This deep and well documented work traces the history of the (increasingly) mathematical treatment of natural phenomena (mixed mathematics, as it is sometimes called) from the time of Plato and Aristotle to Descartes. The introduction summarizes the important issues in current history of science studies clearly and concisely and outlines an ambitious agenda: that of recasting our view of the history of science. Capecchi provides convincing evidence that the emergence of mathematically-based experimentation (as personified by Galileo) was not a revolution, but rather the culmination of a continuous evolution of the roles of observation, experiment, and mathematics in science. This includes the emergence of new experimental techniques and, more importantly, new mathematics including analytic geometry and calculus.

In terms of mathematics, a key concern is the ontological status of mathematics and its resultant epistemological standing. What is mathematics, and what use can science make of it? How trustworthy are its results? Capecchi does a fine job of explicating the ways our evolving view of mathematics (and experimentation) influenced the kind of science which people did. In particular, the question of the status of mathematics receives a great deal of attention. Capecchi provides us with careful descriptions of the role of mathematics in the work of the various actors he considers.

In my view there are (broadly put) three possible positions one can take with regard to the role of mathematics in the sciences. Capecchi does a fine job of teasing out the history of the (continuing) debate as to which of these three (if any) is appropriate.

- Mathematics provides a set of transcendent rules which govern the workings of the universe. As Newton put it, we are to “subject Nature to the laws of mathematics”
- Regardless of its ontological status mathematics is the best (perhaps the only?) tool we can use to describe the observed events in the physical universe.
- Mathematics is a free creation of the human mind and (for whatever reason) has turned out to be a useful means of generating correct predictions of future events. It has no relation to what’s actually going on. This view is often referred to as instrumentalism.

Each of these three positions had several well-known advocates.

In addition to Newton, Johannes Kepler viewed mathematics as the free creation of God which was then used as a sort of blueprint for the construction of the universe. Since mathematics was created prior to the universe, Plato and others saw mathematics as imposing its will on the universe. When interviewed after the first observational confirmation of his theory of relativity, Einstein was remarkably unexcited, stating that should his theory have been proved wrong he would have been sorry for the Lord. Capecchi provides a wonderful quote from Evangelista Torricelli (1608–1647) which perfectly captures the notion of God as mathematician and aesthetician: “That the descents of heavy bodies in equal times are as odd numbers… is so true that nature, even if (it) wanted, could not do otherwise.” (p. 317)

Galileo saw himself as describing how, and not why things work the way they do. He would do so by using experimental methods to deduce mathematical relationships. As Capecchi paraphrases: “Mathematics is the language of the grand book of the universe. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed.” (p.263)

Among the most important of those who espoused the instrumentalist philosophy was Pierre Duhem (1861–1916) who famously spoke of Ptolemy’s epicycles and deferents as mathematical fictions aimed merely at “saving the phenomena.” More recently, the ontological status of mathematics has been critically reevaluated. The most complete treatment of this subject I am aware of is *Plato’s Ghost* by Jeremy Gray. For many philosophers mathematics is more correctly viewed as the free creation of humans, not God. Charles Sanders Pierce (1839–1914) said that “...mathematics deals exclusively with hypothetical states of things, and asserts no matters of fact whatever. The certainty of pure mathematics … is due to the circumstance that it relates to objects which are the creation of our own minds.” (*Plato’s Ghost*, p. 242) If that is the case, we can hardly claim certitude with regard to mathematically based claims about the physical world.

While mathematics is a central theme, this is not a history of mathematics. Mathematicians who are unfamiliar with the historical actors (and the historians of science who study them) described here will find the text heavy going. As an example, on several occasions the Galilean scholar Stillman Drake is referred to merely as “Drake.” For the most part book titles are given in the original language of the author without translation. I think English translations would have been helpful here.

The key story Capecchi tells is that of the emergence of experimental methods aimed at creating what we would now call mathematical models which are subsequently used to predict further results. At a deeper level Capecchi traces our ambivalence with respect to the status of these models. Does a mathematical model represent reality (as Laplace would claim) or merely provide a helpful means of organizing our observations and experiences?

As we were all taught at some point in time, the “Scientific Method” is supposed to provide a simple, straightforward route from experimental results to scientific theory, usually in the form of a mathematical relationship. The actual situation is (as usual) much more complicated. While experiments are often offered in support of theories it is often doubtful at best that the scientist in question created a given theory based solely on experimental data. Capecchi reports on the work of several historians of science who have investigated this matter by attempting to replicate historically reported results using the technology we presume was available at the time. An example of particular interest here is Galileo’s *Two New Sciences* where he reports that the distances traveled by an object on an inclined plane are proportional to the odd numbers. Capecchi reports on recent work which suggests that he could not have obtained the accuracy he claims with the tools he had available to him. For more on this see *Reconstructions: Recreating Science and Technology of the Past* (Staubermann, Klaus, eds. Edinburgh: National Museum of Scotland, 2011)

The discussions of dynamics, cosmology, astronomy, financial mathematics, and strength of materials are likely those which mathematicians without a strong interest in history of science will find most interesting and informative. In all of these areas, the role and status of mathematics was a source of continuing debate during the time period under consideration.

There is a very nice treatment of the various abacus texts including The *Regoluzze *of Paolo dell’ Abbaco (c. 1281 to c. 1367), which was first published by Guglielmo Libri Carucci dalla Sommaja (1803–1869). (Guglielmo is an interesting character in his own right. Appointed Inspector of libraries in France, he stole thousands of books and was forced to flee the country.) Capecchi provides a list of 50 practical problems from *The Regoluzze*. These provide a nice insight into the nature of these books, many of which were based on Leonard of Pisa’s *Liber Abaci*. Here are two examples:

- (# 46, page 82) If you want to subtract a number from another, put the smaller under the greater and then subtract each figure below from the figure above, starting from the right. When the figure below is the greater add ten to the figure above and add one to the figure below
- (#41, page 81) If you divide 72 years by what is lent the hundred per year (that is, the interest rate as a percentage), you will obtain in how many years a given quantity will double.

The breadth of sources and the care with which they are documented should make this work an important source for those with a strong interest in the history of science. The author introduces the work and methods of the famous and the not-so-famous in an engaging, detailed and complete manner. Quotations (many translated by the author himself) and sketches (often redrawn for clarity by the author) provide a window into the thinking of the person whose work is under discussion. At the end of each chapter original language versions are provided for quotations rendered in English within that chapter. In addition, a list of references is provided at the end of each chapter. The index contains only names which prevents searches for specific techniques or results. This is partially mitigated by the table of contents which provides enough information to allow informed browsing. At the end I found myself wishing Capecchi had spent more time on broad trends — perhaps in the form of chapter summaries.

Finally, I would be remiss if I did not report on the disturbing number of typographical errors and awkward sentences — a situation one would not expect to encounter in a Springer text. Here is an example from page 73: “Pisano found some difficulties due to the use on not decimal units of measure and should spent some time to convert units of short measures into units of large measure.” There are also numerous instances of a single word error such as “he was not satisfied by the lake of causal explanations…” (p. 239). On page 184 we are told that 6 is a perfect number because the sum of its factors \(1+2+3\) equals their product \(1\cdot 2\cdot 3\).

Richard Wilders is Marie and Bernice Gantzert Professor of the Liberal Arts and Sciences and Professor of Mathematics at North Central College in Naperville, IL. He teaches undergraduate courses in the history of science and of mathematics.