Today is pi day (March 14) and to celebrate this day I decided to share a chapter from the book The Secret Code of the Psalms explaining my discovery of precise ratios of pi in the numerical Hebrew structure of the Psalms in the Bible. This was an accidental discovery as a result of restoring the poetic structure of the Psalms which is hidden in all published versions of the Bible; my original intent was to properly display the poetry to the common reader in the three volume work, The Symbolism of the Psalms (published here in vol 1, vol 2, vol 3). Once the poetry was restored, it became obvious that the Hebrew scriptures were arranged according to numerical counts of verses, words and even letters. Later these counts were used by the Masorites to ensure the accuracy of the text.
With that introduction, here is the full text of one of the chapters to the book The Secret Code of the Psalms:, the entire works explains the mysterious arrangement and hidden numerical structure of all 150 Psalms:
The number pi (π) is an irrational number that represents the ratio between the circumference and diameter of a circle. It is also a transcendental number, which is a number that is not a root of a polynomial equation with integer coefficients. In the ancient Middle East the Divine was often represented as the sun or a circle, thus it is a likely that the mathematical concept of pi was considered to be a "Divine signature" that was considered sacred and holy. In documenting and analyzing the numerical structure of the Psalms, it was discovered that the design of the Psalms contain several ratios of pi, many to a high degree of accuracy. As the Hebrews were not likely to have known about these accurate ratios for pi, they provide evidence of Divine inspiration in scripture. There is strong evidence that these ratios of pi are directly related to the holiness of the numbers three and seven, the number of letters of the Hebrew alphabet, and the sacred names of God in Hebrew scripture. As pi is a transcendental number that is typically represented by infinite series of numbers, it is thus a fitting number to represent the infinite Divine.
In Babylonian mathematics the number pi was approximated as the number 3, which was good enough for the architecture of the time. This approximation is also found in the dimensions of the temple of Solomon (1 Kings 7:23). The Babylonians knew that the number 3 was an approximation - an Old Babylonian tablet from the 19th-17th century B.C. approximated pi more accurately as 25/8. The Egyptain Rhind Mathematical papyrus which dates to about the same time period approximated pi as 256/81 by approximating the circle with an octagon. In the third century B.C. Archimedes approximated pi as somewhere betweeen 223/71 and 22/7. In the second century B.C. Ptolemy approximated pi as 377/120, which is the first known approximation that is accurate to three decimal places. In the fifth century A.D. the Chinese mathematician Zu Chongzhi calculated pi to an accuracy of seven decimal places, and gave two other approximations for pi: 22/7 and 355/113. The latter is the best rational approximation of pi using less than 5 digits in the numerator and denominator. This approximation was not improved upon until about a thousand years later in by Indian and Persian mathematicians. This was improved upon in the succeeding centuries with increasing levels of accuracy, with many approximations of pi calculated from infinite series. With the advent of computers pi is now calculated with accuracies to trillions of digits.
In modern times pi is represented in decimal form, but any decimal can be represented as a continued fraction of whole numbers. A continued fraction is an iterative process that represents a number as the sum of its integer part and the reciprocal of another number. The continued fraction expansion of a number indicates which rational numbers or fractions are the best approximations. These best approximations are known as convergents, where there is no better approximation with a smaller denominator of the convergent number. The number pi is 3.14159... The first best convergent of pi is the number 3. By taking the reciprocal of the remainder, the remainder can be best approximated as 1/7. Thus the next best convergent ratio for pi is 3 plus 1/7 or 22/7. By taking the remainder and converting it to a fraction, adding it to the previous result, a continued fraction for pi can be created as follows:
This continued fraction is developed by first starting from an accurate decimal representation of pi. In this series, the first few convergent ratios that best represent pi are 3, 22/7, 333/106, 355/113, 103993/33102, 104438/33215, and 208341/66317. Each succeeding convergent ratio is more accurate than the former, as the value converges to the value of pi at infinite. The convergent ratio of 355/113 for pi was derived by Zu Chongzhi in the 5th century A.D. using a different method.
