Category: Sacred Geometry: Language of the Angels

Released in January 2021, this book explores sacred building as the continuation of the megalithic: its astronomical knowledge, geometrical and metrological methodology and knowledge of the size of the Earth. The megalithic was our first numerate culture which discovered the main design design parameters for our existence, organised by higher intelligences called angels.

Jupiter’s gravitational and numerical influence

This post begins a Theme relating to the Trigon event occurring on 21st December 2020, when Jupiter and Saturn are conjunct at dusk in the sky. This touches upon what such synchronicities mean for other long term periods seen from Earth, such as the Moon’s nodal period of 6800 days and even the Precession of the Equinoxes over 25,800 ± 120 years.

Jupiter is the second largest body in the solar system next to the sun itself. In fact, Jupiter is not far short of being a sun itself and, being the closest giant planet to the Earth, our planet is strongly influenced by Jupiter’s gravity which, unlike the Sun’s continuous pull to maintain Earth’s orbit around it, Jupiter pulls upon the Earth and the Moon on an episodic basis when the Earth is passing between the Sun and Jupiter.

The Trigon Period of Jupiter and Saturn

Being a dark, planetary body, the episodic pull of Jupiter follows a different pattern to each of the inner, terrestrial planets; Mercury, Venus, Earth and Mars, since each has a different orbital period which, combined with Jupiter’s orbit, brings each under Jupiter’s influence or absence. The combined episodic pull of Jupiter and Saturn, is visually seen in their conjuction every 20 years, which occurs just over a third of the Zodiac onwards, thus giving a cosmic significance to the equilateral triangle as a sacred geometry.

Figure 1 The series of Trigon conjunctions of Jupiter and Saturn, as will be the case on 21st December 2020

Only Earth’s large moon stops the axial tilt of the Earth from varying significantly, then causing large changes in climate which would have restricted the development of the relatively stable habitats and biomes we enjoy.

361 days: Jupiter and the Zodiac

The combination of Jupiter’s orbital period (of 4332 days) and Earth’s (of 365.2422 days) generates an interesting set of numerical facts since Jupiter passes through each of the twelve signs of the Zodiac in 361 days. This number is 19 times 19 days so that 12 times 361 days equals 4332 days. But these numbers are a product of the solar year of 365.2422 days, since the day length on Earth is 1 year divided by the 365.2422 days due to its rotation. If the day length were less or more then Jupiter’s complete orbit would still be as long but the numbers from Earth would not.

This is a major aspect of what the megalithic astronomy had to learn, that the relative time lengths of the many cosmic periods, counted in days, could be numerically interrelated when quantified. The situation of the earth orbit and its rotation would present Jupiter as a bright moving star which completed its journey through the stars in 12 times 361 days. Jupiter and the Zodiac of 12 constellations would inevitably become fused as seen in the story of Zeus, the Greek god name for Jupiter whose symbol is the twelve-fold circle. The pre-Classical Greeks were matriarchal, following the lunar month of twelve whole lunar months within the solar year and, the solar year only arose as the patriarchal northern tribes occupied Greece after the Bronze Age collapse. The name Zeus is therefore not matriarchal since the Greeks had no “Z”. Zeus arrived in ancient Greece with the tribes displaced from the North escaping the worsening climate at higher latitudes. And, whilst 12-foldness is associated with the Sun being in one of the 12 zodiacal constellations, Jupiter defines these through passing through each sign (on average) in 361 days.

399 days: Jupiter’s synodic period

Twelve-ness is a massively widespread tradition (see John Michell – Twelve-fold Tribes for instance) and the brightest celestial body next to the Sun is the Moon which expresses twelve whole lunar months a year (plus 7/19 of a lunar month). The common lunar year was therefore twelve months long, taking 354.367 days to complete, this countable between thirteen full moons. It is no accident that the 12-ness of the lunar year is connected with Jupiter’s 12-ness of its 361 day years, since the Jupiter synod has a strong grip on our moon: the synod is 9/8 lunar years long – a musical whole tone. And Saturn also has a similar grip, its synod of 378 days being 16/15 lunar years long.

When the Earth passes by Jupiter, the latter goes retrograde or backwards relative to the stars, meaning it appears to travel east night-by-night, rather than the norm for all planets (and the sun and moon) of slowly travelling west in our skies, as they orbit. During this retrograde period, the planet describes a loop in the sky relative to the stellar background, before returning to where it should be in the stars. Between the loops of Jupiter’s synodic period the 398.88 days could be counted in days. This can only mean that over millennia, the Moon became synchronised by the regular proximity of Jupiter to our moon.

