Mathematical Masterpieces: Teaching with Original Sources
Reinhard Laubenbacher & David Pengelley
Mathematics, New Mexico State University,
Las Cruces, NM 88003
[ Vita Mathematica: Historical Research and
Integration with Teaching,
R. Calinger (ed.), MAA, Washington, 1996, pp. 257--260]
Our upper-level university honors course, entitled Great Theorems:
The Art of Mathematics, views mathematics as art and examines selected
mathematical masterpieces from antiquity to the present. Following a common
practice in the humanities, for example in Chicago's Great Books program
and St. John's College curriculum, we have students read original texts
without any modern writer or instructor as intermediary or interpreter.
As with any unmediated learning experience, a special excitement comes
from reading a first-hand account of a new discovery. Original texts can
also enrich understanding of the roles played by cultural and mathematical
surroundings in the invention of new mathematics. Through an appropriate
selection and ordering of sources, students can appreciate immediate and
long-term advances in the clarity, elegance, and sophistication of concepts,
techniques, and notation, seeing progress impeded by fettered thinking
or old paradigms until a major breakthrough helps usher in a new era. No
other method shows so clearly the evolution of mathematical rigor and abstraction.
The end result is a perception of mathematics dramatically different
from the one students get from traditional courses. Mathematics is now
seen as an evolving human endeavor, its theorems the result of genius struggling
with the mysteries of the mathematical universe, rather than an unmotivated,
ossified edifice of axioms and theorems handed down without human intervention.
For instance, after reading Cayley's paper (see below) introducing abstract
group theory, the student is much less bewildered upon seeing the axiomatic
version so devoid of any motivation. Furthermore, Cayley makes the connection
to the theory of algebraic equations, which a student might otherwise never
become aware of. An additional feature of the method is that suddenly value
judgments need to be made: there is good and bad mathematics, there are
elegant proofs and clumsy ones, and of course plenty of mistakes and unsubstantiated
assertions which need to be examined critically. Later follows the natural
realization that new mathematics is being created even today, quite a surprise
to many students.
To achieve our aims we have selected mathematical masterpieces meeting
the following criteria. First, sources must be original in the sense that
new mathematics is captured in the words and notation of the inventor.
Thus we assemble original works or English translations. When English translations
are not available, we and our students read certain works in their original
French, German, or Latin. In the case of ancient sources, we must often
depend upon restored originals and probe the process of restoration. Texts
selected also encompass a breadth of mathematical subjects from antiquity
to the twentieth century, and include the work of men and women and of
Western and non-Western mathematicians. Finally, our selection provides
a broad view of mathematics building upon our students' background, and
aims, in some cases, to reveal the development over time of strands of
mathematical thought. At present the masterpieces are selected from the
In our experience students find the study of original sources fascinating,
especially when combined with readings in the history of mathematics. The
benefits for instructors and students alike are a deepened appreciation
for the origins and nature of modern mathematics, as well as the lively
and stimulating class discussions engendered by the interpretation of original
sources. As part of their assignment students complete a research project
on a topic of their choice, with the only constraint that it be part mathematical
and part historical. Other assignments focus for the most part on mathematical
points in the sources and related topics.
The Greek method of exhaustion for computing areas and volumes, pioneered
by Eudoxus, reached its pinnacle in the work of Archimedes during the third
century BC. A beautiful illustration of this method is Archimedes's determination
of the area inside a spiral. 
An important ingredient is his summation of the squares of the terms in
certain arithmetic progressions. As in all of Greek mathematics, even this
computation is phrased in the language of geometry. Further advance toward
the definite integral did not come until the Renaissance.
The search for algorithms to solve algebraic equations has long been important
in mathematics. After Babylonian and ancient Greek mathematicians systematically
solved quadratic equations, progress passed to the medieval Arab world.
The work of Arab mathematicians began to close the gap between the numerical
algebra of the Indians and the geometrical algebra of the Greeks. Notable
is Omar Khayyam's Algebra of the late 11th or early 12th century.
