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Mathematics in Civilization

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This book is easy and entertaining to read. It puts mathematics in context and motivates to study mathematics. The book also teaches to respect the work of all those mathematicians that have made it possible for us to reach the current level of civilization.

The first half of the book is a fairly interesting discussion of astronomy and its applications to cartography and navigation. Aristarchus calculated the distance to the moon and the sun from three easy measurements: the shadow of the earth is about two moon radii wide at the distance of the moon (observable at lunar eclipses, of course), both the moon and the sun subtend and angle of about half a degree when viewed from the earth, and we can calculate the ratio of the distance to the sun and the distance to the moon by measuring the angle between the sun and the moon at half moon. But to carry out these calculations we need to know values of trigonometric functions, so we go on to tackle this problem from first principles by deducing Aristarchus' approximation 1/60<sin(1)<1/45 from the geometrically obvious inequalities sin(x)/x<sin(y)/y and tan(x)/x>tan(y)/y where x>y applied to a 30-60-90 triangle. The section ends with an interesting exercise: "Could the 'size of the universe' have been reasonably estimated in early times if the earth had no moon?" Astronomical observations are also crucial for navigation, of course, and then we are led to map making. We study the usual Mercator projection (x=longitude, y=log(tan(latitude/2 + 45))), but unfortunately we cannot prove its key property of being conformal (instead the authors present a "slightly involved" proof that stereographic projection is conformal, which they say is a step in the right direction because then the rest can be done by complex function theory). It follows that a constant compass course will be a straight line in the Mercator projection; on the surface of the earth itself it will be a complicated "loxodromic" curve, although later we shall see that it becomes an equiangular spiral when projected onto the equatorial plane. After having seen that both astronomy and navigation calls for quite extensive calculations we look at how the invention of logarithms reduced this burden dramatically. Again there are some interesting exercises along the way, such as "Can a table of values of the function x^2/4 be used to convert division [and multiplication] to addition and/or subtraction?" (There are Babylonian clay tablets relevant in this context.) The second half of the book is a very dumb presentation of the calculus and a few applications. For instance, one supposed application is a better way of calculating logarithms through infinite series expansion. But the series expansion is derived from Taylor's theorem (which is unnecessary and historically backwards) and thus relies on appeal to the "formula" for the derivative of the logarithm function which is stated in a table without the least indication of proof. Even if the authors had told us that we can understand the logarithm as the integral of 1/x this would not have done us any good since the one insight on which all understanding of this and a million other things rest---the fundamental theorem of calculus---is simply stated without any indication of why it is true. Thus the useful series for the logarithm, which would have been very interesting to derive from first principles, is instead derived by three layers of appeal to authority.