Chapter VII: The Pythagoreans (2)

150. The planetary system which Aristotle attributes to “the Pythagoreans” and Aetios to Philolaos is sufficiently remarkable.[817] The earth is no longer in the middle of the world; its place is taken by a central fire, which is not to be identified with the sun. Round this fire revolve ten bodies. First comes the _Antichthon_ or Counter-earth, and next the earth, which thus becomes one of the planets. After the earth comes the moon, then the sun, the five planets, and the heaven of the fixed stars. We do not see the central fire and the _antichthon_ because the side of the earth on which we live is always turned away from them. This is to be explained by the analogy of the moon. That body always presents the same face to us; and men living on the other side of it would never see the earth. This implies, of course, that all these bodies rotate on their axes in the same time as they revolve round the central fire.[818]

Footnote 817:

For the authorities, see R. P. 81-83. The attribution of the theory to
Philolaos is perhaps due to Poseidonios. The “three books” were
doubtless in existence by his time.

Footnote 818:

Plato attributes an axial rotation to the heavenly bodies (_Tim._ 40 a
7), which must be of this kind. It is quite likely that the
Pythagoreans already did so, though Aristotle was unable to see the
point. He says (_de Caelo_, Β, 8. 290 a 24), ἀλλὰ μὴν ὅτι οὐδὲ
κυλίεται τὰ ἄστρα, φανερόν· τὸ μὲν γὰρ κυλιόμενον στρέφεσθαι ἀνάγκη,
τῆς δὲ σελήνης ἀεὶ δηλόν ἐστι τὸ καλούμενον πρόσωπον. This, of course,
is just what proves it does rotate.

It is not very easy to accept the view that this system was taught by Philolaos. Aristotle nowhere mentions him in connexion with it, and in the _Phaedo_ Plato gives a description of the earth and its position in the world which is entirely opposed to it, but is accepted without demur by Simmias the disciple of Philolaos.[819] It is undoubtedly a Pythagorean theory, however, and marks a noticeable advance on the Ionian views then current at Athens. It is clear too that Plato states it as something of a novelty that the earth does not require the support of air or anything of the sort to keep it in its place. Even Anaxagoras had not been able to shake himself free of that idea, and Demokritos still held it.[820] The natural inference from the _Phaedo_ would certainly be that the theory of a spherical earth, kept in the middle of the world by its equilibrium, was that of Philolaos himself. If so, the doctrine of the central fire would belong to a somewhat later generation of the school, and Plato may have learnt it from Archytas and his friends after he had written the _Phaedo_. However that may be, it is of such importance that it cannot be omitted here.

Footnote 819:

Plato, _Phd._ 108 e 4 sqq. Simmias assents to this doctrine in the
emphatic words Καὶ ὀρθῶς γε.

Footnote 820:

The primitive character of the astronomy taught by Demokritos as
compared with that of Plato is the best evidence of the value of the
Pythagorean researches.

It is commonly supposed that the revolution of the earth round the central fire was intended to account for the alternation of day and night, and it is clear that an orbital motion of the kind just described would have the same effect as the rotation of the earth on its axis. As the same side of the earth is always turned to the central fire, the side upon which we live will be turned towards the sun when the earth is on the same side of the central fire, and turned away from it when the earth and sun are on opposite sides. This view appears to derive some support from the statement of Aristotle that the earth “being in motion round the centre, produces day and night.”[821] That remark, however, would prove too much; for in the _Timaeus_ Plato calls the earth “the guardian and artificer of night and day,” while at the same time he declares that the alternation of day and night is caused by the diurnal revolution of the heavens.[822] That is explained, no doubt quite rightly, by saying that, even if the earth were regarded as at rest, it could still be said to produce day and night; for night is due to the intervention of the earth between the sun and the hemisphere opposite to it. If we remember how recent was the discovery that night was the shadow of the earth, we shall see how it may have been worth while to say this explicitly.

Footnote 821:

Arist. _de Caelo_, Β, 13. 293 a 18 sqq. (R. P. 83).

Footnote 822:

Plato, _Tim._ 40 c 1, (γῆν) φύλακα καὶ δημιουργὸν νυκτός τε καὶ ἡμέρας
ἐμηχανήσατο. On the other hand, νὺξ μὲν οὖν ἡμέρα τε γέγονεν οὕτως καὶ
διὰ ταῦτα, ἡ τῆς μιᾶς καὶ φρονιμωτάτης κυκλήσεως περίοδος (39 c 1).