DISK ICONOGRAPHY OF THE ANCIENT MIDDLE EAST
There are figures of speech in the Psalms which are related to the iconography of the Divine in the Middle East, which is associated with a circle. Given this fact the ratio of pi would have been considered sacred. Several Psalms mention Jehovah as having wings:
Keep me as the apple, the daughter of the eye, hide me under the shadow of Your wings (Ps. 17:8).
The Hebrew word for "apple" literally means the black pupil of the eye, thus it is a hidden reference to a circle. The phrase "shadow of Your wings" is next mentioned in Psalm 36:
Jehovah, how precious is Your mercy, God, and the sons of Man in the shadow of Your wings (Ps. 36:7).
From Ps. 17:8 to the end of Psalm 36 there are exactly 355 verses. This is followed by 113 verses in Psalms 37-41, where 355/113 is a convergent ratio for pi, indicating that there was knowledge of this ratio long before its rediscovery in the 5th century A.D. The numeric word count of Psalm 17 references Psalm 61, which contains a similar statement:
I will seek refuge under the cover of Your wings (Ps. 61:4).
Similar statements can be seen in Ps. 57:1 and 63:7. Wings are also associated with the sun (a circle) in Malachi:
But unto you that fear my name shall the Sun of righteousness arise with healing in his wings (Mal. 4:2).
The Hebrew word for "wing" is used to refer to a border as well as the four quarters of the earth (see Deut. 22:12, Isa. 11:12), thus two lines dividing a circle into four quarters would have been called "wings." The image of a winged disk to represent God was common throughout the ancient Near East. Below is the winged sun of Thebes of Egypt:
The winged disk was also adopted in the religion of Zoroastrianism in ancient Persia:
RING AND ROD ICONOGRAPHY OF THE ANCIENT MIDDLE EAST
In the above diagram the Divine figure is holding a ring in his hand. In the iconography of the Middle East, a god, goddess or king was often shown holding a ring and rod. This was an ancient symbol, first found on a cylinder seal in Mesopotamia. On a cylinder seal from the Uruk period (ca. 3500-3000 B.C.), a female worshipper is shown bowing down before a rod and ring (from D.J. Wiseman and Werner Forman, Cylinder Seals of Western Asia, London: Batchworth Press, p. 4):
In later depictions, deities from the Third Dynasty of Ur (ca. 2100 B.C.) as well as Old Babylon (ca. 1800 B.C.) and Assyria (ca. 700 B.C.) are shown with a ring and rod in their hand. In other cases, the deity is shown holding a solitary ring. Yet the meaning of these symbols has remained a mystery among scholars. The ring and rod have been associated with the circular shen symbol of Egypt which was often depicted with a staff (James Hall, Illustrated Dictionary of Symbols in Eastern and Western Art, London: John Murray, 1994, pp. 79-80):
In hieroglyphs the Shen ring is depicted as a stylized loop of rope. A more clear association between the rod and ring can be seen in a depiction of the falcon god Horus which is similar to the winged disk, and also holds two Shen rings in its claws:
The shen symbol means to encircle, and signified eternal protection. It was often elongated to contain a royal name, which is known as a cartouche. It was also often associated with a staff. As for the rod and ring symbols in Mesopotamia, it has been surmised that they have literal and metaphorical meanings related to measurement (Henri Frankfurt, Cylinder Seals, London: Macmillan, 1939, p. 179). This can be seen on the Stele of Ur-Nammu, who built the Ziggurat of Ur. On the stele a variety of building projects are shown, as well as a deity who holds a rope arranged in a circle with a rod:
The rod is thus a unit of measurement that was used in building activities, whose measure was considered Divine. In the Sumerian tale Descent of Inanna Inanna descends into the netherworld but must strip herself of her regalia and clothing as she passes through each gate. Among the items she held was a "gold ring over her wrist, and a lapis measuring rod and line in her hand." In another version of the story the gold ring is gripped in her hand, which resembles the iconography of the rod and ring. In scripture there an angel appeared to the prophet Ezekiel in the same manner to provide measurements for a future temple:
When he brought me there, behold, there was a man whose appearance was like bronze, with a linen cord and a measuring reed in his hand. And he was standing in the gateway (Ez. 40:3).