Our months today have divided the solar year into twelve months of 30 or 31 days, to resemble Jupiter’s 12-fold zodiac and 12-month lunar year, the Roman emperors vying to lengthen a month and name it after themselves (examples being October after Octavius, September after Septimius and August after Augustus). And since a zodiacal sign is traversed after 361 days by its definer, it is inevitable that there are not 12 solar years in a Jupiter orbit but just less (11.86 years). However, the fact that 4332 days is not 12 times 365.2422 days accesses, through its deficit, more subtle possibilities hidden in a numerical world of differences.

Differences between periodicities, especially involving the moon that rotates the Earth, define those periods through the fact that they endlessly repeat so that differences accumulate over longer periods and when these differences are divided into the periods, a new set of numbers are generated. One could call orbital systems differential calculators and modern math would describe them as potentially discrete systems, which form due to gravitational recurrence. This idea that the planetary and lunar systems generate numbers is somewhat hidden by our modern description of such systems as subject to gravitational dynamics. The numbers allowed the ancient astronomers to discover a static numerical view of planetary astronomy through counting days. In contrast, modern astronomy calculates the location of celestial bodies from first principles; especially when trying to visit planetary bodies in spacecraft.

Before, during and after Sacred Geometry

above: Carreg Coetan Arthur portal dolmen in Newport, Pembrokeshire.

The prehistory of sacred geometry was the late stone age, when the stone circles, dolmens, and long alignments to astronomical events on the horizon, used megaliths (large stones) in geometrical ways. Their geometries served their quest to understand the heavens, without telescopes or arithmetic, by using counted time periods as geometrical lines, squares and circles. Geometry, supplemented by the days counted between alignment events, was therefore a prelude to sacred and then secular geometry.

By developing early geometrical methods, they forged an enduring cultural norm lasting millennia, as part (or not) of the more-familiar aspect of the neolithic, innovating an agricultural pastoralism, that could support settlements, cities and, only then, the great civilizations of the middle and far east. It was civilization that generated our earliest written histories; these still powering our historical context and leading the basic notion of economic progress and territorial expansion, as superior to all that went before.

Our surviving megaliths are hence deeply enigmatic, a mysterious and mute presence in a world far less mysterious. The megaliths may have something we have forgotten in a collective way, something pushed out by millennia of later ideas and now relatively recent ones too.

There seems little trace of the megalithic astronomers themselves, their geometricized landscape overlaid by our notions of a primitive Stone Age.  And, as the prelude to world history, their geometry gave birth to sacred geometry and sacred buildings; pyramids, ziggurats, temples and religious complexes. In some way, therefore, geometry obtained its sacredness from the skies or the earth itself, as if these had been built from the harmonious organization of the solar system seen from Earth and given to it by one or more gods or angels.

Sacred geometry the became a secular and analytical geometry, which would become an encyclopedic exploration of all that geometry could do, rather than a set of techniques dreamt up by a band of roaming astronomers. In our schools, many lose interest in having to learn geometry in the abstract and so, in this, the megalithic had an advantage. They could learn geometry as and when they needed it, as their astronomy brought up new questions to solve, learning by finding methods to answer questions.

If one truly travels backwards in time, to discover what the megalithic astronomers had understood, I believe one has to decide which bits of your own skills have to be applied to solve the riddles of the megalithic mind. Each modern researcher must not assume the megalithic could calculate using numbers, use trigonometry, knew Pythagoras’ theorum, and so on. And yet, one can employ modern equipment to help investigate the megalithic. Google Earth, for example, can allow megalithic alignments to be studied, their azimuth, length and interrelation, whilst the context of sites can be seen that may provide clues not available in site plans, written descriptions and so on, which are sometimes difficult to obtain or require a personal expedition. The most basic tool for me has been the Casio scientific calculators, since the megalithic interaction with space (geometry) was blended with the interaction of numerical time counting, numbers which exist in the geocentric world of time.

Finally, one must realise the past is only in the present through our attention to it and, in the absence of much official interest in applied geometry, dimensionality and astronomical intent of the sites, it is left to non-specialists to become new specialists in the sense of recovering and conserving the true achievements of the megalithic, for our present age, while the monuments still exist as living mysteries. In this I advocate the path leading to what this website is about.

