A Mathematical Puzzle

Archimedes and the Square Root of Three

Archimedes states that the square root of three (√3) is less than 1351/780 and greater than 265/153. It is not known how he determined this.

The square root of three (√3) is the height of an equilateral triangle with sides of two units: by Pythagoras' Theorem, √(2² - 1² ) = √3. Furthermore, it is known that when such a triangle is circumscribed, the base cuts the diameter in the ratio 3:1, as at Figure 1.a.

Figure 1: Approximating the square root of three.

It is also widely known that 26/15 is a fair approximation to √3, as at Figure 1.b. As triangle ACD is similar to ABC, the length of CD can be calculated as 15² ÷ 26 = 225/26, by applying the same ratio, as shown at Figure 1.c. The impact of then multiplying throughout by 26 is shown at Figure 1.d. However, as can be appreciated from Figure 1.a., if CD is 225 then BC must be 675 (see Figure 2, left, below), not 676, as Figure 1.d above right, but this suggests combining the two results to provide a further approximation to √3, that is, 1351/780, as at Figure 2, centre.

Figure 2: Refining the approximation to square root of three.

It is now possible to remove the original approximation (26/15), as shown graphically at Figure 2, right. This derives a further approximation to √3, that is, 1325/765, which resolves to 265/153. Thence, it can be shown by squaring that 1351/780 is greater than √3, and that 265/153 is less. QED.

Note: If a/b and c/d are approximations to √3 then (a + c) / (b + d) and (a - c) / (b - d) are two others. It also follows that 3b/a and 3d/c are two more. For example, the fraction 7/4 is an approximation to √3, and by the foregoing reasoning so is 12/7, and it follows that 19/11 is also, leading readily to 26/15. Upon conclusion, above, it should be appreciated that, for √3/1, a² = 3b², but 1351² > (3x780²) and 265² < (3x153²). It is assumed that the approximation of 26/15 could have derived from the sexagesimal system when measuring an equilateral triangle with sides of 2 units (2°). It may have been found that the height was 1° 44' (√3 = 104/60 = 26/15).

A similar process can be applied to other key irrationals, such as √2 (7/5, pertaining to the square) and √5 (9/4, pertaining to the pentagon).

© G.J. Bath Source: Bath G.J. When the Stones Talk Back: An Excursion into Pseudo-Archaeology Vol 1. KeyPress 2015, x-xi