1.1. Review by: Charles V Jones.
Isis 70 (4) (1979), 629-631.

This collection was edited by Carolyn Eisele from Peirce manuscripts, most of which are in the Harvard University libraries. Although most of these have been available on microfilm, with an index, no substantial collection of mathematical papers has been published and made conveniently available before this edition.
...
Professor Eisele has made an enormous contribution by editing these papers for publication. In addition, in her several introductions (totalling about 81 pages in the first three volumes; the last volume is simply introduced by a series of excerpts from various Peirce papers), she has attempted to provide "some cohesive form [to] the major elements of Peirce's mathematical activity." She has provided quite a bit of background information and suggested that a thematic thread through these papers is the concept of continuity. But I do not think she has succeeded in providing cohesiveness. For example, the introduction to the third volume is separated into sections such as Infinitesimals, Existential Graphs, and similar headings recognisable as topics in which Peirce was interested. But these discussions are not clearly linked to the papers printed, because the table of contents uses a different set of headings, and no footnotes or explanations link the introductory material to pertinent parts of the papers following. Further, extracts are quoted or paraphrased often without indicating manuscript sources, names are mentioned with bewildering rapidity or familiarity (Peirce's brother is often referred to by his nickname), and the reader's attention is seldom directed to other discussions of the same ideas. No dates are provided with the manuscripts (instead the reader is referred to Robin's index). Consequently, one finds two papers on probability and induction, for example, placed next to each other which were apparently composed forty-four years apart, but there is no indication of this time lapse. This is an unnecessary inconvenience to the historian and could lead the unwary into unfounded conclusions. Also, the indexes are not adequate, because they fail to give all the locations where entries are discussed, and the entries are too few for the breadth of the manuscripts. I think that some material in the first volume is in the wrong order.

1.2. Review by: Max H Fisch.
Transactions of the Charles S Peirce Society 14 (3) (1978), 200-211.

The editorial matter of The New Elements consists primarily of a preface, a general introduction, and introductions to the several volumes. That to the fourth consists entirely of short extracts from Peirce's own writings.

A measure of the importance the editor attaches to consecutive reading may be found in her having absorbed Peirce's own footnotes into parentheses in the text, and absorbed into her own introductions much of what another editor might have dispersed into editorial footnotes. The footnotes with which she is most comfortable are those in which she gives us parallel passages from other manuscripts which she wants us to read at just those points in our progress through the volumes. The footnotes with which she is least comfortable are those that might tempt us to close the volume and take others from our shelves, or make a trip to the library. The design of The New Elements, she trusts, is such that to most of our questions that are proper to the business in hand and that have not yet been answered, we shall find answers by reading on.

Even a first reading less consecutive and thorough than the editor wishes is enough to persuade us that Peirce was a recognised member of the community of mathematical investigators; that he was a contributor to the mathematical revolution that took place in his lifetime; and that he had an unsurpassed vision of the proper role of mathematics in the growth of mind, the shaping of intelligence, and the advancement of science. Would that that vision had been more influential in the revolution in the teaching of mathematics that followed so tardily upon the revolution in mathematics itself!

The editor is a past-president of the Peirce Society, the secretary-treasurer of the Peirce Foundation, active also in the History of Science Society and in Phi Beta Kappa; and she was the chief organiser of the Peirce Bicentennial International Congress that took place in Amsterdam in 1976. Her teaching career was in mathematics at Hunter College. In her own training she inherited the revolution in mathematics. Some sense of her participation in the revolution in the teaching of mathematics may be got from the preface to Annita Tuller's A Modern Introduction to Geometries (1967);

I am most especially indebted to my former teacher, present colleague and friend of many years standing, Professor Carolyn Eisele, for her continual encouragement and constructive criticisms. Professor Eisele suggested writing the book in the first place and used a mimeographed version in her classes at Hunter College. It is with pleasure that I record here my thanks to her for all the time and energy she spent on this book.

It was only quite late in her researches that Professor Eisele became fully aware of the extent of Peirce's unpublished mathematical writings. Long before that, she was a student of Peirce as scientist and as historian of science. Her most widely read and most influential paper was "Charles S Peirce, Nineteenth Century Man of Science," in 1959. It was that paper, for example, that encouraged the late Victor Lenzen, then just retired from a career in physics, to undertake his studies of Peirce as astronomer, geodesist, mathematical physicist, and mathematician.

Shortly after that paper, Professor Eisele began a study of the manuscripts for Peirce's pioneering Lowell Institute Lectures on the History of Science in 1892-93. From these and numerous other manuscripts, it was her ambition to construct the nearest approach now possible to the history of science which Peirce contracted in 1898 to write for Putnam's but never finished. The volume toward which she was working would contain as appendices his more detailed studies of particular episodes. The more general history of science that would fill most of the volume would have its chief value not as a work of reference, but in the vision of the history of science that it would permit its readers to share.

