first of Euler's 6 volumes on the calculus is his Introductio in analysin infinitorum of 1748. On display is the 1796 French translation Introduction à l'analyse infinitésimale . In this work, Euler makes functions the primary object of study in the calculus, an emphasis
which has continued to this day. It is open to the first page of Chapter VII which deals with the trigonometric functions. Note that Euler takes a circle of radius 1; ever since we have been doing trigonometry on the unit circle. Here he introduces the symbol "π"; it was not known until 1896 that William Jones had introduced the symbol earlier in 1706. Also on display is a 1791 German translation, Leonhard Eulers Einleitung in die Analysis des Unendlichen . It is open to Brouckner's continued fraction for 4/π. Note that while the denominators are 2, the numerators grow. This is why is is clear to Euler that π is irrational.
Leonhard Euler's Vollstandige Anleitung zur Differenzial-Rechnung , three volumes, 1790-1793, is the second of his great works on the calculus. It begins with the study of finite differences, a topic from MA103, and proceeds to present the differential calculus in a form that we use today.
The Letters of Euler on different subjects in natural philosophy: addressed to a German princess (1835) is a masterpiece of popularization. Written in French and published in St. Petersburg in 1768 and 1772 in three volumes, the work became an immediate hit and was translated into the major languages. The 234 letters discuss music theory, philosophy, mechanics, optics, astronomy, and theology. They refute ideas of Berkeley, Hume, and Wolff.
The 1744 original of Euler's Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive Solutio problematis isoperimetrici latissimo sensu accepti is on display. This is Euler's influential work on the calculus of variations. In the ordinary calculus, the variable is a real number, while in the calculus of variations, functions themselves are the variables. The prototype problem in this field is the brachistochrone problem: Find the path (the function) along which a body moves under the influence of gravity from one point to a lower point. The solution is a cycloid.
Observations upon the new principles of gunnery (1777) is a new acquisition for the USMA library and is on display for the first time. It is exceptionally rare; WorldCat lists only 9 other copies! It is probably the most important book ever written on gunnery, for it provides a calculus based explanation for projectile motion where air resistance is taken into account. After Euler moved to Berlin, Frederick the Great asked him what the best book on gunnery was. Although Robins had written a polemical tract attacking Euler's Mecanica , Euler graciously said that Robins has produced the best work. At the behest of Frederick, Euler translated the book into German and added some mathematical ``annotations'' thereby increasing the length of the book from 150 pages to 450. Napoleon Bonaparte read the work in French translation and his notes survive. Carl von Clausewitz, Vom Kriege (1832) wrote ``Bonaparte rightly said that many of the decisions faced by the commander-in-chief resemble mathematical problems worthy of the gifts of a Newton or an Euler.''
Benjamin Robins hoped to become the first professor of Mathematics at the newly founded (1742) Woolwich Military Academy and so prepared a course of lectures on gunnery. But since he had written a polemical tract against a powerful politician he was blocked from the job. Nonetheless, he published his New Principles of Gunnery , a work which provided considerable experimental evidence about the trajectories of projectiles in air. This edition, from his Works (1761) is open to Proposition VI. He argues, that in the absence of air resistance, Galileo predicted that a rifle ball shot at 1700 ft/sec at a 45 degree angle would go 17 miles (get out your calculus and check). However, with air resistance it would go less than half a mile.
Scientia navalis (1749) is the most famous of the several books wrote on ships and navigation. His early work on the masting of ships won an accessit or honorable mention in the 1728 Paris Prize competition. At this time, Euler had not seen a tall ship.
Theoria der Planeten und Cometen is a 1781 translation of Euler's Theory of Planets and Comets, which first appeared in Latin in 1744.