        THE FORTY-SEVENTH PROBLEM
        by D.R. Lane, member of Northern California
                                 Research Lodge
    
    
    THE FORTY-SEVENTH PROBLEM occurs in the first
book of Geometry as compiled by Euclid and is stated as
follows:
    *"In any right angled triangle, the square which is de-
scribed upon the side subtending the right angle is equal
to the squares upon the sides which contain the right
angle".
*Mackey
    To put the meaning of this into plain words, if a right
triangle has for its base a length equivalent to three, and
for its perpendicular a length equivalent to four, then its
hypoteneuse will have a length equivalent to five, for the
square of three is nine and the square of four is sixteen;
their sum is twenty-five and the square root of twenty-
five is five.
    A triangle and squares of these dimensions constitute
the form in which the problem is presented emblematical-
ly in Masonry.
    The geometrical fact represented by  this problem is
often used in the laying out of buildings, and hence this
theorem has been called "The Carpenter's Theorem". Ac-
tually it belongs far more to the operative Mason than to
the carpenter, for the worker in wood uses it in only one
way  to set the corner of a building square--whereas the
operative Masons used it in many ways.
    But even before there were any Operative Masons as
we know of them, this theorem belonged to the survey-
ors of land, to astronomers and to mathematicians.
    In fact, it belongs to all of us.  If man had never sought
out the truths implicit in this great geometrical proposi-
tion, we should not be living in the kind of world we now
enjoy.  On this problem, and on the Thirty-Second Prob-
lem, are rounded the science of algebra.  This, and some
other theorems, formed the basis for the development of
trignometry  and  spherical  geometry.   Hence  it  serves
as ancestor to astronomy, navigation, and to all the high-
er mathematics.  If it be true that all science may be
reduced to a basis of pure mathematics, then this Forty-
Seventh Problem is parent to the whole body of the world's
scientific knowledge.
    This problem, and the facts derived from it, enter
thousand times a day into the life of every civilized
For example, it made possible the boring of the great
Twin Peaks tunnel in San Francisco so accurately that
when the bores from the two ends met, they were only
one-eighth of an inch out of line.  Without this problem,
and what engineers and scientists have derived from it,
we should not have the great bridges which span  the
Golden Gate and San Francisco Bay.  Before the time of
Ptolemy, the ancient Egyptians used this problem,  and
little else, to measure the diameter of t.he world  and
with very fair accuracy, too.  We use it, and its daugh-
ters in mathematics, to find the distances of the stars,
to grind the lenses and mirrors of our great telescopes,
to devise the cunning circuits of our television sets and
to construct our atom bombs.  It is part and parcel of
civilization as we know it---and by no means the smallest
part or lightest parcel.
    To whom, then, shall we credit this gigantic discovery?
We do not know.  Our Monitor says the Forty-Seventh
Problem was an invention of Pythagoras* but this cannot
be taken literally for this theorem was known many cen-
turies before the birth of Pythagoras.**   However,  he
may have learned of it during a visit which he is supposed
to have made to Egypt in the sixth century B.C. and
which indeed, it is probable that he did make,*** for the
Egyptians knew-and used at least two of the mathe-
matical facts embodied in this theorem.  First, it enabled
them to lay out the bases of structures as squares or ob-
longs with great accuracy.   For example,  consider the
great  pyramid of  Gizeh.   The  base  of  this  vast  pile
measures 755 feet 9 inches on each side and covers an
area of no less than thirteen acres. Yet so accurately did
the builders lay out its foundation that the angular error
in closing this great square is less than two-thirds of
an inch.
*Monitor
**Mackey
*** Brittanica
    Second, the Forty-Seventh Problem provided the Egyp-
tians with the angle at which the sides of the pyramids
are inclined, namely, 52 degrees, which is the angle be-
tween the shorter side of the right-angled triangle and its
hypoteneuse.  Within a very slight error, the sides of all
the pyramids but one are inclined at this angle.  In that
single exception that angle is 53 degrees, which was no
doubt an error in construction.  In the cases of two other
pyramids,  those of King Snefcru at Medum and King
Khufu at Gizeh, the error amounts to only eight minutes
of arc* which surely indicates 52 degrees was the goal
at which they aimed.
*Brittanica
    Furthermore, we may assume that this problem was a
key factor in the surveying which had to be done in Egypt
each year after the inundation of the Nile had elimi-
nated all landmarks.  In this surveying it was of course
necessary to establish lines and angles, particularly right
angles, and the principle embodied in the Forty-Seventh
Problem was then and still is the simplest method of set-
ting lines at right angles to each other.
    And beside all this, the Forty-Seventh Problem was the
standard of all the Egyptian measures of extent.*  By the
use of a right triangle, a circle, and certain simple lines,
it is possible to derive a great number of proportionate
measures.  Assuming that the sides of the triangle are
300, 400 and 500 units  feet, inches, yards, cubits, what
you will--the measurements derived will be exactly 480,
320, 180, 144 and 108.  If we suppose these units to be
cubits  (the Egyptian cubit was 20.6 inches**)--then all
of them may he found in the dimensions of well known
Egyptian structures which date from very early times.***
*Mackey
**Brittanica
***Mackey

