125
OBSEEVATIONS
EEGAEDINa PYRAMID
NUMBERS.
By
R.M.
Johnston,
F.L.S.(Diagrams.)
The
ancient structuresof Egypt,especially tlae pyramids,have ever been regarded with the
most
profound interest.Travellers
and
historians find inthem
an everlastingtheme
for description. Geometricians also find in their designs,magnitude
and
dimensions,much
matter for scientificspeculations;
and
the mystic inspired bytheir age, grandeur,and
mystery,isdisposed to gatherfrom
theirevery featuresome more
orless fancifullyconceived revelationormiracle.Nor
canwe wonder
at this.Egypt
is theland ofwonders.Great pyramidscovering acres of land; colossi sitting silent
in granite thrones; obelisksofprodigious heightwonderfully
carved
from
a singlestone;and
temples, sphinxes,and
canals,of stupendousproportions.
When
we
consider that allthesemonuments
were hoary with age at the time of Herodotus,and
that a close study of their worksand
hieroglyphics reveals that their buildershad
attained great knowledge inastronomy, geometry, architecture, engineering,
and
variousarts,
we
may
readily admit that our highestmodern
civilisationwascradledintheland ofthePharaoh's.
Itis not
my
intention,however, to enter into the enquirvofEgyptiancivilisationatpresent.
The
observations whichI haveto
make
are confinedtothepyramid structuresthemselves. Itis
now
wellestablishedthat pyramidal structures werepeculiarly characteristic ofthemost
ancientcivilisations of India, Babylon, Nineveh, Egypt, China,North
America,Mexico, and even in islands of the Pacific;
and
that thewhole or greater part of
them
are associated withsepultures for the dead.
But
while it is most probable that originally suchmonuments
were built solely forcommemoration
and
for the preservation of the remainsof noble persons, there are also good reasons for
supposingthat
some
ofthem
—
such as the GreatPyramid
of Gizeh orCheops
—
fulfilled a double purpose.The
GreatPyramid
ofCheops covers a space of about 13'05 acres. Ifwe make
allowance for slight distui^bance, duetopressure ofthe
enormous
superincumbent weight,we must
assumethatitsdesignerintended itsbase to form a perfect square,
and
itsshapea true pyramid.
The
various measurementsof themost
competent engineers onlyshow
a variation of 11, 13,126 OBSERVATIONS EEGAEDING PYRAMID NUMBERS.
measurements
givea length of 9,137inches,or 36,648 inchesfor the circuit ofthe foursides. Ferguson's,Dufell's, andColonelHoward
Vyse'smeasurementsofheight are themost
reliable,and
they only varybetween 460 feetand
456 feet, or 5,472Englishinches. Broadly speaking therefore its circuit
represents about 100inches for each day in the year,
and
itsheightalmostexactly fibs,of itsside base.
The
orientationoreastward aspect is almost true, being for SouthEast,
4 1 forNorthEast,
+
1 for SouthWest,and
+
•0636forNorth West. Subsequentsettlement or earth tremors
might
easilyaccountfor these minute divergences
from
absolutely trueorientation.While
rejecting themany
fanciful interpretations ofmysticalwriters
drawn from
known
facts with respect to shape, dimensions,measurements,and
orientation,Ihavelong been convinced by the reasoning of soberminded
investigators that the principal characteristics were probablydetermined as a base or fixed standard for measures,
of space
and
capacity;and
if so, the shapeand
dimensions themselves
might
have been suggested to theskilled geometricians of the time
by
reference tosome
astronomicalfact ofimportanceknown
tothem,inconjunction with significant properties ofnumber
and
proportion discoveredby
them
to belong to the structure of cubes inpyramidalform.
That
men who
taught themodern
worldmensuration
and
astronomy, should strive to attain a suremethod
for securing uniformityin standards as applied to weightsand
measurements, is amost
reasonable supposition.That
these standards should be symbolisedby
some
striking or wellknown astronomical fact, is in the highest degree probable,
and
corresponds exactly with the idea of the French astronomers,who
determined the length of their metre in relation to the ascertained length of a.meridianline
drawn from
the Pole to the Equator. (Themetrerepresentingthe tenthmillionth, or 39 '37079 English
inches; thecentimetre being one hundredthofa metre.
