#### gematric calculator

The Saviour's Smile: RISU in Eclogue 4 and the Names MARIA IESUS

The following study developed from a project on Shakespeare's Epitaph, in which, as I am convinced, the names of IESUS and MARIA play a significant gematric role.

I. Aspects of the Decimal System in the names of MARIA and IESUS

#### I. Aspects of the Decimal System in the names of MARIA and IESUS

Introduction into Eclogue 4, 60

1.      The numerical values (NV) of MARIA and IESUS are 40+70 = 110 = 2*55. The sum of the numbers 1-10 is 55. While the 10 numbers can be placed on the points of one tetraktys, the twice 5 letters of MARIA and IESUS will have to be properly arranged on two tetraktyses.

An alternating pattern of the two names splits the numerical sum (NS) 110 into two halves:

 M A R I A I E S U S sm NS1 12 17 1 5 20 55 NS2 1 9 9 18 18 55 110

The count of values starts from the first letter of MARIA and proceeds in two groups of uneven and even numbers from 1-10.

This alternating pattern of uneven and even numbers with the same gematric results reveals the closest possible interrelation between MARIA and IESUS. It shows that MARIA puts her life totally in the service of her son and his work of salvation. It even seems that her share in the work of salvation is equal to her son's.

2.      The letters of MARIA and IESUS have proved to be intertwined in their regular juxtaposition. So it is most plausible to place the 5 letters of each name on the points of two tetraktyses as regularly as possible. The start for each name will be the cornerpoints of the two opposite tetraktyses, Mary's tetraktys running from top to bottom, Jesus' from bottom to top. The second to fifth letters occupy the 4 horizontal points of each tetraktys:

Each tetraktys is now composed of 4 hexagonal and 3 extension letters. There is a total relation of 4 inner to 6 outer letters (lts), whose numerical sum + factoral sum (FS) are:

 4 lts 6 lts NS 64 46 110 FS 40 28 68 178

64 and 46 are inverted numbers, the digits showing accordance with the 4+6 letters. There are two ratios: 40:64 = 8*(5:8); 40:28 = 4*(10:7). The ratio 10:7 corresponds to the 10 points of the tetraktys and 7 points of the hexagon.

3.      The 4 hexagonal letters belong to both tetraktyses, so their values have to be counted twice: 46+64+64 = 174; 68+40+40 = 108. The two sums are to each other as 6*(18:29) = 6*47. The number 6 refers to 6 triangular sides of two tetraktyses, 47 to the ratio of part to whole, i.e. 4 points to the whole of (4 points + 3 lines) of one triangular side. The NS 108 in the product form 12*9 represents the initials of MARIA and IESUS.

These are the values of the two tetraktyses:

 Tetr. M Tetr. I NS FS sm NS FS sm 35 20 55 11 8 19 RISU 64 40 104 64 40 104 99 60 159 75 48 123 159:123 = 3*(53:41)

The ratio of the two FS 60:48 is 12*(5:4) = 12*9 , again confirming the initials of MARIA and IESUS.

4.      The four hexagonal letters, read clockwise, show a Latin noun form in the ablative case, derived from RISUSlaughter, smile. Two letters are taken from MARIA, two from IESUS. So the act of recognition and smile is a mutual one.

If one follows the outer points anticlockwise, starting from the A on the right , one can read AMASYou love. One may suppose a speaker inwardly addressing the Virgin Mary, as he observes her smiling at her child. The remaining letters IE may be complemented like this:

RISU AMAS IE(SUM)You show your love for Jesus by smiling at him.

The letters SUM, which are contained in the two names, complete the 13 points of the tetraktys star, occupying the missing points of the central axis:

5.      Love arises only between persons. It is engendered by acts of mutual recognition. Human love is a divine gift, and its eternal model is the loving relationship between the trinitarian persons of Father, Son, and Holy Spirit.

A SMILE starts from a person's mouth and expands over his whole face. So love transgresses all boundaries. Such a boundary is the hexagon that is expanded to form a hexagram including two tetraktyses. The hexagram is completed by a circle whose area is three times the size of the hexagon area. This expansion can be understood as self-manifestation of the trinitarian persons whose origin is the First person or God Father: The hexagon represents the unity of the Father with the Son and the Holy Spirit, the hexagram their loving relationship.

