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Timing the Circle

The Development of Solar Arc Theory

The Circle. How does this shape, this spatial form, enter consciousness; how does it take on meaning; how does it come to be divided into 360 units; and how is it used as a measurement standard to capture the time of our lives?

Initially, man must have learned to identify the circle from the pupils of maternal eyes, and then from reading the eyes of others. Above, overseeing all, there were the moving disks of the godly Sun and the Moon. There were the intricate centers of so many flowers. Even when worked by a stick into the sand or earth, the circle had uniqueness among forms: with a cross, there was the sense of division (the beginning of the concept of 2); but with the fashioning of a circle, there was the sense of unity, of wholeness, of inviolable symmetry.

We can speculate that, in development, man then began to grasp the concept of magnitude: there were different sizes of trees, rocks, animals, squares, and circles. Thought-tools were developed to express the relationships between things in terms of size, to capture comparisons and set standards. (This phase of evolution is often cited as the dawn of mathematics.)

With the development of language, thoughts became communication. The sense of possession—what defines you and what defines me—was linked with the relativity of magnitude. Counting systems and measurements were then required to describe things accurately, to define location and property, to plan building. And through continuous long-time measurement experimentation, rules emerged—certain measurement practices and relationships that were always valid. The square was understood: all sides were equal! Then, the observation that a square divided by a line drawn diagonally between opposite corners yielded two identical triangles led to understanding triangles; then to the rectangle and other polygons (configured by squares and triangles, a geometry accomplished through subdivision and rearrangement), but with the circle, rules were harder to discover.

The observations about the circle that must have been dominant are, first, that there is no beginning or end to the circle (the concept of constant development, of eternal continuity) and, second, that the wider the circle is, the longer the distance is around it (the concept of containment, of boundary).1

Mathematician-engineer Petr Beckmann gives us an extremely clear suggestion of the development of the properties of the circle, as shown on page 3.

So we find a fairly flat patch of wet sand along the Nile, drive in a stake, attach a piece of rope to it by loop and knot, tie the other end to another stake with a sharp point, and keeping the rope taut, we draw a circle in the sand. We pull out the central stake, leaving a hole O (see drawing below). Now we take a longer piece of rope, choose any point A on the circle and stretch the rope from A across the hole O until it intersects the circle at B. We mark the length AB on the rope (with charcoal); this is the diameter of the circle and our unit of length. Now we take the rope and lay it into the circular groove (inscribed) in the sand, starting at A. The charcoal mark is at C; we have laid off the diameter along the circumference once. Then we lay it off a second time from C to D, and a third time from D to E, so that the diameter goes into the circumference three (plus a little bit) times.2

We discover that the circumference of the circle (the distance around its boundary) is equal to 3 times the diameter, plus a little bit. Experimentation shows that this is always the case no matter what the diameter of the circle is. This coefficient (characteristic relationship) is called a constant, and for the circle it is labeled Pi, shown by the Greek letter .3

Research shows that, some 5,000 years ago, the coefficient of the circle was known in these close terms, in terms accurate enough to allow reliable measurement of the circle: the circumference of a circle was determined by 3+ times the diameter (C=Pi D or, more customarily, C=2 Pi r, where r is the radius, one-half the diameter of the circle).4

The Egyptians and Babylonians pursed the “+”, that little bit over 3, the distance EA in the drawing on page 3. We presume that they tried to define EA in terms of the diameter AB, as a fractional part of that key unit distance. If we mark EA on a rope and lay it off as many times as it will go on (into) the diameter distance AB of any circle, it will go down between 7 and 8 times, i.e., EA is something between 1/7 (0.142857) and 1/8 (0.125) of the unit distance AB. The modern study of Pi does not get much closer; the decimals never end. The basic standard coefficient now used is 3.14159265 (i.e., 3 + .1416).

Learning to calculate the area of a circle (the space within the circumference) was another problem, a complex one indeed.

