# Thinking with Objects: The Transformation of Mechanics in the Seventeenth Century

By Domenico Bertoloni Meli

Baltimore, MD: Johns Hopkins University Press, 2007, illustrated, 389 pages

Reviewed by Michael Nauenberg

Thinking with Objects is a comprehensive book that describes some the outstanding developments in mechanics that occurred during the seventeenth century. The “objects” in the title are the familiar ones found in undergraduate physics labs—the lever, the spring, the pendulum, rolling balls on inclined planes, etc.—that are employed to illustrate the principles of mechanics. Until the seventeenth century, however, most of these principles were not yet known, and experiments with these mundane objects were carried out to find possible mathematical regularities that would describe their observed properties. Any rules found were then applied to understand the properties of a much wider variety of objects, including the motion of celestial bodies.

In recent years, valuable scholarly work has contributed to a deeper understanding of various aspects of the development of mechanics. A professor in the Department of History and Philosophy of Science at Indiana University, Domenico Bertoloni Meli has incorporated the newly found insights in his book, but on some still controversial matters he expresses his own, often strongly held, views. The book contains an excellent and up-todate list of bibliographic references, and the text is often supplemented by reproductions of original diagrams; there are lengthy explanatory captions and informative notes at the end of the book.

In general, physics students learn that the development of mechanics in the seventeenth century began with Galileo and culminated with Newton’s magisterial Principia Mathematica, where the principles discovered in experiments with the pendulum and colliding balls were successfully formulated into precise mathematical laws of motion. But there were also other important figures who contributed to this transformation of mechanics, and Meli discusses their work, too. For example, in the seventeenth, Robert Hooke was undoubtedly the most prolific practitioner in the arts of experimentation. In his investigations he employed not only mundane objects, but also elaborate devices, including the air pump, the microscope and the telescope, which Meli distinguishes here as “philosophical” instruments. In fact, the book cover shows Hooke’s drawing of the spring balance (not identified until it appears in Figure 8.8 on p. 245), which he used to illustrate his famous law, Ut tensio, sic vis [As the extension, so the force].

Recent historical research has also shown that Hooke’s experiments with a conical pendulum and with a rolling ball in an inverted cone led him to fundamental insights on the general principles of orbital motion. In a very fruitful correspondence, Hooke communicated his physical ideas to Newton, who then combined them with his own insights into a precise mathematical formalism of mechanics—without, however, giving Hooke any credit. Another important contributor, Christiaan Huygens, also experimented with the conical pendulum, and developed sophisticated mathematical methods including the use of infinitesimal quantities, predating the invention of the calculus by Newton and Leibniz. This work led Huygens to discover that the centripetal acceleration of a body revolving uniformly around a circle is proportional to the square of its velocity and inversely proportional to the radius of the circle. Thus Huygens— and independently Newton— was able to explain, for the first time, why a body on the surface of our rotating earth does not fly off, and in the process resolved one of the major objections to Copernican astronomy.

Thinking with Objects is however marred by a lack of clarity and some serious errors in explaining the physics and mathematics underlying the historical experiments. For example, in footnote 56 on p. 348, Meli claims that “a correct way to prove isochronism [of a mass oscillating at the end of a spring] would be to use the average speed, which is proportional to AC [the maximum displacement], therefore time [the period] is indeed constant.” In addition, he indicates that this manifestly incorrect argument is to be found in Newton’s proof of Proposition 10 in the Principia, given in corollary 2.

Describing Leibniz’s calculus, Meli explains that “the differential of an incomparably small distance ds gives a distance dds that is twice incomparably smaller than a finite magnitude” (p. 291). This explanation, however, is meaningless mathematically. In fact, at that time, Newton already had given in the Principia a precise mathematical definition to such first- and secondorder differentials by considering the ratios of these quantities, and obtaining the limiting value of these ratios when the differentials vanish, which conforms with modern calculus.

In his introduction, Meli observes that the “choice of endpoints of a historical narrative is crucially important.” For the period under consideration, he bestows the honor for these two nodal points to Guidobaldo dal Monte, who was a patron and collaborator of Galileo, and to Pierre Varignon. Meli describes dal Monte’ s work in considerable detail (the book’s index has about the same number of page references to dal Monte and to Newton), and he makes a good case for attributing to dal Monte a pivotal role at the start of the transformation of mechanics in the seventeenth century. But naming Varignon as an end point of this transformation is not justified. Together with Jakob Hermann and the Bernoulli brothers, Varignon contributed to the translation of Newton’s geometrical formulation of mechanics in the Principia into the differential language of the calculus, first introduced into the continent by Leibniz. But Varignon did not make any original contributions to mechanics, and therefore it cannot be argued that he had any special role as an endpoint to the century-long transformation of this subject.

Some mathematical descriptions in the book are unnecessarily difficult to follow, because Meli often avoids using modern equations. He properly regards such equations as an anachronism that obscures the difficulties of using the mathematics of proportions that was available in the seventeenth century. But why impose these difficulties on modern readers?

In the last chapter of his book, Meli goes a bit overboard by introducing an elaborate “mapping” scheme— drawing lines and arrows connecting objects and systems to illustrate the main thesis of his book. But this mapping does not convey any useful information, nor can I make sense, while referring to these objects, of remarks like “I find their amphibious and ambiguous nature helpful . . . because it reflects the status of mechanics as a mixed mathematical discipline.”

On the whole, however, Thinking with Objects is an excellent historical account that I highly recommend.

Michael Nauenberg is Professor of Physics, Emeritus, at the University of California, Santa Cruz, and a well-regarded scholar on Hooke and Newton.

