sift the matter to the bottom, I have at last resolved to give my ideas
to the world as they stand; crude necessarily, so I must be excused if
in some details I may be found erroneous; there is truth I know in the
general conclusions. I am presumptuous enough to believe that the R.B.
cubit of 1·685 British feet was the measure of the pyramids of Gïzeh,
although there may have been an astronomical 25 inch cubit also. It
appears to me that no cubit measure to be depended on is either to be
got from a stray measuring stick found in the joints of a ruined
building, or from any line or dimensions of one of the pyramids. I
submit that a most reasonable way to get a cubit measure out of the
Pyramids of Gïzeh, was to do as I did:--take them as a whole, comprehend
and establish the general ground plan, find it geometric and harmonic,
obtain the ratios of all the lines, establish a complete set of natural
and even numbers to represent the measures of the lines, and finally
bring these numbers to cubits by a common multiplier (which in this case
was the number eight). After the whole proportions had been thus
expressed in a cubit evolved _from_ the whole proportions, I established
its length in British feet by dividing the base of Cephren, as known, by
the number of my cubits representing its base. It is pretty sound
evidence of the theory being correct that this test, with 420 cubits
neat for Cephren, gave me also a neat measure for Cheops, from Piazzi
Smyth's base, of 452 cubits, and that at the same level, these two
pyramids become equal based.
I have paid little attention to the inside measurements. I take it we
should first obtain our exoteric knowledge before venturing on esotoric
research. Thus the intricate internal measurements of Cheops, made by
various enquirers have been little service to me, while the accurate
measures of the base of Cheops by Piazzi Smyth, and John James Wild's
letter to Lord Brougham, helped me amazingly, as from the two I
established the plan level and even bases of Cheops and Cephren at plan
level--as I have shown in the preceding pages. My theory demanded that
both for the building of the pyramids and for the construction of the
models or subsidiary instruments of the surveyors, simple slope ratios
should govern each building; before I conclude, I shall show how I got
at my slope ratios, by evolving them from the general ground plan.
I am firmly convinced that a careful investigation into the ground plans
of the various other groups of pyramids will amply confirm my survey
theory--the relative positions of the groups should also be
established--much additional light will be then thrown on the subject.
Let me conjure the investigator to view these piles _from a distance_
with his mind's eye, as the old surveyors viewed them with their bodily
eye. Approach them too nearly, and, like Henry Kinglake, you will be
lost in the "_one idea of solid immensity._" Common sense tells us they
were built to be viewed from a distance.
Modern surveyors stand _near_ their instruments, and send their flagmen
to a distance; the Egyptian surveyor was _one of his own flagmen_, and
his instruments were towering to the skies on the distant horizon. These
mighty tools will last out many a generation of surveyors.
The modern astronomer from the top of an observatory points his
instruments direct at the stars; the Egyptian astronomer from the summit
of his particular pyramid directed his observations to the rising and
setting of the stars, or the positions of the heavenly bodies in
respect to the far away groups of pyramids scattered around him in the
distance; and by comparing notes, and with the knowledge of the relative
position of the groups, did these observers map out the sky. Solar and
lunar shadows of their own pyramids on the flat trenches prepared for
the purpose, enabled the astronomer at each observatory to record the
annual and monthly flight of time, while its hours were marked by the
shadows of their obelisks, capped by copper pyramids or balls, on the
more delicate pavements of the court-yards of their public buildings.
We must grasp that their celestial and terrestrial surveys were almost a
reverse process to our own, before we can venture to enquire into its
details. It then becomes a much easier tangle to unravel. That a
particular pyramid among so many, should have been chosen as a favoured
interpreter of Divine truths, seems an unfair conclusion to the other
pyramids;--that the other pyramids were rough and imperfect imitations,
appears to my poor capacity "a base and impotent conclusion;"--(as far
as I can learn, _Mycerinus_, in its perfection, was a marvel of the
mason's art;) but that one particular pyramid should have anything to do
with the past or the future of the lost ten tribes of Israel (whoever
that fraction of our present earthly community may be), seems to me the
wildest conclusion of all, except perhaps the theory that this one
pyramid points to the future of the British race. Yet in one way do I
admit that the pyramids point to our future.
