x 2 y 2 = 12 xy(x 2 y 2 ) = 840 9 + 4T 8 T 9 + 3T 8
Διόϕαντος ὁ Ἀλεξανδρεύς Εὐκλείδες Ερατοσθένης Υπατία Πτολεμαῖος, Ἀρίσταρχος, Ηρων, Πάππος, Θέων, Υψικλῆς, Μενέλαος Ἀριθμητικά βιβλίον βιβλία
ἀριθμός P (x 1, x 2,..., x n ) = 0 P x 1, x 2,..., x n ri Vladimiroviq Mati ceviq ax + by = c x 2 + y 2 = z 2 x 2 Dy 2 = 1 D x n + y n = z n n
Περὶ πολυγώνων ἀριθμῶν Πορίσματα Καὶ ἔχομεν πάλιν ἐν τοῖς Πορίσμοσιν ὅτι Μητρόδωρος Επιτάϕιος Διοϕάντου Επιτάϕιον Διοϕάντου 7
Οὑτός τοι Διόϕαντον ἔχει τάϕος ἆ μέγα θαῦμα. Καὶ τάϕος ἐκ τέχνης μέτρα βίοιο λέγει. Εκτην κουρίζειν βιότου θεὸς ὤπασε μοίρην δωδεκάτην δ ἐπιθείς, μῆλα πόρεν χνοάειν τῇ δ ἄρ ἐϕ ἑβδομάτῃ τὸ γαμήλιον ἥψατο ϕέγγος, ἐκ δὲ γάμων πέμπτῳ παῖδ ἐπένευσεν ἔτει. Αἰαῖ, τηλύγετον δειλὸν τέκος ἥμισυ πατρὸς τοῦδε καὶ ἡ κρυερὸς μέτρον ἑλὼν βιότου. Πένθος δ αὖ πισύρεσσι παρηγορέων ἐνιαυτοῖς τῇδε πόσου σοϕίῃ τέρμ ἐπέρησε βίου. ἕκτην ἕκτη ἑκτός δωδεκάτην δωδέκατη δωδέκατος ἑβδομάτῃ ἑβδόματη ἑβδόματος
ἥμισυ ἥμισυς ἥμισυς πισύρεσσι πίσυρες Αἴολος x x 6 + x 12 + x 7 + 5 + x 2 + 4 = x. x = 84 ἀριθμός μονάς μονάδος συγκοπή ἀριθμός
ὕπαρξις λεῖψις Ψ ὕπαρξις λεῖψις Λεῖψις ἐπὶ λεῖπσιν πολλαπλασιασθεῖσα ποιεῖ ὕπαρξιν, λεῖψις ἐπὶ ὕπαρξιν ποιεῖ λεῖψιν, καὶ τῆς λείψεως σημεῖον Ψ ἐλλιπὲς κάτω νεῦον, Ψ Ψ Λ Ι λεῖψις ς ἀριθμός ἀριθμός x δύναμις x 2 κύβος x 3 10
δυναμοδύναμις x 4 δυναμόκυβος x 5 κυβόκυβος x 6 ἀριθμοστόν 1/x δυναμιστόν 1/x 2 κυβοστόν 1/x 3 δυναμοδυναμιστόν 1/x 4 δυναμοκυβοστόν 1/x 5 κυβοκυβοστόν 1/x 6 x m x n m 6 n 6 m+n 6 χ 1 = α, 2 = β, 3 = γ, 4 = δ, 5 = ε, 6 = ϛ, 7 = ζ, 8 = η, 9 = θ; 10 = ι, 20 = κ, 30 = λ, 40 = μ, 50 = ν, 60 = ξ, 70 = ο, 80 = π, 90 = ϙ; 100 = ρ, 200 = ς, 300 = τ, 400 = υ, 500 = ϕ, 600 = χ, 700 = ψ, 800 = χ, 900 = ϡ; 1000 = α, 2000 = β, 3000 = γ, 4000 = δ, 5000 = ε, 6000 = ϛ, 7000 = ζ, 8000 = η, 9000 = θ. Υ 20000 = β Υ Υ 400000000 = δ Υ ος τετράγωνος ἴσ ι σ ἴσος
x x 3 + 12x 2 + 5x + 3 x 3 6x 2 + 9x 1 = x 3 + 9x (6x 2 + 1) (40x 2 + 13)/(x 4 + 33 4x 2 ) ἐν μορίῳ
ἐν μορίῳ μόριον ϕιλία καλός Φιλοκαλία λλ ου 8x 3 16x 2 = x 3 x = 16/7 ιϝ ϝ ϛ ιϛ πρῶτος, δεύτερος, τρίτος, τέταρτος, α ος β ος γ ος δ ος 50/23 ν κγ ων ων ῶν 1/2 1/3 γ γ γ χ γ 16 9 θ ις θ ις mx 2 + px = q, mx 2 = px + q, mx 2 + q = px.
