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By Reid McInvale

Bro. Reid McInvale, member of Holland Lodge No. 1, Houston, TX and he is a full member of Texas Lodge of Research

The purpose of this paper is to initiate a comprehensive exploration of the philosophy underlying Freemasonry by investigating some nuances of doctrines which in the eyes of this author are implicit in Freemasonry, and by offering explanations for a few Masonic symbols and ceremonies. The specific question to be explored in this paper is: what is the relationship between the practice of circumambulation and Euclid's 47th problem?

Circumambulation is the practice of "...making a circuit about a thing or in an area of reverence..." ENDNOTE (1). In Masonry, circumambulation involves the making of a circuit around the Lodge, while keeping the right hand toward the altar. The official Monitor of the Grand Lodge of Texas provides that "During the circuits of the lodge room, corners should be squared in accordance with the ancient tradition of 'squaring the lodge'". Further, the Monitor provides that the Entered Apprentice during initiation shall proceed to the northeast corner of the Lodge three times during the circumambulation, the Fellowcraft during passing shall proceed there four times, and the Master being raised shall proceed there five times . These numbers which are used in the "squaring of the lodge" will be important to this discussion. ENDNOTE(2)

The 47th proposition of Euclid's first book of the "Elements", also known as "The Pythagorean Theorem", stands as one of Masonry's premier symbols, though it is little discussed and less understood today. That fact is made the more unfortunate, since the 47th proposition may well be the principal symbol and truth upon which Freemasonry is based.

Symbols are used in Masonry to teach. In Anderson's "Constitutions" of 1723, he states that "...[T]he Greater Pythagoras, prov'd the Author of the 47th Proposition of Euclid's first Book, which, if duly observ'd, is the Foundation of all Masonry, sacred, civil, and military...." ENDNOTE(3). Thus, at the beginning of Speculative Masonry as we know it, the 47th proposition was regarded as containing or representing the truth upon which Masonry is based, and the basis of civilization itself.

Today, the 47th problem is somewhat honored in all Lodges in all jurisdictions. The Texas "Monitor" includes what it refers to as the "47th problem" in the information provided relating to the Master Mason degree. The Monitor states that Euclid's 47th problem was chosen to teach us to be "general lovers of the arts and sciences" ENDNOTE (4). That information is worded almost identically to the entry on the 47th problem which is included in my grandfather's "Masonic Manual of Alabama", which was published in 1918 ENDNOTE (5). It is also the same as language included in more ancient manuals and monitors, and is consistent with the information provided in the "Constitutions" of 1723.

The general belief as to the import and meaning of the 47th appears on the surface to have been radically altered since 1723. The 47th appears to have suffered a decline in status, from that of the acknowledged essence of Masonry, to being considered as a simple reminder that art and science are important and are to be respected. That alteration in our view of the 47th reflects a decline in our understanding of the true nature of Masonry.

The key to understanding the significance of the 47th is to obey the words of Anderson, and to properly observe it. To do so, we must first go to the 47th itself and view it. Euclid's 47th problem was set out in Book One of his "Elements". "Elements" is composed of thirteen books, each containing many geometric propositions, and it constitutes the work which is Euclid's contribution to the history of ideas ENDNOTE(6).

Proposition 47 provides that "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Readers might recall having to learn this rule in geometry class at school phrased as "In a right triangle, where A and B are sides of the triangle, and C is the hypotenuse, A squared plus B squared equals C squared."

Given that there are 48 geometric propositions in Book One of "Elements" alone, and a total of at least 465 in the entire work, one must ask 'what it is about proposition 47 that makes it uniquely significant to Masonry?'

To answer that question, we must first look at the history of the particular proposition itself within the context of the history of geometry. While Euclid included problem 47 in his book, he did not discover it. As the "Monitor" states, it was Pythagoras and/or his followers who are generally credited first with having developed the proposition. Indeed, amongst mathematicians, problem 47 is referred to as "The Pythagorean Theorem". Scholarship since the time of Anderson has uncovered the fact that the ancient Babylonians had knowledge of the 47th, well before the Greeks did. Further, it may well have been a student of Pythagoras who discovered it. However, it is Pythagoras whom the world applauds for the discovery.