Here he undertakes the first systematic study of solutions to cubics and
writes: ``Whoever thinks algebra is a trick in obtaining unknowns has thought
it in vain. No attention should be paid to the fact that algebra and geometry
are different in appearance. Algebras are geometric facts which are proved.''
In addition to a general discussion of his view of algebra, an excellent
selection to read is his treatment of the cubic ,
which is solved geometrically via the intersection of a parabola and a
The next major advance toward solving algebraic equations did not come
until the sixteenth century with the work of Cardano and his contemporaries
in Europe. During that time, ancient Greek mathematics was rediscovered,
often via Islamic sources, and old problems were attacked with new methods
and symbols. The general arithmetic solution for equations of degree three
and four essentially awaited Cardano's seminal Ars Magna (1545),
in which Khayyam's equation
now receives a virtually algebraic treatment. 
It is instructive to compare the two texts. The final chapter in the search
for general solutions for the quintic or higher-degree equations was later
written by Niels Abel and Evariste Galois.
By the early seventeenth century the Greek method of exhaustion was being
transformed into Cavalieri's method of indivisibles, the precursor of Leibniz's
infinitesimals and of Newton's fluxions. One of the most astonishing results
of this period was the discovery that an infinite solid can have finite
volume. Torricelli, a pupil of Galileo, demonstrated by the method of indivisibles
that the solid obtained by revolving a portion of a hyperbola about its
axis has finite volume. 
Closed formulae for sums of powers of consecutive integers such as
were already of interest to Greek mathematicians. For instance, Archimedes
used them to determine areas, such as for the spiral above. After much
effort over the centuries, Fermat in the early seventeenth century first
recognized the existence of a general rule, and called this ``perhaps the
most beautiful problem of all arithmetic.'' 
Shortly thereafter Pascal provided a recursive description of the formulae
for sums of powers in an arithmetic progression in Potestatum Numericarum
Summa (Sommation des Puissances Numériques). 
As Pascal mentions, these formulae are connected to the continuing development
of integration techniques at the time.
Improving on the work of Pascal (which he apparently was not aware of),
Bernoulli, in the late seventeenth and early eighteenth centuries, provided
the first general analysis of the polynomial expressions giving the sums
of powers. His Ars Conjectandi (1713) noticed surprising patterns
in the coefficients, involving a sequence of numbers now known as the Bernoulli
Today these Bernoulli numbers are important in many areas of mathematics,
such as analysis, number theory, and algebraic topology.
The eighteenth century was dominated by applications of the calculus, many
of them provided by Euler, who was a master in working with infinite series.
His De Summis Serierum Reciprocarum contains a variety of results
on sums of reciprocal powers, including a recursive analysis of .
Euler's computations are examples of the general formula
which involves Bernoulli numbers and can be derived directly from
the above text of Bernoulli.
The early nineteenth century saw the beginnings of modern number theory
with the publication of Gauss' Disquisitiones Arithmeticae in 1801.
Efforts to prove Fermat's Last Theorem contributed to the development of
sophisticated techniques by mid-century. Before then, however, the only
progress toward a general solution, beyond confirmation of the conjecture
for exponents five and seven (three and four were confirmed by Fermat and
Euler), was provided by Sophie Germain. She developed a general strategy
toward a complete proof, and used the theorems she proved along the way
to resolve Case I of Fermat's Last Theorem for all exponents less than
100. Sophie Germain never published her work. Instead, a part of it appeared
in 1825, in a supplement to the second edition of A. M. Legendre's Théorie
des Nombres, where he credits her in a footnote .
From its beginning, Euclid's parallel postulate 
was controversial. Attempts to prove it from the others led to the nineteenth-century
discovery that it is independent of the rest, allowing for other geometries.
Lobachevsky, the co-discoverer of non-Euclidean geometries along with Gauss
and Bolyai, made several attempts at gaining the attention of the mathematical
world with his ideas. In 1840 he published the very readable book Geometrische
Untersuchungen zur Theorie der Parallellinien, laying out the foundations
of hyperbolic geometry. 
