## 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. 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. 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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