Origins: 3.14159265…

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

Read Kim Jonas’s article “Origins: 3.14159265…” as it originally appeared in Archaeology Odyssey, March/April 2000. The article was first republished in Bible History Daily in 2014.—Ed.


 
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.
 


 
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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.
 

 

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  1. William says

    The Cubit and its relationship to the Aroura is another example of early Egyptian’s understanding of Pi. In ancient Egypt the Aroura was an area measurement expressing the relationship between Pi and the solar year. In Egypt the year was sometimes called a “quarter”. This “quarter” was the geometrical expression of a circle whose circumference was 3652.5” This circle or “quarter” had a diameter of 1162.6” and was equal to a quarter of an Egyptian Aroura. Four of these “quarters” were equal to one Aroura. A simplification of this relationship was expressed by a rectangle Aroura of 1162.6” x 3652.5“ and a square Aroura of 2060.7” x 2060.7”. 1/100th of this square Aroura is the likely origination of the Royal cubit.

    Horapollo in his Hieroglyphics, Bk I, v. stated the following concerning the relationship between the Egyptian year and the Aroura:

    “to represent the current year, they (the Egyptians) depicted the fourth part of an Aroura: now the Aroura is a measure of land of a hundred cubits. And when they would express a year they say a quarter.”

    Hereodotus makes a similar statement:

    “the aroura is a square of a hundred cubits, the Egyptian cubit being the same as the Samian.” (Book II, 168)

    In essence then the quarter Aroura was really a pi/”year” circle, the Royal Egyptian cubit of 20.60” but one expression of that relationship.

    The 18” rod cubit of ancient Egypt may have a similar derivation, being a fractional representation of half the circumference of a quarter Aroura. (i.e. 18.2625” or 1/200th of the 3652.5” circumference of the quarter Aroura.)

    * * *

    To my knowledge this subject was first address in the unusual book written by D. Davidson and C. Adlersmith entitled The Great Pyramid: It’s Divine Message. Ironically the book was born out of an agnostics attempt to disprove some of the early 20th century claims that the Great Pyramid was a so called “Bible in stone”. In any case the above was my own summary of some of the information found in Chapter II, entitled: The Evidences of Scientific Origins in Ancient Egypt. I thought the information was worthy of consideration despite some of the less credible claims with which it was associated. – William Struse

  2. HANS PETER says

    The disparagers of Solomon’s pi who claim the Bible says pi = 3 omit half the evidence. The rest of the parallel passages they cite from 1 Kings 7:23 and 2 Chronicles 4:2 shows their dogma is based on a hit-and-run calculation of a type that would make any undergraduates flunk their exam.

    It seems that none of those experts who so compared the diameter and circumference of Solomon’s Sea of Bronze ever bothered to read on. The next verses, 1 Kings 24 and 26, say that the circumference was measured under the rim, and that this rim was flared:

    “All round the Sea on the outside under its rim, completely surrounding the thirty cubits of its circumference, were two rows of gourds cast in one piece with the Sea itself. (…) Its rim was made like that of a cup, shaped like the calyx of a lily. When full it held two thousand baths.”

    The parallel account in 2 Chronicles 4:3 and 5 reads

    “Under the Sea, on every side, completely surrounding the thirty cubits of its circumference, were what looked like gourds, two rows of them, cast in one piece with the Sea itself. (…) Its rim was made like that of a cup, shaped like the calyx of a lily. When full it held three thousand baths.”

    Obviously, the gourds could not have been under the Sea if they were cast as part of its circumference on every side, and the measuring rope for the circumference would not be stretched around the rim where it would not stay up but only below it. The only practical way to measure such a flared vessel is to stretch the rope around the body below that rim.

    Moreover, only this measure directly around the body is relevant for indicating the volume the vessel could hold, an important part of its description for which the rim diameter is clearly irrelevant. It seems therefore that the scribe of 2 Chronicles 4 was simply as careless in specifying the place of the measurement as in mis-copying the volume of the basin.

    However, both accounts agree that the rim was flared. The ten-cubit diameter measured across its top from rim to rim was therefore larger than that of the vessel’s body which “took a line thirty cubits long to go around it”.

    The circumference and diameter reported were thus not for the same circle, and deducing an ancient pi from these unrelated dimensions would be about as valid as trying to deduce your birth date from your phone number. Yet, so-called scholars have repeated that myth of the Bible’s wrong pi ever since the Enlightenment when it was a convenient way to show the Bible could err, and the same sloppy mathematical malpractice continued during the period of Colonialism when it became important to show the superiority of Western science over the allegedly backwards ancients of the Near East.

