Digitized by the Internet Archive in 2018 with funding from Duke University Libraries https://archive.org/details/pythagoreantrian01oliv THE PYTHAGOREAN TRIANGLE; OR, THE SCIENCE OF NUMBERS. THE PYTHAGOREAN TRIANGLE ii OR, THE SCIENCE OF NUMBERS. BY THE REV. G. OLIVER, D.I). Past M.P.S. Grand Commander S. G. I.G. for England and Wales: Past D.G.M. of the Grand Lodge of Massachusetts : Past D. P. G. M.for Lincolnshire : Honorary Member of numerous Lodges and Literary Societies in every qjiarter of the Globe : AUTHOR OF “THE REVELATIONS OF A SQUARE,” ‘*TH5 BOOK OF THE LODGE,” “ HISTORICAL LANDMARKS,” ETC. ETC. ’A pxv tlov ttolvtwv 77 fjLovas. —HERMIAS. WITH DIAGRAMS. LONDON: JOHN HOGG & CO., PATERNOSTER ROW. MDCCCLXXV. .UVERP00L4CLASC 03.33 04 $ PqQfl 1845 PUBLISHERS’ NOTE. The following posthumous work by the late Rev. G. Olivee, D.D., so widely and deservedly known by his numerous Masonic writings, is printed verbatim et literatim from his hitherto unpub¬ lished MS. The opinion of several distinguished Free¬ masons, competent to advise in such a matter, was, that the work should not be revised, added to, nor in any way altered. This opinion has been the more confidently acted upon, as the MS. bears evidence of having been very carefully revised by the Doctor’s own hand, and there is his own statement to the effect, that both the plan and treatment of the book had been well matured. London, November 1875. PREFACE.ix INTRODUCTION. THE PYTHAGOREAN TRIANGLE EXPLAINED, WITH A DISSER¬ TATION ON THE PECULIARITIES OF MASONIC NUMBER 1 CHAPTER I. THE MONAD OR POINT DISCUSSED AS THE ORIGIN OF ALL CALCULATION. (The Point, Monad, Unity, or the Number One) . . 31 CHAPTER II. THE DUAD OR LINE EXEMPLIFIED. (The Line, Duad, Duality, or the Number Two) . . 53 CHAPTER III. ILLUSTRATION OF THE TRIAD OR SUPERFICE. (The Superfice, or Equilateral Triangle, Triad, Ternary, or the Number Three). CHAPTER IV. PROGRESSIVE GENERATION OF THE TETRAD OR SOLID, REPRESENTING FIRE. (The Solid, Tetrad, Quaternary, or the Number Four) 101 Contents. viii CHAPTER V. GEOMETRICAL APPLICATION OF THE PENTAD OR PYRAMID, REPRESENTING WATER. (The Pyramid, Pentad, Quincunx, or the Number Five) CHAPTER VI. INFINITE DIVISIBILITY OF THE HEXAD OR DOUBLE TRIANGLE, REPRESENTING EARTH. (The Double Triangle, Hexagon, Hexad, or the Number Six). CHAPTER VII. REMARKABLE PROPERTIES OF THE HEPTAD. (The Heptagon, Heptad, Septenary, or the Number Seven) CHAPTER VIII. MYSTERIOUS REFERENCES OF THE OGDOAD OR CUBE, REPRESENTING air. (The Cube, Ogdoad, Octaedron, or the Number Eight) . CHAPTER IX. ANCIENT SUPERSTITIONS ATTACHED TO THE ENNEAD OR TRIPLE TRIANGLE. (The Ennead, Triple Triangle, Nonagon, or the Number Nine). CHAPTER X. THE PERFECT NATURE OF THE DECAD OR CIRCLE, AND THE PLICATION OF THE DODECAEDRON AS A REPRESENTATION OF THE SYSTEM OF THE UNIVERSE. (The Circle, Decad, Panteleia, or the N umber Ten) PAGE 123 145 167 187 199 AP- 219 PREFACE. REEMASONRY is a science, as every brother knows, whose Landmarks are theoretically unalterable, and whose peculiar rites and ordinances are pronounced to be the same yesterday, to-day, and for ever. But, alas! the failure of these conditions proves that Masonry is but a mere human institution after all. It would be easy to produce a host of altered Landmarks for the purpose of showing that no obsolete cere¬ mony or antiquated observance has been able to hold its own against the electric agency of modern progress. .S b X Preface. Accordingly, in tracing the science from the earliest period of its existence, we must prepare ourselves to meet with many changes which have periodically occurred in consequence of improve¬ ments and discoveries in the liberal arts, and the amelioration of manners and customs, as science progressed from ignorance to learning, and from a comparatively savage condition to its present palmy state of refinement and moral culture. To look for a different result would be to ex¬ pect impossibilities. Improvements in art, sci¬ ence, and civilisation, have a mutual dependence on each other. A change in the one will neces¬ sarily produce a modification of all. This is a truism that will not admit of contradiction or dispute. When we hear, therefore, of ancient, quaint phrases, whether in general literature or in Freemasonry, being swallowed up and lost in the undeviating march of scientific and moral improvement, and the substitution of others which are more in accordance with the usages O The Inevitable La w of Mutation. xi of a polished era, we are not to be surprised at such occurrences, nor complain, as many worthy Masons of the old school are apt to do, of modem innovations ; as if the institution were expected to stand still, and remain exempt from the inevi¬ table law of mutation to which all human sci¬ ences are exposed. Thus, for instance, the present race of Masons are ignorant of the reference to a Master Mason’s clothing, as the fraternity understood it a hun¬ dred and fifty years ago, and would scarcely refrain from an incredulous smile when they are told that it consisted of “ a yellow jacket and a pair of blue breeches.” But this was simply a figurative expression to signify the Third Great Light, which was appropriated to the Master, as the instrument by the use of which he drew his designs on the “ Trasel Board ” as a direction for the workmen ; the upper part being of polished brass, and its points blue steel. In like manner, several other peculiar phrases have become obso- xii Preface . lete, and are now imperfectly understood ; as in the instance of tlie “ Broached Thurnel ” for the apprentices to learn their work upon, connected with the Trasel Board