#### gematric calculator

c) The Underlying PRINCIPLES

1.      In chapter b) 21 words proved to be a major gematric unit. At the end of the investigation emerged the underlying system which will be detailed here. The 21 words subdivided into 11+10 were attributed to the MOTHER and FATHER of the expected Saviour Boy:

 lett. NS FS sm lett. NS FS sm SM SM GS MATREM 6 66 56 122 PARENTIS 8 97 77 174 163 133 296 MATRI 5 58 50 108 PARENTI 7 79 69 148 137 119 256 11 124 106 230 15 176 146 322 300 252 552 230:322 = 2*23*(5:7) = 46*12; 552 = 24*23

 lett. NS FS sm lett. NS FS sm SM SM GS PUERO 5 71 48 119 PATRIIS 7 88 65 PUER 4 57 39 96 sm 9 128 87 215 7 88 65 153 216 152 368 368 = 16*23

 lett. NS FS sm lett. NS FS sm SM SM GS ASPICE 6 51 31 82 ASPICE 6 51 31 82 INCIPE 6 54 41 95 INCIPE 6 54 41 95 PARVE 5 58 40 98 PARVE 5 58 40 98 PUER 4 57 39 96 PUER 4 57 39 96 RISU 4 64 40 104 RISERE 6 71 58 129 DEA 3 10 10 20 DEUS 4 47 26 73 CUBILI 6 54 37 91 MENSA 5 49 34 83 34 348 238 586 36 387 269 656 735 507 1242 1242 = 54*23

TOTAL RESULT of 21 words

 lett. NS FS sm lett. NS FS sm SM SM GS mother 54 600 431 1031 father 58 651 480 82 1251 911 2162 2162 = 2*23*47 = 46*47

The 4 violet words belong to the first part of the poem:

 lett. NS FS sm lett. NS FS sm SM SM GS PUERO 5 71 48 119 PATRIIS 7 88 65 PUER 4 57 39 96 PARENTIS 8 97 77 sm 9 128 87 215 15 185 142 327 313 229 542 542 = 2*271

The result of the shortened group of 17 words is:

 lett. NS FS sm lett. NS FS sm SM SM GS MATER 45 472 344 816 PATER 43 466 338 804 938 682 1620 816:804 = (8*102):(4*201); 1620 = 10*(9*18)

2.      The 21 words and 112 letters can be divided into 2 groups of 4-3-4 and 4-2-4:

 lett PUERO PUER PATRIIS PARENTIS 24 MATREM MATRI PARENTI 18 ASPICE INCIPE PARVE PUER 21 ASPICE INCIPE PARVE PUER 21 RISU RISERE 10 DEUS MENSA DEA CUBILI 18

There is a gematric correspondence of the two Groups between the two inner and 4 surrounding word Groups: First, there is a trinitarian ratio of 28:84 = 28*(1:3) letters, second, there are NS:FS-ratios in each word group with the common main NS+FS-factor 47:

 NS FS sm MATREM MATRI PARENTI 203 175 378 RISU RISERE 135 98 233 338 273 611 273:338 = 13*(21:26) = 13*47

 NS FS sm PUERO PUER PATRIIS PARENTIS 313 229 542 ASPICE INCIPE PARVE PUER 220 151 371 533 380 913 ASPICE INCIPE PARVE PUER 220 151 371 DEUS MENSA DEA CUBILI 160 107 267 380 258 638 913 638 1551 638:913 = 11*(58:83) = 11*141 = 3*11*47

The NS:FS-ratio 913:638 is repeated as NS+FS ratio 913:638 between the two upper and lower word Groups.

3.      The innermost PRINCIPLE of this gematric construction unit is contained in the NS+FS of the three words MATREM (l.60) MATRI (l.61) PARENTI (l.62): 203+175 = 378 = 7*(29:25) = 21*18. The geometric reference is the double rhombus (DR) with 4 triangular areas + 7 points and 10 lines:

 sm FW sm NS 4 7 10 21 10 31 FV 4 7 7 18 8 26 sm 39 18 57

57 is the NS of PUER, 39 the FS and 18 the difference between the FS and NS.

The four letters of PUER can be placed on the triangular areas:

The NS+FS of the inner two letters UE correspond to the FS, the outer two letters PR to the NS of the whole word. Morover, the first two letters have the NS+FS 52 = 4*13, the last two letters the NS+FS 44 = 4*11. The factors 13 and 11 are contained in the NS+FS results of the 2 inner and 4 surrounding word Groups

4.      One rhombus connects the hexagonal area with the extended area and so represents 3 area units:

One triangle consists of 7 elements, one rhombus of two triangles, but of 11 elements, three less than 2*7 = 14. If a ratio of elements of one triangle to one rhombus is established, this means 7:11 elements – which is the number of letters of MATREM MATRI and PARENTI – the triangle meant might belong to the hexagonal or the extended part of the tetraktys star. If the triangle belongs to hexagonal part, 7:11 elements represent the ratio 1:3 of circle area, if it is thought to belong to the extended part, the ratio is 2:3.

