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THE PYTHAGOREAN SOURCEBOOK AND LIBRARY |
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APPENDIX I: HOW MANY TETRAKTYS ARE THERE? BY THEON OF SMYRNA THE IMPORTANCE OF THE QUATERNARY obtained by addition (that is to say 1 + 2 + 3 + 4) is great in music because all the consonances are found in it. But it is not only for this reason that all Pythagoreans hold it in highest esteem: it is also because it seems to outline the entire nature of the universe. It is for this reason that the formula of their oath was: "I swear by the one who has bestowed the Tetraktys to the coming generations, source of eternal nature, into our souls." The one who bestowed it was Pythagoras, and it has been said that the Tetraktys appears indeed to have been discovered by him. The first quaternary is the one of which we've just spoken: it is formed by addition of the first four numbers. The second is formed by multiplication, of even and odd numbers, starting from unity. Of these numbers, unity is the first because, as we have said, it is the principle of all the even numbers, the odd numbers and of all the odd-even numbers, and its essence is simple. Next comes three numbers from the odd as well as the even series. They allow for the unification of odd and even because numbers are not only odd or even. For this reason, in multiplication, two quaternaries are taken, one even, the other odd; the even in double ratio, the first of the even numbers being 2 which comes from unity doubled; the odd in triple ratio, the first of the odd numbers being the number 3 which arise from unity being tripled, so that unity is odd and even simultaneously and belongs to both. The second number in the even and double [series] is 2 and in the odd and triple is 3. The third of the order of even numbers is 4, and in the odd series, 9. The fourth among the even numbers is 8, and among the odd numbers, 27.
FIGURE 17. THE PLATONIC LAMBDA The ratios of the most perfect consonances are found in these numbers; even the tone is included. However unity contains the principle of ratio, of limit and of point. The second numbers, 2 and 3, have the side ratio, being prime, incomposite numbers, and measured only by the unit, and are consequently linear numbers. The third terms, 4 and 9, have the power of the squared surface, being equally equal (that is to say square numbers). The fourth terms, 8 and 27, have the power of the cubic solid, being equally equal equally (that is to say, cubic numbers). In this way, by virtue of the numbers from this tetraktys, growth proceeds from the limit and the point up to the solid. In fact, after the limit and the point comes the side, then the surface and finally the solid. It is with these numbers that Plato, in the Timaeus, constitutes the [world] soul. [1] The last of these seven numbers is equal to the sum of all the preceding, as we have 1+2+3+4+8+9=27.There are then two quaternaries of numbers, one which is made by addition, the other by multiplication; and these quaternaries encompass the musical, geometric and arithmetic ratios of which the harmony of the universe is composed. The third quaternary is that which, following the same proportion, embraces the nature of all magnitudes, for the place taken by unity, in the preceding quaternary, is that of the point in this one; and that of the numbers 2 and 3, having lateral (or linear) power, is here that of the line, through its double form, straight or circular, the straight line corresponding to the even number because it terminates at two points (the line and circle are given as examples here), and the circular to the odd, because it is composed of a single line without terminus.And what, in the preceding quaternary, are the numbers 4 and 9, having the power of the surface, the two types of surface, the planar and the curved, are so (surface) in this one. Finally, what, in the preceding are the numbers 8 and 27, which have the power of the cube and of which one is even and the other odd, is constituted by the solid in this one. There are two kinds of solids, one with a curved surface, like the sphere or the cylinder, the other with a plane surface, such as the cube and the pyramid. This is the third tetraktys then, the one having the property of constituting any magnitude, through the point, the line, the surface and the solid. The fourth quaternary is that of the simple bodies, fire, air, water and earth, and it offers the same proportion as the quaternary of numbers. The place occupied by unity in the quaternary of numbers is taken by fire in this one, air corresponds to the number 2, water to the number 3, earth to the number 4; such is indeed the nature of the elements according to their fineness or density, in such a way that fire is to air as 1 is to 2, to water as 1 is to 3, and to earth as 1 is to 4. The other relationships are also equal (that is to say, that air is to water as 2 is to 3, and so forth for the others). The fifth quaternary is that of the shapes of simple bodies, for the pyramid is the figure of fire, the octahedron the figure of air, the icosahedron the figure of water and the cube the figure of earth. The sixth is that of the created things, the seed being analogous to unity and the point. A growth in length is analogous to the number 2 and the line, and a growth in width is analogous to the number 3 and to the surface, and finally a growth in thickness is analogous to the number 4 and to the solid. The seventh quaternary is that of societies. Man is principle and is thus unity. The family corresponds to the number 2, the village to the number 3 and the city to the number 4; for these are the elements which comprise the nation. All of these quaternaries are material and perceptible. The eighth contains faculties by which we are able to form judgment on the preceding, and which are its intellectual part, namely: thought, science, opinion and feeling. And certainly thought, in its essence, must be assimilated to unity; science is the number 2, because it is the science of all things; opinion is like the number 3, because it is something between science and ignorance; and finally feeling is like the number 4 because it is quadruple, the sense of touch being common to all, all the senses being motivated through contact. The ninth quaternary is that which composes the living things, body and soul, the soul having three parts, the rational, the emotional and the willful; the fourth part is the body in which the soul resides. The tenth quaternary is that of the of the seasons of the year, through the succession of which all things take birth, that is, spring, summer, autumn and winter.The eleventh is that of the ages: childhood, adolescence, maturity and old age. There are thus eleven quaternaries. The first is that of the numbers which are formed by addition, the second is that of the numbers formed by multiplication, the fourth is that of magnitudes, the fifth is that of simple bodies, the sixth is that of created things, the seventh is that of societies, the eighth is that of the faculties of judgment, the ninth is that of the living things, the tenth is that of the seasons, and the eleventh is that of the ages. They are proportional to one another, since what is unity in the first and the second quaternary, the point is in the third, fire is in fourth, the pyramid in the fifth, the seed in the sixth, man in the seventh, thought in the eighth, and so forth with the others following the same proportion. Thus the first quaternary is 1, 2, 3, 4. The second is unity, the side, the square, the cube. The third is the point, the line, the surface, the solid. The fourth is fire, air, water, earth. The fifth is the pyramid, the octahedron, the icosahedron, the cube. The sixth is the seed, the length, the width, the height. The seventh is man, the family, the village, the city. The eighth is thought, science, opinion, sense. The ninth is the rational, the emotional and the willful parts of the soul, and the body. The tenth is spring, summer, autumn, and winter. The eleventh is childhood, adolescence, maturity and old age. And the perfect world which results from these quaternaries is geometrically, harmonically and arithmetically arranged, containing in power the entire nature of number, every magnitude and every body, whether simple or composite. It is perfect because everything is part of it, and it is itself a part of nothing else. This is why the Pythagoreans used the oath whose formula we have reported, and through which all things are assimilated to number. From Theon of Smyrna: Mathematics Useful for Understanding Plato, Chapter 38. Translated by Robert and Deborah Lawlor. San Diego, Wizards Bookshelf, 1979. Reproduced with permission of the publisher._______________ Notes: 1. Plato, Timaeus 36BC.
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