In any case, it is wholly incredible that the heaven of the fixed stars should have been regarded as stationary. That would have been the most startling paradox that any scientific man had yet propounded, and we should have expected the comic poets and popular literature generally to raise the cry of atheism at once. Above all, we should have expected Aristotle to say something about it. He made the circular motion of the heavens the very keystone of his system, and would have regarded the theory of a stationary heaven as blasphemous. Now he argues against those who, like the Pythagoreans and Plato, regarded the earth as in motion;[823] but he does not attribute the view that the heavens are stationary to any one. There is no necessary connexion between the two ideas. All the heavenly bodies may be moving as rapidly as we please, provided that their relative motions are such as to account for the phenomena.[824]

Footnote 823:

Arist. _de Caelo_, Β, 13. 293 b 15 sqq.

Footnote 824:

Boeckh admitted a very slow motion of the heaven of the fixed stars,
which he at first supposed to account for the precession of the
equinoxes, though he afterwards abandoned that hypothesis
(_Untersuchungen_, p. 93). But, as Dreyer admits (_Planetary Systems_,
p. 49), it is “not ... necessary with Boeckh to suppose the motion of
the starry sphere to have been an exceedingly slow one, as it might in
any case escape direct observation.”

It seems probable that the theory of the earth’s revolution round the central fire really originated in the account given by Empedokles of the sun’s light. The two things are brought into close connexion by Aetios, who says that Empedokles believed in two suns, while Philolaos believed in two or even in three.[825] The theory of Empedokles is unsatisfactory in so far as it gives two inconsistent explanations of night. It is, we have seen, the shadow of the earth; but at the same time Empedokles recognised a fiery diurnal hemisphere and a nocturnal hemisphere with only a little fire in it.[826] All this could be simplified by the hypothesis of a central fire which is the true source of light. Such a theory would, in fact, be the natural issue of the recent discoveries as to the moon’s light and the cause of eclipses, if that theory were extended so as to include the sun.

Footnote 825:

Aet. ii. 20, 13 (Chap. IV. p. 275, _n._ 609); cf. _ib._ 12 (of
Philolaos), ὥστε τρόπον τινὰ διττοὺς ἡλίους γίγνεσθαι, τό τε ἐν τῷ
οὐρανῷ πυρῶδες καὶ τὸ ἀπ’ αὐτοῦ πυροειδὲς κατὰ τὸ ἐσοπτροειδές· εἰ μή
τις καὶ τρίτον λέξει τὴν ἀπὸ τοῦ ἐνόπτρου κατ’ ἀνάκλασιν
διασπειρομένην πρὸς ἡμᾶς αὐγήν. Here τὸ ἐν τῷ οὐρανῷ πυρῶδες is the
central fire, in accordance with the use of the word οὐρανός explained
in another passage of Aetios, Stob. _Ecl._ i. p. 196, 18 (R. P. 81).
It seems to me that these strange notices must be fragments of an
attempt to show how the heliocentric hypothesis arose from the theory
of Empedokles as to the sun’s light. The meaning is that the central
fire really was the sun, but that Philolaos unnecessarily duplicated
it by supposing the visible sun to be its reflexion.

Footnote 826:

Chap. VI. § 113.

The central fire received a number of mythological names. It was called the Hestia or “hearth of the universe,” the “house” or “watch-tower” of Zeus, and the “mother of the gods.”[827] That was in the manner of the school; but these names must not blind us to the fact that we are dealing with a real scientific hypothesis. It was a great thing to see that the phenomena could best be “saved” by a central luminary, and that the earth must therefore be a revolving sphere like the planets. Indeed, we are almost tempted to say that the identification of the central fire with the sun, which was suggested for the first time in the Academy, is a mere detail in comparison. The great thing was that the earth should definitely take its place among the planets; for once it has done so, we can proceed to search for the true “hearth” of the planetary system at our leisure. It is probable, at any rate, that it was this theory which made it possible for Herakleides of Pontos and Aristarchos of Samos to reach the heliocentric hypothesis,[828] and it was certainly Aristotle’s reversion to the geocentric theory which made it necessary for Copernicus to discover the truth afresh. We have his own word for it that the Pythagorean theory put him on the right track.[829]

Footnote 827:

Aet. i. 7, 7 (R. P. 81). Procl. _in Tim._ p. 106, 22, Diehl (R. P. 83
e).

