The constants of nature

— The constants of nature. A study in the early history of natural law : Philosophia IV 1974 211-246. | L'analyse de la méthode d’Eudoxe ainsi que de ses théories sur la lune, le soleil et le mouvement des planètes montre que | ce sont des proportions du type xy : (x+y) = nTeoTM> : n qui sont à la base des paramètres astronomiques d'Eudoxe, Cette méthode est fondée sur le triangle pythagoricitn, dillérent pour chaque planète. Daus le Tinée, Platon ' utilise cette théorie pour sa propre astronomie fragmentaire. [1441 >27 > MARA ERKKA MAULA, Hauho, Finland € MAA THE CONSTANTS OF NATURE A STUDY IN THE EARLY HISTORY OF NATURAL LAW* The concept of scientific progress is often characterized by saying that «what was previously considered a constant of nature, in later analysis turns out to be a variable» (cf. Ketonen [6], pp. 477-480). Although ambiguous, this maxim nevertheless points towards a number of relevant factors worthy of a closer examination in the history of science. First of all, it refers to the Greek idea of knowledge that is based on invariances discovered in nature and arranged into rational (or, at times, speculative) patterns. Secondly, it refers implicitly to the perpetual oscillation between permanent features in * This paper was read at the Research Center for Greck Philosophy of the Academy of Athens in May, 1974, at the invitation of Professor Johannes Theodoracopoulos. I hereby would like to pay homage to my teacher, Professor Oiva Ketonen, University of Helsinki, on the occasion of his sixticth birthday. His book on the Great World Order [6] kindled my interest in the history and philosophy of the exact sciences. — References to the Bibliography are made in square brackets, references to formulae in round brackets: [1] Cornford, F.M., Plato's Cosmology, London 19561 (2} Dicks, D.R., Early Greek Astronomy to Aristotle, New York 1970 [3] Hare, R.M., Plato and the Mathematicians, in New Essays on Plato and Aristotle, ed. R. Bambrough, London 1965 [4] Heath, T.L., Aristarchus of Samos, Oxford 1913 [5] Heath, T.L., A History of Greek Mathematics, vol. 1-2, Oxford 1921 [6] Ketonen O., Suuri Maailmanjärjestys, Helsinki 1948 [7] Lasserre F., Die Fragmente des Eudoxos von Knidos, Berlin 1966 [8] Maula E., Ancient Shadows and Hours, in «Annales Universitatis Turkuensis», Ser. B, Tom. 126, Turku 1973. [9] Maula E., Studies in Eudoxus’ Homocentric Spheres, Helsinki (Comm. Hum. Litt. vol. 50 - Societas Scientiarum Fennica) 1974 [10] Mittelstrass J., Die Rettung der Phänomene, Berlin 1962 [11] Mittelstrass J., Neuzeit und Aufklärung, New York 1970 [12] Neugebauer O., On the Hippopede of Eudoxus, «Scripta Mathematica» 19 (1953) [13] Neugebauer O., The Exact Sciences in Antiquity, Copenhagen 1957° [14] Schiaparelli, G.V., Le sfere omocentriche di Eudosso, di Callippo e di Aristotele, «Publl. del R. Osservatorio di Brera in Milano» IX (1875).

The Constants of Nature 212: (expressed e.g. in the maxim «like knows like» or any such phrase of the Socthe world-order and permanent features in the formulae that are supposed to express the natural laws, in the search for such invariances. And finaliy it brings into the concept of a scientific paradigm the important methodological aspect. For although the history of science is often done in the narrow sense of recording only the results of science, these to my mind should be contrasted with the contemporary world-view, scientific terminology, epistemological ideal, and analytical means. While the quest of the cosmological and epistemological context of the superseded results justifies the philosophical investigations of a past science, terminological studies undertaken even in complete linguistic isolation do not seem to need any justification. But the modern historian of science still seems to be obliged to give reasons for investigations of past scientific methods, perhaps due to ancient doxographical traditions, which recorded the results but omitted the methods of science. Witness the history of the theory of homocentric spheres, to the beginnings of which this case-study is devoted. But since the bearing force behind the ever renewed historical interpreratic and Atomist schools). Nay, we may start with the admission of changes in nature and nevertheless look for invariances in these changes, expressible in clear-cut formulae. Hence not even Zeno (nor Plato in the Parmenides) has fathered the concept of natural Jaw. although he certainly has done much for the growth of the axiomatic method, What about the Atomists, then? Certainly the Vortex might have implied a distinction between the concepts of a period (T), angular speed (0 = 1 /T) and linear speed (v= rq), although this perhaps never can be documented. Likewise the concept of the resistance of the intermediary substance was capable of explaining a number of natural phenomena, including the Sun’s longitudinal anomaly, the seasonal Mediterranean North-East winds and the brightness of the celestial objects (excluding the Moon). And we happen to know that the mathematician Hippocrates tried to axiomatize the Atomists’ doctrines. But even if we take all these ingredients together, they do not amount to what might be called a natural law. At best, the conglomerate could be called a qualitative description; the descriptive element is simply too strong for the explicit formulation of a natural law, altations of past events is the accruing knowledge of the interdependence of the though it might have been judged adequate for the contemporary concept of historical facts and the (improving) historical methods,1 do not see any jusworld-order. For we can suggest a likely set of axioms which Hippocrates tification for the separation of the results from the methods in the discusprobably pondered: they have included references to the dualism of void and matter; a very great, or even infinite number of atoms: a finite number of kinds of atoms; a very great, or even infinitely great space; a very long, or even infinitely long duration of time; and so forth. We cannot be very far from the birth-place of the concept of natural law, however, for Democritus’ auxiliary concepts include the celestial sphere, and there is the suggestion (VS 2, p. 141) that one of Democritus’ works dealt with «projections of the sion of other sciences cither. True, this combines the interests of the historian of science with those of the philosopher of science, but I believe that both parties will gain from this combination. l The semination of the scientific law of nature. Supposing there is a clear distinction between an invariable world-order and an invariable formula that is meant to describe this world-order, what would the first explicit formulation of a natural law look like? We need not be concerned here with certain modern schools which tend to eliminate the whole distinction in favour of an entirely formal approach to all questions oforder, for this was not the Greek way of thinking. Nor do we need to restrict ourselves to the extreme Parmenidean concept of strict immutability of the world-order even at the expense of explaining all change as illusory, although such a view (in a modified form not alien to some moderns) might be expressed by one invariable formula or diagram. Nor do we armillary sphere on a plane». It is the last point mentioned, the rational element and geometrical design exhibited in the world-order, that makes it futile, in my opinicn, to try to extract any natural laws (in the modern sense of the term) from the Babylonian mathematical astronomy. True, there is an element of prediction and skilful arithmetical techniques, and the practical results are at times even better than the Greek ones. But, as Neugebauer has pointed out ([13]. cf. p. 156), the Babylonian methods «nowhere point to an interpretation through a combination of circular motions or any other mechanical model». And the Mesopotamian mathematical tools, the zigzag and step functions, practicalneed to look for such doctrines alone which assume that the search for truth presupposes a structural isomorphism between the natural processes and the method of investigation (like the dialectics of nature and the dialectical method), or a functional isomorphism between the knower and the known 1. Neugebauer has also shown that Egyptian mathematics simply was not adequate for astronomical calculations of any complexity.

ly exclude such models. I am willing to admit, though, that this is a matter of taste. One can well consider such revisions of the concept of natural law as do not presuppose any underlying geometrical models. The idea of starting from «pure facts» might even appeal strongly to a bold spirit, although in that case the principles of the economy of thought and of the axiomatic system, and the concept of Protophysik, which have already shown their fruitfulness in the interpretation of Greek science, perhaps must be jettisoned. It is not unthinkable, however, that a discovery of one single clay-tablet may drastically change our views on Babylonian science in these respects. For the reasons given above, I would suggest that the first «natural laws» are to be found in Greek science, and in the period after Democritus. This implies Pythagorean astronomy and mathematics as their conceptual environment — not Pythagoras’ own teaching and not eve nthe Philolaic system, but rather the Platonic—Eudoxan research programme of «saving the phenomena». The postulates of uniform circular motion and constant angu- The Constants of Nature 215 my working hypothesis in [9], which has turned out to be quite fruitful, that Plato’s frame of reference was Eudoxus' theory of the homocentric spheres. Now Eudoxus’ theory was developed further mainly through generalization. Hence the scientific progress within this paradigm presumubly could be outlined with reference to the maxim about the constants and variables in the explicit formulation of a natural law. On the other hand, if we start from the later formulations of Eudoxus’ theory — from Torriano in the XVI Century, through Ptolemy, Hipparchus, Apollonius, Eratosthenes, Aristotle and Callippus — and return to Eudoxus again, we must ask whether all later variables are reduced step by step into constants. At least some of them do reduce. But alterations have been made also with respect to the axiomatic element, the models, and the computing techniques. Nonetheless the degeneration of variables into constants is a useful lar velocities, where the diurnal rotation of the sphere of the fixed stars is measure, since it indicates the accuracy of observation and the length of observation series. In Ptolemy and Hipparchus the eccentricity of the solar orbit for instance remains a constant that was included in their laws of nathe swiftest of all (for these assumptions see Mittelstrass, [10], [11}), adumbrature in the planetary theory. In fact the eccentricity is not a constant, but the te the axiomatic element. The celestial sphere accounts for the geometrical period of its regular change was too long (T = 96000 years) for the ancient instruments. Again the obliquity of the eclipticwas considered a constant by the ancients*. Its regular change within certain limits (21939 Se 24°36’) was not observed, because of the long period (T = 41000 years). And to take a third example, the motion of the solstitial and equinoctial points was not discovered before Hipparchus, alt hough it can be observed in far shorter periods (50.26° p.a. or 1° in about 72 years: yet Ptolemy still used a constant: 1° per century, Syntaxis, vii, 2). This means in fact that no earlier observation series could be consulted with respect to star-maps, although observations of the planetary periods (of the Babylonian type, where random errors in the model. And Plato’s insight into the role of mathematics (for which see Hare, [3]), together with the Academicians' and Eudoxus' contributions to, and refinements of, the contemporary (Pythagorean) mathematics, will provide the mathematical techniques and the formulae for the expression of a «natural law»?. I would like to add that although Plato on many grounds can be discussed separately from Eudoxus, the Timaeus fragmentary astronomy can, and should, be discussed with reference to Eudoxus’ cosmology. Indeed it was 2. I would like to remind the reader here that J am using the terms «natural law», «laws of nature» and «law» in rather a modern sense, without claiming that Plato or Eudoxus ever used these terms to denote the concept of natural law. On the other hand, I am inclined to think that, no matter which terms were used, e.g. desmos as in Plato, the concept of the laws of nature was properly understood by Plato and Eudoxus. Perhaps I may quote from a letter of Professor Johathan Hodge, University of Pittsburgh. «My only reservation is about your phraseology, and concerns the use of the word «law». It may well be that the Greeks believed, enunciated and used many propositions that would meet any reasonable criteria for the application of the term «law». However, this would have to be argued for explicitly, surely, given that they very rarely invoked the legal metaphor in asserting and specifying orderliness in natural changes. I have not found the literature on «law», «nalong run balanced each other) may have suggested rounded-off values. As we noted, the progress made within the paradigm of the homocentric spheres touched also features other than periods of the theory. Hence we must look for quite different constants in Eudoxus” system, too, Changes in computing techniques, for instance, may have contributed to the gradual abandoning of the predilection for integers and simple ratios in Pythagorean mathematics. Of this predilection there are clear indications also in Eudoxus. And changes in the model (the celestial sphere with 26 nested planetary spheres) and in the axiomatic element, made it possible for Eudoxus’ followers taral law», «law of nature» etc. very helpful in this regard, since it does not really confront the key issue directly: Were those, like Philo and the Stoics, who first systematically explicated their convictions about nature’s orderliness in terms of the legal, or better, the constitutional analogy, were they simply giving a new gloss on an old doctrine or did they transform it in the process of rephrasing it?» 3. Or do we hear the first voices of doubt in certain explanations of the Milky Way as the Sun's «original route»? (Aëtius III |, 2, cf. Heath [4], pp. 117-118). Professor Holger Thesleff informs me that if these ideas were Pythagorean, they probably derive from the mythological akousma tradition and not from the «mathematical» sect.

