Wikipedia diagram by David Eppstein :
This is an updated text from 2002, called “Finding the Perfect Ruler”
Any number with limited “significant digits” can be and should be expressed as a product of positive and negative powers of the prime numbers that make it up. For example, 23.413 and 234130 can both be expressed as an integer, 23413, multiplied or divided by powers of ten.
What Primes are
Primes are unique and any number must be prime itself or be the product of more than one prime. Having no factors, prime numbers are odd and cannot be even since the number 2 creates all the even numbers, meaning half of the ordinals are not prime once two, the first “number” as such, emerges.
Each number can divide one (or any other number) into that number of parts. In the case of three (fraction 1/3) only one in three higher ordinal numbers (every third after three) will have three in it and hence yield an integer when three divides it.
Four is the first repetition of two (fraction ½) but also the first square number, which introduces the first compound number, the geometry of squares and the notion of area.
Ancient World Maths and Written Language
The products of 2 and 3 give 6, 12, etc., and the perfect sexagesimal like 60, 360 were combined with 2 and 5, i.e. 10, to create the base 60, with 59 symbols and early ancient arithmetic, in the bronze age that followed the megalithic and Neolithic periods.
The first three primes are those most encountered in the ancient world: {2 3 5}. The intervals of normal musical melody are all defined by the first three primes {2 3 5} and the next two primes {7 11} were used to geometrically model pior π: The constant ratio of a circle's circumference to its diameter, approximately equal to 3.14159, in ancient times approximated by rational approximations such as 22/7. as 22/7The simplest accurate approximation to the π ratio, between a diameter and circumference of a circle, as used in the ancient and prehistoric periods.. This set {2 3 5 7 11} was then used as the basis for a fractional metrologyThe application of units of length to problems of measurement, design, comparison or calculation. based upon the foot as number one {1}.
The ten-based decimal arithmetic we use today uses 2 and 5, with negative powers applying to fractional parts. The Bible is an early iron-age compendium in which sacred numbers appear that are largely decimal and its Hebrew language was organised to have each letter equate to numbers {1 2 3 4 5 6 7 8 9} {10 20 30 40 50 60 7 80 90} and then hundreds. In other words, the symbolism of numbers-as-symbols was an invention alongside language that enabled numbers to be like words but also to function within the arithmetic methods relevant to the lives of cities and their markets.
Megalithic Maths and Prime Numbers
Before the ancient world, the history recorded using language is replaced by mythology and material evidence. In the megalithic, there was little or no number notation yet multiplication and division were still possible through repeating a length a number of times and by geometrically dividing lengths as to their prime numbers.
Metrology had to evolve before arithmetic methods were available. Megalithic numeracy stored numbers as lengths, and these were either (a) the fractional measures, made out of primes {2 3 5 7 11}, or (b) the counted lengths of astronomical time that were effectively raw, undifferentiated ordinal numbers.
Numerical information exists within megalithic structures because their builders used a system of measures in their construction. But numbers are also to be found, between points, as undifferentiated lengths but also as lengths processed by measures based upon the manipulation of prime numbers.
To analyse a number for its primal identity, simply remove the decimal place by dividing by powers of {2 3 5}. We should note that, in general, metrological measures can have repeating significant digits but the complexity of their fractional parts seen in decimal is entirely due to the hidden action of prime number denominators, in our base 10 system of notation.
Since metrology is our subject, and since an example is definitely required, we will analyse the tables of Greek Measure given in Ancient Metrology, as John Michell’s Northern and Tropical Values. This is what I began doing on flights to the Netherlands in the early 2000s.
Primal Therapy for Numbers
Whilst spreadsheets and programs can increase productivity, it is useful with smaller primes to follow the calculator method.
Remove Decimals: Use the reciprocal powers of 10, e.g. 1.008 becoming 1008/ 1000
Divide by Successive Primes: Start with the lowest primes, and when the result yields a fractional part, go back and move onto the next prime in order, as in 2, 3, 5, 7, 11, 13, 17, etc [ see Appendix A for list of primes and web references to more]
| Prime | 0 | 1 | 2 | 3 | 4 |
| ¸ 2 | 1008 > | 504 > | 252 > | 126 > | 63 |
| ¸ 3 | 63 > | 21 > | 7 > | a prime! |
| 1008 = > (as above) | 24.32.7 |
| 1000 = 103 > | 23.53 |
Therefore, the Tropical Greek Foot is uniquely,
| 1008 | = | 2.32.7 |
| 1000 | 53 |
It is soon noticed that, when there is a decimal fraction, the numerator has powers of 2 that will simply cancel out with the denominator. Thus the numerator integer is long because of doubling invoked by the decimal system itself. The same is happening with positive powers of ten too, so that our base 10 system of notation is making the metrological values appear ludicrously accurate in their number of significant places. However, it is only the prime powers that really count and these are, in every case, simple.
The ancients had to have used prime notation to represent numbers-as-lengths as powers of prime numbers. Any future study of megalithic monuments will likely benefit from analysing the manipulations of prime numbers using the foot-based metrology recently restored from Historical Metrology of Berriman by John Michell and John Neal.
| in Prime Notation | Primal Value | Tropical (ft) | GREEK lengths | Northern (ft) | Primal Value | Prime Notation |
| 3|2|3|1 | 32.7 /23.53 | 0.063 | Digit | 0.06336 | 2.32.11 55 | 1|2|5|0|1 |
| 1|2|3|1 | 2.32.7 /53 | 1.008 | Foot | 1.01376 | 25.32.11 55 | 5|2|5|0|1 |
| 0|3|3|1 | 33.7 /53 | 1.512 | Cubit3/2 feet of any sort, such as 12/7 {1.714285}, 1.5 Royal feet of 8/7 feet, but sometimes a double foot, such as the Assyrian {9/10} of 1.8 feet. | 1.52064 | 24.33.11 55 | 4|3|5|0|1 |
| 3|2|0|1 | 23.32.7 | 504 | Stade (500 ft) | 506.88 | 27.32.11 52 | 7|2|2|0|1 |
| 4|3|1|1 | 24.33.7 5 | 604.8 | Stade (600 ft) | 608.256 | 28.33.11 53 | 8|3|3|0|1 |
| 1|2|1|1 | 2.32.5.7 | 630 | Furlong | 633.6 | 25.32.11 5 | 5|2|1|0|1 |
| 4|2|1|1 | 24.32.5.7 | 5040 | Mile | 5068.8 | 28.32.11 5 | 8|2|1|0|1 |
* Prime Notation simply shows the prime powers in order separated by lines, in ascending order i.e. 2,3,5,7,11,etc Thus 72 would equal 0|0|0|2 and 72/10 would be shown as 1|1|0|2.
- Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.
- Heath, Robin, and John Michell. Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients. Kempton, Ill.: Adventures Unlimited Press, 2006. Reprint edition of The Measure of Albion.
- Heath, Richard. Sacred Geometry: Language of the Angels. Vermont: Inner Traditions 2022.
- Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
- Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
- —-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pi and the World.
- —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
- —-. Ancient Metrology, Vol. 3, The Worldwide Diffusion – Ancient Egyptian, and American Metrology. The Squeeze Press: 2024.
- Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.