A double triangle + another triangle form a fish shaped figure, 4 more elements add a fourth triangle:

17:4 elements mean a trinitarian ratio of 3:1 triangles. This accounts for the 4 words of the first half of the poem and 17 of the second half.

5.      On a smaller scale FATHER, MOTHER and CHILD have to be looked at more closely gematrically:

 line lett. NS FS sm line lett. NS FS sm PATRIIS 17 7 88 65 153 PUERO 8 5 71 48 119 PARENTIS 26 8 97 77 174 PUER 18 4 57 39 96 PARENTI 62 7 79 69 148 PUER 60 4 57 39 96 PUER 62 4 57 39 96 105 22 264 211 475 148 21 242 165 407 475+407 = 882 = 2*21² 264:242 = 22*(12:11) = 22*23 = 506; 211+165 = 376 = 8*47 153+97 = 250; 250:225 = 25*(9:10); 165:242 = 11*(15:22) = 11*37 71+57+57+57 = 185; 185:222 = 37*(5:6) 105+148 = 253 = 11*23

23 and 47 are the main factors in the total NS+FS result 2162 = 2*23*47. One reason for number 23 is that the NV+FV of P, the initials of PATER and PUER, is 15+8. One can see this when adding 105+60 = 165; 165:88 = 11*(15:8).

25 and 37 refer to the number of elements of the hexagon and the tetraktys, representing area ratio 1:3 of the two tetraktys circles.

The two NS+FS 475+407 equal 882 = 2*21². Two halves can be achieved if the NS+FS of three PUER and of PATRIIS are added: 288+153 = 441 = 21².

There is finally to be added MATREM MATRI RISU RISERE:

 line lett. NS FS sm 7 words 253 43 506 376 882 MATREM 60 6 66 56 122 MATRI 61 5 58 50 108 374 54 630 482 1112 RISU 60 4 64 40 104 RISERE 62 6 71 58 129 122 10 135 98 233 total 496 64 765 580 1345 1345 = 5*269 630:135 = 45*(14:3) = 45*17

269 is the sum of the 4values of PATER MATER:

 NS FS sm FV1 FV2 sm GS PATER 57 50 107 22 12 34 141 MATER 54 49 103 11 14 25 128 111 99 210 33 26 59 269

The result shows an ideal family relationship because father, mother and child love each other by smiling at each other.

6.      The pattern 4-3-4, 4-2-4 explained above fits perfectly for placing the 21 letters on the 21 elements of the DR. How Vergil carried out this task, however, is not easy to find out. After testing different possibilities, I think to have reached a satisfactory solution. It consists of two principles here: First, the letters are placed in a continuous movement from below to top and from top to bottom. Second, the NS and FS are separated and placed in a cross of double rhombi so that the lower rhombus can be combined with the right rhombus:

According to the above pattern there are 6 up and down movements. The placement starts with PUERO on the 4 cross-points and goes down the 3 central points and up again on 4 triangular areas. The second part of the pattern starts from above down on 4 zigzag-lines, then up again on 2 cross lines and finally down on the 4 other zigzag lines.

The surprising result is that the NS+FS of the lower + right and left + upper rhombi are equally 1027 = 79*13 = FW 92 = 4*23 if MATRI in the central point is omitted. 1027 can be referred to the TETRAKTYS and understood as 10 points, 18 lines + 9 triangular areas. On the scale of the basic numbers 1-9 the digits 7 and 9 are complementary to 3 and 1 and so have equally trinitarian meaning. 79 and 97 are the NS of PARENTI (l.62) and PARENTIS (l.26). 7 signifies the number of hexagonal points, 9 the number of points of the tetraktys frame:

In this context 7 to 9 points represent the ratio of 1 to 3 area units of the two concentric circles.

Without MATRI the 8 words of the longitudinal axis have the NS+FS 874 = 19*46. The ratio of 8:13 words in this way is 46*(19:28) = 2162.

The NS+FS 108 of MATRI added to 536 equals 644 = 28*23, thus forming a ratio of 46*(14:33) for the 21 words.

The FS 92 = 4*23 can be associated to a square whose 4 sides consist of 2 units of measure and 3 points. It can be constructed by shifting two angles of an axis cross against each other:

The frame of the square is part of OPERA in the SATOR Square:

The word OPERAwork contains 3 letters of PUER and 4 letters of PUERO. The NS+FS of OPERA is 52+40 = 92. The two equal sums of 1027 coincide with twice OPERA/AREPO of the SATOR-Square.

92 is the NS of SANCTUSHOLY to which PATRIIS shows a certain resemblance of its 4values:

 NS FS sm FV1 FV2 sm GS SANCTUS 92 61 153 27 61 88 241 PATRIIS 88 65 153 17 18 35 188

The axis cross above consists of 1+8+8 = 17 elements.

