The day was fundamental to how ancient astronomers could have had a full understanding of planetary dynamics seen from earth without our modern science. The day on earth is crucial to this capability for the way it simply relates to other time periods seen from the earth; a simplicity based upon number in which the day is a unity between numbers differing by just one unit. Such relationships are called super-particular (fractions) and the moon is the most strongly affected because it orbits the earth just as the sun also appears to, in a way that naturally generates differences of one. The Moon's daily motion provides a direct visualisation of this.

Figure 1

### Judging Angular Motion

Each night, when visible, the moon can be located amongst the stars as the starry sky moves towards the west. The next day, perhaps at the same time, the moon will appear to have moved just over one outstretched fist length towards the east, on its orbit. It has moved about thirteen degrees relative to the stars but the sun moves about one degree every day so, relative to the earth, the moon has moved about twelve degrees. But we cannot measure the sun's motion without measuring how the earth rotates and, without a sidereal clock, this requires a convenient method to measure the rotation of the sky.

 Looking North, the pattern of stars around the Pole rotate in the nightdirectly showing the rotation of the earth

Figure 2

Whilst a sun dial effectively "looks" south to measure the movement of the sun as the earth rotates, the ancient world could look at the circumpolar sky around the north (or south) pole and directly watch it rotating like a clockface. The sidereal day "of the stars" enables one to visualise that in a single day the sun moves one unit out of 365 units of its (average) daily motion so that, the circumpolar clockface moves once around (365) and then one unit more, and this happens every solar day (from noon to noon). Such an act of imagination indicates that the solar day is made up of 365 plus one units. These units are the difference in time between the periods of the sidereal day and the solar day (about four minutes of time) whilst also being the time taken by the rotation of the earth to catch up with the sun's new position after a full sidereal rotation.

The motion of the sun per DAY (in angle) is therefore visible on the circumpolar clockface as its imagined division into the 365 divisions of it, being the sun's path (or ecliptic) naturally conflated, in the ancient imagination, with the circumpolar clockface. This meant that the sun could be visualised as moving around such a clockface (as time does on our mechanical clocks) so that the Pole, in the centre, is also representing both the earth at the centre of the sun's orbit of the earth, within the geocentric frame. In a nutshell, this is why the geocentric frame remains valid and wholly abandoning it as "primitive" has obscured its lessons for ancient astronomy.

Figure 3

This sort of imagination or mapping, of one thing onto another, of two compatible forms, was more psychically active in an age where the visual and verbal carried the traffic of shared experiencing. It is well established that the sky itself became imagined as being like a celestial earth in the later millennia BC if not before, in models based upon constellations whose names and related myths survive today in a culture strongly driven by literary forms. The celestial earth took a familiar form as being like the spherical earth, having lines of longitude (colures), four gates of the sun, two on the Equator at Equinoxes and two north and south at Solstices. In such a visualisation, the circumpolar clock is like the Arctic Circle, of concentric parallels of latitude along which the circumpolar stars travel every earth rotation.

### Early Star Maps

Returning to the Moon's daily motion, it is obvious that there would be 365 "degrees" in the Ecliptic, each a daily motion of the Sun, and these would be best called DAYS of angle. In order to measure the Moon's motion, the stars would need to be calibrated near where the Moon travels in her orbit and so a list of stars would be known whose separations, in DAYS of angle, had been somewhat quantified. Like a star chart, a linear list of star names and the DAYs between them would form a mnemonic of some sort, possibly like a Peruvian knotwork, or versified recitation like our "Thirty day have September, April, June and November, all the rest have thirty one". Such a scheme, of measuring angles between key stars, has survived as the Indian system of Nakshatras now rationalised to the 27 or 28 asterisms, plotted below onto the northern sky. It is also true that full moons and other celestial phenomena were named as occurring with a given nakshatra, suggesting that the system was associated with lunar motion.