The noun form RISU perfectly demonstrates the relationship between the two circle areas by the internal ratio (inr) and external ratio (exr) of its FS:NS:

 FS diff. NS RISU 40 24 64 inr 8*(5:3) exr 8*(5:8)

The two area sizes are represented by the number of radial elements:

If the digits of the two rations 5:3 and 5:8 are added up, a new ratio arises, 8:13, which refers to the elements of the double-rhombus, 8 elements for the extension parts, 13 for the hexagonal part. The two numbers then stand for 2:1 area units:

#### II. Vergil's Eclogue 4, 60-63: Gematric aspects

1.      In his famous 4th eclogue the poet VIRGIL announces the birth of a boy who will bring peace to the world. In line 60 he says:

Incipe, parve puer, RISU cognoscere matrem

Begin, little boy, to recognise your mother with a SMILE

Virgil invites the little boy to reward his mother with a SMILE, while the tetraktys figure rather suggests MARIA is taking the initiative. Virgil wants to say that this child is so extraordinary that a smile is expected as a sign of his divine origin. In reality, however, babies do not smile after their birth at once. They are helpless bundles and have to be cared and caressed by their mothers until their human faculties are gradually elicited.

2.      It is from his overflowing love that God created the world. So the two concentric circles of the tetraktys star can be seen not only as an image of the trinitarian God himself, but also of his creation. Out of love God's Son, his own image, became a human being, and the loving smile between Mother and Son marks the beginning of hope for a future era of reconciliation and peace.

3.         The word RISU is one among 9 words which belong together. This last text unit of Vergil's 4th eclogue consists of lines 60-63. The following text is provided with the NS+FS of each line. The values of single words can be obtained by an online calculator:

 Incipe, PARVE PUER, RISU cognoscere MATREM: 398 295 693 MATRI longa decem tulerunt fastidia menses. 395 313 708 Incipe, PARVE PUER: qui non RISERE PARENTI: 404 305 709 nec deus hunc mensa, dea nec dignata cubilI (E)ST. 342 263 605 1539 1176 2715

Translation:

Begin, little boy, to recognise your mother with a smile:

Ten months caused your mother long inconveniences.

Begin, little boy: Those who have not smiled at their parent:

Neither a god has considered him worthy of his table nor a goddes of her bed.

4.         I have not found any scholarly answer to the prolonged 10 months of pregnancy the little boy's mother had to go through. From the point of view of the decimal system, the number ten is the completion of the basic numbers 1-9. In analogy, the announced extraordinary boy crowns and completes the history of mankind by his birth and the deeds of his life.

The tetraktys consists of units of 10 and 9: 10 points, 9 triangles, 18 lines, together of 37 elements. The poem ends with CUBILI (E)ST. Because of the preceding vowel I, E is dropped when the hexameter is read. The letter S corresponds to the 18 lines and T to the 10 points + 9 triangles. If an E (5) is added to ST, the elements of a single tetraktys are converted to the elements of two tetraktys frames with 7 elements for each of the 6 sides (6*7 = 42).

The total result of NS+FS 2715 exactly reflects the last word of the poem (E)ST: Vergil extended the components 27+10 by another 5. Another meaning of 2715 is to be found further down.

5.         Having a look at the 9 words around RISU, we find that the repetition of PARVE PUER and the references to his mother MATREM MATRI PARENTI consist of equally 18 letters, while RISU RISERE, serving as linking bond between Son and Mother, amounts to 10 letters. So we should find out the values of these 10 letters first:

 RISU RISERE NS 64 71 135 FS 40 58 98 104 129 233

The number 135 especially refers to the 9 triangular areas of the tetraktys, defined by 10 points:

The values of the 4+3 words follow:

 NS FS sm NS FS sm PARVE 58 40 98 MATREM 66 56 122 PUER 57 39 96 MATRI 58 50 108 sm 115 79 194 PARENTI 79 69 148 double 230 158 388 sm 203 175 378 230+203 = 433; 158+175 = 333

Son and Mother correspond to each other by the inverted NS 230 and 203. The digits of the sum 433, a prime number, show the structure of the 10 tetraktys points.

6.         The total results of the three groups (PRM) are:

 P 230 158 388 M 203 175 378 R 135 98 233 621+378 = 999 365 256 621 621:378=27*(23:14)

378 is the sum of the successive numbers 1-27. The boy's mother is of mature perfection, which implies her smiling at him first. It's only by his smiling back that her son lives up to the expectation of his own perfection, represented by the added up result 388+233 = 621. There is a correspondence between the two ratio numbers 23 and 14, not only because their digits have the sum 5, but because 23 is the sum of the NV 14 + its FV 9. The son embodies his mother's nature and is further completed and enhanced by the nucleus of the factoral value. It shows the selfless attitude of the mother, who serves her greater son to fulfil his divine mission on earth.