and the Bough Ashlar— the triad having been changed to the Tracing Board, Bough Ashlar, and Perfect Ashlar. Again, what Mason of the present day under¬ stands the meaning of Irah, which our ancient brethren were so proud of \ And indeed it has puzzled many accomplished Masons of modem times. Being associated with the second degree, it is believed by some to have referred to a per¬ son. But this exposition involves two hostile interpretations, neither of which may be correct; one party holding the opinion that it signified Plirani Abiff; and another that it had a mysteri¬ ous reference to King James III., as he w T as designated by his followers; and if the latter interpretation be correct, it was probably one of the symbolical words of the Order introduced into Masonry by the Jesuits, to express Bediet Peculiar and Obsolete Phrases. xiii (redeat, redibit) Ad Habenas (Hoereditatem) ; according to the anagrammatic form of reading by initials. This was the opinion of Schneider. Some interpret Irah, rtNT, to mean Fear, as the fear of God; while others take it to be m\ he has taught; and lastly, it has been referred to the Temple, as who should say, He has laid the foundation. Utrum Iiorum mavis accipe. Who knows anything at the present day about the obsolete decrees of the Link and Wrestle \ They were formerly connected with the Ark and Mark, the latter having been recently revived; while the Ark, or, as the degree was denomi¬ nated, the Royal Ark Mariners, is in abeyance, and seldom practised amongst the English Masons, although it bears a reference to a legitimate Masonic event. A cowan, or listener, was a character extremely obnoxious to our predecessors, and is not in much more favour amongst ourselves; albeit their quaint method of punishing him, when detected, is now XIV Preface. altogether unknown— e.g., to place him under the eaves of a house in stormy weather until the water ran in at his shoulders and out at his knees. Hence the appellation of eavesdropper, and the origin of the cautionary exclamation, “ It rains !” The modern treatment of a cowan is simply— contempt. The three fixed lights, or windows, subsequently exchanged for our lesser luminaries,were explained one hundred and fifty years ago to signify “ the three Persons, Father, Son, Holy Ghost; ” and were used to find out the meridian, “ when the sun leaves the south, and breaks in at the west window of the Lodge.” While the “ mossy bed,” the ancient signs of disgust and recogni- tion, as well as the primitive name of a Master Mason, are equally obscure at the present day ; having been swept away, along with the original method of characterising chemical bodies by symbols, as being no longer necessary to the system. Even the Masonic cipher, of which our D is used Lcindma rlcs. xv brethren of the last century were justly proud, is now in abeyance, if not obsolete, for it is con¬ sidered by the English fraternity a useless appen¬ dage that may be well dispensed with. In the formula of opening the Lodge before the union of ancient and modern Masons in 1813, it was announced by the chair that “ all swearing, whispering, and unmannerly or profane conversa¬ tion,” were strictly prohibited during Lodge hours, under such penalty as “ the Bylaws shall inflict or a majority think proper.” And the reason publicly assigned for this prudent course was, “ that the business of the Lodge being thus happily begun might be conducted with decency, and closed in harmony and brotherly love.” This formula was discontinued at the above- mentioned period, and a new form substituted, which brought the Christian tendency of the Order more prominently before the Lodge. There are also some passages in the old lectures which the brethren once took for genuine Land- XVI Preface. marks, that have long been disused. In the Entered Apprentice’s Lecture, the following passage occurs : “ What is the day for ?-—To see in. What is the nigbt for 1—To hear in. How blows the wind 1 —Due east and west.” Again : “ITow long do you serve your master 1 ?—From Monday morning till Saturday night. How do you serve him ?—With chalk, charcoal, and earthen pan, &c.” When speaking in the Fellow Craft’s degree of the elevation of the middle chamber, the door was technically said to be “ so high that a cowan could not reach to stick a pin in.” And the illuminated letter, by which it was distinguished, was said to denote “the Great Architect of the Universe, or Him that was taken up to the topmost pinnacle of the Holy Temple at Jerusalem.” As a counterpoise to the abandonment of this group of trivial observances by modern practice, many valuable additions were introduced in the revised lectures during the eighteenth century as The Doctrine of Masonic Number. xvii matters of detail, which, our more ancient brethren would have been proud to acknowledge as mani¬ fest improvements in the system. These were at length collected and embellished by Bro. AY. Preston ; and his Ritual formed the solid basis on which the Lodge of Reconciliation, in the year 1814, constructed the Union Lectures which are now used in the English Lodges. During these gradual improvements, the doctrine of Masonic Number slowly but certainly progressed in every successive formula, until it reached its acme, in the above-named year, by the introduction of all those scientific numeral phenomena which are deduced from a philosophical consideration of the Pythagorean Triangle. In the earlier rituals, Number is but incidentally alluded to. The degrees, steps, lights, ornaments, furniture, and jewels, &c., were arranged on the principle of the Triad; the cardinal virtues, with their appropriate