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Appendix 1—Mathematical and Astronomical Basis of Astrogeographia

David Bowden

According to the Ptolemaic conception for example, out there is the blue sphere, and on it a point (Figure X) - we should have to think of a polar point in the center of the Earth. Every point of the sphere would have its reflected point in the Earth's center.The stars, in effect, would be here (Figure Y). So that in thinking of the sphere concentrated in the center of the Earth, we should have to think of it in the following way:

The pole of this star is here, of this one here, and so on (Figure Y). We come then, to a complete mirroring of what is outside, in the interior of the Earth. Picturing this in regard to each individual planet, we have say Jupiter, and then a 'polar Jupiter' within the Earth. We come to something, which works outward from within the Earth in the way that Jupiter works in the Earth's environment. We arrive at a mirroring (in reality it is the opposite way round), but I will now describe it like this:

a mirroring of what is outside the Earth into the interior of the Earth.1

Figure X Figure Y The basic idea of Astrogeographia, that there is a relationship between the stars and particular locations on Earth, was known to Ptolemy in AD 150 and was probably also known to the ancient Babylonians. The description given above by Rudolf Steiner of the mirroring of each star outside the Earth to a specific point within the Earth provides the essential geometric idea needed to place the new science of Astrogeographia on a clear foundation. From this description a line can be imagined for each star - a line which joins the star to the center of the Earth. This line also passes through the star's mirror point inside the Earth and most importantly it passes through the Earth's surface at a particular location. In this way, each star is uniquely corresponded with its place on Earth.

1 See Rudolf Steiner, The Relationship of the Diverse Branches of Natural Science to Astronomy (Lecture 10).

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It could at first be thought that a single such correspondence between a star and its place on Earth would be enough to solve for a complete map of all the stars and their corresponding locations on Earth. Take for example the correspondence of the star Alnitak with Giza in Egypt. A transparent map of the stars could be overlaid on a map of the Earth and positioned by sliding it in latitude and longitude until Alnitak is lined up over Giza (see Figures 1A, 1B, 1C). This composite map could then be searched to see if it contained any new stellar-terrestrial correspondences. Although this approach is a good starting point for conceptualizing the basic idea of Astrogeographia, it has not in practice led to any new findings of interest.

Figure 1A. Mercator's map of the stars (Fiorenza) Figure 1B. Mercator's map of the Earth

Figure 1C. Map A overlaid on map B - Alnitak in the Orion constellation is lined up over Giza.

An alternative approach is to work with two independent alignments: a separate latitude alignment (or its equivalent in stellar coordinates, declination), as well as a separate longitude alignment. This second approach has been found to lead to numerous new correspondences and insights concerning the relationships between the stars and the Earth. As a result, this approach is the one adopted as the basis for the Astrogeographia model in this book. The separate declination and longitude alignments are described in detail below as the first and second reference alignments for Astrogeographia.

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First and second Astrogeographia reference alignments

The first Astrogeographia reference alignment aligns the 0° declination stars so that they are always directly over the 31N47 latitude line passing through Jerusalem (see Figures 2 and 3). The basis of this alignment is the identification of Jerusalem with "the Middle of the Earth" (see Chapter 9).

Figure 2. The 0° declination stars are those on the Celestial Equator, which is the projection of the Earth's equator outwards onto the sphere of the stars.

Figure 3. The first Astrogeographia reference alignment aligns the 0° declination stars directly over the 31N47 latitude line through Jerusalem - the Jerusalem horizontal axis.

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The second Astrogeographia reference alignment aligns the star Alnitak, in the belt of the constellation of Orion, so that it is always directly over the 31E08 longitude meridian passing through the Great Pyramid of Giza (see Figures 4). Because of the difference in sidereal longitude between Alnitak and Betelgeuse, this means that the star Betelgeuse will be always directly over the 35E13 longitude meridian, which passes through Jerusalem (see Figures 5 and 6).

Figure 4. The second Astrogeographia reference alignment aligns the star Alnitak directly over the 31E08 longitude meridian passing through the Great Pyramid of Giza.

Figure 5. The Jerusalem vertical axis is the 35E13 longitude meridian - the projection of the star Betelgeuse.

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Figure 6. The central importance of Jerusalem for Astrogeographia is indicated here by the exact meeting point of its significant horizontal and vertical axes.

Astrogeographia equations of latitude and longitude A. To find the star which corresponds to a given location on Earth: Equation of Declination

By the first reference alignment:

Δ(declination) = Δ(geographicLatitude) (where Δ means "a change in") declination(Star) - 0° = geographicLatitude(Place) - geographicLatitude(Jerusalem) declination(Star) = geographicLatitude(Place) - 31.77844° .......................................... (1) Golgotha-Jerusalem is identified with "the center of the world" (see Chapter 9). It is traditionally located at the Church of the Holy Sepulchre in Jerusalem at (latitude 31.77844°, longitude 35.22975°). Equation of Longitude

By the second reference alignment:

Δ(siderealLongitude) = Δ(geographicLongitude)

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siderealLongitude(Star) - siderealLongitude(Alnitak) = geographicLongitude(Place) - geographicLongitude(Giza)

siderealLongitude(Star) = geographicLongitude(Place)+siderealLongitude(Alnitak)-geographicLongitude(Giza) siderealLongitude(Star) = geographicLongitude(Place) + siderealLongitude(Alnitak) - 31.13436°

........ (2)

The sidereal longitude of Alnitak is close to constant at 59.93333°. Typically, its variation about this value is 0.07° over the period 3000 BC to AD 3000, being due entirely to the proper motion of the star. So to within 0.1% accuracy, the equation of longitude can be calculated as: siderealLongitude(Star) = geographicLongitude(Place) + 59.93333° - 31.13436°

siderealLongitude(Star) = geographicLongitude(Place) + 28.79897° ..................... (2A) When the declination and sidereal longitude have been calculated from equations (1) and (2), the star in question can be found by searching in an ephemeris that lists declination and sidereal longitude. For an ephemeris that lists sidereal latitude and sidereal longitude, the standard equations for converting from equatorial to ecliptic coordinates can be applied. B. To find the location on Earth which corresponds to a given star: Equation of Latitude From equation (1): geographicLatitude(Place) = declination(Star) + 31.77844° ............. (3) Equation of Longitude From equation (2): geographicLongitude(Place) = siderealLongitude(Star) - siderealLongitude(Alnitak) + 31.13436° ......... (4) Or to within 0.1% accuracy:

geographicLongitude(Place) = siderealLongitude(Star) - 28.79897° .............................. (4A) From Equation 4A, 0° Aries projects to the 28W48 geographic meridian, i.e. 28.79897° west of the 0° Greenwich meridian. The origin of the sidereal zodiac (0AR00, 0N00) projects to geographic (41N22, 28W48). This is close to Fayal Island, one of the main administrative centers of the Azores in the Atlantic Ocean. The 28W48 meridian also passes through the east coast of Greenland (see Figure 7).

