First, perhaps a disclaimer is order – sometimes, I’ve had to do little more than try and explain the logic of my work with numbers, or explain how certain ideas are not tranferrable across diffeent proposed ancient number systems, to give other independent researchers the impression that I intend something other than mere constructive criticism at worst.
Nothing could be further from the truth – I cannot think of another mathematical researcher who isn’t making important discoveries and valuable contributions to our collective understanding of ancient mathematics, often on a frequent basis.
That said, all the same I try to be very careful what I accept as logic. I am certain I have witnessed both of my most helpful mentors go off the deep end with pouring vast effort into ideas that in the long run, cannot be successfully supported – Munck’s “geomathemics” or Morton’s “Archaeo-Sky Matrix” – in what amounts to a phenomenal and tragic squandering of keen human intellect.
Few things are more disheartening to me that to read discussions like some of the ones that involved researcher Don Barone, who posted work on connections between Giza pyramids and planets to a number of “fringe” forums where they’d presumably have been of considerable interest. Don was once told something to the effect of “You’re wrong, the ancient Egyptians didn’t know what decimals are, I just looked it up on Wikipedia” as if one minute’s worth of education somehow beats years worth of education and hands-on experimenting.
Obviously, when alternative researchers like myself suggest that the ancients were using decimal mathematics, we’re talking about something that has not only escaped Wikipedia, but all of history in general. There seems to be little telling what was in the hundreds of thousands of scrolls of ancient writings that went up in flame at libraries like that of Alexandria, to name only one.
Equally tragic and no doubt equally detrimental to a better understanding of human history and capability are the pre-Columbian codices torched en masse by Spanish conquistadors on account of their “superstitious” “heathen” content.
Still, I’m sure none of us are content to rest our arguments on written works that could have existed or even probably existed. When someone insists to me that the ancient Egyptians could only work in unit fractions, I’m well aware that the scant few surviving ancient Egyptian mathematical works have all the earmarks of being generalist works rather than specialist works – they tend to attempt to cover a very diverse range of applications of mathematics with a single style of mathematics that may not even be as simple as it looks at first glance.
I’m also aware of what the alternatives to the idea of ancient use of decimal math look like. One pyramid researcher I encountered was quite devoted to the idea that the ancients possessed all sorts of complicated knowledge from astrophysics to physics outright, yet equally devoted to the orthodox dogmatic proposition that ancients were nonetheless still so simple minded as to be restricted to working with fractional numbers, resulting in pyramid theories saturated with fractions so complicated as to be insoluble for all intents and purposes.
Makes one want to ask if fractions are so well suited for astrophysics, why NASA isn’t using them.
I am rather wary by now of any proposition that the ancients were doing their “rocket science” in fractions, particularly unit fractions, but there’s the rub – how many researchers out there insist on claiming the ancients used fractions because of the historical basis for this, but proceed immediately into elaborate theories using types of fractions for which there is no historical support for them using?
I’m sure I must have unintentionally ruffled someone’s feathers when I attempted to turn around a criticism that alternative researchers often face, on the individual who issued it, to say I wonder if we’d be so keen on the idea of the ancients using nothing but fractions if someone took away our pocket calculators, because that’s what most if not all of us do these days is enter these fractions into our pocket calculators where we don’t really get a good idea how hard they may be to work with in real life.
On a bad day, everyone from metrologist John Neal on down insists the ancients did their calculations in fractions but presents their work in decimal. I have truly never seen anyone actually do their projected “ancient fraction calculations” in the actual manner that the vestiges of historical sources attempt to dictate that ancient math was actually done.
Some of my other misgivings about the ancients having used fractional math to render their astronomical calculations include that while we can get around the reflexive objection that “the ancients didn’t know how to do long division” by asserting that they may have used inverted multipliers instead, which is the same thing without division – for example,
360 / 72 = 360 x (1/72) = 5
In many cases, the simple round numbers that people often like to attribute to the ancients aren’t so simple when they become inverted. Instead, they become complex decimal strings more readily expressed in decimal form. The simple divisor 360 inverts to the inverted multiplier 1/360 = 2.777777777 / 10^n, and so forth.