signs and references, represented the tetrad, which, together with the five points xviii Preface. of fellowship and the seven liberal sciences, ap¬ pear to be all the numerical references which they contain ; and they were not enlarged on or explained until Preston promulgated his Lectures under the sanction of the Grand Lodg;e. And here the numbers 3, 5, 7, and 11 are illustrated rather more in detail in his improved version of the "Winding; Staircase. In the following pages, the doctrines and re¬ ferences which necessarily result from a minute consideration of the Science of Numbers, as enun¬ ciated in the Pythagorean Triangle, will be sub¬ jected to a scientific analysation; for it is a remarkable fact, that although the institution of Freemasonry is based upon it, we have no autho¬ rised lecture to illustrate its fundamental prin¬ ciples, or display its mysterious properties. At every step, we find a triad reference, but the reasons why this occurs are not satisfactorily explained. The monad, the duad, the triad, and the tetrad, meet us at every turn ; and though The Science of Numbers. xix these numbers constitute the foundation of all arithmetical calculations, the candidate is not fully instructed how they operate, or in what manner they ought to be applied. A large portion of the Egyptian philosophy and religion seems to have been constructed almost wholly upon the science of numbers ; and we are assured by Kircher ((Edip. Egypt., tom. ii. p. 2) that everything in nature was explained on this principle alone. The Pythagoreans had so high an opinion of it, that they considered it to be the origin of all things, and thought a knowledge of numbers to be equivalent to a knowledge of God. The founder of the sect received his instructions in this science from the Egyptian priests; who taught him that, while the monad possesses the nature of the efficient cause, the duad is merely a passive matter. A point corresponds with the monad, both being indivisible ; and as the monad is the principle of numbers, so is the point of lines. A line corresponds with the duad, both XX Preface. being considered by transition. A line is length, without breadth, extending between two points. A superfice corresponds with the triad, because in addition to the duad, length, it possesses a third property, viz., breadth; which is effected by setting down three points, two opposite, the third at the juncture of the lines made by the other two. A solid or cube represents the tetrad ; for if we make three points in a triangular form, and set a fourth over them, we have a solid body in the form of a pyramid, which has three dimen¬ sions—length, and breadth, and thickness. In expressing their opinion of the Regular or Platonic bodies, the followers of Pythagoras argued that the world was made by God “in thought, and not in time; ” and that He com¬ menced His work in fire and the fifth element; for there are five figures of solid bodies which are termed mathematical. Earth was made of a cube, Fire of a pyramid, Air of an octaedron, Water of an icosaedron, the Sphere of the A Subject of Surpassing Interest. xxi Universe of a dodecaeclron. And the combina¬ tions of the monad, as the principle of all things, are thus deduced. From the monad came the indeterminate duad; from them came numbers; from numbers, points; from points, lines ; from lines, superfices ; from superfices, solids; from these, solid bodies whose elements are four, viz., fire, water, air, earth; of all of which, under various transmutations, the world con¬ sists. Such dissertations, so far as they are appli¬ cable to the Science of Number, have received some attention in the revised Fellow Craft’s Lecture of Dr Hemming, by an elucidation of the Pythagorean Triangle. The subject is one of sur¬ passing interest to the Free and Accepted Mason, particularly if he be a lover of general Science. At my first initiation, I soon discovered the numerical peculiarity by which the Order is distinguished, and wondered that the Lectures contained such a meagre explanation of this xxii Preface. extraordinary fact. Being somewhat addicted to mathematical studies, I took an intense interest in the pursuit, and, during a course of miscel¬ laneous reading, made various collections on the subject of numbers, simple and compound, which. I found invaluable when I became the Master of a Lodo;e. True, I was sailing in the dark, without either compass or pilot, for the philosophy of Masonry was very imperfectly understood in those days. My researches, however, still made a gradual though slow progress, for what will not perseverance effect ? Whatever I read con¬ tributed to my store, for there is no book so bad but some benefit may be derived from it by an industrious man, as the bee extracts honey from the poisonous flower; and even when I made no notes, I was steadily amassing materials for future use in the peculiar walk of Masonic literature which I was destined to pursue. It will readily be believed, that I had not the The Origin of this Work. xxiii slightest intention of writing for the press at that early period of my career ; for Masonic publica¬ tions at the commencement of the present century were by no means in favour with the English Craft ; and therefore, I threw my acquisitions together as they arose, simply for my own pri¬ vate reference, and without suspecting that they would ever appear in a printed form. But man proposes, and God disposes. From these collections the following Treatise was drawn up. It has been several years in hand, and was not originally intended for publication; but in compliance with an urgent request which has been