## Isaac Schoenberg's *Mathematical time exposures*

In 1983, ten years after he retired, Isaac Schoenberg published

*Mathematical time exposures.*Douglas Quadling reviewed the book and his excellent review is at*The Mathematical Gazette***68**(445) (1984), 233-234. We present here some extracts from that review:There will be some readers who will recall Steinhaus's delightful book

*Mathematical snapshots*, first published in 1939 and re-issued a few years ago. In conscious tribute, Schoenberg has described his 18 more-or-less independent essays as "time exposures", claiming that his "aims are roughly similar, but the pace is more leisurely". Indeed, his first two chapters develop ideas put forward by Steinhaus himself. Others deal with themes as diverse as Fibonacci numbers, convex sets, spline functions, non-differentiable curves, iterative algorithms, the Kakeya problem (the smallest area within which a rod can be turned round in the plane) and geometrical porisms; and there are four each on various aspects of finite Fourier series and König-Szücs polygons, which are described as the paths of (weightless, infinitesimal, perfectly elastic) billiard balls confined within a cube. Schoenberg characterises the contents of his book in terms of the traditional bridal appurtenances "something old, something new, something borrowed, ..." - though the reader can rest assured that there is nothing in it that could remotely be described as "blue"! ...

However, whilst there is plenty to delight the experienced mathematician, it needs to be stressed that this is not a book in the 'popularising' tradition .... Although technically the mathematics rarely goes beyond the level of a first-year undergraduate course, the reader must be prepared to follow arguments which call for stamina and close attention to detail, and to accept definitions and procedures whose justification is not immediately evident. Each essay typically includes two or three theorems whose proofs may involve several pages of ingenious reasoning. Schoenberg offers plenty of geometrical and graphical illustration, but his preferred mode of description is algebraic: indeed, in one essay he acknowledges that "Jakob Steiner, who was a great geometer averse to algebraic calculations, would have heartily disliked our proof of his theorem". ... the ideal recipient would be a young graduate mathematician just considering embarking on research: primarily for the sheer enthusiasm which the author communicates, but also as an excellent introduction to the reading of papers as distinct from textbooks, for the glimpses it offers of some fascinating mathematical by-ways, for the gobbets of potentially usable information lying around to be picked up, and because it offers a challenge to mathematical perseverance on just the right scale.