Footnote 828:

On these points, see Staigmüller, _Beiträge zur Gesch. der
Naturwissenschaften im klassichen Altertume_ (Progr., Stuttgart,
1899); and “Herakleides Pontikos und das heliokentrische System”
(_Arch._ xv. pp. 141 sqq.). Though, for reasons which will partly
appear from the following pages, I should not put the matter exactly
as Staigmüller does, I have no doubt that he is substantially right.
Diels had already expressed his adhesion to the view that Herakleides
was the real author of the heliocentric hypothesis (_Berl. Sitzb._,
1893, P. 18).

Footnote 829:

In his letter to Pope Paul III., Copernicus quotes Plut. _Plac._ iii.
13, 2-3 (R. P. 83 a), and adds “Inde igitur occasionem nactus, coepi
et ego de terrae mobilitate cogitare.” The whole passage is
paraphrased by Dreyer, _Planetary Systems_, p. 311. Cf. also the
passage from the original MS., which was first printed in the edition
of 1873, translated by Dreyer, _ib._ pp. 314 sqq.

[Sidenote: The _antichthon_.]

151. The existence of the _antichthon_ was also a hypothesis intended to account for the phenomena of eclipses. In one place, indeed, Aristotle says that the Pythagoreans invented it in order to bring the number of revolving bodies up to ten;[830] but that is a mere sally, and Aristotle really knew better. In his work on the Pythagoreans, we are told, he said that eclipses of the moon were caused sometimes by the intervention of the earth and sometimes by that of the _antichthon_; and the same statement was made by Philip of Opous, a very competent authority on the matter.[831] Indeed, Aristotle shows in another passage exactly how the theory originated. He tells us that some thought there might be a considerable number of bodies revolving round the centre, though invisible to us because of the intervention of the earth, and that they accounted in this way for there being more eclipses of the moon than of the sun.[832] This is mentioned in close connexion with the _antichthon_, so there is no doubt that Aristotle regarded the two hypotheses as of the same nature. The history of the theory seems to be this. Anaximenes had assumed the existence of dark planets to account for the frequency of lunar eclipses (§ 29), and Anaxagoras had revived that view (§ 135). Certain Pythagoreans[833] had placed these dark planets between the earth and the central fire in order to account for their invisibility, and the next stage was to reduce them to a single body. Here again we see how the Pythagoreans tried to simplify the hypotheses of their predecessors.

Footnote 830:

Arist. _Met._ Α, 5. 986 a 3 (R. P. 83 b).

Footnote 831:

Aet. ii. 29, 4, τῶν Πυθαγορείων τινὲς κατὰ τὴν Ἀριστοτέλειον ἱστορίαν
καὶ τὴν Φιλίππου τοῦ Ὀπουντίου ἀπόφασιν ἀνταυγείᾳ καὶ ἀντιφράξει τοτὲ
μὲν τῆς γῆς, τοτὲ δὲ τῆς ἀντίχθονος (ἐκλείπειν τὴν σελήνην).

Footnote 832:

Arist. _de Caelo_, Β, 13. 293 b 21, ἐνίοις δὲ δοκεῖ καὶ πλείω σώματα
τοιαῦτα ἐνδέχεσθαι φέρεσθαι περὶ τὸ μέσον ἡμῖν ἄδηλα διὰ τὴν
ἐπιπρόσθησιν τῆς γῆς. διὸ καὶ τὰς τῆς σελήνης ἐκλείψεις πλείους ἢ τὰς
τοῦ ἡλίου γίγνεσθαί φασιν· τῶν γὰρ φερομένων ἕκαστον ἀντιφράττειν
αὐτήν, ἀλλ’ οὐ μόνον τὴν γῆν.

Footnote 833:

It is not expressly stated that they were Pythagoreans, but it is
natural to suppose so. Such, at least, was Alexander’s opinion (Simpl.
_de Caelo_, P. 515, 25).

[Sidenote: Planetary motions.]

