Origins: 3.14159265…

Why did the ancients invent increasingly subtle and ingenious methods to arrive at an exact value of pi? Human curiosity.

pi

π, or pi, has a value of 3.14159265…

How do you find the holy grail of mathematics?

You start with a circle, which is the easiest geometric shape to draw (just fix one end of a string in place and swing the other end around it, inscribing a circle). Then measure the circle’s perimeter (also known as the circumference) and the distance across its widest point (the diameter). Divide the circumference by the diameter—and you have that well-known but eternally daunting number, π, or pi, which has a value of 3.14159265…

That is part of the mystique of pi: Whatever the size of the circle, the value remains the same (what mathematicians call a “constant”). Unfortunately, pi is also “irrational,” meaning that it is impossible to calculate its value completely; the decimals go on forever without regular repetition.

Calculating the value of pi has been a puzzle for millennia. One of the earliest implied values is given in a Biblical passage describing the construction of a huge basin for Solomon’s Temple: “Then [Hiram of Tyre] made the molten sea; it was round, ten cubits from brim to brim, and five cubits high. A line of thirty cubits would encircle it completely” (1 Kings 7:23). In other words, pi = 30÷10 or 3.

The Temple craftsmen obviously obtained these numbers through direct measurement—perhaps using a rope—and they came up with a simple approximation of pi. More than a thousand years earlier, the Sumerians had developed a mathematical method for measuring the dimensions of circles, that of inscribed equilateral polygons (a geometric shape with three or more straight sides). The ancient Sumerians realized that the perimeter of a polygon inscribed in a circle would always be slightly smaller than the circle’s circumference. This allowed them to make a fairly accurate measurement of a curved line, which is almost impossible to do with ordinary measuring devices.
 


 
The free eBook Life in the Ancient World guides you through craft centers in ancient Jerusalem, family structure across Israel and articles on ancient practices—from dining to makeup—across the Mediterranean world.
 


 
According to a 4,000-year-old cuneiform tablet discovered in 1936, the Sumerians found the ratio of the perimeter of an inscribed hexagon to that of the circle to be 3456/3600, which factors out to 216/225. The Sumerians could thus measure any circle (by measuring an inscribed polygon and making the adjustment). Then they could measure the circle’s diameter—a simple straight line—and divide it into the circumference, producing an approximation of pi. In this way, the Sumerians found pi to be 3 23/216 (3.1065), a much better calculation of pi than the Biblical value. Why wasn’t this known to the Israelites at the time of Solomon? We’ll never know.

In an ancient Egyptian mathematical treatise known as the Rhind Papyrus (c. 1650 B.C.E.), a scribe named Ahmes states that a certain circular field 9 units across (that is, with a diameter of 9) had an area of 64 units. Today, we know the relations between the diameter, circumference and area of a circle: Area equals pi multiplied by the square of the radius (half the diameter), or a = πr2. Changing this equation around, we find that pi equals the area divided by the square of the radius. The field’s radius is 4.5 (half of nine); the square of 4.5 is 20.25; and 64 divided by 20.25 equals 3.16. Therefore, π = 3.16. Thus some modern commentators have given Ahmes credit for a close approximation of pi. But was our ancient Egyptian scribe aware of this formula? Almost certainly not. He didn’t know he was approximating pi, and I should not like to give him credit for it.

origins

Our next significant player is the Greek philosopher Antiphon. In the late fifth century B.C.E., he realized that if successive polygons were inscribed within a circle, doubling the number of sides each time, the difference between the polygon’s perimeter and the circle’s circumference would diminish toward zero (think of a circle as a polygon with an infinite number of sides). While Antiphon didn’t calculate pi using his method (as far as we know), his idea would be the basis of all improvements in the value of pi until the 17th century C.E.

Two centuries later, Archimedes (c. 287–212 B.C.E.) inscribed a hexagon in a circle; then he doubled the sides until he had a 96-sided polygon inscribed in the circle. At the same time, he superscribed a similar series of polygons outside the circle. By this method, he found that pi was greater than 3.14084 and less than 3.14286—an extremely close approximation of the actual value (3.14159265). Archimedes was the first mathematician to bound pi in this way, by calculating its upper and lower limits. Thus he should be credited with making the search for the value of pi a science.
 


 
Learn about the ancient origins of other inventions—from the calendar to medicinal pills—in The Origins of Things (Or How the Hour Got Its Minutes).
 


 
For almost 2,000 years, no one improved on Archimedes’s method of inscribed and superscribed polygons, though refinements were made in the calculation. The second-century C.E. Alexandrian astronomer Ptolemy, for instance, used Archimedes’s method to reach a value of 3.14167. And the method was invented independently by Indian and Chinese mathematicians. In the fifth century C.E., the Chinese mathematician Tsu Chung-Chih and his son Tsu Keng-Chih, using the polygon method, found that pi falls between 3.1415926 and 3.1415927, which is precise enough for most purposes even today.

The calculation of accurate trigonometric tables in the 16th century made the Archimedian approach much easier to pursue than before. The French lawyer and amateur mathematician François Viète (1540–1603) used trigonometry to calculate the perimeter of a polygon with 393,216 sides, pinpointing p somewhere between 3.1415926535 and 3.1415926537.

But it was Isaac Newton’s development of calculus that reduced the calculation of pi to plain old arithmetic. In 1655, John Wallis published his proof of the infinite product π÷2 = 2 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7… And James Gregory, in 1671, found the infinite sum of π÷4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11… These formulas take hundreds of steps to arrive at even the first few digits of pi, but they demonstrated the feasibility of the new method. Within a few years, Newton found a series of formulas that quickly gave him a 16-digit expansion of pi. From then on, further computation of pi was only a matter of desire and endurance.