Determining the area of a square was easy: subdividing the space bounded by the perimeter gave us a measurement of the space in terms of one side multiplied by the other side, a x b (see diagram, p. 5). With two units per side (2 feet, for example), we get an area of 4 square feet. When the square is increased to a rectangle (oblong), the same rule applies: 3 units on one side and 2 units on the other side gives us an area of six “square” units.

When we subdivide a rectangle or square by a diagonal, we get two triangles, i.e., the area of the triangle in this case is measured as one-half the product of the two sides of the imagined rectangle, ab/2.

The ancients knew that the square had sides that related to each other by a certain measure, 90 units (degrees). They learned that an equilateral triangle (sides of equal length) had the sides meeting each other at 60 units (degrees). In other words, a theorem emerged that the sum of the three angles of a triangle always equaled 180 units (degrees).

But with the circle, there were no sides, there was no beginning or end, no edge, no angle. We postulate that the measurement of the area of a circle was arrived at through increasingly more sophisticated filling-up of the space within the circle, filling the space up with figures whose area could be known. You can almost fill up a circle with a system of equilateral triangles, as we shall see in a moment.

The Babylonians (seeded by the preceding Sumerian culture, in the third millennium b.c.) knew area computation, and it is to them that credit is given for the initial, approximate determination of the area of the circle.5 It is this method of measurement for the area of a circle to which we can trace the origin of circular subdivision into 360 units—those units we now call degrees.

Now: the Sumerian and Babylonian number system was sexagesimal in orientation (based upon 6; ours is decimal, based on 10).6 The three 60-unit angles of the equilateral triangle fit into a neat conceptualization that did indeed come close to defining the area of the circle. Here’s how:

If we inscribe six equilateral triangles within any circle—using up almost all of the circle’s area with areas we can measure—using the circle’s radius as the base of the triangle(s), a hexagon (six-sided figure) is formed, and six 60-unit angles define the center of the figure. Six times 60 gives 360 units. We call them degrees.

The area of the circle became A=Pi r2.7

Indeed, when we use the initial displacement arrangement of the hexagon (six equilateral triangles), there is space left over, as there was with the discovery and derivation of Pi. With exacting Greek thought that followed in time historically, the inscription of more sophisticated polygons within the circle led to more and more precise measurement.8

The Circle and the Year

Throughout centuries, very few cultural constructs can compete with the calendar for complexity and confusion. Yet, we do know for sure that the Moon was the first time regulator—well before the Sun became the central reference for time-keeping. There are two clear reasons for this: first, the Moon was easier to see and study than the Sun and, second, the Moon moved much more quickly than the Sun did against the backdrop of the heavens or in relation to man-made or natural sentinel-markers on earth. The Moon at first glance was a reliable clock: roughly every 29.5 days it passed through its cycle of clear phases, corresponding roughly, over time, to the seasons. The Moon was the beginning of the concept of a year, and for these reasons and others, nearly every culture worshiped the Moon.

The Babylonians observed a lunar year. It is easy and comfortable to see the lunar month as 30 days with twelve cycles (months) coinciding approximately with the movement of the Sun in the same time span. It fit the scheme of things to have 12 months x 30 days to give 360 days within the magic of the 360-unit circle.9

No one really knew what time it was or what day it was until perhaps the late fourteenth century! While there were genius minds working on the problem in England, in Europe, and in India (and undoubtedly the Orient) reforms in the West were held up by the coexistence of two truths: one truth was the truth of the Church, the proclamation of Rome (and so very much hinged on the determination of Easter Sunday, for example), and the other truth was that suggested by nature and mathematical reasoning.

