By George Oliver
Posted by hexnet -

The following is a transcript of Chapter VI of The Pythagorean Triangle: Or, the Science of Numbers by the Rev. Dr. George Oliver, a noted 19th century English Freemason. I have compiled this from several sources, including the Internet Archive, Google Books, and my own copy of the work. I slightly reworked the layout where necessary, including sort of rejiggering the chapter opening here. I've made several minor typographic modifications as well, such as removing the spaces before semicolons (a form of punctuation the good Doctor seems to have had a particular fondness for).

The rest of the book consists of similar musings on the other natural numbers, up to ten. I do not necessarily agree with Dr. Oliver's views, either on the hexad in particular, or on Christian numerology in general. I certainly don't agree with his cryptodecimalist ontology. But it is nonetheless an interesting treatment of the subject, and I offer it here as a noteworthy historical artifact of antiquarian hexagonal thought.

A PDF version of the full book can be downloaded from our Hexagonal Library.



"The second natural division of the circle is made by the radius, the measure of which, being transferred upon the half circumference with the compasses, always cuts it into three, or if transferred upon the whole circle, divides it absolutely into six equal portions, which is an introduction to a multitude of other no less certain divisions, and innumerable proportions between great and small figures." – La Pluche.

"The Hexagon is composed of six equilateral triangles, is equal in all its relations, and retains the quality of being infinitely divisible into similar triangles, according to the geometrical projection observed in the divisions of that trilateral figure, and may, therefore, be considered as the most perfect of all multilateral forms. From a general inquiry it will result, that the three most perfect of all geometrical diagrams are the equilateral triangle, the square, and the equal hexagon." – Hemming's Lectures.

What Dr Wordsworth says about the requisites to enable an author to describe Athens, I would say, with a little alteration, of a writer on the subject of Freemasonry. "To describe Athens, a man should be an Athenian, and speak the Athenian language. He should have long looked upon its soil with a feeling of almost religious reverence. He should regarded it as ennobled by the deeds of illustrious men and have recognised in them his progenitors. The records of its early history should not be to him a science; they should not been the objects of laborious research, but should have been familiar to him from his infancy,—have sprung up, as it were, spontaneously in his mind and have grown up with his growth. Nor should the period of its remote antiquity be to him a land of shadows—a Platonic cave in which unsubstantial forms move before his eyes as if he were entranced in a dream To him the language of its mythology should have been the voice truth."

The masonic writer, however, possesses some advantages over the Athenian topographer. Dr Wordsworth goes on to say: "This, we gladly confess, is not our case. We commence our description of this city with avowing the fact, that it is impossible, at this time, to convey, or entertain an idea of Athens such as it appeared of old to the eyes of one of its inhabitants. But there is another point of view from which we love to contemplate it,—one which supplies us with reflections of deeper interest, and raises in the heart sublimer emotions than could have been ever suggested in ancient days by the sight of Athens to an Athenian. We see Athens in ruins."

On the contrary, we rejoice because we live in times when Masonry is in a palmy and prosperous state—flourishing like a green bay-tree—its principles open to the inspection of every inquirer, and its proud and lofty spirit animating every institution in existence, in every region of the globe. There it stands—a tangible reality—and therefore cannot be misrepresented by unsound theories, or false hypotheses. It occupies a situation on which the ideal cannot be permitted to set her foot; because its ground is holy, and its footstool is truth. And if Athens "issued intellectual colonies into every quarter of the world," as the learned Doctor assures us, Freemasonry has not been backward in imitating so fructifying an example; and has accomplished the very same result which he assigns to the genius of the Athenians—it has become immortal.

With what feelings Freemasonry in ruins might be contemplated, it would be difficult to ascertain, because it stands on too firm a basis ever to be removed. It never will be in ruins, but will last until our system shall be extinguished. If the magnificent buildings of the Acropolis had been, like Freemasonry, animated by the spirit of a true faith, they might still have existed in all their glory, and not have distributed their shattered fragments to enrich the cabinets of modern nations. Genius and intelligence may be transferred; but no people, how brave, rich, and powerful soever they may be; no monument of art, however massive, ponderous, and constructed for durability—if not supported and animated by the power of religion, and the purity of an unsophisticated worship—can escape the universal fiat of annihilation which the lips of Wisdom have pronounced against "all the works of darkness." And as Freemasonry is confessedly a system of light, there is no fear that it will ever be extinguished.