Pythagoras was an Aeonian Greek who moved to Crotona in southern Italy and founded a society with philosophical, religious and political aims. Little is known of the precise doctrine of Pythagoreanism due to a rule of secrecy. It is thought that he and his followers held that philosophy was the use of reason and observation to gain understanding of the universe. The utility of philosophy was that it provided a means to achieve salvation of the soul. Since it was thought that the principal aspect of Divinity was a complete understanding of all things, then by using philosophy to understand things, a man could step closer to Divinity and over the course of numerous lifetimes gain some measure of Divinity himself. In essence, then, Pythagoras and his followers believed that by the exercise of reason and observation the resulting complete understanding of this world would lead to a perfection of the spirit and the attainment of salvation ENDNOTE(7). The discoverer of the Pythagorean Theorem saw a connection between geometry and God, and what we know as the 47th problem played a key role in his thought.

The Monitor of the Grand Lodge of Texas, among other Masonic authorities, holds that Pythagoras was a Master Mason. If one defines a Master Mason as anyone who is familiar with geometry and who both utilizes and admires human reason, then Pythagoras and many other ancients could be called such. Otherwise, there is no evidence to support the assertion so we may safely disregard such claim.. We can also disregard the statement in Anderson's Constitutions of 1723 that Pythagoras, or Peter Gower as the English sometimes refer to him, cried "Eureka" upon discovering the theorem. The famous cry of "Eureka" was in reality shouted by Archimedes upon discovering the method for determining the purity of gold ENDNOTE (8). As others had done in previous times, groups and institutions in the Middle Ages composed legends which claimed for themselves ancient lineages. Medieval scholars in Paris, for example, formulated their legend of translatio studii, wherein it was held that learning originated with the ancient Hebrews, then passed to Egypt, then to Athens, then to Rome and finally to Paris. The scholars conceived of themselves as heirs to an ancient tradition of learning.

The German kings who called themselves the 'Emperors of the Holy Roman Empire' supported their claim to such a lofty title through the tradition that their imperial dignity was transmitted from the Roman Empire to themselves. This tradition is known as the translatio imperii ENDNOTE (9). In such company, Masonry cannot be condemned for formulating legends of its own endowing itself with a long history and glorious tradition.

As we have seen above, Euclid's 47th problem was developed as part of a philisophico-religious system involving the deification of reason. The elevation of reason above other routes to knowledge was first developed in the Greek world. "The supreme contribution of the Greeks was to call attention to, employ, and emphasize the power of reason... Whereas earlier and later civilizations viewed nature as capricious, arbitrary, and terrifying, and succumbed to the belief that magic and rituals would propitiate mysterious and feared forces, the Greeks dared to look nature in the face. They dared to affirm that nature was rationally and indeed mathematically designed, and that man's reason, chiefly through the aid of mathematics, would fathom that design. The Greek mind rejected traditional doctrines, supernatural causes, superstitions, dogma, authority, and other such trammels on thought and undertook to throw the light of reason on the processes of nature... Euclid is the prime example of the power and accomplishments of reason" ENDNOTE (10).

Thus, the concepts involved in geometry and the idea of human reason are inextricably linked. Euclidean geometry is credited with teaching mankind the principles of correct reasoning. "...Euclidean geometry is the father of the science of logic" ENDNOTE (11). The connection between Freemasonry's use of the 47th problem as a symbol and the importance of geometry to reason is important since the Roman Catholic church has attacked Masonry for elevating reason above faith.

Pope Pius IX, in his encyclical, Qui Pluribus, 9 November 1846, attacked those who "[p]ut human reason above faith, and who believe in human progress." That encyclical has been deemed to be an attack on Freemasonry. Assuming that there is some truth to that characterization of Freemasonry, Euclid's 47th problem then has an ideological component which is just as significant to Masonry as its practical application in building.

It is ironic that the term "Great Architect of the Universe" was used as a name for God by the 12th century Roman Catholic clergymen scholars who specialized in teaching geometry and cosmology. They conceived of God as the "Great Architect of the Universe" holding the geometer's compass, and in the words of the scripture ordering "all things in measure and number and weight" (Wisdom 11:20) ENDNOTE (12). Thus the Masonic identification of God and geometry to which, inter alia, the Roman Catholic church objects, was in fact a Catholic invention.

