412 Notes on Explorations.of a polynomial, Amer. Math. Monthly 27 (1920), 299-300 it is shown that,
for a polynomial f(z) with real coefficients, the nonreal roots of f’(z) are in
the closed discs whose diameters are segments joining the pairs of conjugate
nonreal roots of f(z) (J ensen’s theorem). For further results, see J.L. Walsh,
A new generalization of Jensen’s theorem on the zeros of the derivative of
a polynomial, Amer. Math. Monthly 68 (1961)) 978-983.
In the cubic case, the result is especially interesting. Let T be a non-
degenerate triangle in the complex plane whose vertices are the zeros of a
cubic f(t) and let E be the (Steiner) ellipse inscribed in T which touches
the sides of T at their midpoints. Then the zeros of f’(t) are the foci of E.
For an application of this result, see I.J. Schoenberg, A conjectured ana-
logue of Rolle’s theorem for polynomials with real or complex coefficients,
Amer. Math. Monthly 93 (1986)) 8-13.
E.42. Newton’s Method inspired A. Cayley, a nineteenth century British
mathematician, to study the sets of starting points which would yield a
sequence of approximants to a given zero of the polynomial. This has been
recently taken up and integrated with the study of fractals. See, for exam-
ple, the article H.-O. Peitgen, D. Saupe & F. v. Haeseler, Cayley’s problem
and Julia sets, Math. Intelligencer 6 (no. 2) (1984)) 11-20.
E.43. The extract is taken from Newton’s tract, Analysis of equations of
an infinite number of variables (page 320). This has been reprinted in
Volume 1 of The Mathematical Works of Isaac Newton. Assembled with
an introduction by Dr. Derek T. Whiteside, (Johnson Reprint, NY, 1964,
1967).
Readers may also be interested in the facsimile of a 1728 English trans-
lation of another Newton work, Universal Arithmetick, originally written
in Latin in 1684, and reproduced in Volume 2. The last part of the paper
treats solution of equations and location of roots.
E.45 & 46. Continued fractions are of use, not only for approximating
the solutions of equations, but also in the treatment of diophantine equa-
tions and the close approximation of nonrationals by rationals. Irrationals
which are roots of quadratic equations over Z can be characterized by the
periodicity of the numbers occurring in their continued fraction expansion.
For a rich high school level introduction to the topic, consult C.D. Olds,
Continued fractions, (MAA, 1963; New Mathematical Library). A recent
book which provides an historical perspective on continued fractions through
a study of the Greek theory of ratio is D.H. Fowler, The Mathematics of
Plato’s Academy: A New Reconstruction, (Oxford, 1987).
An excerpt of Lagrange’s work along with a brief history of continued
fractions appears in Chapter II, Article 12 (p. 111-115) of D.J. Struik
(ed.), A Source Book in Mathematics, 1200-1800, (Harvard, 1969). See
also the excerpts of work of Bombelli (c. 1526-1573) and Cataldi (1548-
1626) reproduced in translation on pages 80-84 of D.E. Smith, A Source
Book in Mathematics, Volume One (Dover, 1959).