If a scholar espoused the natural, reasoned truth, he would be going against the word of God! For example, the extraordinarily learned and inspired English Franciscan friar Roger Bacon, in the middle of the thirteenth century, sent an urgent appeal to Pope Clement IV to set right time itself. Calculating that the calendar year was some 11 minutes longer than the actual solar year, Bacon informed the supreme pontiff that this amounted to an error of an entire day every 125 years, and that this surplus had already accumulated over the centuries to a measure of nine days. Left this way, the calendar would soon bring March into the dead of winter! Bacon was imprisoned for 14 years.10

But, for our purposes here, to introduce and study the astrological system of Solar Arcs, the key fact for us is that, from the very beginning of systematic observation of the heavens, the concept of a day could be equated with the concept of one degree—extending to the structure of the lunar year—and this equation fit well, mathematically and philosophically, into the archetype of unity, continuity, development, duration, and wholeness—the archetype of the circle.

These enduring Babylonian concepts established astrology’s entire base structure: we have the circle of 360 degrees and we have the zodiac consisting of twelve signs of 30 degrees each (these inventions dated to the early first millennium, b.c.). The zodiac of the Babylonians was created in relation to the fixed stars (a star in Libra would always be in Libra). The Hindus, having learned astrology from the Babylonians and from the earliest Alexandrine astrologers, still use this Babylonian “fixed zodiac.” (The Western world uses the tropical zodiac that was invented by the Greek Hipparchus, in the second century b.c. This zodiac is measured from the vernal equinox and, since this point moves backward through the constellations [1 degree every 72 years], the tropical longitudes of the fixed stars advance at the same rate, completing one sign in 2,160 years.11)

Babylonian astrology, as far as we can tell, dealt with mundane/horary questions: Will there be war? Will it rain next month? Is the king’s reign secure? There were tables for time conversions and international adjustments from one system to another; there were calculated short-range ephemerides. But we have yet to find from that era any diagram of astrological positions. We assume that the Babylonian astrologer looked up the day in his tables and pronounced his reading from there; he could well have made a note on a slab or lump of clay, given it to his client or kept it in his files (these records have been unearthed). The first evidence for personal horoscopes can be dated to about 410 b.c.; the clients had Greek names.12

The earliest Greek charts of planetary positions are simple circles, sometimes with a cross within to denote the angular Houses. It is interesting to note that East was to the left (and South at the top—as we have it today) because this was the way Egyptians drew their geographical maps.13

Scholar James Herschel Holden suggests that these earliest Greek astrol-ogers were content to list only the sign positions of the planets and the ascending sign. “With this information it was easy to visualize the house placements of the planets, so a chart was actually unnecessary. Most likely the use of a chart only became customary when astrologers had abandoned the simple Sign-House (Fixed Equal House) system of house division and adopted one of the later systems that divided the houses irregularly.”

Late into the first century a.d., horoscopes started to emerge drawn in rectangular format. We do not know what happened to the circle.

Also late in the second century, the Egyptian (Alexandrine) geographer, mathematician, astrologer, early physicist Claudius Ptolemy revived strongly the Babylonian concept of “one degree equals one day.” He devised a concept of arcs keyed to the earth’s equator that he thought could extend the natal horoscope into the future. Ptolemy equated the time it takes for one degree to pass the Midheaven—four minutes of sidereal time or 3.9890 minutes of mean solar time—with a year of life. In other words, the first four minutes after birth would symbolize the first year of life; the next four minutes, the second year of life, etc.

However, the problem then became that only six hours of time were needed to form arcs throughout the horoscope to cover development for 90 years of life (4 minutes per year, 60 minutes for 15 years, 3 hours for 45 years, etc.)! The planets and points in the horoscope simply did not move enough symbolically to give astrologers enough to work with developmentally and interpretively. Just two hours of time for the average life span of 30 years at that time! This system was philosophically exacting in concept, computationally tedious in operation, and narrow in results.14

I have always wondered what stimulated Ptolemy to go creatively from the equation of one degree=one day to the extension of one year of life. In the concerted thought for this book, I think I can share a part of the answer to that question.