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Figure 7. Origin of the sidereal zodiac 0° Aries is close to Fayal Island in the Azores on the 28W48 longitude meridian. This meridian also passes through the east coast of Greenland.

A computer program has been written to efficiently carry out the Astrogeographia equations of declination, latitude and longitude to an accuracy of about 0.5° over 5900 years. One degree corresponds to approximately 69 miles (111 kilometers) on the surface of the Earth. This program greatly facilitates ongoing research questions, because it can easily compute what must otherwise be done as many time consuming hand-calculations. From its output, pictures begin to build and insights are gained. When more precise research is suggested, it can be checked out by calculating with full accuracy star data from an ephemeris. For more details and availability of this program, see the section below, at the end of this appendix, entitled Computer Programs for Astrogeographical Research - ASTROGEO & SINEWAVE.

The meridian influence of a star in relation to its historical projection The primary influence of a star is called its meridian influence (see stellar meridians in Chapter 9). This influence acts along its entire geographic meridian at all times. An example of this meridian influence is with the star Betelgeuse, which had meridian influence on Hattusha, the seat of the 14th century BC Hittite empire. At a later time in 968 BC, Betelgeuse had meridian influence on Solomon's building of the Temple in Jerusalem. Both of these locations are on the same 35E13 meridian, which coincides with the longitude projection of the star Betelgeuse (see Figures 3 and 4). Another example is the meridian influence of Sirius, the brightest star in the night sky, on the region of ancient Persia and the work of Zarathustra there from about 6000 BC, in relation to the post-Atlantean cultural age of Persia, of which Zarathustra was the founder. Rudolf Steiner referred to Zarathustra as the founder of the Persian cultural epoch in the age of Gemini, some eight thousand years ago, and he described Sirius as the "heart of Jesus-Zarathustra".2

2 Words of Rudolf Steiner recorded in Rudolf Steiner, The Birth of a New Agriculture: Koberwitz 1924, p. 89.

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Due to the precession of the Earth, there is in every year a unique location on a star's longitude meridian where its meridian influence becomes especially heightened. At this place the star comes into simultaneous latitude and longitude alignment. This is called the star's historical projection for that given year. The increase in the star's influence at this location results from the resonance effect of the two simultaneous alignments. Over the 25920 year precession cycle, the historical projection of a star moves alternately north and south on the star's meridian (see the mid-declination section below). A heightened resonance effect relating to Aldebaran’s movement northward along its meridian is exemplified by its simultaneous latitude and longitude alignment with Northern Africa in 2450 BC (the time of the building of the Great Pyramid in Egypt); with southern Italy in 135 BC (the time of the Romans’ First Servile War occasioned by the revolt of slaves in Sicily); and with Vienna since the middle of the 18th century, when this city became a great cultural center for the arts, sciences, and music.

Example calculation 1

Apply the equations of declination and longitude to find the star that was, in 1371 BC, in latitude and longitude alignment with the Temple Mount in Jerusalem. This is located at (latitude 31.77797°, longitude 35.23581°). From ephemeris data for 1371 BC: sidereal longitude of Alnitak is 29TA52 = 59.86666°

From equation (1):

declination(Star) = geographicLatitude(Temple Mount) - 31.77844° = 31.77797° - 31.77844°

= - 0.00047 = - 0° 00' 02''

From equation (2):

siderealLongitude(Star) = geographicLongitude(Temple Mount)+siderealLongitude(Alnitak)-31.1333°

siderealLongitude(Star) = 35.23581° + 59.86666° - 31.13436° = 63.96811° = 3GE58 From ephemeris data for 1371 BC:

declination of Betelgeuse = -0° 00' 19'' = -0.00516°

sidereal longitude of Betelgeuse = 3GE57 = 63.95000°

Thus the 1371 BC coordinates of (-0° 00' 02'', 3GE58) identify the star as very close to Betelgeuse, at (-0° 00' 19'' , 3GE57). To check this, the projection of Betelgeuse is calculated from equations (3) and (4)

geographicLatitude = declination(Betelgeuse) + 31.77844°

= -0.00516° + 31.77844° = 31.77328° = 31N46 latitude

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geographicLongitude = siderealLongitude(Betelgeuse) - siderealLongitude(Alnitak) + 31.13436° = 63.95000° - 59.86666° + 31.13436° = 35.21770° = 35E13 longitude The great circle distance between this 1371 BC projection of Betelgeuse at (31.77328°, 35.21770°) and the Temple Mount at (31.77797°, 35.23581°) shows a very close alignment - the distance between the Temple and the projection of Betelgeuse in 1371 BC amounting to about 1.8 miles (3 kilometers) on the surface of the Earth. The founding of the Temple was in the fourth year of Solomon’s reign, around 968 BC (see Chapter 7), some 400 years after this very close alignment. Generally speaking, there is a time-lag between the sowing of the seed of a new spiritual impulse and the cultural flourishing that results from the sowing of the seed. An example of this is the period of 1199 years between the beginning of a new astrological age and the start of a new age of human cultural evolution, known as a cultural epoch (see Figure 13). Analogously, there was a time-lag of 403 years (1371 BC to 968 BC) between Betelgeuse’s alignment with Jerusalem and Solomon’s building of the Temple. For a more detailed study of the latitude journey of Betelgeuse alternately north and south on its geographic meridian, see Chapter 8.

Example calculation 2

Apply the equations of latitude and longitude to find the projection of the star Baham onto the Earth in AD 2000. This star forms part of the head of Pegasus. From ephemeris data for AD 2000:

declination of Baham: 6° 11' 52'' = 6.19778°

sidereal longitude of Baham: 12AQ06 = 312.10000°

sidereal longitude of Alnitak: 29TA56 = 59.93333° From equation (3): geographicLatitude = declination(Baham) + 31.77844° = 6.19778° + 31.77844° = 37.97622° = 37N59 latitude From equation (4):

geographicLongitude = siderealLongitude(Baham) - siderealLongitude(Alnitak) + 31.13436° = 312.10000° - 59.93333° + 31.13436° = 283.30103° = 76W42 longitude

These projected geographic coordinates of (37N59, 76W42) identify the vicinity of Washington DC, which is located at (38N54, 77W02).

