CIRCUMAMBULATION AND EUCLID'S 47TH PROPOSITION
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.
ENDNOTES
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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