The
gramme
or standardofweight isderivedfrom
thecentimetre,.i.e.,a cubic centimetre of distilledwater at the temperature of
maximum
density, nearly equal to"0022054 ofan English avoirdupoispound,or 15,438 Englishgrains.)Impressed withthis idea,
and
with the conviction that theEgyptianbuilderswere adeptsin the constructionofmodels,
I soughtto obtain
some
lightupon
thesemattersby studyingthenumerical combinationsof simple cubes built
upon
thepyramid
type. Iwas
guided toa considerable extentinthese investigationsbythewideprevalence of multiples of 7,12,and
10, intheexistingsubdivisionsoftime, space,weightand
value.BY
K. M. JOHNSTON, F.L.S.'
127 of the year(aweek) determined to be seven days,for it
was
in
common
use long beforetliebirthofMoses
?Wliy was
the seventh day originally set apart asthe Sabbath?Why
havewe
theday dividedintotwoparts of 12hourseach,and
why
do multiples of 12 so
commonly
appear in weightsand
measures,especially inastronomicaldivisions?In
many
combinationsconducted withthehoj)e ofthrowinglighton suchmatters, Ifailed to get any remarkable indications, with threeimportantexceptions. Thesethree exceptions possessso
many
remarkableproportionsand numbers
relating to existing subdivisionsofweights,measures,and
values,and
especially with the proportions
and
dimensions of the GreatPyramid, that Ihave been induced to riskthe appellationof
"pyramid
mystic,"and
to lay the remarkable results before themembers
of this Society.The
models which appear before you have eachsome
particularclaimto notice,
and
whether any ofthem
may
offer sufficiently remarkable characteristics or notasbearingupon
theGrreatPyramid,they areallwellworthyofcloseattentionas offering anaturalsolution tothe genesisof particularnumbers
asusedinsubdivisions ormeasuresof time, space,weight, or value.
Pyramid
ofOdb
Numbees,
Having
7 asa
Base.As
shown
indiagram,themost
remarkablecharacteristicisthefactthatthe cube rootof its basal layer,49or 7^, enters into and agrees with all the imjiortant dimensions of the Great Pyramid, including lengthof complete circuit; length
ofside; height; lengthofEgytian cubit;
Enghsh
inch;and
through the latter it harmonises in the
most
obviousand
simple multiples withthesedimensions
and
the days in the year,days in the lunarmonth. Othernatural proportions of the three angled sides of jjyramid connote themonths
in the quarterand
year; while itsaggregatenumber
of cubes,84 or 7 times 12, suggest the alliance of 7
and
12 inmeasurement
oftime.VE "Tt ^ 3^
Demonstration indicatingthat the Great
Pyramid
dimensionswere probably determined
by
the radix of sacrednumber (7), whichin itself has probably been selected because
thecube rootofits squai'e contains nearly the exact figures representing the
known
daysintheyear:—
3
Badix(V7'')
=
'Ei=
36593.1. Circuit of pyramid in inches 36593
=
lOOOOR
2.
Length
ofeachside (4)„ „ 9148=
lOOOOR
S. Heightof
pyramid
„ „ 5488^ =: ^_{(lOOOOB)}=
457•» _{feet}128 OBSERVATJONS KEGAEDING
PYRAMID
NUMBERS.NOTE." ^ Heightproportion
may
have been2 selected because 3 expresses the
number
ofdimensionsin a cube:and
1~
2 exactly expresses the relative elevation surface of a triangleand
square restingupon
acommon
baseand
ofequalvertical height at its
maximum,
in a vertical linedrawn
at right angles to base line.Princijjalunit of
measurement
in inches 25^ =:Nearlyequalto existingcubit
inEgypt.
lOOOR
122_{or}_{144}
NOTE
12'^_{most}
_{probably}_{was adopted}_{as}_{a}divisor, because curiously
enough
the actualnumber
ofsquare cubes contained in apyramid
of evennumbers, which
most
nearlyapproaches the
number
of daysin a year,is 364, andthebase of such pyramidor 1st layercontains 12
X
12 cubes or144: thesecondlayer in importance succeeds it with10
X
10 cubes or 100 (see plan).lOOOR
6. Cubitsin circuit of pyramid, No. 1440 z=^02
lOOOOE
lOOOOR
or
lOOOOEf12? 25
lOOOOR
6. Dittoineach side(mean) 360.

^p^^,^
J
~r 4
» TT •. .
lOOOOR
7. Unitoryearor 1 =1
^
or 3lOOOOR
V7^
V7'
72 49
Days
inWeek
or 7 =: = _{or}I
BY
K, M. JOHNSTON, F.L.S. 1299.
