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Mathematicians Revolutionize the Understanding of Equations

Mathematicians Revolutionize the Understanding of Equations

Overview

The progress of mathematics from its origins in simple counting to its ability to handle and manipulate variables, unknowns, and changing properties accelerated during the 1600s. Mathematics had from earliest times been a supremely practical science, either for enumerating items (counting) or for establishing the relationships among shapes (geometry). Most ancient mathematics took the form either of simple counting or of geometric measurement. The evolution of mathematics in the 1600s was nothing short of explosive, and the evolution of the equation—and the power of the equation as a tool for solving complex problems of many sorts—lay at the heart of the growth of mathematical capability. Equations are mathematical formulae whose purpose is, at least in principle, simple: equations consist of factors that must be balanced or made equal. By introducing systems for accommodating variables and unknowns, and the tools for solving those items, the equation became a powerful mathematical tool, with applications that reached far beyond the realm of numbers, affecting everything from construction (geometry) to theories of the motions of the planets (calculus) and embracing virtually every field of human endeavor. If in the centuries to come mathematics would prove to be the key to the universe, the equation, and the rules for equations that were refined, established, and developed during the 1600s, would prove to be the key to mathematics.

Background

The process of establishing equivalencies, of making both sides of an equation balance out, is the essence of much of mathematics. Put simply, an equation challenges the mathematician to prove that something is equal to something else, that the propositions on both sides of the equal sign can indeed be made to equal each other. From that simple proposition, vastly complex mathematical structures can be constructed and solved.

In order for equations to become effective, however, a large leap of process had to be made. In short, the mathematics of equations had to become formalized—a set of rules had to be established and agreed to. While much of that standardization took place during the 1500s and 1600s (and was aided immeasurably by the fact that printing was by that time well established), equations themselves are at least as old as the ancient Greeks.

Diophantus of Alexandria (c. 210-c. 290) was among the first to extend complex mathematics beyond the more common geometries of his time and into what is today known as algebra. Among other things, Diophantus introduced the use of symbols into equations; previously equations had been written in words. Diophantus's employment of Greek symbols to represent frequently used quantities and other factors played an important role in simplifying the construction (and solution) of complex equations.

It was the Arabian mathematician Muhammad ibn Musa Al-Khwarizmi (c. 780-c. 850) who translated and expanded Diophantus's work, in a book that gave the science of equations its modern name. Al-Khwarizmi's book was called ilm al-jabr wa'l muqubalah (The Study of Transposition and Cancellation), although aljabr (or al-jebr) can also be translated as "the reunion of broken parts." That reunion meant the balancing of both sides of an equation, and aljabr has, of course, been transformed over the centuries into the word algebra, which is the branch of mathematics most closely associated with equations. (In addition to naming algebra, Al-Khwarizmi imported certain concepts, including the zero and numerals, from Hindu sources. When Al-Khwarizmi's work was translated into Latin, those numerals became known as Arabic numerals.)

The introduction of standardized symbology laid the groundwork for modern equations. Another large forward step was taken by French mathematician François Viète (1540-1603), who first used letters of the alphabet to represent unknowns and constants in equations. In Viète's system vowels were used to represent unknowns, and consonants represented constants. Although he is known to this day as the "father of algebra," Viète disliked the term, preferring "analysis" to refer to the process of using equations to solve propositions.

By the turn of the seventeenth century, equations and algebra were in flux with new discoveries and approaches. Among the most dramatic was the work of Albert Girard (1595-1632). A French mathematician, Girard was an engineer as well as a mathematician, and he applied many of his mathematical insights to his engineering work, particularly the development of fortifications. But it was his 1629 book L'invention en Aalgèbre (The Invention of Algebra) that solidified his reputation as a major contributor to the development of equations.

In that book, Girard made a formal approach to what would later become known as the Fundamental Theorem of Algebra. In essence, the Fundamental Theorem states that for any polynomial equation (an equation with at least one algebraic term multiplied by at least one positive variable raised to an integral power) there is a root in the complex numbers. In mathematical terms, this root is expressed as a + bi, with a and b representing real numbers and i representing the square root of -1. Numbers expressed as a + bi are called complex numbers.

