4. The Approximations of DeLagny 5. Halley's Invention of General Formulas 6. Taylor's Application of the Calculus to Halley's Coefficients 4I 44 45 46 I The feature of mathematics which we may call scientific approximation did not receive the recognition among the ancient Greek geometers that has been accorded it by mathematicians of modern times. Since their work in pure mathematics was based on geometry, the Greek scholars evaluated irrational roots as readily as rational roots by the construction of lines. But when it came to applied mathematics, numerical approximation was often not only desirable but absolutely necessary; and so we have some remarkably close approximations by Archimedes (c. 225 B. C.) in his work in mechanics, by Heron of Alexandria (c. 200 A. D.) in his work as surveyor, and by Theon of Alexandria (c. 375 A. D.) in his astronomical computations. Archimedes even used numerical approximations in the domain of pure geometry in his Measurement of the Circle. On the whole, however, the Greek mind did not take to this form of mathematical activity. The Hindus and the Arabs used approximation methods more extensively than the Greeks, and notable work was done by Brahmagupta (c. 510 A. D.), Bhaskara (c. 1150 A. D.), and an unknown Arab computer mentioned by Chelebî. However, it was during the Renaissance, when the newly discovered general formula for solving the cubic was found to be inoperative for the "irreducible case,' that approximation methods became a vital problem, and for two centuries the keenest mathematical minds worked to find smooth, effective methods for approximating cube and higher roots. The purpose of this research is to trace the history of the different methods of approximating roots of numerical higher equations that were used up to 1819, the date of the publication of Horner's method; we purpose to trace their early beginnings in finding the roots of numbers and in solving incomplete equations, and to watch their growth into systematically developed general methods for solving complete equations of any degree. Pure trial and error methods, like some of the methods of the Egyptians and, in modern times, that of Junge, will not be taken up, nor shall we discuss such ephemeral methods as those of Fontaine, Collins, and some of DeLagny's. We shall, moreover, limit the investigation to algebraic methods in algebraic equations of one unknown, and only incidentally shall we touch upon transcendental equations and geometric and trigonometic methods. In searching for literature on this subject the author found only two articles, outside of the encyclopedias, in the nature of surveys: one by Augustus De Morgan in the Companion to the British Almanac (1839), entitled "Notices of the Progress of the Problem of Evolution" and dealing with the extension of the Hindu method of evolution to the solution of numerical equations by Vieta and Horner; and a more extended discussion by F. Cajori in an article entitled "A History of the Arithmetical Methods of Approximation to the Roots of Numerical Equations of one Unknown Quantity," in the Colorado College Publication (Colorado Springs, Colorado) for 1910. In the present survey we purpose to study the origin and growth of the various methods by bringing out the illustrations and the details of explanation as far as possible from the original sources. The writer has had unusual opportunities for studying such source materials in four notable libraries, the Columbia University Library, the Teachers College Library, the New York City Public Library, and the private library of Professor David Eugene Smith, with its unique collection of editions of the Renaissance and early post-Renaissance periods. By this means he has had access to the early editions and standard translations of Euclid, Archimedes, Heron, Theon, Brahmagupta, Mahāvīrā, Bhāskara, Omar Khayyam, as well as the latest brochures and translations of Chinese and Japanese writers on mathematics. He has also been able to consult such works as those of Leonardo of Pisa, Chuquet, Pacioli, Bombelli, Cardan, Stevin, Pitiscus, Vieta, Oughtred, Hume, Hérigone, Wallis, Newton, Raphson, Colson, Rolle, DeLagny, Halley, Taylor, Waring, Euler, Lagrange, and many others who wrote upon the subject previous to Horner. II PRE-ALGORITHMIC METHODS I. EGYPTIAN AND BABYLONIAN WORK IN FINDING ROOTS We formerly thought that the root concept originated among the Greeks, but recent discoveries show us that the finding of roots of numbers and the solution of quadratic equations, and even the approximation methods employed in this work, go as far back as the early Egyptian and Babylonian civilizations. On the Senkereh tablets (c. 2000 B. C.) are tables of squares and cubes, which shows that the Babylonians had at least an indirect notion of square and cube roots.1 Three quadratic equations are known to have been studied in this period. The first one was made known when Griffith (1897) published the mathematical papyrus found by Petrie in Kahun. It deals with areas and requires the solution of the equations xy = 12 and x: y = I: 34, stated, of course, in rhetorical form. In 1900 Schach discovered in a Berlin papyrus a second problem in quadratics, requiring the solution of the equations x2 + y2 = 100 and x: y = 1:34. The third equation was found in the Kahun papyrus by Schach in 1903. It requires the solution of the equations x2 + y2 400 and x: y ancient mathematician solved it by letting x 2, gives x2 + y2 = 64; since √61⁄4 found that x = 2.8 = 16 and y = 12.8 = 12. The same method, commonly called the method of False Position, is also used in solving the other two equations. 3 = = = 22 and 21⁄2 = 2:12. The y = 1; this 1/8 of 20, he No irrational roots occur in these equations. But there are indications that the Egyptians had a definite way of approaching the square root of non-square numbers. However, they seem to have been unaware of their irrational quality. 2. EARLY CONCEPTS OF HIGHER ROOTS AND IRRATIONALS AMONG THE GREEKS That some roots are irrational was first recognized by the Greeks. From measuring the areas of commensurable squares and rectangles 1 M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol. I (hereafter designated Cantor, I), Leipzig, 1907, p. 28. M. Simon, Geschichte der Mathematik im Altertum, Berlin, 1909, pp. 41-42. in terms of unit squares they proceeded to find the areas of nonrectangular and irregular figures, such as the triangle and the circle. This they did by finding an equivalent square (Quadrature). Since they neither possessed algebraic symbols nor used the equational form of statement, they found the unknown side by geometrical construction. Some of the sides were discovered to be incommensurable quantities. As early as 525 B. C. Pythagoras recognized the diagonal length of the unit square to be an irrational number. Theodorus of Cyrene (c. 410 B. C.) and Plato (c. 380 B. C.) discussed the irrational nature of the expressions √3, √5, √7, . √17. In Book X of his Elements Euclid (c. 300 B. C.) gave a comprehensive exposition of the theory of irrationals. Up to the time of Archimedes square roots were almost universally evaluated by geometrical constructions; by him they were often evaluated numerically by a process of approximation. The Greeks seem also to have been the first people to conceive directly the notion of cube root. This also arose from geometrical problems, and, like the square root, was approached geometrically. According to the familiar story the god's demand of an altar twice as large as the old one necessitated the solution of the equation x=2a3. This was probably solved by Menaechmus (c. 365 B. C.) by the aid of conic sections. Archimedes' famous problem of the section of the sphere led to an equation of the type x3- ax2 + b = o, which was also handled geometrically. Heron of Alexandria (c. 200 A. D.) was the first one to make a practise of evaluating cube roots numerically; this he did in the course of his practical work by a process of approximation. 3. THE APPROXIMATIONS OF ARCHIMEDES AND HERON OF ALEXANDRIA In his Measurement of the Circle Archimedes (287-212 B. C.) gives among others the following approximations:5 Nowhere has he disclosed his process, and many conjectures and many ingenious restoration formulas have been advanced as ex 4 Tropfke, Geschichte der Elementar-Mathematik I (1), Leipzig, 1902, pp. 158-61; 207-10; 223-24. Cantor, I (1907), p. 316. |