Months
inYear
or 12=
Angles on4
faces4X3:
also equal to base ofa^
simple
pyramid
of evennumbers whose
aggregate represents 364:alsoj the seventhofthe aggregateofasimple
pyramid
of odd having 7 forits
base.
10.
Lunar
Months
inyearorl3'^nearly(lo""'')=
^
7X4
Square
Pyuamid
ofMixed
Odd
and
Even
Numbbes,
Having
fok
a
Base
(7 x 2)^or
14^=
196.Perhapsthisforms the
most
interesting of all thecombinations. Its natural proportions
and
naturally relatednumbers
aremost
suggestive.The
followingcombinations aremost
striking:—
1. If
we
takeeither the exposed cubes onthemargin
of eachlayer, or the total faces of distinct cubes in the four
sides,theaggregate comes exactly to 365, or the exact
number
of daysintheyear;and
therefore theproportional
number
of cubes on eachtriangularface is 91j,correspondingto days inthe quarterofayear.
2. If
we
now
take the basal layeralone,we
find the exposednumber
ofcubes inthe square to be 52, corresponding to weeksintheyear.3. Ifagain
we
take the aggregateofallcubesinthepyramid,we
find theyamount
to 1,015,and
if thisnumber
bemultiplied
by
36, or4times 9 (the latternumber
representing thenumber
of verticle angles on faces of thefourwedgesorprisms ofwhich thepyramid is built, as
indicated
by
itsdiagonals),we
obtain 36,540, or within 8 inches ofthe best actualmeasurements
of its presentstate, which has no doubt undergone
some
slightsettlement
duetosuperincumbentpressure.4.
A
quarterof this gives 9,135inches, or within 2 inches of themean
ofthe best actual measurements obtainedby
competentinvestigators.
6. If
we now
takethe squareof its basal layer,14 x 14,we
get 196,
and
it is remarkable that if thisnumber
bemultiplied successively
by
half the side,and
by thenumber
ofsides,i.e., 196x 7 x4,we
get5,488, or within 16inchesofthe best estimates of the present height ofthe Great Pyramid, any two of which differfar
more
130 OBSERVATIONS EEGAEDING
PYRAMID
NUMBERS,6.
The
basal layerhas 13 distinct cubes ineacbside,corresponding to
number
ofweeksineach quarter, whicli tbeside typifies naturally; while the three angles of each triangular face
makes
12, corresponding with themonths
intheyear orhoursinthe day.
These combinations are all natural to the particular
structure,
and
are not selected in arbitrary or forcedway
as in
many
suggestions found in works referring to the pyramids.Square
Pyramid
ofEven Numbers Having
12for
itsBase,
The
remarkablecharacteristicofthis pyramidis that—
1.
The
aggregate ofall the cubes,if cappedwith anodd oneasa finishing point,
numbers
365, corresponding to thenumber
ofdaysintheyear. It has 12 cubes along the basal layer of each side, corresponding to months.Thereareexactly 36 cubes in each triangular face,
and
144inbasal layer. If each of these be multiplied
by
thenumber
ofcubes in side of 2nd layer,and
taken as divisor of the circuit and side ofpyramid
they give results which almostexactly correspond with theexisting cubit ofEgypt,The same
result isvery closely attainedbp multiplying the aggregatenumber
of cubes (365) by 7,and
dividing theresultbythe square ofthe secondlayer (100),
2.
But
perhapsthemore
interestingnumbers
inthispyramid
ofeven
numbers
are those of the cubes of the exposedsides of squares, and the aggregates of the cubes in eachlayer.