    For more about this political value of the biblical pi, see http://recoveredscience.com/const100solomonpi.htm. It is time to retire this ignorant myth from discussions of ancient math.

  3. David says

    The author of the Bible wasn’t interested in teaching the exact value of Pi; he was just saying “It was about three times farther around than it was across” so the reader got an idea of the enormous size of it. And he succeeded.

  4. DALLAS says

    Hans Peter is correct. The two circles are not the same, but are separated by the thickness of the rim of the Yam and offset by placing the circumference rope around the basin below the top of the rim. It is time to retire the “Pi = 3″ myth.

    The scribe probably did not have access to the methods used by the craftsmen, which were trade secrets in ancient times. Those methods probably were similar to the older Babylonian methods, already established at the time of the building of Solomon’s Temple (c. 960 BCE). Obviously, without access to a reasonably accurate method of estimating Pi, the craftsmen could not have build in the basin in the first place.

  5. David says

    Hans Pete and Dallas make some excellent points, and I would like to add…given that 3000 years later we will have some error in trying to convert hand reacts to cubits to check the measurement, but when we subtract the two hand breadth width of the rim from the 10 cubit diameter of the sea and use that number divided into the 30 cubit circumference of what is almost certainly going to be the inside circle of the basin, we arrive at a value for pi of 3.11, which yields an error of only 1% from the accepted value of pi. That’s pretty good considering the error involved in trying to guess the conversion from hand breadths to cubits and the apparent imprecision of the methods of measurement.

    Of course the biblical author was not trying to scientifically derive the value of pi, but if measurements the author presents are not verifiable in such a simple manner, then it sheds doubt on the veracity of much of the rest of what the author presents, as evidenced by those who reject the Bible on such flimsy evidence.

  6. Hanan says

    Month 03, day 14th? This is π Day, just because π ≈ 3.14.
    The circumference of a circle is π times the diameter.
    π is irrational number 22/7 which is almost 3.14, yet we can never obtain the real value of π, it goes on like this;
    3.1415926535897932384626433832795028841971693993751058209…
    You may run as far and as fast as you can but you can’t touch the horizon, and you may continue dividing and obtaining as many digits as you wish to break records of people who did it up to over 13.3 trillion decimals, but you will never reach the exact value of π. There will always be infinitely small value missed to us when we try to calculate π.
    Because the definition of π is related to circles, ellipses or spheres, it is found in the most important formulae of our calculations in trigonometry, geometry, cosmology, number theory, statistics, fractals, thermodynamics, mechanics and electromagnetism.
    It is one of G-D’s secrets, and human calculations will never be complete!
    As He made it clear to us in Revelation 22:13;
    “I am the Alpha and the Omega, the First and the Last, the Beginning and the End.”

    Happy π Day!

  7. JoeSnow says

    The Israelites would have to be able to accurately deduce the value of Pi or else they would never have been able to build or create very much. Too many of the things we take for granted in our daily lives depend on an accurate valuation of Pi and since the laws of physics don’t change, it wasn’t any different in those days. To say that they didn’t know what they were doing when calculating Pi is demonstrably false. Civilizations can’t exist without it. Whether they set out with any determination to find it’s value is irrelevant. Engineers, architects, and craftsmen, used Pi whether they were consciously aware of it or not or else everything they made would have fallen apart pretty quickly.

  8. Jennifer says

    Another explanation for the biblical accuracy of Pi can be found here: http://www.khouse.org/articles/1998/158/

  9. Don says

    The passages quoted are not describing Pi, but rather the Sea constructed during the time of Solomon. It could be that the circumference was measured under the rim like some suggest. But even if it is the same circle as the diameter, it poses no mathematical problem. It also would not suggest a Hebrew value of 3 for Pi.
    The two chapters where this story is related do not represent engineering documents for the construction of this sea, which would require great precision. Rather, they are descriptions given for those who would read about this later.
    As a mathematician, I would assume the diameter and circumference given were rounded to the nearest cubit. The concept of decimals is only about four hundred years old. For these ancients to describe portions of a cubit would not be as easy as it would be for us. They would have to mention a secondary unit also, such as a handbreadth which is used to describe the thickness of the Sea in 1st Kings 7:26.
    If the diameter and circumference are rounded to the nearest whole cubit, can the given numbers work? Yes. A diameter of ten cubits would represent 9.5 to 10.5 cubits and a circumference of thirty cubits would represent
    29.5 to 30.5 cubits. Knowing Pi as we do today, we find that if the diameter were from 9.5 to about 9.71, then the circumference would be about 29.8 to 30.5, both dimensions falling within the rounding we expect.


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