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SEVENTEENTH-CENTURY MATHEMATICIAN Blaise Pascal was also a physicist, philosopher, theologian, and inventor, perhaps best known for co-founding the mathematical theory of probability and devising Pascal’s Triangle coefficients of the binomial theorem. To others, he may be remembered for wagering on God’s existence. And, on a more prosaic note, prompted by his father being a tax collector, he invented the first mechanical computing machine. Here are tidbits on Pascal, his thoughts (his Pensées), and the Pascaline.
A Gambling Payoff. In 1654, Pascal and fellow French mathematician Pierre de Fermat began correspondences on gambling. One result was resolution of the Problem of Points, a classic gambling quandary involving the equitable division of pots based on expected values. Probability theory evolved from this initial work of Pascal and Fermat.
Pascal’s Triangle. Binomials are expressions of the form (x+y). For example, (x+y)2 expands to x2 + 2xy + y2. And (x+y)3 = x3 + 3x2y + 3xy2 + y3. But what about, for example, (x+y)7?
Pascal devised an Arithmetical Triangle, today known as Pascal’s Triangle, to determine the coefficients of any such (x+y)n.
Gambling on God. Pascal expanded his idea of a wager’s expected value into a bet involving the existence of a deity. (Actually, reflecting his upbringing, the deity involved was the Christian God.)
As Wikipedia notes, the concept “posits that humans bet with their lives that God either exists or does not.” What’s more, as described in Pascal’s Pensées, 1670, the smart money bets on God’s existence.
The rationale: If God does not actually exist, there is admittedly a finite loss of Earthly pleasures and luxury. But if God does exist, winning the wager offers infinite gains of Heavenly eternity.
A Tax Collector’s Travails. Pascal’s father was a supervisor of tax collection in Rouen, France. It was a painstaking and laborious task involving a stupendous abundance of basic arithmetic, of addition and subtraction.
Hitherto, humans kept track of things numerically by means of tallying, a one-to-one correspondence between counters, tick marks on clay tablets, for example, and the things being counted.
Carrying. In 1642, Pascal devised a mechanical device to perform addition and subtraction and, by means of repeated action, multiplication and division. The theory is straightforward: wheels with an appropriate number of cogs, typically ten for our base-ten system. Pascal’s cleverness, though, was in the design of its “carrying” mechanism: When 1 is added to 9 on a given dial, this dial is zeroed, and the 1 carries on to the next dial.
Specifics of the Pascaline reflect the fact that Pascal’s father dealt with tabulating currency. The French monetary unit of those days, the livre, was divided into 20 sols, each subdivided into 12 deniers. The Pascaline had dials for deniers, sols, and up to 999,999 livres.
Strandh explained, “When one wheel (a) was moved from 9 to 0, a lever (c), which had been lifted by pins (b), fell downwards. The lever was coupled to a pawl (d) which then brought the next wheel (e) one step forward.”
Strandh continued, “The numbers on the drums (f) could be seen in windows (g) on the face of the machine, and by moving a strip (h), the complementary number of each number could be seen instead.”
Pascaline Tidbits. Pascal went through 50 prototypes before he presented the device to the public in 1645. Wikipedia notes that “In 1649, King Louis XIV of France gave Pascal a royal privilege (similar to a patent), which provided the exclusive right to design and manufacture calculating machines in France.”
What began as a means of easing his father’s travails transformed into a machine entertaining the crown heads of Europe. The Pascaline also set a mechanical example for others to follow. ds
© Dennis Simanaitis, SimanaitisSays.com, 2021