Who Invented Pascal's Triangle?

¶ … Pascal's Triangle [...] who really invented Pascal's Triangle. While the mathematical formula known as "Pascal's Triangle" has long been attributed to its' namesake, Blaise Pascal, this is not really the case. The formula was simultaneously discovered centuries before Pascal by the Chinese and the Persians, so it seems, and it was even mentioned by Omar Khayyam centuries before Pascal's existence.

Why has the formula been attributed to Pascal? There are no simple answers, but Pascal, one of the world's most famous mathematicians, was the first "modern" mathematician to realize the true potential of the formula and use it accordingly, and so, it still bears his name. Who Invented Pascal's Triangle? The mathematical formula known as "Pascal's Triangle" has long been attributed to the great mathematician and philosopher, Blaise Pascal, who lived in France during the 17th century.

Pascal only lived to be thirty-nine years old, but during his lifetime, he made significant achievements in mathematics and philosophy, and may be most well-known for the mathematical formula of Pascal's Triangle, which he did not invent, but has long received credit for inventing. Pascal was a bright child, who created the first known type of automatic calculator at the age of nineteen, and invented the modern-day barometer before he turned thirty-one.

He also invented the first syringe, and is credited with many mathematical discoveries in addition to his famous triangle. His work on probabilities however, is still realized as a significant breakthrough in mathematics, as this writer asserts. "His development of the theory of probability, a type of applied mathematics which was to prove of great importance in such fields as biological statistics, was, in its time, what we should now call 'a major break-through'" (Schwartz & Bishop, 1958, p. 351).

So, Pascal many not have truly invented or discovered this probability model, but he developed it and placed it into common usage, which is more than either the Chinese or the Persians managed to accomplish. Pascal created his triangle by searching for a solution to an intriguing problem of probabilities, and once he rediscovered it, he used it extensively, which is one reason it is still called "Pascal's Triangle" today.

However, most historians and mathematicians now commonly accept that while Pascal may have enhanced the triangle formula, it was in existence long before Pascal's rediscovery. One mathematician writes: The triangle of coefficients is quite old, and appears to have been discovered independently by both the Persians and the Chinese. The oldest Chinese reference is in the work of Chia Hsien (ca. 1050) which is no longer in existence. Chia Hsien was using the triangle to extract square and cube roots of numbers.

The Persian mathematician Omar Khayyam (1048?-1131?), the author of the Rubaiyat, probably knew of the triangle since he claimed to have a method for extracting third, fourth, and fifth roots which strongly suggests he was using the triangle. However, the triangle is now known as Pascal's Triangle, named after the French mathematician Blaise Pascal (1623-1662) who made great use of it (Clawson, 1999, p. 133). Initially, Pascal discovered the triangle's formula after a gambler asked him about the probabilities of a dice game.

The man was well-known (at the time) gambler Chevalier De Mere, who wanted to know specifically what numbers could come up in all probability when he threw two dice (Struik, 1948, pp. 145-148). This problem led Pascal to create an "arithmetical triangle' formed by the binomial coefficients and useful in probability appeared posthumously in 1664" (Struik, 1948, p. 148). Therefore, the triangle eulogized Pascal's great mathematical career after his death, and holds his name throughout history not for his invention, but his understanding and furthering of the principles of the triangle.

During his work with the triangle, he discovered many more properties than the Chinese and Persians had recognized, and after his death, when his results were formally published, other mathematicians saw the usefulness of the formula at once, and used it to formulate their own theories and solutions to their own perplexing problems. In fact, the understanding of probabilities the triangle helped mathematicians understand has led to the development of "average gain" or "probable gain" formulas that are still used extensively in business and industry (Borel, 1963, p. 20).

The basic formula for the triangle is simple, as one expert notes. If we assume a fictitious row of noughts prolonging each of these lines to right and left, it is possible to lay down the following rule: each number in any one of these lines is equal to the sum of whatever number lies immediately above it in the preceding line, and whatever number lies immediately to the left of that number.

Thus the third number in the fifth line is 10 = 6 + 4; the fourth number in this same line is 10 = 4 + 6; the fifth number is 5 = 1 + 4 (Borel, 1963, p. 18). There is one problem with Pascal's formula, however. Unfortunately, as the numbers increase, the triangle takes much longer to solve, and the formula becomes ungainly. This created problems with the formula initially, but mathematicians have learned to cope with the formula and have created alternates that let them work with the numbers more effectively, as this expert notes.

"Mathematicians have established certain formulas that allow them to work out the numbers which appear in Pascal's Triangle, as well as the sums of whole rows of these numbers included between fixed limits" (Borel, 1963, p. 18). Thus, Pascal's triangular theory was not perfect, but the formula has lasted through time, been improved, and still makes the study of probabilities cognitive. However, this simple formula has made quite a difference in mathematics circles for centuries for a number of reasons.

First, his treatise on these binomial coefficients later helped contribute to Sir Isaac Newton's eventual invention of the general binomial theorem for fractional and negative powers. In addition, Pascal carried on a long correspondence with Pierre de Fermat, and in 1654, this correspondence helped contribute to the development of the foundation of the theory of.