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The 17th century
The 17th century, the period of the scientific revolution, witnessed the consolidation of Copernican heliocentric astronomy and the establishment of inertial physics in the work of Johannes Kepler, Galileo, René Descartes, and Isaac Newton. This period was also one of intense activity and innovation in mathematics. Advances in numerical calculation, the development of symbolic algebra and analytic geometry, and the invention of the differential and integral calculus resulted in a major expansion of the subject areas of mathematics. By the end of the 17th century, a program of research based in analysis had replaced classical Greek geometry at the centre of advanced mathematics. In the next century this program would continue to develop in close association with physics, more particularly mechanics and theoretical astronomy. The extensive use of analytic methods, the incorporation of applied subjects, and the adoption of a pragmatic attitude to questions of logical rigour distinguished the new mathematics from traditional geometry.
Institutional background
Until the middle of the 17th century, mathematicians worked alone or in small groups, publishing their work in books or communicating with other researchers by letter. At a time when people were often slow to publish, “invisible colleges,” networks of scientists who corresponded privately, played an important role in coordinating and stimulating mathematical research. Marin Mersenne in Paris acted as a clearinghouse for new results, informing his many correspondents—including Pierre de Fermat, Descartes, Blaise Pascal, Gilles Personne de Roberval, and Galileo—of challenge problems and novel solutions. Later in the century John Collins, librarian of London’s Royal Society, performed a similar function among British mathematicians.
In 1660 the Royal Society of London was founded, to be followed in 1666 by the French Academy of Sciences, in 1700 by the Berlin Academy, and in 1724 by the St. Petersburg Academy. The official publications sponsored by the academies, as well as independent journals such as the Acta Eruditorum (founded in 1682), made possible the open and prompt communication of research findings. Although universities in the 17th century provided some support for mathematics, they became increasingly ineffective as state-supported academies assumed direction of advanced research.
Numerical calculation
The development of new methods of numerical calculation was a response to the increased practical demands of numerical computation, particularly in trigonometry, navigation, and astronomy. New ideas spread quickly across Europe and resulted by 1630 in a major revolution in numerical practice.
Simon Stevin of Holland, in his short pamphlet La Disme (1585), introduced decimal fractions to Europe and showed how to extend the principles of Hindu-Arabic arithmetic to calculation with these numbers. Stevin emphasized the utility of decimal arithmetic “for all accounts that are encountered in the affairs of men,” and he explained in an appendix how it could be applied to surveying, stereometry, astronomy, and mensuration. His idea was to extend the base-10 positional principle to numbers with fractional parts, with a corresponding extension of notation to cover these cases. In his system the number 237.578 was denoted
in which the digits to the left of the zero are the integral part of the number. To the right of the zero are the digits of the fractional part, with each digit succeeded by a circled number that indicates the negative power to which 10 is raised. Stevin showed how the usual arithmetic of whole numbers could be extended to decimal fractions, using rules that determined the positioning of the negative powers of 10.
In addition to its practical utility, La Disme was significant for the way it undermined the dominant style of classical Greek geometry in theoretical mathematics. Stevin’s proposal required a rejection of the distinction in Euclidean geometry between magnitude, which is continuous, and number, which is a multitude of indivisible units. For Euclid, unity, or one, was a special sort of thing, not number but the origin, or principle, of number. The introduction of decimal fractions seemed to imply that the unit could be subdivided and that arbitrary continuous magnitude could be represented numerically; it implicitly supposed the concept of a general positive real number.
Tables of logarithms were first published in 1614 by the Scottish laird John Napier in his treatise Description of the Marvelous Canon of Logarithms. This work was followed (posthumously) five years later by another in which Napier set forth the principles used in the construction of his tables. The basic idea behind logarithms is that addition and subtraction are easier to perform than multiplication and division, which, as Napier observed, require a “tedious expenditure of time” and are subject to “slippery errors.” By the law of exponents, anam = an + m; that is, in the multiplication of numbers, the exponents are related additively. By correlating the geometric sequence of numbers a, a2, a3,…(a is called the base) and the arithmetic sequence 1, 2, 3,…and interpolating to fractional values, it is possible to reduce the problem of multiplication and division to one of addition and subtraction. To do this Napier chose a base that was very close to 1, differing from it by only 1/107. The resulting geometric sequence therefore yielded a dense set of values, suitable for constructing a table.
