# Quite Right

In this book, Biggs tells the story of how mathematics has evolved since humans started dealing with quantities and numbers. To some extend, his arguments are that much of the human urge to develop mathematical tools has its origin in socio-economic needs. And of course, much can be argued on these grounds. That includes a fair division of a slaughtered mammoth among the hunters to the cryptographic tools needed to protect our current bank transactions.

In economy, much is related to measuring quantities, values and exchange rates, the computation of interest, and many other financial transactions. Therefore measuring, numismatics, and finance are extensively discussed in the text. The facts from mathematical history can be found in many other books as well, but they are of course also covered in the current one. However it is worth noting that Biggs often adds his reservations to facts and urban legends that are sometimes posited in other books with certainty.

The successive chapters deal with specific aspects in the evolution and they are roughly organized chronologically. So it starts with `the unwritten story' when written language and numbers did not exist. And yet we find tallies on bones or sticks. Some of these seem to list prime numbers, but Biggs definitely claims that assuming that a notion like prime number would be known in prehistory is pure nonsense. On the other hand, money in the form of cowrie shells was indeed used and people measured lengths expressing it in terms of body parts.

Writing and counting started about 4500 years ago when people lived in settlements and the first signs of written language, economic problems and mathematics emerged. The Babylonian cuneiform numbers and the Egyptian arithmetic are well known and also length, weight, area and volume were measured. However, the myth created in the 19th century that the pyramids were built using a cosmological yard and extraterrestial knowledge have been countered later by facts and re-measuring. Gold and silver were used to pay for goods and services. This made precise weighting of precious metals and their alloys was very important.

True coins were issued by kings and emperors and were for example used to collect taxes. The Greek denoted numbers by letters and solved arithmetical problems using pebbles placed in geometric arrangements to obtain results about integers. Geometry was based on using only compass and straightedge and most proofs were geometric constructions. This we know from Eulid's *Elements*, the most reprinted book ever, and which was the basis for mathematical education till rather recently. But again Biggs puts Euclid in perspective. It is not even 100% sure, although very likely, that Euclid was a real person, but whether he wrote his *Elements*as we know it is hard to believe. What we know are transcripts of translations of transcripts etc that were produced many centuries later.

Through the Arabs the Hindu positional number system reached Western civilization and they also introduced algebra and algorithms, although the originals are only remotely resembling their modern counterparts. Money became widespread, but it came in all types and values. For trading and tax collecting, it became more and more important to compute exchange rates, and these computations were done by professionals.

By the end of the Middle Ages, computing interest rates and solving equations were the target mathematical problems and we find here the well known story of Cardano an the Tartaglia-Fior duelling competition in their race to solve the cubic equation. The formulas were important tools in a computational profession and were often kept secret. We learn about the Roman abacus (which was originally not the instrument with beads shifting on wires as we usually think of it, but it consisted pebbles arranged in certain arrangements). It was the *Liber Abbaci* by Fibonacci that promoted the Hindu system over the Roman numerals that finally prevailed and that we are using today. Also combinatorial problems were investigated and also these have Hindu origins.

Mathematics now takes major steps forward with the introduction of logarithms, infinite sums, symbolic notation became usual, Descartes combined geometry and algebra, number theory matured, and calculus was initiated by Fermat and developed by Newton. Because his publication was delayed and because Leibniz had an easier notation the latter was more popular leading to a big controversy between Newton and Leibniz camps (not elaborated in this book). Probability was born from gambling problems and the analogy with financial and insurance problems. The discovery of the bell-shaped normal distribution and the law of large numbers was quite an achievement only possible because of the collection of large numbers of statistical data. In this context, Biggs also elaborates on some game theory and the Pareto optimum. Also fair voting systems or the impossibility of such a system is discussed. He concludes with more modern economic and financial models (Black-Scholes), even Shannon's information theory, and public key cryptography. The latter of course became very important now that money became information stored in bits in a digital computer.

The author of course has a point that economical problems formed an incentive to develop certain mathematical tools. Hence the subtitle *The story of mathematics, measurement and money*, and indeed the reader will probably find less mathematical details and anecdotes than in many other (popular) books on the history of mathematics. On the other hand, we find much more details about the numismatic and financial history than elsewhere. The non-monetary measures are also discussed, but not as detailed as the economic and financial aspects. The text is quite accessible without manipulating a convulsively popularizing style. There is little humor, but a pun usually comes unexpectedly so that it is really funny when it takes you by surprise. The useful illustrations and schemes are in gray-scale and to the point. They are not overdone but are added where they matter.

**Submitted by Adhemar Bultheel |

**13 / Mar / 2016