But in the course of searching for relevant manuscripts, Professor Eisele examined Peirce's unpublished mathematical textbooks and other papers, and became convinced that they should take precedence. After all, his work in mathematics was for the sake of the logic of mathematics, and his work in the history of science was for the sake of the logic of science. The former was logically prior to the latter, and it was more expertly carried out, and with a nearer approach to completeness. So she decided to bring The New Elements to publication first, and then resume work on the history of science. To the latter she is now returning.

Meanwhile, however, she has published about thirty papers on Peirce as scientist, mathematician, and historian of science and of mathematics. Many of these were presented at international congresses and symposia of mathematicians, logicians, historians of science, and philosophers, in such cities as Florence, Pisa, Barcelona, Madrid, Stockholm, Ithaca, Warsaw, Bucharest, Moscow, Amsterdam, and Edinburgh. This year she addresses the philosophers at Dusseldorf on "The Modern Relevance of the Mathematical Philosophy of Charles S Peirce." Fortunately for all of us, these papers are being collected and edited by Richard Martin, and the volume should soon appear. Its tentative title is Studies in the Scientific and Mathematical Philosophy of Charles S Peirce. Wherever in those studies references to the volumes and pages of The New Elements would now be in order, it is to be hoped that such citations will be substituted for, or added to, the original references.

2.1. Review by: Joseph W Dauben.
Isis 73 (1) (182), 143-144.

This collection of thirty essays and papers by Carolyn Eisele is a testimony not just to the work of Charles S Peirce, but to Carolyn Eisele herself. It opens with the substantial contribution on Peirce that she was asked to prepare for the Dictionary of Scientific Biography (1974). Twenty-nine papers then follow (in roughly the chronological order of their publication), including articles, book reviews, reports to learned societies and foundations, as well as lectures delivered at international congresses from Moscow to Edinburgh. The first, "The Liber abaci" (1951), recounts Eisele's own initial contact with Peirce, which was entirely fortuitous and happened as she was reading MS 189 of the Plimpton Collection at Columbia University: "As I continued to handle the volume a sheaf of notes on faded blue-grey paper fell out from under the back cover". When she eventually realised that the pages were not student notes, but a letter by Peirce to Plimpton, she knew that she had made an important discovery. In the letter Peirce revealed himself to be not only a well-informed antiquarian, but a perspicacious historian of mathematics as well. Over the years, Eisele has produced articles on a wide variety of subjects ranging from Peirce's work in probability and logic to map projections, binary arithmetic, the four-colour problem, economics, semiotics, and scientific method. Several articles included analysis of Peirce's work in the history of science, especially his interest in such figures as Duns Scotus, Galileo, Kepler, and Darwin. Studies of Peirce at the Smithsonian Institution, his correspondence with Simon Newcomb, his importance as a "nineteenth-century man of science" and his significance as a philosophical critic of figures like Poincaré and Mach all found their way into lectures, or into print. As her research delved further and further into the manuscripts and mind of Charles Peirce, Eisele began to realise that the key to everything was his mathematics.
...
Carolyn Eisele has laboured unstintingly to bring the message of Peirce's mathematics not just to a wider forum of historians of science, but to philosophers and intellectual historians as well. This handsomely printed volume celebrates her many years of scholarly devotion to the study of Charles Peirce; it should be read and appreciated by Peirce scholars everywhere.

2.2. Review by: I Grattan-Guinness.
American Scientist 69 (2) (1981), 240-241.

The author has been a pioneer in the study of the work of C S Peirce. Indeed, her role in developing the current interest in Peirce is especially significant, for she is one of the very few who studies his contributions to the history of science, geography, mathematics, and education. This volume contains many of the papers that she published originally between 1951 and 1977, together with several previously unpublished items.

The principal historiographical problem in Peirce studies is the mass and complexity of his manuscripts. A full edition of all his writings would require over 100 volumes; the edition now being prepared under the direction of M Fisch is planned to comprise about 25. Eisele's book is a useful guide to several little known aspects of Peirce's output, for she has used (and in some cases published) manuscripts and letters as well as his own published works. Chapter 13, "The Problem of Map Projection," is a particularly good example of insights obtained from supplementing Peirce's printed with his written words.

This volume is a valuable complement to the currently available literature on Peirce, which is largely concerned with his philosophy and logic. A certain amount of repetition is evident, especially from re printing Eisele's introductions to her four-volume edition of Peirce's The New Elements of Mathematics (Mouton, 1976) in Chapter 30, where several of the concerns discussed in the earlier chapters are reworked. The editorial treatment is by and large adequate, as is the production.