In his work of 1619 Napier presented an interesting kinematic model to generate the geometric and arithmetic sequences used in the construction of his tables. Assume two particles move along separate lines from given initial points. The particles begin moving at the same instant with the same velocity. The first particle continues to move with a speed that is decreasing, proportional at each instant to the distance remaining between it and some given fixed point on the line. The second particle moves with a constant speed equal to its initial velocity. Given any increment of time, the distances traveled by the first particle in successive increments form a geometrically decreasing sequence. The corresponding distances traveled by the second particle form an arithmetically increasing sequence. Napier was able to use this model to derive theorems yielding precise limits to approximate values in the two sequences.
Napier’s kinematic model indicated how skilled mathematicians had become by the early 17th century in analyzing nonuniform motion. Kinematic ideas, which appeared frequently in mathematics of the period, provided a clear and visualizable means for the generation of geometric magnitude. The conception of a curve traced by a particle moving through space later played a significant role in the development of the calculus.
Napier’s ideas were taken up and revised by the English mathematician Henry Briggs, the first Savilian Professor of Geometry at Oxford. In 1624 Briggs published an extensive table of common logarithms, or logarithms to the base 10. Because the base was no longer close to 1, the table could not be obtained as simply as Napier’s, and Briggs therefore devised techniques involving the calculus of finite differences to facilitate calculation of the entries. He also devised interpolation procedures of great computational efficiency to obtain intermediate values.
In Switzerland the instrument maker Joost Bürgi arrived at the idea for logarithms independently of Napier, although he did not publish his results until 1620. Four years later a table of logarithms prepared by Kepler appeared in Marburg. Both Bürgi and Kepler were astronomical observers, and Kepler included logarithmic tables in his famous Tabulae Rudolphinae (1627; “Rudolphine Tables”), astronomical tabulations of planetary motion derived by using the assumption of elliptical orbits about the Sun.
Analytic geometry
The invention of analytic geometry was, next to the differential and integral calculus, the most important mathematical development of the 17th century. Originating in the work of the French mathematicians Viète, Fermat, and Descartes, it had by the middle of the century established itself as a major program of mathematical research.
Two tendencies in contemporary mathematics stimulated the rise of analytic geometry. The first was an increased interest in curves, resulting in part from the recovery and Latin translation of the classical treatises of Apollonius, Archimedes, and Pappus, and in part from the increasing importance of curves in such applied fields as astronomy, mechanics, optics, and stereometry. The second was the emergence a century earlier of an established algebraic practice in the work of the Italian and German algebraists and its subsequent shaping by Viète into a powerful mathematical tool at the end of the century.
Viète was a prominent representative of the humanist movement in mathematics that set itself the project of restoring and furthering the achievements of the Classical Greek geometers. In his In artem analyticem isagoge (1591; “Introduction to the Analytic Arts”), Viète, as part of his program of rediscovering the method of analysis used by the ancient Greek mathematicians, proposed new algebraic methods that employed variables, constants, and equations, but he saw this as an advancement over the ancient method, a view he arrived at by comparing the geometric analysis contained in Book VII of Pappus’s Collection with the arithmetic analysis of Diophantus’s Arithmetica. Pappus had employed an analytic method for the discovery of theorems and the construction of problems; in analysis, by contrast to synthesis, one proceeds from what is sought until one arrives at something known. In approaching an arithmetic problem by laying down an equation among known and unknown magnitudes and then solving for the unknown, one was, Viète reasoned, following an “analytic” procedure.
Viète introduced the concept of algebraic variable, which he denoted using a capital vowel (A, E, I, O, U), as well as the concept of parameter (an unspecified constant quantity), denoted by a capital consonant (B, C, D, and so on). In his system the equation 5BA2 − 2CA + A3 = D would appear as B5 in A quad − C plano 2 in A + A cub aequatur D solido.
Viète retained the classical principle of homogeneity, according to which terms added together must all be of the same dimension. In the above equation, for example, each of the terms has the dimension of a solid or cube; thus, the constant C, which denotes a plane, is combined with A to form a quantity having the dimension of a solid.
It should be noted that in Viète’s scheme the symbol A is part of the expression for the object obtained by operating on the magnitude denoted by A. Thus, operations on the quantities denoted by the variables are reflected in the algebraic notation itself. This innovation, considered by historians of mathematics to be a major conceptual advance in algebra, facilitated the study of the symbolic solution of algebraic equations and led to the creation of the first conscious theory of equations.
After Viète’s death the analytic art was applied to the study of curves by his countrymen Fermat and Descartes. Both men were motivated by the same goal, to apply the new algebraic techniques to Apollonius’s theory of loci as preserved in Pappus’s Collection. The most celebrated of these problems consisted of finding the curve or locus traced by a point whose distances from several fixed lines satisfied a given relation.
