7 of 7 people found the following review helpful
By
Alan U. Kennington
- Published on Amazon.com
Format: Paperback
Volume 2 is even more onerous to wade through than Volume 1. I would not recommend reading every word of either of the two volumes. But Volume 2 is right off the scale in mathematical difficulty, both in quality and quantity. Quite often, it is more like a database to be mined by academic historians than a book to be read cover to cover. However, it is of enormous importance, demonstrating even more than volume 1 the depth and dexterity of ancient Greek mathematical thought.
The world must be grateful that a scholar such as Heath has made such Herculean efforts to locate and present the maximum of information about the history of Greek mathematics. Most maths history books gloss over the mathematical nitty-gritty, trying to keep everything as non-technical as possible so as not to scare off the reader. Heath writes as if his subject is important in itself, which it is.
The range of ancient Greek mathematics may seem narrow by modern standards. It is mostly the geometry of lines, circles, ellipses, hyperbolas, parabolas, spirals, cones, cylinders, spheres, and a handful of more adventurous curves, surfaces and solids. The non-integer numerical side of ancient Greek mathematics was mostly expressed in terms of geometry, although in the later, dying centuries of Greek mathematics, Diophantus did develop a purely arithmetical framework free of geometry. Heath makes it quite clear that the later number theory in Greek mathematics was often tedious, repetitive and somewhat shallow compared to what we now call number theory. Heath's exhausting "conspectus" of the "Arithmetica" by Diophantus, on pages 484 to 514, adequately demonstrates this.
Heath assumes that the reader has a good understanding of Euclidean geometry, such as is not available in schools and universities these days. It is nowadays almost a lost craft. It is lucky that Heath was writing at a time when old-style ruler-and-compass Euclidean geometry was still taken seriously as an academic subject. A modern historian would skim over this historical material because of its difficulty and unfamiliarity. Heath also seems to assume that the reader is fluent in Greek, Latin, French and German, because he provides no translations for most text in these languages. Once again, this was usual in his time but is difficult for modern readers. (I do read Latin, French and German, but I had to buy a better dictionary for the Greek.)
My overall impression of ancient Greek mathematicians from this book, especially from Volume 2, is that they were enormously more intelligent than I had ever suspected. Modern mathematicians would have great difficulty reproducing or even comprehending much of ancient Greek geometry. The ancient ingenuity and sophistication were truly impressive. Heath demonstrates this particularly in the case of the three-dimensional geometry of Apollonius and others around his time.
Apart from famous stars such as Archimedes, Apollonius, Ptolemy, Pappus and Diophantus, Heath also gives extensive coverage in Volume 2 of Aristarchus, Nicomedes, Geminus, Nicomachus, Theodosius, Hipparchus, Menelaus and Heron, who also made impressive, and often surprisingly modern, contributions to the progress of mathematics. While reading in this book how much of post-Renaissance mathematics was merely a repeat or continuation of ancient Greek mathematics, I wondered how advanced the world would be now if the Dark Ages had not interrupted the progress of mathematics for more than a thousand years. Heath's descriptions of ancient astronomy and optics in particular made me think that humanity could have reached the Moon 1000 years earlier if Greek mathematics had not been interrupted for so long.
2 of 2 people found the following review helpful
By
Viktor Blasjo
- Published on Amazon.com
Format: Paperback
In continuation of my review of Volume 1, I shall in this review extract the contents of this volume pertaining to the theme of constructions discussed there.
A main theme in this later era of Greek mathematics is the increasing importance of an essentially algebraic point of view. The theory of conic sections is a case in point.
Menaechmus introduced conic sections for the purpose of using them to double the cube (110), but within only a generation or so systematic treatises were written on conics (116), followed later by the more abstract treatise by Apollonius that has come down to us (126). This was clearly a matter of theory for theory's sake, independently of their function in solving construction problems or any other use.
Altogether the volume of writings the Greeks produced on conics outweigh by a mile the scraps they wrote on any other curves except lines and circles, even though, as we saw in Volume 1, they knew many other curves that were about as useful as conics as far as applications were concerned. The only explanation for this is that conics are curves of degree two: they are algebraically the simplest next step beyond line and circle. In an algebraically oriented mode of mathematics conics will be ubiquitous and eminently treatable.
