maybe more than one point of view is possible | The Works of Archimedes

book: The Works of Archimedes | Archimedes

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	• The Works of Archimedes Archimedes Dover Publications, 2002 - 326 pages average customer review:based on 3 reviews view larger image
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Brilliant (but mostly not so _newly_ known)

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.

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maybe more than one point of view is possible

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.)

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Complete works of ancient geometer in highly accessible translation by distinguished scholar. Topics include the famous problems of the ratio of the areas of a cylinder and an inscribed sphere; the measurement of a circle; the properties of conoids, spheroids, and spirals; and the quadrature of the parabola. Informative introduction and 52-page supplement.