The Constants of Nature 217 to abandon certain Eudoxan constants, for instance the equality of the astronomical seasons. Eudoxus’ equalization meant jettisoning Euctemon’s and for the third solar sphere (for the discussion of this feature by other com- Meton's discoveries of the inequality of the seasons (given in Ars Eudoxi as 90, 90, 92 and 93 days; the modern values to the nearest whole day are, 92, 89: 90 and 94 days) made some sixty years earlier. And this is just one example of Eudoxus’ omissions of (direct) observational data, for which he was criticized even in ancient times. On the other hand, considering Eudoxus' achievements in mathematics (see e.g. [12]) and mathematical geography (see e.g. [8]), it is not warranted to simply record these omissions, but we must study to what extent they derive from his method. Likewise we must study how long observation series would have shown that they a re omissionsWith these broader aspects in mind I shall discuss Eudoxus’ method and the considerable number of constant values ascribed to his system in tradition. is at our disposal as soon as we have means of solving a generalized quadramentators see Lasserre [7], pp. 201-203). The computing technique in question tic equation (which in fact gives the periods of two combined spherical motions studied in terms of their plane projections), conceived 0) theory of the irrational numbers draws directly on Eudoxus (Eucl. v, Def. 5), and there is a direct line from Eudoxus to Weierstrass, too. (cf. Heath [5] i., pp. 326-327). Moreover, it is well known that, through Archimedes, Eudoxus has begun anotner line of development, which leads to integral calculus. In these directions Eudoxus' results fertilize much later mathematical thought, because his methods, the theory of proportion and the method of exhaustion, were known, too. It is different with Eudoxus' astronomical results: Schiaparelli [14] had to substitute his own conjectures for Eudoxus' parameters, since he did not know Eudoxus’ method. 2. On the background of _ T Although (1) is superficially similar to Apollonius’ pivotal formula (see Ketonen [6], pp. 130-143), which can be given in the form (2) Tox" Tsid (where Tsyn, Tsid are the synodic and sidereal Te periods of a planet and Tsim the period of the pe = Tsun LE Sun) there is no other explicit connection except this formal resemblance between (1) and (2). We shall not discuss in this case-study whether there is a historical connection between them, discoverable in a more general survey. How is (1) constructed? Leaving aside the problems of the transmission of Babylonian scientific knowledge into Greece about one generation earlier than Neugebauer presumes (sec [13], esp. pp. 150-151) — for in my opinion they will be satisfactorily answered by my results — I submit that the startingpoint has been the so-called «normal forms» of Babylonian mathematics. In these «normal forms» (Neugebauer’s term) two numbers should be found when (a) their product and (b) their sum or difference is given: xy =a @) xty=b Transforming the two dur of (3) into two linear equations af xFy=V Kay b? = da the solution follows (see [13], p. 41) as reconstruction of Eudoxus’ method and computing techniques and see whethindeed is one computing technique which is capable of explaining both the known unproblematic features of Eudoxus’ system and, in addition to these, its known oddities and idiosyncrasies, e.g. the fictitious deviation postulated tion and n an integer or rational fraction) I would now make the additional claim that the proportion (1) is the explicit er these parameter values can be obtained through their medium. For various reasons, which it is unnecessary to repeat here, I suggested that there A » A (where Teomb js the period of the combined mon n formulation of the only «natural law» needed in Eudoxus’ system. Eudoxus' method. I have outlined the reconstruction programme for Eudoxus' theory in my studies in the homocentric spheres [9]. It amounts to claiming that all previous attempts have failed because they have started from the scattered parameter values attached to the system. Instead, one should begin with the ‘comb 4 x+y And lest the more general historical perspectives be forgotten, it is as well to remember the depth of Eudoxus’ mathematical insight’. Dedekind’s of as a proportion: (5) RIE : From the two equationsListe of (3) we EM VO tee — 6) (6) — Xy X+y a = — b 4, For a discussion of Eudoxus’ importance in the history of mathematical analysis see e.g. my paper The Elements of Analysis, Proceedings of the XIVth International Congress commonplace. Alternatively we could start from the concept of a (generalized) harmonof the History of Science, 19-27.8.1974, Tokyo and Kyoto, Japan. ic mean; cf. [7], pp. 175f. 5. There is no obvious term for equation in Eudoxus, but «proportion» of course is a

which is, of course, an obvious step as soon as (3) is known. Moreover, (6) can be conceived of as a proportion. But there is only one solution to (3). In order to gain more freedom of operation, it is quite natural to generalize (3) or (6) by multiplying both a and b by n (where n is an integer or a rational fraction). It is a nat ural step for The Constants of Nature 219 in the sphere», but he must also face the logical consequences of this statement. If any two planetary motions are combined and computed, this means, in Eudoxus’ day, that their projections are drawn on a plane, and the computations are made with reference to their angular speeds. This is a very difficult research programme, unless sufficiently strong analytical and syn- Eudoxus especially, since it can be based on the Elementa, v. 15, which, according to all traditions, belongs to Eudoxus' contribution to Euclid. But making now a/b = Tmb we obtain the proportion (1). That is to say, we have introduced an auxiliary parameter n into the proportion (6). What does (1) express, then? Before the methods of spherical trigonometry were introduced (say between Menelaos' day and Theodosius’ time), either only qualitative results were obtained or else graphical methods were used in the studthetic means can be provided. Let us take the combination of the motions of ies of spherical motions. As Neugebauer says ([13] p. 161), one of these seems to have been based on the discovery that stereographic projection of the sphere maps circles into circles. Hipparchus, who had no spherical trigonometry at his disposal, may have solved spherical triangles by the method of stereographic projection. More modest problems of circular motions in sphere, however, may have been studied even much earlier by means of their I have used here the indices (ind, comb) to denote the individual and combined motions. Since the rotation of the sphere of the fixed stars is not caused projections to a plane. Witness tne work of Autolycus of Pitane®. This is still only one generation after Eudoxus. But then we have the title of Democritus’ work mentioned above, which suggests that problems simitar to those the first and second Eudoxan spheres of any planet as an example. Because the diurnal westward rotation of the fixed stars, represented by the first planetary spheres, is the swiftest motion of the system, and because the second spheres (according to the Eudoxan tradition) are credited with an eastward motion, their combination is of the following type: (7) wind”) — qind(E) = RR by any other outer motion or agent in Eudoxus, the first sphere is credited with the individual motion alone. The directions of rotation are also indicated (W or E), and I reserve the negative sign for eastward motions. Subindices (I, IT) point to the first and second sphere. Now, since =1/T (cycle per period), we obtain from (7) the form: (8) | I Ty" Tes I TT Tp of Autolycus were discussed even before Eudoxus. If indeed «armillary spheres» of any complexity were constructed’, the discovery that stereographic where the directions of rotation have been omitted. But (8) is tantamount to projection maps circles into circles would have been almost inevitable. But we shall see that even far simpler graphical means, viz. the cross sections of part of the proportion (6), which can be generalized so as to obtain (1). the celestial sphere, will suffice in the analysis of Eudoxus' circular motions. stones in [9]. It amounts to the assertion that all second, third and fourth The result of this analysis of combined motions is one of the corner- I am inclined, therefore, to follow Dicks here and to drop too advanced and planetary spheres in Eudoxus, in so far as they are combined, must be credanachronistic instruments from Plato's table— leaving only a sphairion ited with tw o motions (one individual, another combined) and hence also at most (cf. Ep. ii, 312 d). So I assert that this was enough for Eudoxus, and with two periods. But the extant Eudoxan tradition ascribes one motion and that (1) gives the period of the combined motion of two other motions (which one period only to each planctary sphere. Hence either one must credit Euare combined) doxus with (oo advanced mathematical methods, or drop the idea of comcharacterized by their angular speeds, provided we can bined motions, or else be prepared to account for the «extra» motions and «solve» the proportion (1). combined periods. Now there are in the extant Eudoxan traditions certain hints, which and characterized by their angular velocities? It is perfectly in order for a classical scholar to maintain that «Eudoxus combined the planetary motions Eudoxus' lunar and solar theories. Having shown my hand, however, I shall But what does one imply by saying that two motions are it is unnecessary to rally here, at these «additional motions and periods» in simply proceed and see whether the results will warrant my claim. 6. Written perhaps about 330-300 B.C.; cf. The Books of Autolykos: On a Moving Sphere and On Risings and Settings, ed. and tr. F. Bruin and A. Vondjidis, Am, Univ, in Beirut, Beirut 1971. 7. See Cornford's opinion [i], pp. 74, 135; Wilamowitz, Platon, M, p. 390; and Apelt’s Platons Dialoge Timaios und Kritias... n. 89, p. 163; but consult also Dicks [2] p. 120 sq. Before going into a detailed analysis of Eudoxus' method, it is wise to remember that if (1) indeed is the o nl y «natural law» in Eudoxus, it must be capable of explaining a great number of things. Not only the known planetary periods but also others only vaguely referred to, like the «long» Junar