Putting aside the issue of the 47th problem as a symbol, and leaving the question of its religious significance, we turn to its practical application in the process of building, and its relationship to the "squaring of the lodge". The Pythagorean Theorem is important in building, and one of its uses is to square a room. Builders use the theorem to square the corners of rooms by using the ratio of the numbers three, four and five. three squared plus four squared = five squared. Thus, using Euclid's 47th problem, a builder will mark a point, call it point A, three feet down one wall from a selected corner of the room, and then mark another spot, call it point B, four feet from the same corner down the other wall, and then measure the distance between points A and B. If the distance from A to B is exactly five feet, the distance of the predicted hypotenuse, then the corner is an exact right angle and the room is squared.

Applying what we know about the 47th problem, we can readily see that since the Entered Apprentice reaches the Northeast corner of the lodge three times, and that the Fellow craft reaches that corner four times, and that the Master reaches it five times during their respective circumambulations, the candidates are in fact, just as stated in the Texas Monitor, symbolically "squaring the lodge". Any Mason, after having been raised, has reproduced by circumambulation the numbers three, four and five in the most significant corner of the Lodge, the Northeast, and thereby has unknowingly recited with his feet the formula which is contained in the 47th problem, and thereby has "squared the Lodge".

Given what we know about the Pythagorean belief in the Divine nature of numbers, and the use of the Pythagorean Theorem in building, the circumambulation of the Lodge, and the altar, in the prescribed manner takes on a special significance to Masons. The answer to the question which began and which is the focus of this paper is that circumambulation and Euclid's 47th problem are indeed related.

Some will be little surprised by the discovery of a hidden relationship such as has been described. One encounters such relationships often in Masonry. Indeed, the author will go one step further, and state that there are hidden numerical relationships in many, if not all, ceremonies and symbols of Masonry which have yet to be discovered. The challenge is for Masons to agree on the significance of these relationships to us as Masons and how they help explicate the philosophy of Freemasonry.

The discovery of a comprehensive and coherent philosophy of Freemasonry requires a special methodology. The methodology which I propose is the same as that which was used by medieval philosophers in the critical exploration of theology. A brief look at that method is instructive for modern Masons.

The term "summa" refers to a comprehensive treatise which incorporates a method of analyzing doctrines. This method of expounding upon the doctrine of a particular discipline was developed by scholars in the Middle Ages, about the time of the beginning of Operative Freemasonry.

In the Middle Ages, legal scholars known as Romanists applied the dialectical (questioning) method to law. In analyzing Roman law, particularly the code of Justinian, they wrote systematic and comprehensive treatises (summae) which followed the logical order of doctrine rather than the literal order of the text. The summae contained fully reasoned discussion of the major legal topics. Those summae, formed by the application of dialectics to law, provided a rational synthesis of jurisprudence.

The same method was used by theologians. The best known work of St. Thomas Aquinas was entitled the "Summa Theologica". In that work, Aquinas expounded upon religious doctrine divided into units called articles. Articles created a single doctrinal point and followed a set pattern of exposition. The subject of the article was posed as a question to provoke discussion. In the first question presented, the author cited authorities to oppose the question, and then proposed his own solution to the question. Then the author answered the opposing arguments which he had first presented. By this dialectic of pros and cons, conflicts among the authorities were resolved. In effect, this method constituted the formal expression of a master's determination of a disputation. Theology began, like law, to depart from a simple reading of the authoritative text to follow the inner reason of its doctrines.

In helping establish this method, the scholar Abelard listed 158 questions on which he had found divergent authoritative opinion. Examples of the questions he found are "Should God be believed in?" and "Is it permissible to lie?" He then collected all authoritative texts pro and con on each question and named his book "Sic et Non" (For and Against). He challenged his readers to reconcile the authorities through dialectical reasoning. He established rules for harmonizing conflicts. He emphasized the close examination of the meanings of words. Abelard did not answer the questions but only developed the method of arriving at an answer ENDNOTE (13).