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Declination and Geographic Latitude journey of a star over the precession cycle As an overall picture, a star's projection onto the Earth moves alternately north and south on its longitude meridian, as a result of the Earth's precession. From equation (4), a star's projected longitude varies only slightly over thousands of years, typically less than 0.1°, corresponding to 6.9 miles (11.1 kilometers) on the surface of the Earth. This slight variation is due to the proper motion of the particular star, but also to that of the star Alnitak, because it too is involved in the longitude calculations of equation (4). The proper motion of most stars is very small even over thousands of years, so the projected longitude remains remarkably constant. In contrast, the projected latitude of a star varies markedly on the surface of the Earth, as a result of the Earth gently rocking back and forth on its axis over the 25920 year precession cycle. This has the effect that the observed position (declination) of the star varies by + and - the axial tilt of the Earth, or between 23.5° north and 23.5° south of a certain mid-declination point. Mid-declination (or the midpoint of the star's declination journey) occurs when the star is in longitude alignment with the vernal equinox point (e.g. with 5PI16 in AD 2000). This movement of the observed position of a star in the heavens over a 25920 year precession cycle is called its declination journey. This journey returns to its starting point and begins anew at the end of the precession cycle. From equation (3), the projected geographic latitude of a star directly follows its declination, so it too increases and decreases about a certain mid-latitude point by the same 23.5° north and 23.5° south. This movement of the star's projected position on the surface of the Earth over a 25920 year precession cycle is called its latitude journey. Geographic mid-latitude occurs simultaneously with mid-declination of the star, i.e. when the star is in longitude alignment with the vernal point on the ecliptic (5PI16 in AD 2000). Mid-latitude occurs for a second time in the cycle when the star is in longitude alignment with the autumnal point on the ecliptic (5VI16 in AD 2000). Maximum geographic latitude occurs when the star is in longitude alignment with the summer solstice point (5GE16 in AD 2000); and minimum geographic latitude when the star is in longitude alignment with the winter solstice point (5SG16 in AD 2000). The total geographic latitude movement for a star is thus twice the axial tilt of the Earth, or 2 x 23.5° = 47°. This is equal to a distance of 3245 miles (5222 kilometers) on the surface of the Earth - about half the distance between the North Pole and the equator. To find the year in which the mid-declination of a star occurs (year0), apply the condition that the vernal equinox point comes into longitude alignment with the star.

sidereal longitude of the star at mid-declination = sidereal longitude of the vernal point = ( 220.78029 - year0 ) / 72

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The constant in this equation is based on 11th October AD 220 (near to the time of Ptolemy) as the date when the vernal equinox Sun aligned with the First Point of Aries, i.e. 0° Aries. From this equation and a knowledge of the star's sidereal longitude and declination in any given year (call these siderealLongitude1 and declination1 in year1), it is possible to calculate the mid-declination year, and the mid-declination from the sinewave formulas as follows: mid-declination year = year0 = 220.78029 - 72 x siderealLongitude1

mid-declination angle = declination1 - 23.49167° x sin(year1/72 - year0/72)

(where sin = mathematical sine function)

Following this calculation for year0, a more precise value for the mid-declination year can be obtained with the Astrofire astrosophy computer program by manually stepping through the years adjacent to year0 until exact equality is found between the star's sidereal longitude and the sidereal longitude of the vernal point.

In the next section, mid-declination leads to the concept of the VE meridian. The projection of a star at its mid-declination is called its archetypal projection, which is beyond the stream of time. The projection at any other time is called its historical projection, and this occurs within the stream of time.

Concept of the VE meridian in relation to birth horoscope of the world

In any given year, there is only one geographic meridian that is longitudinally aligned with the vernal equinox Sun in that year, and this is called the VE meridian. Its geographic longitude in any given year can be calculated by setting the sidereal longitude(star) = sidereal longitude(VE) in Equation (4A):

geographic longitude of the VE meridian = sidereal longitude(VE) - 28.79897° As an example, in AD 2000: sidereal longitude(VE) = 5PI16 = -24.73333° Thus the VE meridian = -24.73333° - 28.79897° = -53.53230 = geographic 53W32. The concept of the VE meridian is a vital key for understanding the birth horoscope of the world. In any given year, all geographical latitude locations along the VE meridian return momentarily to full alignment with their star of birth, as at the moment of the birth horoscope of the world. This is because all stars on the VE meridian pass simultaneously through their mid-declination positions in that year. Thus the historic projections of these stars are momentarily lifted out of time into their archetypal projections, i.e. the VE meridian in any given year is a picture of the archetypal projections of all of its stars.

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As the observed position of the vernal equinox precesses around the ecliptic over the 25920 year cycle, the VE meridian sweeps in longitude right around the globe, and a whole series of such pictures arises. This series can be assembled into a composite map that shows the archetypal projections for the whole globe. In this way, the birth horoscope of the world emerges (see Figure 8). Each place on Earth celebrates a return to its birth horoscope star twice every 25920 years. The first occurs when the meridian of the given place aligns with the vernal equinox. The second occurs when it aligns with the autumnal equinox. The map of the birth horoscope of the world can be imagined by rotating the axis of the Earth until its axial tilt is 0°. The equator now aligns with the ecliptic plane, so that the precession of the Earth is no longer a part of earthly existence. Every place on Earth is in constant latitude and longitude alignment with the star of its birth, as at the moment of the birth horoscope of the world. The declination of stars no longer continuously changes because there is no longer any precession. Instead the declination of each star is identical with its sidereal latitude. Because the sidereal latitude of stars changes only slightly over the millenia due to their very small proper motion, so every place on Earth now remains in close permanent latitude and longitude alignment with the star of its birth.3 From an astrogeographical perspective, this whole picture points strongly to the birth horoscope of the world being imprinted into the Earth when its axial tilt was 0°, i.e. at a time before the departure of the Moon from the Earth. This would have been a time before the seasons as we know them. This is hinted at in the Italian expression primavera - literally the first spring, in particular in relation to Botticelli's well known painting. From this point of view, the departure of the Moon is the main cause of the Earth's present axial tilt. Since that time, the effect of the axial tilt of the Earth has been for every place to return to full alignment with the star of its birth twice every 25920 years - a kind of cosmic breathing process. As discussed in Chapter 9 and elsewhere, the term thema mundi ("birth horoscope of the world") refers to the imprint of the celestial sphere into the earthly sphere that occurred at the time of the "birth of the world". As suggested above, this birth was some time before the departure of the Moon. In spiritual science the departure of the Moon is described as having occurred during the third epoch of the Earth, called Lemuria4. Also described is how the Moon, after first drawing nearer to the Earth, will fully reunite with it in about five to six thousand years from now5 . This can be understood as a result of human consciousness having become more closely

3 Over long periods of hundreds of thousands of years, for some stars their proper motion could amount to something

significant, possibly even changing the shapes of the constellations. Without an advanced computer program to investigate

this astronomically, this aspect of Astrogeographia has not yet been explored-yet we do see the importance of investigating

this when we have an appropriate computer program to do so. Therefore we have proceeded under the assumption that

the proper motion of stars does not amount to anything of significance in terms of the magnitude of the shifting of the

positions of the stars in latitude and longitude and in relation to one another. This assumption certainly holds for the

historical period that has been the main focus of our attention, since the shapes of the constellations are more or less the

same now as they were for the Sumerians seven thousand years ago.

