I Don't Get the Ordinal Grouping of Powers in Clement of Alexandria's Statement

[Secret Alias]

But surely the shape is deeply significant. Makes me suspect that this isn't a coincidence. https://chs.harvard.edu/CHS/article/dis ... alexandria

[Ben C. Smith]

But surely the shape is deeply significant. Makes me suspect that this isn't a coincidence. https://chs.harvard.edu/CHS/article/dis ... alexandria

I sympathize. It seems potentially artificial.

[Secret Alias]

My son oddly enough scored a wonder goal (golazzo) yesterday which made it 6 - 0. That got me thinking. When he says the last (of six) he means it's the sixth. When he says the last which is really first that essentially means the eighth. As the article notes that's a Clementine trope. 8:6 as Clement and Philo note is also a (D)iatessaron.

[Secret Alias]

I found an ancient source for Pythagorean interest in triangular numbers - Plutarch Platonic questions. Plutarch claimed that if a triangular number is multiplied by 8, and then 1 is added to this product, then the result is a square number. In our case 6 x 8 + 1 = 49. Yes. 3 x 8 + 1 = 25. Yes.

https://books.google.com/books?id=unNXA ... rs&f=false

Question 5:
WHY, SINCE BODIES AND FIGURES ARE CONTAINED PARTLY BY RECTTLINEARS AND PARTLY BY CIRCLES, DOES HE MAKE ISOSCELES TRIANGLES AND TRIANGLES OF UNEQUAL SIDES THE PRINCIPLES OF RECTILINEARS; OF WHICH THE ISOSCELES TRIANGLE FORMS THE CUBE, THE ELEMENT OF THE EARTH; AND A SCALENE TRIANGLE FORMS THE PYRAMID WHICH IS THE SEED OF FIRE, THE OCTAHEDRON WHICH IS THE SEED OF AIR, AND THE ICOSAHEDRON WHICH IS THE SEED OF WATER;—WHILE HE DOES NOT MEDDLE WITH CIRCULARS, THOUGH HE DOES MENTION THE GLOBE, WHERE HE SAYS THAT EACH OF THE AFORE-RECKONED FIGURES DIVIDES A ROUND BODY THAT ENCLOSES IT INTO EQUAL PARTS.1

Is their opinion true who think that he ascribed a dodecahedron to the globe, when he says that God made use of it in delineating the universe? For upon account of the multitude of its bases and the obtuseness of its angles, avoiding all rectitude, it is flexible, and by circumtension, like globes made of twelve skins, it becomes circular and comprehensive. For it has twenty solid angles, each of which is contained by three obtuse planes, and each of these contains one and the fifth part of a right angle. Now it is made up of twelve equilateral and equangular quinquangles (or pentagons), each of which consists of thirty of the first scalene triangles. Therefore it seems to resemble both the Zodiac and the year, it being divided into the same number of parts as these.
Or is a right line in Nature prior to circumference; or is circumference but an accident of rectilinear? For a right line is said to bend; and a circle is described by a centre and distance, which is the place of a right line by which a circumference is measured, this being everywhere equally distant from the middle. And a cone and a cylinder are made by rectilinears; a cone by keeping one side of a triangle fixed and carrying another round with the [p. 434] base,—a cylinder, by doing the like with a parallelogram. Further, that is nearest to principle which is less; but a right is the least of all lines, as it is simple; whereas in a circumference one part is convex without, another concave within. Besides, numbers are before figures, as unity is before a point, which is unity in position. But indeed unity is triangular; for every triangular number2 taken eight times, by adding unity, becomes quadrate; and this happens to unity. Therefore a triangle is before a circle, whence a right line is before a circumference. Besides, no element is divided into things compounded of itself; indeed there is a dissolution of all other things into the elements. Now a triangle is divided into no circumference, but two diameters cut a circle into four triangles; therefore a rectilinear figure is before a circular, and has more of the nature of an element. And Plato himself shows that a rectilinear is in the first place, and a circular is only consequential and accidental. For when he says the earth consists of cubes, each of which is contained with rectilinear superficies, he says the earth is spherical and round. Therefore there was no need of making a peculiar element for round things, since rectilinears, fitted after a certain manner among themselves, do make up this figure.
Besides, a right line, whether great or little, preserves the same rectitude; but as to the circumference of a circle, the less it is, the crookeder it is; the larger, the straighter. Therefore if a convex superficies stands on a plane, it sometimes touches the subject plane in a point, sometimes in a line. So that a man may imagine that a circumference is made up of little right lines.
But observe whether this be not true, that no circle [p. 435] or sphere in this world is exact; but since by the tension and circumtension of the right lines, or by the minuteness of the parts, the difference disappears, the figure seems circular and round. Therefore no corruptible body moves circularly, but altogether in a right line. To be truly spherical is not in a sensible body, but is the element of the soul and mind, to which he has given circular motion, as being agreeable to their nature.