The ring and rod could be representations of the circle and its diameter, as a measuring line was coiled in a rope that was a multiple of the length of the measuring rod. As the rod was likely a standard of measurement it was most likely a measure of the cubit, which was typically determined by measuring the forearm from the elbow to the tip of the middle finger. That the cubit was associated with the diameter of a circle can be seen in the Egyptian cubit, which was divided into seven equal portions known as "palms." If the cubit was used as the diameter of a circle it would represent the convergent ratio 22/7 for pi when measured in palms, with the circumference measured as 3 cubits plus 1 palm. Three cubits would be considered as an approximate representation of the eternal Divine. That this is the case can be seen in the representation of pi in the molten sea of the temple of Solomon, which had a circumference of 30 cubits (as was the height of the temple of Solomon as well as Noah's ark), an exact multiple of 3:
Then he made the sea of cast metal. It was round, ten cubits from brim to brim, and five cubits high, and a line of thirty cubits measured its circumference. Under its brim were gourds, for ten cubits, compassing the sea all around. The gourds were in two rows, cast with it when it was cast (1 Kings 7:23-24).
This is the only direct reference to a measurement of pi in scripture, which uses an approximate value of 3 for pi.
If a cubit represented the diameter of a circle then 3 cubits would be seen as a perfect representation of the circumference. An exact measure of 3 cubits is specified for the altar of the tabernacle of Moses: the altar of burnt offering was 5 cubits wide, 5 cubits long and 3 cubits high (see Ex. 27:1, 38:1). The altar is associated with a circle in the Psalms:
I will wash my hands in innocence, and I will compass Your altar, Jehovah (Ps. 26:6)
The word for "compass" is the same as the word for circle in Hebrew. As the altar was 5 cubits square, the diagonal or diameter of such as circle would almost be exactly 7 cubits. Thus the circumference of a circle that circumscribes the altar would be 22 cubits, representing the convergent ratio 22/7 for pi. Related to this passage the numerical design of Psalm is based on the convergent ratio 355/113 for pi:
|339 = 3 x 113
Psalm 26 was chosen to represent the end of a multiple of 113 for a verse count since 26 is 2 x 13. Also it follows Psalm 25 which is 5 x 5, which is the dimension in cubits of the length and width of the altar. Psalm 27 begins with a reference to light:
Jehovah is my light and my salvation, whom shall I fear? (Ps. 27:1)
This is related to the altar mentioned in Psalm 26, since upon the altar there was a perpetual fire (Lev. 6:13). The altar is associated with the light of the Lord in Psalm 43:
Send out Your light and Your truth, let them lead me... and I will go to the altar of God (Ps. 43:3,4)
The number 27 was chosen to represent the circumference of a circle since it is 3 x 3 x 3, where 3 is the simplest ratio for pi.
CONVERGENT RATIOS OF PI IN SCRIPTURE
The number 318 is 3 x 106, and as 106 is part of the convergent ratio 333/106 for pi this number was regarded as important in the story of Abraham:
And when Abram heard that his brother was taken captive, he drew out his trained servants, born in his own house, three hundred and eighteen, and pursued them unto Dan (Gen. 14:14)
This particular passage is quoted in the Talmud in one of the examples where the Jewish Rabbis made use of Hebrew gematria, which assigned particular numeric values to each Hebrew letter. They noted that the gematria value of Eliezer, Abraham's servant, is 318. From this they explained that Abraham did not call forth 318 servants, but rather just his servant Eliezer. However the number 318 is significant as it is related to the convergent ratio for pi. Evidence of knowledge of this can be seen in Psalms 33-34 which have a total word count of 318, and these two Psalms are related to a series of 333 verses. Psalm 34 also contains a hidden reference to the priest of Melchizedek, who is mentioned in Gen. 14:18 after the victory of Abram and his 318 servants over the enemy.
