If you only ever read one book on mathematics in your life, read this. Paul Lockhart writes so clearly and passionately. (Yes, I used "passion" and "math" together!) He is a brilliant mathematician AND a great writer! This is a very rare combination. He presents math in an intuitive and conceptual way that shows you the beauty of discovery and patterns and symmetry but requires very little calculation or knowledge of the mathematical language. For non-math people this presentation is very accessible and easy to read. He is talking about concepts that are universal. For math-people to presentation style is a reinforcement of why you love math, do math, and a demonstration of how you really think when you are solving a problem. I highly, highly recommend this book!

All numbers _ all shapes and sizes _ time and space _ all are connected. Such relationships are everywhere at all times. Thus, patterns are created; some tangible; some not so much. Deciphering these connections, determining the general pattern, becomes the task of the mathematician. I discovered this concept when, some sixty plus years ago, my 6th grad teacher spent an entire morning detailing how common shapes (squares, rectangles, triangles, trapezoids, etc) were derived from one another by deriving their area formulas from a single equation. They were, in this way, shown to be all one and the same but simply stretched out into this or that shape or, geometrically, as viewed from various perspectives. This revelation came as a major eye-opener for me and changed my entire direction in life's pursuits. (Now that they 'teach to the test,' such revelations are generally and sadly absent from today's classrooms). Measurement as discussed in this wonderful book is not measurement in the classic sense of the word. In other words, one will not find a ruler with the sq rt of 2 on it. In this case, measurement represents a systematic assignment of a value to some subset of a system such that comparisons can be made between the elements of the system and between systems and (consequently) mathematical links between seemingly unrelated areas of the mathematical universe. The book is divided into two main sections. The first concerns what we used to refer to as "Synthetic" geometry. That is: to build up knowledge based on first principles. Numbers as such are conspicuously absent. Only pure reason is utilized much as it was in "ancient" times. In this manner, the author puts us directly into the mind of the mathematician; solving the riddle with nothing more the the one or two elements at hand. The ancient Greeks did this not so much because they lacked the so-called "brilliance" of the modern day Human but by choice, neglecting the more analytical approach. As such synthesis was more in keeping with their predominantly aesthetic view of nature. The second section adds the coordinate system (ala Descartes), thereby providing an Analytical approach and somewhat greater precision to the solutions (and consequently providing some deeper insights into the connections at large!). This is not a How-To book. It is an argument that is detailed and punctuated with an excellent choice of problems to challenge one's perspective. If the reader has no mathematical background, this book will pose a major challenge for sure. Additionally, if the reader lacks a fair background in both Trig. and The Calculus, this too will make forward motion along the line of comprehension rather tedious. That is not to say that one cannot follow the argument but it will be, at minimum, tough going, especially when the author reveals such links as between the Hyperbolic Integral area (dA = dw/w) and the Natural Log (ln). Calculus is, of course, simply a short cut to solving an infinite series of calculations in a few steps. It's use was initially implemented during the so-called "Kerala" Period in Southern India several centuries before Newton or Leibniz, both of whom systemized the science. It's rules are few but rather messy and often difficult (if not impossible) to apply without extensive practice and a good visual sense. Trig is stymied by its extensive use of the ever so transcendental Sin and Cos, etc. and also requires some fairly able visuals. The ability to convert from the geometric to algebraic represents a major breakthrough in extending Mathematics into further realms of exploration and therefore lends a helping hand to those less able to follow the visual signs so to speak. Again, 'measurement' represents only one area of the whole of Mathematics but this book will give anyone who wishes to tread through the mesmerizing tangle of Mathematical mystery an excellent sense of the Mathematical mind.

A wonderful book! It shows clearly the beauty of mathematics at a quite elementary level. The author goes to great lengths to show what is interesting about Mathematics. It is not the complicated formulae, or the algebra, but great, simple, utterly convincing ideas. If the reader is willing to think hard while reading, he/she will be rewarded by many stunning results presented in a completely straightforward manner. The first part, probably the one that best achieves its stated purpose, deals only with geometry. Since the basics objects of geometry (straight lines, circles, angles...) are familiar to anybody, it is really possible to prove beautiful results without using any of the apparatus, such as algebra and calculus, which many people find difficult. The second part is an attempt to introduce algebra and calculus in a very simple and well motivated way. Since I already know the material quite well, I cannot really say whether the attempt actually succeeds, but I certainly found the presentation very striking and quite engrossing.