2.3. Review by: Richard S Robin.
Transactions of the Charles S Peirce Society 18 (4) (1982), 367-370.

Carolyn Eisele has been identified with the scholarly recovery of C S Peirce's work in mathematics and science as well as in the history of science since the early 50's when her articles on Peirce first began to appear. These articles, along with the detailed introductions to the four volumes of Peirce's The New Elements of Mathematics, extensively edited by her, comprise for the most part Studies in the Scientific and Mathematical Philosophy of Charles S Peirce.

The majority of the essays deal with mathematics in one form or another. Some deal with rather specific mathematical problems with which Peirce was concerned, viz., map-projection, the four-colour problem, Fermatian Inference, problems relating to probability and continuity. Others deal with mathematical border areas, viz., mathematical economics and the teaching of mathematics. The New Elements is a collection of Peirce's unpublished papers, including textbook manuscript material, much of which is intrinsically interesting and some of which represents clear advances in mathematical pedagogy.

The scientific essays are much less technical in nature. Rather they are historical in the double sense that (a) they treat Peirce's scientific productivity against the background of 19th century science and (b) they are concerned with his special interests in the history of science and in certain historical figures, especially Galileo and Kepler.
...
Of course Peirce was a logician of immense power and a metaphysician of considerable verve. But it must be acknowledged that Carolyn Eisele has made a persuasive case indeed for thinking that historians of philosophy cannot adequately come to grips with Peircean philosophy if they ignore the range and depth of his mathematical and scientific work. It is impossible to read her Studies without realising how deeply Peirce was committed to mathematics and science. It is inconceivable that a person who worked so intensively on mathematical and scientific problems, beginning at an extremely early age under the tutelage of his father, would not be profoundly influenced by what he knew and how he came to know it.

The Studies is explicitly about Peirce's mathematical and scientific philosophy. Here, I think, its author is less successful, not because she sets out to achieve some end and fails in the attempt but because the problems present are of such magnitude as to preclude easy answers. Carolyn Eisele is fully aware of the magnitude of the problems and certainly regards her efforts as crucial first steps in the right direction. Her intention is to provide an orientation and to offer some suggestions for promising lines of research. One of those suggestions - that we look toward Peirce's doctrine of signs - has considerable merit. She calls special attention to Peirce's role as semiotician, as a philosopher who adopted "mathematical semiotical procedures." If mathematics is not the language of symbols par excellence, what is? Moreover if one takes seriously Peirce's hierarchical ordering of the theoretical sciences, then one has further occasion for reaffirming the significance of mathematics vis I vis the other theoretical sciences. The point of the hierarchical arrangement is to reveal those sciences whose principles are passed down through the ranks. Mathematics is at the top, and stands ahead of philosophy in its several branches and the idioscopic sciences, both physical and social.
...
The majority of the essays (22 out of 27) were published separately before. They are rich in detail, reflecting the fact that their author is herself a mathematician and an avid student of its history. She admits to a mathematical bias. So much the better. She writes about her subject with extraordinary dedication and knowledge. The essays were a revelation to this reviewer when he first came upon them when they originally appeared. Re-reading them in their assembled form, he found them even more of a revelation. The fact remains that present and future Peirce scholars owe and will owe much to the pioneering work of Carolyn Eisele.

2.4. Review by: Z Imre Ruzsa.
Mathematical Reviews MR0569470 (81j:01047).

This book is a collection of 30 papers and lectures (published or read from 1951 to 1977) of the eminent Peirce researcher. Most of these articles are connected with the mathematical philosophy of Peirce. Some main topics of the volume: Peirce as philosopher and historian of science (Chapters 3, 6, 7, 8, 10, 22, 23 and 27; but this topic is touched upon in almost all chapters); Peirce's interest towards Fibonacci, Galileo and others (Chapters 2 and 6); Peirce's critique on the views of some contemporary scientists (Poincaré, Mach, Cantor; Chapters 11, 20, 21, and 26); Peirce on limits and infinitesimals, with reflections to nonstandard analysis (Chapters 5, 23 and 27); his ideas on infinity and continuity (Chapters 14 and 18); Peirce on binary arithmetic, the four-colour problem, and mathematical economics (Chapters 17, 19 and 24); the problems of mathematical education in Peirce's view (Chapters 16, 26 and 30), the mathematical foundations of Peirce's philosophy (Chapters 22, 23 and 27); Peirce as a semiotician in mathematics (Chapter 29).

The book contains numerous quotations from Peirce's letters and other manuscripts. As a whole, it is a very excellent contribution to the elaboration and evaluation of the Peirce heritage.