In addition to Fermat's Last Theorem, another driving force in the development
of number theory was the Quadratic Reciprocity Theorem and the study of
higher reciprocity laws. The theorem, discovered by Euler and restated
by Legendre in terms of the symbol now bearing his name, was first proven
by Gauss. The eight different proofs Gauss published, for what he called
the Fundamental Theorem, were followed by dozens more before the end of
the century, including four given by Gotthold Eisenstein in the years 1844--45.
His article Geometrischer Beweis des Fundamentaltheorems für die
quadratischen Reste 
gives a particularly elegant and illuminating geometric variation on Gauss'
third proof. [12,13]
WILLIAM ROWAN HAMILTON:
After extended efforts, Hamilton's attempts to define a multiplication
on three-dimensional vectors led to his flash of insight in 1843 that this
was possible if one allowed vectors of dimension four. Selections from
his book Elements of Quaternions give an interesting account of
his geometric view of the quaternions. 
They provided one of the first important examples of a non-commutative
number system, thus spurring the development of abstract algebra.
By the mid-nineteenth century, group structures had emerged implicitly
in several branches of mathematics, for instance in modular arithmetic,
the theory of quadratic forms, as permutations in the work of Galois on
algebraic equations, in the work of Hamilton on the quaternions, and in
the theory of matrices. In his paper On the Theory of Groups, as Depending
on the Symbolic Equation
Cayley was the first to investigate the abstract concept of a group and
began the classification of groups of a given order in a purely abstract
In an attempt to explain calculus better to his students, Dedekind constructed
the real numbers through what is now known as Dedekind cuts, from which
their continuity can be deduced rigorously. Together with Cantor's equivalent
construction, this work represents the culmination of the century-long
effort to arithmetize analysis. In 1872 he published these ideas in the
celebrated Stetigkeit und die Irrationalzahlen. 
Mathematics was changed forever toward the end of the nineteenth century
by Cantor's bold embrace of the infinite. Selections from his readable
Beiträge zur Begründung der Transfiniten Mengenlehre develop
the foundations of his theory of transfinite numbers, or what we today
call ordinals and cardinals. 
A vast generalization of a Dedekind cut, combined with ideas from game
theory, led Conway in the 1970s to create, with a single construction,
the so-called surreal numbers, an enormous number system containing both
the real numbers and Cantor's ordinals. Chapter 0 of his book On Numbers
and Games provides a delightful introduction to his surreal world.
Conway's work is one of the rare examples of very recent mathematics that
is deep but can be read with minimal background.
After using the traditional lecture approach for some time, we discovered
the amazing effectiveness of a combination of two pedagogical devices:
the ``discovery approach''; and extensive writing. The discovery method
assumes that students should discover the mathematics for themselves. Hence,
for each source we briefly provide the historical and mathematical context,
alert the students to any difficult points in the text, and then stand
by to answer questions while they work through the source in pairs. A wrap-up
discussion lets everyone share his or her understanding of the material,
and any remaining difficulties are resolved. This method generates tremendous
enthusiasm and a genuine sense of discovery. Strikingly, we see that this
method also leads to a deeper understanding of the sources than the lecture
The students write frequently and about every aspect of the course:
the mathematical details of the sources, their historical context, lecture
notes, thoughts jotted in the throes of problem solving, and their own
ideas about the process that creates mathematics. This writing experience
leads to a more comprehensive view of the great theorems we study as well
as a much better grasp of the mathematical details in their proofs.
For students in the sciences, engineering, and mathematics education,
our course provides both a broad and humanistic view of mathematics, and
for many students it is a breath of fresh air within the traditional mathematics
curriculum. For mathematics majors the course is an enriching capstone
for their entire undergraduate experience.
To main page on
Teaching with Original Historical
Sources in Mathematics
D. Pengelley and R. Laubenbacher
Sun Feb 7 00:38:56 MST 1999