4 Rudolf Steiner describes five ages of the Earth so far: Polarea, Hyperborea, Lemuria, Atlantis, and Post-Atlantis

– this being the present age. See Rudolf Steiner, Cosmic Memory, pp. 69-135.

5 Events in Lemuria become transformed or fulfilled in the present Post-Atlantean age. So the departure of the Moon

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aligned with the cosmic consciousness, and thus bringing about a re-alignment of the Earth's equator with the ecliptic plane. Even wider vistas of human evolution are possible, such as a re-alignment of the ecliptic plane with the galactic plane, occurring in some far future time when human beings are ready for this.

that was necessary for human evolution at that time will be paralleled by its reuniting with the Earth towards the end

of the present age. See Rudolf Steiner, Materialism and the Task of Anthroposophy, p. 263.

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Sinewave formulas for calculating the latitude journey of a star

This section continues from the earlier calculations for the mid-declination year (year0 ), and the mid-declination angle of a star. Imagine now a line drawn from a particular star to the center of the Earth. It will intersect the surface of the Earth at a certain latitude. The effect of the three-dimensional precessional rocking back and forth of the Earth is that this intersection point will move alternately north and south on its longitude meridian in such a way that a sinewave movement in latitude results. The maximum variation in latitude either side of the mid-latitude point is the amplitude of the sinewave. This is equal to the axial tilt angle of the Earth, about 23.5°. declination(year) = mid-declination + (axial tilt) x sin(year/72 - year0/72) declination(year) = mid-declination + 23.49167° x sin(year/72 - year0/72) ......... (5)

geographicLatitude(year) = declination(year) + 31.77844° geographicLatitude(year) = mid-declination + 23.49167° x sin(year/72 - year0/72) + 31.77844°

........ (6)

mid-geographicLatitude = mid-declination + 31.77844° both occurring in year0 .......... (7)

Maximum geographic latitude = mid-declination + 23.49167° + 31.77844° = mid-declination + 55.27011° The condition for the sine function to be a maximum is year/72 - year0/72 = 90°, or year = 6480 + year0 Minimum geographic latitude = mid-declination - 23.49167° + 31.77844° = mid-declination + 8.28677° The condition for the sine function to be a minimum is year/72 - year0/72 = -90°, or year = -6480 + year0 The range of geographic latitude (minimum to maximum) = 55.27011° - 8.28677° = 46.98334° = 2 x (axial tilt) = 2 x 23.49167°

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Figure 9. Sinewave shows vertical movement slower near the peak and valley, faster near middle of the wave. Sinewaves have the interesting property that the movement is slowest near both the peak and valley of the wave, and fastest near the middle part of the wave. This can be seen by observing the variation in the gradient of a sinewave function graph over a full cycle (see Figure 9). Thus the historical projection of a star moves with greatest speed when at its mid-declination, and with slowest (zero) speed when at its furthest north or furthest south. Compare this with the 2000 year non-linear timeline steps in Figure 12 showing the movement of Alnitak's projection across Africa. The sinewave formulas provide a very convenient and straightforward way to track the movement of a star over the millenia. Starting with the declination and sidereal longitude of a star in any given year, a once-only calculation is made for the mid-declination angle and also for the year in which this occurs (year0). Then the formulas are set up and ready for use. The declination journey and the latitude journey of the star can now be easily calculated over a very wide range of years of interest. The range of years is not limited to 25920 years, because the formulas automatically extend into adjacent precession cycles. The constant value of 23.49167° given in the sinewave formulas for the Earth's axial tilt to the ecliptic refers to the average axial tilt over its 41000 year Milankovitch cycle. This is chosen in order to simplify the calculations as much as possible. Of course, the actual axial tilt over this 41000 year cycle does not in fact remain constant . It varies by 0.86° about the average value. The maximum value of 24.35000° last occurred in 8700 BC. The average value was around AD 1550. The next minimum of 22.63333° will be in AD 11800. The AD 2000 axial tilt of 23.43929° is very close to the average value of 23.49167°.

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Figure 10. Luni-solar component of precession gives rise to the picture of the Earth as a spinning top.

The sinewave formulas take into account only the luni-solar aspect of precession. This is the main component of precession and is what gives rise to the picture of the Earth as a spinning top (see Figure 10). Use of this basic model of the precession means that the calculations are simplified as much as possible. Omitted are the many finer cosmic and terrestrial influences that modify the overall precession, for example the planetary components of precession, the nutation due to the Moon's orbital plane being tilted at about 5° to the ecliptic (see Figure 11) and movements occurring within the Earth itself. Even with the omission of these other factors, the sinewave formulas still manage to achieve an accuracy of several degrees over 50000 years.

Figure 11. Rotations of the Earth: P = Precession, R = Daily Rotation, N = Nutation. In summary, the sinewave formulas are very useful tools for astrogeographical research. They can fairly accurately calculate the declination and geographic latitude of stars from as far back as 25000 BC, and as far forwards as AD 25000. Thus it is possible to research historical periods back to early Atlantis (about 20000 BC), or even earlier to Lemuria6. However it is not valid to

6 See Rudolf Steiner, Ancient Myths, pp. 155-156.

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keep extending back indefinitely, because the assumptions upon which the luni-solar component of precession is based become invalid. Again if extending into the future, the assumptions become invalid when the Moon reunites with the Earth. As Astrogeographia continues to evolve, many new questions arise. A computer program has been written to efficiently carry out the calculation of the sinewave formulas to an accuracy of several degrees over 50000 years. One degree of error corresponds to approximately 69 miles (111 kilometers) on the surface of the Earth. This program greatly facilitates ongoing research questions, because it can easily compute what must otherwise be done as many time consuming hand-calculations. From its output, pictures begin to build and insights are gained. When more precise research is suggested, it can be checked out by calculating with full accuracy star data from an ephemeris. For example, improving the accuracy of the results is easy with the Astrofire astrosophy computer program, by manually stepping adjacent to the years indicated by the SINEWAVE program output. For more details and availability of this program, see the section Computer Programs for Astrogeographical Research - ASTROGEO & SINEWAVE.