Some 400 years before Claudius Ptolemy’s time, Egyptian king Ptolemy II Philadelphus (285–246 b.c.)15 had brought seventy-some Jewish translators to Alexandria to translate the Old Testament of the Bible into Greek (certainly as a service to the enormous Jewish population living and working in Greek-speaking Alexandria and in the interest of trade relations with the Israelites to the North-Northeast). The translation was called the Septuagint (sep’too-uh-jint), a word from the Greek for “seventy,” the number of translators working on the project. Through the Bible (the Old Testament), circulating by that time for well over 300 years, broadly accessible in Greek translation, the known world was reading and hearing stirring, epic stories. Those who knew astrology also heard in the text an extraordinary number of astrological references.

The astrological references were there and the Biblical facts were there; imagine their impact, not only on the Jews! In the main, we can look at the beginning of Genesis (particularly 1:14–15, “And God said, ‘Let there be lights in the dome of the sky to separate the day from the night: and let them be for signs and for seasons, and for days and years, and let them be lights in the dome of the sky to give light upon the earth’”); then the beautiful fifteen verses in Ecclesiastes (3:1–15), beginning “For every thing its season, for every activity under heaven its time . . . ”; and the clear astrological reference in the accurate translation of the Lord’s Prayer: “Our Father who lives in the heavens, Let your name be honored, Let your Kingdom come. Let your will be done down here on the earth, as perfectly as it is in the sky.” And very important for our discussion (but not last among so many references throughout the Bible) is the reference in 4 Ezekiel 5–6: “For I assign to you a number of days, three hundred ninety days, equal to the number of years of their punishment . . . [and then] forty days I assign you, one day for each year.”

This pair of verses in Ezekiel (with the earlier, primal reference in Genesis) is often misquoted, often irresponsibly paraphrased as “and a day shall be a year in the eyes of God,” etc. But we must note clearly that the context of these verses in Ezekiel is not a statement of Divine Chronoscopy. There is no equation being established between different units of time. Rather, the sentence is case-specific: it is Ezekiel’s role to bear the weight of the iniquity of Israel and of Judah. This is to be accomplished within a religious (magical) ritual extending itself within his lifetime. With this perspective, there is no way that the verse can be used to justify any system of progressions in astrology—and, for that matter, there is no reference to a day-for-a-year equation anywhere in the Bible.16

But, how were these verses read so long ago? They are very tempting even now, with our imaginations eager to pick up any kind of exalted, authoritative endorsement. Certainly, the learned Ptolemy knew the great book of the Jews. Certainly, he reacted to the celestial (astrological) references, the numerology, the symbolisms. Certainly, he knew the academic/scientific inheritance from Babylon. And how easy it is to paraphrase for one’s own personal edification! We know no more. But we know for sure that Ptolemy revived the degree-day equation and added “the year” to it. And I feel sure that references in the Septuagint—valid or not—were part of the atmosphere absorbed in Ptolemy’s inspiration and creative process.

The academic brilliance of Ptolemy’s System (later to be called Primary Directions) influenced mathematicians for 1,400 years! Great scholars worked overtime to try to simplify what Ptolemy had created so brilliantly: scholarly mathematicians like Antonius Maginus (around 1604) and his contemporary Valentino Naboda (Naibod, to the English) came into prominence. Maginus adapted Primary Directions into a system leading to what we now call Secondary Progressions; he still related symbolic planetary positions to the equator as Ptolemy did, but he advanced in the ephemeris one day for every year of life rather than clocking degree passage over the Midheaven.

(At the same time, during the first half of the seventeenth century, Johannes Kepler [who had been an assistant to Tycho Brahe and was court mathematician, astrologer, and astronomer to the emperor Rudolph II, king of Rumania and Hungary] recognized that planets moved with variable speeds in elliptical orbits and invented some ten minor aspects. He is said to have considered that the number of days after birth that the Sun took to reach a natal planet was equivalent to the number of years of the native’s life that would elapse before the indicated influence would manifest itself.17 This is a reference contemporaneous with the work of Maginus and Naibod, preparing the way for Secondary Progressions and eventually pure Solar Arc theory.)

