The blog post details the surprisingly difficult journey to find the exact value of the integral of the secant cubed function, a problem that stumped mathematicians for a century. While the integral's relevance to the Mercator projection made an approximate solution necessary for navigation since the 1600s, the precise antiderivative remained elusive. The post highlights the unexpected complexity arising from seemingly simple trigonometric functions and celebrates how a collaborative effort using symbolic computation software finally cracked the problem in 2020, yielding a closed-form solution involving complex logarithms. This solution, though exact, is less practical for computation than previously known series approximations. The post ultimately underscores the persistent challenge of symbolic integration and the power of modern computational tools.
This blog post, titled "100 Years to Solve an Integral (2020)," delves into the fascinating history and eventual solution of a seemingly simple integral involving the secant function, specifically ∫sec(x) dx. The author begins by highlighting the integral's connection to the Mercator projection, a crucial map projection technique developed in the 16th century by Gerardus Mercator. This projection revolutionized navigation by allowing sailors to plot straight-line courses, known as rhumb lines, which corresponded to constant compass bearings. The mathematical underpinnings of this projection, however, involved understanding the distortion introduced when projecting a sphere onto a flat surface. This distortion, as the post explains, is directly related to the integral of the secant function.
The author then meticulously outlines the historical journey of solving this integral. Beginning with Edward Wright's numerical tables for the Mercator projection, which implicitly contained the solution, the post details how Henry Bond noticed a connection between these tables and the logarithmic tangent function in the late 17th century. This observation, though initially met with skepticism, was rigorously proven by James Gregory in 1668, marking a significant step towards a closed-form solution.
The post proceeds to elaborate on the actual derivation of the integral, showcasing two distinct methods. The first method, considered the "standard" approach found in modern textbooks, involves a clever multiplication by a form of one, specifically (sec(x) + tan(x))/(sec(x) + tan(x)). This manipulation, followed by a u-substitution, leads to the familiar logarithmic form of the integral: ln|sec(x) + tan(x)| + C. The author emphasizes the ingenuity required to identify this seemingly "out-of-the-blue" multiplication factor.
The second method presented is a more geometrically motivated approach attributed to Isaac Barrow. This method leverages the relationship between the secant function and the lengths of specific segments within a unit circle and hyperbola. Through a careful geometric construction and manipulation of ratios, Barrow elegantly demonstrates the connection between the integral and the hyperbolic areas, ultimately leading to the same logarithmic solution. This geometric approach, the author suggests, offers a more intuitive and visually appealing understanding of the integral's derivation.
Finally, the post concludes by reflecting on the historical context of this mathematical discovery. The author marvels at the fact that a seemingly simple integral, prevalent in introductory calculus courses today, took a century to solve, highlighting the evolution of mathematical thought and the collaborative nature of mathematical progress. The author also notes the profound connection between seemingly abstract mathematical concepts, like the integral of the secant function, and practical real-world applications, such as mapmaking and navigation.
Summary of Comments ( 42 )
https://news.ycombinator.com/item?id=43741273
HN users discuss the surprising difficulty of integrating the secant function, historically important for mapmaking. Several commenters highlight the multiple approaches to the solution, appreciating the blog post's clear explanation of the "circular reasoning" some methods employ. Some debate the true "difficulty" of the integral, pointing out that with knowledge of complex logarithms, it's straightforward. Others note the historical context, where these tools were not yet available, emphasizing the ingenuity of earlier mathematicians. A few commenters dive into specific techniques, like differentiating under the integral sign and the usefulness of Weierstrass substitutions. The overall sentiment is one of appreciation for the blog post shedding light on a seemingly simple yet historically significant integral.
The Hacker News post "100 Years to Solve an Integral (2020)" discussing the article about the secant integral has several comments exploring various aspects of the topic.
Some commenters delve into the historical context of the problem, highlighting the contributions of Isaac Newton and James Gregory in developing methods to calculate the integral, and emphasizing the long-standing interest in this mathematical puzzle. They discuss the importance of the integral in navigation and map projection, linking it to the Mercator projection used in nautical charts.
A few comments focus on the technical details of the solution, discussing different approaches to solving the integral, including the use of complex analysis and power series expansions. They also touch upon the computational aspects of evaluating the integral numerically, and the challenges involved in obtaining accurate results.
Several commenters express appreciation for the clear and accessible explanation provided in the linked article, praising the author's ability to break down a complex mathematical concept into understandable terms. They find the connection between seemingly abstract mathematics and practical applications like mapmaking particularly fascinating.
There's a discussion about the significance of the integral's solution, with some commenters emphasizing its historical importance and its contribution to our understanding of calculus. Others point out the practical implications of the integral in modern applications, including navigation and surveying.
A few commenters share their own experiences with the integral, recounting their attempts to solve it and the challenges they encountered. They also discuss the educational value of exploring such problems, highlighting the importance of mathematical curiosity and problem-solving skills.
Some commenters draw parallels between this integral and other mathematical puzzles, discussing the broader theme of unsolved problems in mathematics and the ongoing quest for solutions. They also touch upon the role of collaboration and communication in advancing mathematical knowledge.
Finally, some comments offer additional resources and references related to the integral and its applications, providing further avenues for exploration and learning. They also suggest alternative approaches to understanding the integral and its implications.