Product Description

The complete works of antiquity's great geometer appear here in a highly accessible English translation by a distinguished scholar. Remarkable for his range of thought and his mastery of treatment, Archimedes addressed such topics as 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. This edition offers an informative introduction with many valuable insights into the ancient mathematician's life and thought as well as the views of his contemporaries. Modern mathematicians, physicists, science historians, and logicians will find this volume a source of timeless fascination. Unabridged reprint of the classic 1897 edition, with supplement of 1912.

Product Details

• Amazon Sales Rank: #23141 in Books
• Published on: 2002-04-16
• Original language: English
• Number of items: 1
• Binding: Paperback
• 326 pages

Features

• ISBN13: 9780486420844
• Condition: NEW
• Notes: Brand New from Publisher. No Remainder Mark.
• Click here to view our Condition Guide and Shipping Prices

Customer Reviews

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.

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.)

A responese to the review by "Muhammad The Egyptian"
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 reality is that Egyptians despite all their accomplishments where culturally frozen in time, as demonstrated by their art. Greek art, on the other hand, tells a very different story. The Greeks had a vibrant and flourishing culture that shows rapid advances in art, philosophy, rhetoric and mathematics.

Lastly, the inflammatory statement about white men having recently emerged from caves on all fours with no knowledge of fire is absolutely ridiculous. Again the author cannot possibly even believe himself. As a student of nutritional anthropology I can produce volumes of research by Archaeologists and Anthropologists that directly disprove those statements (check Peter S. Ungar).

Muhammad "The Egyptian" (On the Mothership) has absolutely no credibility in my eyes. I think he made some interesting points and pointed to some interesting texts. However, between his shown lack of understanding and repeated false assertions we can only assume that everything he said is untrue. At least until we have seen proof with our own eyes that what he has said is true (guess I better get cracking at his "booklist").

Now as for Archimedes himself, he was an absolute genius and I don't understand how anyone can argue otherwise. Archimedes most important known work "The Method" was last transcribed in the 10th century, but tragically the work was "scrubbed clean" and boldly written over in the 13th century by a monk creating a "prayer book." In 1906 Heiberg discovered this work in Istanbul. He photographed the book and using a magnifying glass he translated as much of the faintly visible text as he could. That is what is contained in this book being reviewed. That is not the end of the story though. "The Method" disappeared during World War I and was presumed lost forever until it resurfaced in 1998 when it was auctioned off in Paris for two million dollars. Soon William Noel, the curator at Walters Art Museum in Baltimore heard of this great discovery and requested the volume in order to restore and fully transcribe and translate the work. This project has not yet been completed.

What has been translated from these writings has revealed that Archimedes had discovered and was employing calculus, approximately two millennium before its discovery. I feel pretty confident that Archimedes broke that ground. I also seriously doubt that Archimedes "stole" any of his discoveries from other scholars. We all build on the hard work and discoveries of our predecessors, constantly striving to advance the frontiers of our fields. "There's nothing new under the sun" is an apt quote. The fact is no one "owns" these ideas. Mathematics is a human endeavor that belongs to all of us. Archimedes was the first one to write it down, therefore he goes down in history as the first one to bring these truths to the public. Bravo Archimedes!