7. arithmetic & number theoretic recreations a. Fibonacci numbers



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F. 91r, p. 182. Ship with 2 sails: (12, 18).

F. 91v, p. 183. Three men in prison: (6, 12, 18). (Tropfke 520 reproduces this in B&W.)

F. 93r, p. 186. Emptying a cask: (6, 8).

F. 95v, p. 191. Ship with three sails: (6, 10, 12). (Coloured plate opp. p. 120 of the text volume.)

F. 96v, p. 193. Emptying a cask: (8, 12, 16).

F. 97v, p. 195. Lion, wolf & fox eating a goat: (2, 3, 5). (Tropfke 581 reproduces this in B&W.)

Ff. 98v-99r, pp. 197 198. Furnace with 3 fires: (10, 15, 20).


Johann Widman. Op. cit. in 7.G.1. 1489. (On pp. 131-132, Glaisher mentions the following.) Ff. 136r 138v: Eyn fasz mit dreyen Czapfen; Von der Mulen; Leb, wolff, hunt; Schiff. (Cistern problem; 3 mills; lion, wolf, dog eating a sheep; ship with 3 sails.)

Calandri. Arimethrica. 1491.


F. 68v. Ship with two sails. (12, 15). Woodcut of ship with indeterminate number of sails.

F. 69r. Cask with two taps. (4, 6). Woodcut of cask with two taps.

F. 70r. Ship with three sails. (10, 12, 15). Same woodcut as on f. 68v.

F. 70r. Cask with three taps. (4, 6, 8). Same woodcut as on f. 69r.

F. 70v. Three masters build a house. (10, 12, 15). Woodcut of two builders. (H&S 70 gives Italian and English and says it also occurs in the Treviso Arithmetic (1478) [but that has a very different type!], Pacioli, Cataneo, Tartaglia, Buteo (1559), Clavius, Tonstall.)

F. 70v. Three masters doing a job. (30, 40, x) in 15.

F. 71v. Cistern. (4,  11). Woodcut of cistern. (Rara, 48 is a reproduction.)

F. 72v. Lion, leopard & wolf eating a sheep. (1, 2, 3) days. Nice woodcut. (H&S 70 gives Italian and English, says there is a remarkable picture and says it occurs in Fibonacci [again, there it occurs in a different form] and Cataneo.)


Pacioli. Summa. 1494. See also Buteo.

F. 99r, prob. 6. Building a house, (8, 10, 4). Says one can have more builders and it is similar to dog, wolf & lion eating a sheep.

F. 99v, prob. 16. Three mills, (6, 8, 3) days.

F. 99v, prob. 17(not printed). Three mills, (10, 5, 4) days.

PART II.

F. 66v. prob. 91. Cask with four taps. Volume above highest tap is 1/3 of the cask. Volume between highest and second highest is 1/4; volume between second and third highest is 1/5; volume between third highest and lowest tap is the rest of the cask. Each tap can empty the section just above it in 1, 2, 3, 4 days. How long to empty with all taps? He assumes the cask holds 60 so the rates are 20, 15/2, 12/3, 13/4 per day. Answer is 80/139 + 60/59 + 48/29 + 4, but he gives the sum as 7 245235/2959139. Clearly the denominator denotes 29·59·139 = 237829, but the correct sum is 7 58901/237829 and I cannot see how his expression relates to the answer. The answer is not 7 + 24/29 + 52/59 + 35/139, nor any similar expression that I can think of.

Ff. 66v-67r, prob. 92. Basin has inlets which fill it in 1, 2, 3 hours and outlet which empty it in 2, 3, 4 hours, i.e. (1, 2, 3, -2, -3, -4). How long to fill? He follows with remarks that all such problems can be done similarly. Cf della Francesca.


Blasius. 1513. F. F.iii.r: Decimatertia regula. Three rivers can water a field in (1, 2, 3) days. Gets 13 1/11 hours for all three -- so he is using 24 hour days.

Tagliente. Libro de Abaco. (1515). 1541.


Prob. 117, f. 58r. Ship with two sails -- (12, 15).

Prob. 119, f. 58v. Cask with two taps -- (4, 6).


Tonstall. De Arte Supputandi. 1522.

Quest. 26, pp. 157-159. Three mills can do at rates of 18, 13, 8 per day. How long to do 24?

Quest. 27, pp. 159-161. Cistern, (1, 2, 3) and (4, 6, 8).

Quest. 28, pp. 161-162. Cistern, (1/4, 1/2, 1).

Quest. 29, pp. 162-163. Cistern, (4, -11).

Quest. 32, pp. 164-165. Four architects building a house, (1, 2, 3, 4) years. Says it is similar to a cistern problem.

Quest. 33, p. 166. Three architects building a house, (30, 40, x) in 15.


Riese. Die Coss. 1524.

No. 117, p. 55. Cask with three taps, (1, 2, 3).

No. 118, p. 56. Three windmills can grind 20, 17, 15 per day. How long to do 24?


Cardan. Practica Arithmetice. 1539.

Chap. 47, ff. L.iii.r - L.iii.v (pp. 70-71). Simple example -- 5 mills grind 7, 5, 3, 2, 1 per hour, how long will they take to grind 500?

Chap. 66, section 125, ff. kk.vi.r - kk.vi.v (pp. 180-181). Cask with four taps located at levels 1/3,  1/3 + 1/4,  1/3 + 1/4 + 1/6,  1 from the top and which empty their respective portions in 4, 3, 2, 1 hours. How long to empty the cask with all taps?

Chap. 66, section 126 (misprinted 123), ff. kk.vi.v - kk.vii.r (p. 181). Cistern:  (1, 2, 3,  4, -5, -3/4).


Gemma Frisius. Arithmetica. 1540. (20, x) in 14 -- man & wife drinking a cask of wine. ??NYS -- Latin given in H&S, p. 71.

Recorde. Second Part. 1552. 1668, pp. 329-330: A question of water, the eighth example. (6, 8, 9, 12).

Tartaglia. General Trattato, 1556, art. 74, p. 248v; art. 176 177, p. 261v; art. 187 188, pp. 262r 262v.

Art. 74: 120 per 40 and 15 ½ per 6 to do 120.

Art. 176: (16, 20).

Art. 177: (60, 80, x) in 30.

Art. 187: 1 per 8, 1 per 6 and 1 per 3 to do 25.

Art. 188: (10, 5, 4).


Buteo. Logistica. 1559.

Prob. 6, pp. 205-206. Three mighty drinkers drinking an amphora of wine in (24, 12, 8) hours. Cites Pacioli. (H&S 71)

Prob. 7, pp. 206-208. Three architects build a house: (x, x/2, x/3) in 2 months. Says Pacioli gives (x, x+6, x+8) in 2 and solves it wrongly.

Prob. 8, pp. 208-209. Ship with two sails, (8, x) in 5.

Prob. 61, pp. 266-268. Cask with three taps 1/4, 2/3, 1 of the way down which could empty the whole cask in (6, 3, 3) hours.

Prob. 62, pp. 268-269. Cistern, (+2, -3).


Gori. Libro di arimetricha. 1571.

F. 74v (pp. 82 83). Cistern empties in (4, 6) hours. Ship with three sails, (3, 4, 5) days.



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