Let us, then, as good and worthy Masons, ornament our Order with deeds of virtue, truth, and brotherly love, and remember the advice of one who was inspired by wisdom, although not enlightened by revelation—

Stat sua cuique dies; breve et irreparabile tempus,
Omnibus est vitae; sed fainam eztendere factis,
Hoc virtutis opus. – Virgil.

The hexad was considered by all nations a sacred number, because the world was created in six days; and six of the properties of nature only are said to belong to the active dominion, to good and evil; and the planetic orb is the figure of the six properties of the spiritual world. It was represented by the double triangle, because it has six points which, amongst the Pythagoreans, denoted health, and was defined, "the consistence of a form;" while sickness was considered the violation of it.

This figure was used by the heathen as a charm against the influence of evil demons. The Arabs believe that "communicable or contagious diseases are six: smallpox, measles, itch, putridity, melancholy, and pestilential maladies; and that diseases engendered are also six: leprosy, hectic, epilepsy, gout, elephantiasis, and phthisis." The double triangle constituted one form of the seal of Solomon, which was so celebrated in the fictions of Arabian romance; and is used by Christians to express the two natures of Christ. With this reference, it was introduced into the cathedrals and monastic edifices of the middle ages as a conspicuous symbol; and is still to be seen in painted windows, altar screens, and other decorative parts of these sacred buildings. These two intersecting triangles were also emblems of creation and redemption, fire and water, prayer and remission, repentance and forgiveness, life and death, resurrection and judgment.

The number six signified perfection of parts, because it is the only number under ten which is whole and equal in its divisions; and produces a hexagon by extending the measure of the radius of a circle six times round the circumference. The above proposition is beautifully illustrated "in the edificial palaces of bees, those monarchal spirits, who make their combs six-cornered, declining a circle, whereof many stand not close together, and completely fill the area of the place; but rather affecting a six-sided figure, whereby every cell affords a common side unto six more, and also a fit receptacle for the bee itself, which, gathering into a cylindrical figure, aptly enters its sexangular house, more nearly approaching a circular figure than either doth the square or triangle. And the combs themselves are so regularly contrived, that their mutual intersections make three lozenges at the bottom of every cell; which, severally regarded, make three rows of neat rhomboidal figures, connected at the angles, and so continue three several chains throughout the whole comb."

Nature herself seems to affect a partiality for the hexad, in the formation of crystals; all of which are hexangular or six-cornered; for which Pliny and other ancient naturalists endeavoured in vain to assign a reason. There are three different forms, however, which crystals appear to assume: 1. The perfect columnar crystal is composed of eighteen planes, in an hexangular column, terminated by an hexangular pyramid at each end. 2. Crystals without a column are composed of two hexangular pyramids, connected at the base. 3. Imperfect crystals have usually an hexangular column, irregularly affixed to some solid body, showing also an hexangular or pentangular pyramid. "Which regular figuration," as Dr Brown observes, "hath made some to opinion, that it hath not its determination from circumscription, or as conforming unto contiguities, but rather from a seminal root and formative principle of its own, even as we observe in several other concretions."

The sceptics used to amuse themselves by such arguments as these: If something be detracted from another, either an equal is detracted from an equal, a greater from a lesser, or a lesser from a greater. But none of these—therefore detraction is not possible. That detraction is not made by any of these ways is manifest. That which is detracted from another must be contained in it; but an equal is not contained in an equal, as six in six; for that which containeth ought to be greater than that which is contained. Neither is the greater contained in the lesser, as six in five; that were absurd. Neither is the lesser contained in the greater; for if five were contained in six, by the same reason, in five will be contained four; in four, three; in three, two; and in two, one. Thus six shall contain five, four, three, two, and one, which being put together make fifteen, which must be contained in six, if it be granted that the lesser is contained in the greater. In like manner, in the fifteen which is thus contained in six, will be contained thirty-five; and so by progression, infinite numbers; but it is absurd to say that infinite numbers are contained in the number six; therefore it is absurd to say that the lesser is contained in the greater.

