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6.BF.3. WELL BETWEEN TWO TOWERS
The towers have heights A, B and are D apart. A well or fountain is between them and equidistant from the tops of the towers. I denote this by (A, B, D). Vogel, in his DSB article on Fibonacci, says the problem is Indian, and Dold pointed me to Mahavira. Pseudo dell'Abbaco introduces the question of a sliding weight or pulley -- see Pseudo dell'Abbaco, Muscarello, Ozanam-Montucla, Tate, Palmaccio, Singmaster.

I have just found that Bhaskara I gives several unusual variations on this.

See Tropfke, p. 622. See also 10.U.
INDEX of A, B, D problems, with A  B.
0 4 8 Chaturveda

0 9 27 Bhaskara II

0 12 24 Bhaskara I

0 18 81 Bhaskara I

5 6 12 Bhaskara I

10 10 12 Bhaskara I

13 15 14 Mahavira

18 22 20 Mahavira

20 24 22 Mahavira

20 30 30 Gherardi?

20 30 50 Perelman

30 40 50 Fibonacci, Muscarello, Cardan

30 50 100 Bartoli

30 70 100 Tate

40 50 30 Muscarello

40 50 70 Pseudo-dell'Abbaco

40 50 100 della Francesca

40 60 50 Lucca 1754

60 80 100 Calandri c1485

70 100 150 Columbia Algorism, Pacioli

80 90 100 Calandri 1491
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 2 and 3.

These are 'hawk and rat problems'. A hawk is sitting on a wall of height A and a rat is distance D from the base of the wall. The rat tries to get to its hole, in the wall directly under the hawk. The hawk swoops, at the same speed as the rat runs, and catches the rat when it hits the ground. Hence this is the same as our two tower problem, but with B = 0, so I will denote this version by (A, 0, D). Bhaskara I attributes this type of problem to unspecified previous writers. Shukla adds that later writers have it, including Chaturveda and Bhaskara II, qqv.

Ex. 2: (12, 0, 24).

Ex. 3: (18, 0, 81). Bhaskara I explains the solution in detail and Shukla gives an English precis of it. Let ABOC be the horizontal diameter of a circle and let LBM be a vertical chord. LB is our pole, with the hawk at L, and the rat is at C and wants to get to B. The point of capture is O, because LO = OC. From LB² = AB x BC, we can determine AB and hence the other values.

Looking at Chaturveda (below), I now see that turning this sideways gives the same diagram as the broken bamboo problem -- the tree was BC and breaks at O to touch the ground at L. So the broken bamboo problem (H, D) is the same as the two towers or hawk and rat problem (D, 0, H).

Bhaskara I. 629. Ibid. Examples 8 and 9 are 'crane and fish problems'. A fish is at the NE corner of a rectangular pool and a crane is at the NW corner and they move at the same speeds. The fish swims obliquely to the south side, but the crane has to walk along the edge of the pool. The fish unfortunately gets to the side just as the crane reaches the same point and gets eaten. This again like our two tower problem, but with one pigeon unable to fly, so it has to walk down the tower and across. Because the pool is rectangular, the two values A and B are equal.

Ex. 8: (6, 6, 12).

Ex. 9: (10, 10, 12). The meeting point is 3 3/11 from the SW corner.

Chaturveda. 860. Commentary to the Brahma sphuta siddhanta, chap. XII, section IV, v. 41, example 4. In Colebrooke, p. 310. Cat and rat, where the cat behaves like the hawk of Bhaskara I: (4, 0, 8).

Mahavira. 850. Chap. VII, v. 201-208, pp. 249-251.

He gives several problems, but he usually also asks for the equal distance from the top

of each tower to the fountain.

v. 204. Two pillars, with a rope between them which touches the ground but with equal lengths to the tops. (13, 15, 14).

v. 206. Two hills with mendicants who are able to fly along the hypotenuses. (22, 18, 20)

v. 208. Same context. (20, 24, 22).

Bhaskara II. Lilavati. 1150. Chap. VI, v. 149 150. In Colebrooke, pp. 65 66. Peacock and snake version of the hawk and rat problem: (9, 0, 27).

Fibonacci. 1202. De duabus avibis [On two birds], pp. 331 332 (S: 462-463). (40, 30, 50). He does the same problem differently on pp. 398 399 (S: 544-545).

Gherardi?. Liber habaci. c1310. P. 139. (20, 30, 30).

Lucca 1754. c1330. F. 54v, pp. 120-121. (60, 40, 50).

Columbia Algorism. c1350. No. 136, pp. 139 140. (70, 100, 150).

Bartoli. Memoriale. c1420. Prob. 10, f. 76r (= Sesiano 138-139 & 148-149, with reproduction of the relevant part of f. 76r on p. 139). (50, 30, 100).

Pseudo-dell'Abbaco. c1440.

Prob. 80, p. 72, with picture on p. 71. (50, 40, 70).

Prob. 158 159, pp. 129 133, with illustrations on pp. 130 & 132, deal with the related problem where a rope with a sliding weight hangs between two towers, and the diagram clearly shows the weight in the air, not reaching the ground, so that the resulting triangles are similar. [I found it an interesting question to determine when the rope was long enough to reach the ground, and if not, how much above the ground the weight was -- see Muscarello, Ozanam-Montucla, Singmaster below.]