Astronomy 3: Understanding Time Cycles

above: a 21-petal object in the Heraklion Museum which could represent the 21 seven-day weeks in the 399 days of the Jupiter synod. [2004, Richard Heath]

One of the unfortunate aspects of adopting the number 360 for calibrating the Ecliptic in degrees is that the megalithic counted time in days and instead saw the ecliptic as divided by the 365¼ days. In transferring to the number 360, with all of its easy factors, 8 x 9 x 5, moderns cannot exploit a key advantage of 365¼ days.

If the lunar orbit takes 27.32166 days then each day the moon moves by 1/27.32166 of the ecliptic every day. For this reason, after 27.32166 days the orbit completes because the Moon’s “year” then equals one as the angular motion has been 27.32166/ 27.32166 = 1.

The same is true of the lunar nodes, which retrograde to the east along the ecliptic in 18.618 years. For this reason one can say, the lunar nodes move by 1/18.618 DAYS (in angle) every day and to travel one DAY in angle, the nodes take 18.618 DAYS per day (needing the new term “node day” equal the 18.618 days.*** A solar year takes 19.618 node days (since 365¼ equals 18.618 x 19.618) and an eclipse year takes 18.618 x 18.618 – 346.62 days

*** These are average figures since the moon comes under variable gravitational influences that are episodic.

A general rule emerges in which the larger, whole cycles, lead to reciprocals which can be numerically characterized by knowing the number of the days in the larger period.

For instance, Jupiter has a synodic excess over the solar year of 398.88 days and this means its angular motion is 1/ 398.88 DAYS per day while Saturn’s synod is 378.09 days and its angular motion is 1/ 378.09 DAYS per day. These synods are, by definition, differential to the Sun at 1/ 365.2422 DAYS per day.

Without seeing astronomy as calibrated to day and year cycles, one is robbed of much chance to appreciate the megalithic view of time and the time-factored buildings that came to be built in pursuit of quite advanced knowledge.

Looking from the relatively large cycles to the extremely small, daily angular changes of celestial bodies seen from Earth, reveals a further obscuration created, in this case, by the heliocentric view of the solar system, rather than the geocentric view which is obviously founded on days and years seen from the surface of the planet.

The largest cycle the megalithic could see using their techniques, reverses the direction from large-to-small to small-to-large since the precessional cycle (of the equinoctal nodes of the earth’s obliquity) is around 25,800 ± 100 years long. A star or constellation on the ecliptic appears to move east, like the lunar nodes, and using the angular measure of DAYS, it is possible to estimate that the equinoctal points move by a single DAY, in a given epoch, something like 71 years. The precessional cycle is therefore 71 years multiplied by the 365.2422 DAYS of the whole ecliptic.

The most important benefit of using DAY angles is that knowledge of a few celestial periods opens up a realm in which different scales of time can be derived from first principles. And added to that, the celestial periods appear related to one another so that so-called sacred numbers emerge such as the seven day week which divides into both the Saturn synod (54 weeks), Jupiter synod (57 weeks), the 364 day saturnian year (52 weeks) and others.

To understand the full scope of megalithic astronomy requires a geocentric calibration of the ecliptic as having 365¼ angular DAYS.

Geometry 7: Geometrical Expansion

above: the dolmen of Pentre Ifan (wiki tab)

In previous lessons, fixed lengths have been divided into any number of equal parts, to serve the notion of integer fractions in which the same length can then be reinterpreted as to its units or as a numerically different measurement. This allows all sorts of rescaling and exploitation of the properties of integer numbers.

Here we present a megalithic method which extended two or more fixed bearings (or alignments), usually based upon a simple geometrical form such as a triangle or a rectangle. This can be how the larger geometries came to be drawn on the landscape (here called landforms) of separated megaliths and natural features which appear to belong together. For example,

Outliers: Alexander Thom found that British stone circle were often associated with single outliers (standing stones) on a bearing that may correspond to horizon event but equally, appears to give clues to the metrology of the circle in the itinerary length to the outlier from the circle’s centre.

Figure 1 Stone circle plans often indicate nearby outliers and stone circles

Stone circles were also placed a significant distance and bearing away (figure 1), according to geometry or horizon events. This can be seen between Castle Rigg and Long Meg, two large flattened circles – the first Thom’s Type-A and the second his Type-B.

Figure 2 Two large megalithic circles appear linked in design and relative placement according to the geometry of the double square.