Thirty-six centuries ago, they, already venerable with antiquity, looked
proudly down on living labouring Israel, in helpless slavery, in the
midst of an advanced civilization, of which the history, language, and
religion are now forgotten, or only at best, slightly understood.
Thirty-six centuries hence, they may look down on a civilization equally
strange, in which our history, language, and religion, Hebrew race, and
British race, may have no place, no part.
If the thoughts of noble poets live, as they seem to do, old Cheops,
that mountain of massive masonry, may (like the brook of our Laureate),
in that dim future, still be singing, as he seems to sing now, this
idea, though not perhaps these words:
"For men may come, and men may go,
But I go on for ever."
"Ars longa, vita brevis." Man's work remains, when the workman is
forgotten; fair work and square, can never perish entirely from men's
minds, so long as the world stands. These pyramids were grand and noble
works, and they will not perish till their reputation has been
re-established in the world, when they will live in men's memories to
all generations as symbols of the mighty past. To the minds of many now,
as to Josephus in his day, they are "_vast and vain monuments,_" records
of folly. To me they are as monuments of peace, civilization and
order--relics of a people living under wise and beneficent
rulers--evidences of cultivation, science, and art.
§ 16. THE PENTANGLE OR FIVE POINTED STAR THE GEOMETRIC SYMBOL OF THE
GREAT PYRAMID.
From time immemorial this symbol has been a blazing pointer to grand and
noble truths, and a solemn emblem of important duties.
Its geometric significance, however, has long been lost sight of.
It is said to have constituted the seal or signet of King Solomon (1000
B.C.), and in early times it was in use among the Jews, as a symbol of
safety.
It was the Pentalpha of Pythagoras, and the Pythagorean emblem of health
(530 B.C.).
It was carried as the banner of Antiochus, King of Syria (surnamed
Soter, or the Preserver), in his wars against the Gauls (260 B.C.).
Among the Cabalists, the star with the sacred name written on each of
its points, and in the centre, was considered talismanic; and in ancient
times it was employed all over Asia as a charm against witchcraft. Even
now, European troops at war with Arab tribes, sometimes find, under the
clothing, on the breasts of their slain enemies, this ancient emblem, in
the form of a metal talisman, or charm.
The European Göethe puts these words into the mouth of Mephistopheles:
"I am hindered egress by a quaint device upon the threshold,--that
five-toed damned spell."
I shall set forth the geometric significance of this star, as far as my
general subject warrants me, and show that it is the _geometric emblem
of extreme and mean ratio_, and the _symbol of the Egyptian Pyramid
Cheops_.
A plane geometric star, or a solid geometric pyramid, may be likened to
the corolla of a flower, each separate side representing a petal. With
its petals opened and exposed to view, the flower appears in all its
glorious beauty; but when closed, many of its beauties are hidden. The
botanist seeks to view it flat or open in its geometric symmetry, and
also closed, as a bud, or in repose:--yet judges and appreciates the one
state from the other. In the same manner must we deal with the five
pointed star, and also with the Pyramid Cheops.
In dealing with so quaint a subject, I may be excused, in passing, for
the quaint conceit of likening the interior galleries and chambers of
this pyramid to the interior whorl of a flower, stamens and pistil,
mysterious and incomprehensible.
Figure 67 (page 101), is the five pointed star, formed by the unlapping
of the five slant sides of a pyramid with a pentagonal base.
Figure 70 (page 106), is a star formed by the unlapping of the four
slant sides of the pyramid Cheops.
The pentagon GFRHQ, (_Fig._ 67) is the base of the pyramid "_Pentalpha_"
and the triangles EGF, BFR, ROH, HNQ and QAG, represent the five sides,
so that supposing the lines GF, FR, RH, HQ and QG, to be hinges
connecting these sides with the base, then by lifting the sides, and
closing them in, the points A, E, B, O, and N, would meet over the
centre C.
Thus do we close the geometric flower Pentalpha, and convert it into a
pyramid.
In the same manner must we lift the four slant sides of the pyramid
Cheops from its star development, (_Fig._ 70) and close them in, the
four points meeting over the centre of the base, forming the solid
pyramid. Such transitions point to the indissoluble connection between
plane and solid geometry.