7.         The number 27 has several meanings. First, as the cube of 3 it has trinitarian relevance. With regard to the two concentric circles of the tetraktys star, 13 points represent 3 area units, and 7+7 points 1+3 = 4 area units of the inner and outer circle (each circle with a central point). Then it can refer to the 3*9 diametrical elements of the double-rhombus, and finally to 9+9 lines and 9 triangles of the tetraktys.

If the 10 points and 9 triangles of the tetraktys form one unit and the 18 lines the second unit of 37 elements, the FV of 19+18 are 19+8 = 27. So the product 27*37 = 999 represents the FS+NS of the two groups of elements.

The meaning of 27, as specifically relevant in the present context, however, consists in the NV of P+M = 15+12 = 27, the initials of PUER and MATER. The FV of the two numbers are 8+7 = 15. So it seems plausible to associate these two numbers with the total NS+FS 2715 of lines 60-63.

8.         If 27 is composed of the values of P+M, Vergil will have spent special attention to the 9 initials. We can group them into two pairs of P+M, three further P, and two R:

 2(P+M) 3 P sm 2 R tot NV 54 45 99 34 133 FV 30 24 54 34 88 sm 84 69 153 68 221 54:99 = 9*(6:11) = 9*17 153:68 = 17*(9:4) = 17*13

The trinitarian number 153 forms a ratio with the NS+FS of the 2 R 68, which shows that the mutual smile between Mother and Son flows from their common and united nature. The numbers 11, 13, 17 refer to the number of elements of 3 geometrical figures to be found both in the tetraktys and the double-rhombus:

9.         The poet attributes great significance to the little boy's smile. Two aspects already mentioned have to be considered: The son has his own nature and has adopted that of his mother, in whom his body was formed. It seems he strongly claims his life as his own at the expense of his mother. It's perhaps for this reason that the poet calls upon the boy to smile at his mother in order to give her something back as his due.

The perfect number 378 suggests that his mother smiles at him first. The NS of RISU and RISERE is 135 = 5*27. If he answers her smile, he gives her 135 of his values so that the NS+FS ratio between Son and Mother is now (621-135):(378+135) = 486:513 = 27*(18:19). In this way the 37 elements of the tetraktys are most equally and justly divided up.

The numbers 18+19 are the NV of ST, the last two letters of the poem. Furthermore, the NS of the last line is 18*19 = 342.

10.    Also divisible by 27 are the NS+FS of the first two + the last three of the nine words so that successive groups of 2, 4, 3 words come about, which assembled as a 3-digit number 243 is 9*27.

 NS FS sm NS FS sm PUER 57 39 96 PARVE 58 40 98 RISERE 71 58 129 PUER 57 39 96 PARENTI 79 69 148 sm 115 79 194 sm 208 177 373 194+373 = 567 = 21*27

The 5:4 words thus have the NS+FS ratio 27*(21:16). The ratio can refer to the frame of a double-rhombus numbered by 1, 2, and 3, 21 being the sum of the hexagonal part, and the 16 the sum of the extension part:

The digits 243 refer to the the middle part of a double-rhombus: 2 crosslines, 4 triangles, 3 points. These 9 elements also apply to the two diametrical zig-zag lines so that the NS+FS result 999 shows 27 elements if the three parts are counted separately, whereas as a united figure the double-rhombus consists of 21 elements.

Two groups of words provide divisibility by 37 and the result 333: First, the FS of the 7 mother-son words (s. table above), second the NS+FS of the words PUER (96) MATRI (108) RISERE (129), all divisible by 3. The ratio to the rest of values is 1:2.

11.        The 9 words give some clues to further gematrical constructions: First, the 4 words referring to the Son (2x parve puer) and 3 to the Mother (matrem, matri, parenti) consist of equally 18 letters. The 10 letters of RISU RISERE add 5 letters to both sides. In fact, the NS+FS of each half of 23 letters is divisible by 27 with the ratio 19:18 again:

 PARVE PUER RISU MATREM MATR sm I PARVE PUER RISERE PARENTI sm NS 58 57 64 66 49 294 9 58 57 71 79 274 FS 40 39 40 56 44 219 6 40 39 58 69 212 98 96 104 122 93 513 15 98 96 129 148 486 513:486 = 27*(19:18)

Second, the numbers 5 and 18 are to be found in the numbered tetraktys, with 5 placed in the centre and 1+7+10 = 18 in the corners. The 1+3 numbers (standing also for 1:3 area units) were given a trinitarian meaning in the Roman system: The letter equivalents are ESYou are. The 4+6 letter words RISU RISERE are chosen for this purpose:

RISERE occupies the 6 hexagonal arc points.