recently made upon me, I have allowed the manuscript to be put to press, in the hope that it will afford amusement and instruction to the assiduous Mason who consults its pages with the sober intention of improving his knowledge, by acquiring a store of additional facts which may assist his investigations into the more abstruse arcana of Masonic Numbers. xxiv Preface. It has been well observed, that such specula¬ tions are by some considered as trifling and use¬ less ; but perhaps they judge too hastily; for few employments are more innocent, none more ingenious, nor, to those who have a taste for them, more amusing ; and mathematical amuse¬ ments sometimes lead to important and useful discoveries. GEO. OLIVER. / N TROD UCTION. THE PYTHAGOREAN TRIANGLE EXPLAINED , WITH A DISSERTATION ON THE PECULIARITIES OF MASONIC NUMBER. THE PYTHAGOREAN TRIANGLE. INTRODUCTION. HAVE often wondered how it could happen that our ancient brethren should have omitted to work out the details of Freemasonry in a more particular and perfect manner than we find accomplished in the publica¬ tions of the last century; because it was generally believed, even then, that such discussions would be extremely advantageous, by dissipating the mists and prejudices which biassed the minds of men, and indisposed them for the reception of truth. Numerous evidences of this fact are scattered over the writings of the few masonic authors which distinguished that period. “ The best way,” says Lawrie in his preface, “of re¬ futing the calumnies which have been brought against the fraternity of Freemasons, is to lay before the public a correct and rational account 4 The Pythagorean Triangle. of the nature, origin, and progress of the insti¬ tution, that they may be enabled to determine whether or not its principles are in any shape connected with the principles of revolutionary anarchy, and whether or not the conduct of its members has ever been similar to the conduct of traitors.” And from the publication of such sentiments, it must be evident to every brother’s experience, that the feeling against Freemasonry, which displayed itself so openly, only a few years ago, has assumed a much milder form, if it be not entirely removed. It will not however be difficult to account for the dearth of masonic writers in a preced¬ ing age. Before the eighteenth century, sym¬ bolical masonry, being limited to the simple ceremonial, needed few illustrations ; because, as the science was chiefly operative, the most valu¬ able secrets would be those which had a reference to building—to the scientific ornaments and de¬ corations of each particular style of architecture as it flourished in its own exclusive period ; and these mysteries were communicated gradually, as the candidate rose through the different stages of his order or profession. There appears to have been one general prin¬ ciple, which extended itself over every style from the early English to the florid, decorated, and perpendicular, and constituted one of the most ineffable secrets of the Masonic Lodges. It is now known to have been the hieroglyphical device styled Vesica Piscis ; “ which may be traced Prevailing Secrets in Early Times. 5 from the Church of St John Lateran, and the old St Peter’s at Rome, to the Abbey Church at Bath, which is one of the latest Gothic buildings of any consequence in England. It was formed by two equal circles cutting each other in the centres, and was held in high veneration, having been invari- ably adopted by master masons in all countries. In bas-reliefs which are seen in the most ancient churches, over doorways, it usually circumscribes the figure of our Saviour. It was indeed a prin¬ ciple which pervaded every building dedicated to the Christian religion, and has been exclusively attributed to the scientific acquirements of Euclid.” 1 The prevailing secrets of the Lodges in these early times, were the profound dogmata of Geo¬ metry and Arithmetic, by the use of which all their complicated designs were wrought out and perfected. These sciences are inseparable from the system ; and accordingly have been faith¬ fully transmitted to our own times. “ The secret meetings of master masons within any particular district, -were confined to consultations with each other, which mainly tended to the communica¬ tion of science, and of improvement in their art. An evident result was seen in the general uni¬ formity of their designs in architecture, with respect both to plan and ornament, yet not without deviations. We may conclude that the craft or mystery of architects and operative masons was involved in secrecy, by which a 1 Kerrich in Arcbseol., vol. xvi. p. 292. 6 The Pythagorean Triangle. knowledge of their practice was carefully ex¬ cluded from the acquirement of all who were not enrolled in their fraternity. Still, it was absolutely necessary that when they engaged in contracts with bishops or patrons of ecclesiasti¬ cal buildings, a specification should be made of the component parts, and of the terms by which either contracting party should be rendered con¬ versant with them. A certain nomenclature was then divulged by the master masons for such a purpose, and became in general acceptation in the middle ages.” 1 The abstruse calculations which accompanied the sciences of geometry and arithmetic, are no longer necessary to Freemasonry as an institution purely speculative ; and they were accordingly omitted in the revised system as it was recom¬ mended to the notice of the fraternity by the Grand Lodge in 1717, and we retain only the beautiful theory of these sciences, with their application to the practice of morality, founded on the power and goodness of T.G.A.O.T.U. It would be an injustice to our brethren of the last century to believe that they did not enter¬ tain a profound veneration for the principles of the masonic order. But the customs and habits of the people of England, living in that day, differed materially from our own. They were times when conviviality and a love of social harmony prevailed over the more sedate pursuits and investigations of science, in which such an 1 Dallaway, Arckit., p. 410. The Organisation of Freemasonry. 