A great many of these are of course numbers that the “conventional wisdom” insists that the ancients weren’t able to manage.
Setting aside fractions for a moment, another pitfall of attributing simplistic math to the ancients is that it refuses to stay simple very long. Even for the amount of academic works that I’ve read that discuss Mayan calendar math, I’m still left with little idea what academia in general thinks was the ancient protocol for dealing with this fact.
Some of these academic works simply exclude calculations that don’t work out perfectly in nice round whole numbers, virtually gutting the broader applicability of these calendar systems, while others proceed as if the Maya simply knowingly rounded off these unruly decimal numbers that often result from the breakdown of accepted calendar numbers and “supernumbers”. It seems as if the question of the Maya would have dealt with numbers like 346.62 days, the Eclipse Year, may remain an open one that still begs for consideration of whether it would have required decimal math to overcome some of these obstacles.
Sure, we’d make that into 346 and 2/3 in two seconds flat – but have you ever heard anyone give the Maya credit for even knowing what a fraction is?
While I don’t have a particular example prepared, this is something that anyone can discover for themselves in a hour simply by picking any accepted Mayan calendar number and attempting to break it down by the obvious subordinate cycles of the moon, or the Orbital and Synodic Cycles of the planets, and observing how simple some of the resulting numbers aren’t.
I continue to have great concerns that what some researchers may be doing is conflating ancient marketplace math – the “common math” – with the mathematics required for many astronomical calculations. There is no doubt that historical examples do point to astronomy having been simplified for the “average person” – our own calendar of 365 days rather than the true ~365.25 day year is itself an example of this – yet even while we make and use such mathematically egregious approximations, our own astronomy requires knowing the difference between a Tropical Month of 27.321582 days and a Sidereal Month of 27.321661554 days!
Importantly, just because most of us including myself still use the 365 day year plus the Leap Year when required, doesn’t mean that none of us can tell the difference between the Tropical and Sidereal month.
In my own work, the working premise is that ancient man had means, motive and opportunity to develop complex math. “He” (or quite possibly she) was able to deal with math to develop calendars in the first place, and was motivated to do so beginning with knowing when winter was approaching and how much goods needed to be stored up to survive until the next harvest season.
As far as opportunity goes, we find examples of what we believe are calendars that are a minimum of three times as old as the oldest pyramids, and even older.
The opportunity part of this equation then seems to be that we had thousands or tens of thousands of years to develop the complex math required to keep complex calendar systems on track without a veritable encyclopedia of necessary corrections even if the scattered remains that survive now are thus far exclusively devoted to math for the “average person”.
More than one published researcher has asserted that developing base-ten mathematics is an eminently logical development for ten-fingered beings such as ourselves. Likewise, it seems a rather logical development if the earliest need for mathematics is to effectively divide a ~365-day year into smaller, more manageable units such as months, since the only real factors of 365 are 73 and (10/ 2) = 5.
Yet 365 is otherwise so indivisible by nice round numbers as to provide great impetus to find constructive ways to deal with remainders even as early as the first discovery that there were 365 days in a year.
I nearly fell out of my chair when I first started working with calendar math as the possible original intended application of the mathematics that Carl Munck was teaching. Again, we have only to try to divide the 365 day year into months just like our very own – of 30 or 31 days – to be confronted by uncanny approximations of numbers I already knew like the back of my own hand.
365 / 30 = 12.16666666; Remen = 1.216733603 ft
365 / 31 = 1.177419355; Munck “Alternate Pi” = 1.177245771; Harris-Stockdale Megalithic Foot (HSMF) = 10 x (sqrt 2) / 12 = 1.178511302 ft (?)