A more accurate convergent ratio for pi is 355/113. The number 355 is implicitly mentioned in scripture in the book of Exodus, in the description of the construction of the tabernacle:
The silver from those of the congregation who were recorded was a hundred talents and 1,775 shekels, by the shekel of the sanctuary: a beka a head (that is, half a shekel, by the shekel of the sanctuary), for everyone who was listed in the records, from twenty years old and upward, for 603,550 men. (Ex. 38:25-26)
The number 1775 is 5 x 355. One can readily see that 603,550 is 600,000 + 3,550, where 3,550 is 10 x 355. That this is by design can be seen in other passages, where the Israelite males are numbered elsewhere as 600,000 (see Ex. 12:37, Num. 11:21), and in other places as 603,550 (see Num. 1:46, 2:32). Elsewhere, the number of firstborn Levites are numbered as exactly 22,000 (Num. 3:46) which corresponds to the convergent ratio 22/7 for pi. In the Psalms, there are 1775 or 5 x 355 verses in Psalms 42-142. The number 142 is 2 x 71 similar to how 355 is 5 x 71. To indicate the relationship between Psalms 42 and 142, Psalms 42-141 contain 1764 verses which is 42 x 42. The number for the 603,550 men in Exodus and Numbers is represented in the Psalms in the following verse structure:
Psalm 112 is followed by Psalm 113 which thus represents the convergent ratio 355/113 for pi. As discussed previously in the chapter The Hidden Chronology of the Psalms these 600 verses also represents the 600 years of Noah and Psalm 95 corresponds with both the flood of Noah and the Exodus out of Egypt.
That the 1,775 shekels for the extra 3,550 men represent the circumference of a circle is shown by the fact that these 1,775 shekels of silver were used for the hooks of the pillars that were used to create the external boundary of the tabernacle. This relationship is confirmed by the amount of gold collected for constructing the tabernacle:
All the gold that was used for the work, in all the construction of the sanctuary, the gold from the offering, was twenty-nine talents and 730 shekels, by the shekel of the sanctuary (Ex. 38:24)
The number 29 is 22 + 7 which are the two numbers for the convergent ratio 22/7 for pi. The number 730 is 2 x 365, where 365 is the number of days in the solar year. The number 355 is a close approximation for the number of 354.37 days for the lunar year. Thus the gold represents the Sun and the silver represents the moon, and the cycle of a year would correspond to the circumference of a circle. Also for a circle with a circumference of 365 the diameter would be 116 which is 4 x 29, a multiple of the number of talents of gold.
THE CONVERGENT RATIO OF PI AND THE HEBREW ALPHABET
A more obvious connection can be seen between the 22 letters of the Hebrew alphabet and the convergent ratio 22/7 for pi. The denominator is seven, and thus the number 7 is considered holy throughout scripture, especially the Sabbath of the seven day week. As the 22 letters are consonants, the number 7 may be a representation of Hebrew vowels. Vowels are not recorded in scripture, and notations were added later by the Masorites to indicate how the Hebrew words should be vocalized. In Greek there are 7 specific vowels, and in later Greek and Gnostic traditions the seven vowels of the Greek alphabet were said to correspond with the seven heavens, to which was assigned one of the seven known planets. As the heavens were worshipped as idols in the ancient world, the Jews completely divorced themselves from such symbolic associations, but in place of the seven planets the law instituted the seven branched menorah candlestick. This association is mentioned by the Jewish historian Josephus in describing the table of shew-bread in the temple:
"Now, the seven lamps signified the seven planets... the twelve loaves that were upon the table signified the circle of the Zodiac and the year..." (Antiquities, v. 5; 5)
Here the number 7 is associated with a circle, in this case the circle of the zodiac or the year. This provides indirect confirmation that the 22 letters of the Hebrew alphabet can be arranged in a circle to represent the convergent ratio 22/7 for pi. The seven candlesticks of the menorah are also represented as seven pillars:
Wisdom has built her house, she has hewn out her seven pillars (Prov. 9:1)
The same word for pillars are mentioned in Psalm 75: The earth and all who dwell in her are dissolved, I balance the pillars of it (Ps. 75:3). The "earth" was probably represented as a circle of the known world. Verse 3 demarcates the first and second stanzas of Psalm 75, which each contain exactly 22 words. Psalm 75 is the exact middle of the 150 Psalms, and 2 x 150 is 300 which a multiple of 3, the simplest convergent ratio for pi.
