The Renaissance began with the reintroduction of Aristotle’s thought into the
West, especially by Thomas Aquinas. The resulting focus on the natural world and on
reason as the means to understand it revolutionized European thought and led to major
mathematical and scientific discoveries, including advances in algebra. Philosophers,
mathematicians, and scientists differed in their view of the nature of reason, but by the
end of the Renaissance, they had accepted its crucial role in the acquisition of knowledge.
The result was the next major period of Western history, the Age of Reason. Although
there is not uniform agreement on the exact dates of this era, we will consider the
seventeenth and eighteenth centuries, beginning especially in the latter half of the
seventeenth.
Let us begin with the key development, mathematically speaking, that provided
the transition from the Renaissance to the Age of Reason: the integration of algebra and
geometry into analytic geometry, independently and almost simultaneously by Rene
Descartes (1596-1650) and Pierre de Fermat (1601-1665). Historically, geometry and
algebra had developed as separate branches of mathematics. In the third century B. C.,
the ancient Greek mathematician, Apollonius, associated curves with equations in his
monumental work on the conic sections, but he did not recognize that equations could be
represented by curves. This was the crucial insight of Descartes and Fermat that led to
analytic geometry.
In the sixteenth century, the Frenchman, Francois Viete (1540-1603), contributed
to abstract mathematical thinking by representing coefficients in equations by consonants
and variables by vowels. He then inaugurated the study of the theory of equations by
establishing relationships among the roots of types of equations rather than by merely
solving specific ones. Descartes used his insights into analytic geometry to do extensive
work in the theory of equations, including his well-known rule of signs. He modified
Viete’s approach to notation by using letters toward the first of the alphabet to represent
constants and ones toward the end to represent unknowns, a practice we still follow today.
Fermat used his achievements in analytic geometry to take steps toward the
development of calculus. In a treatise entitled Method of Finding Maxima and Minima,
he essentially took the derivative of a polynomial function and set it to zero to find its
relative extrema. By extending his technique, he calculated the slope of a curve at any
point on it. He did not have the explicit concept of a limit, but otherwise his method is
the one we use today. He also approximated the area under a curve by means of
circumscribed rectangles. Although he left no record that he recognized the inverse
relationship between differentiation and integration, his work strongly hinted at the
Fundamental Theorem of Calculus.
The one person most responsible for launching the Age of Reason mathematically
and scientifically was Isaac Newton (1642-1727). His revolutionary discoveries
dramatically demonstrated the power of reason to grasp the nature of reality. Although
he was reluctant to publish, at the urging of the astronomer, Edmund Halley, he
composed Mathematical Principles of Natural Philosophy (also referred as the Principia,
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part of its name in Latin). It appeared in 1687 and is considered to be the most important
scientific work ever written.
During the years 1665 and 1666, Newton made three major discoveries: calculus,
the law of gravitation, and the nature of color. He developed differentiation
systematically, and he inverted the process to find an area. Isaac Barrow (1630-1677),
Newton’s teacher at Cambridge, evidently understood the possibility of such an approach,
but Newton was the first person to formulate a general method of solving the area
problem and thereby to grasp the Fundamental Theorem of Calculus. He also gave the
first correct statement of the binomial series for rational exponents, and he came to
realize that infinite series were more than tools of approximation: they provided a means
of representing functions. He nearly succeeded in defining the concept of a limit
although a satisfactory definition was not formulated until the nineteenth century.
In the Principia, Newton discussed his work in calculus and applied it to his three
laws of motion and the law of gravitation to obtain profoundly important results,
including the derivation of Kepler’s three laws and the deduction of the path of motion of
one body around a second, fixed body. By studying the motion of bodies in resisting
media such as air and water, he launched the field of hydrodynamics. His analysis of the
solar system included the calculation of the mass of the sun, the average density of the
earth, the variation of the earth’s gravitational attraction across its surface, the precession
of the equinoxes, and the cause of the tides.
The ancient Greeks believed that different principles explained motion near the
earth versus motion beyond the moon. Newton’s law of gravitation united Kepler’s work
on the movement of the planets and Galileo’s discoveries about bodies falling to the earth
to provide strong evidence that nature is uniform in its behavior.
Newton’s work on the nature of light and color was also of profound importance.
The Atomist philosophers in ancient Greece held that the concept of color is subjective
and cannot be quantified. By decomposing white light into its spectrum, Newton
changed the understanding of color fundamentally by showing that it is based on
wavelengths, which, of course, are quantifiable. The combined impact of his
achievements fostered an intellectual revolution in Europe.