In spite of the immense respect as I have for Peter Harris, that difference of some 12/10000 of a foot between “Alternate Pi” and the HSMF isn’t likely to be recognizable through any field measurements. To me, for all intents and purposes they are virtually the same thing and I have 20 years experience with 1.177245771 and its properties to tell me that it is the superior number for practical purposes.
Once again, I will express my belief that what the ancient Egyptians were working with for metrological units, were probably believed by them to be the oldest heirloom numbers possible. Anything else would be just too much of a coincidence, and on top of the years squandered by Munck and Morton there are the years I squandered myself by looking time and again at the equation 365.0200808 / 300 = 1.216733603 thinking “What an interesting coincidence” and never thinking to ask “What if it isn’t a coincidence?” Been there, done that, OOPS…
Again, some 20 years now I have been working with Munck’s numbers, constantly testing their capability for expressing important data, and their value as a networked system of numbers as a whole. The entire time I’ve felt literally apologetic about the numbers that, per se, end up excluded from the system, constantly asking “Why can’t we have Phi in this system? Why can’t we have sqrt 2, sqrt 3, or sqrt 5 verbatim? Why can’t we use 7 or 11 or 17 or 22, etc etc.”
The past several years more than ever, I’ve seen why, and why we use approximations of things like this where necessary to create a system that is even more versatile than ordinary numbers, yet ultimately predictable enough to be understood. It may not immediately so (few if nay learn a new language overnight), but picking the best parameters immediately helps to take most of the endless guesswork out of things. Although the ancients continue to surprise us, 50-80% of the time my first guess is right because, well what else would it be? If it’s near to Pi, Pi is obviously the best first guess.
If there’s anything David Kenworthy’s work has shown me, it’s that numbers like 7 and 11 are mutually exclusive to the actual Pi ratio – if you use 22/7 as Pi for a moment, you’ve begun creating a system that is going to be hostile to Pi. Pi and Phi are also mutually exclusive, which is something I should have accepted long ago, especially since I seem to have discovered most of our alternatives to Phi myself.
Because Pi is the stuff of circles and Phi is the stuff of pentagons, and because of the prevalence of ancient circular architecture and the general absence of ancient pentagonal architecture, Pi obviously won out over Phi a long time ago.
There’s probably little point in feeling apologetic about this math having rejected prime numbers – it’s probably what anyone who worked with fractions would have done soon enough. Please correct me if I’m wrong, but operating fractions involves reducing numbers to common denominators, and of course prime numbers aren’t reducible because they lack factors by which they can be reduced thus.
The great irony that motivates this post, though, is that for all the talk about how “the ancients didn’t know what decimals are”, is that except for a few instances such as root functions (anyone remember the “root rules”?), the ancients didn’t need to care what decimals were.
If you’re working to ten digits, simply move the decimal points 20 places to the right, and now you don’t need to know what decimals are because now for intents and purposes, you’re working with whole numbers. Move the decimals 20 places back to the left when you’re finished if you prefer – or if you’re the ancient Maya breaking down the calendar number 819 (not 81900!) into 364 x 2.25 (not 364 x 225!) perhaps you needn’t even bother putting the decimals back where you found them.
I’m tempted to throw out decimals myself, although it’s a little bit late now. Imagine how it plays out searching my files for references to various numbers. How much easier life would be and how much faster progress might occur if all I had to search my files for was 1177245771, rather than 0.1177245771 and 1.177245771 and 11.77245771 and 117.7245771 and 1177.245771 and etc.
That’s what the argument about ancient decimals really seems to be – little more than semantics, and usually brandished by those who’ve apparently given little forethought to the matter.
Anyway, if someone is trying to convince me that before a Sumerian can sell a Babylonian a fish, the fish has to be weighted and the weight converted to volume, and the volume has to be converted from cubic Sumerian units to cubic Babylonian units by an elaborate series of maneuvers including a shift from base 10 math to base 77 that takes an hour to work out on papyrus in unit fractions or some such, no, I’m probably not buying, sorry.