Example calculation 3

Apply the sinewave formulas to calculate the movement in declination and geographic latitude over the 25920 year precession cycle for the historical projection of Aldebaran. This star has a declination of 16.50778° and a sidereal longitude of 45.04803° in AD 2000. declination1 = 16.50928° siderealLongitude1 = 45.05000° year1 = 2000 From AD 2000 ephemeris data: sidereal longitude of Alnitak = 29TA56 = 59.93333° geographicLongitude = siderealLongitude(Aldebaran) - siderealLongitude(Alnitak) + 31.13436° = 45.05000° - 59.93333° + 31.13436° = 16.25103° = 16E15 longitude Aldebaran will continue to track this 16E15 longitude meridian to a high degree of accuracy over the millenia. year0 = 220.78029 - 72 x siderealLongitude1 = 220.78029 - 72 x 45.05000 = -3023 = 3024 BC Mid-declination = declination1 - 23.49167° x sin(year1/72 - year0/72) = 16.50928° - 23.49167° x sin(2000/72 + 3023/72) = -5.53237° Mid geographic latitude = mid-declination + 31.77844° = -5.53237° + 31.77844° = 26.24607° = 26N15

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At mid-declination in 3024 BC, Aldebaran is in longitude alignment with the vernal point. Its projection to (26N15, 16E15) identifies the vicinity of Tmassah in Libya.

Maximum declination = mid-declination + 23.49167° = -5.53237° + 23.49167° = 17.95930° Maximum geographic latitude = maximum declination + 31.77844° = 17.95930° + 31.77844° = 49.73774° = 49N44

This maximum latitude or most northerly point at (49N44, 16E15) identifies the vicinity of Policka in the Czech Republic. The year for this will be 6480 + year0 = 6480 - 3023 = AD 3457. Or one cycle before in 3457 - 25920 = -22463 or 22464 BC.

Minimum declination = mid-declination - 23.49167° = -5.53237° - 23.49167° = -29.02404°

Minimum geographic latitude = minimum declination + 31.77844° = -29.02404° + 31.77844°

= 2.75440° = 2N45

This minimum or most southerly point of (2N45, 16E15) identifies the vicinity of Bayanga in the Central African Republic. The year for this was -6480 + year0 = -6480 - 3023 = -9503 or 9504 BC

Equation (5): declination(year) = mid-declination + 23.49167° x sin(year/72 - year0/72)

= -5.53237° + 23.49167 x sin(year/72 + 41.98611°)

Equation (6): geographicLatitude = declination(year) + 31.77844° With the sinewave formulas (5) and (6), both the star's declination journey, and the movement of its projected geographic latitude alternately north and south on the 16E15 longitude meridian, can be calculated for any year in the 25920 year precession cycle. Table 1 gives examples of the calculations for the epoch years of the twelve cultural ages.

year declination geographic latitude (declination + 31.77844°)

AD 5734 14.5° 46N15 AD 3574 18.0° 49N44 AD 1414 15.1° 46N55 747 BC 6.8° 38N34

2907 BC -4.9° 26N55 5067 BC -16.7° 15N05 7227 BC -25.5° 6N15 9387 BC -29.0° 2N46

11547 BC -26.2° 5N35 13707 BC -17.8° 13N56 15867 BC -6.2° 25N35 18027 BC 5.6° 37N25 20187 BC 14.5° 46N15

Table 1. Projection of Aldebaran - movements in declination and in geographic latitude on the 16E15 meridian.

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Aldebaran projects to within 30 miles (48 kilometers) of Vienna since the middle of the 18th century, during the time when this city became a great cultural center for the arts, sciences and music.

Example calculation 4

Apply the sinewave formulas to calculate and draw the timeline for the historical projection of Alnitak as it moves up the continent of Africa over a half precession cycle. From AD 2000 ephemeris data:

declination of Alnitak: -1° 56' 34'' = -1.94278°

sidereal longitude of Alnitak: 29TA56 = 59.93333°

declination1 = -1.94278° siderealLongitude1 = 59.93333° year1 = 2000 From equation (4):

geographicLongitude = siderealLongitude(Alnitak) - siderealLongitude(Alnitak) + 31.13436°

= 59.93333° - 59.93333° + 31.13436°

= 31.13436° = 31E08 longitude Alnitak will continue to track this 31E08 longitude meridian to a high degree of accuracy over the millenia.

year0 = 220.78029 - 72 x siderealLongitude1 = 220.78029 - 72 x 59.93333 = -4094 or 4095 BC mid-declination = declination1 - 23.49167° x sin(year1/72 - year0 /72) = -1.94278° - 23.49167° x sin(2000/72 + 56.86111°) = -25.33169°

Mid geographic latitude = mid-declination + 31.77844° = -25.33169° + 31.77844° = 6.44675° = 6N27

At the mid-declination date of 4095 BC, Alnitak is in longitude alignment with the vernal point. Its projection to (6N27, 31E08) identifies a location between Madbar and Bor in the far south of Sudan.

Equation (5): declination(year) = mid-declination + 23.49167° x sin(year/72 - year0/72) = -25.33169° + 23.49167 x sin(year/72 + 56.86111°)

Equation (6): geographicLatitude = declination(year) + 31.77844°

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The results of the calculations for particular years of interest in the 12960 half precession cycle are shown in Table 2, with the corresponding timeline projection across the continent of Africa is shown in Figure 12.

year

declination

geographic latitude

(declination + 31.77844°)

Alnitak

AD 2386 -1.8° 29N56 - closest to Giza, within 3.7 miles (6 kilometers) - most northerly point, turns to move south again after this - vernal equinox Sun enters Aquarius in AD 2376

AD 33

-5.6°

26N13

- vicinity of Thebes (Luxor)

2450 BC -16.2° 15N34 - building of the Great Pyramid of Giza - 93 miles (150 kilometers) west of Khartoum in Sudan

2907 BC -18.7° 13N07 - beginning of Egyptian cultural age (beginning of Kali Yuga was 3102 BC) - 186 miles (300 kilometers) south of Khartoum in Sudan

5067 BC -30.8° 0N57 - beginning of the Persian cultural age - source of the Nile (meeting of Blue Nile & Lake Victoria)

7227 BC

-41.5°

9S44

- beginning of the Indian cultural age - Northern Zambia

10574 BC

-48.8° 17S03

- most southerly point, turned to move north after this - the "First Time" of the ancient Egyptians (about 3300 years before the end of Atlantis) - 62 miles (100 kilometers) north of Harare in Zimbabwe - vernal equinox Sun enters Leo in 10,764 BC

Table 2. Historical projection of Alnitak as a timeline of its movement up the 31E08 Nilotic meridian of Africa.

