Robert E. Bradley

Alternating sequences and induced operators

We show that when a positive Lp contraction is equipped with a norming function having full support, then it is related in a natural way to an operator on any other Lp space, 1 < p < oo. This construction is used to generalize a theorem of Rota concerning the convergence of alternating sequences.

Polar Ordinates in Bernoulli and L’Hôpital

Priority for the invention of polar coordinates should probably be accorded to Newton, although Jakob Bernoulli has priority of publication in 1691, because Newton’s results were only published posthumously, almost half a century later. However, it was not until the middle of the 18th century that polar coordinates took on the form that is used today. Earlier versions all featured ordinates emanating from a single point or pole, with some geometric object playing the role that now belongs to an angular coordinate. The largest and most accessible collection of these early schemes of polar ordinates is probably to be found in the Marquis de L’Hopital’s Analyse des infiniment petits (1696), based on the lessons given to the Marquis by Johann Bernoulli. We describe Bernoulli’s approaches to polar ordinates, as presented in l’Hopital’s textbook.

Fontenelle’s Eulogy for the Marquis de L’Hôpital

When l’Hopital died in 1704, he was a member of the Paris Academy of Sciences. Bernard de Fontenelle was the secretary of the academy at that time and wrote a eulogy for l’Hopital. Most of what little is known about l’Hopital prior to 1690 comes from this document. This chapter contains the first English translation of the entire eulogy.

Selected Letters from the Correspondence Between the Marquis de L’Hôpital and Johann Bernoulli

This chapter contains a substantial portion of the correspondence between l’Hopital and Bernoulli, available in English translation for the first time. Included are more than fifty pages of the correspondence, covering the period from 1692 until the publication of l’Hopital’s Analyse. Included is the famous letter of March 17, 1694, where l’Hopital offers Bernoulli an annual stipend of three hundred French pounds in return for his services, including the rights to publish some of Bernoulli’s discoveries. There is also the letter of July 22, 1694, where L’Hopital’s Rule appears for the first time. There are also many details of a personal or professional nature that illuminate the complex friendship between these two men.

Use of the Differential Calculus for Finding Evolutes

The first four chapters of the Analyse followed the general outlines of the Lectiones de calculo differentialis, the notes on the differential calculus that Bernoulli had provided to l’Hopital when he tutored him in 1691–92. Chapter 5 is the first of six chapters that l’Hopital had a more independent role in composing. It concerns finding the evolute of a given curve, which may be defined as the locus of the centers of curvature of that given curve. The study of these curves originated with Huygens. L’Hopital determines the formula for finding the center of curvature at any point on a curve, whether in rectangular coordinates or in the case where ordinates all emanate from a single point. He then finds the evolutes of many different curves, including the cycloid, which Huygens had shown is congruent to its evolute. The chapter concludes with l’Hopital’s description of the cusp of the second kind. Chapter 5 is one of the two longest chapters in the Analyse.

Bernoulli’s Lectiones de Calculo Differentialis

L’Hopital met Bernoulli in November 1691 and almost immediately hired him as a tutor, to teach him the new calculus. In 1691–92, Bernoulli gave him lessons on the subject, including handwritten notes. Bernoulli kept copies of these notes for himself and long after the publication of the Analyse, he published the second part of these notes, on the integral calculus. However, his notes on the differential calculus, which form the basis of the first four chapters of l’Hopital’s Analyse, remained unknown until a copy was discovered in Basel in 1922. These notes, originally written in Latin, appear here for the first time in English translation. When compared to the corresponding chapters of the Analyse, we see clearly that Bernoulli provided the major results and the structure for that portion of l’Hopital’s book, but that l’Hopital also contributed significantly, especially as a lucid expositor. This chapter contains the full text of Bernoulli’s lessons on the differential calculus, including reproductions of the hand drawn figures.

Concerning Induced Operators and Alternating Sequences

When a positive L p -contraction T has a semi-invariant function, then it is related in a natural way to a contraction of L r , where p and r are arbitrary indices in the range (1, ∞). General theorems concerning the pointwise convergence of alternating sequences have been obtained using these induced operators (see [AB,B]). In this note, we study the norm convergence of such sequences, and investigate the existence and uniqueness of induced operators when T is an L 1 -contraction.