Fogure 4.  Position of the Hindu Nakshatras as per the coordinates specified in Surya Siddhantha [Kishorekumar 62]

Such a system of references would enable more accuracy, in DAYs than having an extended digit or fist held up to the sky. It would transform the starry path of the Moon into a journey along a measuring ruler, which is continually being "eaten" by the West during the year, as new "helical" constellations arise in the East just before the Sun rises. The helical arising of new constellations correlate with where the sun is on the ecliptic during the year and each such constellation corresponds to a given angle of the circumpolar clockface as it appears. Zodiacal constellations rising in the East would become associated with the clockface of the North. (see my Academia Article about the Kercado Roof Axe)

### Quantifying the Moon's Motion

It would then be possible, using a calibrated ecliptic of reference stars, to measure the Moon's motion per day as 13 1/3rd DAYS of motion (probably by seeing 40 DAYS of motion every THREE days) but also it would be intuitively obvious that one DAY per day of the Moon's motion was being lost each day relative to an advancing Sun, which my definition moves one DAY per day. Thus the Moon relative to the Sun moves 12 1/3rd DAYS per day (37 DAYS in three days). When we look at the three-day whole numbers of DAYS, namely 40 and 37 DAYS, these form a ratio (40/37) very close (one part in 4660) to the actual ratio of lunar month over lunar orbit (29.53089 days / 27.32166 days) and hence in proportion to their differential speed relative to the stars and the Sun, as below.

Figure 5

This suggests that combining observations of the motion of the moon, the circumpolar stars, the solar year's day count of 365 days, and a zodiac of stellar backgrounds calibrated in DAYS of angle, one can achieve a very accurate whole number approximation to the ratio held between the lunar month and its orbit. One needs to develop standardised units of measure which can then articulate numbers as lengths of time, visualised in right triangles and compared to observation. Combining numerical counts (365) to calibrate the stars in both the Ecliptic and circumpolar "clockface" (by angle), the sun within the year and a reference star in the circumpolar region provide a reading of the sun and moon in sidereal and solar time.

Once known, the difference of lunar orbit and its illumination (the lunar month) will have an impact on the longer time period of the solar year (12.368 months) because the slower cycle of moon's illumination makes the moon's orbit faster to complete (as shown above). After a solar year it is intuitively obvious that the number of orbits will be one greater than the the number of lunar months, because both the moon and the sun must be returned to the same point of the ecliptic.

### Relating the Moon's illumination to the Solar Year

The differential lunar motion, to the stars and to the sun, came to form the basis of how lunar and solar years were visualised by Megalithic astronomers: by means of a right triangle, and initially using day-inch counts to study that relationship. It was soon noticed that the shortest side was 1/4 the baseline of the lunar year, so that the triangle was half of a four-square triangle is each square three units long. The next diagram shows how Robin Heath's speculative Lunation Triangle (see his Sun, Moon and Stonehenge. 1998.) was found to be encoded (Heath and Heath. 2010.) in Le Manio's Quadrilateral in near Carnac, Brittany. Before that Robin Heath had thought the megalithic always counted such triangles in megalithic yards but at Le Manio (circa. 4000 BC) we found day-inches had been used to count over the three year near-anniversary of 37 lunar months and solar years.

The count at Le Manio yielded a differential length between three lunar years (36 months) and three solar years (37.1 months) equal to the megalithic yard of 32 plus 5/8ths (32.625) day-inches (an excess per year of 10 plus 7/8ths ([10.875) day-inches.

Note how days provided the counting concept for the megalithic, this then realised by using a constant unit of length to count days, these being necessarily small like a digit or thumb (in French an inch is called the thumb, from its width).

Figure 6

The one third excess per year (seen in 12 plus 1/3rd lunar months) is now improved as a new approximation of one third of the megalithic yard, this then being 10.875 day-inches whilst one third of a lunar month is 9.84 days. This highlights the importance of the ratio between the lunar month and the megalithic yard, a subject doubly interesting since Robin Heath had initially thought British megalithic monuments (generally built after the Quadrilateral) often appear coded in megalithic yard although the idea of counting days in inches remained. The transition, from counting months in day-inches and counting them in megalithic yards, appears to have been more than "Hey, let's count in months and use this megalithic yard we just made" as can be seen when this ratio is scutinised megalithically.