As the _geometric emblem of extreme and mean ratio_, the pentangle
appears as an assemblage of lines divided the one by the others _in
extreme and mean ratio_.
To explain to readers not versed in geometry, what extreme and mean
ratio signifies, I refer to Figure 65:--
Fig. 65.
Let AB be the given line to be divided in extreme and mean ratio,
_i.e._, so that the whole line may be to the greater part, as the
greater is to the less part.
Draw BC perpendicular to AB, and equal to half AB. Join AC; and with BC
as a radius from C as a centre, describe the arc DB; then with centre A,
and radius AD, describe the arc DE; so shall AB be divided in E, in
extreme and mean ratio, or so that AB: AE:: AE: EB. (Note that AE is
equal to the side of a decagon inscribed in a circle with radius AB.)
Let it be noted that since the division of a line in mean and extreme
ratio is effected by means of the 2, 1 triangle, ABC, therefore, as the
exponent of this ratio, another reason presents itself why it should be
so important a feature in the Gïzeh pyramids in addition to its
connection with the primary triangle 3, 4, 5.
Fig. 66.
To complete the explanation offered with figure 65, I must refer to Fig.
66, where in constructing a pentagon, the 2, 1 triangle ABC, is again
made use of.
The line AB is a side of the pentagon. The line BC is a
perpendicular to it, and half its length. The line AC is produced
to F, CF being made equal to CB; then with B as a centre, and
radius BF, the arc at E is described; and with A as a centre, and
the same radius, the arc at E is intersected, their intersection
being the centre of the circle circumscribing the pentagon, and
upon which the remaining sides are laid off.
We will now refer to figure 67, in which the pentangle appears as the
symbolic exponent of the division of lines in extreme and mean ratio.
Thus: MC : MH :: MH : HC
AF : AG :: AG : GF
AB : AF :: AF : FB
while MN, MH or XC: CD:: 2: 1--being the geometric template of the work.
Thus every line in this beautiful symbol by its intersections with the
other lines, manifests the problem.
Note also that
GH = GA
AE = AF
DH = DE
I append a table showing the comparative measures of the lines in Fig.
67, taking radius of the circle as a million units.
Fig 67.
Table Showing the Comparative Measures of Lines.
(_Fig. 67._)
ME = 2000000 = diameter.
AB = 1902113 = AD ÷ DB
MB = 1618034 = MC + MH = MP + PB
AS = 1538841·5
EP = 1453086 = AG + FB
AF = 1175570 = AE = GB
MC = 1000000 = radius = CD + DX = CH + CX
AD = 951056·5 = DB = DS
PB = 854102
QS = 812298·5
MP = 763932 = CH × 2 = base of Cheops.
AG = 726543 = GH = XH = HN = PF = FB = Slant
edge of Cheops. = slant edge of Pent. Pyr.
DE = 690983 = DH = XD = apothem of Pentagonal Pyramid.
{apothem of Cheops.
MH = 618034 = MN = XC = {altitude of Pentagonal Pyramid.
{side of decagon inscr'd in circle.
MS = 500000
{mean proportional between MH and HC
485868 = {
{altitude of Cheops.
OP = 449027 = GF = GD + DF
HC = 381966 = half base of Cheops.
SO = 363271·5 = HS
CD = 309017 = half MH
PR = 277516
GD = 224513·5
SP = 263932
The triangle DXH represents a vertical section of the pentagonal
pyramid; the edge HX is equal to HN, and the apothem DX is equal to DE.
Let DH be a hinge attaching the plane DXH to the base, now lift the
plane DXH until the point X is vertical above the centre C. Then the
points A, E, B, O, N of the five slant slides, when closed up, will all
meet at the point X over the centre C.
We have now built a pyramid out of the pentangle, whose slope is 2 to 1,
altitude CX being to CD as 2 to 1.
Apothem DX = DE
Altitude CX = HM or MN
Altitude CX + CH = CM radius.
Apothem DX + CD = CM radius.
Edge HX = HN or PF
Note also that
MP
-- = CH
2
OP = HR
Let us now consider the _Pentangle as the symbol of the Great Pyramid
Cheops_.