152. We must not assume that even the later Pythagoreans made the sun, moon, and planets, including the earth, revolve in the opposite direction to the heaven of the fixed stars. It is true that Alkmaion is said to have agreed with “some of the mathematicians”[834] in holding this view, but it is never ascribed to Pythagoras or even to Philolaos. The old theory was, as we have seen (§ 54), that all the heavenly bodies revolved in the same direction, from east to west, but that the planets revolved more slowly the further they were removed from the heavens, so that those which are nearest the earth are “overtaken” by those that are further away. This view was still maintained by Demokritos, and that it was also Pythagorean, seems to follow from what we are told about the “harmony of the spheres.” We have seen (§ 54) that we cannot attribute this theory in its later form to the Pythagoreans of the fifth century, but we have the express testimony of Aristotle to the fact that those Pythagoreans whose doctrine he knew believed that the heavenly bodies produced musical notes in their courses. Further, the velocities of these bodies depended on the distances between them, and these corresponded to the intervals of the octave. He distinctly implies that the heaven of the fixed stars takes part in the concert; for he mentions “the sun, the moon, and the stars, so great in magnitude and in number as they are,” a phrase which cannot refer solely or chiefly to the remaining five planets.[835] Further, we are told that the slower bodies give out a deep note and the swifter a high note.[836] Now the prevailing tradition gives the high note of the octave to the heaven of the fixed stars,[837] from which it follows that all the heavenly bodies revolve in the same direction, and that their velocity increases in proportion to their distance from the centre.

Footnote 834:

The term οἱ μαθηματικοί is that used by Poseidonios for the Chaldæan
astrologers (Berossos). Diels, _Elementum_, p. 11, n. 3. As we have
seen, the Babylonians knew the planets better than the Greeks.

Footnote 835:

Arist. _de Caelo_, Β, 9. 290 b 12 sqq. (R. P. 82).

Footnote 836:

Alexander, _in Met._ p. 39, 24 (from Aristotle’s work on the
Pythagoreans), τῶν γὰρ σωμάτων τῶν περὶ τὸ μέσον φερομένων ἐν ἀναλογίᾳ
τὰς ἀποστάσεις ἐχόντων ... ποιούντων δὲ καὶ ψόφον ἐν τῷ κινεῖσθαι τῶν
μὲν βραδυτέρων βαρύν, τῶν δὲ ταχυτέρων ὀξύν. We must not attribute the
identification of the seven planets with the seven strings of the
heptachord to the Pythagoreans of this date. Mercury and Venus have in
the long run the same velocity as the sun, and we must take in the
earth and the fixed stars. We can even find room for the _antichthon_
as προσλαμβανόμενος.

Footnote 837:

For the various systems, see Boeckh, _Kleine Schriften_, vol. iii. pp.
169 sqq., and Carl v. Jan, “Die Harmonie der Sphären” (_Philol._ 1893,
pp. 13 sqq.). They vary with the astronomy of their authors, but they
bear witness to the fact stated in the text. Many give the highest
note to Saturn and the lowest to the Moon, while others reverse this.
The system which corresponds best, however, with the Pythagorean
planetary system must include the heaven of the fixed stars and the
earth. It is that upon which the verses of Alexander of Ephesos quoted
by Theon of Smyrna, p. 140, 4, are based:

γαῖα μὲν οὖν ὑπάτη τε βαρεῖά τε μέσσοθι ναίει·
ἀπλανέων δὲ σφαῖρα συνημμένη ἔπλετο νήτη, κ.τ.λ.

The “base of Heaven’s deep Organ” in Milton’s “ninefold harmony”
(_Hymn on the Nativity_, xiii.) implies the reverse of this.

The theory that the proper motion of the sun, moon, and planets is from west to east, and that they also share in the motion from east to west of the heaven of the fixed stars, makes its first appearance in the Myth of Er in Plato’s _Republic_, and is fully worked out in the _Timaeus_. In the _Republic_ it is still associated with the “harmony of the spheres,” though we are not told how it is reconciled with that theory in detail.[838] In the _Timaeus_ we read that the slowest of the heavenly bodies appear the fastest and _vice versa_; and, as this statement is put into the mouth of a Pythagorean, we might suppose the theory of a composite movement to have been anticipated by some members at least of that school.[839] That is, of course, possible; for the Pythagoreans were singularly open to new ideas. At the same time, we must note that the theory is even more emphatically expressed by the Athenian Stranger in the _Laws_, who is in a special sense Plato himself. If we were to praise the runners who come in last in the race, we should not do what is pleasing to the competitors; and in the same way it cannot be pleasing to the gods when we suppose the slowest of the heavenly bodies to be the fastest. The passage undoubtedly conveys the impression that Plato is expounding a novel theory.[840]