It is singularthat in the first series the sequence 1, 4, 12, 20, shouldexactlycorrespond with the sequence of English standards of
money
value,viz.: Farthing, farthingsin penny, penniesin a shilling,and
shillings inapound.The
figures of the base,12and
144, are associated withsubdivisions of square measured multiples or subdivisions of 28, as 7, 14,28, 56, 112, 2,240 as insubdivisions ofweight; andin thesecond seriesof aggregates
we
haveinthe second layer the
numbers
10, 220,and
in the basal exposedmarginofcii'cuit44,allsuggestiveofsome
connection with reasons whichoriginally entered intothedetermination ofsubdivision of 44, 220,440, 1,760, intheEnglishmile.Conclusion,
Taken by
themselves the remarkable coincidences withknown
factsrelating tomeasurement
oftimeand spacemight
only be construed as simple examples of the facility withPyffAM/DOFEV£^J/^UMBERS
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BY
R. M. JOHNSTON, F.L.S. 131measuremeutsor proportionals relatingto theearth'sdiameter,
circumference, distance
from
the sun, annual period of revolution, etc.; forit iseasyby
slightvariations ofanyroot, arbitrarilymade, and ai'bitrarily selected multiples, tomake
any
number
approximate tosome
important terrestrialmeasurement, provided that the computer is himself
previouslyaware of the proportional,size,ormeasurement, with
which a
show
of correspondenceis de^fired.Much
ofthesocalledremarkablecoincidences of mystical writersare of this class; forit not unfrequently happens that the
same
rootmeasurement, by slightalteration,is
worked
uptobringaboutcoincidences with very different things.
Thus
Mr. PiazziSmith,
by
taking the height of the niche of the Queen'sChamber
ofthe GreatPyramid
as 182'62,and
multiplyingitby
2, he obtains 365242, equivalentto the daysin the year;and
again byarbitrarily taking thesame
dimensionsas 185,and
multiplyingit successivelyby
3*1416and
10, he obtains 5,812, which he arbitrarily concluded to be the heightof the Great
Pyramid
in inches.But
curiouslyenough
the
same
dimensions, 18262, multipliedby
10and
divided by 2 (whynotatonce multiplyby
5'^) ismade
toshow an
approximateto lengthofone of the sides in inches. These
are
common
examples ofthefacility with whichmany
fancy the discovery ofpurposeful design innumbers
or dimensions,when
dealt withinafancifuland
arbitraryway.It seemstohave beenforgotten
by
such persons thatany
root figure,
by
thearbitrary selectionofa multiplieror divisor,may
bemade
to coincide exactly with any othernumber
provided the manipulator knows hefcrehand the numher or proportional with
which
correspondence is sought to beestablished.
But making
allallowance for the vagaries of the mystics, there aremany
legitimate subjects of enquiry,upon
which
some
lightmight
bethrown bythecarefulinvestigation of ancient structuresAt
the present day it is remarkablehow
largelythenumbers
7, 12,and
10, or simple multiples of these enterinto standards of space, time,weight,and
value.Itis easy toimagine
how
10was
seizedupon
sofrequently asa standard of
measurement;
for counting bymeans
of the digitsof thetwo hands so universaland
sonaturalat once suggests a probable reason; but thereasonsforthe originalselection of 7
and
12 fora similar purpose are not so easily conceived.What,
forexample, were the determining causes for the selection ofthemany
subdivisions of weights,values, time, linealandsquaremeasure?Why
havewe
a sequenceof 4, 12, 20, in Englishmoney
insubdivisions ofthepenny, shilling
and pound
; of 14,28,132 OBSERVATIONS REGARDING
PYRAMID
NUMBERS.1,760, in subdivisions of theEnglish mile; of multiples of
12 in square measure; of either 8 or 7as a root of wine
measure?
8x1
8x84
7x96
ffalL:8x42
7x48
1 8X126
puncheon
:7^^44
tierce
pipe:
8x62
7x72
8 X252
7x288
hogshead
tun
Then
going tothe survivals of ancient systems of linearmeasurement,
how
canwe
account for the origin of linealmeasures, suchas
—
The
Englishfoot ... Equivalentto 12The
ancient "Pied deEoi"
ofPranceThe
German
elleThe
Italian pie ... _{„}The
common
guerze ofPersia ... ... ,,
The
pic ofTurkey
... ,,The
braccio ofAncona
,,The
shortpichaofGreece ,,The
long_ „
The
existing derah orcubitof
Egypt
... .,Jewishcubits ... l „
May
itnot bepossiblethereforethatthe ancientdraftsmen or modellers of pyramidshad
seizedupon
many
of these characteristicsshown
in the formsand
figures referred to,bothfor subdivisions of measures
and
weights,and
also to typifyin theirimportantfixed standardssome
of themore
remarkablefacts of astronomy then