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_With the respectful compliments of the author._

        Address,
           Mr. R. BALLARD,
                     Malvern,
                         ENGLAND.

    Illustration: Plumb Line over Pyramids




  THE SOLUTION OF THE PYRAMID PROBLEM
  OR, PYRAMID DISCOVERIES.

  WITH A NEW THEORY AS TO THEIR ANCIENT USE.

  BY ROBERT BALLARD, M. INST. C.E., ENGLAND; M. AMER, SOC. C.E.
  CHIEF ENGINEER OF THE CENTRAL AND NORTHERN RAILWAY DIVISION
  OF THE COLONY OF QUEENSLAND, AUSTRALIA.


  NEW YORK:
  JOHN WILEY & SONS.
  1882.




  Copyright,
  1882,
  By JOHN WILEY & SONS.


  PRESS OF J. J. LITTLE & CO.,
  NOS. 10 TO 20 ASTOR PLACE, NEW YORK.




NOTE.


In preparing this work for publication I have received valuable help
from the following friends in Queensland:--

     E. A. Delisser, L.S. and C.E., Bogantungan, who assisted me in my
     calculations, and furnished many useful suggestions.

     J. Brunton Stephens, Brisbane, who persuaded me to publish my
     theory, and who also undertook the work of correction for the
     press.

     J. A. Clarke, Artist, Brisbane, who contributed to the
     Illustrations.

     Lyne Brown, Emerald,--(photographs).

     F. Rothery, Emerald,--(models).

     and--A. W. Voysey, Emerald,--(maps and diagrams).




CONTENTS.


                                                                  PAGE

     §1. The Ground Plan of the Gïzeh Group                          9

         Plan Ratios connected into Natural Numbers                 13

     §2. The Original Cubit Measure of the Gïzeh Group              15

     §3. The Exact Measure of the Bases of the Pyramids             19

     §4. The <DW72>s, Ratios and Angles of the Three Principal
           Pyramids of the Gïzeh Group                              20

     §5. The Exact Dimensions of the Pyramids                       23

     §6. Geometrical Peculiarities of the Pyramids                  28

     §6A. The Casing Stones of the Pyramids                         32

     §7. Peculiarities of the Triangles 3, 4, 5 and 20, 21, 29      35

     §8. General Observations                                       38

     §9. The Pyramids of Egypt the Theodolites of the Egyptian
           Land Surveyors                                           41

     §10. How the Pyramids were made use of                         44

     §11. Description of the ancient Portable Survey Instrument     51

          Table to explain Figure 60                                64

     §12. Primary Triangles and their Satellites;--or the Ancient
            System of Right-angled Trigonometry unfolded by a
            Study of the Plan of the Pyramids of Gïzeh              64
          Table of some "Primary Triangles" and their Satellites    74

     §13. The Size and Shape of the Pyramids indicated by the
            Plan                                                    77

     §14. A Simple Instrument for laying off "Primary Triangles"    79

     §14A. General Observations                                     80

     §15. Primary Triangulation                                     82

     §16. The Pentangle or Five-pointed Star the Geometric Symbol
            of the Great Pyramid                                    91

          Table showing the comparative Measures of Lines           96

     §17. The manner in which the <DW72> Ratios of the Pyramids
            were arrived at                                        104




LIST OF WORKS CONSULTED.


     _Penny Cyclopædia. (Knight, London. 1833.)_

     _Sharpe's Egypt._

     _"Our Inheritance in the Great Pyramid." Piazzi Smyth._

     _"The Pyramids of Egypt." R. A. Proctor. (Article in Gentleman's
     Magazine. Feb. 1880.)_

     _"Traite de la Grandeur et de la Figure de la Terre." Cassini.
     (Amsterdam. 1723.)_

     _"Pyramid Facts and Fancies." J. Bonwick._


    Illustration: General Plan of Gïzeh Group




THE SOLUTION OF THE PYRAMID PROBLEM.


With the firm conviction that the Pyramids of Egypt were built and
employed, among other purposes, for one special, main, and important
purpose of the greatest utility and convenience, I find it necessary
before I can establish the theory I advance, to endeavor to determine
the proportions and measures of one of the principal groups. I take that
of Gïzeh as being the one affording most data, and as being probably one
of the most important groups.

