On the forty-second page of “Pi: A Biography of the World's Most Mysterious Number” authors Ingmar Lehmann & Alfred S. Posamentier wrote (emphasis added):
were compared for various circular objects. This was likely the beginning of the establishment of comparison between the two measurements that seem related to each other. Was there some sort of common difference or common ratio between their lengths? Each time this comparison showed that the circumference was just a bit more than three times as long as the diameter. The question that perplexed individuals over the millennia was how much more than three times the diameter was the circumference? That would indicate that the relationship was one of a ratio. The history of Π is the quest to find the ratio between the circumference of a circle and its diameter.
The Ancient Egyptians
Frequent measurements probably showed that the part exceeding three times the diameter appeared to be about one-ninth of the diameter. We can assume this from the famous Rhind Papyrus, written by Ahmes, an Egyptian scribe, about 1650 BCE.1 He said that if we construct a square with a side whose length is eight-ninths of the diameter of the circle, then the square's area will be equal to that of the circle. At this point, you can see there was no reason to find the ratio of the circumference to the diameter. Rather, the issue was to construct a square using the classical tools (an unmarked straightedge and a pair of compasses), with the same area as that of a given circle. This became one of the three famous problems of antiquity.2 Although we know today
- This was a mathematical practical handbook, containing eighty-five problems copied by the scribe Ahmes from previous works. Alexander Henry Rhind, a Scottish Egyptologist, purchased this eighteen-foot-long (one-foot-wide) manuscript in 1858, which is now in the collection of the British Museum. This is one of our primary sources of information about the Egyptian mathematics of the times.
- The other two famous problems of antiquity are using only an unmarked straightedge and a pair of compasses to construct a cube with twice the volume of a given cube using these same tools to trisect any angle.