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http: //en. wikipedia. org/wiki/Archimedes_Palimpsest

http: //datapeak. net/mathematics. htm

Archimedes determined that the area of the parabola was 4/3 that of the inscribed triangle y = x 2 4 3 2 1 -2 -1 0 1 2

Archimedes determined that the area of the parabola was 4/3 that of the inscribed triangle y = x 2 4 3 2 1 -2 -1 0 1 2 Area of triangle = 8

Archimedes determined that the area of the parabola was 4/3 that of the inscribed triangle y = x 2 4 3 2 1 -2 -1 0 1 2 Area of triangle = 8 ∫y dx = ∫ x 2 dx + const

Archimedes determined that the area of the parabola was 4/3 that of the inscribed triangle y = x 2 Area of triangle = 8 ∫y dx = ∫ x 2 dx + const Area under the parabola 4 = 2[x 3/3]0+2 3 2 1 -2 -1 0 1 2

Archimedes determined that the area of the parabola was 4/3 that of the inscribed triangle y = x 2 Area of triangle = 8 ∫y dx = ∫ x 2 dx + const Area under the parabola 4 = 2[x 3/3]0+2 3 = 2 x 8/3 = 16/3 2 1 -2 -1 0 1 2

Archimedes determined that the area of the parabola was 4/3 that of the inscribed triangle y = x 2 Area of triangle = 8 ∫y dx = ∫ x 2 dx + const Area under the parabola 4 = 2[x 3/3]0+2 3 = 2 x 8/3 = 16/3 2 Area of parabola is thus 16 - 16/3 = 32/3 1 -2 -1 0 1 2

Archimedes determined that the area of the parabola was 4/3 that of the inscribed triangle y = x 2 Area of triangle = 8 ∫y dx = ∫ x 2 dx + const Area under the parabola 4 = 2[x 3/3]0+2 3 = 2 x 8/3 = 16/3 2 Area of parabola is thus 16 - 16/3 = 32/3 1 = 8 x 4/3 -2 -1 0 1 2

y = x 2 4 3 2 1 -2 -1 0 1 2

Enlarge Image Archimedes showed that the area of this section of a parabola is four-thirds the area o the enclosed triangle (red). He did it using a straight-lined approximation (blue).

The top layer of writing in this 700 -year-old book describes Christian prayers. But underneath, almost obliterated, are the only surviving copies of many of the works of the ancient Greek mathematician Archimedes. The most remarkable of the above works is The Method, of which the palimpsest contains the only known copy. In his other works, Archimedes often proves the equality of two areas or volumes with Eudoxus' method of exhaustion, an ancient Greek counterpart of the modern method of limits. Since the Greeks were aware that some numbers were irrational, their notion of a real number was a quantity Q approximated by two sequences, one providing an upper bound and the other a lower bound. If you find two sequences U and L, with U always bigger than Q, and L always smaller than Q, and if the two sequences eventually came closer together than any prespecified amount, then Q is found, or exhausted, by U and L. Archimedes used exhaustion to prove his theorems. This involved approximating the figure whose area he wanted to compute into sections of known area, which provide upper and lower bounds for the area of the figure. He then proved that the two bounds become equal when the subdivision becomes arbitrarily fine. These proofs, still considered to be rigorous and correct, used geometry with rare brilliance. Later writers often criticized Archimedes for not explaining how he arrived at his results in the first place. This explanation is contained in The Method. http: //en. wikipedia. org/wiki/Archimedes_Palimpsest

Sometime after Johan Heiberg examined the book in 1906, someone painted gold -leaf images over four of the pages (left). Multispectral imaging couldn't peer beneath the reflective metal paint, but x-ray fluorescence imaging revealed the underlying text (right).

Archimedes wrote his manuscript on a papyrus scroll 2, 200 years ago. At an unknown later time, someone copied the text from papyrus to animal-skin parchment. Then, 700 years ago, a monk needed parchment for a new prayer book. He pulled the copy of Archimedes' book off the shelf, cut the pages in half, rotated them 90 degrees, and scraped the surface to remove the ink, creating a palimpsest—fresh writing material made by clearing away older text. Then he wrote his prayers on the nearly-clean pages. What happened to the monk's book after that is unclear, but in 1908, Johan Ludwig Heiberg, a Danish philologist, discovered it in a library in Constantinople. He was astonished to find that the book contained previously unknown texts by Archimedes. He studied the book in detail, puzzling out the faint letters with a microscope. His efforts brought the works to the attention of scholars around the world, but after he had completed his transcription, the book again disappeared until nearly a decade ago, when it was auctioned off at Christie's. Enlarge Sometime after Johan Heiberg examined the book in 1906, someone painted gold-leaf images over four of the pages (left). Multispectral imaging couldn't peer beneath the reflective metal paint, but x-ray fluorescence imaging revealed the underlying text (right). The owner of the Archimedes Palimpsest. The book's anonymous buyer has funded an enormous research project on the volume. First, intensive conservation and restoration stabilized the condition of the book itself. Then the researchers took digital pictures of it in different wavelengths of light, creating a multi-spectral image that could be manipulated to reveal the text by Archimedes. On four of the pages, forged paintings covered the entire text, so the researchers used x-ray fluorescence imaging to peek beneath the paintings and decipher the obscured text

For seventy years, a prayer book moldered in the closet of a family in France, passed down from one generation to the next. Its mildewed parchment pages were stiff and contorted, tarnished by burn marks and waxy smudges. Behind the text of the prayers, faint Greek letters marched in lines up the page, with an occasional diagram disappearing into the spine. The owners wondered if the strange book might have some value, so they took it to Christie's Auction House of London. And in 1998, Christie's auctioned it off—for two million dollars. For this was not just a prayer book. The faint Greek inscriptions and accompanying diagrams were, in fact, the only surviving copies of several works by the great Greek mathematician Archimedes.

After imaging a page from the palimpsest, the original Archimedes text is now seen clearly

A typical page from the Archimedes Palimpsest. The text of the prayer book is seen from top to bottom, the original Archimedes manuscript is seen as fainter text below it running from left to right http: //en. wikipedia. org/wiki/Archimedes_Palimpsest

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