Mathematics Magazine Contents—February 2014

In this issue we have a window into the eighteenth century, as Antonella Cupillari shows us how Maria Gaetana Agnesi graphed rational and algebraic functions in her influential textbook of 1748. We also have a rare hat trick in the Notes section: three new theorems. There is a new way to characterize parabolas that Archimedes would have appreciated, a new way to fill in a rectangle with digits so that each L-shaped triple occurs exactly once, and—illustrated on the cover—a new construction involving a star, a circle, and five concurrent lines. Or, if you prefer, you can start at the back of the issue with the 2013 Putnam solutions. —Walter Stromquist

Vol. 87, No. 1, pp.2-78.

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ARTICLES

Maria Gaetana Agnesi’s Other Curves (More Than Just the Witch)

Antonella Cupillari

Maria Gaetana Agnesi’s name is commonly associated with the curve known as “the Witch of Agnesi.” However, the versiera (or versoria), as Agnesi called it, is only one of many curves she introduced in her mathematical compendium, the Instituzioni Analitiche (1748). Some of the other curves are much more interesting and complex than the versiera, and they are grouped in the lengthy section (pp. 351–415) titled “On the construction of Loci of degree higher than second degree.” This article showcases Agnesi’s presentation of some of them, made without using calculus. Instead, her tools of choice were geometry, algebra, and the method of using easier and already-known curves to build more challenging ones.

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Maclaurin’s Inequality and a Generalized Bernoulli Inequality

Iddo Ben-Ari and Keith Conrad

Maclaurin’s inequality is a natural, but nontrivial, generalization of the arithmetic-geometric mean inequality. We present a new proof that is based on an analogous generalization of Bernoulli’s inequality. Applications of Maclaurin’s inequality to iterative sequences and probability are discussed, along with graph-theoretic versions of the Maclaurin and Bernoulli inequalities.

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When Can You Factor a Quadratic Form?

Brian G. Kronenthal and Felix Lazebnik

Consider the problem of determining, without using a computer or calculator, whether a given quadratic form factors into the product of two linear forms. A solution derived by inspection is often highly nontrivial; however, we can take advantage of equivalent conditions. In this article, we prove the equivalence of five such conditions. Furthermore, we discuss vocabulary such as “reducible,” “degenerate,” and “singular” that are used in the literature to describe these conditions, highlighting the inconsistency with which this vocabulary is applied.

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NOTES

Defining Exponential and Trigonometric Functions Using Differential Equations

Harvey Diamond

This note addresses the question of how to rigorously define the functions exp(x), sin(x), and cos(x), and develop their properties directly from that definition. We take a differential equations approach, defining each function as the solution of an initial value problem. Assuming only the basic existence/uniqueness theorem for solutions of linear differential equations, we derive the standard properties and identities associated with these functions. Our target audience is undergraduates with a calculus background.

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Some Logarithmic Approximations for $$\pi$$ and $$e$$

Poo-Sung Park

We offer the approximations $$\pi\approx\log_{5}157$$ and $$e\approx\log_{8}285$$, which may be useful where other logarithms are involved.

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A 5-Circle Incidence Theorem

J. Chris Fisher, Larry Hoehn, and Eberhard M. Schröder

We state and prove a surprising incidence theorem that was discovered with the help of a computer graphics program. The theorem involves sixteen points on ten lines and five circles; our proof relies on theorems of Euclid, Menelaus, and Ceva. The result bears a striking resemblance to Miquel’s 5-circle theorem, but as far as we can determine, the relationship of our result to known incidence theorems is superficial.

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A Simple Proof that e^p/q Is Irrational

Thomas Beatty and Timothy W. Jones

Using a simple application of the mean value theorem, we show that rational powers of e are irrational.

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A Unique Area Property of the Quadratic Function

Connie Xu

Suppose that a function $$f$$ defined on the real line is convex or concave with $$f''(x)$$ continuous and nonzero for all $$x$$. Let $$(x_{1}$$, $$f(x_{1}))$$ and $$(x_{2}, f(x_{2}))$$ be two arbitrary points on the graph of $$f$$ with $$x_{1} < x_{2}$$. For $$i = 1, 2$$, let $$L_{i}$$ denote the tangent line to $$f$$ at the point $$(x_{i},  f(x_{i}))$$ and let $$A_{i}$$ be the area of the region $$R_{i}$$ bounded by the graph of $$f$$, the tangent line $$L_{i}$$, and the line $$x =\hat{x}$$, the $$x$$-coordinate of the intersection of $$L_{1}$$ and $$L_{2}$$. It is proved that $$f$$ is a quadratic function if and only if $$A_{1} = A_{2}$$ for every choice of $$x_{1}$$ and $$x_{2}$$.

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de Bruijn Arrays for L-Fillings

Lara Pudwell and Rachel Rockey

We use modular arithmetic to construct a de Bruijn L-array, which is a $$k\times k^{2}$$ array consisting of exactly one copy of each L-shaped pattern (a $$2\times2$$ array with the upper right corner removed) with digits chosen from $$\{0,\dots,k-1\}$$.

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PROBLEMS

Proposals 1936-1940
Quickies 1037 & 1038
Solutions 1911-1915

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Escape the Square

Mark Dalthorp

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REVIEWS

Algebra: imperative? STEM: crisis? Alive: in 10 years?

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NEWS AND LETTERS

74th Annual William Lowell Putnam Mathematical Competition

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