This book gives the ancient theories and proofs of math. It is thrilling to read, for example, proofs by Descartes and Archimedes showing the area of a circle and how to use geometry to depict the product of two numbers or their square root long before the modern tools of calculus were developed. In so doing you can learn how the area of a circle is πr^2. (Archimedes said that a circle with radius r and circumference 2πr has the same area as a right triangle with height r and baseline 2πr. The area of this triangle is 1/2 * base * height = 1/2 * 2πr * r = πr^2). What I just clearly explained here is not so clearly explained by Stephen Hawking.

Hawking famously explained to the laymen in his book on the universe the theories of Einstein. But his editors in this book fell short perhaps unable to comprehend what they read. This book would be of better use to the laymen if Hawking proceeded as does David Foster Wallace in his own book on math to walk the reader through these proofs one small step at a time. Instead Hawkins tosses out these ideas in tersely worded passages that do not assist those who are less clever in understanding what it means.

This book could take one a lifetime (or even longer) to distill. Its ideas so important, clever, and yes God-like in its beauty that each time I am able to understand one proof completely I plan to post my own explanation on the Internet for other students to read. BTW this book is in it's nth edition. Each subsequent edition 1,2,3,..., n-1 must have been a revision made to correct logical and technical errors in the mathematics. Might I suggest that version n+1 include some improvement in the prose as well. Put some graduate students on the task and make wholesale revisions.