mx 2 = px, mx 2 = q, px = q. x 2 Επιτετάχθω δὴ τὸν ρ διελεῖν εἰς δύο ἀριθμοὺς ὅπως τὸ τοῦ α ου ἀριθμοῦ γ ον καὶ τὸ τοῦ β ου ε ον ἐπὶ τὸ αὐτὸ συντεθέντα ποιῇ λ x 1 x 2 x 1 /3+x 2 /5 = 30. x = x 2 /5 x 2 = 5x x 1 /3 = 30 x x 1 = 90 3x x 1 + x 2 = 5x + (90 3x) = 2x + 90 = 100 x = 5 x 1 = 75 x 2 = 25
Εὑρεῖν δύο ἀριθμοὺς ὅπως ἡ ὑπεροχὴ αὐτῶς καὶ ὁ πολλαπλασιασμὸς ποιῇ δοθέντας ἀριθμούς. x y x y = 4 xy = 96 x + y = 2t x = t + 2 y = t 2 xy = (t + 2)(t 2) = t 2 4 = 96 t 2 = 100 t = 10 x = 12 y = 8 Εὑρεῖν δύο ἀριθμοὺς ὅπως ὁ ἀπὸ ἑκατέρου αὐτῶς τετράγωνος, προσλαβὼν τὸν λοιπὸν, ποιῇ τετράγωνον. x y x 2 + y y 2 + x y = 2x + 1 x 2 + (2x + 1) = (x+1) 2 (2x+1) 2 +x = 4x 2 +5x+1 = (2x 2) 2 +13x 3 x = 3/13 y = 6/13 + 1 = 19/13 3/13 19/13 y = 2kx + k 2 k x 2 + y = x 2 + (2kx + k 2 ) = (x + k) 2 (2kx + k 2 ) 2 + x n (2kx + k 2 ) 2 + x = (2kx + k 2 n) 2 + (4kn + 1)x n 2 + 2k 2 n. x = n(n 2k 2 )/(1+4kn) x = n(n 2k2 ) 1 + 4kn, y = k(2n2 + k) 1 + 4kn, k n x y k = 1 n = 3
Τὸν ἐπιταχθένα τετρ+αγωνον εἰς δύο τετραγώνους. x y x 2 +y 2 = 16 Oxy x 2 + y 2 = 16 (0, 4) y = kx 4 k x 2 + (kx 4) 2 = 16 x = 8k k 2 + 1, y = 4(k2 1) k 2 + 1. k = 2 x = 16/5, y = 12/5 Εὑρεῖν τρεῖς ἀριθμοὺς ἴσους τετραγώνῳ, ὅπως σὺν δύο λαμβανόμενοι τοῦ λοιποῦ ὑπερέχωσι τετραγώνῳ.