‘διὰ τί, τῶν μὲν εὐθυγράμμων τῶν δὲ κυκλικῶν σωμάτων καὶ σχημάτων ὄντων, τὰς τῶν εὐθυγράμμων ἀρχὰς ἔλαβε τὸ ἰσοσκελὲς τρίγωνον καὶ τὸ σκαληνόν: ὧν τὸ μὲν τὸν κύβον συνέστησε γῆς στοιχεῖον ὄντα, τὸ δὲ σκαληνὸν τήν τε πυραμίδα καὶ τὸ ὀκτάεδρον καὶ τὸ εἰκοσάεδρον, τὸ μὲν πυρὸς σπέρμα τὸ δ᾽ ἀέρος τὸ δ᾽ ὕδατος γενόμενον: τὸ δὲ τῶν κυκλικῶν ὅλως παρῆκε, καίτοι μνησθεὶς τοῦ σφαιροειδοῦς, ἐν οἷς φησι τῶν κατηριθμημένων σχημάτων ἕκαστον σώματος περιφεροῦς εἰς ἴσα διανεμητικὸν εἶναι;’
πότερον, ὡς ὑπονοοῦσιν ἔνιοι, τὸ δωδεκάεδρον τῷ σφαιροειδεῖ προσένειμεν, εἰπὼν ὅτι τούτῳ ‘πρὸς [p. 130] τὴν τοῦ παντὸς ὁ θεὸς κατεχρήσατο φύσιν ἐκεῖνο διαζωγραφῶν.’ καὶ γὰρ μάλιστα τῷ πλήθει τῶν στοιχείων ἀμβλύτητι δὲ τῶν γωνιῶν τὴν εὐθύτητα διαφυγὸν εὐκαμπές ἐστι, καὶ τῇ περιτάσει καθάπερ αἱ δωδεκάσκυτοι σφαῖραι κυκλοτερὲς γίγνεται καὶ περιληπτικόν: ἔχει γὰρ εἴκοσι γωνίας στερεάς, ὧν ἑκάστην ἐπίπεδοι περιέχουσιν ἀμβλεῖαι τρεῖς ἑκάστη γὰρ ὀρθῆς ἐστι καὶ πέμπτου μορίου: συνήρμοσται δὲ καὶ συμπέπηγεν ἐκ δώδεκα πενταγώνων ἰσογωνίων καὶ ἰσοπλεύρων, ὧν ἕκαστον ἐκ τριάκοντα τῶν πρώτων σκαληνῶν τριγώνων συνέστηκε: διὸ καὶ δοκεῖ τὸν ζῳδιακὸν ἅμα καὶ τὸν ἐνιαυτὸν ἀπομιμεῖσθαι ταῖς διανομαῖς τῶν μοιρῶν ἰσαρίθμοις οὔσαις.