Expanding geometrically

The site plan of Castle Rigg (bottom left, fig. 1) can have the diagonal of a double square (in red) emerging between two stones which then bracket the chosen direction. This bearing could be maintained by expanding the double square so that west-to-east and south to north expand as the double and single length of a triangle while the hypotenuse then grows towards the desired spot according to a criteria such as, a latitude different to that of Castle Rigg. That is, at any expansion the eastings and northings are known as well as the distance between the two circles while the alignments, east and northeast in this example, are kept true by alignment to previous established points. Indeed, one sees that the small outlier circle of Long Meg, to Little Meg beyond, was again on the same diagonal bearing, according to the slope angle of the cardinal double square.*** One can call this a type of projective geometry.

***This extensive double square relation between megalithic sites was first developed by Howard Crowhurst, in Ireland between Newgrange and Douth in same orientation as figure 2, and by Robin Heath at https://robinheath.info/the-english-lake-district-stone-circles/.

It seems impossible for such arrangements to have been achieved without modern equipment and so the preference is to call these landforms co-incidental.  But, by embracing their intentionality, one can see a natural order between Castle Rigg and, only then, Long Meg’s outlying Little Meg circle, and through this find otherwise hidden evidence of the working methods in the form of erratics or outliers, whose purpose is otherwise unclear.

Equilateral Expansion

The work of Robin Heath in West Wales can be an interesting challenge since not all the key points on his Preseli Vesica are clearly megalithic, perhaps because megaliths can be displaced by settlements or be subsumed by churches, castles and so on. (see Bluestone Magic, chapter 8). First, for completeness, how is a vesica defined today? In his classic Sacred Geometry, Robert Lawlor explains the usual construction and properties of the vesica :

Drawing 2.3. Geometric proof of the √3 proportion within the Vesica Piscis. from Sacred Geometry by Robert Lawlor.

Draw the major and minor axes CD and AB. Draw CA, AD, DB and BC. By swinging arcs of our given radius from either centre A or B we trace along the vesica to points C and D, thus verifying that lines AB, BC, CA, BD and AD are equal to one another and to the radius common to both circles.

We now have two identical equilateral triangles emerging from within the Vesica Piscis. Extend lines CA and CB to intersect circles A and B at points G and F. Lines CG and CF are diameters of the two circles and thus twice the length of any of the sides of the triangles ABC and ABD. Draw FG passing through point D.

Sacred Geometry by Robert Lawlor

Primitive versus later geometry

Lawlor’s presentation have the triangles appearing as the conjuction of two circles and their centers. However, the points and lines of modern geometry translate, when interpreting the megalithic, into built structures or significant features, and the alignments which may join them. The alignments are environmental and in the sky or landscape.

  • A is Pentre Ifan, a dolmen dating from around 3500 BC.
  • B is located in the Carningli Hillfort, a mess of boulders below the peak Carningli (meaning angel mountain). Directly East,
  • C is the ancient village, church and castle of Nevern.
  • D is a recently excavated stone circle, third largest in Britain at around 360 feet diameter, but now ruinous, call Waun Mawn.

The two equilateral triangles have an average side length around 11,760 feet but, as drawn, each line is an alignment of azimuth 330, 0, 30, and 90 degrees and their antipodes. 

The Constructional Order

Relevant here is how one would lay out such a large landform and we will illustrate how this would be done using the method of expansion.

North can be deduced from the extreme elongation of the circumpolar stars in the north, since no pole star existed in 3200BC. At the same time it is possible to align to plus and minus 30 degrees using Ursa Major. This would give the geometry without the geometry so to speak, since ropes 11760 feet long are unfeasible. It seems likely that the Waun Mawn could function as a circumpolar observatory (as appears the case at Le Menec in Brittany, see my Lords of Time).

If the work was to start at Carningli fort, then the two alignments (a) east and (b) to Waun Mawn could be expanded in tandem until the sides were 11760 feet long, ending at the circle to the south and dolmen to the east. The third side between these sites should then be correct.

Figure 3 Proposed use of equilateral expansion from Carnigli fort to both what would become the dolmen of Pentre Ifan (az. 90 degrees) and Waun Mawn (azimuth 150 degrees).

The vesica has been formed to run alongside the mountain. The new eastern point is a dolmen that points north to another dolmen Llech-y-Drybedd on the raised horizon, itself a waypoint to Bardsey Island.

