A Mathematician among the Molasses Barrels: Maclaurin\u27s Unpublished Memoir on Volumes

Suppose we are given a solid of revolution generated by a conic section. Slice out a frustum of the solid [14, diagrams pp. 77, 80]. Then, construct a cylinder, with the same height as the frustum, whose diameter coincides with the diameter of the frustum at the midpoint of its height. What is the difference between the volume of the frustum and the volume of this cylinder? Does this difference depend on where in the solid the frustum is taken? The beautiful theorems which answer these questions first appear in a 1735 manuscript by Colin Maclaurin (1698–1746). This manuscript [14], the only original mathematical work by Maclaurin not previously printed, is published here for the first time, with the permission of the Trustees of the National Library of Scotland. (An almost identical copy [15] exists in the Edinburgh University Library.) In this work, Maclaurin proved that the difference between the cylinder constructed as above and the frustum of the given solid depends only on the height of the frustum, not the position of the frustum in the solid. When the solid is a cone, Maclaurin showed that its frustum exceeds the corresponding cylinder by one fourth the volume of a similar cone with the same height. For a sphere, the cylinder exceeds the frustum by one half the volume of the sphere whose diameter is equal to the height of the frustum; this holds, he observed, for all spheres. He derived analogous results for the ellipsoid and hyperboloid of revolution. Finally, for the paraboloid of revolution, he proved that the cylinder is precisely equal to the frustum

How to Teach Your Own Liberal Arts Mathematics Course

The article encourages Mathematics faculty members to design their own Liberal Arts Mathematics courses by using their own interests and expertise to link mathematics to the world of their students. The author argues that any such course should be guided by these five principles: Draw on the interests of each individual student; teach important mathematics; go slowly enough so students have a sense of mastery; encourage the students to use the mathematics they already know; and let students create projects on topics they choose and then share their projects with the class. The author describes how she implements these principles in two of her own Liberal Arts courses, Mathematics, Philosophy, and the ‘Real World and Mathematics in Many Cultures. The article includes examples of the materials used in these courses, and provides an extensive bibliography. It also lists a set of actual student projects from each course. It concludes that courses designed according to its principles result in students being able and willing to do mathematics, and knowledgeable and enthusiastic about the role mathematics plays in the wider world

Was Newton\u27s Calculus a Dead End? The Continental Influence of Maclaurin\u27s Treatise of Fluxions

We will show that Maclaurin\u27s Treatise of Fluxions did develop important ideas and techniques and that it did influence the mainstream of mathematics. The Newtonian tradition in calculus did not come to an end in Maclaurin\u27s Britain. Instead, Maclaurin\u27s Treatise served to transmit Newtonian ideas in calculus, improved and expanded, to the Continent. We will look at what these ideas were, what Maclaurin did with them, and what happened to this work afterwards. Then, we will ask what by then should be an interesting question: why has Maclaurin\u27s role been so consistently underrated? Thse questions will involve general matters of history and historical writing as well as the development of mathematics, and will illustrate the inseparability of the external and internal approaches in understanding the history of science

Computers and the Nature of Man: A Historian\u27s Perspective on Controversies about Artificial Intelligence

The purpose of the present paper is to provide a historical perspective on recent controversies, from Turing\u27s time on, about artificial intelligence, and to make clear that these are in fact controversies about the nature of man. First, I shall briefly review three recent controversies about artificial intelligence, controversies over whether computers can think and over whether people are no more than information-processing machines. These three controversies were each initiated by philosophers who, irrespective of what the programs of their time actually did, viewed with alarm the argument that if a machine can think, a thinking being is just a machine. I will then turn to the major business of this paper: to contrast two developments from within the field of AI which have been interpreted by some as successful steps toward simulating human thought, and also to contrast some reactions to that claimed success. Finally, we will look at some recent developments in the field of AI that suggest that the whole discussion about machine intelligence is at best premature and at worst irrelevant

Mathematics in America: The First Hundred Years

There are two main questions I shall discuss in this paper. First, why was American mathematics so weak from 1776 to 1876? Second, and much more important, how did what happened from 1776-1876 produce an American mathematics respectable by international standards by the end of the nineteenth century? We will see that the weakness -at least as measured by the paucity of great names- co-existed with the active building both of mathematics education and of a mathematical community which reached maturity in the 1890\u27s

“Notation, Notation, Notation” or Book Review: Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers, by Joseph Mazur

This review describes Mazur\u27s engaging popularization of an interesting and important topic, the history of mathematical symbols and notation. The reviewer only wishes that some of the history had been done better

“It’s All for the Best”: Optimization in the History of Science

Many problems, from optics to economics, can be solved mathematically by finding the highest, the quickest, the shortest—the best of something. This has been true from antiquity to the present. Why did we start looking for such explanations, and how and why did we conclude that we could productively do so? In this article we explore these question and tell a story about the history of optimization. Scientific examples we use to illustrate our story include problems from ancient optics, and more modern questions in optics and classical mechanics, drawing on ideas from Newton’s and Leibniz’s calculus and from the Euler-Lagrange calculus of variations. A surprising role is also played by philosophical and theological ideas, including those of Leibniz, Maupertuis, Maclaurin, and Adam Smith

Newton, Maclaurin, and the Authority of Mathematics

Sir Isaac Newton revolutionized physics and astronomy in his Principia. How did he do it? Would his method work on any area of inquiry, not only in science, but also about society and religion? We look at how some Newtonians, most notably Colin Maclaurin, combined sophisticated mathematical modeling and empirical data in what has come to be called the Newtonian Style. We argue that this style was responsible not only for Maclaurin’s scientific success but for his ability to solve problems ranging from taxation to insurance to theology. We show how Maclaurin’s work strengthened the prestige of Newtonianism and the authority of mathematics in general, and close with some observations about the authority of mathematical methods throughout history

Is Mathematical Truth Time-Dependent?

Another such mathematical revolution occurred between the eighteenth and nineteenth centuries, and was focused primarily on the calculus. This change was a rejection of the mathematics of powerful techniques and novel results in favor of the mathematics of clear definitions and rigorous proofs. Because this change, however important it may have been for mathematicians themselves, is not often discussed by historians and philosophers, its revolutionary character is not widely understood. In this paper, I shall first try to show that this major change did occur. Then, I shall investigate what brought it about. Once we have done this, we can return to the question asked in the title of this paper