To compare two lengths that are proximate to each other the megalithic used right triangles and in Le Manio: How ancient maths manipulated factors the relationship between the lunar month and megalithic yard was seen as another integer ratio 29:32, shown again below.

 Figure 7.The 29:32 triangle as relating the lunar month to the megalithic yard

The slope angle of a right triangle defines a scaling so that anything on the base is increased, in this case, by 32/29. Placing 29.53059 on the base gives a hypotenuse of 32.625. In fact it gives slightly less, 32.585 which is the astronomical megalithic yard rather than the three year megalithic yard, a yard one would generate over nineteen years of the Metonic period which is a more accurate anniversary. In fact, the 1063.1 days of three lunar years (on the baseline of the Quadrilateral between points P and Q shown in figure 5.) is the product of 32.625 and 32.585, due to the properties of N;N+1 triangles.

The 29:32 scaling can be reapplied, to the megalithic yard, by arcing it down to the extended baseline of the triangle. The hypotenuse then becomes 36 inches long or three feet and the third of it is therefore one foot of twelve inches so that, the third of the hypotenuse will be the yearly excess if and when the megalithic yard is used to count lunar months instead of the the actual day-inch count of 29.53059 days. 36 inches times 29/32 is exactly 32.625, the three year megalithic yard. The three year count will automatically generate a 32.625 day-inch length that enabled the megalithic astronomers to see, using the geometry below, that counting months in those yards would produce a yearly excess of one foot (rather than 10.875 inches).

Figure 8

### Showing the Three Year period is very fortunate

It is the integer-part of the lunar month day-count and megalithic yard, which give the 29:32 ratio, whilst their integer parts are necessarily also 29:32 with respect to each other. Thus, 0.53059 times 32/29 equals 0.585478 In contrast, the English yard of 36 inches has no integer-part and so its interaction with 29/32 remains a purely integer affair. The megalithic yard generated over three years is therefore 36 times 29/32 = 0.9063 = 261/8 = 32.625 inches.

The 32.625 inch megalithic yard and the (actual) astronomical ratio of 32.585, when multiplied together, give the three lunar year day-count.  Where 32.625 x 32.585 = 1,063.0856 days. This factoring of a difference and a super-particular ratio is always the case for any period captured by means of a right angled triangle. Any normalised right triangle has a differential of one, being N:N+1. In the case of the lunar and solar years N equals 32.585, the astronomical megalithic yard and such a 32.585:33.585 triangle has the invariant slope angle of 14 degrees between the longest sides.

If we take the case of four years rather than three, four lunar years last 1417.468 days and four solar years 1460.9688 days, a difference of 43.5 days which, when divided into four solar years is reduced to the invariant 32.585, so that 1417.468 is 43.5 times 32.585, the difference times the invariant ratio. The remarkable (and helpful) thing about the three years period is that the difference is very close to the 29:32 relationship, being proximate as 29.53059 days times 32.625 (the three-year megalithic yard).

If the number of days within a lunar month and hence between lunar and solar years over three years had been numerically different due to a different day length or lunar orbit, then the lunar month would not have rested near the lower limit of 29 in the ratio 29:32 so as to project upwards to 32.585 in such a way that 36 counting units, in this case day-inches, would closely meet the downward projection from 36 inches to form the three-year megalithic yard. The megalithic transition of moving to counting months using the latter megalithic yards would not have yielded a one foot difference over one year and a yard of 36 inches over three.

We know that later "ancient" metrology adopted the foot as a standard measure equalling one for its ability to then do proportional mathematics of this kind until arithmetical methods emerged in the ancient near east, these then running alongside the older metrological tradition as often reserved for the sacred functions of architecture and geomancy (placing upon the earth). In this sense day-inch counting formed the cornerstone of metrology, and had the three year period not have been so numerically favourable, megalithic astronomy may not have developed human metrology as a pre-arithmetic mathematics.

The stone which the builders rejected,
the same is become the head of the corner:
this is the Lord's doing,
and it is marvellous in our eyes?Matthew 21:42