Gottfried Wilhelm Leibniz (1646-1716) formulated his ideas concerning calculus
with no knowledge of Newton’s work. He also developed the techniques of
differentiation and integration, and he understood the inverse relationship between the
two. His methods of solving separable and homogeneous first order differential
equations still appear in modern textbooks. In addition to his work in calculus, he
introduced the term “function” and used it essentially as we do today. Another of his
contributions was the introduction of the notion of determinants in the context of systems
of equations.
Leibniz was quite adept at devising excellent mathematical symbolism. He is
responsible for the notation we employ to denote the differential and the integral, and he
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was the first mathematician to use the dot consistently for multiplication. Other
innovations of his include the colon for division and the similarity and congruence
symbols.
In 1642, Blaise Pascal (1623-1662) invented the first mechanical calculating
machine, one that would add and subtract. In 1671, Leibniz improved on Pascal’s
achievement by designing a device that would also multiply and divide. Their work
launched advances in the automation of computation that continue to this day.
Newton’s work on calculus preceded that of Leibniz by about ten years, but the
latter published his first. A bitter dispute arose between the two concerning who should
receive priority. Today it is recognized that they made their discoveries independently
and deserve equal credit, just as with Descartes and Fermat in the case of analytic
geometry. The dispute had a very negative effect on British mathematicians, however,
because it isolated them from their counterparts on the European continent.
The development of calculus unleashed a torrent of mathematical and scientific
activity, including the contributions of the Bernoulli family across multiple generations.
The most prominent members were Jacob (also known as Jacques) Bernoulli (1654-1705)
and his brother Johann (Jean) (1667-1748), both of whom were influenced by Leibniz.
Jacob determined the equation of the catenary, studied the equation r^2 = a cos 2x
(named the lemniscate of Bernoulli) and extensively investigated the logarithmic spiral.
He was so fascinated by this last curve he requested that it be inscribed on his tombstone.
He also published the polar coordinate system essentially in its modern form although it
had been used earlier by Newton, and he suggested the term “integral” to Leibniz, who
adopted it.
Fermat and Pascal founded the theory of probability when they corresponded on a
problem related to a game of dice. Jacob Bernoulli wrote the first significant work on
probability, including a general theory of permutations and combinations. Using
mathematical induction, he provided the first rigorous proof of the binomial theorem for
positive integer exponents, and he considered the expansion of (1 + 1/n)^n. He
determined that the limit of this expression as n approaches infinity is less than 3 and he
proposed finding it in order to calculate continuously compounded interest. Another of
his major contributions to probability was the law of large numbers.
Both Jacob and Johann Bernoulli advanced differential equations by solving y’ +
P(x) y = Q(x) y^n for any real number n, known today as Bernoulli’s equation. The
latter’s solution was based on the substitution z = y^(1-n), an approach that appears in
modern textbooks. The younger brother is also credited with developing the calculus of
exponential functions. He studied ones of the form y = b^x, as well as the more general y
= x^x. Another of his contributions was the introduction of the partial fractions
integration technique.
In 1692, Johann Bernoulli tutored a French marquis, G. F. A. de ĽHôpital (16611704), in calculus, and they signed an agreement in which Bernoulli granted ĽHôpital use
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of his mathematical discoveries in exchange for a regular salary. In 1696, ĽHôpital
published the first textbook on differential calculus, and it included one of Bernoulli’s
major results, the technique that we call ĽHôpital’s Rule. The book was well written and
enjoyed considerable success throughout the eighteenth century, as did another textbook
by ĽHôpital on analytic geometry.
When we hear the name of Abraham de Moivre (1667-1754), we naturally think
of his famous theorem and the calculation of the nth roots of a real or complex number.
However, he did significant work in probability as well, addressing actuarial problems
and questions related to life annuities. As we have seen, Jacob Bernoulli was also
interested in the mathematics of finance. An early contributor to this area was the Dutch
mathematician, Jan De Witt (1629-1672), who wrote a tract entitled A Treatise on Life
Annuities. In addition, he completed what is often considered to be the first textbook on
analytic geometry, in which he used the focus-directrix ratio to define conic sections.
Indeed, he introduced the term “directrix” into the vocabulary of mathematics.
Newton’s extensive use of infinite series stimulated the Englishman, Brook
Taylor (1685-1731), to study the series that bears his name. Taylor was also quite
interested in the concept of perspective and wrote two books on the subject. Another
contributor to the topic of series was the Scottish mathematician, Colin Maclaurin (16981746), whose series, of course, is a special case of Taylor’s. Maclaurin also obtained
results on conic sections, and he published the method known as Cramer’s Rule before
Gabriel Cramer (1704-1752) himself.