3.1. Review by: Stephen H. Levy.
Transactions of the Charles S Peirce Society 18 (4) (1982), 367-370.

This issue of the internationally renowned journal Historia Mathematica comprises the papers presented at the colloquium held at Hunter College in May, 1981, in honour of Peirce's work and Carolyn Eisele's invaluable contributions to Peirce scholarship, in particular her 1976 publication of Peirce's New Elements of Mathematics. The publication of these essays in 1982 also marks Carolyn Eisele's 80th birthday. The issue, therefore, constitutes a Festschrift in her honour, as James B Freeman, one of the organisers of the conference, points out in his preface. There are six distinguished speakers represented - Max H Fisch, Hilary Putnam, Ernest Nagel, Joseph W Dauben, Kenneth L Ketner, and Carolyn Eisele herself. With such a powerful line-up, we can expect to find the essays most stimulating and informative. Our expectations are indeed fulfilled.
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In her paper, "Mathematical Methodology in the Thought of Charles S Peirce," Carolyn Eisele develops her theme that "Peirce hoped to create an exact philosophy by applying the ideas of modern mathematical exactitude." She lends credence to the indubitable influence of mathematics upon his thought with an erudite display of Peirce's mathematical activities, one rivalled only by that of Fisch. She stresses the need to appreciate the mathematical advances of the day in order to follow Peirce's thought properly. In addition to those discussed earlier, these include "the Riemannian sheeted surface, ... invented to represent the character of multiple-valued analytic functions ..." in application to Peirce's Existential Graphs.

As an example of her thesis, Eisele argues that:

.. to understand pragmatism in a Peircean sense one must approach the concept in terms of Peirce's scientific methodology - in terms of the mathematical procedure that he called 'Theorematic Reasoning', (emphasis in original).

She explains that this involves the modification of a diagram - perhaps by introducing auxiliary lines as in geometry - representing the given premises and the resultant ability the modification affords for deducing the conclusion. The specific change stems from an ingenious guess, Peirce's abduction. She remarks that "investigating the truth of any scientific assumption is analogous." The scientific hypothesis comprises a modification of "our expectations in regard to future experience." She quotes Peirce saying that the pragmatic maxim "is simply the experimental method ... . the invariable procedure of all science, ... even of mathematics." The transition here from analogy to identity is perhaps made if we regard expectations as requiring, or being a kind of, diagram.

Eisele emphasises the importance of Peirce's "mathematical substance" as well as his methodology. She takes as one example Peirce's application of non-Euclidean or extra-Euclidean ideas to the problem of the nature of space. She first proceeds through Peirce's representation of a one-dimensional line in which one "passes through infinity" in traversing a line which "returns into itself" or is such that starting at a given point one can move in either direction and eventually return to the starting point. (One can think of a great circle on a sphere.) She next cites Peirce's discussions of the 2-dimensional Möbius strip, which permits similar travel over any of its points since the surface is in effect one-sided, not two. Finally, she mentions Peirce's description of a 3-dimensional space whose Mobius-like twist would give it a shape different from our own, but one which "a man would accustom himself to ..." The similarities and differences here with contemporary conceptions of physics merit further research. Each would likely illuminate the other.

Another example Eisele mentions is Peirce's long letter to James, in which he argues that Möbius's Barycentric Calculus (1827), introducing homogeneous coordinates into geometry, supplies a notation for analysing "the modes in which concepts are, or should be, represented as compounded in definitions. ..." She remarks that "the extent of the influence of these non-Euclidean elements on Peirce's philosophy has yet to be fully determined," and encourages us to take up the task. Let us return to the device of Fisch and imagine the task to be a tour of Peirce's thought. Thanks to Eisele's tireless scholarship, then, a sure guide is available.

4.1. Review by: Richard S Robin.
Transactions of the Charles S Peirce Society 23 (2) (1987), 318-321.

In the last ten years alone Carolyn Eisele has served the Peirce community extraordinarily well both as an editor and essayist. Her edited New Elements of Mathematics by Charles S Peirce, the four volumes of which are devoted in one way or another to Peirce's mathematical writings, appeared in 1976. Three years later there followed a collection of her pathbreaking essays, edited by Richard Martin, entitled Studies in the Scientific and Mathematical Philosophy of Charles S Peirce. Now we have her faithfully edited and handsomely produced two-volume work on Peirce's contributions to the history of science and its logical underpinnings. All these publications are of a piece; together they provide us with a much overlooked but much needed perspective on Peirce's career and thought.

Historical Perspectives is not your ordinary history of science, and readers coming to it in expectation of finding something like the usual historical presentation will be disappointed. But if they approach the work with altered - not necessarily diminished - expectations they will find much that is rewarding, for the simple reason that Peirce brings to his historical studies his aptitude for science and his genius for logic.