Naibod established the mean-motion of the diurnal Sun (59'08") as the key increment in progression—an extremely important step on the way to revelation of Solar Arc theory. Naibod’s mean arc symbolically equaled one year of life. This mean-motion is the rate of motion that divides the interval between 57' and 61' minutes of arc, the span of possible Sun motion, at a point with the same number of incidences of divergence above and below the measure, with a constant, graduated development between the two extremes.18

The great William Lilly, writing in the middle of the seventeenth century, recognized these three great systems: “But now we come to handle the measure of time in Directions, wherein there are at this day [1647] three severall opinions, yet no such as doe make any great difference in the matter.”19 I think Lilly is saying here that we have three great theoretical systems but none of them is a clear winner.

As we have seen, the first was Ptolemy and his Primary Directions. The second measure of time was that propounded by Maginus (published in 1604), who linked himself to the recently deceased, brilliant Danish astronomer/astrologer Tycho Brahe and Doctor John Dee, an original Fellow of Sir Isaac Newton’s own Trinity College at Cambridge, an occultist and astrologer for Queen Elizabeth’s coronation.

The third measure was Naibod’s mean motion arc.

It is interesting that Maginus himself, in 1619, published Naibod’s work and evaluated it with Lilly as “In my owne judgement, the most exactest measure that hitherto hath been found out.”20 I think the popularity Naibod enjoyed was due to the simplicity of his measure: all that was required was an elementary one-reference table.

It seems at this point in time, in the middle of the seventeenth century, that horoscopes—through Lilly’s great public exposure especially—were being drawn less and less in square format and more and more in circular format, but we do not know specifically why. I would submit, however, that occultism was a growing study among astrologers at that time, and the symbolism of the circle was of extreme importance and power in divinatory thought.

The prevailing House system had settled into the Regiomontanus system,21 but this gradually began to change with the rise in fame and importance of Placido de Titis (1603–1668), who directly followed Brahe, Kepler, Maginus, Naibod, and Lilly in prominence in the development of astrological systems.

Placidus was an Italian nobleman, who became a monk and a professor of mathematics at the University of Pavia (1657–1668). He also served as an astrology consultant to the Archduke Leopold William of Austria. Placidus wrote expert treatises on House division, Primary Directions, and the day-for-a-year system of Progressions that were “Secondary” to Ptolemy’s Primary Directions. The Placidus material changed astrology dramatically: almost all the West adopted the Placidian House system, and Secondary Progressions came into full bloom, with simplified orientation to the ecliptic.22

Two hundred and forty-some years later, at the close of the nineteenth century, the British astrologer Sepharial23 joined the list of astrologers who had tried to simplify Ptolemy and refine Naibod’s refinements, all of them searching for an even more facile way and, indeed, a more accurate way, to capture the symbolic significance of the Sun’s movement from day to day (from year to year in terms of degrees) and apply that to the entire horoscope in terms of life time. Sepharial surveyed the scene in his The Science of Foreknowledge24: giving much credit to “Bonattis,” an Italian mathematics professor who published (in 1687) a treatise espousing Placidus’ methods, including Secondary Progressions.25 Sepharial then presents pure Naibod progression, adding the mean increment of the Sun’s motion to each planet and point in the horoscope, arcing them forward this amount in relation to the age of the individual. He then tried a “companion” technique of taking the mean daily motion of the Moon (13°10') and applying this increment to all planets and points in the natal horoscope. And then, he confuses history by calling these Moon projections “Secondaries.”

At this time, we must take pause to realize how inaccurate so much of the astrology was in these times—even with all the intricate trigonometric mathematics connected with Ptolemy. There were so few recorded birth data, astrologers worked without full knowledge of the three outer planets, and they all operated under the greatest of expectations from an avidly interested public, enthusiasms fanned by the intense self-promotion by the astrologers of the day.