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Figure 12. Alnitak's projection traces out a non-linear timeline on the 31E08 meridian of Africa. The 2000 year steps show slower movement near the top and bottom, and faster near the middle.

Why was the Great Pyramid built at Giza ?

No satisfying answer to this question seems to come from archaeology or archaeoastronomy. If it is only a matter of the Pyramid's shafts being aligned to certain stars, then the Pyramid does not need to be located at Giza for this. To shed some light on this unanswered riddle we can consider the declination journey of Alnitak as described in the previous section. According to most researchers the Great Pyramid was built around 2450 BC. Much earlier in 10574 BC (during the last millenia of Atlantis) Alnitak on its journey had reached its most southerly declination of -48.8°, as a result of the precessional movement of the Earth. It then turned to move north again, and has since been rising upwards on the 31E08 meridian of Africa (see Table 2 in Example calculation 4). The beginning of the rising of Alnitak in 10574 BC can be taken as what the ancient Egyptians called the First Time. It was then that Orion, a constellation of great significance for them, was at its lowest point in the southern sky, and seen for only a few hours each night.

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From the First Time onwards, the precession of the equinoxes has gradually carried Orion (Osiris) and his consort Sirius (Isis) higher and higher in the sky - in effect a return and resurrection of Osiris. The ancient Egyptians oriented themselves towards the south as the direction of Upper Egypt and the source of the Nile. And the south was also the direction from which they observed Orion and Sirius ascending in the sky over the centuries. Later in the Egyptian cultural age, the resurrection of Osiris was experienced in the annual cycle as well, as the much awaited heliacal dawn rising (rebirth) of Orion and Sirius from their 70 day annual absence (death) in the night sky. The ancient Egyptians greatly welcomed this yearly return as the sign of the imminent flooding of the Nile and the return of abundant life. The carrying of Orion and Sirius higher in the sky by the precessional movement of the Earth continues into modern times, and is yet to culminate in its full return in AD 2386. It is then that Alnitak will reach its most northerly point in the sky, with a declination of -1.8°. From an astrogeographical perspective, it is remarkable that as the northwards journey of Orion and Sirius comes to its end in AD 2386, the projection of Alnitak simultaneously comes into closest alignment with the Great Pyramid, within 3.1 miles (5 kilometers), after a journey of 3245 miles (5222 kilometers). This occurrence in AD 2386 is very close to AD 2375, when the vernal equinox Sun leaves the constellation of Pisces and enters the constellation of Aquarius, marking the beginning of the astrological age of Aquarius (see Figure 13). Perhaps these coincidences can bring some new understanding to the riddle of why the Great Pyramid was built at Giza. The riddle could be rephrased as to why the Pyramid was built exactly where the projection of Alnitak comes into full latitude and longitude alignment with it, and just at the time of the beginning of the astrological age Aquarius.

Figure 13. The astrological ages based on the precession of the equinoxes. The cultural ages follow 1199 years later7.

Rudolf Steiner described how events that occurred during the Egyptian cultural age of Taurus8 reappear transformed as their fruit or fulfillment9 in the cultural age of Pisces. One hypothesis for this riddle could thus be that the Pyramid was built to convey a message (as fruit of the Egyptian

7 See Robert Powell, Hermetic Astrology, vol. I: Astrology and Reincarnation, p. 63.

8 Rudolf Steiner describes the first five cultural ages of Post-Atlantis: India, Persia, Egypt, Greece-Rome, and Europe

as under the signs of Cancer, Gemini, Taurus, Aries and Pisces respectively. See Rudolf Steiner, Ancient Myths, pp. 80-87.

9 See Rudolf Steiner, Egyptian Myths and Mysteries, pp. 13-14, 20. "Glancing at the immediate implications of our theme,

we see a large domain. We see the gigantic Pyramids, the enigmatic Sphinx. The souls that belonged to the ancient Indians

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age) across the millenia to the humanity of AD 2386 in the cultural age of Pisces (our own age). This message will proclaim to Piscean humanity that with the full return of Orion and Sirius to their most northerly points in the sky, the great evolutionary work of the Egyptian age of transforming the astral body into sentient soul has come to a certain fulfillment (see the legend of the Golden Fleece in Chapter 8). Then following the entry of the vernal equinox Sun into Aquarius there will be a period of 1199 years, which is needed for the seed of a new spiritual impulse to come into cultural flourishing. After this gestation period will come the Russian cultural age of Aquarius under the guidance of Sophia. It is during this age that the evolutionary work begun by the Egyptians will continue with the further transformation of sentient soul into spirit self (see Chapter 8). A related riddle regarding the Great Pyramid is the question of why it was built next to the Sphinx, which according to some researchers predates it by at least 6000 years. The Sphinx is the largest monolithic sculpture in the world and has a lion's body and a human head. One hypothesis for this is that the Sphinx was built close to 10586 BC when the vernal equinox Sun entered Leo the Lion. This time was very close to the First Time of the ancient Egyptians of 10574 BC mentioned above, when Orion and Sirius began rising from their lowest point in the night sky. Thus the picture arises of the building of the Lion-Sphinx in 10586 BC at the beginning of the astrological age of Leo to commemorate the First Time. And then the later building of the Great Pyramid next to the Sphinx in 2450 BC to prefigure what could be called the Last Time. This will be the time of AD 2386 near the beginning of the astrological age of Aquarius (as described above), when Alnitak comes into full alignment with the Great Pyramid and, at the same time, Orion and Sirius rise to their most northerly points in the sky. Then will be proclaimed from the heights the fruit and fulfillment of the great evolutionary work of the Egyptian age.

How close is the projected ecliptic to an exact mathematical circle ?

This can be determined by choosing any three points, 120° apart, on the projected ecliptic. These three points define a unique plane in three-dimensional space. This plane can be taken as a representation of the projected ecliptic.

Ecliptic projected ecliptic

sidereal latitude

Sidereal Longitude

geographic latitude

geographic longitude

0N00 0AR00 41N22 28W48

0N00 0LE00 45N03 91E12

0N00 0SG00 8N27 148W48

Table 3. A plane representing the projected ecliptic - defined by three points 120° apart.

were also incarnated in Egypt and are again incarnated today. If we follow our general line of thought into detail, we will

discover two phenomena that show us how, in super-earthly connections, there are mysterious threads between the

Egyptian culture and that of today."