7 astonishing progress distinguishes the present times. In the seventeenth and eighteenth cen¬ turies, London was an atmosphere of clubs, and a society of this kind existed in every street for the peculiar use of its inhabitants, besides those which were exclusively frequented by persons possessing similar tastes or habits of amusement. And it will be no disparagement to masonry, if we believe that its private Lodges did not sustain a much higher rank than some of these celebrated meetings ; for the Kit-Cat, the Beefsteak, and other clubs, were frequented by the nobility and most celebrated literary characters of that polished era. It was the organisation of Freemasonry that gave it the distinctive character which elevated its pretensions above the common routine of club-life; and although it is admitted that the members of the latter entertained a strong attachment to their several institutions, yet none were so enthusiastic as those who had enlisted in the cause of masonry, as we may learn from the few testimonies which remain. A mason of high standing, more than a century ago, thus expresses his feelings respecting the order: “ Masonry is the daughter of heaven ; and happy are those who embrace her. By it youth is passed over without agitation, the middle age without anx- iety, and old age without remorse. Masonry teaches the way to content, a thing almost un¬ known to the greater part of mankind. In short, its ultimate resort is to enjoy in security 8 The Pythagorean Triangle. the things that are, to reject all meddlers in state affairs or religion, or of a trifling nature; to em¬ brace those of real moment and worthy tendency, with fervency and zeal unfeigned, as sure of being unchangeable as ending in happiness. They are rich without riches, intrinsically pos¬ sessing all desirable good ; and have the less to wish for by the enjoyment of what they have. Liberty, peace, and tranquillity, are the only ob¬ jects worthy of their diligence and trouble.” 1 But this, as well as almost all the testimonies of that period to its superior excellence, is con¬ fined exclusively to the practice and rewards of Christian morality. Modern revision has however extended the limits of scientific investigation in the order of Freemasonry beyond what was intended by those who decreed that “the privileges of masonry should no longer be restricted to operative masons, but extend to men of various professions, provided they were regularly approved and initi¬ ated into the order.” And Dr Hemming and his associates, in the year 1814, thought it expe¬ dient to introduce some peculiar disquisitions from the system of Pythagoras, on the combina¬ tions of the point, the line, the superfice, and the solid, to form rectangular, trilateral, quadrila¬ teral, multilateral figures, and the regular bodies; the latter of which, on account of their singu¬ larity, and the mysterious nature usually ascribed to them, were formerly known by the name of the 1 Pocket Companion, p. 296. Temperance and Early Hours. 9 five Platonic bodies; and they were so highly re¬ garded by the ancient Geometricians, that Euclid is said to have composed his celebrated work on the Elements, chiefly for the purpose of displaying some of their most remarkable properties. These disquisitions usually conclude with an explana¬ tion of the forty-seventh problem of Euclid, which is called the Eureka of Pythagoras. Our transatlantic brethren have improved upon this still further. Some of their Grand Lodges have given a public sanction to the introduction of literary and scientific subjects, not contained in the usual lectures, and the open discussion of them at the private meetings of the society. And a committee of the Grand Lodge of New York, in their report for the year 1842, decided that “ masonic periodicals, if judiciously conducted, are calculated to accomplish a vast amount of good, by diffusing more extensively those sound, moral, and benevolent principles, which so emi¬ nently characterise this venerable institution; your committee therefore recommend those publi¬ cations to the liberal patronage of the fraternity.” To promote this laudable purpose, the Grand Lodges have recommended temperance and early hours ; a general observance of which, I am per¬ suaded, would not only afford ample leisure for scientific investigations, but would also operate very favourably both for the welfare and credit of the society ; and it is much to be wished that such a system of discipline could be established by a similar authority in the English Lodges; for 10 The Pythagorean Triangle. a, laxity of practice in these particulars is calcu¬ lated to introduce loose and incorrect habits which cannot fail to prove injurious to their popularity. If a Lodge be opened beyond the prescribed time, its labours may be protracted, particularly if its members are too much attached to refreshment, to a late hour, which will be in¬ consistent with domestic comfort, and may pro¬ bably create dissatisfaction and hostility in their families. There is a delicate sensibility in the female mind which is easily excited, and an im¬ pression may be made in