Footnote 838:

The difficulty appears clearly in Adam’s note on _Republic_, 617 b
(vol. ii. p. 452). There the ἀπλανής appears rightly as the νήτη,
while Saturn, which comes next to it, is the ὑπάτη. It is
inconceivable that this should have been the original scale. Aristotle
touches upon the point (_de Caelo_, Β, 10. 291 a 29 sqq.); and
Simplicius sensibly observes (_de Caelo_, p. 476, 11), οἱ δὲ πάσας τὰς
σφαίρας τὴν αὐτὴν λέγοντες κίνησιν τὴν ἀπ’ ἀνατολῶν κινεῖσθαι καθ’
ὑπόληψιν (ought not the reading to be ὑπόλειψιν?), ὥστε τὴν μὲν
Κρονίαν σφαῖραν συναποκαθίστασθαι καθ’ ἡμέραν τῇ ἀπλανεῖ παρ’ ὀλίγον,
τὴν δὲ τοῦ Διὸς παρὰ πλέον καὶ ἐφεξῆς οὕτως, οὗτοι πολλὰς μὲν ἄλλας
ἀπορίας ἐκφεύγουσι, but their ὑπόθεσις is ἀδύνατος. This is what led
to the return to the geocentric hypothesis and the exclusion of earth
and ἀπλανὴς from the ἁρμονία. The only solution would have been to
make the earth rotate on its axis or revolve round the central fire in
twenty-four hours, leaving only precession for the ἀπλανής. As we have
seen, Boeckh attributed this to Philolaos, but without evidence. If he
had thought of it, these difficulties would not have arisen.

Footnote 839:

_Tim._ 39 a 5-b 2, especially the words τὰ τάχιστα περιιόντα ὑπὸ τῶν
βραδυτέρων ἐφαίνετο καταλαμβάνοντα καταλαμβάνεσθαι (“they appear to be
overtaken, though they overtake”).

Footnote 840:

Plato, _Laws_, 822 a 4 sqq. The Athenian says of the theory that he
had not heard of it in his youth nor long before (821 e 3). If so, it
can hardly have been taught by Philolaos, though it may have been by
Archytas.

[Sidenote: Things likenesses of numbers.]

153. We have still to consider a view, which Aristotle sometimes attributes to the Pythagoreans, that things were “like numbers.” He does not appear to regard this as inconsistent with the doctrine that things _are_ numbers, though it is hard to see how he could reconcile the two.[841] There is no doubt, however, that Aristoxenos represented the Pythagoreans as teaching that things were _like_ numbers,[842] and there are other traces of an attempt to make out that this was the original doctrine. A letter was produced, purporting to be by Theano, the wife of Pythagoras, in which she says that she hears many of the Hellenes think Pythagoras said things were made _of_ number, whereas he really said they were made _according to_ number.[843] It is amusing to notice that this fourth-century theory had to be explained away in its turn later on, and Iamblichos actually tells us that it was Hippasos who said number was the exemplar of things.[844]

Footnote 841:

Cf. especially _Met._ Α, 6. 787 b 10 (R. P. 65 d). It is not quite the
same thing when he says, as in Α, 5. 985 b 23 sqq. (R. P. _ib._), that
they perceived many likenesses in things to numbers. That refers to
the numerical analogies of Justice, Opportunity, etc.

Footnote 842:

Aristoxenos _ap._ Stob. i. pr. 6 (p. 20), Πυθαγόρας ... πάντα τὰ
πράγματα ἀπεικάζων τοῖς ἀριθμοῖς.

Footnote 843:

Stob. _Ecl._ i. p. 125, 19 (R. P. 65 d).

Footnote 844:

Iambl. _in Nicom._ p. 10, 20 (R. P. 56 c).