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For example, if the three points 0AR00, 0LE00 and 0SG00 are chosen (as in Table 3), the equation of the plane is:

0.58727 x + 0.45123 y - 1.85071 z + 1 = 0 In this equation, the radius of the Earth is scaled to 1, and: - positive x-axis through (0° latitude, 0° longitude), i.e. Greenwich meridian meeting with the equator - positive y-axis through (0° latitude, 90° longitude), i.e. 90° meridian meeting with the equator - positive z-axis through (90° latitude, 0° longitude), i.e. the North pole

The other nine zodiac reference points on the ecliptic can now be checked to see if they lie in this plane. If this is found to be so, then the projected ecliptic is a perfect plane circle, i.e. the circle that results from the intersection of the plane with the sphere of the Earth. As shown in Table 4, when the other nine points are calculated it is found that they do not exactly lie in the representative plane. They deviate in latitude from the exact plane by as much as 3.3°. It can be concluded from this that the projected ecliptic, although very close to a circle, is not a perfect plane circle. It is a circle-like curve embedded in the surface of the Earth. Closer study of Table 4 reveals the presence of a gentle peak (3.3°) and a gentle valley (-1.3°) in the otherwise perfect circle of the projected ecliptic.

ecliptic projected ecliptic exact plane projected ecliptic

sidereal latitude

sidereal longitude

geographic latitude

geographic longitude

geographic latitude

deviations in latitude

from an exact plane

0N00 0AR00 41N22 28W48 41N22 0.0° 0N00 0TA00 50N44 1E12 48N49 1.9° 0N00 0GE00 55N07 31E12 51N50 3.3° 0N00 0CN00 52N57 61E12 50N37 2.4° 0N00 0LE00 45N03 91E12 45N04 0.0° 0N00 0VI00 33N52 121E12 35N12 -1.3° 0N00 0LI00 22N12 151E12 23N07 -0.9° 0N00 0SC00 12N50 178W48 13N05 -0.3° 0N00 0SG00 8N27 148W48 8N27 0.0° 0N00 0CP00 10N36 118W48 10N21 0.3° 0N00 0AQ00 18N30 88W48 18N23 0.1° 0N00 0P I00 29N41 58W48 30N08 -0.4°

Table 4. The projected ecliptic - showing deviations in latitude from an exact plane. So a clearer mathematical description of the projected ecliptic is a circle-like curve with 3.3° peak of undulation. The undulations arise when the sidereal latitude of a star is converted into declination, by application of the standard astronomical equations. As these equations are based

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on rotating the plane of the ecliptic into the plane of the equator, the results show larger changes in the converted angle for stars near the summer and winter solstice points of the ecliptic (5GE16 and 5SG16 in AD 2000). And smaller changes for stars near the autumnal and vernal points of the ecliptic (5VI16 and 5PI16 in AD 2000). Figures 15 and 16 show the projected ecliptic within the wider family of constant sidereal latitude contour lines from -70° to +80°. As described in Chapter 8, the projected ecliptic is tilted towards the plane of equator at the same angle as the Earth's axial tilt of 23.5°. The projected ecliptic rotates as a whole around the Earth in a westwards direction every 25920 years, or through 30° (the width of one constellation) in 2160 years. This can be understood by considering firstly the summer solstice point on the ecliptic, which projects to the most northerly point. This rotates slowly westwards right around the Earth, which corresponds to the movement of the summer solstice point in reverse order through the zodiac constellations. It is important to note that the stars, as fixed points in the heavens, cannot when projected rotate around the Earth, because they remain during the whole precession cycle fixed on their particular geographic longitude meridian, apart from very small shifts in this meridian due to their proper motion. Rather it is a moving point in the heavens, (the moving summer solstice point) which moves right around the ecliptic, and under conditions that result in it always projecting to the most northerly point on the projected ecliptic - it is the projection of this moving point in the heavens that is found to be rotating around the Earth.

Secondly, consider that the observed positions of the vernal equinox, winter solstice and autumnal equinox likewise move around the ecliptic in the same constant latitude way as the summer solstice. They track each other 90° apart on the ecliptic and move as a whole progressively backwards through the stars, traversing the zodiac constellations in reverse order (see Figure 8 in Chapter 8). These four cardinal points on the ecliptic project to the Earth as four points on the projected ecliptic, 90° apart in longitude: the most northerly point (summer solstice); the most southerly point (winter solstice); and in between, the two intersections of the projected ecliptic with the 31N47 latitude line (the vernal and autumnal equinoxes). Each of these four points moves westwards all the way around the Earth in the same constant latitude way, whilst keeping a constant 90° spacing in geographic longitude. In fact, a similar analysis to the above applies to every point on the ecliptic, thus the projected ecliptic rotates as a whole westwards around the Earth in the same constant latitude way.

Constant sidereal longitude and latitude lines

The family of constant sidereal longitude lines

The family of constant sidereal longitude meridians can be found by applying the equation of longitude to the sidereal meridians passing through the initial points of each of the zodiac constellations: Aries (0AR00), Taurus (0TA00), Gemini (0GE00), and so on (see Table 5 and Figure 14). Because the proper motion of stars over long periods of time is very small, the constant sidereal longitude meridians remain close to constant across the millenia.

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zodiacal sidereal

longitude

sidereal longitude

angle

geographic longitude

0AR00 0° 28W48 0TA00 30° 1E12 0GE00 60° 31E12 0CN00 90° 61E12 0LE00 120° 91E12 0VI00 150° 121E12 0LI00 180° 151E12 0SC00 210° 178W48 0SG00 240° 148W48 0CP00 270° 118W48 0AQ00 300° 88W48 0PI00 330° 58W48

Table 5. Constant sidereal longitude meridians and their corresponding geographic longitude meridians.

Figure 14. The family of constant sidereal longitude lines 0° Aries, 0° Taurus, 0° Gemini, etc. 0° Aries aligns with the 28W48 geographic meridian, i.e. 28.8° west of the 0° Greenwich meridian.

