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In the nineteenth century, another familiar face was discovered in Pascal’s triangle: the Fibonacci sequence. Perhaps this was inevitable, as the method of constructing of the triangle was recursive – we repeatedly performed the same rule, which was the adding of two numbers on one line to produce a number on the next line. The recursive summing of two numbers is exactly what we do to produce the Fibonacci sequence. The sum of two consecutive Fibonacci numbers is equal to the next number in the sequence.

Pascal’s triangle with only squares divisible by 3 in white.

Pascal’s triangle with only squares divisible by 4 in white.

Pascal’s triangle with only squares divisible by 5 in white.

The gentle diagnals in Pascal’s triangle reveal the Fibonacci sequence.

Fibonacci numbers are embedded in the triangle as the sums of what are called the ‘gentle’ diagonals. A gentle diagonal is one that moves from any number to the number underneath to the left and then along one space to the left, or above and to the right and then along one space to the right. The first and second diagonals consist simply of 1. The third has 1 and 1, which equals 2. The fourth has 1 and 2, which adds up to 3. The fifth gentle diagonal gives us 1 + 3 + 1 = 5. The sixth is 1 + 4 + 3 = 8. So far we have generated 1, 1, 2, 3, 5, 8, and the next ones are the subsequent Fibonacci numbers in order.

Ancient Indian interest in Pascal’s triangle concerned combinations of objects. For instance, imagine we have three fruits: a mango, a lychee and a banana. There is only one combination of three items: mango, lychee, banana. If we want to select only two fruits, we can do this in three different ways: mango and lychee, mango and banana, lychee and banana. There are also only three ways of taking the fruit individually, which is each fruit on its own. The final option is to select zero fruit, and this can happen in only one way. In other words, the number of combinations of three different fruits produces the string 1, 3, 3, 1 – the third line of Pascal’s triangle.

If we had four objects, the number of combinations when taken none-at-a-time, individually, two-at-a-time, three-at-a-time and four-at-a-time is 1, 4, 6, 4, 1 – the fourth line of Pascal’s triangle. We can continue this for more and more objects and we see that Pascal’s triangle is really a reference table for the arrangement of things. If we had n items and wanted to know how many combinations we could make of m of them, the answer is exactly the mth position in the nth row of Pascal’s triangle. (Note: by convention, the leftmost 1 of any row is taken as the zeroth position in the row.) For example, how many ways are there of grouping three fruits from a selection of seven fruits? There are 35 ways, since the third position on row seven is 35.

Now let’s move on to start combining mathematical objects. Consider the term x + y. What is (x + y)2? It is the same as (x + y) (x + y). To expand this, we need to multiply each term in the first bracket by each term in the second. So, we get xx + xy + yx + yy, or x2 + 2xy + y2. Spot something here? If we carry on, we can see the pattern more clearly. The coefficients of the individual terms are the rows of Pascal’s triangle.

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2y + 3xy2 + y3

(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

The mathematician Abraham de Moivre, a Huguenot refugee living in London in the early eighteenth century, was the first to understand that the coefficients of these equations will approximate a curve the more times you multiply (x + y) together. He didn’t call it the bell curve, or the curve of error, or the normal distribution, or the Gaussian distribution, which are the names that it later acquired. The curve made its first appearance in mathematics literature in de Moivre’s 1718 book on gaming called The Doctrine of Chances. This was the first textbook on probability theory, and another example of how scientific knowledge