x 1, x 2, x 3 x 1 + x 2 + x 3 x 1 + x 2 x 3 x 2 + x 3 x 1 x 3 + x 1 x 2 t x 1 + x 2 + x 3 = (t + 1) 2, x 1 + x 2 x 3 = 1. x 1 = t + 1 2, x 2 = 1 + t2, x 3 = t + t2 2 2. x 3 + x 1 x 2 = 2t t = 8 x 3 + x 1 x 2 = 4 2 x 1 = 17 2, x 2 = 65 2, x 3 = 40. t = 32. Εὑρεῖν τέσσαρας ἀριθμοὺς ὅπως ὁ ἀπὸ τοῦ συγκειμένου ἐκ τῶν τεσσάρων τετράγωνος, ἐάν τε προσλάβῃ ἕκαστον, ἐάν τε λείψῃ, ποιῇ τετράγωνον. x 1, x 2, x 3, x 4 (x 1 +x 2 + x 3 + x 4 ) 2 ± x k k = 1, 2, 3, 4 (3, 4, 5) (5, 12, 13) (39, 52, 65) (25, 60, 65) 65 2 = (8 2 + 1 2 ) 2 = (8 + i) 2 (8 i) 2 = (63 + 16i)(63 16i) = 63 2 + 16 2, 65 2 = (7 2 + 4 2 ) 2 = (7 + 4i) 2 (7 4i) 2 = (33 + 56i)(33 56i) = 33 2 + 56 2. (16, 63, 65) (33, 56, 65) (a, b, c) a 2 + b 2 = c 2
c 2 ± 2ab = (a ± b) 2 x 1 + x 2 + x 3 + x 4 = 65λ x 1, x 2, x 3, x 4 λ 2 x 1 = 2λ 2 39 52, x 2 = 2λ 2 25 60, x 3 = 2λ 2 16 63, x 4 = 2λ 2 33 56. 2λ 2 (39 52 + 25 60 + 16 63 + 33 56) = 12768λ 2 = 65λ. λ = 65/12768 x 1 = 714025 6792576, x 2 = 528125 6792576, x 3 = 4225 80864, x 4 = 46475 485184. Εὑρεῖν δύο ἀριθμοὺς ὄπος ὁ ἀπὸ τοῦ μείζονος κύβος προσλαβὼν τὸν ἐλάσσονα ἀριθμὸς ἴσος ᾖ τῷ ἀπὸ τοῦ ἐλάσσονος κύβῳ προσλαβόντι τὸν μείζονα ἀριθμόν. x y x > y x 3 + y = y 3 + x. (x y)(x 2 + xy + y 2 ) = x y x 2 + xy + y 2 = 1 y = kx k x 2 (1 + k + k 2 ) = 1 k 1 + k + k 2 c 1 + k + k 2 = (1 c 2 ) + (1 2c)k + (k + c) 2, k c k = (1 c 2 )/(2c 1) 1 + k + k 2 = ((c 2 c + 1)/(2c 1)) 2 c 1/2 x = 2c 1 c 2 c + 1, y = 1 c2 c 2 c + 1.
c = 3/4 x = 8/13 y = 7/13 k = 7/8 (x, y) Oxy x 2 + xy + y 2 = 1 Εὑρεῖν τρεῖς ἀριθμοὺς ἐν τῷ ἀορίστῳ, ὅπως ὁ ὑπὸ δύο ὁποιωνοῦν μετὰ μονάδος μιᾶς ποιῇ τετράγωνον. x, y, z xy + 1, yz + 1, zx + 1 xy = ξ 2 + 2ξ y = ξ ξ xy +1 = ξ 2 + 2ξ + 1 = (ξ + 1) 2 xξ = ξ 2 + 2ξ x = ξ + 2 zξ + 2z + 1 z = 4ξ + 4 zξ + 2z + 1 = (4ξ + 4)ξ + 2(4ξ + 4) + 1 = 4ξ 2 + 12ξ + 9 = (2ξ + 3) 2
yz + 1 = ξ(4ξ + 4) + 1 = 4ξ 2 + 4ξ + 1 = (2ξ + 1) 2 x = ξ + 2, y = ξ, z = 4ξ + 4, ξ Εὑρεῖν τρεῖς ἀριθμοὺς ἐν τῇ γεωμετρικῇ ἀναλογίᾳ, ὅπως ἕκαστος αὐτῶς λείψας τὸν δοθέντα ἀριθμὸν ποιῇ τετράγωνον. x 1, x 2, x 3 x 2 = x 1 x 3 x 1 x 3 x 1 12, x 2 12, x 3 12 x 1 = 169/4 x 1 12 = 169/4 12 = 121/4 = (11/2) 2 x 2 x 3 x 3 = t 2 t x 2 = 13t/2 13t/2 12 t 2 12 16(13t/2 12) = 104t 192 = (104t 361) + (361 192) = (104t 361)+169 t = 361/104 t 2 = 130321/10816 130321/10816 12 = 529/10816 = (23/104) 2 x 1 = 169 4, x 2 = 361 16, x 3 = 130321 10816. x 1 = x x 2 12 y 2 y (x, y) Oxy x 2 y 2 = 12 (x y)(x+y) = 12 x y = p x+y = 12/p x = 12 + p2, y = 2p 12 p2. 2p