Baltimore, MD: Johns Hopkins University Press, 2007, illustrated, 389 pages

Reviewed by Michael Nauenberg

Thinking with Objects is a comprehensive book that describes some the outstanding developments in mechanics that occurred during the seventeenth century. The “objects” in the title are the familiar ones found in undergraduate physics labs—the lever, the spring, the pendulum, rolling balls on inclined planes, etc.—that are employed to illustrate the principles of mechanics. Until the seventeenth century, however, most of these principles were not yet known, and experiments with these mundane objects were carried out to find possible mathematical regularities that would describe their observed properties. Any rules found were then applied to understand the properties of a much wider variety of objects, including the motion of celestial bodies.

In recent years, valuable scholarly work has contributed to a deeper understanding of various aspects of the development of mechanics. A professor in the Department of History and Philosophy of Science at Indiana University, Domenico Bertoloni Meli has incorporated the newly found insights in his book, but on some still controversial matters he expresses his own, often strongly held, views. The book contains an excellent and up-todate list of bibliographic references, and the text is often supplemented by reproductions of original diagrams; there are lengthy explanatory captions and informative notes at the end of the book.

In general, physics students learn that the development of mechanics in the seventeenth century began with Galileo and culminated with Newton’s magisterial Principia Mathematica, where the principles discovered in experiments with the pendulum and colliding balls were successfully formulated into precise mathematical laws of motion. But there were also other important figures who contributed to this transformation of mechanics, and Meli discusses their work, too. For example, in the seventeenth, Robert Hooke was undoubtedly the most prolific practitioner in the arts of experimentation. In his investigations he employed not only mundane objects, but also elaborate devices, including the air pump, the microscope and the telescope, which Meli distinguishes here as “philosophical” instruments. In fact, the book cover shows Hooke’s drawing of the spring balance (not identified until it appears in Figure 8.8 on p. 245), which he used to illustrate his famous law, Ut tensio, sic vis [As the extension, so the force].

Recent historical research has also shown that Hooke’s experiments with a conical pendulum and with a rolling ball in an inverted cone led him to fundamental insights on the general principles of orbital motion. In a very fruitful correspondence, Hooke communicated his physical ideas to Newton, who then combined them with his own insights into a precise mathematical formalism of mechanics—without, however, giving Hooke any credit. Another important contributor, Christiaan Huygens, also experimented with the conical pendulum, and developed sophisticated mathematical methods including the use of infinitesimal quantities, predating the invention of the calculus by Newton and Leibniz. This work led Huygens to discover that the centripetal acceleration of a body revolving uniformly around a circle is proportional to the square of its velocity and inversely proportional to the radius of the circle. Thus Huygens— and independently Newton— was able to explain, for the first time, why a body on the surface of our rotating earth does not fly off, and in the process resolved one of the major objections to Copernican astronomy.

Thinking with Objects is however marred by a lack of clarity and some serious errors in explaining the physics and mathematics underlying the historical experiments. For example, in footnote 56 on p. 348, Meli claims that “a correct way to prove isochronism [of a mass oscillating at the end of a spring] would be to use the average speed, which is proportional to AC [the maximum displacement], therefore time [the period] is indeed constant.” In addition, he indicates that this manifestly incorrect argument is to be found in Newton’s proof of Proposition 10 in the Principia, given in corollary 2.

Describing Leibniz’s calculus, Meli explains that “the differential of an incomparably small distance ds gives a distance dds that is twice incomparably smaller than a finite magnitude” (p. 291). This explanation, however, is meaningless mathematically. In fact, at that time, Newton already had given in the Principia a precise mathematical definition to such first- and secondorder differentials by considering the ratios of these quantities, and obtaining the limiting value of these ratios when the differentials vanish, which conforms with modern calculus.

In his introduction, Meli observes that the “choice of endpoints of a historical narrative is crucially important.” For the period under consideration, he bestows the honor for these two nodal points to Guidobaldo dal Monte, who was a patron and collaborator of Galileo, and to Pierre Varignon. Meli describes dal Monte’ s work in considerable detail (the book’s index has about the same number of page references to dal Monte and to Newton), and he makes a good case for attributing to dal Monte a pivotal role at the start of the transformation of mechanics in the seventeenth century. But naming Varignon as an end point of this transformation is not justified. Together with Jakob Hermann and the Bernoulli brothers, Varignon contributed to the translation of Newton’s geometrical formulation of mechanics in the Principia into the differential language of the calculus, first introduced into the continent by Leibniz. But Varignon did not make any original contributions to mechanics, and therefore it cannot be argued that he had any special role as an endpoint to the century-long transformation of this subject.

Some mathematical descriptions in the book are unnecessarily difficult to follow, because Meli often avoids using modern equations. He properly regards such equations as an anachronism that obscures the difficulties of using the mathematics of proportions that was available in the seventeenth century. But why impose these difficulties on modern readers?

In the last chapter of his book, Meli goes a bit overboard by introducing an elaborate “mapping” scheme— drawing lines and arrows connecting objects and systems to illustrate the main thesis of his book. But this mapping does not convey any useful information, nor can I make sense, while referring to these objects, of remarks like “I find their amphibious and ambiguous nature helpful . . . because it reflects the status of mechanics as a mixed mathematical discipline.”

On the whole, however, Thinking with Objects is an excellent historical account that I highly recommend.

Michael Nauenberg is Professor of Physics, Emeritus, at the University of California, Santa Cruz, and a well-regarded scholar on Hooke and Newton.