If that’s the kind of thing we get for insisting that the ancients only knew how to use simple fractions, I hope readers can see my issue with the premise.
I even have misgivings about whether legal standardization of ancient measures results in more, or less, standardization. Ideally, yes, by Royal decree ancient unit “x” shall be equal to umpteen barleycorns so that all trade in the kingdom shall be fair and honest, but it’s also an incentive and an opportunity for any dishonest persons to tamper with the standards to earn a little extra.
To put it simply, in my opinion, the standards of the marketplace make poor standards indeed for ancient metrological values, and obviously there is little if any need to apply the kind of math it takes to distinguish the Tropical Month from the Sidereal Month for the village fishmonger to peddle a fish or the local baker to follow a pastry recipe. The mathematics of the marketplace or kitchen is simply NOT the mathematics of astronomy, and vice-versa.
This might help explain to readers why I gravitate toward ancient metrological values that find standardization though geometry, numbers that will always be exactly the same according to the equations that generate them, like 54 / 10 Pi = Royal Cubit 1.718873385 ft, or sqrt 15 x (Pi / 10) = Remen 1.216733603 ft – constants we call them, because that’s what they are, constant. Predictable. Fathomable. Recognizable. Reliable – or as the ancients may have thought of them, numbers that are as old as time itself, literally.
Mix them with numbers that they clash with, however, and they may not stay that way long.
Again, I’m really not meaning to knock anyone else’s work here, I am greatly indebted to a staggering number of researchers – probably every one of them that I can name – for insights and inspiration, but I thought that in case some of my logic seems hard to follow, it might help readers for me to gather up some of my thoughts on why I approach numbers the way that that I do, and why I’ve chosen to work with a proposed ancient system of numbers that rejects many numbers that other researchers would see nothing against working with.
I’m not sure how to find it now, but there was a post on the GHMB from forum member “loveritas” that described a similar preference for certain whole numbers that were described by this poster as factors of 360 – numbers by which 360 is evenly divisble. Naturally, 7, 11, 17, 19, 21 etc aren’t on that list – what is on the list, are factors of 360 (and for us many of their simple products and dividends as well — 9 is a factor of 360: 360 / 4 = 9, so 9 x 2 and 9 / 2 and 9 x 3 etc are also valid).
This was a good way for me to see these numbers described, because it highlights the compatibility of these numbers with sexigesimal (base 60) math the way we use it nowadays to measure latitude or longitude or azimuths or or time – 360* (60 x 60) degrees in a circle, 60 minutes per degree, 60 seconds per minute.
I doubt it’s any coincidence that the most powerful combination of numbers I know of is 60 and sqrt 60 – this combination gets sqrt 60 working at data retrieval at least as high as the 33rd power – an unprecedented thirty-three pieces of data for the “price” of two!
How many examples of ancient architecture can we find by the usual suspects where we can’t find sqrt 60 as part of the design? It’s one of those numbers they understandably seemed to want to build into virtually everything, as if they actually wanted and intended for us to be able for us to find it along with other critical data retrieval keys like 1.177245771, 1.622311470, Pi, or (Pi / 3) so that with the very first architecture we attempt analyze we can begin to see what’s going on.
Again, that is what any of us who are looking for anything in the proportions of ancient architecture are working with, is the inherent proposition of data storage and retrieval. The more the better, obviously.
It certainly seems to highlight the sense of heritage that seems to go with the idea of recording astronomy data in architectural proportions generation after generation, generally in particularly durable form, and the idea that this may represent a sort of guarantee that we are inheriting the very same world our ancestors lived in, with the very same cycles and the very same seasons.
I’ve probably strayed far from the point of this by now, the point being that whether or not the ancients knew what decimals were really isn’t the question, not at all.
–Luke Piwalker
Luke Piwalker 