### N. ARIGA's Studies on Eighteenth-Century Mechanics

#### OVERVIEW

My major research field is history of mechanics in the eighteenth century. Especially I'm interested in its conceptual dimensions: concept of force, the relationship between mechanics and mathematical analysis, and so on.

It is well known that in this period, roughly from Newton through Euler to Lagrange, the science of mechanics became "analyticalized" and "systematized": Newly invented differential and integral calculus were more and more in common use, and mechanical laws and principles were integrated into theoretical systems. Mathematicians on the Continent, including the Bernoullis, d'Alembert and Euler, made much contribution to these developments.

In the course of the eighteenth century, however, foundational concepts of mechanics themselves were recast into those familiar to us, and this is especially the case for the concept of force. (I mean the generic concept of force which appears in what we call equations of motion (F = ma), rather than Newton's force of attraction etc.) Some studies have treated this aspect of eighteenth-century mechanics, but seemingly not enough attention has been paid to it.

I began my academic research with the principle of least action, which was the theme of my master's thesis. After that, I began to study on the vis viva controversy and related subjects: Leibnizian ideas of vis viva and vis mortua, his idea of "dynamics" and its reception (or transformation), the problem of collision of bodies, and criticisms on the concept of "force of bodies in motion." My suggestion is that the modern concept of mechanical force and the modern usage of the word "force" emerged in these discussions.

Also I've been interested in the relationship between mechanics and mathematical analysis. I made a study about contemporary ideas on the foundation of calculus and, for the case of Lagrange's early years, tried to show how his understanding of analysis and mechanics was interconnected.

Before completing my PhD thesis, I obtained a job at the National Museum of Nature and Science (in Japan). Since my official work is basically about the history of science and technology in Japan, I continue my PhD study at home. In the thesis I consider the reform of the concept of force and the emergence of modern science of mechanics.

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#### DESCRIPTIONS OF MY ARTICLES

Since I've written my articles mainly in Japanese, I'll here give descriptions about them.

##### Euler's variational mechanics (Japanese, 2006)

This is my first article, but now I'm inclined to call it as a kind of "étude". It analyses Euler's two main works on the principle of least action, i.e. the second appendix to his book on variational calculus (1744) which discusses motion of projectiles, and his paper on the "harmony" between principles of rest and motion (1751). I maintained that Euler's argument had a contradiction, for he adhered to minimum rather than extremum.

##### What was the vis viva controversy? (Japanese, 2009)

This is a "survey article". Since I could find neither summary of the vis viva controversy nor a survey of historical studies on this subject, I tried to give an overview. It consists of two sections: section one summarizes the views and arguments of some historical studies (from 1960s on), while section two gives a chronicle of the controversy based on secondary works.

##### Maupertuis's "action" and Euler's "effort":

Two principles of least action in the mid-eighteenth century (Japanese, 2009)

This article stresses the difference between Maupertuis's and Euler's conception of the principle of least action. My point is that they employed the term "quantity of action" in different meanings. They departed (as previous studies correctly point out) from different problems independently, and (this is my argument) even when they came to know their works each other, they didn't change their views on what this principle meant. I showed these points by referring to their articles and correspondence. The conclusion is that we cannot regard their works as two different versions of the same principle, but that there were two different principles.

##### Young Lagrange meets the "metaphysics" of mathematics:

In the context of French infinitesimal controversy (Japanese, 2010)

This article attempts to evaluate Lagrange's conceptions of differential calculus in his early years. His lecture note in the latter half of 1750s shows that at that time Lagrange considered a kind of limit procedure as the foundation of the calculus. I explored its background in debates in France, for Lagrange appeared to have been familiar with main works written in French. After examining works by L'Hôpital, Fontenelle and Maclaurin (all cited in Lagnrange's lecture note), I concluded that his lecture was an attempt to present the system of calculus in a novel manner and that he got this idea possibly from d'Alembert's article in Encyclopédie.

##### Variational mechanics at dawn:

Maupertuis, Euler, Lagrange and the principle of least action (Japanese, 2011)

This is an extended version of my lecture aimed at physical scientists. As the title suggests, it describes the emergence of the principle of least action in the eighteenth century. I presented the history, however, not as a linear development. Rather I stressed the difference between the three persons' works by paying attention to what they did with that principle. Since this paper is not a reviewed article (published in proceedings, basically without notes), I'd like to recompose an article on this subject.

##### Theories of collision as an example of Rational Mechanics, from 1720 to 1730 (Japanese, 2012)

This paper is, as far as I know, the first serious attempt to analyze theories of collision in the eighteenth century. Collision of bodies is a popular topic in seventeenth-century mechanics (or mechanical philosophy at large) but has been forgotten in eighteenth-century mechanics. I explored some theories proposed from 1720 to 1730, finding that this subject was indeed popular and that especially Johann Bernoulli and Euler showed interest in reducing the theory into general principles of mechanics. Thus we may regard collision of bodies as an important problem for the development of eighteenth-century mechanics.

##### The emergence of the dynamique in the Paris academy of sciences: From a science of force to a science of motion (English, 2013)

This piece, presented in a French-Japanese workshop, is my first article written in English. It discusses how and when the word "dynamique" became popular in French literatures. "Dynamica" was invented by Leibniz at the end of the seventeenth century and originally meant a science on "vis viva." Around 1740, however, members of the Paris Academy of Sciences started to use the term "dynamique" for a science about a special kind of motion. I showed that this new connotation was due to what I call Bernoullian connections, including Maupertuis and Clairaut as main figures.

The whole article is available --> [PDF file].