Mathematics of Levi ben
Gershon
Solving Proportion Problems
Using Matrix Algebra
Rabbi Levi ben Gershon 1288-1344
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Philosopher
Biblical Exegete
Inventor
Scientist
Mathematician
Levi’s Mathematics
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Maaseh Hoshev (1321)
Commentary on Euclid (1320’s)
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.
Levi’s 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.
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…
a
c
x   y  z  w  y   x  z  w 
b
d
e
g
z   x  y  w  w   x  y  z 
f
h
Levi’s Matrix Algebra Lemma
x+y+z = 6
x+y+ w = 7
x+ z+w = 8
y+z+w = 9
Subtract each of the smaller from the largest:
w – x = 3, z – x = 2, y – x = 1
Adding these gives (y+z+w) – 3x = 6,
and therefore: 9 – 3x = 6, and x = 1.
Example
5
1
x   y  z  w  y   x  z  w 
9
2
3
1
z   x  y  w  w   x  y  z 
7
3
Levi adds the constraint: = x+y+z+w - (9-5)(2-1)(7-3)(3-1)
Levi Claims:
y+z+w = 9×1×4×2 = 72
x+z+w = 4×2×4×2 = 64
x+y+w = 4×1×7×2 = 56
x+y+z = 4×1×4×3 = 48
Levi Proves his Claim
x+y+z+w - (9-5)(2-1)(7-3)(3-1) = x+ (5/9)(y+z+w)
y+z+w - (9-5)(2-1)(7-3)(3-1)
= (5/9)(y+z+w)
y+z+w - (5/9)(y+z+w)
= (9-5)(2-1)(7-3)(3-1)
(9-5)/9 (y+z+w)
= (9-5)(2-1)(7-3)(3-1)
y+z+w
= 9(2-1)(7-3)(3-1)
Example
5
1
x   y  z  w  y   x  z  w 
9
2
3
1
z   x  y  w  w   x  y  z 
7
3
y+z+w = 9×1×4×2 = 72
x+y+w = 4×1×7×2 = 56
x+z+w = 4×2×4×2 = 64
x+y+z = 4×1×4×3 = 48
By Levi’s lemma:
w – x = 24
z – x = 16
x = 8,
y = 16,
z = 24,
y–x=8
w = 32
5
1
x   y  z  w  y   x  z  w 
9
2
3
1
z   x  y  w  w   x  y  z 
7
3
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.
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)