THE PYTHAGOREAN TRIANGLE AND THE PYTHAGOREAN ABACUS

   The numerical array to the left may be termed the Pythagorean Triangle. In each row the sequence of numbers ascend and descend about the numbers in the central column, these are the counting numbers starting at the top with one.

. This mid-sequence number of each row is the first occurrence of that number in the array. All subsequent occurrences of that number will fall in diagonal columns descending left and right from this mid-point. We can think of the sequence of numbers in each row as a special instance of a group of sequences. These all begin with one and ascend to a given counting number, which is repeated (n) times, then the sequence descends to one. . In the array rows  of sequences with (n) greater than one, the number of times the mid-sequence number is repeated, the ascending and descending numbers are connected by a diagonal column. The sum of the numbers included in these sequences will equal the product of the mid-sequence numbers of the connected rows. In the special case where (n) equals  there is no diagonal displacement of the sequence, it is included completely in a given row, and the sum of the sequence equals the mid-sequence number times it self , it's square.

It so happens that the sequences revealed by the array of the Pythagorean Triangle map the rectangular arrays of the Pythagorean Abacus. As seen here to the right, the rows of beads on the rods of  the rectangular array revealed between two triangular arrays, with bases of three beads, corresponds to the row of numbers in the Pythagorean Triangle with the mid-sequence number three.

To the left it is shown that the rows of beads on the rods of the rectangular array, on the abacus, revealed between two triangular arrays, with bases of four and six beads corresponds to the sequence formed by connecting ascending and descending numbers of rows with mid-sequence numbers  four and six with a column of fours.               

               CONFIGURING MID-NUMBER SEQUENCES TO ADD AND CONTRACT TO ODD AND EVEN SEQUENCES                                        

The numbers of  mid-number sequences can be configured and added together to reveal odd and even sequences. The arrows in the figure to the right indicate that numbers should be configured in rolls beginning in the top roll placing numbers left to right, from one until reaching the mid-sequence number, then continuing in subsequent rolls right to left, through the sequence back to one. Adding the columns contracts the sequences to odd or even sequences. Such sequences will begin with a number equal the number of times the mid-sequence number is repeated. For example in the first sequence the three occurs only one time so the contracted sequence begins with one, in the next sequence the four occurs three times so the contracted sequence begins with three, and in the last sequence the three occurs two times so the contracted sequence begins with two. The extension of the contracted sequence will equal the mid-sequence number. Thus the contracted sequence can be determined with out the above configuration. The product of any two numbers can be expressed as the sum of an odd or even sequence of numbers. To see how these odd and even sequences or revealed on the abacus Link Here