In the work Sefer Yetzirah, an early work of Jewish esotericism perhaps dating to the first century A.D., the 22 letters of the Hebrew alphabet are arranged into a circle as follows:
In the system of the Sefer Yetzirah, the 22 letters or divided into three "mother" letters (aleph, mem and shin), seven "doubles" (bet, gimel, daleth, kaph, pe, resh, tau) and twelve "elementals" (he, vav, zayin, cheth, teth, yod, lamed, nun, samekh, ayin, tsade, qoph). This division is probably a later invention, and is based on the three letters in the name of Jehovah, the seven planets, and the 12 signs of the zodiac. By the time it was written knowledge of the convergent ratio 22/7 for pi was lost, but in place of it a different system was invented attempting to associate the circle with the zodiac. Instead of recognizing the number 7 as representing the diameter, a separate category of 7 letters were created. Scriptural support for arranging the letters in a circle can be found in the book of Revelation, where the Lord declares Himself to be the Alpha and Omega, which are the first and last letters of the Greek alphabet. In Hebrew the letters are aleph and tau, which are adjacent to each other on the circle:
I am Alpha and Omega, the beginning and the ending, says the Lord, which is, and which was, and which is to come, the Almighty (Rev. 1:8)
I am Alpha and Omega, the first and the last (Rev. 1:11)
In the Old Testament, it is Jehovah who declares Himself to be the first and the last:
I, Jehovah, the first, and with the last; I am he (Isa. 41:4)
Thus says Jehovah, the King of Israel and his Redeemer, Jehovah of hosts: I am the first and I am the last; besides me there is no god (Isa. 44:6)
Listen to me, O Jacob, and Israel, whom I called! I am he; I am the first, and I am the last (Isa. 48:12).
THE HEBREW LETTER CIRCLE AND THE NAMES OF GOD
In the diagram of the Hebrew letter circle as described in the Sefir Yetzirah the 22 letters are on the edge of a circle, with 231 lines drawn between every possible letter combination. It is described as follows:
"These twenty-two letters, the foundations, He arranged as on a sphere, with two hundred and thirty-one modes of entrance. If the sphere be rotated forward, good is implied, if in a retrograde manner evil is intended. For He indeed showed the mode of combination of the letters, each with each, Aleph with all, and all with Aleph. Thus in combining all together in pairs are produced these two hundred and thirty-one gates of knowledge. And from Nothingness did He make something, and all forms of speech and every created thing, and from the empty void He made the solid earth, and from the non-existent He brought forth Life."
Of the 231 lines between the letters, the one of most interest is the one between the first letter aleph (א) and the letter lamed (ל) which bisects the circle. These two letters form the word "God" (Heb. El). The name "God" is thus represented by the diameter of a circle. The letter aleph (א) is derived from an Egyptian hieroglyph that depicts the head of an ox and the name of the letter comes from a western Semitic word that means "ox." Thus for the molten sea of the temple of Solomon was placed on top of 12 oxen:
It stood on twelve oxen, three facing north, three facing west, three facing south, and three facing east. The sea was set on them, and all their rear parts were inward (1 Kings 7:25).
Each opposing pair of oxen bisect the circle. In the convergent ratio 333/106 for pi, the number 106 would represent the length of the diameter of a circle which would divide the sea of the temple of Solomon into two halves. Psalm 106 ends the fourth book of the Psalms, and it mentions the parting of the Red Sea:
He rebuked the Red sea and it was dried up, and He made them to walk through the depths as through the wilderness (Ps. 106:9)
This is followed by the false worship of a calf:
They made a calf in Horeb, and bowed down to the molten image. And they changed their glory, into the pattern of an ox that eats herbs (Ps. 106:19-20).
The sacred bull was worshipped throughout the ancient Middle East. In ancient Egypt the bull god was known as Apis, and before that the cow goddess Hathor was worshipped. Both of these gods were portrayed with the horns of a bull that supported a sun disk. As Psalm 106 mentions both the parting of the Red Sea along with the calf of Horeb, this indicates an ancient knowledge that the name "God" (Heb. El) was a symbolic depiction of a diameter of a circle. There are 999 or 3 x 333 verses in Psalms 54-106, which indicates that the number for Psalm 106 numerically represents the convergent ratio of 333/106 for pi.
