Salvatore J. Petrilli

Using Capture-the-Flag to Enhance the Effectiveness of Cybersecurity Education

Incorporating gamified simulations of cybersecurity breach scenarios in the form of Capture-The-Flag (CTF) sessions increases student engagement and leads to more well-developed skills. Furthermore, it enhances the confidence of students in their own abilities. Our argument is supported by a study in which undergraduate students taking a cybersecurity class were surveyed before and after participating in a CTF.

Analysis and Translation of Raffaele Rubini's 1857 'Application of the Theory of Determinants: Note'

Scholars in the Kingdom of Two Sicilies, which united the island Kingdom of Sicily and the mainland Kingdom of Naples (the southernmost regions in modern-day Italy) from 1815 to 1860, were exposed to some of the works of French mathematicians after translations of these works were completed. However, until Francesco Brioschi (1824 − 1897) published his works of algebraic theory, beginning in 1854 with his Teoria dei determinanti (Theory of Determinants) [O’Connor and Robertson, 2006], these scholars had limited knowledge of algebra, specifically determinants. Their ignorance of this algebraic theory was due not only to the kingdom’s geographic isolation, but also to an academic schism between two branches of mathematical thought, synthetic and analytic. This schism had consequences for several Italian mathematicians, including Raffaele Rubini (1817− 1890), and led Rubini to publish his 1857 article, “Application of the Theory of Determinants: Note.” The article, “Analysis and Translation of Raffaele Rubini’s 1857 ‘Application of the Theory of Determinants: Note’,” available online in MAA Convergence, provides context for and analysis of Rubini’s “Note,” along with some biographical information about this obscure mathematician and ideas for using his “Note” about determinants in college and high school courses. The present document contains the first English translation of Rubini’s article, “Application of the Theory of Determinants: Note.”

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.

Use of the Differential Calculus for Finding the Points of Curved Lines That Touch An Infinity of Lines Given in Position, Whether Straight or Curved

In Chapters 6 through 8 of the Analyse, l’Hopital studies various kinds of envelopes: curves that are tangent to all the members of some family of lines or curves. In Chapter 8, l’Hopital studies various envelopes that do not belong to the category of caustics by reflection or refraction. L’Hopital considers envelopes of parabolas, of circles, and of various families of straight lines.

Use of the Differential Calculus for Finding Caustics by Reflection

In Chapters 6 through 8 of the Analyse, l’Hopital studies various kinds of envelopes: curves that are tangent to all the members of some family of lines or curves. In Chapter 6, l’Hopital studies caustics by reflection, or catacaustics. This problem derives from optics and is essentially the study of envelopes made by light rays reflected in a mirror. L’Hopital considers a variety of shapes of mirrors and of sources of light rays.

The Solution of Several Problems That Depend upon the Previous Methods

In Chapter 9 of the Analyse, l’Hopital considers some miscellaneous problems based on the methods of Chapters 1– 4, that do not belong to the topics covered in Chapters 5– 8 This chapter begins with the celebrated theorem that now goes by the name L’Hopital’s Rule. This rule is actually due to Bernoulli, and the version given here only covers the case in which y takes the indeterminate form \(\frac{0} {0}\) at a finite value of x. Much of the rest of the chapter is taken up with a study of the epicycloid.

Use of the Differential Calculus for Finding Inflection Points and Cusps

In Chapter 4 of the Analyse, l’Hopital defines higher order differentials and describes how second order differentials may be used to locate inflection points and cusps on a curve. In addition to the usual rectangular coordinates, l’Hopital also considers the case where ordinates all emanate from a single point. Although these are not the polar coordinates that came into use in later centuries, because there is no accompanying angular coordinate, they are nevertheless useful in this and subsequent chapters for describing certain curves. L’Hopital finds the inflection points of the prolate cycloid, the Conchoid of Nicomedes and of a curve that is essentially the same as the “Witch of Agnesi.”