The Constants of Nature 221 and solar periods implied by the «slow» motions of the respective third spheres. And in addition to these the directions of rotation known from trathird dition (from Simplicius on), the axial inclinations of the third and fourth spheres, and so forth. These I call surface parameter values. They Plato’s «great ha rmonia»we obtain, as duly pointed out in [9], a rightangled triangle which determines an excellent value for the Eudoxan obliquiin turn depend on what J call structural parameter values. The ty of the ecliptic. Its hypotenuse is equal to 39 = 3 + 9 + 27 or the sum of structural parameter values include the Eudoxan value for the obliquity of gnomon observathe ecliptic (which cannot have been based on direct the sum of Plato’s «double intervals», and 36. The triangle in question aptions, since the Sun was credited with a fictitious additional deviation from the ecliptic). That obviously restricts the choice of the axial inclinations menpoint reinforces the second one. On an additive interpretation of Plato's «triple intervals», and its shorter sides equal 15 = 1 + 2+4+ 8 or pears in the cross-section of the celestial sphere as follows. tioned. The traditional periods, too, may depend on one or several other periods (considered more fundamental for some reason) — be that the (unknown) Eudoxan luni-solar cycle, or Plato’s «perfect year» (Tim. 39), should eo | (wherec = 39,a = 15, b = 36, and a = 2237) my working hypothesis about the frame of reference in the Timaeus hold good, or yet something else. And also the pattern followed in combining planetary motions belongs to the structural features essential in the reconstruction, of Eudoxus' system. Now my claim that (1) is the sole «natural law» in Eudoxus implies that whatever hierarchy of parameters we take, it is crowned by (1), on which all structural parameter values depend. Since these parameter hierarchies are discussed in more detail in [9], I give just one example of their interesting connections, Ît is a fact that the Eudoxan obliquity of the ecliptic can be computed starting from T=30 days for the month, and T= 360 days for the year, on which all other traditional planetary periods in turn depend. What are these particular connections? First, T= 360 days is the calendaric year and T = 30 days is the length of the calendaric «full» month. The sidereal periods are «one year» for Venus and Mercury, two for Mars, 12 for Jupiter and 30 for Saturn. Second, all synodic periods known from the tradition, can be conceived of in the following way. Tgr, = 570 days = 360 + 780 i ; period. T5... = 110 days which, since the Greeks rounded off by simply cancelling the fractions (cf. [13], p. 68), can be made 11029. days = 2. Fr days or double the harmonic mean of the calendaric month and year. Tu. = 260 days = 1/3 . 780 days = = 47/3. 30 + 360 2 days or four thirds of the arithmetical mean of the calendaric month and year. And Tr. Jupiter = Tr Saturn = 390 days = 2. tropic, CD a diameter of the winter tropic, equator, CBa diameter of the ecliptic, N EF a diameter of the the North 30 + 360 2 or double the arithmetical mean of the calendaric month and year. The Pole, S the South Pole, and a the obliquity of the ecliptic. In determining a the relative lengths of the sides of the triangle are needed. Hence we can equally well multiply all sides by ten and obtain (a = 150, b = 360, and c = 390). The full importance of this triangle will be seen later, but suffice it to say that b = 360 and c = 390 can be conceived of as geomettical representations of the Eudoxan periods for the Sun, ‘ A “am days or the arithmetical mean of the calendaric year and the true Martian synodic Ss Fig. 1. Cross section of the celestial sphere. ABis a diameter of the summer Jupiter and Saturn, while (c - b) = 30 represents the lunar period of thirty days. But the same triangle, the preconceived geometrical scheme which made it necessary to postulate the fictitious solar motion, can be obtained also through the application of the rule for generating Pythagorean triples (see [13], p. 39). (9) ¢ = p° + q° (where p and q are arbitrary integers which a =p? — q? are relatively prime and not simultaneously b = 2pq odd, and p>q) Several ancient commentators have attested that there are Pythagorean triples in Plato, who knew either (9) or some more particular rule. If we take Plato's basic «triple» and «double» intervals (3 and 2), and make (p = 3, q =2), we obtain (c = 13, a =5, b =12) from (9). It may be noted that this is the simplest case where (q Z£ 1). Multiplying the sides (a, b, c) by thirty we

The Constants of Nature 223 obtain the same triangle (c = 390, b = 360, a = 150) again. Hence there is in obtain the equation n? + 120 n — a? = 0. Hence n =— 60 + / 607+ af, fact both a «Pythagorean» and a «linear» or additive genesis for it. This means, among other things, that at least some Eudoxan periods can be interpreted in terms of Pythagorean triples (and hence in terms of p, q). and also in the geometrical terms of line segments (a, b, c), and the same holds for at least some acute angles. In fact, we shall see that a 11 Eudoxan periods can be so interpreted. This amounts to a synthesis of the Pythagorean and Hip- Here again (60? + a?) must be a square; let it be denoted by c?. Now the auxiliary variables a and c are obtained from the following Pythagorean trianpocratic mathematics. Perhaps these remarks will suffice for a discussion of the background of Eudoxus’ method. 3. Eudoxus’ method of analysis and synthesis. I shall now discuss the mathematical character of the computing techniques needed in solving (1), taking Eudoxus’ lunar theory as my startingpoint. This might well correspond to the actual order of ancient studies, as suggested by Plato’s educational programme for astronomy (cf. Epin. 990b, 990c-d, Rep. vii, Tim. 39c-d). Because (7) holds for the Moon, too, the comgle: c =p*4 q? a =p?— q?, and b = 2pq = 60 (see (9) ). Since 2pq = 60, we must consider the following cases: Pa 30 Pp 4 6 | i 5 790 37° . (10) an + agi = aggnb (11) apro — aid = afm (where Th > Tyr) (12) womb (Mm — wind (E) = — fem. (E) (where Tit? < Tee) If tradition (from Simplicius on) is correct about the westward rotation ascribed to the third lunar sphere, however, only (10) or (11) will apply. The lunar period T = 30 days, which we have discovered, may belong either to the second or to the third lunar sphere. Let us discuss first the case Tomb = 30 days. If so, tradition implies that Tird is «long», since wit! must be «slow». Whether this will be so, can be seen by respect to Tomb — 30 days. We obtain the proportion xy solving (1) with 30n ofSee onl. ans determined by p and q in the following 03 xty — n © 3 | ae 15 2 15 LI” 15/2 phrase can always be given in terms 30 30 1 3949 30/1 of a/p = tan(—) na ui dun The lowermost case (p = 30, q = 1) implies indeed that particular form of (9) which Proclus (On Eucl. I, p. 487, 7-21; cf. Heath [5], i, p. 81) ascribed to Plato. If we choose it, we can draw the corresponding Pythagorean triangle diagrammatically as follows. b= os 349 © piq = 30/1 a= p'- q’ = 699 2pq = 60 Fig. 2. A Pythagorean triangle for the tentative discussion of the third lunar (combined) motion in Eudoxus’ system The construction of this Pythagorean triangle is the culmination of the analysis of the lunar motions. The auxiliary parameter n has given rise to an auxiliary drawing of a synoptical character. We can now «turn backwards» and calculate first » (which could easily be illustrated by a diagram like Fig. 1). Itis ny =— 60-+c = 841 = 29? (while n = — 961, being a negative integer, may be omitted; cf. Heath [5], ii, p. 464), Hence from (14) x =Tind = = 870 days (which could well be the «long» lunar period), and y = Tomb — = 29 days (which is the «hollow» calendaric month). And Téomb = 30 = p/q = 30/1. This is the end of the synthetic part. Now even the last vestiges of the auxiliary parameter », introduced in the first step of the analy- _n+y ni 1208 tical part, have disappeared $. The geometrical by-product of the procedure NN_ EnFy ?nn? = 120n is the acute angle a = 3° 49”. [t may be characterized in terms of q /p; q/p= y = 2 Let us discuss the negative sign case of (13) first. (i) If we are to obtain rational solutions, (n? + 120 n) must be a square. Let it be a?. From this we way: 30 30 days (or the «full» calendaric month). In the last step we form the ratio x /y According to (5), the solution of the «normal forms» corresponding to (13) is ca degrees and minutes for the sake 6/5 bined motion of the second and third lunar spheres comes from one of the following formulae: (where the acute angle a is given in p/q = 1/30 = tan (=>). Presumably « is the axial inclination of the third lunar 8. Hence it is literally true that Eudoxus' system could not have been reconstructed from his results, but his method must be known, too.