I shall first try to set forth the results of my investigations into the
peculiarities of construction of the Gïzeh Group, and afterwards show
how the Pyramids were applied to the national work for which I believe
they were designed.


§ 1. THE GROUND PLAN OF THE GIZEH GROUP.

I find that the Pyramid Cheops is situated on the acute angle of a
right-angled triangle--sometimes called the Pythagorean, or Egyptian
triangle--of which base, perpendicular, and hypotenuse are to each other
as 3, 4, and 5. The Pyramid called Mycerinus, is situate on the greater
angle of this triangle, and the base of the triangle, measuring _three_,
is a line due east from Mycerinus, and joining perpendicular at a point
due south of Cheops. (_See Figure 1._)

    Fig. 1. Cheops, Mycerinus

I find that the Pyramid Cheops is also situate at the acute angle of a
right-angled triangle more beautiful than the so-called triangle of
Pythagoras, because more practically useful. I have named it the 20, 21,
29 triangle. Base, perpendicular, and hypotenuse are to each other as
twenty, twenty-one, and twenty-nine.

The Pyramid Cephren is situate on the greater angle of this triangle,
and base and perpendicular are as before described in the Pythagorean
triangle upon which Mycerinus is built. (_See Fig. 2._)

    Fig. 2. Cheops, Cephren

    Fig. 3. Cheops, Cephren, Mycerinus

_Figure 3_ represents the combination,--A being Cheops, F Cephren, and D
Mycerinus.

Lines DC, CA, and AD are to each other as 3, 4, and 5; and lines FB,
BA, and AF are to each other as 20, 21, and 29.

The line CB is to BA, as 8 to 7; the line FH is to DH, as 96 to 55; and
the line FB is to BC, as 5 to 6.

The Ratios of the first triangle multiplied by forty-five, of the second
multiplied by four, and the other three sets by twelve, one, and sixteen
respectively, produce the following connected lengths in natural numbers
for all the lines.

    DC      135
    CA      180
    AD      225
    -----------
    FB       80
    BA       84
    AF      116
    -----------
    CB       96
    BA       84
    -----------
    FH       96
    DH       55
    -----------
    FB       80
    BC       96

Figure 4 connects another pyramid of the group--it is the one to the
southward and eastward of Cheops.

In this connection, A Y Z A is a 3, 4, 5 triangle, and B Y Z O B is a
square.

    Lines YA to CA are as  1 to 5
          CY to YZ     as  3 to 1
          FO to ZO     as  8 to 3
    and   DA to AZ     as 15 to 4.

I may also point out on the same plan that calling the line FA radius,
and the lines BA and FB sine and co-sine, then is YA equal in length to
versed sine of angle AFB.

This connects the 20, 21, 29 triangle FAB with the 3, 4, 5 triangle AZY.

I have not sufficient data at my disposal to enable me to connect the
remaining eleven small pyramids to my satisfaction, and I consider the
four are sufficient for my purpose.

    _Fig. 4._

                                           _At level of_  _At level of_
                                            _Own base_   _Cephren's base_

    _Of these natural_   }   _Cheops_          56½       52½
    _numbers the bases_  } = _Cephren_         52½       52½
    _of the pyramids are_}   _Mycerinus_       26¼       27¼
    _as follows._        }


I now establish the following list of measurements of the plan in
connected natural numbers. (_See Figure 4._)


Plan Ratios connected into Natural Numbers.