7.      MATRI is justified to occupy the centre of the DR-cross because its NS+FS 108 is connected with the NS+FS 1620 of 17 words by the ratio 108*(1:14), so it is rightly the centre of 15 frame elements of the DR.

The FS of the NS+FS of the 21 words and the 3 subsums again lead to the prime number 269, the 4values of PATER MATER:

 sm FV NS 1027 1027 108 2168 72 FV 92 92 13 197 197 sm 269

An NS+FS ratio of the two zigzag lines is obtained if MATRI is assigned to the right zigzag line:

728:560 = 7*8*(13:10). 7*8 is related to 7 points and 8 lines of the DR-frame. 13:10 could mean 13 points of the tetraktys star and 10 points of the tetraktys. But it could specially refer to the DI, the plural of DEUS. The NV of the two letters are 4+9 = 13 and their FV 4+6 = 10. 13 can be seen in a trinitarian sense as One God in Three Persons. As a two-digit number 49 is the sum of elements of the tetraktys star. Vergil's Eclogue 4 consists of 409 words.

8.      PUERO (l.8), MATRI (l.60) and PARENTI (l.62) are three dative cases which, as we confidently imagine, the poet Vergil chose to dedicate his poem to these unique venerable divine persons by whom mankind would be presented with a felicitous future. He did not do so without making sure of their combined special gematric properties. Besides the NS+FS their FV are necessary:

 NS FS sm FV1 FV2 sm GS PUERO 71 48 119 71 11 82 201 MATRI 58 50 108 31 12 43 151 sm 129 98 227 102 23 125 352 PARENTI 79 69 148 79 26 105 253 total 208 167 375 181 49 230 605 253:352 = 11*(23:32) = 11*55 = 5*11² 208+181 = 389; 167+49 = 216 = 6³ 208 = 16*13

605 is the NS+FS of line 63, the last line of the poem.

The inverted sums 253 and 352 show the relationship between father, mother and child: A mother conceives her child by a father and carries him in her womb until he is born. In the numerated tetractys the sum of numbers 1-10 can be grouped to sums of 23 and 32:

The digits of 216 can describe the structure of points on the tetraktys frame: 2 bottom corner points, 1 top corner point, 6 hexagonal points. In this context can mean 3*6 elements of the tetractys frame: 9 points, 9 lines. On the other hand each tetractys side consists of 4 points and 3 lines, 7 elements together. If the points are numerated with 2 and the lines with 3, the sum is 8+9, for three sides 3*(8+9), which could be read as the prime number 389:

Equally or more important are three geometrical "fish figures" (cf. above) in the tetractys, each consisting of 17 elements:

The cube number also points to the three-dimensional figure of hexahedron with its 6 corners. This context is indicated by the prime numbers 151 and 181 both contained in the DR which in the form of a DR-cross is a construction element for the octahedron. 151 refers to the structure of DR points: 5 hexagonal points and 2 extension points, 181 to 10 units of measure, 2 cross lines 8 frame lines. 181 is of DIVINE significance as it shows 2 units of measure – limited by uncountable 0 – in a circular circumference beyond the basic numbers 1-9:

This above model is characterised by 3+17 elements leading to the following calculation: 3+17 = 20; 17*3 = 51; 20+51 = 71. 3, 17 and 71 are factors of the NS of the whole poem: 25347 = 3*7*17*71. 181 is also the main NS+FS factor of the 4 last lines: 2715 = 15*181.

This basic circular model can be represented in linear extension:

The linear model ends with a second point so that the calculation is 2+17+2 in contrast to the circular model 2+17+1. As a 4-digit number 2172 = 12*181 the linear model points to the circular model of 1+8+1 units of measure.

The circular model substantiates a special numeration of the DR-points in loop shape so that 9 positions delimit 8 units of measure including the two cross lines:

The right DR is an attempt to illustrate the palindromic number 181.

Vergil obviously also considered the verse numbers of the three names 8+60+62 = 130 which turns attention to the SATOR-Square: SATOR TENETThe Creator Holds has the NS 69+61 = 130 = 2*65 = FV 20. NS+FV equal 150 = 6*25. The single digits of the two multiplication numbers are the same, they represent the elements of a double triangle with 2 triangular areas (F), 6 lines and 5 points:

The three lines lead to 181 again:

 sm NS 8 60 62 130 FV 6 12 33 51 sm 181

In the SATOR Square 181 is the sum of the prime number NS:

 T E R N A sm NV 19 5 17 13 1 55 frequ. 4 4 4 1 4 17 NS 76 20 68 13 4 181

TERNA is neuter in the plural meaning a Set of Three, which can be theologically applied to the three divine persons. The 5 letters have the average NS 11. Another anagram is ENTRAEnter! expressing an invitation to learn about the wisdom of the SATOR-Square. A third anagram is REANTthey shall calculate, expressing a wish to recognize the numeral system behind the 25 lettes of the square.

Written: January 2022