THE BUILDER FEBRUARY 1916

THE FORTY-SEVENTH PROBLEM OF EUCLID
BY BRO. C. C. HUNT, GRINNELL, IOWA

The Master Mason will readily recognize this proposition as one of
the emblems of the Third Degree. He will also recall the monitorial
explanation of it there given, and possibly feel that it is an
explanation which does not explain. He may not question the
legendary history of it as given to him, but he does not understand
why it should have been selected as a Masonic emblem, nor how it
teaches Masons to be lovers of the arts and sciences. In fact there
are many Masons who are not mathematicians and do not even know
what the proposition is, and on this point the monitor is silent.

It is the object of this paper to briefly consider the history of
the proposition and offer a few suggestions as to its Masonic
significance. In doing this we may reach the conclusion that some
of the monitorial statements are not historically true, or at least
that they have not been proven. We will find, however, that the
value of its symbolism does not depend on the truth of the
historical statements given in the monitors, but is inherent in the
proposition itself.

This will be hard for many Masons to understand. Through
association of ideas, we are accustomed to think that the
traditions which cluster around a central truth, are essential
parts of that truth, and when critical investigation attacks the
truth of the tradition, we feel it is an attack upon the truth
itself. It is this trait of human nature which is the underlying
cause of all religious persecution, and we are by no means free
from it as Masons, though it is contrary to the fundamental
principles of Masonry.

As members of the Masonic Research Society, it is our duty to
search for the truth, no matter how much it may conflict with our
preconceived notions or with traditions. If we but search aright,
we will find that these traditions are but the outer garments with
which time has clothed the truth, and that they are not its
essential essence.

In our associations with each other we meet a kindred soul whom we
learn to love and honor. We are told that he is the descendant of
a great and honored name in history, and we say that the spirit of
his forefathers has fallen upon him. Then some critic appears and
shows that there is no proof of his illustrious ancestry, or
perhaps entirely disproves it. What of it? Is he not the same
friend we knew before? Has his soul lost any of its greatness? May
not the spirit of a great soul have descended upon him, though his
physical blood does not literally flow in his veins? We are told
that the spirit of the prophet Elijah descended upon Elisha and
centuries later appeared in John the Baptist. Yet there was no
blood relationship between them. So it is with the proposition we
are now studying. Its tradition and its history are both
interesting, but its truth and the richness of its symbolism are
not affected thereby.

In Euclid's Elements of Geometry there are thirteen books, and the
subject we are considering is the forty seventh proposition of the
first book. It is not a problem but a theorem, and is so called by
Euclid. A problem in geometry is something to be done, as a figure
to be drawn, while a theorem is something to be proved. This
proposition is to prove, as Euclid states it, that "In any
right-angled triangle, the square which is described on the side
subtending (opposite) the right angle is equal to the square
described on the sides which contain the right angle." The sides
containing the right angle are called respectively the base and
perpendicular, while the side opposite the right angle is called
the hypothenuse.

Our monitors state that "This was the invention of our ancient
friend and brother the great Pythagoras." This statement has been
denied by many students of the subject. It has been claimed that
this proposition was known to the Egyptians long before the time of
Pythagoras, and that he learned it from them and carried it into
Europe and Asia. We have no proof either for or against this claim.
Pythagoras himself wrote nothing, and we know of his teachings only
through the writings of his disciples. Vitruvius, a celebrated
Roman architect of the time of Augustus Caesar, attributes the
discovery of this proposition to Pythagoras. Plutarch quotes
Apollydorus, a Greek painter of the 5th century B.C., as authority
for the statement that Pythagoras sacrificed an ox on the discovery
of this demonstration. Proclus credits Pythagoras with the first
demonstration, but asserts that his proof was different from that
given in Euclid. In fact so many writers, both ancient and modern,
have attributed this proposition to Pythagoras that it is commonly
called by his name, "The Theorem of Pythagoras."

