Pascal's Triangle

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Construction

     As any current researcher of Pascal’s Triangle would notice, there is a slight difference in Blaise Pascal’s representation of the triangle in 1654 [Fig. 1] and present-day representations [Fig. 2]. 

Fig. 1


 
 

Fig. 2



In these figures it can be seen that triangles found in today’s textbooks are shifted to allow for a downward flow of numbers, while the numbers in Pascal’s Triangle of 1654 extended diagonally from left to right.  Despite differences in their representations, today’s arithmetic triangle contains the same numerical qualities as Pascal’s seventeenth century triangle.  To construct ‘his’ triangle Blaise Pascal began by drawing two lines perpendicular to each other, as shown in Fig. 1 as GV and Gµ.  He, then, divided each of those lines into a number of equal and continuous parts and extended then either vertically or horizontally, which created cells.  Next, Pascal joined the two points of the first division to form a triangle of which this line is the base.  Continuing in this manner, Pascal created many triangles, bases, and cells.  Finally, to complete his construction, Pascal drew in a line of symmetry, which began at the uppermost left point and extended down in a diagonal right direction.  Distinguishing between cells of the same parallel rank, the same perpendicular rank, and the same base in his Traité du triangle arithmétique, Pascal was able to develop his findings of the triangle in clear terms.  He defined cells of the same parallel rank to be cells which run horizontally from left to right (such as A, B, and C), while cells of the same perpendicular rank were defined to be cells which run vertically from the top downward (such as A, D, and H).  In addition, cells of the same base were those that the same base runs diagonally (such as P, M, and F) (Calinger 349-50).  At this point, as he stated, Pascal was finally able to place numbers into his triangle. 

                                      “We now place numbers in each cell
                                      and this is done in the following way:
                                      the number of the first cell which is
                                      in the right angle is arbitrary, but
                                      once it has been placed all the other
                                      numbers are determined, and for
                                      this reason it is called the generator
                                      of the triangle.  And every one of the
                                      other numbers is specified by this
                                      sole rule: The number of each cell is
                                      equal to that of the cell preceding it
                                      in its perpendicular rank plus that of
                                      the cell which precedes it in its
                                      parallel rank” (350-1).
 

He further went on to state, “in every arithmetic triangle all the cells of the first parallel rank and of the first perpendicular rank are equal to the generator” (351).  A person of today may be quite confused by the complicated terms Pascal used to construct and describe ‘his’ triangle, yet his triangle contained the same numerical qualities as the triangle of today. 

     Instead of describing features of the triangle by terms such as parallel rank and perpendicular rank, today’s triangle is described in terms of rows.  For example, Pascal’s generator would be considered today to be in row number zero, which is the number 1.  The first row would consist of the numbers in the line directly below row number zero, and so on.  Also, the numbers in each row of the triangle are described in terms of element numbers.  For example, in Fig. 2, the third row contains the numbers 1 3 3 1.  The first 1 would be defined as element number zero, while the first 3 would be defined as the first element, and so on.  Furthermore, the zeroth and last elements in every row of the triangle are always 1, and, to form any other number of the triangle one would just have to add the two numbers immediately above it.  For example, in Fig. 2, the second element in the fifth row, a 10, is formed by finding the sum of the 4 and the 6 from the fourth row.  Now, rather than struggling through Pascal’s steps for construction, the triangle can be developed by following these simple rules.