PASCAL: Treatise on the Arithmetical Triangle

Treatise on the Arithmetical Triangle by Pascal

Proportions and Consequences of the Arithmetical Triangle

  • First Consequence: “In every arithmetical triangle all the cells of the first parallel row and of the first perpendicular row are the same as the generating cell.”
  • Second: “In every arithmetical triangle each cell is equal to the sum of all the cells of the preceding parallel row from its own perpendicular row to the first, inclusive.”
  • Third: “In every arithmetic triangle, each cell is equal to the sum of all the cells in the preceding column that are included between its row and the first row inclusively.”
  • Fourth: “In every arithmetic triangle, each cell diminished by one is equal to the sum of all those which are included between its row and its column exclusively.”
  • Fifth: “In every arithmetical triangle, each cell is equal to its reciprocal.”
    • The reciprocal is the corresponding cell in the same base that is equidistant from the ends of that base.
    • Sixth: “In every arithmetic triangle, a row and a column which have the same exponent are composed of cells which are all the same, the ones to the others.
    • Seventh: “In every arithmetic triangle, the sum of the cells of each base is double that of the preceding base.”
    • Eighth: “In every arithmetic triangle, the sum of the cells of each base is a number of the double progression that begins with 1 and whose exponent is the same as that of the base.”
    • Ninth: “In every arithmetical triangle, each base diminished by one is equal to the sum of all the preceding.”
    • Tenth: “In every arithmetical triangle, the sum of as many consecutive cells in a base as you wish, starting at one end, is equal to the sum of as many cells of the preceding base, plus the sum of one fewer cells.”
    • Eleventh: “Each diagonal cell is double that which precedes it in its row or column.”
      • The cells that are traversed by the line that bisects the right angle are diagonal cells.
      • Twelfth: “In every arithmetic triangle, for two consecutive cells on the same base, the higher is to the lower as the number of cells between the higher and the top of the base is to the number of cells between the lower and the bottom of the base, inclusively.”
        • In this consequence, Pascal clearly introduces the use of mathematical induction.
        • Thirteenth: “In every arithmetical triangle, for two consecutive cells in the same column, the lower is to the upper as the exponent of the base of the upper is to the exponent of its row.”
        • Fourteenth: “In every arithmetical triangle, for two consecutive cells in the same row, the bigger is to its predecessor as the exponent of the base of this predecessor is to the exponent of its column.”
        • Fifteenth: “In every arithmetic triangle, the sum of the cells of any row is to the last of this row as the exponent of the triangle is to the exponent of the row.”
        • Sixteenth: “In every arithmetic triangle, any row is to the previous row as the exponent of the previous row is to the number of its cells.”
        • Seventeenth: “In every arithmetic triangle, any cell together with all the cells of its column, is to the same cell together with all the cells of its row, as the number of cells taken in each row.”
        • Eighteenth: “In every arithmetic Triangle, two parallel ranks equally distant from the extremities, are between them as the number of their cells.”
        • Nineteenth and Last: “In every arithmetic Triangle, two contiguous cells being in the divide, the inferior is to the superior taken four times, as the exponent of the base of that superior to a number greater by the unit.”
        • To find the number in any cell without using the arithmetical triangle, one needs only the exponents of the column and row of the cell. If the sought for number is located in column 5 and row 3, then take all exponents which precede 5 – i.e. 1, 2, 3, 4 – and as many natural numbers beginning at 3 – i.e. 3, 4, 5, 6. Multiply the first three numbers in each group together to get 360. Multiply the last two numbers to get 24. Divide 360 by 24 to get 15, which is the sought for number.

Usages of the Arithmetical Triangle of which the Generator is Unity

  • Numbers of the first order: 1, 1, 1, 1, etc.
  • Second order: 1, 2, 3, 4, 5, etc.
  • Third order: 1, 3, 6, 10, etc.
    • Third triangular is the sum of the first three natural numbers – 1, 2, 3 – which is 6.
    • Fourth order: 1, 4, 10, 20, etc.
      • Adds a triangular from the third order
      • Fifth order: 1, 5, 15, 35, etc.
        • Sum of fourth order
        • Sixth order: 1, 6, 21, 56, etc.
          • Sum of fifth order
          • Etc.
          • The orders of numbers correspond to the rows of the arithmetical triangle.
          • Combinations are all the ways of taking a certain number of cells from a set. For example, given the set A, B, C, D, there is a certain number of ways to remove two letters from the group – these ways are called combinations.
          • The number of combinations of 1 in 4 is 4; 2 in 4 is 6; 3 in 4 is 4; and 4 in 4 is 1.
          • The sum of all combinations that we are able to make of four is 15 – i.e. 4 plus 6 plus 4 plus 1.
          • “If there are any four numbers, the first such as we will wish, the second greater by unity, the third such as we will wish, provided that it is not smaller than the second, the fourth greater by unity than the third: the number of combinations of the first in the third, added to the number of combinations of the second in the third, equals the number of combinations of the second in the fourth.”
          • “In every arithmetic Triangle, the sum of the cells of any parallel rank equals the number of combinations of the exponent of the rank in the exponent of the triangle.”
          • “The number of any cell that it be equals the number of combinations of a number less by unity than the exponent of its parallel rank, in a number less by unity than the exponent of its base.”
          • “Two numbers being proposed, to find how many times the one is combined in the other by the arithmetic Triangle.” Solve for the numbers 4 and 6.
            • The first way to solve this problem is to calculate the sum of the cells of the fourth rank of the 6th triangle – i.e. 1 plus 4 plus 10 = 15.
            • The second way is to take the 5th cell of the 7th base – i.e. 15.
            • To determine how much each player is owed in a game of chance, one must add what a player will earn if he loses and what a player will earn is he wins and then divide it in half if there is an equal chance of winning and there are only two players.
            • For example, a player stands to earn 2 if he loses and 8 if he wins. 2+8 is 10, and half of ten is 5. Therefore, if the players agreed to discontinue the game of chance, then the player would receive 5.
            • The triangle can also be used to find the powers of binomials.

In Pascal’s Treatise on the Arithmetical Triangle, he describes what constitutes an arithmetical triangle and then proceeds to enumerate consequences of such a figure. These consequences are essentially patterns found in the figure that arise out of necessity. The process of discerning patterns and cause and effect is a useful exercise; for it enhances our understanding of the universe.

In primitive times, men did not understand the causes of the weather; and thus they attributed hurricanes, tornadoes, blizzards, etc. to the wrath of the gods. However, the men who were able to discern patterns and the causes of these natural phenomena were better able to organize their farms in accordance with anticipated weather; and thus these men were more likely to survive and pass on their knowledge to posterity. Recognition of patterns is an important faculty to improve, and math affords us such an exercise to achieve this goal.

Treatise on the Arithmetical Triangle by Pascal

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