ἢ πρότερόν ἐστι κατὰ φύσιν τὸ εὐθὺ τοῦ περιφεροῦς, μᾶλλον δ᾽ ὅλως; πάθος τι τῆς εὐθείας ἡ περιφερής; κάμπτεσθαι γὰρ λέγεται τὸ ὀρθόν καὶ ὁ κύκλος γράφεται κέντρῳ καὶ διαστήματι: τοῦτο δ᾽ ἐστὶν εὐθείας τόπος, ὑφ᾽ ἧς καὶ μετρεῖται: τὸ γὰρ περιέχον ἐκ τοῦ μέσου πανταχόθεν ἴσον ἀφέστηκε. γεννᾶται δὲ καὶ κῶνος καὶ κύλινδρος ἀπ᾽ εὐθυγράμμων, ὁ μὲν τριγώνου περὶ μίαν πλευρὰν μένουσαν τῇ ἑτέρᾳ πλευρᾷ καὶ τῇ βάσει περιενεχθέντος: ὁ δὲ κύλινδρος παραλληλογράμμου ταὐτὸ τοῦτο παθόντος. ἔτι τῆς μὲν ἀρχῆς ἐγγυτέρω τὸ ἔλαττον, ἐλαχίστη δὲ πασῶν ἡ εὐθεῖα: τῆς γὰρ περιφεροῦς τὸ μὲν ἐντός ἐστι κοῖλον κυρτὸν δὲ τὸ ἐκτός. ἔτι [p. 131] τῶν σχημάτων οἱ ἀριθμοὶ πρότεροι, καὶ γὰρ ἡ μονὰς τῆς στιγμῆς: ἔστι γὰρ ἡ στιγμὴ μονὰς ἐν θέσει. καὶ μὴν ἡ μονὰς τρίγωνός ἐστι: πᾶς γὰρ τρίγωνος ἀριθμὸς ὀκτάκις γενόμενος καὶ μονάδα προσλαβὼν γίγνεται τετράγωνος: τοῦτο δὲ καὶ τῇ μονάδι συμβέβηκε: πρότερον οὖν τοῦ κύκλου τὸ τρίγωνον: εἰ δὲ τοῦτο, καὶ εὐθεῖα τῆς περιφεροῦς. ἔτι τὸ στοιχεῖον εἰς οὐδὲν διαιρεῖται τῶν συνισταμένων ἐξ αὐτοῦ, τοῖς δ᾽ ἄλλοις εἰς τὸ στοιχεῖον ἡ διάλυσις. εἰ τοίνυν τὸ μὲν τρίγωνον εἰς οὐδὲν περιφερὲς διαλύεται, τὸν δὲ κύκλον εἰς τέτταρα τρίγωνα τέμνουσιν αἱ δύο διάμετροι, πρότερον ἂν τῇ φύσει καὶ στοιχειωδέστερον εἴη τοῦ κυκλικοῦ τὸ εὐθύγραμμον. ὅτι τοίνυν προηγούμενον μέν ἐστι τὸ εὐθύγραμμον, τὸ δὲ κυκλικὸν ἐπιγιγνόμενον καὶ συμβεβηκός, αὐτὸς ὁ Πλάτων ἐνεδείξατο: τὴν γὰρ γῆν ἐκ κύβων συστησάμενος, ὧν ἕκαστον εὐθύγραμμοι περιέχουσιν ἐπιφάνειαι, σφαιροειδὲς αὐτῆς γεγονέναι τὸ σχῆμά φησι καὶ στρογγύλον. ὥστ᾽ οὐδὲν ἔδει ποιεῖν τῶν περιφερῶν ἴδιον στοιχεῖον, εἰ καὶ τοῖς εὐθυγράμμοις πρὸς ἄλληλά πως συναρμοττομένοις ὁ σχηματισμὸς οὗτος ἐπιγίγνεσθαι πέφυκεν.
ἔτι, εὐθεῖα μὲν ἥ τε μείζων ἥ τε μικροτέρα τὴν αὐτὴν εὐθύτητα διατηρεῖ, τὰς δὲ τῶν κύκλων περιφερείας, ἂν ὦσι σμικρότεραι, καμπυλωτέρας καὶ σφιγγομένας τῇ κυρτότητι μᾶλλον ὁρῶμεν: ἂν δὲ μείζους, ἀνειμένας: ἱστάμενοι γοῦν κατὰ τὴν κυρτὴν [p. 132] περιφέρειαν οἱ μὲν κατὰ σημεῖον οἱ δὲ κατὰ γραμμὴν ἅπτονται τῶν ὑποκειμένων ἐπιπέδων: ὥστ᾽ ὑπονοήσειεν ἄν τις εὐθείας κατὰ μικρὰ πολλὰς συντιθεμένας τὴν περιφερῆ γραμμὴν ἀποτελεῖν.
ὅρα δὲ μὴ τῶν ἐνταῦθα κυκλικῶν καὶ σφαιροειδῶν οὐδέν ἐστιν ἀπηκριβωμένον, ἀλλ᾽ ἐντάσει καὶ περιτάσει τῶν εὐθυγράμμων ἢ μικρότητι τῶν μορίων τῆς διαφορᾶς λανθανούσης ἐπιφαίνεται τὸ στρογγύλον καὶ κυκλοειδές ὅθεν οὐδὲ κινεῖται φύσει τῶν ἐνταῦθα σωμάτων ἐγκυκλίως οὐδὲν ἀλλ᾽ ἐπ᾽ εὐθείας ἅπαντα: τὸ δ᾽ ὄντως σφαιροειδὲς; οὐκ ἔστιν αἰσθητοῦ σώματος ἀλλὰ τῆς ψυχῆς καὶ τοῦ νοῦ στοιχεῖον, οἷς καὶ τὴν κυκλοφορικὴν κίνησιν ὡς προσήκουσαν κατὰ φύσιν ἀποδίδωσιν.