The Constants of Nature 235 sphere to the second one, which is equal to the Moon’s maximum deviation (n=x— y =c—b > (p — q)* from the (Eudoxan) ccliptic. For although (22°37’-+- 3°49’) is too small in AL pp BE = arbre > q(p — q) the proportion Y msi; a— bc comparison with the correct value in Eudoxus’ day? (23° 44° + 5°08"), the negative maximum deviation (22° 37° — 3°49’) is a bit too great (as compared - (16)< with the real value 23°44’ — 5°08"), and since the Eudoxan obliquity of the a b—c Solution of the «normal forms» — mb x—y =n constituting vonb e ecliptic was not defined in terms of the Sun's third motion, all planetary deviations from it necessarily have been compromises. If we choose any other combination of p and q, neither the axial inclinations nor the periods obtainx /y = ua p /q x—y n Solution of the proportion c—a Hence the final solutions are invariable with respect to the ratio p /q, which ed will do. The solution of the positive sign case of (13) follows the same pattern. is a special ratio for each planet. In both cases the two acute angles can be characterized also by means (ii) If we are to obtain rational solutions, (n° — 120 n) must be a square. Let it be a®. From this we obtain the equation n° — 120 n — a? =0, Hence n = of their trigonometrical functions, i.e. as ratios of the gnomon = + 604/60? + a?. Here again (602 + a?) must bea square; let it be denotshadow (see [8]). ed by c?. The auxiliary variables a and c are obtained from the same Pytha- (17) tana == = b/a; and tan (90%— a) DE 202 to its = à /b gorean triangle as above (Fig. 2). Turning again backwards from this culmi= 96! =31* (while n, =— 841 but this is unnecessary, because the paraphrase in terms of q /p is enough. may be omitted). Hence from (14) x = Tijf = 930 days (which could well If Tomb = pin (1) and q = 1, the Pythagorean Solution and the Linear be the «long» lunar period), and y = T#"? = 31 Solution are equal (as in the case of the Moon). If not, they differ, but the nating point we compute n. It is n, =60-+ 901 days which, however, was not a length of a month in Eudoxus’ time. Yet no other combination of p and q will give a better solution, either. The point is, however, that the solutions (i) and (ti) of the «normal forms» corresponding to (13) can be generalized so as to apply to (I), and furthermore, in two ways. One generalization is in terms of p and q, and it might be called the Pythagorean Solution. The other generalization is in terms of the sides (a, b, c) of the right-angled triangle which is the culmination of the analysis, and it might be called the Linear Solution. I give first the two generalizations for a positive sign case of (1), and next for the negative sign case of (1). fn =x+y=c+b — (p+q) Solution of the «normal forms» te, Loto 2 De pi x+y=n b+c— a (15)< y = > 5 . è g(p +q ) a constituting xy Teomb y the proportion ts = = triangle needed in the Linear Solution is always reducible to the triangle pertaining to the Pythagorean Solution (of the acute angles)" by simply multiplying all sides by a coefficient. Indeed, this was seen earlier when we discussed the triangles based on Plato’s «great harmonia». Since the Moon is the simplest case from the methodological point of view (in other planets the two solutions do not coincide), it is reasonable to assume. as we did, that it has been the first target in Eudoxus’ studies. Considering the computing technique sufficient for solving (1), it seems warranted to divide the whole procedure into two parts. The first part consists in finding out the two triangles starting from the fact that double the period of a combined motion (Tomb) equals the side b of the triangle pertaining to the Linear Solution. This part might be called the Method of Analysis. Its culminating point is the construction of the two triangles. When these have been found out, the rest can be computed as shown in (15) and (16). The second part of the procedure might be called the Method of Synthesis. It ends with the forming of the ratio x /y = p/q, which we call the he - ° =p/q Solution ofthe proportion solution of the proportion. This use of language in my opinion corresponds to Hare’s results in [3] and offers an interesting opening for future discus- L 9. See Dicks [2], n. 240. 10. It may be noted that the solutions of x and y in terms of p and q are reminiscent of the well-known ancient Greek problem called «the application of the arca». it. Eudoxus might have had access to tables listing or characterizing the angles in terms of q fp, but also direct observation will suffice, since q /p= tan (a /2), when tana = == 2pq /{p® — q?), according to Eucl. vi. 3. PIAOXOGIA

sions of the influence of Eudoxus’ (und Plato’s) mathematical methods on the contemporary philosophical methods of analysis and synthesis. As an example one might mention Plato’s methods of the One and the Great-andSmall. I think that the above outlined Eudoxan methods, together with the Pythagorean method of approximating the surds, will throw light on them. After having read carefully Proclus and Pappus, I would also say that their accounts of the methods of analysis and synthesis are compatible with my The Constants of Nature + 33 days (which were used in ancient times, but, as far as we know. not before Callippus; for more detail see [9], Table 2 and notes). But no likely results are obtained: n will not remain an integer, and the axial inclinations computed remain unlikely. (The relatively best result ensues from T=29 = days or the mean synodic month; n = 59? /2 and p/q = 60/1 or u = 1°55’). reconstruction of Eudoxus, which might stimulate the current discussion. In this paper, however, I shall continue with an application of (15) and (16) to 227 (vi) If T=30 days does not belong to Tim, Tind, or Teom but to Tied Eudoxus’ planetary theories}. in (8), it can be shown that there will be no «long» lunar period at all for the third lunar sphere (all lunar motions being combined). For these rea- 4. sons I conclude that (18) gives the correct solution, viz. The Moon. Starting from the traditional Eudoxan synodic periods we have discovered a lunar period of 30 days. Above, one alternative was discussed, viz. Tomb = 30 days. It remains to investigate the other alternatives, in which either Tit, Tromb, or Tind is equal to 30 days. I refer to the previous alterna- I The directions of rotation as in (10); Tist = | 3 days — from (8); (19) p/q =29 /1; n = 900; x = 870 days = Tim; y = 30 days = Tyem?; Tomb — 29 days; and a = 3057" tives by (i) and (ii). (iii) What about making Tip"? = 30 days in (1)? If nothing more is known, the positive sign case is reduced to (i) and the negative sign case to (ii). It is easy to see, however, that the values of Te and ee can be in- It may be noted that the periods of 30 and 29 days need not be justified by calendaric reasons only. For since the Moon’s motion in fact is not uniform (contrary to the Platonic-Eudoxan postulate), the length of the synodic month terchanged, if the sign is changed, t00. Consequently we must discuss also (i.e. the period between two consecutive new-moons) is not always equal to the proportion 29 > days either. In fact the difference of two synodic months may be about 08) 870 . 30 Fao ps = 29 n n° thirteen hours. Hence the periods T= 30 days and T= 29 days may reprewhich gives an even better value for the axial inclination: p/q = 29/1 or a = 357 (while 930 . 30 “930 — 30 = 31 days gives a worse value a = 3°42’). (iv) There is one more possible solution which employs the integer lengths viz. 30 . 29 30 — 29 Tim. 39c, where Plato indicates the synodic month. The two triangles corresponding to (19) are equal and can be represented I think that this is the correct solution. of the calendaric months, sent the upper and lower limits of the synodic month, accurate enough for practical (e.g. calendaric) purposes. Therefore (19) is compatible also with diagrammatically in the following way. = 870 days, but this implies an altogether implausible axial inclination: p/q = 30/29 or a = 1°57. (And the positive sign case does not yield an integer solution). b= 2 pg = 2 29? = 58 (v) Next we may try to apply the «normal form» technique and experiment with the period T = 30 days, taking instead of the other lunar period a = 357 of 29 days some other length of a month, e.g. T = m 291 or 29 + 8» p’q’? 2209! - 17 = 840 & p/q = 29) Fig. 3, The lunar triangle for the combination of the second and third lunar motions. 12. In the Appendix my former student and present collegue, Mr. Eero Kasanen discusses my reconstruction of Eudoxus’ method from an algebraic point of view. It can easily be checked that (15) holds good.

The Constants of Nature 229 The Sun. portant results required. Eudoxus’ method We have discovered a solar period of 360 days. Since in the Eudoxan deviation postulated for the third solar sphere. Wher it is added to the Eutradition (from Simplicius on) the third solar motion is credited with an ea s timplies the obtained fictitious doxan obliquity of the ecliptic, quite an accurate positive deviation in latitude ward motion, the combination of the second and third solar motions canis obtained (22° 37’ + 1° 15.5" as compared with the real value e = 23°44’). not be of the type (10) if the tradition is correct. Hence the solutions do not And the negative (fictitious) deviation is still within the limits of Eudoxus” come from the proportion observational accuracy (sec [2], p. 155). xy pis. x+y 360 n Furthermore, the seemingly superfluous axial inclinations postulated nn for the third planetary spheres now also make sense. For starting from (8) we which implies westward motions only. But the combination cannot be of the obtain for the Sun by the same method as in the previous cases discussed, type (12) either, because in that case T;em® would become «long», in conan axial inclination. But Tío"? = 360 tradistinction to Eudoxus’ lunar theory, and an implausible axial inclination (in a geocentric system)!5. Hence the axial inclinations of their third spheres, would be implied (a = 89°22’). Starting from the combination of the type the poles of which are on the ecliptical plane, can be understood to be the same as that of the Sun. This (11), we obtain the solution from the proportion (21) xy X — y days for Venus and Mercury strange feature from the oldest tradition (Arist. Met. A 8, 1073b 30—32), too, is implied by Eudoxus’ 364 n also method. For the other planets, these axial inclinations will be different in each case, and n solving different from those of the Moon and the Sun. As for the periods obtained, (20) )?* might be called the «seasonal year». For it will be remembered that no better value for the year will do. We cannot make, e.g., Toom’ = 365 Eudoxus omitted Euctemon’s and Meton’s discoveries and equalized or 365 > days (just as this type of operation was impossible in the case of where the period Teomb — 364 days (which has been found by the astronomical seasons. The two triangles pertaining to the Linear and Pythagorean Solutions the Moon). But it is as well to remember that Eudoxus complemented his are different from each other in the case of the Sun. They can be represented astronomical work with calendaric studies’*. Hence the «cosmological condiagrammatically as follows. stants» obtained do not seem implausible. coefficient 1/4 reduces to 6. The fourth motions of planets proper, Since ahippopede was created for each planet proper by the third and fourth planetary spheres, rotating in opposite directions in the synodic period (except for Mars which is credited with a period exactlyone thir d of the b - 2-364 true synodic period: for an explanation of this see [9], pp. 82-83), their motions m= 115,5" <> p/q = 91/1 2 a= 33120 2 a-p-q = 91 2 -1 2 cannot be combined in the sense of affecting each other's periods. However, 8240 Fig. 4. The two triangles for the third solar motion; the Linear Solutions come from the left, the Pythagorean Solutions from the right the motions of the fourth and second planetary spheres can be so combined (see [9], p. 112-113). What is more, due to the Eudoxan arrangement where The solution is: p/q = 91/1, n = 32400, x = Tin? = 32760 days or 14. Eudoxus’ observational accuracy can be estimated, thanks to Hipparchus’ comthe «long» solar period, y = Tromb — 360 days or the calendaric year, and mentary (Comm. in Arat.) As regards the tropics, equator, arctic and antarctic circle, the a = 1915.5’, It may be noted that we now have reached one of the most imerror ranges from 1° - 3° in the majority of the cases. See Dicks [2], p. 155-156, but also [8]. It may be noted that x = 32760 days = 91 years of 360 days each, implies a fictitious solar motion (not to be confused with precession) of about 4° a year, and indeed there are 13. Censorinus associates a year of 364 1 {2 days with Philolaus { De Die Nat.). Being based on a progression of 3, just as Plato’s triple intervals, this may tell about the contemindications of such a motion in Eudoxus, The period of 91 years meets our expectations concerning the length of Eudoxus’ observation series: they are far shorter. porary discussions; 364 is the nearcst integer if fractions are simply cancelled. — The so- 15. Cf. Dicks [2], n. 345. lution cf. (20) with p /a = 90/1 gives u = 1°16". 16. See Dicks [2], pp. 188-189.

