The Works of Archimedes

I must first in the spirit of full disclosure tell you that I have not as yet read this book, nor any of the works of Archimedes. That being said, I am a student of the classics and have some basic understanding of the historical and scientific discoveries relevant to these works. Truly my main impetus is simply to respond to the other reviewer, who has posted this same review on at least a half a dozen other works by Archimedes. I have two main points and some comments on the significance of Archimedes work. The first point is that although the reviewer is correct in that original discoverers don't always receive credit for their discoveries, the actuality of the history is drastically more complex. My second point is simply that the previous reviewer has made a gross use of hyperbole as well as minced facts and history. It is true that many whites have historically received credit for discoveries first made in non-white societies. Our modern understanding of the motions of the heart, a discovery attributed to William Harvey in the 17th century, was actually first discovered by Ibn Al-Nafis of Damascus in the 13th century. Ibn Al-Nafis also described the anatomical anomalies of Galenic medical writings, about 4 centuries before Vesalius. It is interesting to note that Ibn Al-Nafis's works were first brought to Europe and translated to latin only years before Vesalius published his De Humani Corporis Fabrica, and decades before Harvey's treatise on the circulation of the blood. However, to say that these Western scientists "stole" these discoveries from Nafis is inaccurate. Both Vesalius and Harvey will go down in history because they 1) wrote treatises on their works which were so thorough and precise as to leave no scientific room for refutation 2) and they fought for their discoveries at a time in history when it was possible to overthrow a 1400 year old doctrine. I don't think the previous reviewer realizes the debt that they owe to these "white" scientists. If not for their life works it is possible that Ibn Al-Nafis's discoveries would have never reached widespread acceptance. We would still be bloodletting and purging in order to balance the four "humors." Trust me, not a world you want to get sick in =) The fact is that originators don't always receive the credit they deserve. No matter the art or the time, the pattern is that highly creative individuals are the pioneers of their fields. However, their knowledge remains largely esoteric. It takes another set of individuals to bring that esoteric knowledge into the common place. I would show this model through more examples but don't want to bore the reader. If you doubt my assertion, meditate on it for a bit. I believe you will find endless examples. Now, as for the reviewers assertions that the "the master builders" of Egypt had already discovered all the sciences and arts. That's flagrant hyperbole and we all know it. Just take a look at Egyptian statuary vs Greek statuary. The rea

I enjoyed the previous review, but do not wholly agree. It seemed to me the method of centers of gravity was the one by which Archimedes discovered, rather than proved, his results. His proofs do seem to me to involve limiting arguments which are at least reminiscent of riemann sums. Indeed even the method of centers of gravity involved slicing up solids in a way that to me suggests again riemann sums. Perhaps i have not read as carefully as the previous reviewer. I agree however that the works are startlingly wonderful and inspiring. The key to Archimedes' geometry solutions was the principle of parallel slices, that two figures all of whose slices parallel to a given reference line or plane have equal areas, or lengths, themselves have equal volume, or area. This is of course the fundamental theorem of calculus for equating areas, and the cavalieri principle, for equating volumes. Note it does not suffice to calculate them, merely to equate two such areas. thus Archimedes had to bootstrap up from one known area or volume to another. Thus starting from an actual decomposition of a cube into three pyramids, one sees that a right pyramid has volume 1/3 of cube. Then by parallel slices one sees the same for any pyramid or cone. then by taking complements one computes the volume of a sphere, by showing that horizontal slices of a cone and a sphere add up to the slice of a cylinder. Knowing cylinder and cone volume thus gives a sphere's volume. Finally one of the hard problems we give students is finding the volume of a bicylinder, the intersection of two transverse cylinders. After seeing Archimedes' solution of the volume of a sphere, by the principle of parallel slices, equating the volume of a sphere, slice by slice, with that of the complement of a (double) cone in a cylinder, one easily intuits his (still lost) solution of the volume of a bicylinder, as that of the complement of a square based (double) pyramid in a block! (of course reading further one sees it was rediscovered by Zeuthen 100 years ago, but so what, it is fun to do it oneself.)

Again I feel I must post a review to counter misleading information in an earlier review. The author of the previous review seems to think these works were _not_ available to scholars during the Renaisance. In fact, the famous "Archimedes Palimpsest" that resurfaced in the 1990s is only a small part of the works of Archimedes found in this book. Moreover, this book is a reprint of the translation published in 1897 by Thomas L. Heath. Heath _did_ have access to the Palimpsest (or maybe to a translation into German or to a copy--of this I am unsure) and did include a translation in this book in 1897. It is true that after the Palimpsest resurfaced in the 1990s and began to be examined by modern methods, some lacunae were filled in. But that's not even most of the Palimpsest, let alone most of the contents of this book. Moreover, the newly discovered material is not in this book (but Heath's translation of the Palimpsest is). The previous reviewer is _extremely_ confused about the history. Now to the contents of the book. The famous Palimpsest actually is my favorite part. Prepare to be dazzled. Many 20th-century calculus texts, saying that integral calculus was anticipated by Archimedes in the 3rd century BC, are so phrased that they may give their readers the impression that Archimedes worked with something similar to Riemann sums, or similar nonsense. The truth is far more interesting. Archimedes used infinitesimals explicitly. His proofs were amazingly efficient. If you think that a brilliant proof by an ancient mathematician having only relatively primitive methods at his disposal must be longer and more complicated than a proof by modern methods, think again. Modern methods are indeed more efficient, but not because one writes _shorter_ proofs; rather it is because (at least in the present case) one writes _fewer_ proofs. Archimedes introduced the concept of center of gravity. In the Palimpsest, he finds not only areas and volumes, but centers of gravity (that of a solid hemisphere of uniform volume is 5/8 of the way from the "north pole" to the center of the sphere, Archimdes shows in one of his startlingly efficient proofs--just one example). It was not only by the use of infinitesimals that Archimedes solved problems that would now be treated by integral calculus. For example, one of the methods (just one of them) by which Archimedes found the area between a parabola and one of its secant lines involved dividing that area into an infinite sequence of triangles, the sum of the areas of which is a geometric series. Many other examples are in these pages.