A student of Johann Bernoulli, Leonhard Euler (1707-1783) became one of the
dominant mathematicians of the eighteenth century. Like Leibniz, he was quite adept at
devising fruitful mathematical notations. He was the first to use the letters e and i to
represent the base of the natural logarithm system and the imaginary unit respectively,
and he popularized the Greek letter pi to stand for the ratio of the circumference of a
circle to its diameter. It is to Euler that we owe the practice of using lower case letters for
the sides of a triangle opposite angles denoted by the corresponding upper case letters, as
well as the Greek letter sigma to indicate a summation. His abbreviations, sin, cos, tang,
cot, sec, and cosec, for the six trigonometric ratios are almost identical to those we use
today. Perhaps his most important contribution of all to mathematical symbolism was the
use of f(x) to represent a function of x.
Far more important than Euler’s introduction of functional notation, however, was
his injection of the concept of a function into the heart of mathematics. In addition to
representing algebraic expressions as functions, he defined the concept of radian measure
of angles, gave a strictly analytic presentation of trigonometric functions, treated
logarithms as exponents in the manner we do today, and extended the idea of a function
to ones that are piecewise defined. He clearly formulated what is now known as Euler’s
formula, e^(ix) = cos x + i sin x, which is used, for example, to represent the solution of
certain linear differential equations with constant coefficients. Zero, one, e, pi, and i, are
considered to be five of the most important numbers in mathematics. If x is assigned the
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value of pi in Euler’s formula, all five appear in the resulting equation, for it can be
written as e^(i pi) + 1 = 0.
Many of the methods taught today in differential equations courses originated
with Euler. He was primarily responsible for the concept of an integrating factor, the
distinctions between homogeneous and nonhomogeneous equations and general and
particular solutions, and the systematic methods for solving linear equations of higher
order with constant coefficients. He contributed to the solving of exact equations by
showing that the continuous mixed partials of a function of two variables are equal, and
he also advanced work with functions of several variables in another respect by using
iterated integration to evaluate multiple integrals.
Euler’s work with infinite series was not rigorous, but he used the concept to
obtain important results, for example, in number theory. He developed a proof of the
infinitude of the prime numbers based on the divergence of the harmonic series, and he
showed that the series whose terms are the reciprocals of the perfect squares converges to
pi/6. This last result was a problem of particular interest to Jacob Bernoulli. Another of
Euler’s contributions to number theory was his phi function, which represents the number
of positive integers, including one, that are less than and relatively prime to a given
positive integer.
Euler also had a fundamental impact on the teaching of analytic geometry, both
two- and three-dimensional. Although he did not originate the use of polar coordinates,
as we have seen, he gave a thorough, systematic treatment of them that included
conversion equations between rectangular and polar forms in modern trigonometric terms,
and he used both positive and negative numbers for the radius coordinate.
Another of his innovations was the representation of curves by means of parametric
equations, and he introduced the use of arc length as a parameter.
The extension of the ideas of analytic geometry to three dimensions had been
suggested by several mathematicians, including Johann Bernoulli and one of his students,
Jacob Hermann (1678-1733). However, Euler was evidently the first to observe that the
quadric surfaces form a family analogous to the conic sections. He gave a systematic
classification of these types, including the ellipsoid, the hyperboloids of one and two
sheets, the cone, and the elliptic and hyperbolic paraboloids.
When Euler was at the Berlin Academy, where he spent twenty-five years, one of
his colleagues was Johann Heinrich Lambert (1728-1777). Just as Euler had developed
the trigonometric functions analytically, Lambert did the same for the hyperbolics, using
the notations sinh x, cosh x, and tanh x in the process. Another of his contributions was
to prove that pi is irrational. He also attempted to derive Euclid’s parallel postulate from
the other axioms of Euclidean geometry until he realized that there was a flaw in his
proof. No mathematician prior to the nineteenth century came closer to developing a
non-Euclidean geometry than he.
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Another leading mathematician of the eighteenth century was Joseph-Louis
Lagrange (1736-1813). His Analytical Mechanics gave an axiomatic treatment of
Newtonian mechanics, and although his attempt to develop calculus on the basis of
infinite series was not successful, in the process he launched the study of the theory of
functions of a real variable. As part of this effort, he introduced the term “derived
function,” from which our word “derivative” comes, and he devised the prime notation
for derivatives. Among his other accomplishments were the first formulation of the
Mean Value Theorem for derivatives, the technique of Lagrange multipliers, and the
method of variation of parameters to solve nonhomogeneous, linear differential equations.
In algebra, Lagrange’s work on the theory of equations prepared the way for the
concept of a group and the results of Abel and Galois on the unsolvability by radicals of
the general quintic and higher degree polynomial equations. Of course, the theorem that
the order of a subgroup of a finite group divides the order of the group is named for him.
In number theory, he studied congruences, and he proved that every positive integer is the
sum of at most four perfect squares.