Constantly, in studying old astrologers’ work, we find error after error after error; not large necessarily in astronomical reckoning, especially in the late sixteenth century and thereafter thanks to Brahe’s instruments, inventions, and observational skill, and the discoveries by Kepler and Galileo; but grossly large in the dating and timing of births and astrologers’ fundamental interpretive assessments. Citations of birthdates constantly turn up different in different sources, and reckless commentaries and analyses abound.

Error was still haunting every prediction system. The Naibod application of the mean motion of the Sun was easy to use; it had the “large-stroke” feel about it that signaled something “right.” But after about 30 years of age, the correspondence between events and planetary significators in an individual’s horoscope, i.e., that which was predicted, seemed to be about a year off, especially for people born between March and September. As those people—and everyone else eventually—got older, discrepancies between events and predicted ages got bigger. This system did not work reliably either.

What was happening was that the mean Naibod measurement was not taking into account—“did not respect the symbolism of” is perhaps a better way of saying this—the very prominent changes in daily Sun motion throughout the year. In the name of simplification—simply using the mean daily motion of the Sun—Naibod had done away with the individualized measures of time linked to the different rate of movement of every person’s Sun, determined by the month of birth. The whole-degree-for-a-year system, when applied to everyone, did not work well; Naibod’s adjustment using the mean motion of the Sun might have improved results for some nativities but, as a rule, it did not work either.

In the ephemeris, the Sun moves much more quickly during the months October to February. In the months March through September, the Sun “takes its time”; the daily motion is conspicuously slower.

A person born with a slow diurnal Sun motion (less that one degree or less than the small Naibod adjustment), as we have seen, would gradually accumulate a discrepancy away from the one-day/one-year equation. By about age 30, this discrepancy would accumulate about one degree or one day in the ephemeris or one whole year in life! It would therefore take 31 years of life to live out the more slowly accumulated arc of 30 degrees.

It appears that this problem did not get smoothed out in astrological practice until early in the twentieth century! Exactly when and by whom are questions we can not answer. We would think that the solution of such a tremendously long, drawn-out problem would have occasioned lots of ballyhoo, but there is none discernable in the literature. Much like the circular formatting of the horoscope itself, the one-degree=one-year equation in a reliable technique was distilled through experimentation over time and, by the early twentieth century it was in refined practice.

England was still the world center for astrological thought a hundred years ago. The great Alan Leo, with his expert astrological knowledge and superb marketing ability, popularized astrology as never before. I think it is fair to say that, at the turn of the twentieth century, natal astrology had securely overcome the tremendous emphasis that had long been placed upon horary (since birth times were rarely known but times of questions could be recorded). Leo’s textbooks, including his fine, important work on the Secondary Progressed Horoscope, are still vital and sellable a hundred years later. But we see that the problem with Secondary Progressions—that there is so little movement (astrological development) of the planets Jupiter, Saturn, Uranus, Neptune, and Pluto (unknown before 1930), thus forcing so much delineation responsibility upon Mercury, Venus, and Mars, in addition to the Sun and Moon—was pushing Secondary techniques beyond symbolic limits.

We see minor aspects coming into SP theory, aspects to internal Placidian cusps (which change, of course, with every House system, while the Angles remain constant), and even transits related to Secondary Progressed planetary positions. The system was clearly trying to catch up with life as life became busier and more complex, as lives became longer.

The Solution

So often, great problems, after they are understood and resolved, can be illuminated in the simplest terms. The interrelationship of material mass and the speed of light to create energy—a huge concept indeed—becomes through Einstein simply E=mc2. The distance around a circle—an elusive concept for thousands of years—can be reasoned out to measure 2Pi r. Religious teachings present simple “golden rules” to guide complexities of human behavior. Darwin’s insight about evolution was elementary and epic at the same time.

