Aspects - Origins Kepler College ES-174 ©2001, 2009 J. Lee Lehman

References Addey, John. 1977. Harmonics in Astrology. Cambridge Circle: Green Bay. Barker, Andrew. 2000. Scientific Method in Ptolemy's 'Harmonics.' Cambridge University Press: New York. Burnett, Charles, Jan P. Hogendijk, Kim Plofker and Michio Yano, Ed. 2004. Studies in the History of the Exact Sciences in Honour of David Pingree. Brill: Leiden.

References Lilly, William. 1647. Christian Astrology. Reprinted in 1985 by Regulus: London. McEvilley, Thomas. 2002. The Shape of Ancient Thought: Comparative Studies in Greek and Indian Philosophies. Allworth Press: New York. Taylor, Thomas. 1816, 1983. The Theoretic Arithmetic of the Pythagoreans . Samuel Weiser, Inc.: York Beach, ME.

References Godwin, Joscelyn. 1995. Harmonies of Heaven and Earth. Mysticism in Music . Inner Traditions International: Rochester, VT. Hamblin, David. 1983. Harmonic Charts. Aquarian Press: Wellingborough.

Pythagoras I Consider the number 12. 12 is the number of signs of the zodiac. The following numbers divide 12: 1, 2, 3,4 and 6

Pythagoras II 1 is the monad. 2 is the 1st even number and the 1st prime number. 3 is the 1st odd number. 4 is the 1st evenly-even number. 6 is the 1st evenly-odd number and the first perfect number.

Pythagoras III Of all of the number series that the Pythagoreans developed, using arithmetic (adding equal differences), geometric (adding ratios), and harmonic (relative to harmony) series, none of these number series included the number 360. The aspect series can only be understood through the integers representing the division of the circle.

Pythagoras: The Monad 1 is the monad. One can be considered to be the 1st number, the 1st prime number, and the 1st odd number. But it is mainly the symbol of unity, or the world undivided.

Pythagoras: The Monad II "Unity therefore, or the monad, which is in arithmetic what a point is in geometry, is the principle of interval and length; but itself is neither capacious of interval or length; just as a point is the principle of a line and of interval, but is itself neither interval nor line." (page 63)

Pythagoras: The Monad III "The monad, as we learn from the extracts preserved by Photius from Nicomachus, was called by the Pythagoreans intellect, male and female, God, and in a certain respect matter. They also said that it in reality mingled all things, is the recipient and capacious of all things, is Chaos, confusion, commixtion, obscurity, darkness, a chasm... and void of mixture... It is likewise called by them the axis, the Sun,... The tower of Jupiter, and spermatic reason. Apollo likewise, the prophet, and ambiguous.“ (page 169)

Pythagoras: the Duad I 2 is the 1st even number and the 1st prime number. Each sequence, the even and odd numbers, was again subdivided into three subgroups, according to its properties of division. An even number is defined as being able to be halved, while still producing an integer result. Feminine. Line. Diversity. Number of excess and defect. Duality.

Pythagoras: the Duad II "The duad was called by the Pythagoreans, as we learn from Nicomachus,' audacity, matter, the cause of dissimilitude, and the interval between multiple and the monad. This alone produces equality from composition and mixture, on which account it is equal. But it is likewise unequal, defect, and abundance, and is alone unfigured, indefinite, and infinite. It is also alone the principle and cause of the even.... It is also harmony, patience, though not in a certain respect in energy.'" (page 173)

Pythagoras: the Triad I 3 is the 1st odd number and the first perfect number. The odd numbers do not produce an integer result when halved. Masculine. Plane. Unity and diversity are restored to unity. Another way, simply put, is that the triad is the triangle, the first plane figure.

Pythagoras: the Triad II “Nichomachus ... observes... ‘The triad ... causes the power of the monad to proceed into energy and extension. But it is also the first of numbers, and is properly a system of monads. Hence afterwards, the Pythagoreans refer this number to physiology. For it is the cause of that which has triple dimensions, gives bounds to the infinity of number, is similar and the same, homologous and definite. The triad also is intellect, and is the cause of good counsel, intelligence and knowledge.’” (page 178)

Pythagoras: the Triad III "Nichomachus ... observes... 'The triad ... is also the most principal of numbers, and is the mistress of geometry, possesses authority in whatever pertains to astronomy and the nature and knowledge of heavenly bodies, and connects and leads them into effect. Every virtue also is suspended from this number, and proceeds from it'" (page 178)

Pythagoras: the Triad IV Why does Nicomachus refer to 3 as the first number? "... number is more increased by multiplication than by addition, as we have before observed from Proclus, and this is the case with the triad, but is not so with the duad or monad." (p 179) "That it is also the first perfect number is evident from this, that three things, as Aristotle observes, are all, and the all is perfect from having a beginning, middle, and end." (p 179)

Pythagoras: the Triad V 3 quaternities (cardinal, fixed, mutable) 3 faces and decans per sign 3 Triplicity rulers per sign (Dorothean-style) "According to the Chaldeans likewise, there are three ethereal worlds prior to the sphere of the fixed stars."

Pythagoras: the Tetrad I 4 is the 1st evenly-even number and the 1st square number. An evenly-even number is divisible by an even number, and then by even numbers again until finally the number 1 is reached. Feminine. Solid. Justice. Steadfastness. Simply put, it's the tetrahedron, the first solid.