Nicolas de Condorcet (1743-1794) exhibited the strong interest of numerous
mathematicians during the Age of Reason in the application of mathematics to social
questions, as well as the physical sciences. In addition to publishing books on both
probability and integral calculus, he developed the pairwise comparison voting method
that bears his name. He was also a member of the French committee that established the
metric system, as was Lagrange. When some groups in France opposed inoculation for
smallpox, Condorcet used his knowledge of probability and statistics to defend the
practice.
Pierre Simon de Laplace (1749-1826) made major contributions both to
mathematics and mathematical physics. His work on the theory of probability was more
significant than that of any other mathematician, and by completing the gravitational
portion of Newton’s work in his monumental Celestial Mechanics, he showed that the
solar system is stable. He also improved Newton’s calculation of the speed of sound. Of
course, he was responsible for the method of Laplace transforms in differential equations.
We have seen that several mathematicians of the Age of Reason had an interest in
the social sciences. A strong proponent of this breadth of outlook was the British
philosopher, John Locke (1632-1704), who was a friend of Newton and who understood
in a general way what Newton had accomplished mathematically and scientifically. In
fact, Locke advocated the application of reason to all questions, both scientific and
humanistic. His influence led to the designation of the eighteenth century portion of the
Age of Reason as the Enlightenment.
Locke’s advocacy of reason to address nonscientific questions, as well as
scientific ones, had a profound cultural impact, especially on political philosophy. He
held that individuals possess rights by their nature and that the purpose of government is
to protect those rights. His view that ultimate sovereignty rests with the individual, not
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the state, was a direct challenge to the monarchies of his day. In fact, at one point he fled
to Amsterdam and lived under an assumed name to escape persecution and possible
execution in England. His work had a fundamental effect on Thomas Jefferson and the
other Founding Fathers of the United States. The Declaration of Independence is a
thoroughly Lockean document: In the famous second paragraph, Jefferson stated his
political axioms, from which he derived the nature of government. In doing so, he was
following Locke, who had written that “in all sorts of reasoning every single argument
should be managed as a mathematical demonstration.” As he was about to become the
first secretary of state following the ratification of the Constitution and the beginning of
the Washington administration, Jefferson again emphasized the nation’s debt to Locke
when he said: “It rests now with ourselves alone to enjoy in peace and concord the
blessings of self-government so long denied to mankind: to show by example the
sufficiency of human reason for the care of human affairs….”
Indeed, Jefferson embodied Locke’s commitment to apply reason to all questions.
He was well versed in the mathematical advances of his time, and it was at his urging that
the United States placed its monetary system on a decimal basis. Always interested in
agriculture, he used his knowledge of calculus and surfaces of least resistance, learned by
studying Newton’s Principia and related works, to invent an improved plow. Among his
vast multitude of achievements, he was a self-taught architect. When he founded the
University of Virginia and designed the Rotunda, its main building, he placed the library
on the top floor. Illumination by means of skylights, an architectural device he frequently
used, represented the enlightenment of the mind. Concerning the University, he wrote:
“For here we are not afraid to follow truth wherever it may lead nor to tolerate any error
as long as reason is free to combat it.” His three intellectual heroes were Locke, Newton,
and Francis Bacon, who advocated the experimental method as the fundamental
technique by which science advances. Because of Jefferson’s unswerving commitment to
reason and his insatiable pursuit of knowledge, he came to symbolize the American
Enlightenment.
One of the goals of mathematicians and scientists during the Age of Reason was
to analyze the nature of motion. To concretize the cultural impact of their work, let us
compare two famous sculptures, one by Michelangelo (1475-1564) and the other by
Gianlorenzo Bernini (1598-1680). Both depict the Biblical duel between David and
Goliath. Michelangelo, who died the year Galileo was born, portrays a stationary David
as he prepares for battle. Bernini, who was contemporary with Galileo, Kepler, and
Newton, communicates a sense of motion as David attacks. Both are magnificent works,
which symbolize the shift in intellectual focus from the static to the dynamic during the
Age of Reason.
Isaac Newton’s revolutionary work demonstrated the power of the human mind to
grasp the laws of nature and to express them mathematically. The Age of Reason was an
extraordinarily creative mathematical period, but questions of rigor were not addressed
systematically. For example, as we have mentioned, the idea of a limit was never
satisfactorily defined. As the eighteenth century came to a close, mathematicians
received increasing criticism for their imprecise concepts and loose reasoning. The
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resolution of such issues was to be one of the major mathematical themes of the
nineteenth century.
Bibliography
Boyer, C. (1968). A History of Mathematics, New York: John Wiley & Sons, Inc.
Kline, M. (1972). Mathematical Thought from Ancient to Modern Times, New York:
Oxford University Press.
The PowerPoint presentation is available via email: mathprofs@sbcglobal.net.
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