a moment which will be found difficult to eradicate. The members of a Lodge, therefore, ought to be ever on their guard lest an unfavourable prejudice against the craft be thus created ; because, in such a case, every little deviation, which, under extraordinary cir¬ cumstances, may be unavoidable, will be magni¬ fied into a serious fault. And when transgressions, even though they be imaginary, are multiplied in the opinions of those who ought to be most dear to the fraternity, and whose happiness it is their duty to promote by every attention in their power, an estrangement of heart may be occa¬ sioned, which will embitter domestic comfort, and produce misunderstandings and disagree¬ ments, for which the pleasure^and enjoyments of Freemasonry will be in vain expected to com¬ pensate. Nothing can supply the loss of domestic com¬ fort, which is the one great source of happiness that an all-wise Creator has provided for us on The Great Source of Happiness. 11 earth. If therefore a fear of injuring the inter¬ ests of Freemasonry should fail to induce the observance of decorous hours in the conduct of a Lodge, let this consideration be superadded—let an attention to the comforts, and a respect for the prejudices of their families prompt the frater¬ nity to avoid late sittings. It is a practice which answers no one good purpose, secures no valu¬ able end, conveys no true gratification in the enjoyment, and embitters the reflections of the ensuing day. And beyond all this, it places in jeopardy those fireside comforts, those domestic virtues, which the religion we profess, the masonry we practise, and the reason with which the Most High has endowed us, alike concur in stimulating us to cultivate and adorn. To carry out all these points, and to bear harmless the order during the process, much depends on the knowledge and judgment of the Master ; and it is of such importance to the pro¬ sperity of Freemasonry that this officer be judici¬ ously selected, that it behoves every candidate to consider well his capabilities for the office before his election. It is not enough that he is au fait at the openings and closings of the several de¬ grees, and well acquainted with all other routine ceremonies ; he ought also to be conversant with the history, the antiquities, and the philosophy of the order, and the tendency of its mysteries and pursuits to promote the practice of Christian morality; for on this knowledge the success of his administration will, in a great measure, de- 12 The Pythagorean Triangle. pend. In these days, bodies of men meet together for other purposes than to hear the repeated reci¬ tation of a series of commonplace maxims, which soon lose their interest and become as sounding brass and a tinkling cymbal. Even an acquaint¬ ance with the traditions of Freemasonry is not without its utility^. They lead to something of a higher character, and are intimately connected with its philosophy. The most minute legend, although abstractedly it may be considered trifling and unmeaning, is not without its use, and if traced to its elements, will be found to bear a relation to facts or doctrines connected with our best and dearest interests. It appears to me, that in the revision of the English Ritual at the Union, a great omission occurs which it would be well to supply ; and in the present taste for scientific lectures and inves¬ tigations, nothing would tend to elevate the cha- racter of Freemasonry more than to afford an opportunity for its indulgence by furnishing the means of carrying out its references in the intro¬ duction of a higher range of science. Free- masonry, to be completely successful, should take precedence in science as it does in morals and the exercise of charity: for there are few institutions which equal it in the walks of benevolence. Its charities are unrivalled. It cherishes the orphan —it supports the widow—it relieves the destitute —and provides for the worthy aged brother an asylum from the storms of penury and indigence, at that helpless period of life when his strength A Higher Range of Science. 13 fails him, and he is no longer able to wrestle suc¬ cessfully with adversity and want. It is true the seven liberal sciences are referred to in the second degree ; but, with the exception of Geometry, they occupy no important place in the lecture. And for this reason, I suppose, that in ancient times the order was denominated Geo¬ metry. On this science, with its application to architecture, our disquisitions are abundant and powerfully interesting ; and why should not a lec¬ ture on the elementary principles of other sciences be equally gratifying to the members of a Lodge ? Arithmetic, or the science of Number, is nearly allied to Geometry; we patronise Music in prac¬ tice but hear nothing of it in theory; and of As¬ tronomy we are merely told that it is an art by which we are taught to read the wonderful works of God in those sacred pages the celestial hemi¬ sphere, and that while we are employed in the study of this science, we perceive unparalleled instances of wisdom and goodness, and through the whole of the creation trace the glorious Au- thor by His works. That great philosopher Pythagoras, who,by the superiority of his mind, infused a new spirit into the science and learning of Greece, and founded the Italic sect, taught his disciples Geometry, that they might be able to deduce a reason for all their thoughts and actions, and to ascertain cor¬ rectly the truth or falsehood of any proposition by the unerring process of mathematical demon¬ stration. Thus being enabled to contemplate the 14 The Pythagorean Triangle. reality of tilings, and to