Old Yard’s Mastery of the Square Root of 2

The old yard was almost identical to the yard of three feet, but just one hundredth part smaller at 2.87 feet. This gives its foot value as 99/100 feet, a value belonging to a module very close to the English/Greek which defines one relative to the rational ratios of the Historical modules.

So why was this foot and its yard important, in the Scottish megalithic and in later, historical monuments?

If one forms a square with side equal to the old yard, that square can be seen as containing 9 square feet, and each of those has side length 99/100 feet. This can be multiplied by the rough approximation to 1/√ 2 of 5/7 = 0.714285, to obtain a more accurate 1/√ 2 of 99/140 = 0.70714285.

Figure 1 Forming a Square with the Old Yard. The diagonal of the foot squares is then 7/5, the simplest approximation to √ 2.

This is possible because 99/100 x √ 2 = 1.4 = 7/5 almost exactly. Therefore 140/99 x 7/5 must equal √ 2! We can therefore see that, in a square with side length a full yard of 3 feet, the diagonal will be √ 2 = 100/99 x 7/5 = 140/99 = 1.41 feet long.

Figure 2 The reciprocal foot, of 100/99, couples with the “bad” approximation of root 2, to maintain rationality to the accuracy required for practical metrology in the megalithic.

This reveals the usefulness of the old yard’s foot of 99/100, with respect to the regular yard: Times 7/5, its reciprocal lands close to root 2 and maintains rationality between the side lengths of foot squares and their diagonals. And the reciprocal of 1 / √ 2 = 99/140 is just as useful when using Quadrature to shrink a square as 140/99 is to double a square that way.

The foot of 99/100 derives from a root foot of 63/64 feet, being 176/175 of that foot and hence the root canonical value. The inverse root measure of 64/63 feet, times 126/125, gives the Byzantine foot (used in dome of Hagia Sophia, Istanbul of 100 feet of 128/125 feet** – hence a standard canonical variant of 64/63. One is tempted in fact to call the module with root 64/63 feet, Byzantine and that of 63/64 the inverse Byzantine**. Both the Byzantine module and its inverse give access to the square root of two in accurate rational approximation to squares involving English feet.

** unfortunately inverse is not as good a term but in John Neal’s work “reciprocal” is already a micro-variation of the root by 175/176. Whilst the meaning of reciprocation is to swap numerator and denominator of a rational fraction, inversion will also do.

**the ratio 128/125 is the minor diesis of music where three major thirds fall short of an octave doubling leaving that ratio. 100/99 is the reciprocal measure of 64/63 whilst the Byzantine is the standard canonical. The standard value is 56/55 feet, also of known interest and having applications.

References

  • John R. Hoyle analyzed Alexander Thom’s data to gain further certainty that the megalithic yard was employed in the British megalithic. He also found the old yard of 2.97 in Thom’s data for large Scottish rings – see his Megalithic Matters. page 26. Matador 2014.
  • The standard reference for Ancient Metrology is resolved in three volumes of that name, two recently published by John Neal (Squeeze Press).
  • The method of removing one hundredth part of the crude approximation to √ 2 of 7/5 was Biblical, sometimes equated to circumcision. It generates 140/99, the diagonal of the English foot square, of 1.41 feet whilst the inverse Byzantine double foot 198/140 gives 1.4142857 feet. Both are accurate rational approximations to √ 2, to one part in 19600.5, the first smaller and the second larger than the √ 2. A square with sides half an English foot has the inverse foot of 99/140 as diagonal worth
    ~1/√ 2.

One thought on “Old Yard’s Mastery of the Square Root of 2”

Leave a Reply

Your email address will not be published. Required fields are marked *