Most helpful positive review
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Brilliant (but mostly not so _newly_ known)
on February 10, 2007
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
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.