The solution to extending the natal horoscope into time through a dynamic symbolism that was reliably individual lay not with simplifying the impressive Ptolemaic system. Rather, it rested upon the symbolism of the Sun itself, in its development upon the ecliptic: the Sun—the pacemaker of our life—had a Sign location and speed that defined each individual. Progressing this individual Sun forward in the ephemeris the magical one day for each year of life would establish the key passage of development. It would not matter if the speed were fast or slow, because the speed would be reckoned individually, at the individual’s own pace. The predictive formula would be personalized through the individual’s actual Sun speed.

This was the increment that would work. This was each individual’s Solar Arc. If we then applied this individual Solar Arc to every planet and point in the horoscope, we would symbolically advance our natal potentials uniformly in time, and we would create over 1,000 dynamic new relationships with the natal figure to support occurrences within individual life development. No predictive system could match this.

Following the development of the Babylonian degree-day formulation through Ptolemy, putting aside the enormous detour of daunting mathematics, following other Arab scholars, and a classroom of Italian mathematicians especially, in the seventeenth century, we arrive at the simple fact that the Sun is key, and its individualized progressed motion approximating a day-for-a-year captures one’s progress throughout life time.

In technique, nothing could be simpler: we calculate the Secondary Progressed Sun position (your computer goes directly into a Solar Arc program that does just what we are describing now), that is, the Sun’s position at the birth time a specific number of days after birth, to correspond to the same number of years in life. [A birth on June 3 will reveal its 20th year, for example, through a Sun position calculated at the birth time on June 23, 20 days after the actual birth date; the 41st year would be a Sun position calculated at the birth time on July 14, 41 consecutive days later into the next month following the actual birth date).26

The next step is a simple subtraction: from the Secondary Progressed Sun position—calculated using the natal birth time—we subtract the natal Sun position. The answer is the Solar Arc increment to the birthday month (or specified date) in the selected year of life.

The last step is to advance every planet, the nodal axis (in counterclockwise motion),27 the Midheaven and Ascendant axes the distance of the Solar Arc. Simply add (this is what the computer does) the Solar Arc to every planet and sensitive point in the horoscope and note the new positions into the outer ring on your chart form, surrounding the natal horoscope. The primal symbol of life development, the Sun, is being shared with every symbol in the horoscope.

Life development is bringing the birth horoscope forward in time, while maintaining the individualistic relativity of the natal positions. Dynamic new aspect relationships are then created to reflect the developmental tensions in the process of becoming.

Bridging into the twentieth century, two German astrologers dramatically confirmed the efficacy of Solar Arcs as a streamlined and exacting prediction technique. Alfred Witte (1878–1941) founded the well-known Hamburg School of Astrology and pioneered a complex system of astrology involving eight hypothetical planets. We know this system today as Uranian Astrology. The Hamburg School used/uses the Solar Arc technique as its measure for developmental time.

Reinhold Ebertin (1901–1988) founded the Cosmobiological School of Astrology, working strongly with midpoint synthesis and Solar Arcs. In the pre-computer era, Ebertin created an elegant system (a kit) of plastic and paper wheels which allowed an outer ring of Solar Arcs to rotate around a natal horoscope and to display direct and indirect arcs measured quickly and easily. Ebertin’s Combination of Stellar Influences was a brilliant analysis of all possible arc measurements and was a major contribution to the evolution of astrology.28

Perhaps the last gasp given to Ptolemaic/Placidian Primary Directions was taken by Swiss astrologer Heinrich Kuendig, born early in the 1900s. His working of the complex system caught on very strongly in Denmark—and still is practiced widely there—through Danish astrologer Irene Christensen’s leadership and textbook, The Precise Astrological Prediction: A Presentation of Kuen-dig’s Method (self-published, Copenhagen, 1974).

And now, at the beginning of a new century, we have the excitement ahead—not just as you read this book but ahead in your work for the rest of your life as astrologers—of seeing Solar Arcs in action, forward and backward in time. Solar Arcs have come of age and indeed are invigorating the astrology world everywhere.