Bryant, John and Chris Sangwin. How Round is Your Circle? Where Engineering and Mathematics Meet. Princeton, N.J.:  Princeton University Press, 2008. Pp. xix +306. Hardcover, $29.95.

A simpler problem than the roundness of a circle is how do you know if an angle is really a right angle?  An experiment to check is to make a triangle of wood or cardboard including the angle in question.  Set one side of the triangle along a straight line, mark the position of the ostensibly right angle, mark the position of the other corner of the triangle, then flip the triangle over, set the same side against the same straight line with the ostensibly right angle in the same position, and see if the other corner is at the same point.  Much of the book is about making models to “prove” mathematical ideas.  Most of the mathematics illustrated is geometry and trigonometry, with some attention to infinite series.  The latter comes up in considering a stack of dominoes, where each domino partly overlaps the domino beneath it, the stack appearing to lean in one direction.  The question is how far in the horizontal direction can the stack extend?  The answer is as far as one likes but the height of the stack increases rapidly as more horizontal distance is spanned—the sum of the terms of the series does not converge.  Other intriguing questions, amenable to a mathematical analysis, are discussed.  How is a drill that makes a square hole designed?  How does a planimeter, a device that measures the area of a shape by tracing its boundary, work?  How is a linkage designed so one vertex moves in a circle and another moves in a straight line?  The linkage is useful in various pumps and oil well mechanism.  If the number of teeth in meshing gears are relatively prime, then wear caused by one misshapen tooth will be uniformly distributed over the teeth of the other gear.  The authors point out that many practical engineering problems could be real challenges to mathematicians.  That point may have been made before, but the context presented was unfamiliar to me and intriguing.  The examples are fun, and so is the authors’ perspective.