When this view is uppermost in his mind, Aristotle seems to find only a verbal difference between Plato and the Pythagoreans. The metaphor of “participation” was merely substituted for that of “imitation.” This is not the place to discuss the meaning of Plato’s so-called “theory of ideas”; but it must be pointed out that Aristotle’s ascription of the doctrine of “imitation” to the Pythagoreans is abundantly justified by the _Phaedo_. The arguments for immortality given in the early part of that dialogue come from various sources. Those derived from the doctrine of Reminiscence, which has sometimes been supposed to be Pythagorean, are only known to the Pythagoreans by hearsay, and Simmias requires to have the whole psychology of the subject explained to him.[845] When, however, we come to the question what it is that our sensations remind us of, his attitude changes. The view that the equal itself is alone real, and that what we call equal things are imperfect imitations of it, is quite familiar to him.[846] He requires no proof of it, and is finally convinced of the immortality of the soul just because Sokrates makes him see that the theory of forms implies it.

Footnote 845:

Plato, _Phd._ 73 a sqq.

Footnote 846:

_Ibid._ 74 a sqq.

It is also to be observed that Sokrates does not introduce the theory as a novelty. The reality of the “ideas” is the sort of reality “we are always talking about,” and they are explained in a peculiar vocabulary which is represented as that of a school. The technical terms are introduced by such formulas as “we say.”[847] Whose theory is it? It is usually supposed to be Plato’s own, though nowadays it is the fashion to call it his “early theory of ideas,” and to say that he modified it profoundly in later life. But there are serious difficulties in this view. Plato is very careful to tell us that he was not present at the conversation recorded in the _Phaedo_. Did any philosopher ever propound a new theory of his own by representing it as already familiar to a number of distinguished living contemporaries? It is not easy to believe that. It would be rash, on the other hand, to ascribe the theory to Sokrates, and there seems nothing for it but to suppose that the doctrine of “forms” (εἴδη, ἰδέαι) originally took shape in Pythagorean circles, perhaps under Sokratic influence. There is nothing startling in this. It is a historical fact that Simmias and Kebes were not only Pythagoreans but disciples of Sokrates; for, by a happy chance, the good Xenophon has included them in his list of true Sokratics.[848] We have also sufficient ground for believing that the Megarians had adopted a like theory under similar influences, and Plato states expressly that Eukleides and Terpsion of Megara were present at the conversation recorded in the _Phaedo_. There were, no doubt, more “friends of the ideas”[849] than we generally recognise. It is certain, in any case, that the use of the words εἴδη and ἰδέαι to express ultimate realities is pre-Platonic, and it seems most natural to regard it as of Pythagorean origin.[850]

Footnote 847:

Cf. especially the words ὃ θρυλοῦμεν ἀεί (76 d 8). The phrases αὐτὸ ὃ
ἔστιν, αὐτὸ καθ’ αὑτό, and the like are assumed to be familiar. “We”
define reality by means of question and answer, in the course of which
“we” give an account of its being (ἧς λόγον δίδομεν τοῦ εἶναι, 78 d 1,
where λόγον ... τοῦ εἶναι is equivalent to λόγον τῆς οὐσίας). When we
have done this, “we” set the seal or stamp of αὐτὸ ὃ ἔστιν upon it (75
d 2). Technical terminology implies a school. As Diels puts it
(_Elementum_, p. 20), it is in a school that “the simile concentrates
into a metaphor, and the metaphor condenses into a term.”

Footnote 848:

Xen. _Mem._ i. 2, 48.

Footnote 849:

Plato, _Soph._ 248 a 4.

Footnote 850:

See Diels, _Elementum_, pp. 16 sqq. Parmenides had already called the
original Pythagorean “elements” μορφαί (§ 91), and Philistion called
the “elements” of Empedokles ἰδέαι. If the ascription of this
terminology to the Pythagoreans is correct, we may say that the
Pythagorean “forms” developed into the atoms of Leukippos and
Demokritos on the one hand (§ 174), and into the “ideas” of Plato on
the other.

We have really exceeded the limits of this work by tracing the history of Pythagoreanism down to a point where it becomes practically indistinguishable from the earliest form of Platonism; but it was necessary to do so in order to put the statements of our authorities in their true light. Aristoxenos is not likely to have been mistaken with regard to the opinions of the men he had known personally, and Aristotle’s statements must have had some foundation. We must assume, then, a later form of Pythagoreanism which was closely akin to early Platonism. That, however, is not the form of it which concerns us here, and we shall see in the next chapter that the fifth-century doctrine was of the more primitive type already described.