The line MP = the base of Cheops.
The line CH = half base of Cheops.
The line HM = apothem of Cheops.
The line HN = slant edge of Cheops.
Thus: Apothem of Cheops = side of decagon.
Apothem of Cheops = altitude of pentagonal pyramid.
Slant edge of Cheops = slant edge of pentagonal pyramid.
Now since apothem of Cheops = MH
and half base of Cheops = HC
then do apothem and half base represent, when taken together, extreme
and mean ratio, and altitude is a mean proportional between them: it
having already been stated, which also is proved by the figures in the
table, that MC : MH :: MH : HC and apoth: alt :: alt : half base.
Thus is the four pointed star _Cheops_ evolved from the five pointed
star _Pentalpha_. This is shown clearly by Fig. 68, thus:--
Fig. 68.
Within a circle describe a pentangle, around the interior pentagon of
the star describe a circle, around the circle describe a square; then
will the square represent the base of Cheops.
Draw two diameters of the outer circle passing through the centre square
at right angles to each other, and each diameter parallel to sides of
the square; then will the parts of these diameters between the square
and the outer circle represent the four apothems of the four slant sides
of the pyramid. Connect the angles of the square with the circumference
of the outer circle by lines at the four points indicated by the
diameters, and the star of the pyramid is formed, which, when closed as
a solid, will be a correct model of Cheops.
Calling apothem of Cheops, MH = 34
and half base, HC = 21
as per Figure 6. Then-- MH + MC = 55
and 55 : 34 :: 34 : 21·018, being only in error a few inches in the
pyramid itself, if carried into actual measures.
The ratio, therefore, of apothem to half-base, 34 to 21, which I ascribe
to Cheops, is as near as stone and mortar can be got to illustrate the
above proportions.
Correctly stated arithmetically let MH = 2.
Then HC = √5 - 1
MC = √5 + 1
and altitude of Cheops = √(MH × MC)
Let us now compare the construction of the two stars:--
Fig. 69.
TO CONSTRUCT THE STAR PENTALPHA FIG. 69.
Describe a circle.
Draw diameter MCE.
Divide MC in mean and extreme ratio at H.
Lay off half MH from C, to D.
Draw chord ADB, at right angles to diameter ECM.
Draw chord BHN, through H.
Draw chord AHO, through H.
Connect NE.
Connect EO.
Fig. 70.
TO CONSTRUCT THE STAR CHEOPS, FIG. 70.
Describe a circle.
Draw diameter MCE.
Divide MC in mean and extreme ratio, at H.
Describe an inner circle with radius CH, and around it describe the
square a, b, c, d.
Draw diameter ACB, at right angles to diameter ECM.
Draw Aa, aE, Eb, bB, Bd, dM,
Mc, and cA.
The question now arises, does this pyramid Cheops set forth by the
relations of its altitude to perimeter of base the ratio of diameter to
circumference; or, does it set forth mean proportional, and extreme and
mean ratio, by the proportions of its apothem, altitude, and half-base?
The answer is--from the practical impossibility of such extreme accuracy
in such a mass of masonry, that it points alike to all, and may as
fairly be considered the exponent of the one as of the others. Piazzi
Smyth makes Cheops 761·65 feet base, and 484·91 feet altitude, which is
very nearly what he calls a [Pi] pyramid, for which I reckon the
altitude would be about 484·87 feet with the same base: and for a
pyramid of extreme and mean ratio the altitude would be 484·34 feet.
The whole difference, therefore, is only about six inches in a height of
nearly five hundred feet. This difference, evidently beyond the power of
man to discover, now that the pyramid is a ruin, would even in its
perfect state have been inappreciable.
It appears most probable that the star Pentalpha led to the star Cheops,
and that the star Cheops (_Fig_. 70) was the plan used by the ancient
architect, and the ratio of 34 to 21, hypotenuse to base, the template
used by the ancient builders.
Suppose some king said to his architect, "Make me a plan of a pyramid,
of which the base shall be 420 cubits square, and altitude shall be to
the perimeter of the base as the radius of a circle to the
circumference."--Then might the architect prepare an elaborate plan in
which the relative dimensions would be about--
R. B. CUBITS
{Base 420
Base angle 51° 51′ 14·3″ {Altitude 267·380304 &c.