    +------------+------------+------------+-------------+
    |BY  1}    48|BC  6}    96|DC  45}  135|FB   5}    80|
    |     } 48   |     } 16   |      }3    |      } 16   |
    |YZ  1}    48|FB  5}    80|BC  32}   96|BY   3}    48|
    +------------+------------+------------+-------------+
    |DN 61}   183|DN 61}   183|CY   3}  144|FH  96}    96|
    |     } 3    |     } 3    |      }   48|      } 1    |
    |NR 60}   180|NZ 48}   144|BC   2}   96|DH  55}    55|
    +------------+------------+------------+-------------+
    |CY 16}   144|PN 61} 146·4|JE   3}   72|YX   7}    63|
    |     } 9    |     } 2·4  |      } 24  |      } 9    |
    |DC 15}   135|PA 48} 115·2|EX   2}   48|AY   4}    36|
    +------------+------------+------------+-------------+
    |BA 21}    84|CA  4}   180|BC  32}   96|EA   7}   105|
    |     } 4    |     } 45   |      } 3   |      } 15   |
    |FB 20}    80|DC  3}   135|EB  21}   63|AZ   4}    60|
    +------------+------------+------------+-------------+
    |CB  8}    96|YZ  4}    48|FO  32}  128|AB   7}    84|
    |     } 12   |     } 12   |      } 4   |      } 12   |
    |BA  7}    84| AY 3}    36|OR  21}   84|BO   4}    48|
    +------------+------------+------------+-------------+
    |ED  8}   120|BA  4}    84|FT  84}   84|BC   8}    96|
    |     } 15   |     } 21   |      } 1   |      } 12   |
    |AE  7}   105|EB  3}    63|ST  55}   55|AC  15}   180|
    +------------+------------+------------+-------------+
    |VW 55}    55| GE 4}    96|VW  55}   55|ND  61}   183|
    |     } 1    |     } 24   |      } 1   |      } 3    |
    |FW 48}    48|DG  3}    72|SV  36}   36|NO  32}    96|
    +------------+------------+------------+-------------+
    |SJ  7}    84|HN  4}   128|BJ  45}  135|PA  48} 115·2|
    |     } 12   |     } 32   |      } 3   |      } 2·4  |
    |SU  6}    72|FH  3}    96|AB  28}   84|AZ  25}    60|
    +------------+------------+------------+-------------+
    |GX   2}  144|GU   5}  180|EO  37}  111|SR  61}   183|
    |      } 72  |      } 36  |      } 3   |      } 3    |
    |DG   1}   72|DG   2}   72|AY  12}   36|RZ  12}    36|
    +------------+------------+------------+-------------+
    |SU   2}   72|HW 144}  144|HT  36}  180|FH  96}    96|
    |      } 36  |      } 1   |      } 5   |      } 1    |
    |SV   1}   36|DH  55}   55|DH  11}   55|FE  17}    17|
    +------------+------------+------------+-------------+
    |TW  36}   36|FO   8}  128|DA  15}  225|EA 105}   105|
    |      } 1   |      } 16  |      } 15  |      } 1    |
    |TU  17}   17|OZ   3}   48|AZ   4}   60|EF  17}    17|
    +------------+------------+------------+-------------+
    |SR  61}  183|JB  45}  135|AC  15}  180|WH 144}   144|
    |       } 3  |      } 3   |      } 12  |      } 1    |
    |RO  28}   84|BY  16}   48|CN   4}   48|HG  17}    17|
    +------------+------------+------------+-------------+
    |YW  20}   80|FW  48}   48|YV  15}  135|TH 180}   180|
    |      } 4   |      } 1   |      } 9   |      } 1    |
    |AY   9}   36|FE  17}   17|AY   4}   36|HG  17}    17|
    +------------+------------+------------+-------------+
    |MY   9}  108|AC  20}  180|VZ  61}  183|             |
    |      } 12  |      } 9   |      } 3   |             |
    |ZY   4}   48|CG   7}   63|ZO  16}   48|             |
    +------------+------------+------------+-------------+
    |AC   9}  180|EA  35}  105|EU  84}   84|             |
    |      } 20  |      } 3   |      } 1   |             |
    |CH   4}   80|AY  12}   36|FE  17}   17|             |
    +------------+------------+------------+-------------+
    |NZ  12}  144|CY   3}  144|CA   5}  180|             |
    |      } 12  |      } 48  |      } 36  |             |
    |ZA   5}   60|YZ   1}   48|AY   1}   36|             |
    +------------+------------+------------+-------------+

    The above connected natural numbers multiplied by eight become
    R.B. cubits.               R.B.C.

            (Thus, BY  48 × 8 =  384
                   GX 144 × 8 = 1152).


§ 2. THE ORIGINAL CUBIT MEASURE OF THE GIZEH GROUP.