The NS of lines 60-63 are 1539 = 81*19 >FV 31 and the FS 1176 = 24*49 >FV 23. The two FV 23+31 = 54 can be seen in connection with the numbered tetraktys and the specific grouping of 4+6 numbers. The sum 54 as 2*27 relates to the tetraktys star with its two concentric circles. On the assumption of each circle having its own central point, both the hexagon and its extension part consist of 1+24 elements. The 2 other elements are the circle arc and the circle area.

Third, if the frame of the double-rhombus is numbered loop-like, number 17 comes next to 1. The unoccupied middle part consists of 6 elements so that there is room for 23 numbers or letters. Twice 23 letters then call for a cross of double-rhombi:

The NS+FS of the left and upper rhombi amount to 500, the bottom and right rhombi to 499. The two numbers are constitutive of 999. Their FVs are 19+499 = 518. Split up into 5+18, this number exactly represents the sum of the 1+3 numbers of the tetraktys and the word ES. So 999 – in the wisdom of the decimal system – must especially refer to the tetraktys with 9 triangles 9 frame lines and 9 interior lines. It seems that Vergil's special ambition was to achieve these two numbers 500 and 499.

12.        The four central letters of the figure read – anticlockwise from left – VESV. If the last letter V is replaced by TA, the word is VESTA, the goddess and code name of the Roman gematric system.

The FV of double V (20) is 9 = I, the FV of ES (5+18) is 13 = N. In this way we get 3 further letters, INI = FS 31, which is another indication of the 3:1 aspect. INI as an imperative form means begin, start. The seven letters VESVINI can be rearranged to the word IUVENISyoung man, a reference to the little boy's later life and task.

13.         The ratio 3:1 has a natural basis in 3 elements of the middle part of a rhombus + 1 central point. The four rhombi of the cross provide 12+4 elements. The NS of the 12 letters are divisible by 13 and enable a ratio if two pairs of rhombi are joined, as the above figure shows: 52:78 = 13*(4:6). The FS is 117 = 13*9, which gives a FS:NS ratio of 13*(9:10) = 13*19 = 247. The equivalent letters are NT.

The product of two NS (top and right rhombi) 37*41 is 1517, which comes about if the FS 518 (from 500 and 499) is added to 999. The equivalent letters for 1517 are PR, which can be aptly associated with the initials of PUER and RISU.

If the NS of the 4 central letters 63 and 247 are added, the result 310 is divisible by the inverted number 31. The FS of the central letters is 31 itself, which gives a total result of 31*(1+10) = 341.

#### III. The last sentence

 QUI non risere parenti: 235 185 420 nec deus HUNC mensa, dea nec dignata cubili (E)ST. 342 263 605 577 448 1025

1.      The last two lines show a grammatical anomaly that has caused much controversy in the history of classical philology: The plural of the relative pronoun QUI is followed by singular HUNC. The shift from plural to singular was considered so unusual that later copyists changed QUIPARENTI into CUIPARENTES. Fortunately the two lines are correctly quoted and commented by Quintilian 9,3,8.

What has become clear so far is that the poet has no doubt that the little boy will smile at his mother. So he expresses in negative terms what he means positively: All children who show affection for her mothers will be rewarded in some way. But the one little boy who is praised in the poem will rise far above all the others, attaining divine honours for his merits.

The motive why Vergil calls upon the boy to smile at his mother is not because he is in any doubt about the boy, but because he loves and admires his mother. He wants the boy to share his enthusiasm and to participate himself in the relationship between mother and son.

2.      The imagery of the last line, as may be supposed by now, serves special gematric concepts. Most striking is the parallelism DEUS MENSADEA CUBILI. Both word pairs consist of 9 letters (4+5, 3+6), half the number of the Son-Mother letters. The NS show a clear gematric relationship:

 NS FS sm NS FS sm DEUS 47 26 73 DEA 10 10 20 MENSA 49 34 83 CUBILI 54 37 91 96 60 156 64 47 111 96:64 = 32*(3:2); 160+107=267=3*89

The product 32*5 deserves some attention. The natural constituents of 32 are 15+17 with the letter equivalents of P+R and initials of PUER and RISU. The letter equivalent of 5 is E. The letter group PER, however, is the repetitive component of the inner frame of the SATOR-Square with the NS 37, which is the main factor of 999:

The NS+FS of the initials of PARVE PUER (2x) and RISU RISERE is also 160.