Pythagoras: the Tetrad II "The tetrad, as we learn from Nicomachus, was called by the Pythagoreans, 'the greatest miracle, a God after another manner (than the triad,) is manifold, or father, every divinity. It is also the fountain of natural effects, and is the key-bearer of nature. It is the introducer and cause of the constitution and permanency of the mathematical disciplines." (page 181)

Plato: the Tetractys The tetractys is the set of the first four numbers (1,2,3,4). The sum of the tetractys is the decad (i.e., 1+2+3+4=10), and thus, the decad is derived from the tetractys. This was considered the symbolism behind the evolution of the physical world from nonphysical unity (McEvilley, pp 160-161)

Pythagoras: Hexad 6 is the 1st evenly-odd number. 6 is the second perfect number. A number is analyzed by determining all its divisors (except itself). In the case of 6, that's 1, 2, and 3. These divisors are then summed: 1+2+3=6 If this sum is equal to the original number, then that number is perfect.

Pythagoras – the Dodecagon What's so special about 12 in the first place? 12 is the number of zodiacal signs. 12 is the first superperfect number. A number is analyzed by determining all its divisors (except itself). In the case of 12, that's 1, 2, 3, 4 and 6. These divisors are then summed: 1+2+3+4+6=16 If this sum is greater than the original number, then that number is superperfect, or superabundant.

Pythagoras - the Dodecagon 12 is the first superperfect number. 12 is the number of zodiacal signs. All of its divisors define aspect types All of its divisors themselves are the first of a number series. This is also the number of tones in the chromatic octave in music.

Pythagoras - the Dodecagon "In the system of the Pythagorean Philolaus, for example, the Cosmic Person is correlated to a shape and to the astronomical totality: 'The dodecagon, which corresponds to Zeus, is the whole zodiac with its twelve signs.*'" * Burkert, Lore and Science in Ancient Pythagoreanism, p 349, cited in McEvilley, page 46.

Pythagoras - the Dodecagon It's worth mentioning that, in the context of this quote, Zeus was invoked by the Orphics as a symbol for cosmic unity: all other gods and goddesses were part of the body of Zeus. There were Orphic communities surrounding Pythagoras' Italian community in Croton. Pythagoras or one of his direct students was said to have written some of the Orphic works.

Pythagoras: 28 28 is the 2nd or 3rd perfect number (1+2+4+7+14=28). Represents the lunar cycle. The sum of the first seven numbers (1+2+3+4+5+6+7) = 28

Pythagoras VI "Indeed, that numbers are participated by the heavens, and that there is a solar number, and also a lunar number, is manifest according to the adage, even to the blind. For the restitutions of the heavenly bodies to their pristine state would not always be effected through the same things, and in the same manner, unless one and the same number had dominion in each." (page 167)

Aspects, by Ptolemy "We may learn from the following why only these intervals [i.e., aspects] have been taken into consideration. The explanation of opposition is immediately obvious, because it causes the signs to meet on one straight line. But if we take the two fractions and the two superparticulars most important in music, and if the fractions one-half and one-third be applied to opposition, composed of two right angles, the half makes the quartile and the third the sextile and trine.

Ptolemy, Aspects... continued "Of the superparticulars, if the sesquialter and sesquitertians be applied to the quartile interval of one right angle, which lies between them, the sesquialter makes the ratio of the quartile to the sextile and the sesquitertian that of the trine to the quartile. Of these aspects trine and sextile are called harmonious because they are composed of signs of the same kind, either entirely of feminine or entirely of masculine signs; while quartile and opposition are disharmonious because they are composed of opposite kinds." (Robbins, pp 73-75)

Ptolemy, Aspects... continued "Related to sight, and to the movements in place of the things that are only seen – that is, the heavenly bodies – is astronomia; related to hearing and to the movements in place, once again, of the things that are only heard – that is, sounds – is harmonics. They employ both arithmetic and geometry, as instruments of indisputable authority, to discover the quantity and quality of the primary movements; and they are as it were, cousins, born of the sisters, sight and hearing, and brought up by arithmetic and geometry as children most closely related in their stock." (Harmonics, 94.13-20; cited by Barker)

Sesqui "Of the superparticulars, if the sesquialter and sesquitertians be applied to the quartile interval of one right angle, which lies between them, the sesquialter makes the ratio of the quartile to the sextile and the sesquitertian that of the trine to the quartile.” Sesqui - one and one-half Sesquiquadrate = 90 degrees + one-half (45 degrees) = 135 degree aspect One & one-third: 90 degrees + one-third (30 degrees) = 120 degree aspect or the trine

Swerdlow on Ptolemy's Harmonics “[Ptolemy's] aspects divide the zodiac into arcs in simple numerical ratios, and the same ratios are found in numerical concords. In this way, the efficacy of the recognized aspects is explained on the basis of harmonics as well as why signs separated by arcs not in the ratios of concords do not have effective aspects. These harmonic causes of aspects are retained in the Tetrabiblos, which, however, omits the explanation in the Harmonics, without which they are barely intelligible.” Swerdlow, p 174 in Burnett Et Al, Ed.