THE LEIBNIZ HARMONIC TRIANGLE

The German polymath Gottfried Leibniz discovered a remarkable set of numbers in the form of a triangle. The Leibniz numbers have a symmetry relation about the vertical line. But unlike Pascal’s triangle, the number in one row is obtained by adding the two numbers below it. For example 1/30 + 1/20 = 1/12. To construct this triangle we can progress from the top and move from left
to right by subtraction: we know 1/12 and 1/30 and so 1/12 − 1/30 = 1/20, the number next to 1/30. You might have spotted that the outside diagonal is the famous harmonic series

but the second diagonal is what is known as the Leibnizian series

which by some clever manipulation turns out to equal n/(n + 1). Just as we did before, we can write these Leibnizian numbers as B(n,r) to stand for the nth number in the rth row. They are related to the ordinary Pascal numbers C(n,r) by the formula:

In the words of the old song, ‘the knee bone’s connected to the thigh bone,and the thigh bone’s connected to the hip bone’. So it is with Pascal’s triangle and its intimate connections with so many parts of mathematics – modern geometry, combinatorics and algebra to name but three. More than this it is an exemplar of the mathematical trade – the constant search for pattern and harmony which reinforces our understanding of the subject itself.