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## TAYLOR,B.(1685-1731) and MACLAURIN,C.(1698-1746)

 Emery student of the calculus is familiar with the name of the Englishman Brook Taylor (1685-1731) and the mane of the Scotsman Colin Maclaurin (1698-1746), through the very useful Taylor's expansion and Maclaurin's ex pansion of a function.  It was in 1715 that Taylor published (with no consider ation of convergence) his well-known expansion theorem. In 1717, Thaylor applied his series to the solution of numerical equations as follows: Let a be an approximation to a root of f(¥ö) = 0; set f(a)=k, f(a)=k, and ¥ö=a £« h; expand 0 = f(a £« h)by the series; discard all powers of h above the second; substitute the values of k, k, k, and then solve for h.   By successive applications of this process, closer and colser approximations can be obtained.  Some work done by Taylor in the theory of perspectiove has found a modern application in the mathematical treatment of photogrammetry, the science of surveying by means of photographs taken from an airplane.   Recognition of the full importance of Taylor's series awaited until 1755, when Euler applied them in his differntial calculus, and still later, when La grange used the series whith a remainder as the foundation of his theory of functions.     Taylor was educated at St.  John's College of Cambridge University and early showed great promise in mantematics.  Hi was admitted to the Royal Society and became its secretary, only to resign at the age of thirty- four so that he might deote his time to writing. Maclaurin was one of the ablest mathematicians of the eighteenth century.   The wo-called Maclaruin expansion is nothing but the case where a = 0 in the Thylor expansion above and was actually explicitly given by Taylor and alse by James Stirling (1692-1770) some years before Maclaurin used it, with acknowl edgment, in his Treatise of Fluxions (two volumes, 1742).  Maclaurin did very notable work in genmetry, particularly in the study of higher plane curves, and he showed great poser in applying classical geometry to physical problems.     Among his many papers in applied mathematics is a prizewinning menoir on the mathematical theory of tides.   In his Treatise of Fluxions appears his investi gation of the mutual attraction of two ellipsoids of revolution.     Maclaurin probably knew as early as 1729 the rule for solving systems of simultaneous linear equations by determinants that today is called Cramer's rule. The rule first appeared in print in 1748 in Maclaurin's posthumous Trea tise of Algebra.  The Swiss mathematician Gabriel Cramer (1704-1752) inde pendently published the rule in 1750 in his Introduction a l;analyse des lignes courbes algebriques, and it is probably his superior notation that led the general mathematical world to learn the rule from him rather than form Maclaurin. Maclaurin was a mathematical prodigy.  He matriculated at the University of Glasgow the at the age of eleven.  At fifteen, he took his master's degree and gave a remarkable public defense of his theses on the power of gravity.  At nineteen, he was elected to the chair of mathematics at the Marischal College in Aberdeen; at twenty-one, he published his first important work, Geometria organica.  At twenty-seven, he became deputy, or assistant, to the professor of mathematics at the University of Edinburgh.  There was some difficulty in obtaining a salary to cover his assistantship, and Newton offered to bear the cost sdpersonally so that the university could secure the services of so outstanding a young man.  In time, Maclaurin succeeded the ddman he assisted.  His treatise on fluxions appeared whem he was forty-four, only four years before he died; this was the first logical and systematic exposition of Newton's method of fluxions and was written gy maclaunn as a roply to Bishop Berkeley's anack on the pronciples of the calculus.