Fermat adopted Viète’s notation in his paper “Ad Locos Planos et Solidos Isagoge” (1636; “Introduction to Plane and Solid Loci”). The title of the paper refers to the ancient classification of curves as plane (straight lines, circles), solid (ellipses, parabolas, and hyperbolas), or linear (curves defined kinematically or by a locus condition). Fermat considered an equation among two variables. One of the variables represented a line measured horizontally from a given initial point, while the other represented a second line positioned at the end of the first line and inclined at a fixed angle to the horizontal. As the first variable varied in magnitude, the second took on a value determined by the equation, and the endpoint of the second line traced out a curve in space. By means of this construction Fermat was able to formulate the fundamental principle of analytic geometry:
Whenever two unknown quantities are found in final equality, there results a locus fixed in place, and the endpoint of one of these unknown quantities describes a straight line or a curve.
The principle implied a correspondence between two different classes of mathematical objects: geometric curves and algebraic equations. In the paper of 1636 Fermat showed that, if the equation is a quadratic, then the curve is a conic section—that is, an ellipse, parabola, or hyperbola. He also showed that the determination of the curve given by an equation is simplified by a transformation involving a change of variables to an equation in standard form.
Descartes’s La Géométrie appeared in 1637 as an appendix to his famous Discourse on Method, the treatise that presented the foundation of his philosophical system. Although supposedly an example from mathematics of his rational method, La Géométrie was a technical treatise understandable independently of philosophy. It was destined to become one of the most influential books in the history of mathematics.
In the opening sections of La Géométrie, Descartes introduced two innovations. In place of Viète’s notation he initiated the modern practice of denoting variables by letters at the end of the alphabet (x, y, z) and parameters by letters at the beginning of the alphabet (a, b, c) and of using exponential notation to indicate powers of x (x2, x3,…). More significant conceptually, he set aside Viète’s principle of homogeneity, showing by means of a simple construction how to represent multiplication and division of lines by lines; thus, all magnitudes (lines, areas, and volumes) could be represented independently of their dimension in the same way.
Descartes’s goal in La Géométrie was to achieve the construction of solutions to geometric problems by means of instruments that were acceptable generalizations of ruler and compass. Algebra was a tool to be used in this program:
If, then, we wish to solve any problem, we first suppose the solution already effected, and give names to all the lines that seem necessary for its construction—to those that are unknown as well as to those that are known. Then, making no distinction in any way between known and unknown lines, we must unravel the difficulty in any way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are together equal to the terms of the other.
In the problem of Apollonius, for example, one sought to find the locus of points whose distances from a collection of fixed lines satisfied a given relation. One used this relation to derive an equation, and then, using a geometric procedure involving acceptable instruments of construction, one obtained points on the curve given by the roots of the equation.
Descartes described instruments more general than the compass for drawing “geometric” curves. He stipulated that the parts of the instrument be linked together so that the ratio of the motions of the parts could be knowable. This restriction excluded “mechanical” curves generated by kinematic processes. The Archimedean spiral, for example, was generated by a point moving on a line as the line rotated uniformly about the origin. The ratio of the circumference to the diameter did not permit exact determination:
the ratios between straight and curved lines are not known, and I even believe cannot be discovered by men, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact.
Descartes concluded that a geometric or nonmechanical curve was one whose equation f(x, y) = 0 was a polynomial of finite degree in two variables. He wished to restrict mathematics to the consideration of such curves.
Descartes’s emphasis on construction reflected his classical orientation. His conservatism with respect to what curves were acceptable in mathematics further distinguished him as a traditional thinker. At the time of his death, in 1650, he had been overtaken by events, as research moved away from questions of construction to problems of finding areas (then called problems of quadrature) and tangents. The geometric objects that were then of growing interest were precisely the mechanical curves that Descartes had wished to banish from mathematics.
Following the important results achieved in the 16th century by Gerolamo Cardano and the Italian algebraists, the theory of algebraic equations reached an impasse. The ideas needed to investigate equations of degree higher than four were slow to develop. The immediate historical influence of Viète, Fermat, and Descartes was to furnish algebraic methods for the investigation of curves. A vigorous school of research became established in Leiden around Frans van Schooten, a Dutch mathematician who edited and published in 1649 a Latin translation of La Géométrie. Van Schooten published a second two-volume translation of the same work in 1659–1661 that also contained mathematical appendixes by three of his disciples, Johan de Witt, Johan Hudde, and Hendrick van Heuraet. The Leiden group of mathematicians, which also included Christiaan Huygens, was in large part responsible for the rapid development of Cartesian geometry in the middle of the century.