[Secret Alias]

Next point. I know how to say 'triangular number' in Greek from Plutarch - τρίγωνος ἀριθμὸς. Did Clement know or make reference to this concept? Yes he did. Stromata 6.11:

"The days of men shall be," it is said, "120 (ρκʹ) years." And the sum is made up of the numbers from r to 15 added together (κατὰ σύνθεσιν). And the moon at 15 days is full. On another principle, 120 is a triangular number (τρίγωνος ἀριθμὸς), and consists of the equality of the number 64 (ξδʹ), [which consists of eight of the odd numbers beginning with unity], the addition of which (1, 3, 5, 7, 9, 11, 13, 15 αʹ γʹ εʹ ζʹ θʹ ιαʹ
ιγʹ ιεʹ) in succession generate squares ; and of the inequality of the number 56, consisting of seven (ἑπτὰ) of the even numbers beginning with 2 (2, 4, 6, 8, 10, 12, 14 βʹ δʹ ʹ ηʹ ιʹ ιβʹ ιδʹ.), which produce the numbers that are not squares (κατ' ἄλλο πάλιν σημαινόμενον συνέστηκεν ὁ
ἑκατὸν κʹ ἀριθμὸς ἐκ τεσσάρων)

Again, according to another way of indicating. the number 120 consists of four numbers -- of one triangle, 15; of another, a square, 25; of a third, a pentagon, 35; and of a fourth, a hexagon, 45. The 5 is taken according to the same ratio in each mode. For in triangular numbers, from the unity 5 comes 15; and in squares, 25; and of those in succession, proportionally. Now 25, which is the number 5 from unity, is said to be the symbol of the Levitical tribe. And the number 35 depends also on the arithmetic, geometric, and harmonic scale of doubles -- 6, 8, 9, 12; the addition of which makes 35. In these days, the Jews say that seven months' children are formed. And the number 45 depends on the scale of triples -- 6, 9, 12, 18 -- the addition of which makes 45; and similarly, in these days they say that nine months' children are formed.

[Secret Alias]

Interestingly another number from the Gospel of John is a triangular number = 153

n Δ
1 1
2 3
3 6
4 10
5 15
6 21
7 28
8 36
9 45
10 55

n Δ
11 66
12 78
13 91
14 105
15 120
16 136
17 153
18 171
19 190
20 210

n Δ
21 231
22 253
23 276
24 300
25 325
26 351
27 378
28 406
29 435
30 465

n Δ
31 496
32 528
33 561
34 595
35 630
36 666

Clearly (or possibly) Clement's conviction that 6 = 8 may derive from the triangular number 8 = 36 and the triangular number of 36 = 666

[Secret Alias]

So to sum up again.

1. Clement's grouping of 'teachers' or influence forms an isosceles triangle
2. The isosceles triangle appears to be the shape of the triangular number 3
3. We know that other gnostics took a great interest in the triangular number 4 (= tetractys)
4. We know that Philo described the triangular number 2 (the dyad) as the the female hypostasis Wisdom cf ARCO DEN HEIJER

If God the Father was the Monad and Wisdom the Dyad then it would seem as if the Triad was not only masculine but corresponded to the 'Son' or some sort of 'second God.' Furthermore it would seem that the Tetractys was a kind of second daughter which corresponded - possibly - with the fallen Sophia.

[Ben C. Smith]

How are 2 and 4 "triangular" numbers?

Also, a nitpick: the triangles are generally also envisioned as equilateral, not just isosceles.

[Ben C. Smith]

How are 2 and 4 "triangular" numbers?

Oh, wait. I think you just wrote that confusingly. You must mean a triangle with 2 lines and a triangle with 4 lines, right?

[Secret Alias]

Yes sloppy as always.