## A History of Mathematics

by Carl B. Boyer and Uta C. Merzbach

This is a classic work, and a very thorough one at that. It starts with the earliest inklings of mathematical thought in the river civilizations of Babylonia, Egypt, and China. These efforts were primarily directed at solving problems in geometry arising from land ownership issues. The Egyptians developed some impressive techniques to cope with the destruction of land markers by the annual floods of the Nile; they were able to reconstitute land boundaries each year with pretty good accuracy. Taxation issues also drove much early geometry, because usually taxes were based on the amount of land owned by a farmer, which could only be determined by calculation. So long as the parcel of land was rectangular, things were easy, but in practice, there were just too many odd-shaped parcels to permit such simple approaches. Especially difficult were cases requiring a partition of a parcel into several smaller parcels with specified amounts of land.

These geometric problems also spawned an interest in equations involving squares, which in turn required systems for figuring square roots. However, in no case did these people develop what we would call algebra. They instead developed recipes for solving various problems. The standard textbooks were really just listings of practical problems along with their solutions; the student learned to solve any problem by finding its closest match in the textbook and substituting his numbers for those in the textbook. No general theory was taught.

They also got involved in calculating circumferences and areas of circles, which led them into the problem of determining the value of pi. For this, they used various semi-empirical methods, breaking a circle up into lots of tiny line segments and calculating the lengths of the segments, then adding them all up. The Egyptians and Babylonians ended up with values good to three or four decimal places, but the Chinese seemed content to use 3 or, in some cases, 22/7.

My main reason for reading the book was to learn about the difference between Greek mathematics and that of earlier cultures, and the difference really is striking. For earlier cultures, mathematics was nothing more than a set of procedures for solving practical problems. For the Greeks, geometry was a philosophical pursuit. Their approach was entirely abstract and theoretical. This confirms my speculations regarding the radical new style of thinking that the Greeks introduced. The big surprise for me was how early this developed. Pythagoras promulgated his theorem around 500 BCE. We suspect that the Egyptians, Babylonians, and Indians were familiar with the Pythagorean theorem before Pythagoras came along. Indeed, it seems likely that Pythagoras picked up the idea while traveling in the east. But Pythagoras did something entirely new: he proved the theorem. Before he came along, the theorem was nothing more than a rule that seemed to work well. But he added the concept of proof and developed a proof – all around 500 BCE. This is much earlier than Aristotle, whom I had thought to be the inventor of the concept of rigorous logic. This book showed that rigorous logic was in Greek used 200 years before Aristotle.

I read all the material on the history of mathematics up to the period of the Renaissance, and stopped there; I had gotten everything I needed.

I do not recommend this book for any but a mathematician. It is loaded with arcane mathematical terminology such as
equant, evolute, deferent, and mensuration. I’ve got a goodly amount of math under my belt, but I was befuddled by his nomenclature. Moreover, in attempting to represent ancient mathematical thought, the author mixes the terminology of the time with modern terminology. It’s quite confusing.