The reason for building the vesica appears wrapped up in the fact that its alignments are only three, tightly held within a fan of 60 degrees pointing north and back to the south. But the building of the double equilateral cannot be assumed to be related to the circular means of its construction given by Lawlor above. That is, megalithic geometry did not have the same roots as sacred geometry which has evolved over millennia since.

Geometry 6: the Geometrical AMY

By 2016 it was already obvious that the lunar month (in days) and the PMY, AMY and yard (in inches) had peculiar relationships involving the ratio 32/29, shown above. This can now be explained as a manifestation of day-inch counting and the unusual numerical properties of the solar and lunar year, when seen using day-inch counting.

It is hard to imagine that the English foot arose from any other process than day-inch counting; to resolve the excess of the solar year over the lunar year, in three years – the near-anniversary of sun and moon. This created the Proto Megalithic Yard (PMY) of 32.625 day-inches as the difference.

Figure 1 The three solar year count’s geometrical demonstration of the excess in length of 3 solar years over 3 lunar years as the 32.625 day-inch PMY.

A strange property of N:N+1 right triangles can then transform this PMY into the English foot, when counting over a single lunar and solar year using the PMY to count months.

The metrological explanation

If one divides the three-year excess (here, the PMY) into the base then N, the normalized base of the N:N+1 triangle. In the case of the sun and moon, N is very nearly 32.625, so that the lunar to solar years are closely in the ratio 32.625:33.625. Because of this, if one counts 

  • months instead of days,
  • using the three-year excess (i.e. the PMY) to stand for the lunar month,
  • over a single year,

the excess becomes, quite unexpectedly, the reciprocal of the PMY;

One has effectively normalized the solar year as 12.368 PMYs long. This single year difference, of 0.368 lunar months cancels with the PMY; the 0.36827 lunar months becoming 12.0147 inches. Were the true Astronomical Megalithic Yard (AMY of 32.585 inches) used, instead of the PMY, the foot of 12 inches would result. Indeed, this is the AMYs definition, as being the N (normalizing value) of 32.585 inches long, unique to the sun-moon cycle. The AMY only becomes clear, in feet, after completion of 19 solar years. This Metonic anniversary of sun and moon over 235 lunar months, is exactly 7 lunar months larger than 19 lunar years (228 months).

But this is all seen using the arithmetical methods of ancient metrology, which did not exist in the megalithic circa 4000BC. Our numeracy can divide the 1063.1 day-inches by 32.625 day-inches, to reveal the AMY as 32.585 inches long, but the megalithic could not. Any attempt to resolve the AMY in the megalithic, using a day-inch technology***, without arithmetical processes, could not resolve the AMY over 3 years as it is a mere 40 thousandths of an inch smaller than the PMY. So arithmetic provides us with an explanation, but prevents us from explaining how the megalithic came to have a value for the AMY; only visible over long itineraries requiring awkward processes to divide using factorization. However, by exploiting the coincidences of number built in to the lunar and solar years, geometry could oblige. 

***One can safely assume the early megalithic resolved
eighths or tenths of an inch when counting day-inches.

The geometrical explanation

In proposing the AMY was properly quantified, in the similarly early megalithic cultures of Carnac in France and the Preselis in Wales, one must turn to a geometrical method

  1. One clue is that the yard of 3 feet (36 inches) is exactly 32/29ths of the PMY. This shows itself in the fact that 32 PMYs equal 29 yards.
  2. Another clue is that the lunar month had been quantified (at Le Manio) by finding 32 months equalled 945 day-inches. By inference, the lunar month is therefore 945 day-inches divided by 32 or 945/32 (29.53125) day-inches – very close to our present knowledge of 29.53059 days.

From point 1, one can geometrically express any length that is 32 relative to another of 29, using the right triangle (29,32). And from point 2, since the 945 day period is 32 lunar months, as a length it will be in the ratio 29 to 32 to a length 32 PMYs long, the triangle’s hypotenuse.

Point 1 also means that 32 PMY (of 32.625 inches) will equal 1044 inches, which must also be 29 x 36 inches, and 29 yards hence handily divides the 32 side of the {29 32} right triangle into 29 portions equal to a yard on that side. One can then “mirror the right triangle about its 29-side so as to be able to draw 29 parallel lines between the two, mirrored, 32-sides, as shown in figure 1. The 945 day-inch 29-side which already equals 32 lunar months (in day-inches), now has 29 megalithic yards in that length, which are then an AMY of 945/29 day-inches!