detect imposture and deceit, they were pronounced to be on the road to perfect happiness. Such was the discipline and teaching of the Pythagorean Lodges. It is related, that when Justin Martyr applied to a learned Pythagorean to be admitted as a candidate for the mysterious dogmata of his philosophy, he was asked whether, as a preliminary step, he had al¬ ready studied the sciences of Arithmetic, Music, Astronomy, and Geometry, which were esteemed the four divisions of the mathematics; and he was told that it was impossible to understand the perfection of beatitude without them, because they alone are able to abstract the soul from sensibles, and to prepare it for intelligibles. He was further told that in the absence of these sciences no man is able to contemplate what is honest, or to deter¬ mine what is good. And because the candidate acknowledged his ignorance of them he was re- fused admission into the society. Above all other sciences or parts of the ma¬ thematics, however, the followers of Pythagoras esteemed the doctrine of Numbers, which they believed to have been revealed to man by the celestial deities. And they pronounced Arith¬ metic to be the most ancient of all the sciences, because, being naturally first generated, it takes away the rest with itself, but is not taken away with them. For instance, animal is first in nature before man ; for by taking away animal we take away man ; but by taking away man we do not take away animal. They considered numbers ex- The Doctrine of Numbers. 15 tending to the decad, to be the cause of the essence of all other things; and therefore esteemed the creation of the world as nothing more than the harmonious effect of a pure arrangement of num¬ ber. This idea has been adopted by Dry den— From harmony, from heavenly harmony, This universal frame began; From harmony to harmony, Through all the compass of the notes it ran, The diapason closing full in man. Pythagoras had another idea, as we are informed by Censorinus, respecting the creation of the world, and taught that it was fashioned according to the principles of musical proportion; that the seven planets which govern the nativity of mortals have a harmonious motion, and intervals corre¬ sponding to musical diastemes, and render various sounds according to their several distances, so perfectly consonant that they make the sweetest melody, but “ inaudible to us by reason of the greatness of the noise, which the narrow passage of our ears is incapable of receiving.” And further, he esteemed the monad to repre¬ sent the great and good Creator, under the name of Dis, or Zeus, or Zau ; and the duad he re¬ ferred to the evil and counteracting principle or daemon, “ surrounded,” as Plutarch expresses it . 1 “ with a mass of matter.” And Porphyry adds , 2 that the monad and duad of Pythagoras seem to have been the same with Plato’s irepa ? and a-nupov, his finite and infinite in his Philebus ; the former 1 De Placitis Placitorum, 1. i. c. 7. 2 Vit. Pyth,, p. 47. 1G The Pythagorean Triangle. of which two only is substantial, that first most simple Being, the cause of all unity and the mea¬ sure of all things. According to the above doctrine, the monad was esteemed the father of Number, and the duad its mother : whence the universal prejudice in favour of odd numbers, the father being had in greater honour than the mother. Odd numbers being masculine were considered perfect, and applicable to the celestial gods, while even numbers, being female, were considered imperfect, and given to the terrestrial and infernal deities. Virgil has recorded several instances of this predilection in favour of odd numbers. In his eighth Eclogue, he says, Terna tibi hrec primum triplici diversa colore Licia circumdo; terque h£ec altaria circum Effigiem duco; Numero dcos impure gaudet. Thus translated by Dryden— Around his waxen image first I wind Three woollen fillets of three colours join’d ; Thrice bind about his thrice-devoted head, Which round the sacred altar thrice is led. Unequal numbers please the gods. The Eastern nations of the present day appear to reverse this principle. When two young per¬ sons are betrothed, the number of letters in each of their names is subtracted the one from the other,- and if the remainder be an even number, it is considered a favourable omen, but if it be odd, the inference is that the marriage will be unfor¬ tunate. Aritliomancy. 17 Some curious superstitions of this character were still in existence during the last century amongst ourselves, and may not at the present time be altogether obsolete. A Scottish minister, who wrote a treatise on witchcraft in 1705, says, “Are there not some who cure diseases by the charm of numbers, after the example of Balaam, who used magiam geometricam, Build me here seven altars, and prepare me seven oxen and seven rams. There are some witches who enjoin the sick to dip their shirts seven times in water that runs towards the south.” Sir Henry Ellis has collected many instances of the use of odd numbers, in his notes on Brand’s “ Popular Anti¬ quities,” to which the curious reader is referred. 