Mr. J. J. Wild, in his letter to Lord Brougham written in 1850, called
the base of Cephren seven seconds. I estimate the base of Cephren to be
just seven thirtieths of the line DA. The line DA is therefore thirty
seconds of the Earth's Polar circumference. The line DA is therefore
3033·118625 British feet, and the base of Cephren 707·727 British feet.

I applied a variety of Cubits but found none to work in _without
fractions_ on the beautiful set of natural dimensions which I had worked
out for my plan. (_See table of connected natural numbers._)

I ultimately arrived at a cubit as the ancient measure which I have
called the R.B. cubit, because it closely resembles the Royal Babylonian
Cubit of ·5131 metre, or 1·683399 British feet. The difference is 1/600
of a foot.

I arrived at the R.B. cubit in the following manner.

Taking the polar axis of the earth at five hundred million
geometric inches, thirty seconds of the circumference will be
36361·02608--geometric inches, or 36397·4235 British inches, at nine
hundred and ninety-nine to the thousand--and 3030·0855 geometric
feet, or 3033·118625 British feet. Now dividing a second into sixty
parts, there are 1800 R.B. cubits in the line DA; and the line DA
being thirty seconds, measures 36397·4235 British inches, which
divided by 1800 makes one of my cubits 20·2207908 British inches, or
1·685066 British feet. Similarly, 36361·02608 geometric inches
divided by 1800 makes my cubit 20·20057 geometric inches in length.
I have therefore defined this cubit as follows:--One R.B. cubit is
equal to 20.2006 geo. inches, 20·2208 Brit. inches, and 1·685 Brit.
feet.

I now construct the following table of measures.

    +---------+----------+--------+--------+--------+
    |  R. B.  |PLETHRA OR|        |        |        |
    | CUBITS. | SECONDS. |STADIA. |MINUTES.|DEGREES.|
    +---------+----------+--------+--------+--------+
    |    60   |     1    |        |        |        |
    |   360   |     6    |    1   |        |        |
    |  3600   |    60    |    10  |    1   |        |
    | 216000  |   3600   |   600  |    60  |     1  |
    |77760000 | 1296000  | 216000 |  21600 |    360 |
    +---------+----------+--------+--------+--------+

Thus there are seventy-seven million, seven hundred and sixty thousand
R.B. cubits, or two hundred and sixteen thousand stadia, to the Polar
circumference of the earth.

Thus we have obtained a perfect set of natural and convenient measures
which fits the plan, and fits the circumference of the earth.

And I claim for the R.B. cubit that it is the most perfect ancient
measure yet discovered, being the measure of the plan of the Pyramids of
Gïzeh.

The same forgotten wisdom which divided the circle into three hundred
and sixty degrees, the degree into sixty minutes, and the minute into
sixty seconds, subdivided those seconds, for earth measurements, into
the sixty parts represented by sixty R.B. cubits.

We are aware that thirds and fourths were used in ancient astronomical
calculations.

       *       *       *       *       *

The reader will now observe that the cubit measures of the main
Pythagorean triangle of the plan are obtained by multiplying the
original 3, 4 and 5 by 360; and that the entire dimensions are obtained
in R.B. cubits by multiplying the last column of connected natural
numbers in the table by eight,--thus--

                   R. B.
                  CUBITS.
    DC   3 × 360 = 1080
    CA   4 × 360 = 1440
    DA   5 × 360 = 1800

            or,


       NATURAL    R. B.
       NUMBERS.   CUBITS.
    DC   135 × 8 = 1080
    CA   180 × 8 = 1440
    DA   225 × 8 = 1800
                   &c., &c.