the axes of the third spheres of planets proper are situated on the ecliptic, The Constants of Nature 231 The two triangles pertaining to the solutions are as follows. the inclinations between the third and fourth spheres can be computed. The attached Fig. 5 will illustrate the arrangement. bs 2pq: 2-19-? Fig. 6. The two triangles for the fourth motion of Venus The solution is: p/q = 19/7, n =617-—, x= 977, y = 360 days, a = Fig. 5. A cross section of the celestial sphere. EE = the equator, LL = ecliptic = the axis of a planet’s third sphere, BB = the axis of the ecliptic = the equator of a planet's third sphere, AA = the equator of a planet’s fourth sphere (inclination to the ecliptic appears as a), and CC = the axis of the same planet’s fourth sphere. Hence the inclinations ¡ of the axis and equator of the fourth sphere to the axis and equator of the third sphere appear as the complement of a. = 40°27’. Hence i = 49 33° (Schiaparelli in [14] gave i = 46°). Mercury The proportion is (for a westward rotation) y (25) 0 Ia = n = The computation of the periods of the combined motion of the planets’ second and fourth spheres is made easier by the fact that both the sidereal The two triangles pertaining to the solution are as follows. and the synodic periods are known. Otherwise the computations follow the coefficient = 13/40 same pattern as in the cases of the Moon and the Sun. If the tradition (from Simplicius on) about the westward rotation of the fourth spheres is correct, the combined motions of the second and fourth spheres must be of the be 2-110 10 following types (and the lengths of the synodic and sidereal periods will de- 13 termine which one applies for a given planet): (22) cogomb (W) + wind (W) — wsomb (W) (23) cafone (W) — qind (E) — ont (W) a = 200 (where Find > Tomb) Fig. 7. The two triangles for the fourth westward motion of Mercury Because the combined period of the fourth spheres in principle is observa- The solution is: p/q = 9/4, n = 520, x = 360 days, y = 160 days, anda = ble, it is clear that it is equal to the planet’s synodic period. Likewise the si- = 47° 55°. Hence i == 42° 05° (Schiaparelli gavei = 23°). dereal period is equal to the combined period of the second sphere. Hence the proportions, the corresponding two triangles, the solutions, and the in- It will be seen that, in contradistinction to Venus, the inclination i differs considerably from Schiaparelli’s (conjectural) value, Since he had startclinations of the fourth spheres to the third spheres are obtained as Venus The proportion is (24) xy x—y __ 570n on follows. ed from the assumption that i for Venus and Mercury represents these planets’ maximum elongation, we may consider an alternative in the case of the direction of rotation of the fourth sphere of Mercury. If Simplicius is not right in crediting the fourth sphere of Mercury with a westward motion, but was led to do so because of the analogy of other planets, the combination of

Erkka Maula the motions of the second and fourth planetary spheres of Mercury may have been of the following type (cf. (12) )!”: (26) pomo Wi _ cond (E) — — wromb (E) (where TE < Te If so (or, in case (12) was the combination of the third and second spheres and i was computed from it), the solution will be: p/q = 17/4, x = 360 days, y= 347 days, and a = 26°29". Because the positions of the sides (a and b) can be interchanged (cf. (17) ), it may have been that è = 26°29’, which The Constants of Nature where the period 233 T= 260 days, instead of the true Martian synodic period Ty" = 780 days, probably is the period of tre actual loop (see the argument in [9] pp. 81-84). Besides, Schiaparelli has shown that if the true synodic period is taken, Mars will have no retrograde motion at all, no matter what value / is given (see also Dicks [2], pp. 186-187, and Lasserre [1], pp. 205-206). The two triangles pertaining to the solution are as follows: would make a fairly good approximation to Mercury’s true maximum longation (e = 27°45’; for Venus e = 47° 30°). It must be remembered, however, that due to the eccentricities of orbits, the elongations vary. In Mercury (17° 50° < e S 27°45’) and in Venus (45.5° < e S 47.5°). In ancient times the elongations were generally given as e = 20° N. or S. for Mercury and e = 50° b = 2pq- 2:23 13 E. or W. for Venus (cf. Chalcidius, Comment. § 70, p. 138, ed. Wrobel). The fact that Mercury’s elongation was given North or South, may indicate a peculiarity in the treatment of Mercury, like the one suggested above. But even a=p’-q? = 237-137=360 i = 42°05’ is possible, for Hipparchus finds errors up to 23° (and in one Fig. 8. The two triangles for the fourth motion of Mars. case nearly 60°) in Eudoxus' description of the colures (Comm. in Arat. i, 11, 9-21; ii, 11, 21; cf. [2], p. 155). Mars The proportion is n OQ xy __ TE = 260n - 17. Whatever arrangement we accept for the computation of the inclination f for Mercury and Venus, we must compare their directions of rotations to those of the Sun For in the Timaeus Plato speaks about motions in opposite directions. Tm. 36 d may be a very concise summary of the motions of the seven sets of planetary spheres in Eudoxus. But at 38d the motions of Venus and Mercury are contrasted with the motion of the Sun. Three The solution is: p/q > x == 720 days, y = 406 + days, and a = 58057. Hence i = 31903" (Schiapatelli gavei = 34°, but in all outer planets his inclinations arc pure guessing, guided by modern spherical trigonometry, for the outer planets may be at any angular distance from the Sun). (IfTrem® = 780 days and the solution comes from (26), i == 35°09, or p/q = 25/13). Jupiter possibilities may be discerned here. (a) If the ¿was computed from (12) or (26) for Mercury, it possesses a dynamis contrary to the dynameis of Venus and the Sun,viz. the power (whatever it was) causing the eastward combined motion of the fourth sphere, in contradistinction to the westward combined motions of the innermost spheres of Venus and the Sun. (b) If the fwas computed from (12) or (26) for Venus, too (= 63°, which is still possible; see the note on Eudoxus’ observational accuracy in the description of the colures), both Venus and Mercury possess this contrary dynamis.(c) The simplest explanation is, however, = 23/13, n= 1126 (28) The proportion is xy — 390 n that since all three have the same period (ibid.) of one year in the zodiacal motion, Venus and Mercury are contrasted with the Sun on the account that they have four motions in Eudoxus' system. The motions of their third spheres of course are contrary to the Sun’s ecliptical (or near-ecliptical) motion and so are their retrograde motions. — For adifferent explanation sec Cornford [1], pp. 105-106, but consult also [9], p. 32. n. Heath [4], pp. 165-169, discusses at some length certain ancient explanations. I must add that Harold Cherniss in a letter to me doubts any reference in the Timaeus to Eudoxus’ systems. Be that as it may, the only things that I really have deduced from Plato are the foxotes and the clue into Pythagorcan triples. Fig. 9, The two triangles for the fourth motion of Jupiter.

The solution is: p/q = 131/13, n = 12.360.144/131, x = 12.360 days, y 92 = 428 131 days, and a = 78°40’. Hence i = 11°20° (Schiaparelli gave i = 13). The Constants of Nature 235 all projections equal zero and the particular solutions are superfluous. If they were studied in equatorial projections, the northward motions mentioned in the tradition will appear as having a direction opposite to the fourth motions (since hippopedes are formed). Saturn The proportion is (29) xy _ 390n QA 8% The hippopede constructions. Schiaparelli’s conjectural hippopedes have been studied in detail by sev- The two triangles pertaining to the solution are as follows. eral generations of ancient and modern commentators. In the Bibliography coefficient = 347/30 of [9] I give the titles of the most representative investigations, and it is unnecessary to dwell on the topic here. Suffice it to say that Schiaparelli”s hippopedes are close enough to mine to give an approximative summary. But it must be emphasized that although his inclinations (i) are rather near to mine, mine are implied by Eudoxus’ reconstructed method, and do not ensue from modern observation and mathematics (cf. Neugebauer [12]). 9 Conclusion. Fig. 10. The two triangles for the fourth motion of Saturn. Lest the somewhat cumbersome figures seem suspicious, it may be advisable to do the computations necessary in detail, taking Saturn as an example. From (15) we obtain x = 30 . 360 days Ad and b = 2.390 days. I have tried to show that all astronomical parameters, known either accurately or in principle from the Eudoxan tradition, and characterizing the Eudoxan theory of the homocentric spheres, can be obtained as soon as means are discovered for dealing with a generalized proportion of the following type: Hence (30) a+b+c= 60. b 360 Here x,y are two planetary periods (of which one is known or postulated in =60.13 a+c = 60. xy: (x + y) =n Tomb: n advance), n is a positive integer or a positive rational fraction needed in the 347 Now pjq= 2 72 = 347/13 (a paraphrase for i = 4" 15°; Schiaparelli gave generalization (cf. Eucl. v. 15) and Tomb js a planet's synodic or sidereal period, known in advance from observation. The main body of my paper consists in the reconstruction of Eudoxus’ method, which, when applied to the i = 6°). And y= q/p . x = 13/347 . 30 . 360 = 390/347 . 360 days (tan observational planetary data («the phenomena to be saved»), produces the >= y /x = q/p). This is all. astronomical parameter values. These include both real, observational values (of which some are rather accurate) and entirely fictitious ones. The success 7. The third of the reconstruction may be judged from the fact that also the fictitious motions of planets proper. values, such as the inclination of the Sun’s third sphere, can be obtained. The main types of parameter values obtained are the planetary periods It is, in my opinion, a master-stroke of methodological economy that the combinations of the second and third motions of the planets proper all (c.g. Tird = the period of the individual motion of a planet's second sphere, are of the type (12) which has not been used elsewhere. And even if (26) was Tio" = the period of the combined motion of a planet’s second sphere, used for the combination of the second and fourth spheres of Venus and Meretc.), the directions cury, (10) and (12) will suffice for the third motions of the planets proper. westward individual motion of a planet’s fourth sphere, etc.), the inclina- If they were studied in their ecliptical projections, as one may well assume, tions of the spherical rotations (c.g., DI = the of the spherical axes and, being deducible from these, the maximum