x 2 y 2 = 12 p t = 1 (13/2, 11/2) x 1 = (13/2) 2 = 169/4 Εὑρεῖν τρεῖς ἀριθμοὺς ὅπως ὁ ἀπὸ ἑκάστου αὐτῶν τετράγωνος, ἐάν τε προσλάβῃ τὸν συγκείμενον ἐκ τῶν τριῶν, ἐάν τε λείψῃ, ποιῇ τετράγωνον. x 1, x 2, x 3 x 2 1 ± (x 1 + x 2 + x 3 ), x 2 2 ± (x 1 + x 2 + x 3 ), x 2 3 ± (x 1 + x 2 + x 3 ) (40, 42, 58), (24, 70, 74), (15, 112, 113). (a, b, c) c 2 ± 2ab = (a ± b) 2 2ab = 3360
x 1 + x 2 + x 3 λ x 1 = 58λ, x 2 = 74λ, x 3 = 113λ 58 2 ± 2 40 42λ 2 x 1 + x 2 + x 3 = 2 40 42λ 2 245λ = 2 40 42λ 2 λ = 7/96 x 1 = 406 96, x 2 = 518 96, x 3 = 791 96. (a, b, c) a = 2mn, b = m 2 n 2, c = m 2 + n 2, m n p = ab/2 = mn m 2 n 2 p mn m 2 n 2 = p xy(x 2 y 2 ) = p y < x Oxy x = r φ, y = r φ r 4 φ φ( 2 φ 2 φ) = 1 2 r4 2φ 2φ = 1 4 r4 4φ = p. r 4 = 4p 4φ. p = 840 Εὑρεῖν τρίγωνον ὀρθογώνιον ὅπως ὁ ἐν τῇ ὑποτεινούσῃ λείψας τὸν ἐν ἑκατέρᾳ τῶν ἀρθῶν ποιῇ κύβον. a = 2xy b = x 2 y 2 c = x 2 + y 2 x y x > y
xy(x 2 y 2 ) = 840 y = 2 a = 4x, b = x 2 4, c = x 2 + 4 c a = x 2 + 4 4x = (x 2) 2 c b = 8 = 2 3 x 2 = 8 8 2 = 64 = 4 3 x = 10 a = 40, b = 96, c = 104 x = 66 a = 264, b = 4352, c = 4360 Εὑρεῖν τρίγωνον ὀρθογώνιον ὅπως ὁ ἐν τῇ περιμέτρῳ αὐτοῦ ῇ τετράγωνος. καὶ προσλαβὼν τὸν ἐν τῷ ἐμβαδῷ αὐτοῦ ποιῇ κύβον. a b c a + b + c ab/2 + a + b + c a = 2ξ b = ξ 2 1 c = ξ 2 + 1 ξ > 1
2ξ 2 + 2ξ ξ(ξ 2 1) + 2ξ 2 + 2ξ = ξ 3 + 2ξ 2 + ξ 2ξ 2 +2ξ = m 2 ξ 2 m ξ = 2/(m 2 2) ξ 3 +2ξ 2 +ξ = 2m 4 /(m 2 2) 3 m 2m 4 /(m 2 2) 3 2 < m 2 < 4 m = 27/16 2m = 27/8 = (3/2) 3 2m 4 /(m 2 2) 3 ξ = 512/217 a = 1024 217, b = 215055 47089, c = 309233 47089. m = 29791/16000 Περὶ πολυγώνων ἀριθμῶν 1 + 2 + 3 + 4 = 10 δεκάς δεκάδος
τετρακτύς μονάς μονάδος δυάς δυάδος τριάς τριάδος ἁρμονία τετράς τετράδος κόσμος 4 : 3 3 : 2 2 : 1
a 1, a 2, a 3,... a k+1 a k d a 1 d a 1, a 1 + d, a 1 + 2d,..., n a n = a 1 + (n 1)d, n = 1, 2, 3,.... a, a, a,... a 1 = a d = 0 1, 2, 3,... a 1 = 1 d = 1 n a 1, a 2, a 3,... n S n = a 1 + a 2 +... + a n 1 + a n. S n = a n + a n 1 +... + a 2 + a 1. S n + S n = 2S n = (a 1 + a n ) + (a 2 + a n 1 ) +... + (a n 1 + a 2 ) + (a n + a 1 ). k a k +a n+1 k = (a 1 +(k 1)d)+(a 1 +(n k)d) = a 1 +(a 1 +(n 1)d) = a 1 +a n. 2S n = n(a 1 + a n ), S n = n 2 (a 1 + a n ).