From the harmonious movements of the planets, the hexad was considered an apt symbol of harmony; although the Pythagoreans ascribed it to a different cause. They explained it in reference to musical proportions; because 6 to 12 produced a diapason concord which contains semi-tones; and 6 to 8 a diatessaron or fourth; whence the hexad was sacred to Venus, as the patroness of harmony. Macrobius, Boethius, and others, give a curious account of the accident by which Pythagoras found out these proportions; which may class with his discovery of the 47th proposition of Euclid, for which he is said to have sacrificed a hecatomb. It is thus related by Nicomachus: At one part of his life he was particularly anxious to discover some infallible instrument of music, by the use of which the entire system might be enunciated. Accidentally passing by a blacksmith's shop, he took notice of the hammers striking on the anvil; and after listening attentively for some time, he observed that the sound formed three perfect concords. Going into the shop, he made various trials himself, and found that the difference in the sounds was produced by the weight of the hammers, and not according to the force of those who struck. On this hint he tied four strings across his private room of the same substance, to each of which he hung a different weight. Then striking the strings he discovered all the concords; and that to which the greatest weight was suspended, he found to be a diapason. By the same process, he found out all the intervals.

Aristotle has used some elaborate arguments to prove that there are no figures capable of filling a place about one point, except the triangle, the square, and the hexagon; viz., by six equilateral triangles, four squares, and three hexagons. But in solids, the pyramid and cube will do the same. In this process, he shows that for equilateral triangles to fill space, it is requisite that some angles of such triangles composed about one point should make four right angles. But six equilateral triangles make four right angles; for one makes 2/3 of one right angle, and therefore six make 12/3 of one right, i.e., four right angles. The four angles of a square, and the three angles of a hexagon make each four right angles. But no other figure can effect this, as will clearly appear, if, its angles being found, it is multiplied by any number; for the angles will always be less than, or exceed four right angles.

In ancient music, a sixth was called hexachords, of which Guido divided his scale into seven; three by B quardo, two by B natural, and two by B flat. It was on this account that he disposed his gamut in three columns. In these columns were placed the three kinds of hexachords, according to their order.

A famous symbol in the Egyptian mythology, which has exercised the ingenuity of many commentators, was the globe, serpent, and wings, of which there were six various ways of disposing the several parts: (1.) From the lower part of an annulus surmounted by two wings rising perpendicularly, two serpents issue in opposite directions. The whole is enclosed within a circle. (2.) The winged globe alone without the serpent. The wings expanded. This figure might be intended to represent the rays of the rising sun, which are, poetically, his wings. From this sacred figure, which represented the deity of the Gentiles, was probably borrowed the sublime metaphor of Malachi—The Sun of Righteousness shall arise with healing in His wings. (3.) A simple globe without wings, from which issue two serpents. (4.) A winged globe, through the lower part of which passes a serpent. (5.) A plain globe, over which passes the serpent. (6.) The same as the first without the circumscribing circle.

In the Theosophic or Rosicrucian systems of Freemasonry, the number six was referred to the rainbow, because it displays that number of prismatic colours; and from this principle they deduced the following argument, which was used in their lectures: "The rainbow is a token of God's Covenant, a representation to man of all the three principles out of which he was created—viz., the red and dark brown betoken the first principle, i.e., the dark, fire world, the kingdom of God's anger. The white and yellow show the second principle, the majestic colour, the holy world, God's love. The green and blue is the third principle's colour—the blue from chaos, the green from saltpetre, where, in the flagrat, the sulphur and mercury do sever, and produce various colours, which betoken the inward worlds hidden in the four elements. The rainbow is a further symbol of Christ appearing in the three principles, as the Judge of mankind. In the first, or fiery, all evil things shall be swallowed up. In the second, or that of light, He will defend the good, in love and meekness, from the flames of fire. In the third, or kingdom of nature, the humanity of the Judge is typified, and shows His impartiality in passing sentence on every man according to his works."

In one of the degrees of ineffable Masonry, the same number is denoted by the double equilateral triangle, which is there said to refer to the six peculiar branches of the noblest office in the Temple—viz., (1) To survey the constitutional rolls previous to their being deposited in the archives of Masonry, or hollow pillars of the temple; (2) to see that the stones fitted into each other with perfect exactness and geometrical truth; (3) to inspect the Holy Place, and (4) the Sanctum Sanctorum; (5) the ark of the covenant; and (6) all the other utensils thus emblematically pointed to by the double equilateral triangle.