The Constants of Nature 237 deviations of a planet’s sphere from the equator (including the Eudoxan solar period, Tip"? = 364 days = the «shorter» period = the «seasonal value for the obliquity of the ecliptic) and from the ecliptic. year» 4 . 91 days, and u = 1°15.5' In essence, the reconstructed Eudoxan method is based on Pythagorean Venus: Tomb — 360 days, = appr. half an el li (p/q = 91/1). DE), incl. = the same as for Mercury mathematics, and pains have been taken to show that it can be backed by propand ositions belonging to Eudoxus’ known contribution to the Elementa. Eudo- Tigo = Tomb = 570 days, and incl.,,= 49933’ (p /q = 19/7) . (Note also the xus’ method consists of two parts: an analysisandasynthesis. Because Eudoxus' importance is acknowledged by almost all subsequent promotors of mathematical analysis (for more detail see my paper The Elements of Analysis, «Proceedings of the XIV International Congress of the History other alternatives discussed above). of Science», 1974, Tokyo and Kyoto), and because Eudoxus’ method has affected much contemporary philosophical analysis and synthesis, my reconstruction may be of some interest to the historian and philosopher of the exact sciences. Besides, it may illustrate the logic of scientific discovery. At the heart of the reconstructed method (discussed from the algebraic the Sun, while for other planets this will be different for each, Mercury: Tomb = 360 days, Dina) , Trent = Tyomb = | 10-13 days, and incl.y = 42"0S’ (p/q = 9/4). (Note also the other alternatives discussed above). Mars: Tomb — 2 , 360 days, Digi , Teomb — Tromb— 260 days (which is the period of the actual loop, while Ts = 780 days), and incliv = = 31° 03’ (p/q= 23/13). (Note also the alternative Tremb = 260, Tromb = point of view in the Appendix) lies in each case a Pythagorcan triangle, dif- 780 days, i = 35° 09’). ferent for each planet. Its sides are generated from two relatively prime, unequal integers p. q in the usual way, and the triangle itself has the role of an auxiliary drawing in a geometrical proof, In terms of this triangle both the auxiliary parameter n and the solution to (30) can be given, and its angles determine the axial inclinations and maximum deviations. The solutions to (30) are invariable with respect to p, q (which may be called «the mathematical constants of nature») and can be obtained in each case in the form: x /y = Jupiter: and incl.py Tomb = 12 . 360 days, Dind(w) , Tcomb — Teomb — 390 days, = 11° 20° (p /q = 131 /13). Saturn: Tim = 30 . 360 days, Dit , Tomb — Tiomb — 390 days, and inclv = 417" (p /q = 347 /13). Loxotes: a = 2237 (p/q = 3/2); the real value in Eudoxus' day e = = 239 44’, p /q. The solution gives rise to the tangent (= ratio of the gnomon to its shad- It will be seen that calendaric considerations indeed played a role in Euow) of the angles of inclination, say i, and maximum deviation, say a. In fact, doxus’ concept of empirical data, and that the Aippopede constructions realit also suggests the main features of the astronomical instrument (arachne) ly aimed at approximative (or idealized) descriptions of the observed planewhich Eudoxus has used in measuring angular distances. Moreover, the intary motions. But it must have been something of a shock for Eudoxus and verse of p/q = tan + or= tan > (cf. Eucl. UT. 18, VI. 3), which leads to his followers to discover that one and the same method of computation indeed produces parameter values that were well known from the past. It must the theory of stereographic projection, expedient in the calculation of anguhave appeared as if in (1) a «law of nature» had been invented. A «law» which, lar velocities, paraphrased in termes of periods in (30). These further implialbeit simple, nonetheless indicates in an economical interpretation the «concations of the reconstruction, however, will be discussed elsewhere. Suffice stants of nature». it to say that Eudoxus’ method can be understood as a geometrical treatment The inner beauty of Eudoxus' system, even in the barest outline drawn of time, and that his main heuristic aid has been the Pythagorean numerical in this paper, is gripping. The simplicity of the mathematical tools, the very analysis (cf. lambl., Jn Nicom. arithm. 10, 17), concrete, yet strikingly effective methods of analysis and synthesis, the ex- I list here the most important parameter values explained. (D = directreme parsimony of the explanans and the ingenious interpretation, tion of rotation). contrasted with the multitude of the phenomena explained. are to the full The Moon: Tyomb = 30 days, Dist) , Tid — 870 days = the «long» credit of Eudoxus’ genius. His solution to the cosmological problem is characterized by strong methodological monism, which Philipp of Opus problunar period, Tromb = 29 days = the «shorter» period = the «hollow» month, and a = 3°57’ (p/q = 29/1), y/x =q/p = 1/29 = tan (a 2). ably refers to in"the Epinomis 991e-992a. Other details may be omitted here, The Sun: Tomb = 360 days, DISSE , Tind = 32760 days = the «long» but suffice it to say that if for instance the sum and difference of p and q in

each particular solution are considered, it will be noted that the same numbers (12, 13, 30, 36, 90, and 360) occur at all levels of the explanation, strong- The Constants of Nature 239 remains that Eudoxus’ model was in full correspondence with reality at given times only. For surely no physical body can rotate on three or four spheres ly suggesting that one has seen into the depths of the celestial architecture. At the same time, however, Eudoxus’ system was capable of further development by his followers. Hence it is not a system of sterile rigidity like Aristotle's, but the beginning of a dynamic paradigm. of different Eudoxus continued active studies on Cnidus until the end of his life. Strabo (C 119), following Poscidonius, mentions that he observed Canopus (a Carinae): this is probably connected with his attempt at an estimate of the diameter of the Earth. But the mathematical foundations of the Greek astronomy (and ours) were laid, as cogently as it was possible for Eudoxus, in his doxus’ model with the «saved phenomena», as well as his strikingly concrete theory of the homocentric spheres. If the reconstruction of Eudoxus’ method submitted above is correct, the philosophical discussion of such themes as the real meaning of the principle of «saving the phenomena», the concept of time and its mathematical treatment, and the heuristic value of the methods of analysis and synthesis in other, more general problems, may begin. When viewing Eudoxus’ theory of the homocentric spheres as a whole. radii at the same time. And different radii they had, as Aristotle clearly implies (notwithstanding what modern commentators have claimed), for that is essential if stereographic projections are used in the calculation of angular velocities. Nevertheless, the «temporary» agreement of Eumethod of analysis (or should we say, in view of A’ rpad Szab6’s results concerning the origins of terms for «proof» and «showing», Eudoxus’ optical method of analysis), are extremely effective theoretical tools. It is with great delight, therefore, that I can end this paper by pointing out that the attitude towards models and their agreement with reality was exactly of the type required in Plato's Academy (cf. Ar. Met. 990a). For my working hypothesis even in this paper has been that Plato in the Timaeus is using Eudoxus’ theory as the frame of reference for his own fragmentary astronomy. It is from the Timaeus that I have discovered the clues to Eudoxus' Pythagorean triples and Eudoxus’ obliquity of the ecliptic (and one notes the striking discrepancy between the geometrical model and the is insufficient physical reality. Not only is it so that a geometrical framework geometrical in dealing with the computation of angular velocities, and the results obtain- 37c Plato uses the term agalma for the relation between the World-Soul and ed approximative. The model, albeit in motion, is far from the modern ideal of physical models which try to preserve the possibility of checking at any the planets, and [ have shown elsewhere (Plato’s Agalma of the Eternal Gods, time. Eudoxus’ model, on the other hand, «saved» a fixed number of phein Plato is a pregnant philosophical metaphor, exceptionally apt for the denomena at a given time, and it had predictive value in the case of cach scription of the relationship between «everlasting» and «temporal» beings. In Plato’s great harmonia provides the arachne with two scales, musical and harmonic scales for angular measurement). «Yearbook of the Philosophical Society of Finland», 1969) Now, that at Tm. agalma planet at a given time only, viz. at the time when a planet’s maximum devia- Plato’s hands, agalma has undergone a change from a religious concept into tion was observable. With respect to these «saved phenomena», however, the model is in full correspondence with reality. That is to say, the language of Eudoxus’ theory exhibits the idea of logical or semantical atomism at certain a philosophical one. But it has preserved some of the original religious confixed times. and alive», (iii) it is the object of the Demiurge’s delight, and (iv) it seems to One can understand this view of language against the contemporary semantical theories in Greek philosophy (see e.g. my paper The Semantics of notations. The agalma at Tm. 37c is (i) a likeness of its paradigm and the Demiurge tries to make it «yet more like its pattern», (ii) it is «set in motion have a function similar to cult-statues of which the divinities are supposed to discuss — at least during the exalted moments of actual worship. Time in Plato's Timaeus, «Acta Academiae Aboensis», vol. 38, No. 3, 1970). It is in this intellectual atmosphere that Eudoxus presented a geometri- For the Pythagoreans «the whole universe is filled with numbers» (cf. Ar. cal model that «saved the phenomena», but fully corresponded to reality at Met., 1090a 20 sq.), but for Aristotle already the contents of the two boxes, given times only. Like so many later mathematicians, Eudoxus has created one containing the language and linguistic models, the other containing reality, are different. We can almost put our finger on the point of divergence: ever is done with the model will suggest particular solutions to problems disat De Caelo 289b Aristotle discusses the possibility of motions of stars and their circles independently from each other. Hitherto, a planet and its sphere had formed a unity, one physical body if we wish to put it so. But even if we grant this somewhat alien idea of physical bodies, the fact a visible model, the heuristic value of which lies precisely in the fact that whatcovered in reality. Is this not the same perennial ideal that also philosophers like Leibniz and Wittgenstein have aimed at with their theories anchored to semantical atomism?

