Chapter III: Part 3

I claim, therefore, that the theory I have set up, that the pyramids were the theodolites of the Egyptians, is sound. That the ground plan of these pyramids discloses a beautiful system of primary triangles and satellites I think I have shown beyond the shadow of a doubt; and that this system of geometric triangulation or right-angled trigonometry was the method practised, seems in the preceding pages to be fairly established. I claim, therefore, that I have discovered and described the main secret of the pyramids, that I have found for them at last a practical use, and that it is no longer "_a marvel how after the annual inundation, each property could have been accurately described by the aid of geometry._" I have advanced nothing in the shape of a theory that will not stand a practical test; but to do it, the pyramids should be _re-cased_. Iron sheeting, on iron or wooden framework, would answer. I may be wrong in some of my conclusions, but in the main I am satisfied that I am right. It must be admitted that I have worked under difficulties; a glimpse at the pyramids three and twenty years ago, and the meagre library of a nomad in the Australian wilderness having been all my advantages, and time at my disposal only that snatched from the rare intervals of leisure afforded by a arduous professional life.

After fruitless waiting for a chance of visiting Egypt and Europe, to 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″.

I took _Cheops_ out, first as a [Pi] pyramid, and made his lines to a base of 420 cubits, as follows--

Half-base 210
Altitude 267·380304
Apothem 339·988573 (_See Fig_. 73.)

_Fig. 73. (Cheops) _

But to produce the building ratio of 34 to 21, as per diagram Figure 6 or 9, I had to alter it to--

Half-base 210
Altitude 267·394839
Apothem 340°

Thus the theoretical angle of Cheops is 51° 51′ 14·3″, and the probable angle at which it was built, is 51° 51′ 20″, as per figure 15.

Cheops is therefore the mean or centre of a system--the slopes of Mycerinus being a little flatter, and those of Cephren a little steeper, Cheops coming fairly between the two, within about 10 minutes; and thus the connection between the ground plan of the group and the slopes of the three pyramids is exactly as one might expect after examination of Figure 3, 4 or 5.