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Episodes from the Early History of Mathematics (Anneli Lax New Mathematical Library)

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While mathematics has a long history, in many ways it was not until the publication of Euclid's Elements that it became an abstract science. Babylonian mathematics, the topic of the first chapter, largely dealt with counting and the focus in this book is on the notations the Babylonians used to represent numbers, both integers and fractions. Although their notation had its' limits, we still use it today for time and angle measure.
And then there was Euclid, and all was ordered. There is no reason to believe one way or another that Euclid was the first to prove the theorems in his classic work, but there is no doubt as to his organizational genius. His "rigorous" setting down of the principles of geometric thought was truly a turning point in abstract mathematics, If you are not impressed when reading the material of the second chapter, taken from Euclid, then you have no aesthetic appreciation for what mathematics is. While the mathematics has been cleaned, the beauty has never been topped.
The next chapter is about the greatest genius before Newton, Archimedes. In fact, had he been blessed with better notation, it is possible that he would have invented, or at least pre-invented calculus. If even half of the legends about his mechanical skill are true, they are still amazing. Apparently, entire armies and navies were terrified at the rumor that one of his mechanical devices was about to be used. The crispness of his theorems and the logical progression will be just as instructive thousands of years from now.
The final chapter describes how Ptolemy was able to construct trigonometric tables. Using the chords of circles, he was able to construct tables that can still be used today. Civilization improves and mathematicians continue to expand the mathematical field and refine earlier work. However, the elegance of earlier work still shines through, and in this book you can experience some of the earliest mathematical diamonds, hewn from thought and destined to survive as long as humans do.

It is very rare for any intellectual discipline today to be built on the foundation that is thousands of years old. The discipline for which this observation holds most unequivocally is mathematics: the discoveries and tools that have been created well over two thousand years ago are still as valid and relevant today as they were when they first appeared.

This book begins with the Babylonian mathematics and explores their use of the number system which had number sixty as its base. The author uses images of the original cuneiform clay tablet and through a series of intuitive steps shows how we can deduce what their number system looked like and how arithmetic operations were carried out. It is interesting to see how to do arithmetic in the base sixty in its own right, since it is not a number system that is used often. Nonetheless, the Babylonian number system is the source of our own way of dividing time and measuring angles in terms of minutes and seconds, and the book makes a persuasive case that this is actually a very compact way of writing down very small numbers and working with them efficiently. Unfortunately, after some interesting early developments Babylonian mathematics did not progress too far and remained on a relatively rudimentary level.

The bulk of the book deals with Greek mathematics. This is really where the story of mathematics as we understand it today begins, and Greeks already showed a remarkable level of mathematical sophistication. The author presents a few of the most important discoveries of Greek mathematics, primarily in geometry, although Greeks did make many other important contributions. Several important theorems are worked out following the original presentation as much as possible. Nonetheless many concessions were necessary in order to make the text legible for the modern reader.

One of the beast features of this book is that it's not just a description of ancient mathematics - there are numerous exercises throughout the text that aim to engage the reader and draw him or her in into the actual mathematical practice. It is quite remarkable in a way to be having the same thought processes that Euclid or Pythagoras might have been having all those centuries ago. In this limited sense we are able to achieve a sort of union of minds that is hard to imagine in any other sphere of human endeavor.