 4P 2R sm NV 60 34 94 FV 32 34 66 sm 92 68 160

The prefix RE means back, which in the present context refers to the boy's smiling back at his mother. The letters ERE in RISERE coincide with the same sequence of letters in the inner frame of the SATOR Square, representing action (ER) and reaction (RE) of the mother's and son's smile. In fact, it is a perpetual repetition of mutual affection that is symbolised by the act of smiling. The NS of the 3 letters is 27, contained as a factor in 999.

The next point of interest should be the frequency of these three letters in the 9 words. There are 5 P, 8 E, and 10 R, 23 letters in all and relatable to the 4 numbered tetraktys points whose some is 23: 5 for the centre, 8 as 1+7 and 10 for the corners. The NS+FS is the palindromical number 535.

Of course, we mustn't forget the 3 M of the mother. The NS+FS is 3*(12+7) = 57. The new sum 592 is divisible by 37 and provides the ratio of 26:20 letters = NS+FS 37*(16:11).

1.      We obtain the main syntactical structure of the relative clause and the main clause if we connect QUI (3) and HUNC (4) with their finite verb forms RISERE (6) and DIGNATA EST (10). The numbers of letters of each pair correspond to each other according to the following equations: 1+2+3 = 6; 1+2+3+4 = 10.

The NS+FS of the pronominal words are:

 NS FS sm QUI 45 23 68 HUNC 44 31 75 89 54 143 143=11*13

The adjacing values 45+44 harmonise perfectly. The model for 89 is a measure of 8 units delimited by 9 points. These 17 elements are concentrically numbered from the centre to the edge:

The FS 23+31 = 54 are identical with the FV of the FS and NS of the total values 1176 and 1539, as explained above (with following figure).

The values of the verb forms are:

 NS FS sm RISERE 71 58 129 DIGNATA 54 51 105 EST 42 32 74 sm 167 141 308 308=11*28

Both the NS+FS of the pronominal words and verb forms are divisible by 11. The total result is 11*41.

The NS+FS of the whole sentence is 1025 = 25*41. So the difference between the two results is 14*41. The corresponding numbers of letters are 23+34 = 57, which is referable to the 2+3 and 3+4 elements of a hexagon diameter and a tetraktys side. If they are numbered, the diameter has the result 11 and the tetraktys side (according to the right counting mode) 11 (4-3-4) for the lines and 14 (5-2-2-5) for the points:

Finally, the relative clause consists of 19, the main clause of 38 letters, forming the ratio 1:2.

In conclusion, the gematric evidence clearly excludes the text variant CUI RISERE PARENTES as Vergil's original version. Still, the internet offers four times more the wrong text.

2.      If the poet proposes to the boy the privilege of sharing the table with a god and the bed with a goddess as reward for smiling at his mother, he is expected to give proof of such a blissful prospect by performing a perfect gematric construction.

The precondition for the boy's extraordinary reward is his smiling back at his mother. So the words RISU and RISERE should be assigned to the twice 9 letters in the last line:

 13 NS FS sm 15 NS FS sm RISU 64 40 104 RISERE 71 58 129 DEUS MENSA 96 60 156 DEA CUBILI 64 47 111 160 100 260 135 105 240 260+240 = 500 = 20*(13:12) 160:100 =20*(8:5); 135:105 =15*(9:7)

The three products gained by ratios are 20*25; 20*13, 15*16.

All single values are divisible by 5. The model aimed at is especially the hexagon.

The ratio of letters of the two word groups is 13:15, also recognisable in the components of two ratios. It refers to three hexagon axes with 3*5 = 15 elements, which diminish to 13, with only one centre point given for all three axes. If the numbers of 20*13 and 15*16 are added, the ratio 33:31 again represents three hexagon axes, this time numbered:

A NS/FS balance calculation produces as a result the 37 elements of the tetraktys from the two constituent numbers 19+18:

 FV sm NS 295 64 FS 205 46 500 110 FV 19 18 37

3.      The NV of the NS+FS 500 (5*10*10) is composed of 5+7+7 and so agrees in meaning with the NS 577 of the last sentence. The three digits represent 1+3+3 area units (F) of the two concentric circles of the tetraktys star, as is shown in the following figure:

The three digits can be understood as 5+7 = 12 with FV 7. In this sense 500 and 577 refer to the letter M and the boy's mother (MATREM, MATRI).

The above double-rhombus shows a 1-5-1 structure of points. This is also reflected in the NS+FS of the lines 60-63 2715 = 15*151.

4.      There may be different reasons why Vergil chose exactly 9896 hexameters for his epos AENEIS. But as the NS+FS of PARVE PUER is 98 and 96, we may confidently believe that he dedicated his great epic work to the longed-for saviour of the world.

Written: March 2009

Last entry: December 2012