Figure The 29:32 relationship of the PMY to the yard as 32 PMY = 29 yards whilst 32 lunar months (945 days) is 29 AMY.

Comparing the two AMYs and their necessary origins

Using a modern calculator, 945 divided by the PMY actually gives 28.9655 PMY and not 29, so that 945 inches requires a unit slightly smaller than the PMY and 945/29 gives the result as 32.586 inches, which length one could call the geometrical AMY. This AMY is 30625/30624 of the AMY in ancient metrology which is arrived at as 2.7 feet times 176/175 equal to 32.585142857 inches. By implication therefore, the ancient AMY is the root Drusian step whose formula is 19.008/7 feet whilst the first AMY was resolved by the megalithic to be 945/29 inches.

This geometrical AMY (gAMY?) obviously hailed from the world of day-inch counting, which preceded the ancient arithmetical metrology which was based upon fractions of the English foot. The gAMY is 32/29 of the lunar month of 29.53125 (945/32) day-inches, since 945/32 inches × 32/29 is 945/29 inches.

Using ancient metrology to interpret the earliest megalithic monuments may be questionable in the absence of a highly civilised source which had, in an even greater antiquity, provided it; from an “Atlantis”. In contrast, the monumental record of the megalithic suggests that geometrical methods were in active development and involved less sophisticated metrology, on a step-by-step basis.  From this arose the English foot which, being twelve times larger than the inch, could provide the more versatile metrology of fractional feet, to provide a pre-arithmetical mechanism, to solve numerical problems through geometrical re-scaling. This foot based, fractional metrology then developed into the ancient metrology of Neal and Michell, which itself survived to become our historical metrology [Petrie and Berriman].

The two types of AMY, geometrical and the metrological, though not identical are practically indistinguishable; the AMY being just over one thousandths of an inch larger. The geometrical AMY (945/29 inches) is shown, by figure 2, to be geometrically resolvable, and so must have preceded the metrological AMY, itself only 40 thousandths of an inch less than the PMY.

The two AMYs, effectively identical, reveal a developmental history starting with day-inch counting, and division of 945 inches by 29 was made easy by exploiting the alternative factorisation of 32 PMV as 36 × 29 yards using geometry. The AMY of ancient metrology was the necessary rationalization of 945/29 inches into the foot- based system.

Bibliography for Ancient Metrology

  1. Berriman, A. E. Historical Metrology. London: J. M. Dent and Sons, 1953.
  2. Heath, Robin, and John Michell. Lost Science of Measuring the Earth: Discovering the Sacred Geometry of the Ancients. Kempton, Ill.: Adventures Unlimited Press, 2006. Reprint edition of The Measure of Albion.
  3. Michell, John. Ancient Metrology. Bristol, England: Pentacle Press, 1981.
  4. Neal, John. All Done with Mirrors. London: Secret Academy, 2000.
  5. —-. Ancient Metrology. Vol. 1, A Numerical Code—Metrological Continuity in Neolithic, Bronze, and Iron Age Europe. Glastonbury, England: Squeeze, 2016 – read 1.6 Pi and the World.
  6. —-. Ancient Metrology. Vol. 2, The Geographic Correlation—Arabian, Egyptian, and Chinese Metrology. Glastonbury, England: Squeeze, 2017.
  7. Petri, W. M. Flinders. Inductive Metrology. 1877. Reprint, Cambridge: Cambridge University Press, 2013.

Geometry 5: Easy application of numerical ratios

above: Le Manio Quadrilateral

This series is about how the megalithic, which had no written numbers or arithmetic, could process numbers, counted as “lengths of days”, using geometries and factorization.

My thanks to Dan Palmateer of Nova Scotia
for his graphics and dialogue for this series.

The last lesson showed how right triangles are at home within circles, having a diameter equal to their longest side whereupon their right angle sits upon the circumference. The two shorter sides sit upon either end of the diameter (Fig. 1a). Another approach (Fig. 1b) is to make the next longest side a radius, so creating a smaller circle in which some of the longest side is outside the circle. This arrangement forces the third side to be tangent to the radius of the new circle because of the right angle between the shorter sides. The scale of the circle is obviously larger in the second case.

Figure 1 (a) Right triangle within a circle, (b) Making a tangent from a radius. diagram of Dan Palmateer.

Figure 1 (a) Right triangle within a circle, (b) Making a tangent from a radius.

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