Refined and extended by mathematicians more gifted than Girard, and proved in 1799 by German mathematician Johann Karl Friedrich Gauss (1777-1855), the Fundamental Theorem remains central to algebra today.

The greatest of all contributors to the evolution of algebra and equations (up to his lifetime at least) was the French philosopher René Descartes (1596-1650). In the course of his life, Descartes so revolutionized mathematics and made so many large contributions to philosophical thought that aspects of both disciplines are still referred to as Cartesian in his honor. (And so influential was his writing that his theory of the working of the universe, although false, was accepted as accurate until disproved by Isaac Newton [1642-1727.])

Descartes's philosophy—and his approach to mathematics—rested upon his belief in absolute fact, in the "mechanical" nature of the universe. By beginning with an absolute fact, he thought, one could progress to an understanding of the whole of the universe. This search for absolutes, and absolute accuracy, guided his great mathematical works. (He even found a way to overcome any doubts: the very act of doubting proved the existence of the doubter. From this he derived his famous maxim, Cogito, ergo sum (I think, therefore I am).

In applying his search for absolutes to mathematics, Descartes began by further formalizing the nature of equations. Adapting Viète's work with alphabetical symbols for numbers, and further focusing it, Descartes used the early letters of the alphabet to represent constants, and letters at the end of the alphabet to represent variables. It is to Descartes that we owe the familiar use of x and y variables in algebra. In addition he devised a system for displaying exponents, and he was the first to use the square root symbol.

This systematic approach served Descartes well as he undertook the great mathematical work of his life, the unification of algebra and geometry. According to legend, Descartes was restricted to bed rest as a result of ill health, and thus confined amused himself by watching the movements of a housefly around his sickroom. Insight struck as he watched the fly flit about—every position of the fly's constantly varying motion could be expressed as a point that could be located in three dimensions by determining the coordinates of three intersecting lines, representing east/west, north/south, and up/down.

From that insight it was another step for Descartes to develop an equation that could translate any point into an equation, and conversely any equation composed of representations of points into a geometrical curve. Descartes then separated the world of curves into two types: geometric curves were those that were mathematically pure and could be expressed as equations; mechanical curves cannot be expressed in equations.

Descartes's fusion of algebra with geometry transformed both, and set in motion a wave of mathematical progress that led directly to Newton's development of calculus, which further extended Descartes's insight by applying algebraic equations to variables that are constantly changing, such as objects in motion.

Because of the great power and usefulness of Descartes's application of equations to analyzing geometric functions, the resultant discipline became known, and is still known, as analytic geometry, and the coordinates represented in its equations are known as Cartesian coordinates. His insistence that virtually all natural phenomena could be expressed as equations was a vital contribution to the development of modern science.

Impact

Without a formalized system for expressing mathematical variables, equations would have remained cumbersome and unwieldy, expressed in different ways by different mathematicians. The accretion of standardized approaches to equations, though, played an enormous role in transforming mathematics into a collaborative effort, one in which researchers and thinkers throughout the world shared a common means of expression, however different their approach to similar problems might be. Mathematics began to develop its now-familiar set of rules expressed in a common language. (All of this, it should be repeated, was aided immeasurably by the printing press and mass distribution of ideas and treatises.) Thus one generation could build upon the work of previous generations without having to re-discover or reinvent basic principles, approaches, and proofs. Descartes's combination of algebraic equations with geometric points, quite simply, provided the basis for Isaac Newton to develop calculus, which in turn altered and refined our understanding of how the universe works.

KEITH FERRELL

Further Reading

Dunham, William. Journey through Genius: The Great Theorems of Mathematics. New York: John Wiley & Sons, 1990.

Katz, Victor. A History of Mathematics: An Introduction. 2nd ed. New York: Addison-Wesley, 1998.

Swetz, Frank J., ed. From Five Fingers to Infinity: A Journey through the History of Mathematics. Chicago and Lasalle: Open Court, 1994.

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