The Hebrew name for God - El - is composed of aleph and lamed which bisects the Hebrew letter circle in half. In terms of ordinal values, the letter aleph has a value of 1 and lamed has a value of 12, for a total of 13. Also the majestic plural form of God - Elohim - has a total ordinal value of 41 which is the 13th prime number. This was perhaps based on the convergent ratio 355/113 for pi, where 13 represents the number 113. The Hebrew word agol means "round" or "circular" (Heb. agol) begins with aleph and ends with lamed and is mentioned in the description of the measurements of the temple of Solomon (see 1 Kings 7:23,31,35, 10:19). A close word using similar letters is the word for "calf" (Heb. egel or eglah) which is significant as the word for God (Heb. El) represents the diameter or a circle. The letter aleph which is the first letter of these words originated from the head of a bull, thus there were 12 bulls surrounding the molten sea of the temple of Solomon.
Just as the name "God" or Hebrew El is can be represented as the diameter of a Hebrew letter circle, so the name Jehovah can be represented by the circumference. The name Jehovah originates from a root word meaning "to be" (Heb. hawah) which is composed of the letters he (ה) and vav (ו). The letter yod was originally a prefix for the third person, thus the name Jehovah means "He is." On the Hebrew letter circle the letters he and vav are adjacent to each other, and if one proceeds from the letter he to the letter vav and then all the way round back to the letter he, one would have proceeded round the entire circumference of the circle. This is implied by the meaning of the letter vav, which in Hebrew conjoins words together with the meaning of "and." Two letters of vav spells the word "hook" in Hebrew and the hooks of the pillars of the tabernacle were made from the 1,775 or 5 x 355 shekels of silver. Just as the name "God" can be represented by the diameter of a circle, so the name "Jehovah" can be represented by the circumference of a circle. Moreover the letter he is the fifth letter and the letter vav is the sixth letter and 5 + 6 is 11 which is half of 22. If the letter vav is doubled, as in the Hebrew word for "hook," then the Hebrew word "to be" would add up to 22. In scripture the name of Jehovah and the title of God are are often combined as "Jehovah God," and this title could be thus represented by a circle with its diameter.
In the convergent ratio 355/113 for pi the number 113 is 100 + 13. As stated earlier, in the Hebrew letter circle the letters aleph (א) and the letter lamed (ל) form the word "God" (Heb. El) which bisects the circle. These two letters have ordinal values of 1 and 12 which total 13. Similarly, the circumference of the circle which is 355 is 300 + 55. The circumference is represented by the name Jehovah. The name Jehovah has two letters he which have an ordinal value of 5, which were perhaps used to represent the number 55. The name Jehovah begins with the letter yod which has an ordinal value of 10 and the following letter is he which has a value of 5. The letter vav can be treated as the word "and" which is then followed by the final letter he. The name Jehovah can thus be rendered as "(10 x 5) and 5" which is 55. Thus the name "Jehovah God" can be symbolically represented by the ratio 355/113 for pi. In a similar manner, the verse structure of Psalms 1-71 were modeled after the numeric values of the letters in the name of Jehovah; Psalm 71 was chosen to end the series since 5 x 71 is 355 (see the chapter, The Verse Structure of the Psalms).
THE NAME OF JEHOVAH IN THE PSALMS
The relationship between the name of Jehovah and pi is indicated not only in the ordinal values of the letters in the name of Jehovah but also in the number of instances the name of Jehovah appears in the Psalms. In the first 22 Psalms there is a direct relationship between the number of the Psalm and the number of instances of the name of Jehovah, as follows:
|Names of Jehovah
|11 = 10 + 1
|6 = 2 x 3
|16 = 2 x 8
In most cases the number of times that the name of Jehovah is mentioned matches the last digit of the Psalm number; in a few instances the number of times Jehovah is mentioned matches the Psalm number exactly. This pattern ends at Psalm 22 which matches the 22 letters of the Hebrew alphabet. However there is another pattern which indicates the names of Jehovah are arranged after the convergent ratio 22/7 for pi as follows:
|Names of Jehovah
Before Psalm 15, there are two series where the names of Jehovah follow the pattern of 11, 22 and 7. Psalm 7 and 14 are used in the group of 7 since their numbers are a multiple of 7. Following Psalm 15, there are two groups of 22 with a group of 7 in the middle. Psalm 15 demarcates two groups that have equal verse counts:
The total number of verses is 318 which is 3 x 106 where 106 is part of the convergent ratio 333/106 for pi. Similar to the name of Jehovah in Psalms 1-22, the title Most High (Heb. Elyown) appears in the Psalms 22 times. Its first occurrence is in Ps. 7:17 and the last occurence is in Ps. 107:11, where the Psalm numbers correspond with the number 7 in the convergent ratio 22/7 for pi.
