Simanaitis Says

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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. 

Blaise Pascal, 1623–1662, French polymath.

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.

Pascal’s Triangle, in which the next row is derived from the row above it. See also “Holmes and Maths” here at SimanaitisSays.

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.

Pensées de M. Pascal sur la Religion er sur Quelques Autres Sujets, By Blaise Pascal, 1670.

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. 

A Pascaline, approximately 14 in. x 5 in. x 3 in. This one, signed by Pascal in 1652, is in the Musée des Arts et Métiers in Paris.

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.

The “carrying” function. This and the following image from A History of the Machine, by Sigvard Strandh, A&W, 1979.

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.”  

Strandh wrote, “Descartes brought one with him when he left for the royal court in Stockholm in 1649, and he presented it, and an elevated dedication by Pascal, to Queen Christina. ” 

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,, 2021 


  1. Pingback: BLAISE PASCAL—COUNTING ON THINGS – Glyn Hnutu-healh: History, Alchemy, and Me

  2. Tom Phillips
    March 2, 2021

    Hi Dennis,
    A discussion of Pascal’s Principle, without which jacking up a car would be much harder, would be worth a post.

  3. simanaitissays
    March 2, 2021

    Yes,Tom. P = F/A, but, as my grandkids would say, “What’s ‘jacking up a car,’ Grandpa?” “Uh, it’s to fix a flat.” “What’s a ‘flat,’ Grandpa?”

  4. sabresoftware
    March 3, 2021

    The Odhner mechanical calculator ( works on a similar concept. My father had one for running his business. It now resides in my personal calculator museum. It’s a great fun machine, and if you ever drop it on your foot you’ll be crippled for life. The website says 11 lb. but it feels heavier than that.

    My calculator museum has that machine, an old HP25 (50 step programmable but no permanent storage) from college days, and my TI-59 programmable from early work days.

    Also a Casio pocket computer (programmable in Basic) which might still be operational had I not swung my jacket while putting it on with it in the pocket and hit a table edge destroying the display (not cracked but the LCD “pixels” inside are dead).

    A Sharp IQ-8300M Organizer. Not sure when I stopped using it but probably around the time I got my first iPhone. Also have a tiny Radio Shack LCD calculator 2″x3.25″x1/8″ (the buttons were starting to fail, but sure was handy).

    One that is not in there yet is a simple Casio 4 function plus memory & percentage that is just slightly larger than the Radio Shack. What makes it worthy of the museum is that I bought it in a Walmart out of town when going to visit a project site and realized that I had left my calculator behind at the office, for the princely sum of $4.95! In 1995. It is still running now, 26 years later on the original battery! Buttons are starting to go now.

    But what is missing and has me very sad is my pocket sized slide rule (6″) that my father gave me and which I used throughout university days and even at work. I know that I wouldn’t have given it away or thrown it out (intentionally), but so far I have not been able to track it down. Yet.

    Reminds me of a co-worker who lost his HP-21 at the office and found it months later inside a catalogue binder in the office library where he had used it as a bookmark and forgot it.

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