Like the pentad, the number six was an ancient symbol of marriage, being formed by the multiplication of 3, the male, with 2, the female number; and from this cause it was named Conciliation, because it links or conciliates, by such involution, male and female into one body, husband and wife. And the Pythagoreans extended the influence of this number to the periods of gestation. They contended that "generally there are two kinds of births; one lesser, of seven months, which comes into the world 207 days after conception; the other greater, of ten months, which is brought forth in the 274th day. The first and lesser is chiefly contained in the number six; for the two first periods of 6 and 8 days make the first concord, diatessaron; the third period is of 9 days, in which time it is made flesh: these to the first 6 are in sesquialtera proportion, and make the second concord, diapente. Then follow 12 days more, in which the body is fully formed; these to the same 6 consist in duple proportion, and make the diatessaron concord These four numbers, 6, 8, 9, 12, added together, make 35 days. It is not without reason, therefore, that the number six is the foundation of generation, for the Greeks call it Teleion, or perfect, because its three parts, 1/6, 1/3, and 1/2 (i.e., 1, 2, 3), make it perfect. The above 35 being multiplied by 6 make 210 days, in which the maturity is fulfilled."

The cabalistic theologists say that this number affects the operation of the senses during sleep; because they consider sleep to be the sixtieth part of death. "The soul," they say, "being pure and holy, ascends in contemplation by degrees to the communication with angels, by which future events are often revealed to it; whence descending, after being perfectly purified, it brings down, unmixed, the knowledge that has been manifested to it—these are prophetic dreams; for, from imagination not entering into them, they deviate in nothing from the truth. If the soul be not perfectly pure, it meets with nothing but mixed and vain phantoms; and if impure, it does not ascend at all, but remains confused by demons and unclean spirits."

The number 666, or the hexad thrice repeated, has engaged the attention of cabalistical theologians for 1800 years, as the mysterious Apocalyptic number; and many and various have been its interpretations. "It has greatly perplexed the curious," says Calmet, " to know whether the name of the beast should be written in Hebrew, Syriac, Greek, or Latin; whether his name be that of his person, or of his dignity, or that which his followers should give him; or , that which he will deserve by his crimes. There are many conjectures in this matter; and almost all commentators have tried their skill, without being able to say, positively, that any one has succeeded in ascertaining the true mark of the beast, or the number of his name." Calmet has enumerated fourteen different interpretations, and concludes by saying, " Since the number 666 is found in names the most sacred, the wisest and safest way is to be silent." I subjoin five instances of the application of this number, to show how uncertain it may be—viz., Dioclesian, Julian the Apostate, Luther, Abinu Kadescha Papa, our holy father the Pope, and Elion Adonai Jehovah Kadosh, the Most High, the Lord, the Holy God. And a recent masonic writer (F. M. Mag., 1857, p .706) says, that by means of a rational interpretation of this number, the mysteries of the triangle and square are united in one masonic symbol, typical of the chief essential attributes of the Great Geometrician of the Universe.

"There was an ancient and almost immemorial tradition among the Jews that the world was to last only 6000 years. They divided the ages, during which it was to continue, in the following manner: Two thousand years were to elapse before the law took place; two thousand were to be passed under the law; and two thousand under the Messiah. Indeed, this sexmillennial duration of the world was, it is probable, too much the belief of the ancient fathers, who conceived that, as the creation was formed in six days, reckoning according to that assertion in the Psalms, that every day is with God as a thousand years, and was concluded by a grand Sabbath, or day of Almighty rest; so the world was ordained to last only during the revolution of six thousand years." Some visionaries, however, have been bold enough to name the precise periods when these six chiliads commence and terminate, and have made each of them correspond with some great historical epoch: 1. From the creation to the flood. 2. To the promise made to Abraham. 3. To the commencement of David's kingdom. 4. To the Babylonish captivity. 5. To the advent of Christ. 6. To the day of judgment.

But a reference to facts will prove this calculation erroneous. It is true the hypothesis that the duration of the world will continue six ages may be quite consistent with analogy and the revealed will of God, but the length of the intermediate periods vary considerably; for the first period, from the creation to the deluge, contains 1656 years; the second, from the flood to Abraham, if it be considered to terminate at the commencement of his peregrination, has only 427 years; the third, to the beginning of David's kingdom at the death of Saul, has 866 years; the fourth, to the Babylonish captivity, 448 years; the fifth, to the advent of Christ, 602 years; and the sixth is now incomplete.