That this is the reason why they are so prominent in Greek mathematics is seen also by the uses the Greeks made of them. For instance, Archimedes derived the rotational volumes of conics (56), the centers of gravity (78) and areas (85) of parabolic segments, and the hydrostatics of paraboloids (94). There is no need or motivation for any of these results. Rather Archimedes is simply doing it because he can, and he can because they are second-degree curves which makes them susceptible to quasi-algebraic treatment and thus feasible to work with.
Another important and thoroughly algebraic use of conics is for solving cubic equations. Archimedes came upon a problem equivalent to a cubic equation in the context of studying the volumes of sections of spheres (43). He solved it by the intersection of two conic sections (45), as did Dionysodorus (46) and Diocles (47).
Now, as we recall from Volume 1, construction by conics was evidently not held is very high regard, since Menaechmus's use of them to double the cube was followed by a barrage of later methods that used other means. This suggests clearly that when they are used for Archimedes's cubic problem this is very much algebra through and through.
It is in this context, I think, that we must understand Pappus's famous division of problems:
"Those problems which can be solved by means of a straight line and a circumference of a circle may properly be called plane; for the lines by means of which such problems are solved have their origin in a plane. Those, however, which are solved by using for their discovery one or more of the sections of the cone have been called solid; for their construction requires the use of surfaces of solid figures, namely cones. There remains a third kind of problem, that which is called linear; for other lines (curves) besides those mentioned are assumed for the construction, the origin of which is more complicated and less natural, as they are generated from more irregular surfaces and intricate movements." (117)
This division seems the be essentially algebraic in character, since it gives a central place to conics while ignoring the various ruler-tool-based constructions that were evidently preferred for the duplication of the cube. The latter are not constructions by intersections of curves, so they do not fit Pappus's classification scheme. It seems thus that Pappus has allowed increasing algebraic awareness to trump more intuitive means of judging solutions. Why else would he insist only on constructions by intersections of curves, as so many earlier mathematicians doubling the cube evidently did not?
This interpretation also fits with the fact than Pappus's algebraically oriented notion of purity of method clashes with actual usage even by such luminaries as Archimedes and Apollonius:
"It seems to be a grave error into which geometers fall whenever any one discovers the solution of a plane problem by means of conics or linear (higher) curves, or generally solves it by means of a foreign kind, as in the case e.g. with the problem in the fifth Book of the Conics of Apollonius relating to the parabola, and when Archimedes assumes in his work on the spiral a neusis of a 'solid' character with reference to a circle; for it is possible without calling in the aid of anything solid to find the proof of the theorem given by Archimedes." (68)
Despite this last point, Pappus actually does precisely what he says is not needed: he proves the construction using conics instead of the neusis (386). It seems clear that the point of this is to prove that Archimedes's solution is in fact "solid," i.e., to map Archimedes's use of neusis into Pappus's algebraic hierarchy of methods. Rather than postulating that Archimedes was a fool, it seems more reasonable to conclude that Archimedes was working with a pre-algebraic, more intuitive way of judging construction methods than that used by Pappus.
Pappus's remark that Archimedes used "a neusis of a 'solid' character" hints at the fact that some uses of neusis can be reduced to ruler and compass, others not. This can perhaps be seen as another indication of the mismatch between intuitive and algebraic classifications of constructions. Apollonius in fact wrote a lost treatise discussing at some length which instances of neusis are reducible to ruler and compass constructions (189), which was possibly a step toward the more algebraic conception, although it also makes sense within an intuitive framework.
In the interest of full disclosure I must note a few items that do not fit my algebra-oriented interpretation. Perseus studied sections of a torus by analogy with sections of a cone (203), and Dionysodorus found the volume of a torus by analogy with Archimedes's determination of volumes of solids of revolution of conics (219). There were no applications of this. One must admit that here it is the geometrical generation of the conics rather than their algebraic properties that are at the forefront. In a somewhat similar vein, Pappus finds the quadratix by projections of 3D curves: once using a cylindrical helix and once using a cylinder with spiral base (380). This peculiar way of legitimising the quadratrix is in odd contrast with the algebraic modes of thought that were becoming so central, and together with the work of Perseus it seems instead to suggest an odd notion that the conics were somehow legitimated because they were part of a three-dimensional surface.