On the other hand, the properties of the triangle whose sides are
respectively, 3, 4, and 5, were certainly known to the Egyptians
and were made the basis of all their measurement standards. We find
evidence of this in their important buildings, many of them erected
before the time of Pythagoras. We also find that this triangle was
to them the symbol of universal nature. The base 4, represented
Osiris, the male principle; the perpendicular 3; Isis, the female
principle; and Horus, their son, the product of the two principles,
was represented by the hypothenuse 5.

May we not find an explanation of this apparent discrepancy in the
statement of Plutarch that Pythagoras discovered the demonstration
of the general proposition, but that the particular case in which
the lengths of the sides are 3, 4, and 5, was earlier known to the
Egyptians? Plutarch also thinks that the case in which the base and
perpendicular are equal (as in the sides of a square) was likewise
known to the Egyptians. This is called the classical form in
Masonry and is the form usually found on the Master's carpet. Both
these forms are rich in symbolism, and if known to the Egyptians,
as they probably were, would naturally lead to the belief that the
general demonstration was also known. Nevertheless it may be true,
as claimed by so many writers, that to Pythagoras we owe the
demonstration of the general proposition, which proved the theorem
true for all possible cases. It was the delight of this philosopher
to discover a universal principle underlying a concrete fact, and
he must have attached a deeper meaning to the general truth than
the Egyptians did to the special cases known to them. With him the
science of numbers was the essence of all truth, and having
discovered a proof for the general proposition, he set himself the
task of finding right triangles whose sides can be expressed in
numbers. Heron of Alexander and Proclus are authority for the
statement that Pythagoras discovered the following method: Take any
odd number for the shortest side; subtract one from the square of
that number and divide the result by two; this will give the medium
side; add one to the medium side and the result will be the
hypothenuse or longest side. This is true as far as it goes, but it
does not give all the right triangles which can be expressed in
numbers.

The numerical symbolism of Pythagoras is an interesting study in
itself and is closely allied to much of our Masonic symbolism, but
that is outside the province of the present paper. It is simply
mentioned here, because, while it is probably not true that he was
raised to the sublime degree of a Master Mason as stated in our
monitors, yet there is so much resemblance between his teachings
and that of Masonry, that we can understand how the error might
have occurred.

The monitor also states that Pythagoras celebrated his triumph in
the discovery of this proposition by the sacrifice of a hecatome
(one hundred oxen). We can see how this may have been an outgrowth
of the statement attributed to Apollodorus above. Ovid denies it
and Hegel laughs at it, saying, "It was a feast of spiritual
cognition, at the expense of the oxen." The strongest argument
against it, however, is the fact that Pythagoras taught the
doctrine of the transmigration of souls and forbade animal
slaughter. However, when we consider that among many of the
ancients the sacrifice of a number of oxen was their method of
expressing their gratitude for a great triumph, we can understand
how the tradition arose, and accept the fact of the joy without
caring for the truth of the sacrifice.

Why should the discovery of this demonstration have been considered
a great triumph? Because it is of the utmost importance to the
science of geometry. Dionysius Lardner, in his edition of Euclid,
quoted by Mackey, says, "Whether we consider the 47th problem with
reference to the peculiar and beautiful relation established in it;
or to its innumerable uses in every department of mathematical
science, or to its fertility in the consequences derivable from it,
it must certainly be esteemed the most celebrated and important in
the whole of the elements, if not in the whole range of
mathematical science. It is by the influence of this proposition
and that which establishes the similitude of equiangular triangles
(in the sixth book) that geometry has been brought under the
dominion of algebra; and it is upon the same principle that the
whole science of trigonometry is founded." The Encyclopedia
Britannica calls it "One of the most important in the whole of
geometry, and one which has been celebrated since the earliest
times ;" and adds, "On this theorem almost all geometrical
measurement depends, which cannot be directly obtained."