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The family of constant sidereal latitude lines The family of constant sidereal latitude circles can be found by applying the equation of latitude to each sidereal latitude, in steps of say 10° from 70°S to 80°N, including the projected ecliptic 0°N (see Table 6 and Figures 15 to 16). This family is analogous to the geographic latitude circles on maps of the Earth. Each member of this family has the same basic form as described for the projected ecliptic in the previous section, i.e. they are undulating curves on the surface of the Earth. However, as evident in Figures 15 and 16, the further north or south that a constant sidereal latitude line is from the projected ecliptic, the greater will be its peak-to-valley undulation (deviation in latitude from an exact plane). For example, Table 6 shows that the 35° constant sidereal latitude line has a deviation in latitude of 10.1°, i.e. an undulation three times larger than on the ecliptic. The year to year movement of the peaks and valleys in any of the constant sidereal latitude lines over the course of the Earth's precession cycle is the same as described above for the peaks and valleys in the projected ecliptic. Again, the speed of rotation takes any peak or valley fully around the Earth every 25920 years, or through 30° (the width of one constellation) every 2160 years.

sidereal latitude

contour line

maximum deviation in latitude from an exact plane

40° 10.1°

20° 6.7°

0° ecliptic 3.3°

-20° 1.5°

-40° 4.7°

-60° 10.3°

Table 6. Constant sidereal latitude contour lines - showing maximum deviation in latitude from an exact plane.

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Figure 15. The family of constant sidereal latitude lines (AD 2000). The summer solstice or most northerly point of the projected ecliptic, is at (55N13, 36E28) which is 56 miles (90 kilometers) south-west of Moscow.

Figure 16. The family of constant sidereal latitude lines (AD 2000) - viewed from the North Pole. Longitudes on ecliptic: SS = summer solstice, WS = winter solstice,VE and AE = autumnal and vernal equinoxes

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The apparent Ecliptic North Pole and Ecliptic South Pole

Viewed from directly above, the projected ecliptic is seen as a complete circle (see the view in Figure 3 of Chapter 10). The center of this circle is aligned over geographic (66N34, 144W47), 145 miles (234 kilometers) northeast of Fairbanks, Alaska. This point can be thought of as the apparent Ecliptic North Pole. It projects to sidereal (58N13, 4SG01), in the constellation of Hercules. The apparent Ecliptic South Pole is the exact opposite point on the celestial sphere at (58S13, 4GE01). This projects to (3S00, 35E13), located on the Jerusalem meridian in the Serengeti National Park, Tanzania, about 160 miles (257 kilometers) south-west of Nairobi (see Figure 4 in Chapter 10).

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Computer Programs for Astrogeographical Research - ASTROGEO & SINEWAVE

These two programs are intended to be used as astrogeographical research tools in conjunction with the book Astrogeographia - Correspondences between the Stars and Earthly Locations - A Bible of Astrology and Earth Chakras by Robert Powell and David Bowden.

Finding the answers to Astrogeographia research questions is greatly facilitated by these programs, because it can easily compute what must otherwise be done as many time consuming hand-calculations. From their output, pictures begin to build and insights are gained. When more precise research is suggested, it can be checked out by calculating with full accuracy star data from an ephemeris.

The two programs are available from the Sophia Foundation-follow the links from the homepage of the Sophia Foundation: www.sophiafoundation.org > Astrosophy > Star Wisdom > Astrogeo and Sinewave CDs available for purchase. Please note that these two programs run under Windows (32-bit), but not Windows (64-bit). 1. ASTROGEO program

This program calculates a star's projection onto the Earth, or the reverse calculation of which star corresponds to a given place on Earth. It is based on the Astrogeographia equations of declination, latitude and longitude, as described in the book. It can calculate these equations for any year in the range from 2950 BC to AD 2950. Accuracy over the 5900 years is about 0.5°, which corresponds to 35 miles (56 kilometers) on the surface of the Earth. There are four possible inputs that can be chosen from a menu:

1. Choose from a list of 35 stars (for AD 2000) 2. Enter ephemeris star data as sidereal latitude & sidereal longitude, e.g. (25S18, 29TA56) for Alnitak 3. Enter ephemeris star data as declination & right ascension, e.g. (-1° 56' 34'', 5:40:45.5) for Alnitak 4. Enter place data as geographic latitude & longitude, e.g. (29N59, 31E08) for Giza

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Three examples of output from the ASTROGEO program : 1. When (25S18, 29TA56) are entered for Alnitak's sidereal latitude & sidereal longitude, the program calculates the other two, which in this case are declination & right ascension, and geographic latitude & longitude.

2. When (-1° 56' 34'', 5:40:45.5) are entered for Alnitak's declination & right ascension, the program calculates the other two, which in this case are sidereal latitude & longitude, and geographic latitude & longitude.

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3. When (29N59, 31E08) are entered for Giza's geographic latitude & longitude, the program calculates the other two, which in this case are declination & right ascension, and sidereal latitude & longitude.

2. SINEWAVE program

This program calculates a star's declination journey and also its latitude journey on the surface of the Earth over and beyond the 25920 years of the Earth's precession cycle. It is based on the sinewave formulas described in the book, which can be calculated for any year from 25000 BC (or earlier) to AD 25000. Accuracy is to within several degrees over the 50000 years. One degree of error corresponds to approximately 69 miles (111 kilometers) on the surface of the Earth. Improving on the accuracy of the results is easy with the Astrofire astrosophy computer program by manually stepping adjacent to the years indicated by the SINEWAVE program output. There are four possible inputs that can be chosen from a menu:

1. Choose from a list of 35 stars 2. Enter ephemeris star data as sidereal latitude & sidereal longitude, e.g. (25S18, 29TA56) for Alnitak 3. Enter ephemeris star data as declination & right ascension, e.g. (-1° 56' 34'', 5:40:45.5) for Alnitak 4. Enter ephemeris star data as declination & sidereal longitude, e.g. (-1° 56' 34'', 29TA56) for Alnitak

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For the star data entered, the program calculates all necessary parameters for the sinewave formulas, and then outputs the results as a range of declinations, together with the corresponding geographic latitudes and the years in which they occur.

An example of output from the SINEWAVE program

When the star Betelgeuse is chosen as No. 15 from the list of 35 stars, the program calculates the projected geographic coordinates of Betelgeuse, for its most northerly declination, most southerly declination, its mid-declination, and the declination for the specific year of enquiry. In this case, -1370 or 1371 BC was entered.

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A timeline for the declination journey and the geographic latitude journey of Betelgeuse is then displayed for approximately 17000 years either side of the mid-declination point. This provides an overall picture of the movement of the star's projection over and beyond the 25920 years of the Earth's precession cycle.

Following this timeline, special questions can be researched for any year in the range from 25000 BC to AD 25000, e.g. the timeline can be changed to zoom in on the movement of the star's projection in 5 year steps about Betelgeuse's most northerly point (39N11, 35E13), which occurs in AD 2091.