On this subject, I remember reading a pamphlet many years ago, which interested me by its ingenuity; and as the events which it commemorates' are most of them comprised in the historical lectures of Masonry, I will give a brief outline of it, so far as my recollection will bear me out. The author commenced by instituting a comparison between the days of the week and the millenaries of the world, in illustration of the text, " One day is with the Lord as a thousand years, and a thousand years as one day." The first day of the week, Sunday, or the first thousand years, was, according to this author, opened by the creation of the world, and closed with the translation of Enoch; an event, one would think, which could scarcely fail to strike a wicked race with wonder, awe, and reverence, and produce the effect of turning them from their wickedness to worship the living God. It failed to do so; and the prevalence of fraud and violence brought on its destruction by an universal deluge.

The next chiliad, or Monday, the second day of the week, he terminates with the mission of Abraham; and opens the third millennium, or Tuesday, with a series of gracious revelations which heralded the establishment of the Jewish Church, a type of a more perfect dispensation which would ultimately be revealed from on high. During this period, the Mosaic dispensation was promulgated, and the law firmly established for the civil and religious government of the Hebrew nation, who were delivered from their cruel bondage in Egypt, and received possession of the Promised Land as an inheritance.

The next millennial period, corresponding with Wednesday, commenced with the construction of Solomon's Temple, and the attainment of that exalted summit of prosperity and power which, as had been promised to Abraham, his posterity should attain. This period commenced gloriously for the Jews; but its progress was marked with calamity. Their kingdom was taken from them, and they were deprived of the power of governing their own people. At the period when their sufferings were the most severe, and the sceptre had for ever departed from Judah, a still more refulgent era dawned upon the world. The dayspring from on high called the bright Morning Star—Orieus—the Sun of Righteousness—the Messiah so long promised to the Jews—enlightened the benighted atmosphere at the commencement of the fifth millennium, or Thursday. Now the fulness of time was come; the prophecies of the Messiah or Shiloh were fulfilled; the everlasting gospel was preached; and the work of redemption completed by Him of whom Moses and the prophets did write—Jesus of Nazareth, the King of the Jews.

The sixth chiliad, corresponding with Friday, opened in darkness. Literature and religion were both at a low ebb. Emperors and potentates were ignorant of letters, and the nobility of every Christian country were few of them able to read or write their own names. Charlemagne and Barbarossa, famous for conquest, were neither of them capable of writing their own despatches, or reading them when written; and some of the popes were equally illiterate. Under such circumstances, religion would necessarily be exchanged for superstition. The controlling spirits were barbarous and ferocious; for they had no mitigating principles to fall back upon. The yoke of superstition was always burdensome; and by its corrective policy—such as it was—the iron chiefs of every Christian nation were overawed, and subjected to the influence of an hierarchy more ambitious and insatiable than themselves.

Such was the opening of the sixth millennial period, in the year 1000 of the Christian era; about which time vast improvements in ecclesiastical architecture were on the eve of being accomplished; and the Freemasons spread over the face of every country where the religion of Jesus was professed, the proudest specimens of human taste and genius which have distinguished any age or nation since the world was made. From the commencement of the period in which we live, science and learning have rapidly increased, and the day of perfect civilisation has arrived. We are drawing near to the close of this period, and the opening of a glorious millennium—prefigured by the Jewish Sabbath—the day on which God rested from His labours at the creation of the world. This period will also continue a thousand years, when Christ will reign in glory over the whole society of the redeemed, and Satan be cast, bound, into the bottomless pit.

Many of the primitive Christians, and particularly Barnabas, the companion of St Paul, maintain this opinion. The latter writer, in his Catholic Epistle, says: " God made in six days the works of His hands, and He finished them on the seventh day; and He rested on the seventh day, and sanctified it. Consider then, my children, what that signifies—He finished them in six days. The meaning of it is this—that in 6000 years the Lord God will bring all things to an end; for with Him one day is a thousand years, as Himself testifieth. Therefore in six days shall all things be accomplished. And what is this that He saith—and He rested on the seventh day? He meaneth this—that when His Son shall come, and abolish the season of the wicked one, and judge the ungodly, and change the sun and the moon and the stars, then He shall gloriously rest on that seventh day."