{Apothem 339·988573 &c.
The king then orders another pyramid, of the same base, of which
altitude is to be a mean proportional between apothem and half-base--and
apothem and half-base taken as one line are to be in mean and extreme
ratio.
The architect's plan of this pyramid will be the simple figure
illustrated by me (_Fig_. 70), and the dimensions about--
R. B. CUBITS.
{Base 420
Base angle 51° 49′ 37-42/471″ {Altitude 267·1239849 &c.
{Apothem 339·7875153 &c.
But the builder practically carries out _both_ plans when he builds to
my templates of 34 to 21 with--
R. B. CUBITS.
{Base 420
Base angle 51° 51′ 20″ {Altitude 267·394839 &c.
{Apothem 340
and neither king nor architect could detect error in the work.
The reader will remember that I have previously advanced that the level
of Cephren's base was the plan level of the Gïzeh pyramids, and that at
this level the base of Cheops measures 420 R.B. cubits--same as the
base of Cephren.
This hypothesis is supported by the revelations of the pentangle, in
which the ratio of 34 to 21 = apothem 340 to half-base 210 R.B. cubits,
is so nearly approached.
Showing how proportional lines were the order of the pyramids of Gïzeh,
we will summarise the proportions of the three main pyramids as shewn by
my dimensions and ratios, very nearly, viz.:--
_Mycerinus. Base : Apothem :: Altitude : Half-Base._
as shown by the ratios, (_Fig_. 13), 40 : 32 :: 25 : 20.
_Cephren. Diagonal of Base : Edge :: Edge : Altitude._
as shown by ratios, (_Fig_. 12), 862 : 588 :: 588 : 400.
_Cheops. (Apothem + Half Base): Apoth. :: Apoth. : Half Base._
as shown by the ratios, (_Fig_. 9), 55 : 34 :: 34 : 21.
and--_Apothem : Altitude :: Altitude : Half-B._
Similar close relations to other stars may be found in other pyramids.
Thus:--_Suppose NHO of figure 69 to be the NHO of a heptangle instead of
a pentangle_, then does NH represent apothem, and NO represent base of
the pyramid Mycerinus, while the co-sine of the angle NHM (being MH
minus versed sine) will be equal to the altitude of the pyramid. The
angle NHM in the heptangle is, 38° 34′ 17·142″, and according to my
plan of the pyramid Mycerinus, the corresponding angle is 38° 40′ 56″.
(_See Fig_. 19.) This angular difference of 0° 6′ 39″ would only make
a difference in the apothem of the pyramid of _eight inches_, and of
_ten inches_ in its altitude (apothem being 283 ft. 1 inch, and
altitude 221 ft.).
§ 17. THE MANNER IN WHICH THE SLOPE RATIOS OF THE PYRAMIDS WERE ARRIVED
AT.
The manner in which I arrived at the Slope Ratios of the Pyramids, viz.,
32 _to_ 20, 33 _to_ 20, and 34 _to_ 21, for _Mycerinus_, _Cephren_, and
_Cheops_, respectively (_see Figures_ 8, 7 _and_ 6), was as follows:--
First, believing in the connection between the relative positions of the
Pyramids on plan (_see Fig_. 3, 4 _or_ 5), and their slopes, I viewed
their positions thus:--
Mycerinus, situate at the angle of the 3, 4, 5 triangle ADC, is likely
to be connected with that "primary" in his slopes.
Cephren, situate at the angle of the 20, 21, 29 triangle FAB, and
strung, as it were, on the hypotenuse of the 3, 4, 5 triangle DAC, is
likely to be connected with _both_ primaries in his slopes.
Cheops, situate at the point A, common to both main triangles, governing
the position of the other pyramids, is likely to be a sort of mean
between these two pyramids in his slope ratios.
