Appendix B - The parameter coefficients revisited.
A number of additional comments may be made with reference to the parametric formula that generates G (the first eight words of the Bible), thus:
 - The three parameters (500, 105, and 99) have substantial geometrical links with the first two perfect numbers (6 and 28), and with one another - as Figure A5 confirms:
 - For convenience, the parametric equation and table of parameter coefficients are reproduced here:
We observe that over the range (ie omitting the triplet associated with the first, 913)
the p-values (coefficients of the first parameter, 500) are constant
the q-values (coefficients of the second parameter, 105) vary from -3 to 1
the r-values (coefficients of the third parameter, 99) vary from -3 to 0
It follows that the eight G-values (highlighted) represent a subset of the set of 17 positive integers obtained by allowing the coefficients to move freely within these limits, thus:
86, 92, 98, 185, 191, 197, 203, 290, 296, 302, 308, 395, 401, 407, 500, 506, 605
Given the limiting values 86 and 605, the probability of a number chosen at random achieving a 'hit' is 17/520, or 0.03269. To select seven of the set in succession is therefore associated with a probability of (0.03269)^7 - equivalent to odds of 25 billion to one against! It is therefore abundantly clear that these relationships could hardly have arisen by chance!
 - And what of 913 - the number represented by the first Hebrew word of Holy Scripture? As has already been observed, it is different in kind from the following seven - its strategic position being peculiarly marked in this manner. So what probability can be associated with this? We take our lead from the geometrical considerations which attend G: the sum of the first seven, and the sum of the eight, were intended (most appropriately, in view of the triangular pedigree of the parameters!) to yield triangular numbers. This matter is now considered. Because
we seek a positive integer (k, say) such that both (k + 1788) and (k + 2090) are triangular. A GW-BASIC search reveals k = 913 as the only solution - and along with it the eye-catching features of the triangles, 2701 and 3003, and the triplet of the uniquely-triangular, 666 - all matters which have been dealt with in earlier pages! Clearly then, this particular number is an essential feature of the overall design.
 - The table of coefficients reveals a further interesting feature, viz the triplet of unit magnitude relating to 296 - the 7th Hebrew word word and last of Genesis 1:1. It follows that any multiple of 296 will be represented by a triplet of the form, (m, -m, -m) - where m is the multiplier. Additional features of interest follow:
attention has already been drawn to the fact that the name "Jesus" - as rendered in NT Greek, nominative case - is also the number 888, or 3 x 296; it would therefore appear in our table of coefficients as (3, -3, -3);
similarly, since the title "Christ" is 1480, or 5 x 296, it would appear as (5, -5, -5);
and finally, their combination, (8, -8, -8), would represent 2368 - "Jesus Christ"; interestingly, this repetition of eights is an image of 888, "Jesus"!
 - There is a further extra-biblical observation: it is to do with the 'friendly' or 'amicable' number pair, 1184/1210 - strangely overlooked for many centuries by those searching for further examples of the phenomenon. The feature that binds such numbers together is the fact that each is the sum of the proper divisors of the other. Now the proper divisors of a given number are all the numbers, smaller than itself (including 1), which divide it exactly. Thus, the proper divisors of 1184 are (1, 2, 4, 8, 16, 32, 37, 74, 148, 296, and 592 - their sum being 1210; those of 1210 are (1, 2, 5, 10, 11, 22, 55, 110, 121, 242, and 605) - their sum, 1184. This number, 4 x 296, is the arithmetic mean of the Lord's name and title, 888 and 1480. In our table of parameters it would appear as (4, -4, -4).
 - It may be noted that the entry (1, 1, 1) represents the straight sum of the parameters, ie 500 + 105 + 99 = 704, or 11 x 64. Intriguingly, this number leads us to 407 (the 6th of G) in three distinct ways: (a) by writing its digits in reverse; or (b) by summing the cubes of its digits; or (c) by subtracting 297 (= 3 x 99 - the length of an A4 sheet in mm). Factorising 407 we obtain 11 x 37 - and we observe that the factors accompanying 11 in these expansions, viz 64 and 37, are those of 2368, "Jesus Christ".
 - We began this particular line of investigation with the number 1260. Observing that this number is 12 x 105, we can write it as the triplet of coefficients (0, 12, 0). Its close companion in the Book of Revelation, 666, is likewise (0, 12, -6).
 - Finally, it is worth observing that the foregoing parametric equation is capable of generating an infinite number of copies of G! This may be deduced from the evaluation of the coefficient triple (3, -19, 5), ie (3 x 500 - 19 x 105 + 5 x 99) = 0! Clearly therefore, following vector principles, the addition/subtraction of this triple to/from any other can have no effect on the value generated. As an illustration of this, consider the triple (1, 1, 1) which as we have seen generates the value 704. The addition of (3, -19, 5) yields (4, -18, 6) - this also generating 704!
Return to Appendix 1
Return to 'Exceptional Measures'