The sides of the Tetractys have the values 1 2 3 4 9 8 27, which are called "the seven boundaries of all numbers" (Mead 164); arranged in this way they are called the Lambdoma or Platonic Lambda. Their significance is described below.

In addition to the Tetractys that increases by addition, 1 2 3 4, the Pythagoreans say there is another that increases by multiplication, that is, in geometric proportion. Plato (Tim. 31C) says that a continuous geometric proportion is the most perfect bond, and so we find such a proportion along both sides of the Lambdoma: 1 2 4 8 and 1 3 9 27.

The duple progression (1 2 4 8) represents "the evolution of the vehicle" proceeding out of Unity, that is, the differentiation and division that constitute the physical body. The triple progression (1 3 9 27), in the order of involution (27 9 3 1), represents "the development of consciousness" as a return to unity, that is, the unification and integration of the psyche. (Note that the Pythagoreans considered 1 to be neither even nor odd, nor even a number properly speaking, but the source of both the even and odd numbers; 2 was the first even and 3 the first odd.) In general, the duple axis represents the physical, temporal, divisible and perishable; the triple axis represents the incorporeal, eternal, indivisible and imperishable.

(Aristides III.24; Mead 165-8; Theon II.38)

(Cornford, Pl. Cosm. 43-52; Plato, Tim. 31B-32C; Theon II.38)

Empedocles (fr. 17) taught that there are two primal forces in the universe, Strife (Neikos) and Amity (Philotes), which are the motive power behind the motion of the elements and the progress of the soul. Strife is the power of division (2) and therefore governs the duple progression; Amity is the power of union (3) and therefore governs the triple progression. Both are necessary to the existence of the universe. Indeed, Aristides (II.17) assigned the duple progression to Ares and the triple progression to Aphrodite, but he also assigned the interior numbers (6 12 18) of the Tetractys to Hephaistos, the Divine Craftsman (Demiourgos) who bound together Ares and Aphrodite when they had once united. Therefore, we will consider the planes again and see how they are bound together by the interior numbers for, in addition to the sides of the Tetractys, its bottom, 8 12 18 27, is bound together by geometric proportion, as is its center: 4 6 9, 2 6 18, and 3 6 12.)

The Pythagoreans considered the even numbers female and the odd numbers male, genders which can be interpreted as suggested by Aristotle (De Gen. & Cor. 716a14): the female is that which generates in itself, the male is that which generates in another. The value 6, corresponding to alchemical Mercury, represents the androgynous psyche as the mean between the female (4) and male (9) psyches, which correspond to alchemical salt and sulphur, respectively. The existence of a mean between the extremes represents good proportion (the Golden Mean), so the corresponding virtues are beauty (4) in the body and moderation (9) in the soul.

Since 2 represents the material principle and 3 the spiritual principle, we find that 8 is the number of earth and 27 is the number of fire, for Plato (Tim. 31B) said that physical objects presuppose three dimensions and are characterized by tangibility, which requires earth, and visibility, which requires fire. However, they cannot be united without a mean to join them, and indeed two means are required to join two cubic numbers. Thus the elements water (12) and air (18) are both necessary to physical unity.

The four elements, earth water air fire (8 12 18 27), are progressively less dense (material) and more subtle (spiritual). The extremes are absolutely dense (2.2.2) and absolutely subtle (3.3.3); the means are relatively dense (2.2.3) and relatively subtle (2.3.3). The subtle elements (air=2.3.3, fire=3.3.3) are both hot (separating), due to an excess of sulphur (3); the dense elements (earth=2.2.2, water=2.2.3) are both cold (coagulating), due to an excess of salt (2). The intermediate elements, water (12) and air (18), both contain mercury (6) as an exact divisor.

The elements satisfy two proportions: earth is to water as air is to fire (8:12 :: 18:27) - that is, as wet is to dry; and earth is to air as water is to fire (8:18 :: 12:27) - that is, as opposites. The four parts in double opposition (hot/cold, wet/dry) represent balance among parts (among the four there are six 3s and six 2s), and so the associated virtues are health (8) in the body and justice (27) in the soul.

(Aristides III.24; Cornford, Pl. Cos. 43-52; Plato, Tim. 31B-32C)

As we've seen, Plane I is the Ineffable One. It is the pivot around which rotate the Axis of the Different (the duple progression) and the Axis of the Same (the triple progression), which rotate in opposite directions and together drive the universe and human experience. The Axis of the Different is the Erratic Cause, which is under the control of Nature (Physis) and moves by Chance (Tyche) and Necessity (Anangke); its rotation (counterclockwise from the Pole) causes cyclic generation and destruction; with it turn the Seven Planetary Spheres (for the planets have different motions). The Axis of the Same is the Inerratic Cause, which is under the control of Art (Techne), that is Reason and Design; its rotation (clockwise from the Pole) causes perpetual generation; with it turns the Astral Sphere (for the fixed stars all have the same motion). (Cornford, Pl. Cosm. 127-34, 141, 160-73; Plato, Tim. 47E-48B)

The three principles (4 6 9 = salt quicksilver sulphur) on the plane of the soul show its ways of apprehending the world. Along the axis of the divisible (1 2 4 8) it apprehends Intermediate Difference (represented by Salt = 4), that is, the mean between Formal Difference (2) and Physical Difference (8), which allows it to distinguish things. Along the axis of the indivisible (1 3 9 27) it apprehends Intermediate Sameness (represented by Sulphur = 9), that is, the mean between Formal Sameness (3) and Physical Sameness (27), which allows it to apprehend the identity of things. By combining the two it apprehends Intermediate Existence (represented by Quicksilver = 6), that is, existence in the psychic world. (Aristides III.11; Cornford, Pl. Cosm. 59-66; Plato, Tim. 35A)

Any geometric progression A:B:C:D generates a second geometric progression A : A+B : A+2B+C : A+3B+3C+D. In this way the "stationary progression" 1:1:1:1, representing the Unus Mundus, generates the duple progression 1:2:4:8, which in turn generates the triple progression 1:3:9:27. (Theon II.51)

Cornford, F. M.,
*Plato's Cosmology: The* Timaeus *of Plato Translated with a
Running Commentary*, London: Routledge & Kegan Paul, 1937.

Theon Smyrnaeus, *Mathematics Useful for Understanding Plato*,
tr. R. & D. Lawlor, San Diego: Wizards Bookshelf, 1979.

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