# Mathematics Magazine - December 2005

### ARTICLES

The Lost Calculus: Tangency and Optimization without Limits
Jeff Suzuki
339-353
In the years between 1637 and 1670, a set of techniques for solving the problems of tangents, extrema, curvature, and area emerged. Unlike the calculus of Newton and Leibniz, which required the use of limits, implied or otherwise, the "lost calculus" relied only on algebraic and geometric properties. The foundations of this limit-free calculus were set down by Descartes, but the keys to unlocking its potential could be found in two algorithms developed by the Dutch mathematician Jan Hudde. We look at this lost calculus, and conclude that a calculus of algebraic functions could have been developed entirely independently of the notion of a limit.

Matt Richey and Paul Zorn
354-367
A straightforward probability problem from the 2002 Putnam exam concerns predicting a free throw shooter's future success under hypotheses that suggest a Bayesian approach to probability and statistics. The original problem is generalized and solved, and also "embedded" in a Bayesian context that helps clarify several features of and surprises in the original problem. A short general discussion of Bayesian statistics is included for completeness.

The Least-Squares Property of the Lanczos’ Generalized Derivative
Nathanial Burch, Paul Fishback and Russell Gordon
368-378
The Lanczos derivative is attributed to Cornelius Lanczos and is an integral-based, proper extension of the usual derivative. Its peculiar formula arises both as the solution to a simple linear regression problem as well as from the minimization of a certain mean-square-error. These facts give the Lanczos derivative both statistical and Fourier-analytic flavors. We show how this derivative may be expressed in probability terms using the uniform random variable. We also show how the Lanczos derivative and its higher-order analogs may be viewed as limits of integrals involving Legendre polynomials.

### POEM

Triangles
378

They are widely admired, and for good reason. Like handsome shop windows or smartly dressed museum guides, they invite our warm inspection, and never disappoint. They are tranquil on the page.

### NOTES

Honey, Where Should We Sit?
John A. Frohliger and Brian Hahn
379-384
A classic calculus problem asks us to determine far away one should stand from a painting in order to maximize the vertical viewing angle. In most textbooks, the person is standing on level ground. In this note, we use geometry to generalize the problem and discuss maximizing the angle when he viewer is standing along a horizontal line, a slant line, and the graph of an arbitrary continuous function.

PROOF WITHOUT WORDS

Alternating Sums of Odd Numbers
Arthur T. Benjamin
385
We present a visual proof that (2n - 1) (2n 3) + (2n - 5) (2n - 7) + ... +/ - 1 = n.

A Short Proof of Chebychev’s Upper Bound
William Staton and Kimberly Robertson
385-387
In 1852 Chebychev proved that for n at least 2, the number of primes not exceeding n is between An/log(n)] and B[n/log(n)] for suitable constants A and B. We provide a short derivation of the upper bound with B=log(8). Our B is not as small as Chebychev's, but is considerably smaller than what is presented in some textbooks.

Recounting the Odds of an Even Derangement
Arthur T. Benjamin, Curtis D. Bennett, and Florence Newberger
387-390
Let D(n) denote the number of derangements ofn elements, i.e., the number of permutations of n elements without any fixed points. The formula for D(n) is well known, but how many of these derangements are even permutations? Let E(n) and O(n) denote the number of derangements of n elements that are even and odd permutations, respectively. Thus E(n) + O(n) = D(n). To solve for E(n) and O(n), we provide two proofs that E(n) O(n) = ( 1){n - 1} (n - 1). The first proof uses determinants. The second proof is a combinatorial proof that finds an almost one to one correspondence (or bijection or involution) between the even derangements and the odd derangements.

Volumes of Generalized Unit Balls
Xianfu Wang
390-395
We give a unified formula for volumes of generalized unit balls. Two methods are given, one is by induction and the other by Laplace transform.

### PROOF WITHOUT WORDS

A Triangular Sum
Roger B. Nelsen
395
A proof without words of the identity where tn denotes the nth triangular number.

Partitions into Consecutive Parts
M. D. Hirschhorn and P. M. Hirschhorn
396-397
The number of partitions of the number n into consecutive parts is equal to the number of odd divisors of n. We provide an elementary proof of the following refinement of this result. The number of partitions of n into an odd number of consecutive parts is equal to the number of odd divisors of n less than , while the number of partitions of n into an even number of consecutive parts is equal to the number of odd divisors of n greater than .

Means Generated by Integral
Hongwei Chen
397-399
Using the ratio of the power integral , we present a unified interpretation for the various means, including arithmetic, geometric, and logarithmic means, and for inequalities among them.

Nonattacking Queens on a Triangle
Gabriel Nivasch and Eyal Lev
399-403
The n-queens problem asks for placing n non-attacking queens on an n x n chessboard. Here we analyze a variant of this problem for a triangular board: How many queens can be placed on a triangular board of side n, with no two queens attacking each other? We derive an upper bound on the maximum number of queens, and we show that our bound is tight by giving an explicit solution that matches with the bound for all n.