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6.BF.1 THE BROKEN BAMBOO
A bamboo (or tree) of height H breaks at height X from the ground so that the broken part reaches from the break to the ground at distance D from the foot of the bamboo. In fact the quadratic terms drop out of the solution, leaving a linear problem. This may be of Babylonian origin?? The hawk and rat problems of 6.BF.3 are geometrically the same problem viewed sideways.

In all cases below, H and D are given and X is sought, so I will denote the problem by (H, D).

See Tropfke, p. 620.
Chiu Chang Suan Ching (Jiu Zhang Suan Shu). c 150? Chap. IX, prob. 13, p. 96. [English in Mikami, p. 23 and in Swetz & Kao, pp. 44 45, and in HM 5 (1978) 260.] (10, 3).

Bhaskara I. 629. Commentary to Aryabhata, chap. II, v. 17, part 2. Sanskrit is on pp. 97 103; English version of the examples is on pp. 296-300. The material of interest is examples 4 and 5. In the set-up described under 6.BF.3, the bamboo is BOC which breaks at O and the point C reaches the ground at L.

Ex. 4: (18, 6). Shukla notes this is used by Chaturveda.

Ex. 5: (16, 8).

Chaturveda. 860. Commentary to the Brahma sphuta siddhanta, chap. XII, section IV, v. 41, example 2. In Colebrooke, p. 309. Bamboo: (18, 6).

Mahavira. 850. Chap. VII, v. 190-197, pp. 246-248.

v. 191. (25, 5), but the answer has H - X rather than X.

v. 192. (49, 21), but the answer has H - X rather than X.

v. 193. (50, 20), but with the problem reflected so the known leg is vertical rather than horizontal.

v. 196. This modifies the problem by imagining two trees of heights H and h, separated by D. The first, taller, tree breaks at height X from the ground and leans over so its top reaches the top of the other tree. If we subtract h from X and H, then │X - h│ is the solution of the problem (H - h, D). Because the terms are squared, it doesn't matter whether X is bigger or smaller than h. He does the case H, h, D = 23, 5, 12.

Bhaskara II. Lilavati. 1150. Chap. VI, v. 147 148. In Colebrooke, pp. 64 65. (32, 16).

Bhaskara II. Bijaganita. 1150. Chap. IV, v. 124. In Colebrooke, pp. 203 204. Same as Lilavati.

Needham, p. 28, is a nice Chinese illustration from 1261.

Gherardi. Libro di ragioni. 1328. Pp. 75 76: Regolla di mesura. (40, 14).

Pseudo-dell'Abbaco. c1440. No. 166, p. 138 with B&W reproduction on p. 139. Tree by stream. (60, 30). I have a colour slide of this.

Muscarello. 1478. F. 96v, pp. 224-225. Tree by a stream. (40, 30).

Calandri. Aritmetica. c1485. Ff. 87v-88r, pp. 175-176. Tree by a stream. (60, 30). = Pseudo-dell'Abbaco.

Calandri. Arimethrica. 1491. F. 98r. Tree by a river. (50, 30). Nice woodcut picture. Reproduced in Rara, 48.

Pacioli. Summa. 1494. Part II, f. 55r, prob. 31. (30, 10). Seems to say this very beautiful and subtle invention is due to Maestro Gratia.

Clark. Mental Nuts. 1897, no. 78. The tree and the storm. (100, 30). [I have included this as this problem is not so common in the 19C and 20C as in earlier times.]

N. L. Maiti. Notes on the broken bamboo problem. Gaņita-Bhāratī [NOTE: ņ denotes an n with an underdot] (Bull. Ind. Soc. Hist. Math.) 16 (1994) 25-36 -- ??NYS -- abstracted in BSHM Newsletter 29 (Summer 1995) 41, o/o. Says the problem is not in Brahmagupta, though this has been regularly asserted since Biot made an error in 1839 (probably a confusion with Chaturveda -- see above). He finds eight appearances in Indian works, from Bhaskara I (629) to Raghunath-raja (1597).