Ten Decimals of π Are Sufficient To Give the Circumference of the Earth To the Fraction of an Inch

Simon Newcomb? John Casey? George McC. Robson? Apocryphal?

Dear Quote Investigator: The number π is a fundamental mathematical constant which equals the ratio of a circle’s circumference to its diameter. Trillions of digits of the decimal expansion of π have been calculated using electronic computers and innovative algorithms. Yet, this precision is not needed in the realm of practical measurement. Simon Newcomb, the prominent astronomer and mathematician, once said something like this:

Ten decimal places of π are sufficient to give the circumference of the earth to a fraction of an inch.

Would you please help me to find a citation?

Quote Investigator: In 1881 Simon Newcomb published “Elements of Geometry” which was based on the foundational tome by Euclid. Here is a passage discussing the calculation of π. The word “computer” refers to a human calculator. Emphasis added to excerpts by QI: 1

This number π is of such fundamental importance in geometry that mathematicians have devoted great attention to its calculation. . . . Dase, a German computer, carried the calculation to 200 places of decimals. The following are the first 36 figures of his result:

3.141 592 653 589 793 238 462 643 383 279 502 884.

The result is here carried far beyond all the wants of mathematics. Ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope.

The emphasized text above differs slightly from the modern quotation because it does not repeat the term π.

Below are additional selected citations in chronological order.

For completeness, here is an image showing the pertinent passage from Newcomb’s 1881 book “Elements of Geometry”.

In 1885 “The First Six Books of the Elements of Euclid: And Propositions I-XXI of Book XI” was published by John Casey who provided “copious annotations and numerous exercises”. The passage about the accuracy of π was printed without attribution: 2

The result is here carried far beyond all the requirements of Mathematics. Ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope.

In 1899 an article by George McC. Robson in “Home Study Magazine” included the statement without attribution: 3

Ten places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty places would give the circumference of the visible universe to a quantity imperceptible with a microscope.

In 1911 an article in “The School News and Practical Educator” credited Newcomb and presented a student exercise to illustrate his point: 4

Professor Simon Newcomb said of the value of π, “ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch.” The equatorial radius of the earth is 3963.296 miles Show that the values for the circumference of the earth obtained by using for π the value 3.1415926536 or 3.14159265359 do not differ by 1 inch.

In conclusion, Simon Newcomb should be credited with the passage he wrote in “Elements of Geometry” in 1881. The passage has appeared in later books with and without proper attribution.

Image Notes: Image depicting Earth sunrise see from space by qimono at Pixabay.

(Great thanks to Donald A. Byrd whose inquiry led QI to formulate this question and perform this exploration.)

Notes:

  1. 1881, Elements of Geometry by Simon Newcomb (Professor of Mathematics, United States Navy), Book VI: Regular Polygons and the Circle, Problem VII, Quote Page 235, Henry Holt and Company, New York. (Google Books Full View) link
  2. 1885, The First Six Books of the Elements of Euclid: And Propositions I-XXI of Book XI, With Copious Annotations and Numerous Exercises by John Casey L.L.D F.R.S., Third Edition Revised, Note G, Start Page 308, Quote Page 310, Hodges, Figgis, & Company Dublin, Ireland. (Google Books Full View) link
  3. 1899 January, Home Study Magazine, Volume 3, Number 12, Calculating π Without Mathematics by George McC. Robson, Start Page 573, Quote Page 574, The Colliery Engineer Company, Scranton, Pennsylvania.(Google Books Full View) link
  4. 1911 October, The School News and Practical Educator, Volume 25, Number 2, Arithmetic: Eighth Year by E. H. Taylor (Instructor in Mathematics, Eastern Illinois State Normal School), Taylorville and Chicago, Illinois. (Google Books Full View) link