Blaise Pascal · Mathematics
Pascal’s <em>Treatise on the Arithmetical Triangle</em>: how one simple table of numbers unifies the binomial coefficients, combinations, and figurate numbers - and how, to prove one of its properties for infinitely many rows at once, Pascal gave one of the first fully explicit proofs by <em>mathematical induction</em>.
The arithmetical triangle is one of the most fertile objects in all of mathematics, and its rule could not be simpler. Start with 1 at the top. Each new number is the sum of the two numbers just above it (with 1s running down the edges). So the rows grow: 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1; and on forever. Pascal did not invent this table - versions of it appear centuries earlier in India, Persia, and China, in the work of scholars like al-Karaji and Yang Hui. What Pascal did, in his Treatise on the Arithmetical Triangle (composed in 1654, published in 1665), was to give it its first unified, systematic treatment: he drew out its many hidden properties, proved them, and revealed that a single table quietly governs combinations, the expansion of binomials, the figurate numbers, and the odds in games of chance. In the West the table has borne his name ever since.
The power of the triangle is that it turns hard counting problems into simple table lookups. How many different five-card hands can be dealt from fifty-two cards? How many committees of three can be formed from ten people? How many routes run through a grid? Each is a question about combinations, and each answer sits waiting in the arithmetical triangle. Pascal showed how to read them off and how the rows relate - why, for example, the numbers in a row sum to a power of two, and why the triangle is symmetric (choosing which three to include is the same as choosing which seven to leave out). He also used the triangle to crack the problem of points from the previous lesson, because counting the ways an unfinished game can end is, once again, a problem of combinations. One modest table, it turned out, was a master key to the whole mathematics of the possible - of how many ways things can be arranged, chosen, and combined.
In his treatise Pascal states a whole series of properties of the triangle, which he calls its ‘consequences.’ The twelfth of these concerns the ratio between two neighbouring numbers in a row, and Pascal wants to prove it holds in every row of the endless table. He pauses to acknowledge the difficulty head-on: the proposition has an infinite number of cases. Then he gives his remedy. He will prove just two lemmas - that the property holds in the first relevant row, and that whenever it holds in one row it holds in the next - and from these two, he says, the result follows necessarily in all the rows, out to infinity. It is a strikingly modern moment: a mathematician confronting the infinite, and taming it not by checking cases but by the airtight logic of the inductive step. Proof by induction would become one of the central techniques of all mathematics, the standard way to establish a truth about every whole number at once.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Pascal’s arithmetical triangle collects the binomial coefficients (the numbers of combinations) in one table, and that in proving its ‘Twelfth Consequence’ he laid out proof by induction: a base case plus a step that carries the property from each row to the next. Explain the triangle and the logic of…
Leads to Gottfried Leibniz.
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