Reasoning thus, with the addition of the knowledge I possessed of the
angular estimates of these slopes made by those who had visited the
ground, and a useful start for my ratios gained by the reduction of base
measures already known into R.B. cubits, giving 420 as a general base
for Cheops and Cephren at one level, and taking 210 cubits as the base
of Mycerinus (half the base of Cephren, as generally admitted), I had
something solid and substantial to go upon. I commenced with Mycerinus.
(_See Fig_. 71.)
_Fig. 71. (Mycerinus)_
LHNM represents the base of the pyramid. On the half-base AC I described
a 3, 4, 5 triangle ABC. I then projected the line CF = BC to be the
altitude of the pyramid. Thus I erected the triangle BFC, ratio of BC to
CF being 1 to 1. From this datum I arrived at the triangles BEA, ADC,
and GKH. GK, EA, and AD, each represent apothem of pyramid; CF, and CD,
altitude; and HK, edge.
The length of the line AD being √(AC² + CD²), the length of the line HK
being √(HG² + GK²), and line CH (half diagonal of base) being
√(CG² + GH²). These measures reduced to R.B. cubits, calling the line
AC = ratio 4 = 105 cubits, half-base of pyramid, give the following
results:--
R. B. BRITISH
CUBITS. FEET.
Half-base LA = 105·000 = 176·925
Apothem EA = 168·082 = 283·218
Edge HK = 198·183 = 333·937
Altitude CD = 131·250 = 221·156
Half diag. of base CH = 148·4924 = 250·209
and thus I acquired the ratios:--
Half-base : Altitude :: Apothem : Base.
= 20 : 25 :: 32 : 40 nearly.
To place the lines of the diagram in their actual solid position--Let
AB, BC, CA and HG be hinges attaching the planes AEB, BFC, CDA and HKG
to the base LHNM. Lift the plane BCF on its hinge till the point F is
vertical over the centre C. Lift plane CDA on its hinge, till point D is
vertical over the centre C; then will line CD touch CF, and become one
line. Now lift the plane AEB on its hinge, until point E is vertical
over the centre C, and plane HKG on its hinge till point K is vertical
over the centre C; then will points E, F, D and K, all meet at one point
above the centre C, and all the lines will be in their proper places.
The angle at the base of Mycerinus, if built to a ratio of 4 to 5
(half-base to altitude), and not to the more practical but nearly
perfect ratio of 32 to 20 (apothem to half-base) would be the complement
of angle ADC, thus--
4 165″
--- = ·8 = Tan. < ADC = 38° 39′ 35---
5 477
312″
∴ < DAC = 51° 20′ 24---
477
but as it is probable that the pyramid was built to the ratio of 32 to
20, I have shown its base angle in Figure 19, as 51° 19′ 4″.
Figure 72 shows how the slopes of _Cephren_ were arrived at.
_Fig. 72. (Cephren)_
LHNM represents the base of the pyramid. On the half-base AC, I
described a 3, 4, 5 triangle ABC. I then projected the line CF (ratio 21
to BC 20), thus erecting the 20, 21, 29 triangle BCF. From this datum, I
arrived at the triangles BEA, ADC, and GKH; GK, EA and AD each
representing apothem; CF and CD, altitude; and HK, edge. The lengths of
the lines AD, HK and CH being got at as in the pyramid Mycerinus. These
measures reduced to cubits, calling AC = ratio 16 = 210 cubits
(half-base of pyramid) give the following result.
R. B. BRITISH
CUBITS. FEET.
Half-base 210·00 353·85 = LA
Apothem 346·50 583·85 = EA
Edge 405·16 682·69 = HK
Altitude 275·625 464·43 = CD
Half-diag. of base 296·985 500·42 = CH
thus I get the ratios of--Apothem : Half-Base :: 33 : 20, &c. The planes
in the diagram are placed in their correct positions, as directed for
Figure 71.
The angle at the base of Cephren, if built to the ratio of 16 to 21
(half-base to altitude), and not to the practical ratio of 33 to 20
(apothem to half-base), would be the complement of < ADC, thus--
16 16″
-- = ·761904 = Tan. < ADC = 37° 18′ 14--
21 46
30″
∴ < DAC = 52° 41′ 45--
46
but as it is probable that the pyramid was built to the ratio of 33 to
20, I have marked the base angle in Fig. 17, as 52° 41′ 41″.