The Constants of Nature 241 blem (4) be generalized accordingly? We introduce a parameter n. Let x; + xs * APPENDIX be equal to n, then x, . Xy = Teomb . n. Thus the characteristic features of An algebraic the original problem have been maintained. Eudoxus’ assumed contribution view. 10 Elementa includes In Eudoxus' cosmological construction, angular velocities (inverses of periods) can be combined by projecting them pairwise fiom one sphere to (5) I another. From the mathematical point of view, such combinations can be which gives evidence of the preceding mode of reasoning. Problem (2) is alexpressed by the following formula: (1) Ti + zT = a an == ====5 En (v1, . 15) omb tered into an equivalent form DE Xo — Coll. and so the respective Ta T, — Teomb #1 2 normal forms acquired will be ra In order to master the combined motion Eudoxus thus had to be capable of X» Xa XE 2) (6) = Teomb With the known mathematical tools — n O n ° ‘comb => + V(=) FTom.n Xp. X_ == Tomb y solving the algebraic problem: Given Tomb give x, and x, such that ide . XG solution 5 maFY ts 5) re r of that age the problem was unsolvable. Hence we try to find out a possible reconstruction by the anal- When testing this result the following conclusion can be immediately drawn: ogy of which Eudoxus could have solved (2) usingthe methods of analysis and synthesis. Although the following algebraic formalism is allowing parameter n to take several values an infinite set of pairs Xy, Xe perhaps anachronistic, contemporary mathematicians, however, would have meter n which logically corresponds to auxiliary designs in the geometrical been capable of making essentially the same reasoning. satisfying (2) is obtained. The only flaw is the permanent presence of paraanalysis. To begin with let us consent to Erkka Maula’s working hypothesis that Eudoxus knew the (far older) Babylonian method of normal forms to solve sis. Partitioning is compensated for by building the ratio anew. The soluquadratic equations. tions x}, Xy in (6) are moulded into the ratio x, : x, which is called the so] ub X (3) problem b )* solution n = «#70 Y (2 7+. ) When looking for the characteristic features of (2) in the synthesis restoring the ratio split in the analytion of the original problem. Calculating + tab We proceed to the ., x analysis we n n a FT com 2+ÿ (3) n n n n E FTcom 2.0 JFTcom b (3+/ (5) y? Tromb Y (5) Frs realise that there are in fact the equated ratios of left and right sides of the = normal forms, i | (4) Xy Xy =Teomb X1 + X2=1 | solution | xy =1 pryl AT Teomb x=>23+12F VI/4F Tem | | After this observation it is natural to examine the solution of (4) — and to our pleasure it satisfies the original problem (2), Heuristic «optical» partition yielded a fruitful result — let us continue in the same manner. Problem (2) has an infinite set of solutions and basically (2) is a ratio. How can pro* By Eero Kasanen, University of Turku. Teomb + we witness the striking disappearance of parameter n. All pairs of number X1, X» Satisfying the condition (8) X, __ XF Temb “= a also satisfy the original problem. The value of parameter n= x, + xs is seen in (6). What have we gained? The problem was divided into parts by using the above outlined method of analysis. Parameter n was introduced to maintain @AOZODIA 4

the characteristic features. The pair of simultaneous equations was solved by The Constants of Nature 243 "EôG x,y eivur S00 riavntixés nepiodor (Gnd tig Sxoies À piu siva yrwoth the method of normal forms. The solution was defined and calculated in the i) mapudsdeypévyn dc afloja ¿x tOv apotépov), n eivar Betixdc axépatos synthesis in which parameter n was also eliminated. 9) Oetixdg pytds rAucputixòg GpiOuds, énupairntos oth yevixevon (xpBA. Let the following figure illustrate the situation. Na a YX EúxA. V - 15), «ui Tomb eivan à cuvodiKki) solution x problem 2 Tram _- - - - — - a X1 EX Xp | ! x, F Tomb um dotpiKi] neplodog Evög xAa- VATH, YVO0OTH tpoxataPoaAtkas And Tv rapathpyon. Tò kúpio pépos tod GpOpov xpuypatedetat tiv úvakotackevh this ue068ov tod Evsdtov, ñ ônoiu, Stav ¿pupuocóR otà rAuvntikà and nmapatynpiosic Ôeboué va («yd va owdobv tà parvopeva»), diver tic adotpovojuKés TULÈG THV rupapétpov. analysis 2” abtéc nepıAunßavovran Kai moayuarikéc, ÉURELPIKÈG TUUÈS (pepixes and tic Önoieg Eivar paAAOV áxprBeis Xz « Xe = Tomb Kai GAXEG teheios nAucnurırkg). H Enttugia tig Gvaxatacxeviic propet và kp18f and tó Sti uropodv va ¿EuxBobv axdpa Kai of rAaouatiIKÈG TiLÉG, synthesis & parameter n analysis eliminated ónos 7.x. fi rokALON TAG tpitng opaipas tod “HAiov. Oi küpiot tTÓTO: TiuGv napapétpwv rod éEjyOnoav siva of tikég neplodor TA av n- (x.y. Tid = ñ nepiodog fig dronikfig xtvijoeas continued & tig debrepng opaipas Evög nAavnim, Tomb = A nepiodog cvvévacpévns parameter n Kıvfaeag TS debrepng opaípas Evdc rAavitn KAr.), of introduced popes tHv opaipixOv repiotpopüv (1.x. Did W = à dtopuc) xivnon tij¢ tétaptyns opai- Y X1:X2 TT. N the method X,+x,=n nj 3 “ofnormal forms =z "Fr Tomb | n n pas ¿vos nAavitn rpoc Svopas, xAr.), ol ATOKAITELG THY CMApIKOy úatóvov Kai, abtd rod propet và ouvay0n ax’ abréc, of péytotes a pe KxAicetg tig opalpas Evög TAUVATH and Tov ionuepivó (6nou repidapPavetat kai ñ tum tig Aodórntoc ris ExAeintixfi¢ tod EVSSEOv) kai and tiv ExAELNTLKT. ‘H pédodos tod EvdédéEou nov dvaxatacxevdoape Pacileta: obctacti- Ka otà rudayopixa Maßnpatıa xataBúdane npoonüdeıss và SeiEwpe Gti propel va Ünootnpix0f pè rpotüoeis, Tod avijKovv ott yvooti ouußoAn OI ETAOEPEZ THE PYZENZ MIA MEAETH IIA THN IIPQ:I-MH [ZTOPIA TOY ®YEIKOY NOMOY Avon Kai pid oùvôeon. “Eredi à onuaoiu tod Edô6Eov dvayvwpiterat and Ghous oxedov tods pEtayevéotepoug dnadodc ts pabnpatikfic ävéAuonc (na nepiooôtepes Aentopépetes PAéne tò GpOpo pov The Elements of Anal- Ileptanyn.* ysis, «Proceedings of the XIV International Congress of the History of Età pedéty adri yiverar tpoondGera và Serx0r, Sti GAec of dotpovopixég rapàpetpor, rod elvas yvwotés site éxnaxptfds cite Kat’ dpxtv and civ rapúdoon tod Evddtov Kui tod yapaxtnpitouv th Oempía tov yiú tic 6uÔKevtpeg opaipes. uropoüv va ¿¿ay0odv Stav Bpe0r tpónos và pelerndobv pè pid Yevixevpévn a&varoyia tod torov: (30) tod Evddtou ota Zrorgeia. ‘H pébodos ünotekeitu And dúo pépn: più avaxy: (x + y) =n Terb:n. Science», Tokyo /Kyoto, 1974), Kai éxe.di) à pé0050¢6 toò Evddtov Exnpéace TI) ovyxpown piAocopırı) avadvoy kai cbvOecn cè peydàn Extacn, À dvaxaTAGKEVT] propei và rapovoiáln Kémroio Évôtapépov Yiù tov iotopixd Kai TO piAdoopo TÓV BetiKdv ¿mortnuóv. “Axdpa, propri va diapotion Th Ào- YUKT] TG Eriotnpovikiis avaxdAvyns. X16 xévipo tíic uc066ov rob GvaxatacKsvdcape (Kai nob dnd thy GAyePpixy droyn ousnteitu otd napdptnpa tod dpOpov) Ppioxetar Eva nu- * Metégpaon àrò TO ayyAıxö M. Apayhva-Movézov. dayopırö tpiyovo yiù «Ads repintwon, StagopetiKd ya xáde nAuvim. Ot

