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The chapter provides an overview of Elyot’s compilation based upon his own comments, and then concentrates on the lexicographical principles that characterize his particular approach. His word selection, his treatment of homographs and their lexicographical arrangement are described. The study of his grammatical framework reveals the use of some terms which, according to the OED, are first found in his dictionary. His descriptive analyses of word-formations are highlighted and a detailed account is given of his sources.

This chapter covers the lexicographical ordering of lower Christoffel words, which is equivalent to the ordering by their slopes (Borel and Laubie). Lower Christoffel words are particular Lyndon words. They are maximum for the lexicographical order among Lyndon words of a given slope (Borel and Laubie). They are, together with the upper Christoffel words, the only unbordered finite Sturmian words (Chuan). They are exactly the Lyndon words which are Sturmian words (Berstel and de Luca). The standard factorization of a lower Christoffel word is obtained by cutting before the smallest lexicographical suffix. Finally, they are exactly the Lyndon words which are equilibrated (Melançon).

What is truth? How do we form our judgements as to what is true and what is untrue about the world? Are we simply following some algorithm - no doubt favoured over other less effective possible algorithms by the powerful process of natural selection? Or might there be some other, possibly non-algorithmic, route - perhaps intuition, instinct, or insight - to the divining of truth? This seems a difficult question. Our judgements depend upon complicated interconnected combinations of sense-data, reasoning, and guesswork. Moreover, in many worldly situations there may not be general agreement about what is actually true and what is false. To simplify the question, let us consider only mathematical truth. How do we form our judgements - perhaps even our ‘certain’ knowledge - concerning mathematical questions? Here, at least, things should be more clear-cut. There should be no question as to what actually is true and what actually is false - or should there? What, indeed, is mathematical truth? The question of mathematical truth is a very old one, dating back to the times of the early Greek philosophers and mathematicians - and, no doubt, earlier. However, some very great clarifications and startling new insights have been obtained just over the past hundred years, or so. It is these new developments that we shall try to understand. The issues are quite fundamental, and they touch upon the very question of whether our thinking processes can indeed be entirely algorithmic in nature. It is important for us that we come to terms with them. In the late nineteenth century, mathematics had made great strides, partly because of the development of more and more powerful methods of mathematical proof. (David Hilbert and Georg Cantor, whom we have encountered before, and the great French mathematician Henri Poincaré, whom we shall encounter later, were three who were in the forefront of these developments.) Accordingly, mathematicians had been gaining confidence in the use of such powerful methods. Many of these methods involved the consideration of sets with infinite numbers of members, and proofs were often successful for the very reason that it was possible to consider such sets as actual ‘things’ - completed existing wholes, with more than a mere potential existence.

Basic theory of continued fractions: finite continued fractions (for rational numbers) and infinite continued fractions (for irrational numbers). This also includes computation of the quadratic number with a given periodic continued fraction, conjugate quadratic numbers, and approximation of reals and convergents of continued fractions. The chapter then takes on quadratic bounds for the error term and Legendre’s theorem, and reals having the same expansion up to rank n. Next, it discusses Lagrange number and its characterization as an upper limit, and equivalence of real numbers (equivalent numbers have the same Lagrange number). Finally, it covers ordering real numbers by alternating lexicographical order on continued fractions.

This chapter offers an overview of words and quadratic numbers, and in particular ordering the conjugates of a Christoffel word. Within this topic the reader learns that the reversal of a Christoffel word is a conjugate, and that the lower and upper Christoffel words of the same slope are the smallest and the largest in their conjugation class. The chapter discusses computation in terms ofMarkoff numbers of the quadratic real number which has a periodic continued fraction with periodic pattern equal to a Christoffel word written on the alphabet 11, 22. It also reviews computation of theMarkoff supremum of a periodic biinfinite sequence, and of theLagrange number of a periodic sequence, both having a periodic pattern as above.