THE GOLDEN RATIO: The Story of Phi, The World's Most Astonishing Number, by Mario Livio, (Broadway Books, New York: 2002), p.40-41
“Most of our knowledge about the familiarity of the ancient Egyptians with fractions, for example, comes from the Rhinds (or Ahmes) Papyrus. This is a huge (about 18 feet long and 12 inches high) papyrus that was copied around 1650 B.C. From earlier documents by a scribe named Ahmes. The papyrus was found at Thebes and bought in 1858 by the Scottish antiquary Henry Rhind, and it is currently in the British museum (except for a few fragments, which turned up unexpectedly in a collection of medical papers, and which are currently in the Brooklyn Museum). The Rhind Papyrus, which is in effect a calculator's handbook, has simple names only for unit fractions, such as ½, 1/3, ¼, etc., and for 2/3. A few other papyri have a name also for ¾. The ancient Egyptians generated other fractions simply by adding a few units unit fractions. For example, they had ½ + 1/3 +1/10 to represent 4/3 and 1/24 + 1/58 +1/174 + 1/232 to represent 2/29. To measure fractions of a capacity of grain called hekat, the ancient Egyptians used what were known as 'Horus-eye' fractions. According to legend, in a fight between the god Horus, son of Osiris and Isis, and the killer of his father, Horus' eye got torn away and broken into pieces. The god of writing and of calculation, Thoth, later found the pieces and wanted to restore the eye. However, he found only pieces that correspond to fractions ½, ¼, 1/8, 1/16, 1/32, and 1/64. Realizing that these fractions only add up to 63/64, Thoth produced the missing fraction of 1/64 by magic, which allowed him to complete the eye.
“Strangely enough, the Egyptian system of unit fractions continued to be used in Europe for many centuries. For those during the Renaissance who had trouble memorizing how to add or subtract fractions, some writers of mathematical textbooks provided rules written in verse. An amusing example is provided by Thomas Hylles's The Art of Vulgar Arithmetic, both in Integers and Fractions (published in 1600):
Addition of fractions and likewise subtraction
Requireth that first they all have like bases
Which by reduction is brought to perfection
And being once done as ought in like cases,
Then add or subtract their tops and no more
Subscribing the base made common before.
“In spite of, and perhaps (to some extent) because of, the secrecy surrounding Pythagoras and the Pythagorian Brotherhood, they are tentatively credited with some remarkable mathematical discoveries that may include the Golden Ratio and incommensurably. Given, however, the enormous prestige and successes of ancient Babylonian and Egyptian mathematics, and the fact that Pythagoras himself probably learned some of his mathematics in Egypt and Babylonian, we may ask: Is it possible that these civilizations or others discovered the Golden Ratio even before the Pythagoreans? This question becomes particularly intriguing when we realize the literature is bursting with claims that the Golden Ratio can be found in the dimensions of the Great Pyramid of Khufu at Giza. To answer this question, we will have to mount an exploratory expedition in archaeological mathematics.”
“Most of our knowledge about the familiarity of the ancient Egyptians with fractions, for example, comes from the Rhinds (or Ahmes) Papyrus. This is a huge (about 18 feet long and 12 inches high) papyrus that was copied around 1650 B.C. From earlier documents by a scribe named Ahmes. The papyrus was found at Thebes and bought in 1858 by the Scottish antiquary Henry Rhind, and it is currently in the British museum (except for a few fragments, which turned up unexpectedly in a collection of medical papers, and which are currently in the Brooklyn Museum). The Rhind Papyrus, which is in effect a calculator's handbook, has simple names only for unit fractions, such as ½, 1/3, ¼, etc., and for 2/3. A few other papyri have a name also for ¾. The ancient Egyptians generated other fractions simply by adding a few units unit fractions. For example, they had ½ + 1/3 +1/10 to represent 4/3 and 1/24 + 1/58 +1/174 + 1/232 to represent 2/29. To measure fractions of a capacity of grain called hekat, the ancient Egyptians used what were known as 'Horus-eye' fractions. According to legend, in a fight between the god Horus, son of Osiris and Isis, and the killer of his father, Horus' eye got torn away and broken into pieces. The god of writing and of calculation, Thoth, later found the pieces and wanted to restore the eye. However, he found only pieces that correspond to fractions ½, ¼, 1/8, 1/16, 1/32, and 1/64. Realizing that these fractions only add up to 63/64, Thoth produced the missing fraction of 1/64 by magic, which allowed him to complete the eye.
“Strangely enough, the Egyptian system of unit fractions continued to be used in Europe for many centuries. For those during the Renaissance who had trouble memorizing how to add or subtract fractions, some writers of mathematical textbooks provided rules written in verse. An amusing example is provided by Thomas Hylles's The Art of Vulgar Arithmetic, both in Integers and Fractions (published in 1600):
Addition of fractions and likewise subtraction
Requireth that first they all have like bases
Which by reduction is brought to perfection
And being once done as ought in like cases,
Then add or subtract their tops and no more
Subscribing the base made common before.
“In spite of, and perhaps (to some extent) because of, the secrecy surrounding Pythagoras and the Pythagorian Brotherhood, they are tentatively credited with some remarkable mathematical discoveries that may include the Golden Ratio and incommensurably. Given, however, the enormous prestige and successes of ancient Babylonian and Egyptian mathematics, and the fact that Pythagoras himself probably learned some of his mathematics in Egypt and Babylonian, we may ask: Is it possible that these civilizations or others discovered the Golden Ratio even before the Pythagoreans? This question becomes particularly intriguing when we realize the literature is bursting with claims that the Golden Ratio can be found in the dimensions of the Great Pyramid of Khufu at Giza. To answer this question, we will have to mount an exploratory expedition in archaeological mathematics.”