The study of Freemasonry requires a similar methodology to the study of law and theology. In all of these disciplines, ancient texts are available which are assumed to be authoritative. In the fields of theology and law, men have worked for centuries, and are still working, to develop a comprehensive statement of principles, a foundation upon which future scholars can rely to base their own research and conclusions. In the case of Masonry, however, many persons have written of its history and symbols, but few have written of the inner reasoning of its doctrines and ceremonies in a way that sets forth comprehensively the underlying philosophy of Freemasonry. Quite to the contrary, Masons are invited, indeed encouraged, to form their own private opinions of the meaning of Masonry and its symbols. It is the task of this paper, and of papers to come, to provoke discussion with the intent of persuading Masons at large collectively to assume the duty of developing an overt, comprehensive philosophy which can serve as a basis for understanding, and explaining, Freemasonry.

By focusing research and discussion on specific points of Freemasonry, we might generate as a byproduct an increased interest in Masonic research. Some of the Brethren do not participate actively in Masonic research and the authoring of papers because they do not know of an important topic upon which to write. As a consequence, we have many papers on individual Masons and on the history of Masonry in certain locations, but few on the speculative aspects of Masonry. Perhaps it is time to focus more attention on the fact that we are engaged in Speculative Masonry. In doing so we may encourage our Brethren in the study of Masonry.

I have proposed a question regarding a point and then proposed my own answer. By doing so in this paper and in subsequent papers I hope to explore the underlying rationale of certain Masonic doctrines and ritual and help to uncover the philosophy of Freemasonry. I also seek to elicit a response from other Masons who are interested in exploring the underlying philosophy of Freemasonry. Their responses will serve to more fully develop our understanding of Freemasonry and will act as a catalyst for further exploration of our Craft.

By using the dialectical (back and forth, pro and con) method  of exploring Masonic concepts, perhaps a synthesis of divergent ideas of Freemasonry can be achieved. This method is in complete accord with Masonic principles, for it seeks to develop a uniform explanation for the tenets of Freemasonry "upon which all men can agree", and does so by inviting all men to discuss. In this paper, I have proposed one article of a 'Summa Masonica'. In future papers I hope to address other questions regarding aspects of Freemasonry.

Having proposed this first problem and then answered it, I call upon the reader to tender counter arguments, or even better, new articles of their own devising.


1. "Circumambulation", Coil's Masonic Encyclopedia, (New York.: Macoy, 1961), 128.

2. Monitor of the Lodge, Grand Lodge of Texas, A.F. & A.M. (Waco, Tx.: Waco Printing Co., 1982), 16,46,74.

3. Little Masonic Library, [rev.ed.], 5 Vols. (Richmond, Va.: Macoy, 1977), 1:203-204

4. Monitor of the Lodge, Grand Lodge of Texas, A.F. & A.M. (Waco, Tx.: Waco Printing Co., 1982), 91-92

5. Masonic Manual, Grand Lodge F. & A.M. of Alabama, (Birmingham, Ala.: Press of Dispatch Printing Co., 1918), 81.

6. The Thirteen Books of Euclid's Elements, vol. 11 of Great Books of the Western World, 54 vols. ( Chicago: Encyclopedia Britannica, Inc., 1987), 28.

7.'Pythagoras and Pythagoreanism', Encyclopedia of Philosophy, (New York: Macmillan Publishing Co. and the Free Press, 1967), 7:37-39.

8. "Eureka", Webster's Third International Dictionary, (Springfield, Mass.: Merriam-Webster Inc., 1981),784

9. Baldwin, John W., The Johns Hopkins University. The Scholastic Culture of the Middle Ages, 1000-1300,(Lexington, Mass.: D.C. Heath and Co., 1971), 56.

10. Kline, Morris., Mathematics for the Nonmathematician , (New York: Dover Publications, Inc., 1967), 15.

11. Ibid., 149

12.Baldwin, John W., The Johns Hopkins University. The Scholastic Culture of the Middle Ages, 1000-1300,(Lexington, Mass.: D.C. Heath and Co., 1971), 107.

13. Ibid., 83.

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