What is its significance in Masonry? Our monitors tell us that it
teaches Masons to be lovers of the arts and sciences. Since it is
so important a proposition in the science of mathematics, we can
understand why it should be adopted as a symbol of scientific
investigation, and to such an investigation all Masons are pledged
in their search for truth, the great object of Masonic study.
But has it not a deeper meaning? Dr. Lardner says it is the basis
of the application of algebra to geometry. Algebra is the
application of symbols to mathematics, and Masonry is the
application of symbolism in character building. The Britannica says
that mathematical measurements which cannot be directly obtained
depend on this proposition. Yes, and as applied to Masonry, the
highest truths of morality cannot be directly obtained. They must
come to us indirectly through the medium, principally, of
symbolism.

There is no apparent relation between the numbers 3, 4, and 5 and
5, 12, and 13, for instance; but when we raise these numbers from
the first to the second power (that is, square them), we obtain 9,
16, and 25 in the first case, and 25, 144, and 169, in the second.
In this form we notice in each case that the sum of the first two
squares is equal to the third, and that the numbers in which we
could at first see no relation are the sides of right angled
triangles. So it is in life. Measured on the level of our lower
natures, there is no relation between our own desires and our
brother's needs. We are connected, it is true, as the sides of a
triangle are connected, but there is no reason why we should not
use him for the accomplishment of our own selfish purposes,
irrespective of his welfare. It is only when we square our lives by
the square of virtue, and our selfish desires are raised to
spiritual purposes, that we perceive that our own welfare is
intimately connected with that of our brother. His misfortunes are
our misfortunes, and we can no more injure him and not be ourselves
harmed thereby, than we can strike off our right hand and be none
the worse by reason thereof.

We are traveling upon the level of time to our eternal destiny. We
cannot stand still, but must constantly go forward. Shall we also
go upward ? All the time there is a spiritual force striving to
lift us to higher levels. We may refuse to avail ourselves of it
and remain in the depths of our lower nature; or we can accept it
and allow its divine influence to shine in our lives. The base
represents our earthly nature on the level of time; the
perpendicular is the divine spirit striving to manifest itself
through us. When these forces are squared to each other, their
union becomes a constant onward and upward movement to the throne
of God Himself. Pythagoras himself recognized this symbolism when
he said that early in life he came to the place where two ways
parted. One was easy and pleasant traveling; the other was rugged
and tended upward. It necessitated hard climbing. Which was the way
that led to life ? All who travel there and find these two paths,
know that he should choose the upward path, but the other seems so
much more pleasant, and many are inclined to walk therein. They
will try it a little while, and then return to the better way. But
there is no turning back on the level of time. The farther they go
on the lower level, the wider apart become the two ways, and the
harder to cross from one to the other.

How often we have heard Masons say that there is no moral lesson to
be derived from the 47th proposition of Euclid, and that it is not
to be described as the symbol of any moral truth. Have they
forgotten that there is not an observance or symbol of Masonry
which has not a deep significance? Significance for what? Certainly
as Masons it would have no especial significance for us unless it
aided us in attaining the great purpose of our Order, "the
uprearing of that spiritual temple, that house not made with hands,
eternal in the heavens." It may well be that the significance is
not recognized by us, but that by no means proves its nonexistence.
It may be buried in the rubbish of preconceived opinions, and it
only needs diligent digging to bring it to light.

We have here suggested but a few of the many applications of this
symbol in the hope that it will stimulate others to more diligent
research.

A GRIP

The clasp of two hands is literally a physical contact of two
pieces of human flesh. Woefully secular and lifeless it can be! We
all know the flabby, the clinging, the nervous, the icy hand grasp.
Yet who has not sometimes rejoiced in the grasp of a hand that
conveys life and love? Two souls are here united by a physical
contact which gives birth to new aspirations and new certainties.
Two human beings are here linked hand to hand in mutual respect,
mutual trust, and mutual encouragement.
--Richard C. Cabot.

