Mathematics of Levi ben Gershon

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Mathematics of Levi ben Gershon

• Solving Proportion Problems
• Using Matrix Algebra

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Rabbi Levi ben Gershon 1288-1344

• Philosopher
• Biblical Exegete
• Inventor
• Scientist
• Mathematician

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Levis Mathematics

• Maaseh Hoshev (1321)
• Commentary on Euclid (1320s)
• De Sinibus, Chordis et Arcubus (1342)
• De Numeris Harmonicis (1343)
• A harmonic number, as defined by Phillipe de
  Vitry (1291 1361), is a number of the form
  2n3m. Levi proves that (1,2), (2,3), (3,4),
  (8,9) are the only pairs of such numbers that
  differ by 1.

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Levis Maaseh Hoshev

• Theory Over 60 theorems and proofs on
  arithmetic, algebra, sums, combinatorics,
  proportions.
• Early use of proof by mathematical induction
• Practice Algorithms for addition, subtraction,
  multiplication, division, square roots, cube
  roots, certain sums, proportion problems,
  permutations, and combinations.

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Proportions and Matrix Algebra

• Here is another easy way of finding numbers
  where the first plus one part or parts of the
  rest equals the other plus one part or parts of
  the rest

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Levis Matrix Algebra Lemma

• xyz 6
• xy w 7
• x zw 8
• yzw 9
• Subtract each of the smaller from the largest
• w x 3, z x 2, y x 1
• Adding these gives (yzw) 3x 6,
• and therefore 9 3x 6, and x 1.

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Example

• Levi adds the constraint xyzw -
  (9-5)(2-1)(7-3)(3-1)
• Levi Claims
• yzw 9142 72
• xzw 4242 64
• xyw 4172 56
• xyz 4143 48

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Levi Proves his Claim
xyzw - (9-5)(2-1)(7-3)(3-1) x
(5/9)(yzw) yzw - (9-5)(2-1)(7-3)(3-1)
(5/9)(yzw) yzw - (5/9)(yzw)
(9-5)(2-1)(7-3)(3-1) (9-5)/9 (yzw)
(9-5)(2-1)(7-3)(3-1) yzw
9(2-1)(7-3)(3-1)
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Example

• yzw 9142 72 xzw 4242 64
• xyw 4172 56 xyz 4143 48
• By Levis lemma
• w x 24 z x 16 y x 8
• x 8, y 16, z 24, w 32

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• x 8, y 16, z 24, w 32
• Any multiple of this solution is also a
  solution to the original problem.
• Levi has machinery to avoid the absurdity of
  negative solutions.

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Conclusions

• Levi implicitly considers matrix algebra in the
  context of under-determined systems of equations
  derived from problems on proportions.
• Levi has ad-hoc methods to solve particular kinds
  of systems of n equations with n variables.
• Levi has no general method akin to Gaussian
  elimination for solving an arbitrary system of
  equations. Jiuzhang suanshu (179 CE)