n n(n + 1) 1 + 2 + 3 +... + n =. 2 a 1 = 1 k 2 k a (k) n = 1 + (k 2)(n 1). n k V n (k) = n (2 + (k 2)(n 1)). 2 V n (k) n k n k Υψικλῆς ὁ Ἀλεξανδρεύς k V (k) 1 = 1, V (k) 2 = k. k = 3 V (k) n+1 = V (k) n + (k 2)n + 1. T n = V (3) n = n(n + 1) 2 ( ) n + 1 =. 2 T 0 = 0 n n 1 + 2 + 3 +... + n = T n. n + 1 n 0 T n+1 = T n + (n + 1).
k = 4 Q n = V (4) n = n 2. k = 5 P n = V (5) n = n(3n 1), 2 k = 6 H n = V (6) n = n(2n 1). H n = T 2n 1. V n (k) (n 1)n = n + (k 2), 2 n 1 k = 6 V (k) n = n + (k 2)T n 1. ( ) V (k) n = n + (k 2)T n 1 = (n + T n 1 ) + (k 3)T n 1 = T n + (k 3)T n 1, V (k) n = T n + (k 3)T n 1. ( ) k = 6 n k + 1 k V (k+1) n = T n + (k 2)T n 1 = (T n + (k 3)T n 1 ) + T n 1 = V (k) n + T n 1.
k\n V (k+1) n = V (k) n + T n 1. ( ) V (k) m+n = V m (k) + V n (k) + (k 2)mn. V m (k) + V n (k) + (k 2)mn = m + (k 2)T m 1 + n + (k 2)T n 1 + (k 2)mn = = (m + n) + (k 2)(T m 1 + T m 1 + mn) = (m 1)m + (n 1)n + 2mn = (m + n) + (k 2) 2 = (m + n) + (k 2) (m + n)2 (m + n) = 2 (m + n 1)(m + n) = (m + n) + (k 2) 2 = (m + n) + (k 2)T m+n 1 = V (k) m+n. V n (k) k =
k T n = V n (3) n a (3) n = n T 1 = 1 O T 2 = 1 + 2 = 3 P 1 Q 1 O O n P i Q i i n T n k O O n i O i k 2 (k 2)(i 1) a (k) i V (k) i k k O O π/k k
2π/k π/k O k 2 m k k > 2 m = 1 m
9 + 4T 8 m > 1 k n + (k 2)T n 1 = m n(2 + (k 2)(n 1)) = 2m. 2m
T 9 + 3T 8 d k d > 1 n = d, 2 + (k 2)(n 1) = 2m d. k = 2 + 2(m d) d(d 1). d = m k = 2 d = 2m k = 2 1/(2m 1) m = 36 2m = 72 2, 3, 4, 6, 8, 9, 12, 18, 24 d = n = 2 k = 36 d = n = 3 k = 13 d = n = 6 k = 4 d = n = 8 k = 3 k 8 n(n + 1) 2 + 1 = (2n + 1) 2. ( )
n m n(n + 1)/2 = m 2 4n 2 + 4n = 8m 2 (2n + 1) 2 2(2m) 2 = 1 x = 2n + 1 y = 2m x 2 2y 2 = 1 x 1 = 3, y 1 = 2 x 0 = 1, y 0 = 0 x n, y n x 2 2y 2 = 1 [ 2, 3, 4, 5 x n y n ] = [ 3 4 2 3 ] n [ x 2 = 17, y 2 = 12; x 3 = 99, y 2 = 70; x 4 = 577, y 4 = 408; x 5 = 3363, y 5 = 2378, n 2 = 8, m 2 = 6; n 3 = 49, m 3 = 35; n 4 = 288, m 4 = 204; n 5 = 1681, m 5 = 1189. T 1681 1189 2 x 2 Dy 2 = 1 D x 0 = 1, y 0 = 0 D πρόβλημα βοεικόν D D x 1, y 1 8(k 2)V (k) n + (k 4) 2 = (2 + (2n 1)(k 2)) 2, ( ) 1 0 ]
