In March, Peirce began a discussion by correspondence with his former Johns Hopkins student Allan D. Risteen about how to conduct experiments to measure the curvature of space. On 3 March, Risteen wrote that the application of Peirce’s method for determining “the constant of Non-Euclidean space” was “very beautiful,” and on the following day he sent Peirce a list of twentythree “double (or triple) stars to which the spectroscopic method might perhaps be applied.” About three weeks later, on the 24th, Peirce wrote that he intended to go ahead with the investigation: “I propose to see what the evidences of the curvature of space may be. Probably there is no argument on the subject not open to objection. Yet if they all tend one way, it will come to something.” Peirce’s “Methods of Investigating the Constant of Space” (sel. 36), would guide his experiments. On 16 February, at Peirce’s request, Mendenhall had shipped a crate of instruments to him including a theodolite, a wye level, a plane table and alidade, and two telemeters. Presumably, Peirce needed these instruments for his curvature observations.
It was also around this time when Ernst Schröder and his work in logic began to reenter Peirce’s stream of thought. Peirce and Schröder had corresponded during Peirce’s Johns Hopkins years but lost touch afterward. Schröder reestablished contact in February 1890 when he announced to Peirce that the first volume of his Vorlesungen über die Algebra der Logik (exakte Logik) would soon be published and that he would have a copy sent to Peirce. The volume did not come out until early in 1891, however, and Peirce would have presumably received his copy by March, or would have seen it at the Astor Library, and would have known that Schröder had built on his foundations.35 As Christine Ladd-Franklin observed in her January 1892 review in Mind: “The plan of Dr. Schröder in his book follows closely upon that of Mr. Peirce as set forth in Vol. III. of the American Journal of Mathematics; that is to say, all the formulae are established by analytical proofs based upon the definitions of sum, of product, and of the negative, and upon the axiom of identity and that of the syllogism…. The proofs are, for the most part, the same as those given by Prof. Peirce.” Of course there was a lot more Schröder than Peirce in Schröder’s Logik, and some notable differences of opinion, but Peirce must have been pleased with the promise of Schröder’s book for the future of his line of logical thought. It would not be until 1893, after Peirce had received three more letters from Schröder, that they finally resumed a regular correspondence (by then Peirce had also received the second volume of Schröder’s Logik).
In the spring of 1891 Peirce resumed work on his own “exact logic,” specifically on the algebra of the copula (sels. 31–35). This work was the successor to Peirce’s well-known treatment of the algebra of the copula in his famous American Journal of Mathematics papers of 1880 and 1885,36 papers that had much influenced Schröder. Peirce might have been working up a presentation paper, possibly stimulated to resume his study of the copula because of Schröder’s rejection of the copula as the preferred logical connective. Peirce may also have been working toward a logic book; his 1894 “How to Reason,” more commonly known as his “Grand Logic,” would include a substantial chapter on “The Algebra of the Copula.” It is noteworthy that Peirce began sel. 32 with what is in effect the truth table for the copula of inclusion—or the copula of consequence, as he called it in sel. 33. In these selections, Peirce showed some movement toward a graphical approach that would gain momentum with his recognition in his summer 1892 “Critic of Arguments” papers that there are significant diagrammatic features in logical algebras. Peirce’s attention to operations with parentheses and rules for inserting or omitting propositional variables into or from parenthetical spaces bears some resemblance to later existential-graph operations.37
It was likely in March 1891 that the Peirces decided to name their estate Arisbe. In December 1890, they had settled a claim made against their property by Levi Quick and at last gained clear title to all of their estate. They had finished a first round of home renovations the previous year and now that Charles had begun construction of his philosophical mansion in the pages of the Monist, he and Juliette returned to making grand designs for their house and their estate. They made a survey of Wanda Farm, they unsuccessfully petitioned the township to reroute the public road that passed nearby, and they received confirmation that they held the timber and quarry rights to their property. With the return of hope for the possibilities of life in their Milford home, and perhaps with the idea of symbolizing that the estate was ready to assume a new character, they must have felt that the time was right to settle on a new name. How they came to choose the name “Arisbe” is unknown and has given rise to a lot of speculation. Max H. Fisch for example suggested that Peirce named his Arisbe after the Milesian colony northeast of Troy along the Hellespont in ancient Greece,38 because Miletus was the home of the first Greek philosophers, and the scientific philosophy they aspired to was of a kind with the philosophical program Peirce had inaugurated with his “Guess at the Riddle.” Or perhaps it was because Peirce was familiar with Book IV of Homer’s Illiad, which tells the story of Axylus, whose home on the road through Arisbe was known as a place of welcome to all who passed by. Whatever the source of the inspiration (other explanations exist), the estate became known as “Arisbe” in 1891 and by the fall, the Peirces had begun using “Arisbe” regularly in their correspondence.
The fifth volume of the Century Dictionary was published in May, so sometime before that Peirce turned to the proofs for the sixth and final volume. Under much work pressure, he arranged in mid-May for the Century Company to hire Allan Risteen to help him with the research for his remaining definitions. Risteen worked as a safety engineer for an insurance company in Hartford, Connecticut, but he was able to spend much of June and July helping Peirce, often traveling to New York or Cambridge for library research. Surviving correspondence indicates that Peirce used Risteen to help with specialized mathematical or scientific terms, working up material for the constellations in the S to Z letter range, for example, but much of Risteen’s effort focused on three words given encyclopedic treatment: theorem, transformation, and triangle. By the middle of July, Risteen had finished his work as Peirce’s special assistant and Peirce must have concluded his work by the end of August. The 18 September 1891 issue of Science carried a notice that the Century Dictionary “is at last completed; the sixth and concluding volume will soon be brought out, the final pages being now on the press.”
In early June, Peirce asked Risteen to add “trees” to his research list of mathematical subjects. It had occurred to Peirce that Arthur Cayley’s diagrammatic method of using branching trees to represent and analyze certain kinds of networks based on heritable or recurrent relations would be useful for his work on the algebra of the copula and his investigation of the permutations of propositional forms by the rearrangement of parentheses. In “On the Number of Dichotomous Divisions: A Problem in Permutations” (sel. 35), Peirce sought to determine how many propositional forms there were given a certain number of copulas (or any other non-associative connective) and a continuous supply of parentheses, and used Cayley’s method of trees to work out his solution.39 When Risteen wrote back on 10 June, his report, illustrated with diagrams, dealt