Struik: A Concise History of Mathematics: The Seventeenth Century

by Carson Reynolds

The production of machines and tools in the Seventeenth Century spurred the developments of mechanics and motion. Several authors provided early writings on machines: Kyeser (c. 1400), Leon Battista Alberti (c. 1450), and da Vinci (c. 1500). Tartigula’s Nuova scienzia looked at clocks and the orbit of projectiles. Much interest revolved around statics, and computing the center of gravity. Aided by translations of Heron and Archimedes, Sion Stevin, Luca Valerio and Paul Guildin pioneered methods to compute centroids, paving the way for the likes of Kepler, Cavalieri, and Torricelli.

Departing from Greek notions of rigor in favor of quick results 17th century astronomy made contributions. Kepler looked at volume computation in Stereometria doliorum vinorum (1615). Gilileo devised a mechanics for freely falling bodies, a start of a theory of elasticity. As an mathematical experimentalist, Galileo studied motion and presented his ideas in Discorsi (1638). He also provided parabolic orbits, and tables to compute height and range given initial conditions. Toriricelli and Cavalieri were his pupils. Cavalieri provided the first systematic exposition on calculus in Geometria indivisibilibus continuorum (1635). He conceived an indivisible point which generated a line, and a line generating a plane, line segments generating a volume. His principle of Cavalieri allowed him to perform integration of polynomials.

Descartes provided a unification of algebra and geometry in his Geometrie (1637), which fostered the development of analytical geometry. “Cartesian coordinates” and the application of algebra to geometry had already appeared before Geometrie was published. The book’s merit lies in it’s application of 16th century algebra to ancient geometry, as well as a rejection of homogeneity restrictions (viz. x^2 is considered a line). Descarte’s notation was more modern. Fermat’s Isagoge (1679), John Wallis’ Tractatus de sectionibus conicis (1655), John De Witt’s Elementa curvarum linearum (1659), and L’Hospital’s Traite analytique des sections coniques (1707) provide applications of algebra to Apollonius’ work. All these authors hesitated to accept negative coordinates.

Cavalieri’s book brought the tangency problem to the forefront. Chritian Huygens, Torricelli, and Isaac Barrow tried to follow ancient geometrical methods towards non-rigorous solutions. Fermat, Descartes, and John Wallis instead used algebra. Much of the work from 1630-1660 concerned itself with algebraic curves and their integration. Pascal and Descartes looked at cycloids in addition to algebraic curves. Fermat and Johannes Hudde looked at primitive methods to determine maxima and minima. Barrow related integration and differentiation as inverse problems. Pascal anticipated Newton by dropping terms of lower dimensions. The scholastics Gregorie de Saint-Cincent, Paul Guldin and Andre Tacquet developed and coined the “exhaustion method” following Archimedes and Eudoxus. Pascal was influenced by Tacquet’s On Cylinders and Rings (1651). Without periodicals, many formed discussion groups, which in turn led to the founding of Academies.

Wallis’ Arithmetica infinitorum (1655) was a bold extension of Cavalieri’s work. He introduced infinite series, infinite products, and looked at imaginaries, negative and fractional exponents in a crude but bold manner. Christian Huygens explored evolutes and involutes of a plane curve by studying pendulum clocks in his Horologium oscillatorium (1673). Huygens influenced Newton and studied tractrix, logarithmic curves, cantenary, and the cycloid as a tautochronous curve. Both Wallis and Huygens really anticipated calculus, but still belonged to the era before.

Fermat and Desargues work on classical topics provided new results, and even opened new fields. In Fermat’s notes on Diophantus, we find his Last Theorem. Fermat related another theorem (a^(p-2)-1 is divisible by p when p is prime and a is prime to p) was appeared in 1640 in a letter. Fermat and Pascal were founders of probability theory, interest in which was motivated by the insurance industry and gambling. The probleme des points led to a correspondence between Fermat and Pascal that established the foundations of probability theory (1654). Huygens’ De ratiociniis in ludo aleae (1657) was the first treatise on probability, followed by tables of annuities computed by DeWitt and Halley (1671, 1693). Pascal known for his triangle, but also is theorem concerning a hexagon inscribed in a circle. He was also notable for formulating a principle of complete induction and building a computing machine. Gerard Desargues wrote a book on perspective (1636), made contributions to projective geometry and a theorem on perspective triangles.

General methods for differentiation and integration required a fusion of Cavalieri’s work with the algebras of Descartes and Wallis; this was provided by Newton and Leibniz. Newton discovered calculus first, but Leibniz published first, and eventually dominated the field. Newton’s Philosophaiae naturalis principa mathematica (1687) provided an axiomatic mechanics, the law of gravity, a solution to the two-body problem for spheres, and a theory of the moon’s motion. Newton called calculus the “theory of fluxions” which was developed more generally between 1665-66. Newton was led to fluxions by study of Wallis’ Arithmetica. His discovery of binomial series allowed him to apply his theory to algebraic and transcendental functions. Newton expressed velocities as “pricked letters” which were applied to positions or “fluents.” His description and notation were difficult to understand because of ambiguities and confusion, for which he was taken to task by Bishop Berkeley in 1734. Newton also studied conics and plane cubic curves, exceeding his contemporaries simple reworkings of Apollonius with new theorem relating cubics and divergent parabola. Newton was always reluctant to publish his work, much if appearing decades after initially proven.

Leibniz was motivated by broad interests and in the search for general and universal linguistic, logical, and computational methods. He contributed heavily to the development of symbolic logic and math notation. Leibniz’s calculus was developed between 1673-1676 in Paris under the influence of Huygens and through the study of Descartes and Pascal. Leibniz’s approach was more geometrical than Newton’s. He thought in terms of a characteristic triangle which appeared in Pascal and Barrow’s works. Leibniz published his calculus in 1684 entitled Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irrationales quantitates moratur, et singulare pro illis calculi genus. His notation (dx, dy and the integration symbol) is still used, as are the terms calculus, differentiation, integration, function, and coordinates. In 1687 the Bernoulli brothers began to make use of his methods. In 1696, l’Hospital published a textbook entitled Analyse des infiniment petits, containing l’Hospital’s rule. The so-called Liebniz series were first discovered by James Gregory, a Scotch mathematician. He also found the binomial series and Taylor’s series (1670-1671). Leibniz’s vagueness brought criticism from Nieuwentijt, similar to Berkeley’s criticism of Newton.