    The stadium of Ptolemy was 480 cubits in length.

    The base of the great pyramid of Memphis measures
500 cubits on each side.
    The Egyptian stadium was 400 cubits in length.
    The Hebrew and Babylonian stadium was 320 cubits in
length.
    The length of the stadium of Cleomedes was 360 cubits.
or twice the 180 derived from the problem.
    The lesser Egyptian stadium was 216 cubits in length,
and 216 of course is just twice the 108 derived from this
problem.
    Since the structures date from as early as 4,750 B.C.
and Pythagoras was born some time between 600 and 590
B.C.,* it is evident that this great philosopher was not
the first to know of the Forty-Seventh Problem, though
to do him justice we must admit that he is generally
credited with being the first to develop a logical mathe-
matical proof for it.**  Many others have been developed
since, including one by our own President Garfield.***
*Hall
**Plane
***Plane


    Naturally, our operative forbearers must have had full
knowledge  of  this  problem  and  its  wide mathematical
properties from a very early date.  It was essential to
that understanding of geometry without which they could
not have built such structures as the aqueducts of Rome,
the Gothic churches and the great Cathedral of Cologne.
    It was the key to their great trade secret, Geometry.
This was their monopoly from the time of the Roman
Colleges and it was so much a part of Masonry that in
the Old Constitutions Geometry and Masonry were held
to be synonymous.*  And so it is with little surprise that
we find the Forty-Seventh Problem, which is the heart
and soul of geometry listed among the heiroglyphical em-
blems of Masonry today.
*History Vol. 3, Page 865

    We do not know when this problem was first invested
with symbolic values.  It is a matter of record, however,
that the ancient Egyptians regarded the right angled tri-
angle formed in demonstrating the problem as the symbol
of  universal  nature.*   To  them,  the  base  represented
Osiris, or the male principle; the perpendicular represent-
ed Isis, or the female principle, and the hypoteneuse was
Horus, their son, the product of the male and female
principles.

*Mackey, Hall, Pike

    Between  their  time  and  the  beginnings  of  modern
Masonry it may have had other meanings and it may not.
    However, some students hold that our Operative breth-
ren gave it a symbolic significance,* though what meaning
they ascribed to it has admittedly long been lost to us.
The learned Doctor Mackey says,  "All  the geometrical
symbols of Freemasonry, such as . . . the Forty-Seventh
Problem . . . may be considered as debris of what has
been called the Lost Secrets of the old Freemasons".*

*Speth
**History Vol. 3, Page 865

    We do know that the representation of the problem
which we use today appears on the frontispiece of Ander-
son's  Constitutions  of  1723  where  it  is  one  of  only
three symbols displayed.  The others are gloves and pil-
lars representative of the five orders of architecture.