Prob. 158 has A, B, D = 40, 60, 40 and a rope of length L = 110, so the rope is more than long enough for the weight to reach the ground, but all he does is show that the two parts of the rope are 66 and 44, which is a bit dubious as there is slack in the rope. The diagram clearly shows the weight in the air. I have a colour slide of this.

Prob. 159 has A, B = 40, 60 with L = 120 such that the weight just touches the ground -- find the distances of the weight to the towers.

Prob. 160, p. 133. (40, 30, 50).

Muscarello. 1478.

Ff. 95r-95v, pp. 222-223. A, B, D = 50, 40, 30. Place a rope between the towers just long enough to touch the ground.

Ff. 95v-96r, pp. 223-224. A, B, D = 30, 20, 40. A rope of length L = 60 with a sliding weight is stretched between them -- where does the weight settle?

F. 99r, pp. 227-228. Fountain between towers for doves: (40, 30, 50).

della Francesca. Trattato. c1480. F. 22r (72). (40, 50, 100). English in Jayawardene.

Calandri. Aritmetica. c1485. Ff. 89r-89v, pp. 178 179. (60, 80, 100). (Tropfke, p. 599, shows the illustration in B&W.)

Calandri. Arimethrica. 1491. F. 100v. Well between two towers. (80, 90, 100). Nice double size woodcut picture.

Pacioli. Summa. 1494. Part II.

F. 59v, prob. 62. (70, 100, 150) = Columbia Algorism.

Ff. 59v-60r, prob. 63. A, B, D = 30, 40, 20 with rope of length 25 between the towers with a sliding lead weight on it. How high is the weight from the ground.

Ff. 61r-61v, prob. 66. Three towers of heights A, B, C = 125, 135, 125, with distances AB, AC, BC = 150, 130, 140. Find the point on the ground equidistant from the tops of the towers.

Cardan. Practica Arithmetice. 1539. Chap. 67.

Section 9, f. NN.vi.r (p. 197). (40, 30, 50).

Section 10, ff. NN.vi.r - NN.vii.v (pp. 197-198). Three towers of heights A, B, C = 40, 30, 70, with distances AB, AC, BC = 50, 60, 20. Find the point on the ground equidistant from the tops of the towers. Same idea as Pacioli, prob. 66.

Ozanam Montucla. 1778. Vol. II, prob. 7 & fig. 5, plate 1. 1778: 11; 1803: 11-12; 1814: 9 10; 1840: 199. Rope between two towers with a pulley on it. Locate the equilibrium position. Uses reflection.

Carlile. Collection. 1793. Prob. XLV, p. 25. Find the position for a ladder on the ground between two towers so that leaning it each way reaches the top of each tower. (80, 91, 100). He simply states how to do the calculation, x = (D² + A² - B²)/2D for the distance from the base of tower B.

T. Tate. Algebra Made Easy. Chiefly Intended for the Use of Schools. New edition. Longman, Brown, Green, and Longman, London, 1848. P. 111.

No. 36. A, B, D = 30, 70, 100. Locate P such that the towers subtend the same angle, i.e. the two triangles are similar. Clearly P divides D as 30 to 70.

No. 37. Same data. Locate P so the distance to the tops is the same. This gives 70 to 30 easily because A + B = D.

No. 38. Same data. Locate P so the difference of the squares of the distances is 400. Answer is 68 from the base of the shorter tower.

Perelman. MCBF. 1937. Two birds by the riverside. Prob. 136, pp. 224-225. (30, 20, 50). "A problem of an Arabic mathematician of the 11th century."

Richard J. Palmaccio. Problems in Calculus and Analytic Geometry. J. Weston Walch, Portland, Maine, 1977. Maximum-Minimum Problems, No. 3, pp. 9 & 70-71. Cable supported pulley device over a factory. A, B, D = 24, 27, 108 with cable of length L = 117. Find the lowest point. He sets up the algebraic equations corresponding to the the two right triangles, assumes the distance from one post and the height above ground are implicit functions of the length of the cable from the pulley to the same post and differentiates both equations and sets equal to zero, but it takes half a page to get to the answer and he doesn't notice that the two triangles are similar at the lowest point.

David Singmaster, proposer; Dag Jonsson & Hayo Ahlburg, solvers. Problem 1748: [The two towers]. CM 18:5 (1992) 140 & 19:4 (1993) 125-127. Based on Pseudo-dell'Abbaco 158-159, but the solution by reflection was later discovered to be essentially Ozanam Montucla.

David Singmaster. Symmetry saves the solution. IN: Alfred S. Posamentier & Wolfgang Schulz, eds.; The Art of Problem Solving: A Resource for the Mathematics Teacher; Corwin Press, NY, 1996, pp. 273-286. Gives the reflection solution.

Yvonne Dold-Samplonius. Problem of the two towers. IN: Itinera mathematica; ed. by R. Franci, P. Pagli & L. Toti Rigatelli. Siena, 1996. Pp. 45-69. ??NYR. The earliest example she has found is Mahavira and it was an email from her in about 1995 that directed me to Mahavira.