This third edition of a popular, well-received text offers undergraduates an opportunity to obtain an overview of the historical roots and the evolution of several areas of mathematics.

The selection of topics conveys not only their role in this historical development of mathematics but also their value as bases for understanding the changing nature of mathematics. Among the topics covered in this wide-ranging text are: mathematics before Euclid, Euclid's *Elements, *non-Euclidean geometry, algebraic structure, formal axiomatics, the real numbers system, sets, logic and philosophy and more. The emphasis on axiomatic procedures provides important background for studying and applying more advanced topics, while the inclusion of the historical roots of both algebra and geometry provides essential information for prospective teachers of school mathematics.

The readable style and sets of challenging exercises from the popular earlier editions have been continued and extended in the present edition, making this a very welcome and useful version of a classic treatment of the foundations of mathematics. "A truly satisfying book." — Dr. Bruce E. Meserve, Professor Emeritus, University of Vermont.

hypothesis) (by N2) whence (by N8) Theorem 4 If e is a natural number such that ae = a for some natural number a, then e = 1. For (by hypothesis) (by N6) whence (by N8) Definition 1 If a + x = b, we say that a is less than b or b is greater than a, and we write a < b or b > a. Theorem 5 If a and b are any two natural numbers, then one and only one of the following situations holds: a = b, a < b, a > b. This is an immediate consequence of N9 and

relations of Problem 3.5.3. (c) This is an immediate consequence of (a) and (b). (e) K =–(1/QP) (1/QT) =–1/(QF)2 =–1/k2. 3.5.5 This follows readily from the definition of the tractrix. 3.5.8 (a) One might interpret abba as “committee” and dabba as “committee member,” and assume that there are just two committees and that no committee member serves on more than one committee. (b) One might interpret a dabba as any one of the three letters a, b, c, and an abba as any one of the three

85 See, for example, D. Hilbert [1], H. G. Forder [1], and W. Sierpinsky. 86 E. H. Moore [2], p. 82. 87 For a simple example, see O. Helmer. 88 H. G. Forder [1]. 89 See E. R. Stabler [4], p. 158. 90 See R. L. Wilder [3], p. 49. 91 A. N. Whitehead, and B. Russell. 92 W. V. Quine. 93 Perhaps the chief problems of metamathematics center around the permissible proof paraphernalia of the axiomatic method so that consistency will be ensured. For an elementary discussion of some

the trisection of an arbitrary angle, and the quadrature of a circle—illustrating the principle that the growth of mathematics is stimulated by the presence of outstanding unsolved problems. Also, some time during the first three hundred years of Greek mathematics, there developed the Greek notion of a logical discourse as a sequence of statements obtained by deductive reasoning from an accepted set of initial statements. Certainly, if one is going to present an argument by deductive procedure,

three integers 1, 2, 3 and of the four integers 1, 2, 3, 4. Each of these interpretations verifies Postulates P1, P2, and P3, but the interpretations are not isomorphic, since it is impossible to set up a one-to-one correspondence between the three elements of the one interpretation and the four elements of the other interpretation. There are advantages and disadvantages in having a postulate set complete. Perhaps the most desirable feature of an incomplete postulate set is its wide range of