8(k 2)V (k) n + (k 4) 2 = 4n(k 2)(2 + (k 2)(n 1)) + ((k 2) 2) 2 = = 8n(k 2) + 4n(k 2) 2 (n 1) + (k 2) 2 4(k 2) + 4 = = 4 + (8n 4)(k 2) + (4n(n 1) + 1)(k 2) 2 = = 4 + 4(2n 1)(k 2) + (2n 1) 2 (k 2) 2 = (2 + (2n 1)(k 2)) 2. N k 8(k 2)N + (k 4) 2 = Q 2 Q Q = 2 + (2n 1)(k 2) 2n 1 = (Q 2)/(k 2) n = Q + k 4 2(k 2). N = 1225 k 8(k 2)1225 + (k 4) 2 = 9800(k 2) + (k 4) 2 Q 2 k = 3, 4, 6, 29, 60, 124, 1225 Q 99, 140, 198, 515, 756, 1100, 3671 n 49, 35, 25, 10, 7, 5, 2 j k V (j) n V (j) n + V (k) n = 2V ((j+k)/2) n. + V n (k) = n + (j 2)T n 1 + n + (k 2)T n 1 = 2n + (j + k 4)T n 1 = ( ( ) ) j + k = 2 n + 2 T n 1 = 2V n ((j+k)/2). 2 0 j k 3 V (k+j) n + V (k j) n = 2V (k) n.
Εκαστος τῶν ἀπὸ τῆς τριάδος ἀριθμῶν αὐξομένων μονάδι, πολύγωνός ἐστι πρῶτον ἀπὸ τῆς μονάδος, καὶ ἔχει γωνίας τοσαύτας ὅσον ἐστὶν τὸ πλῆθος στ τῶν καὶ
Π ΠΕΡΙ ΠΟΛΥΓΩΝΩΝ πρῶτος, πρῶτον ἐστι, ἐστὶν k k > 2 G k (x) = n=1 V n (k) G k (x) = V (k) n x n. (T n + (k 3)T n 1 )x n. n=1 G 3 (x) = T n x n = 1 n(n + 1)x n. 2 n=1 n=1 x < 1 H(x) = x 0 x 0 H(ξ) ξ 2 dξ = 1 2 G 3 (ξ) dξ = 1 2 x n = n=1 nx n+1. n=1 x 2(1 x). H(x) x 2 = H(x) = 1 2(1 x) 2, x 2 2(1 x) 2.
G 3 (x) = H (x), G k (x) = = G 3 (x) = (T n + (k 3)T n 1 )x n = n=1 n=1 x (1 x) 3. T n x n + (k 3) T n 1 x n = n=1 T n x n + (k 3)x T n x n = G k (x) = n=1 n=1 V (k) n x n = n=2 x (k 3)x2 + (1 x) 3 (1 x) 3 x + (k 3)x2 (1 x) 3. V n (k) E k (x) = n=1 V n (k) x n. n! e x = n=0 1 n! xn. E 3 (x) = n=1 T n n! xn. x E 3(x) = n=1 T n (n 1)! xn 1 = n=0 T n+1 x n = n! n=0 n + 1 + T n x n = n!
= n=1 1 (n 1)! xn + n=0 1 n! xn + E 3 (x) = x = xe x + e x + E 3 (x). n=0 1 n! xn + e x + E 3 (x) = E 3 (x) y y = (x + 1)e x y(0) = 0 y = (C + x + x 2 /2)e x. C = 0 x 0 E k (x) = n=1 E 3 (ξ) dξ = = E 3 (x) = x(1 + x/2)e x. n=1 x 0 T n (n + 1)! xn+1 = n=1 (ξ + ξ 2 /2)e ξ dξ = x 2 e x /2, x 2 e x /2 = n=1 T n 1 x n. n! T n 1 x n = n! T n + (k 3)T n 1 x n = x(1 + x/2)e x + (k 3)x 2 e x /2. n! k V n (k) E k (x) = x n = 1 n! 2 x((k 2)x + 2)ex. V (k) n n=1 V (k) n = 1 n! d n G k (k) (0), V dxn n = dn E k dx (0). n f f (n) (0)x n /n!
