There is a common belief that the method of mathematical induction is a product of Western civilisation, discovered by Maurolyco (1495-1575) and that the first clear account of the method was given by Pascal. The term `mathematical induction' was first used by the English mathematician Auguste de Morgan in 1838. In his `Justification Theorem', Dedekind proved that this procedure does not lead to logical complications. (See [1], [2].)
I believe that Pappus of Alexandria (c. 300 A.D.) deserves more credit than is commonly attributed to him in this matter. Here is a short account of Pappus' proof of one of his theorems.
On the same line segment \overline{AB}, three semicircles are drawn so that they are tangent in pairs. In the region bounded by the three semicircles, a chain of tangent circles is inscribed in the manner shown below.
The diameters of these tangent circles are denoted by
d_1, d_2, \ldots, d_n and the distances of their centres to the baseline
\overline{AB} by h_1, h_2, \ldots, h_n respectively. Pappus' Theorem
states that
h_n = nd_n.
The steps taken by Pappus to prove this theorem are essentially the steps in the method of mathematical induction. He knew that the distance of the centre of the first circle from \overline{AB} was equal to its diameter. This had been proved by Archimedes and appears in his Book of Lemmas as Proposition 6.
Pappus also knew about the formula h_n = nd_n as he refers to
it as ``an ancient proposition''. To verify this statement he showed that
for n=1 it was true. To complete the proof he had to show that
if
h_{n-1} = (n-1)d_{n-1}
is true, then
h_n = nd_n
is also true.
He did this by considering two consecutive tangent circles
and showing that
\frac{h_n}{d_n} = \frac{h_{n-1} + d_{n-1}}{d_{n-1}}.
(See [3] for a complete proof.) Pappus then continues as follows.
\frac{h_n}{d_n} = \frac{h_{n-1} + d_{n-1}}{d_{n-1}}
= \frac{h_{n-1}}{d_{n-1}} + 1
h_1 = d_1
\frac{h_2}{d_2} = \frac{h_1}{d_1} + 1 = 2, \quad h_2 = 2d_2
\frac{h_3}{d_3} = \frac{h_2}{d_2} + 1 = 3, \quad h_3 = 3d_3
...........................................
\frac{h_n}{d_n} = \frac{h_{n-1}}{d_{n-1}} + 1 = n, \quad h_n = nd_n.
Thus, it seems to me that Pappus understood the essential ideas in the mehod of mathematical induction.