‘O Eöôoëos EEuxohoúlnoe Evepyds tig pedétes tov otiv Kvido ds 16 The Constants of Nature TEAOG T6 Gais tov. 'O Erpáfwv (c 119), dxodovOdvtus tov Tlocsıd@vıo, üva- Erkka Maula pBjtods p, q HÈ td GuvnMicpévo tpdz0, Kui TÔ 1810 TÔ tpiywvo Éyet TÔ pbrAO nAevpts tov rapáyovrat dnd dbo oyetiKas MPWToUG dvicous axepaiovs dpéper Stt rapuripnoe tiv Kávoro (a Curinae). Tobto npopavög ouvôéetat HÈ tiv npoozáderd tov và bnodoylon tH Sidpetpo T6 Tic. “AAAG ta padnpoTika Oepédra tijs “"EAAnvixijg “Aotpovoplas (kai tic löikfig prac) téOnouv 500 . 244 Bondntikoö oyediov où pu yewperpikid ródeiën. Téoo à PonOyntixh zapéuetpos n 660 xai Avon otòv túro (30) propoöv và SoBoöv GE ouvéprnon npög TÒ tplywvo adtd, Kui of Ywvisg tov KaBopibovv tic &Eovikës àromid ioyupä yıyörav ytd tov EvdoEo pé th Oewpia tov tv ópoKévrpov opaipav. Otav Gewpijon Kaveic tH Bempia tv SpoKévtpwv opaup@v tod Evs6- Kai elyav rpäyparı Siagopetixés drives, Sra caps rpobroBétel 6 ’ApiorotéAnc (napa tots Stapopetixots ioxuptopobs vewtépwv oxoltaotüv), ati udrd elvar odoiddes, Stav ylverar yphon otepeoypapixdv npoßoAäv vo o& tpets i] téooepses opatpes HE StL Aa MOPETIKES Akılveg ovyypdvac. KOv copátov, yeyovög mapapéver Sti 14 npótvno Tod EvddEou Bpioxétuv où nAñpn avrandxpion HÈ thy TpaypatiKdtyta póvo ab dedopévous Xp6vous. Tıari Beßaiag xavéva puGIKÒ cpa Sév propel và neprotpépetar nütig tpoxtäg tous üvekäprnta and ti pEtTaEd Tou oxéon, Evo péxpr tore Evac navaAtys koi fj cpaipa tov oxnparCav Evörnta, Eva puoixd cou, Où Aéyope. "AAAG Kai üv dxépa Sex8obpe abri Tv kéros Goxetn 1Séa Tüv puottiká. Mrofoüpe oxeöòv và ynAupiowne và onueta vijg Suaqopac. td epi obpavod 6 ’ApiorotéAns ovtnté tn Övvaróenta tic Kivnons Töv Gotpav Kai Kai td YAwooïk npOtvna Kai TÔ GAO Tv xpaypatixdrtnta, elvar Siapops- TEA Ta nepiegópeva Kai tv 860 «Kovrıöv», nob TO Eva nepiéger tt YA@oou pes àprOp@v (npBA. ‘Apiot. Metag. 1090 a 20 én.), GAAG Hôn yıd tov ’Aptoto- YOVIUK@V taxotHtov Kal pè Te Kata Tpooéyyion EnttevySévta anotehéopata. TO npéturo, dv xai kivypatixd, ànéyer TOAL And TO vebrepo löuvıkö tOV vaikGv rpotbrav, nob npoonulobv và diatnphoouv tH Svvardtyta Enurndevong (EAÉygov) GE Ónotovönrorte xpôvo. Td npórvro tod Eùöóbov Gpws «Eowoer Eva orudepd Apıdud puivouévov oé Se 50 p évo xp Ovo xai ele npopntikn déla ott nepintmon Ka0e nAuvñTn póvo GE dedouévo ypovo, 5nAGd} otd ypoviKd oypeto örov à peyiorn napéKxKAron Évôs KAuvytH ftav Suvatov va napatnpnôf. Mavtws dvapoptkò pe ubTa Ta «owbévru HalvopEve» TO RpdtvTO àvranoxKpiverar TAñpos npòs Tv apuyuutikétntu. ‘H yA@oou tis Oewpiac tod Edô6Eou éxOétet tiv lôéa tod Aoyikod À «onpavtikod» droptopoù o& bpiouévous otaBepoùs xpévou. Mropei xaveig va KataAGBy tiv äroyn abın yıa Tv yA@oou of cûyKplon HÈ obyxpovés TNG Bewpieg «Lnpavtixiicn oliv EAAnviki PiAocopia (xpBA. The Semantics of Time in Plato's Timaeus, «Acta Academiae Aboensis», vol. 38, No 3, 1970). Pid toùs IIvduyopeioug 16 640 obprav elvar nAfj- TpIkd nAaioto sival Üvenupkes, Stav Eyn va kávn Kavels pé tov Üroloyioud Eou WG oúvodo, nupatnpet thy EvtunmoraKt) ôtupopà TOD yewpetpixo zpoTOTO And TH QVOIKH Kpaypaticdtyta. “Ox. pÓvo adtd GAAG Kai TÔ YewpeEkAioeic Kai tig péytotec nupexxkioeic. Ol Aboeic ord (30) sivat GpetéBAnTeg GvapopiKa npds TÔ p, q (ta Sroiu uropobv va ôvouaoloëv «of hEdnHuTIK&g otabepsc tig PÜGERG») Kai propoüv va EaxyOodv pé tov thro: x /y = p/q pé káde nepintwon. ‘H Avon ovverdyerar thy épanrouévn (= dvadoYia tod yrduovos Tpôs TI] oki Tov) THY ywvidy Anökdıong, 7.x. i Kai péyıoın (tu) nupékklions mey. a. “YrobnAover axdpa, oty mpaypatixdtnTa, Ta Kopie XapaKınpıorıkü Tod dotpovopiKod ôpyévou (dpdxvy), Tob xproiuoroinoe 6 EöôoEoe yıd va merphon yaviaxés drootéoeic, Ki &Kouy 16 ävriotpogpo tot p /q= tan > ñ = tan 5 (BR. EdKA. IT 18, VI 3), nod öönyei ott dewpia tis otepeoypagixils npoßoAfig, rpécpopn ye tov broAOYLOLO YOVLAKOV TUXUTITWV, Tob Tapappatovtar of cuvéprnon npòs tic repıödoug atö (30). ‘Qotdco0 of TEPULTEPW adrès ovvénetes TG dvacbvOeons Bà ovönndodv GAAOD. "ApxeT pbc TÔ nupdv và nodps Sti À pÉBOoöog tod Evd6Eov uropeï và von ds Ye@petpixt Eppnveia tot ypdvov Kai örı fi xüpıa «ebpetixt Borôeié tov brijpEe i zuBayópera avaAvon Tv pOur. ‘O nivakas «OV onpavtikwtépwv napapetpiKdv Tuv é€nyettar otd Kep. 9, Où pav Etor Sti rpéyuott Auepokoywarò Inrnnara Exabuv Eva póAo ori) Bewpla tHv éuneipikdv Sedopéveov tod Eb8dEov Kal ÖTI of KUTUOKELEG pé Baon tv innonddy elyav ottv npayyatikétyta oxond va mEeptypayouv katà mpooéyyion (i éiSavixevpéva) tig mpoortég otiv zapatfpnen mhavytixés Kıvnosıc. IIpéret pws và Arav künag ovyxAoviotikd ya tov EbS0Eo Kai tobs dnadovcs tov và dvaxadrbwovv 6tt pia Kai fj adri pEBOÔoG ÜrcAoyıcnod êEáyer npéyuati napopetpixés TULÈG nob Trav Öpkerù yvwortés and 10 mapedOdv. Kai à ténog (1) zpérer va pavnxe do avaxdAvyn Evög vönov tig pboews, évèc vopov, nod äv Kai anid Seizver Ev TOÜTOIG HÈ più «oixovopikt» épunveiu tic otabepic tic pooeuc. TS Eowrepikö KGALOG Tod ovathatos tod Evédétou, Evronworard axdpn Kot otty Gdpopeph oxtaypaola nob Emxeiphoane ot GpOpo aùró, h ankörmra tHv nadnnurıkav Epyu- Agiov, fj Örepßoiı«n) però Tüv EEnynréov öpov kot à ueyalopuis épunveia, of dvriöraotoAr mè TO RATOOG Tv Patvopévav nob HEnyel, mpéner àvertpikarkta và Eyypapfi otd Evepyntixd tig peyahopviag tod Edd6Eov. “H Avon tov otd Koopokoyikò npófAnga xupaktnpiberar and Eva ioyvpò peBoöokoytkò Lovioué, otdv Önoto nıdavös dvapéperor 6 Dikınnog 6 ’Onoüvtiog oti "Exivoutôa (991 e-992 e).

a. À «otıypıuia» oupotov bnoAofioud TOV yaviaKOv Tayuritwv. *Ev robrotg van, Sag éxions gavia tod nporbnov Tod EvBóEov pé ta «owdtvru parvópe Ev Swer Kai TOV evtunaciane GUYKEKPLHÉVT] uE00865 tov évéluons (7 Kal ärapxèg av Spav dànorekeouérov tod A’rpad Szabé ävapopikà LE tig òvákvens tot noéeitic Kai deific, Où Enpene va rodpe ñ dur pé8oS0g Eoöótov) elvar Ebauperikù dnotedeonatixd Gewprtixd Epyakel Erkka Maula 16 äpôpo aùElvai ovverög peyahn À xup& pov rod propa va Kheiow TH Gvpoavia TOUG HE 16 &moqpaivovras dr À otor rpög ta mpörura Kal ro avayxalog tv TPOYHUTIKÉTNTE Arav üxpıBög tod tÚrov, nob Edewpei a). Teti N «Ünóorûv ’Axaônuia tod IIAátovos (xpBA. "Apıot. Metag. 990 6 HAútov oröv TiGeon Epyaciag» pov oTÒ äpdpo abrò Arav äxpıBög St üg yid Tv ünoKaro ypncıponotet Tú Geopia tod EùöáEov de rAuicıo dvayop tig vobets yıd tù [vonaouatikh tov Gotpovopia. Eróv Tinaro avexddvwa tod Büöódov (kai à Oayopxà zpizAä Kai yıd TH koEórnra tig Ewdeınrıkfig apdyvy, T HOUGIgeydÂn äppovia tod TAërwvoc Siver S60 KAipoxes GTV ÈS kerphoeig). ‘O Ki] Kal th yeoperpuch Gppovir), xAipuka yi tig YOVIUK yd th oxéon Thatav Spas otdv Tigaro 31 c ypnawporotel tov Spo dyadpa Kai Eyw deiksr GA Où petaEd tig «yuyfig Tob KÉGHOU» rai tov rAavnrv, phical Socie(Plato’s Agalma of the Eternal Gods, «Yearbook of the Philoso IKI] HEty of Finland» 1969) dt1 à Spoc äyadua elvar Bapvannaven PIAOGOP tj ayéan Hetapopa cröv MMärava, EEnıpericd zpóopopn va neptypayy wv. To «äyaApar Tae) iv «alwviwv» Kai «npooxaipwy» ypovixdy Ôvrorhr Evvota. Atuota xépia tod MÀátovog and Opn OKEUTIKT] Erıve pihocopikt oeig ts. To thpnoe öpoc pepikécg and tig äpxikëc Opnoxevtikés Groxpd aid An«äyaXpa» otov Tigaio 37 c elvat: 1) sixova tod nupubsiypatds tov, tov, 2) ktploupyds rpoonußel va Tù kávn àkópa Spordtepo 1d apzétund Kai 4) paivobpevo Kai ÉLWUXO, 3) avtixeipevo xapüc tod Anjuoupyod tov, brorideran verat va Asırovpyfj Sms th Aarpevrikdà üyülnara, ota Óroïu ig Autpeiug. örı petéyouv of Bedrntes TovAdylotov où ottypic tEGpoews yenIrv wvevpati«t abtyy atpdoopatpa napovoiace 6 EüdoËos Eva HÈ mv petpikd mpoturo nob «Éouwbe td puwvópevan, GAAG o& dvranóK pion HETOYEVEGTEPOL RPUYUUTIKÉTNTE Lovo OË SeSonévous xpévouc. “Onws tocot À ebpeuxù naßnparıcoi 6 Eöbokos Önttebpynoe Eva óparò mpdétvzo, rob ote Yi(neBodoAoyıct)) GEla tov EyKeitat axpips 015 yeyovög, Sti b,nbnr rob Gvuvera pé Tô KpdToRo sEvmakovet Ext pEpoug koets oë npoßAfnure, , or Kahontovtat otiv npeynatıxörmru. Aév elvai td (to ui@vio iSavixd stein pé énoio Eniong oxérevuv piAöcoyoı Snag 6 Leibniz kai 6 Wittgen tig RPOGKOAANHEVES STOV KANPAVTLKOY atopicpd Bewpies Tous; Hauho /Finland