Α α Ι ι Ρ ρ Β β Κ κ Σ σ ς Γ γ Λ λ Τ τ Δ δ Μ μ Υ υ Ε ε Ν ν Φ ϕ Ζ ζ Ξ ξ Χ χ Η η Ο ο Ψ ψ Θ θ Π π Ω ω ὁ ἡ τό οἱ αἱ τά τοῦ τῆς τοῦ τῶν τῶν τῶν τῷ τῇ τῷ τοῖς ταῖς τοῖς τόν τήν τό τούς τάς τά τώ τώ, τοῖν, τοῖν, τώ τά τά, ταῖν, ταῖν, τά δύο ἀριθμοὺς Ομηρος δύο ἀριθμώ
1 αʹ εἷς, μία, ἕν πρῶτος ἅπαξ 2 βʹ δύο δεύτερος δίς 3 γʹ τρεῖς, τρία τρίτος τρίς 4 δʹ τέτταρες, τέτταρα τέταρτος τετράκις 5 εʹ πέντε πέμπτος πεντάκις 6 ϛʹ ἕξ ἕκτος ἑξάκις 7 ζʹ ἑπτά ἕβδομος ἑπτάκις 8 ηʹ ὀκτώ ὄγδοος ὀκτάκις 9 θʹ ἐννέα ἔνατος ἐνάκις 10 ιʹ δέκα δέκατος δεκάκις 11 ιαʹ ἕνδεκα ἑνδέκατος ἑνδεκάκις 12 ιβʹ δώδεκα δωδέκατος δωδεκάκις 13 ιγʹ τρεῖς καὶ δέκα τρίτος καὶ δέκατος τρὶς καὶ δεκάκις 14 ιδʹ τέτταρες καὶ δέκα τέταρτος κ. δ. τετράκις κ. δ. 15 ιεʹ πέντε καὶ δέκα πέμπτος κ. δ. πεντάκις κ. δ. 16 ιϛʹ ἑκκαίδεκα ἕκτος κ. δ. ἑξάκις κ. δ. 17 ιζʹ ἑπτὰ καὶ δέκα ἕβδομος κ. δ. ἑπτάκις κ. δ. 18 ιηʹ ὀκτὼ καὶ δέκα ὄγδοος κ. δ. ὀκτάκις κ. δ. 19 ιθʹ ἐννέα καὶ δέκα ἔνατος κ. δ. ἐνάκις κ. δ. 20 κʹ εἴκοσι εἰκοστός εἰκοσάκις κ. δ. δεύτερος τρίτος πρῶτος
10 1 δέκα 10 2 ἑκατόν 10 3 χίλιοι 10 6 μέγας 10 9 γίγας 10 12 τέρας 10 15 πετάννυμι 10 18 ἕξ 10 6 μ μικρός 10 9 νάνος μ μ
x 8 x 9 x 7 x 7, x 8, x 9 δυναμοκυβόκυβος x 8 κυβοκυβόκυβος x 9 δυναμοκυβοκυβοστόν 1/x 8 κυβοκυβοκυβοστόν 1/x 9 Οὕτως γὰρ εὐόδευτα γενήσεται τοῖς ἀρχομένοις, καὶ ὴ ἀγωγὴ αὐτῶν μνημονευθήσεται, τῆς πραγματείας αὐτῶν ἐν τρισκαίδεκα βιβλίοις γεγενημένης.
Διονύσιος Διόνυσος
Τὴν εὕρεσιν τῶν ἐν τοῖς ἀριθμοῖς προβλημάτων, τιμιώτατέ μοι Διονύσιε, γινώσκων σε σπουδαίως ἔχοντα μαθεῖν, [ὀργανῶσαι τὴν μέθοδον] ἐπειράθην, ἀρξάμενος ἀϕ ὧν συνέστηκε τὰ πράγματα θεμελίων, ὑποστῆσαι τὴν ἐν τοῖς ἀριθμοῖς ϕύσιν τε καὶ δύναμιν. Ισως μὲν οὖν δοκεῖ τὸ πρᾶγμα δυσχερέστερον, ἐπειδὴ μήπω γνώριμόν ἐστιν, δυσέλπιστοι γὰρ εἰς κατόρθωσίν εἰσιν αἱ τῶν ἀρχομένων ψυχαί, ὅμως δ εὐκατάληπτόν σοι γενήσεται, διά τε τὴν σὴν προθυμίαν καὶ τὴν ἐμὴν ἀπόδειξιν ταχεῖα γὰρ εἰς μάθησιν ἐπιθυμία προσλαβοῦσα διδαχήν.