After the convergent ratio 22/7 the next best convergent ratio for pi is 333/106. The title Lord (Heb. Adonai) is mentioned a total of 53 times in the Psalms and 2 x 53 is 106. The name Jehovah is mentioned for a total of 678 times in the Psalms and 678 is 6 x 113, indicating that it is arranged after the convergent ratio 355/113 for pi. Following the 112 names of Jehovah in Psalms 1-22 the names of Jehovah can be arranged in the following pattern:
|Names of Jehovah
|98 = 2 x 49
This is the only instance in the Psalms where 113 names of Jehovah are directly followed by 355 names of Jehovah. The 355 names of Jehovah begin with Psalm 35 which is a digital permutation of 355. It ends with Psalm 127 which is the middle Psalm in the 15 Songs of Ascent in Psalms 120-134.
The number 7 is considered holy in scripture as it is part of the convergent ratio 22/7 for pi. After the 355 verses there are 98 or 2 x 49 names of Jehovah which corresponds to Psalm 149 since 149 is 100 + 49. The name "Jehovih", where the vowels for Elohim are used instead of the vowels for Adonai, is mentioned only 8 times and if combined with the name "Jehovah" there would be a total of 686 which is 2 x 7 x 7 x 7.
THE SPIRITUAL MEANING OF THE CIRCLE AND THE DIVINE NAME
The circumference of a circle is a representation of the name of Jehovah, and the diameter is a representation of God which in Hebrew is El or Elohim. The spiritual significance of this is explained in the writings of Swedenborg, where the name Jehovah represents the Divine good or mercy, but the title God represents the Divine truth:
"In the Word the Lord is sometimes named Jehovah, sometimes Jehovah God, also the Lord Jehovah, and sometimes God, and this always for a hidden reason in the internal sense. Where love or good and the celestial church are treated of, He is called Jehovah, but when faith or truth and the spiritual church are treated of, He is called God, and this constantly" (Heavenly Arcana, n. 2001).
In terms of geometric symbolism, in his heavenly visions Swedenborg saw that truth is represented by angles, but good is represented by round forms:
"When truths and goods are presented in visible form, as they are in the other life manifestly to the eyes of spirits and angels, then truth is presented in definite quantity, consequently as much or as little, according to the quality of the truth; truth is also presented as angular in various forms, and it is also presented as white. But good is there presented in continuous quantity, thus not as much or as little; good is also presented as round, which is continuous in form, and in color as blue, yellow, and red..." (Heavenly Arcana, n. 8458).
In modern math, angles are expressed in terms of radians, which is the length of the radius on the circumference of a circle; twice the radius is the diameter which is represented by God (Heb. El or Elohim) as God is the title used when Divine truth is the subject in scripture. In another passage Swedenborg explains that internal spiritual good is represented by round forms, external good by a square, and truth by the forms of triangles as triangles are a geometric form of an angle:
"That foursquare means what is just originates in the representatives in the other life. In these representatives goods are presented as round, and the goods of the external man, which are called just, are presented as foursquare; truths however and things that are right are presented there as linear and triangular" (Heavenly Arcana, n. 9717.2).
The evidence suggests that the ratios of pi were considered to be a Divine signature, a mathematical representation of the name of Jehovah. This is confirmed by the iconography of the rod and ring in the ancient Middle East, which were symbols always associated with the deity, and were perhaps representations of a circle with its diameter. These symbols are also associated with measurement, especially that of the cubit. As this measurement was considered sacred, so the ratios of pi were used in the numerical design of the verses, words and sometimes even letters of the Psalms. The geometric representation of the name Jehovah God as the circumference and diameter of a circle is confirmed by the insights offered in the writings of Swedenborg, and the truth of some of revelations given to Swedenborg is confirmed by the analysis of the numeric values of Hebrew letters of the name of God relative to the circle. As shown here as well as the next chapter, much of the numerical structure of the Psalms is based on the convergent ratios of pi for this spiritual reason.