**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.