When it comes to endurance, nothing can beat a computer. In 1949, the primitive ENIAC computer, the first of the “giant brains,” was fed an algorithm for calculating pi. Three days later, it arrived at an answer 2,037 digits long. Today programs are available that allow you to calculate a billion digits of pi on your Pentium computer over the weekend.

What’s the point of computing pi out that far? There is none. If we knew the diameter of the universe, the first 30 digits of pi would theoretically enable us to calculate its circumference to within a millimeter. That’s closer than we would ever need to come; the rest is just showing off.

——————

“Origins: 3.14159265…” by Kim Jonas originally appeared in the March/April 2000 issue of Archaeology Odyssey. The article was first republished in Bible History Daily on March 14, 2014.
 


 
Kim Jonas, a former college math professor, is currently a statistician for the U.S. Census Bureau.
 


 

Related reading in Bible History Daily:

Babylonian Trigonometry Table: The World’s Oldest?

Mesopotamian “Receipts” Illuminated by 3D Technology

Computer Program Learning to Read Paleo-Hebrew Letters
 


 
The free eBook Life in the Ancient World guides you through craft centers in ancient Jerusalem, family structure across Israel and articles on ancient practices—from dining to makeup—across the Mediterranean world.
 


 

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  • Michael says

    Kim,

    It is a welcome pleasure to come across someone who is into ancient mathematics especially where it delves into the biblical value of pi. You referred to the value of pi in the Book of Kings. However, there was another example of pi in the Old Testament and it was presented as a deeper challenge in the mythological story of Noah’s Ark, which has it roots in the Babylonian Epic of Gilgamesh. The details in the Book of Genesis laid out the precise dimensions of the Ark as 300 cubits long by 50 cubits wide and 30 cubits high. But then there came the mystery where there was only one measurement given for the window of the Ark and that was with one cubit.
    Coming from an engineering background I took the one measurement of one cubit to be a diameter and therefore viewed the window of the Ark to be like a porthole of circular shaped design. It meant that the circumference of the window would have been pi at 3.1416 etc…
    This would have been seen as an artistic impression of the diameter of the window of the Ark and its relationship with pi if it not for an interesting observation. Look at the result when we take all of the cubit measurements in the flood story into account and present them as a ratio just the same as we do to express the value of pi by using the simple equation of dividing 22 by 7 to get 3.142857 etc.

    The equation with the cubits involved in the ark requires the biggest number of 300 to be divided by the total of all of the smaller cubit numbers, which were listed in the flood story. But the smallest number would no longer involve the one cubit as the diameter of the window but instead, a half cubit as its radius because in the Bible things move on. The equation is thus as follows:
    300 divided by 50 + 30 + 15 + 0.5 = 3.14136 etc.

    It looks like the biblical writers had expressed the value of pi in a very strange way. It is strange because we are so far removed today from the notations and numerical symbols and the mindset of those bygone biblical mathematicians.

    If you find this contribution interesting then let me know and I will outline an even more significant relationship with pi and it is from particular equations from the copper scroll that was found in a cave in Qumran by the Dead Sea in 1952.

    In the meantime, can you acquaint yourself with a description of the copper scroll because you will surely recognise the values of pi and a circle on column seven.

    Michael Hearns
    Dublin, Ireland.
    e mail hearns.michael@gmail.com

  • Zoltan says

    While it is possible that a round number is used, a much more interesting solution exists. The text reveals that the laver had a rim. The wall of the molten sea was a handbreadth in thickness. If they measured the outside diameter and the inside circumference, a very precise result emerges. A cubit is equal to 45 cm (17.72 in). That means that the outside diameter is 450 cm (177.2 in). A handbreadth is 10.16 cm (4 in), which makes the inside diameter 450-2x10x16=429.68 cm (169.165 in). The inside circumference is 1350 cm (531.496 in). If one divides 1350 by 429.68 (or 531.496 by 169.165), the astonishingly precise value of 3.1418 is the result (accurate to the third decimal digit).

  • James says

    The text quoted in Kings doesn’t necessarily mean that they actually used 3 as the value for pi. Pi is approximately 3 which is all the writer may have been saying, they may have used a more accurate value when actually constructing something.

  • Udo says

    In the paragraph starting with “According to a 4,000-year-old cuneiform tablet ….” there is an arithmetic error: From the ratio of perimeters reported by the Sumerians, 3456/3600, one obtains a value of 3 27/216 or 3.1250 for ???? (and not 3 23/216).

    It is also curious why the ratio originally given is reduced to only 216/225 since there is another common factor of 9 in both numerator and denominator; the ratio is simply 24/25. It is also curious why the Sumerians would have chosen such a high value for the denominator (that is, 3600 units of length around a large circle) and then not measure the hexagon’s perimeter with the same length unit to a better accuracy, namely 3438 to the nearest integer (instead of the reported 3456). However, if a small circle was measured with a unit length that fit 25 times going around then that same unit length would fit just ~24 times around the six hexagon sides … Incidentally, applying the same consideration to the denominator of 225 the closest integer to fit the hexagon is 215 (and not 216 which is apparently just derived from 24) for which a value for ???? of 3.139 would be (or could have been) obtained.
    Is there any more detail available about the initially reported values for the measured perimeters of hexagon and circle?

  • John says

    Wow Paula, with that kind of numerology, I bet you could calculate the day of the end of the world!

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