CALCULUS WITH ANALYTIC GEOMETRY.

By Profs. Richard H. Crowell andWilliam E. Slesnick (Mathematics). NewYork: W. W. Norton, Inc., 1968. 727 pp.$9.95.

This book aims to impart the fundamental facts of the calculus of functions of one real variable, together with the standard lore of elementary special functions and a little geometry of plane curves. The level is a sensible one, not neglecting the fine points but not insisting on them too much either A good instance is the discussion of the Riemann integral which I thought especially well done. This notion is introduced using upper and lower approximating sums in a very clear geometrical way. The fact that an increasing function has an integral is proved and it is noted that a continuous function does too, but the proof is only sketched.

The discussion of differentials (a point upon which many books come to grief) is also very well done.

The only thing I found objectionable was a big production about the identity function, seducing the authors into writing such stuff as (f(X))(2) = f(X(2)) = f(2): perfectly clear, but I should think it would cause more tears than not.

I did miss any serious applications to science; for instance, having spoken of plane curves, acceleration, and the like, an ideal opportunity presents itself for deriving Kepler's laws of planetary motion. I think a realization that the cheap-looking ideas of calculus can do such deep things is a moral that should be insisted upon. But this is my only real complaint.

The text contains many worked examples, and the exercises are well suited to selfstudy (not too easy and not too hard). The style is pleasantly informal but still precise. I am sure the book will be popular with students and teachers alike.

Son-in-law of Stearns Morse, DartmouthProfessor of English, Emeritus, Mr. McKean,formerly Professor of Mathematics at theMassachusetts Institute of Technology, isnow at Rockefeller University.