1 The superstition of divination by number, called Aritliomancy, was so firmly planted in the mind of man by the observances of ancient times, that it appears impossible entirely to eradicate it. An old writer quaintly remarks, “I will not be superstitiously opinionated of the misteries of numbers, though it bee of longe standing amongst many learned men; neither will I positively affirm that the number of six is fatall to weomen, and the numbers of seaven and nine to men; or that those numbers have (as many have written) magnum in tota rerum natura potestatem, great power in kingdoms and commonwealths, in families, ages, of bodies, sickness, health, wealth, losse, &c.; or 1 See also on this subject Censor, de die nat., e. xii. Philo de leg., i. Bodin de Repub., 1. iv. c. 2. Varro in Cell., 1. iii. Jerom in Amos, v. Practice of Piety, fol. 410, &c. &c. B IS The Pythagorean Triangle. with Seneca and others, Septimus quisque annus, &c. Each seaventh yeare is remarkable with men, as the sixth is with weomen. Or, as divines teach, that in the number of seaven there is a misticall perfection which our understandinge cannot at- taine unto ; and that nature herself is observant of this number.” Every tyro knows that odd numbers are mason¬ ic ; and if he be ignorant of the reasons why 3, 5, 7, and 11, have been adopted as landmarks, let him apply to the Master of his Lodge for infor¬ mation, and he will then be satisfied of the wisdom of the appropriation, because number forms one of the pillars which contribute to the support of scientific masonry, and constitutes an elementary principle of Geometry. Thus, in the celebrated Pythagorean triangle, consisting of ten points, the upper single dot or jod is monad or unity, and repre¬ sents a point, for Pythagoras con¬ sidered a point to correspond in proportion to unity ; a line to 2 ; a superjice to 3 ; a solid to 4; and he defined a point as a monad having position, and the beginning of all things ; a line was thought to correspond with duality, because it was produced by the first motion from indivisible nature, and formed the junction of two points. A superfice was com¬ pared to the number three, because it is the first of all causes that are found in figures ; for a circle, which is the principal of all round figures, comprises a triad, in centre—space—circumfer- Application of Numbers to Physical Things. 19 ence. But a triangle, which is the first of all rectilineal figures, is included in a ternary, and receives its form according to that number ; and was considered by the Pythagoreans to be the author of all sublunary things. The four points at the base of the Pythagorean triangle correspond with a solid or cube, which combines the princi¬ ples of length, breadth, and thickness, for no solid can have less than four extreme boundary points. Thus it appears that in applying number to physical things, the system of Pythagoras termi¬ nated in a tetrad, while that of Aristotle, by omitting the point, limited the doctrine of mag¬ nitude to a triad, viz., line—surface—body. In divine things, however, the former philosopher pro¬ fusely used the numberthree, because it represented the three principal attributes of the Deity. The first whereof, as we are informed by Cudworth, is infinite with fecundity; the second infinite knowledge and wisdom ; and the last active and perceptive power. From wdiicli divine at¬ tributes the Pythagoreans and Platonists seem to have framed their trinity of archical hypostases, such as have the nature of principles in the uni¬ verse, and which, though they be apprehended as several distinct substances gradually subordinate to one another, yet they many times extend the to Qhov so far as to comprehend them all wdthin it. While employed in investigating the curious and unique properties which distinguish many of 20 The Pythagorean Triangle. the dibits, we no longer wonder that the inhabi- tants of the ancient world, in their ignorance of the mysterious secrets of science, and the abstruse doctrine of causes and effects, should have ascribed to the immediate interposition of the Deity those miraculous results which may be produced by an artful combination of particular numbers. Even philosophy was staggered; and the most refined theorists entertained singular fancies, which they were unable to solve without having recourse to supernatural agency. Hence the pseudo science of Arithonnancy, or divination by numbers, became very prevalent in the ancient world; and was used by Pythagoras as an actual emanation of the Deity. By this means, according to Tzetzes, he not only was able to foretel future events, but reduced the doctrine to a science, governed by specific rules, which he transmitted to posterity in his “Book of Prognostics.” The ancients had a kind of onomantic arith¬ metic, the invention of which was in like manner ascribed to Pythagoras, whether truly or not is of no importance here, in which the letters of the alphabet, the planets, the days of the week, and the twelve zodiacal signs, were assimilated with certain numbers; and thus, by the use of pre¬ scribed tables, constructed astrologically accord¬ ing to the aspects, qualities, dignities, and debili¬ ties of the planets relatively towards the twelve signs, &c., the adept would authoritatively pro¬ nounce an opinion on questions affecting life and death, good and evil fortune, journeys, detection The Jewish Cabcilists. 21 of theft, or the success of an enterprise. It must be confessed, however, that these predictions were not always correct; for the rules laid down in different systems varied so essentially, that the wisest magician was frequently puzzled to select an appropriate interpretation. The numeral sys¬ tem has been introduced into the modern practice of astrology, and very important results appear to depend on the trine, quartile, and sextile aspect of the planets in the horoscope. Something of this sort was used by the Jewish cabalists ; and hence one of the rules of their cabala was called gemetrici, or numeration, which was chiefly confined to the interpretation of their sacred writings. The letters of the Hebrew lan- (ruao;e beins; numerals, and the whole Bible bein