    We also know that Anderson, in these same Constitu-
tions,  says  among  other  references  to  geometry  that
"The Forty-Seventh Problem  .  .  .  if duly observed,  is
the foundation of all Masonry, sacred, civil ,and military,"
and again:   "Geometry is the basis of true Masonry, and
its rule".
    In what degree  this  notable symbol  appeared  at the
time of Anderson's  Constitutions  is  uncertain.   It  may
have been either the Entered Apprenlice or Fellowcraft
but hardly the degree of Master Mason, for Mackey, our
best source, thinks  the Fellowcraft  degree was devised
about  1719  and  the  Master's  degree  not  until  1723,*
whereas Anderson's Constitutions were approved in 1722,
though not published until the following year.
*History

    Since that time, the Forty-Seventh Problem has been
relegated to a considerably less imposing place.   From
the Irish work it has disappeared entirely.* In the En-
glish work it has been completely removed from the ritual
but reassigned a place as the jewel of the Immediate
Past Master, where it is accorded the following explana-
tion:
*Harris
    "As this figure depends upon several lines, angles and
triangles, which form the whole, so Freemasonry depends
upon its several members. and the principles upon which
the society is established." To this is appended an im-
plied injunction toward charity.*
*Guide
    The eminent Doctor Waite, who is surely entitled to
speak with authority,  says  this explanation  was  taken
from the Old Lectures.  That is, lectures used before the
formation of Grand Lodge in 1717.  If so, it antendates
by some time the explanation given in our rituals.*
*Waite
    In these, the Forty-Seventh Problem is mentioned only
once, and then briefly.  This single reference occurs in
the monotorial portion of  the Master's  lecture  in the
Third degree,  in a section which  the candidate rarely
hears, being merely told where the explanation of this and
other emblems is to be found and admonished to make
himself familiar with them.  The explanation accorded it
is as follows:
    "It teaches Masons to be general lovers of the arts and
sciences."*
*Monitor

    The entire monitorial exposition of this emblem is vir-
tually word for word as it was in the earliest publication
to  which  access  is  readily  available -,an  "Anderson's
Manual" of 1805.  It may be a more satisfying explana-
tion than appears at first glance, for a true lover of the
arts and sciences can hardly be other than an earnest and
sincere seeker after Truth  and hence a good Mason.
NOTE: Measurements derived from Forty-Seventh Prob-
   lem
    "If we inscribe within a circle a triangle, whose perpen-
dicular shall be 300 parts, whose base shall be 400 parts.
and  whose hypoteneuse  shall  be  500  parts,  which,  of
course, bear the same relation to each other as 3, 4 and
5; then if we let a perpendicular fall from the angle of
the perpendicular and base to the hypoteneuse and extend
it through the hypoteneuse to the circumference ofthe
circle, this chord or line will be equal to 480 parts,and
two segments of the hypoteneuse, on each side of it,will
be found equal, respectively, to 180 and 320.  From the
point  where  this  chord  intersects  the  hypoteneuse  let
another line fall perpendicularly to the shortest side of
the triangle, will be equal to 108 parts.  Hence we de-
rive the following measures from the diagram: 500, 480,
400, 320, 180, 144 and 108, and all of these without the
slightest fraction".  -Mackey, P. 271.

Bibliography To Forty-Seventh Problem--D. R. Lane
Encyclopedia  of  Freemasonry-Mackey,   Hughan   and
Hawkins, edition of 1921, cited as "MACKEY".
Encyclopedia Brittaniea,  14th  edition,  cited  as  "BRIT-
TANICA".
Secret  Teaching  of  All  Ages-Manley  Hall,  cited  as
    "HALL".
New Plane Geometry---Welchons & Krickenberger, cited
     as "PLANE".
History of Freemasonry--Mackey and Singleton, cited as
   "HISTORY".

Installing Master's Guide--Emulation  Working,  cited  as
   "GUIDE".
MONITOR  OF  GRAND  LODGE  OF  CALIFORNIA,
   F.& A.M. -cited as "MONITOR".
Symbolism  of  Masonry--paper  by Loudoun  Harris  in
Transactions of Lodge of Research No. CC, Dublin,
for year 1929, cited as "HARRIS"
Transactions  of  Quatuor  Coronatii Lodge  No.  3076
   paper by G.W.  Speth.  Vol.  3,  Page  27,  cited as
   "SPETH"


Anderson's Constitutions---facsimile reproduction in Little
Masonic Library, cited as "ANDERSON".
"ANDERSON'S   MANUAL," -- edition   1805,   cited   as
  "MANUAL,"

Morals and Dogma -Albert Pike, cited as "PIKE".
New   Encyclopedia   of   Freemasonry--Arthur   Edward
  Waite, cited as "WAITE"