            (_See Figure 5, p. 18._)

According to Cassini, a degree was 600 stadia, a minute 10 stadia; and a
modern Italian mile, in the year 1723, was equal to one and a quarter
ancient Roman miles; and one and a quarter ancient Roman miles were
equal to ten stadia or one minute. (_Cassini, Traite de la grandeur et
de la Figure de la Terre. Amsterdam, 1723._)

    _Fig. 5_

       _R.B.Cubits._

    FB =  640       EB =  504
    BA =  672       BA =  672
    AF =  928       AE =  840

    DC = 1080       NO =  768
    CA = 1440       OF = 1024
    AD = 1800       FN = 1280

    DG =  576       AY =  288
    GE =  768       ZY =  384
    ED =  960       ZA =  480


                                                _R.B. Cub._

    _At level of Cephren's_  } {_Cheops' Base_      420
    _Base which is the plane_} {_Cephren's Base_    420
    _or level of the plan_.--} {_Mycerinus' Base_   218


Dufeu also made a stadium the six hundredth part of a degree. He made
the degree 110827·68 metres, which multiplied by 3·280841 gives
363607·996+ British feet; and 363607·996+ divided by 600 equals
606·013327 feet to his stadium.

I make the stadium 606·62376 British feet.

There being 360 cubits to a stadium, Dufeu's stadium divided by 360,
gives 1·6833 British feet, which is the exact measure given for a Royal
Babylonian Cubit, if reduced to metres, viz.: 0·5131 of a metre, and
therefore probably the origin of the measure called the Royal Babylonian
cubit. According to this measure, the Gïzeh plan would be about 1/1011
smaller than if measured by R.B. cubits.


§ 3. THE EXACT MEASURE OF THE BASES OF THE PYRAMIDS.

A stadium being 360 R.B. cubits, or six seconds--and a plethron 60 R.B.
cubits, or one second, the base of the Pyramid Cephren is seven plethra,
or a stadium and a plethron, equal to seven seconds, or four hundred and
twenty R.B. cubits.

Mycerinus' base is acknowledged to be half the base of Cephren.

Piazzi Smyth makes the base of the Pyramid Cheops 9131·05 pyramid (or
geometric) inches, which divided by 20·2006 gives 452·01 R.B. cubits. I
call it 452 cubits, and accept it as the measure which exactly fits the
plan.

I have not sufficient data to determine the exact base of the other and
smaller pyramid which I have marked on my plan.

The bases, then, of Mycerinus, Cephren, and Cheops, are 210, 420 and 452
cubits, respectively.

But in plan the bases should be reduced to one level. I have therefore
drawn my plan, or horizontal section, at the level or plane of the base
of Cephren, at which level or plane the bases or horizontal sections of
the pyramids are--Mycerinus, 218 cubits, Cephren, 420 cubits, and
Cheops, 420 cubits. I shall show how I arrive at this by-and-by, and
shall also show that the horizontal section of Cheops, corresponding to
the horizontal section of Cephren at the level of Cephren's base,
occurs, as it should do, at the level of one of the courses of masonry,
viz.--the top of the tenth course.


§ 4. THE <DW72>s, RATIOS, AND ANGLES OF THE THREE PRINCIPAL PYRAMIDS OF
THE GIZEH GROUP.

Before entering on the description of the exact <DW72>s and angles of the
three principal pyramids, I must premise that I was guided to my
conclusions by making full use of the combined evolutions of the two
wonderful right-angled triangles, 3, 4, 5, and 20, 21, 29, which seem to
run through the whole design as a sort of dominant.

From the first I was firmly convinced that in such skilful workmanship
some very simple and easily applied templates must have been employed,
and so it turned out. Builders do not mark a dimension on a plan which
they cannot measure, nor have a hidden measure of any importance without
some clear outer way of establishing it.

This made me "go straight" for the slant ratios. When the Pyramids were
cased from top to bottom with polished marble, there were only two
feasible measures, the bases and the apothems;[1] and for that reason I
conjectured that these would be the definite plan ratios.

    Footnote 1: The "_Apothem_ is a perpendicular from the vertex of a
    pyramid on a side of the base."--_Chambers' Practical Mathematics, p.
    156._

       *       *       *       *       *

Figures 6, 7 and 8 show the _exact_ <DW72> ratios of Cheops, Cephren, and
Mycerinus, measured as shown on the diagrams--viz., Cheops, 21 to 34,
Cephren, 20 to 33, and Mycerinus, 20 to 32--that is, half base to
apothem.

    Fig. 6 Cheops.            _Note. The Ratios of Bases
                                      to Altitudes are very
    Fig. 7 Cephren.                   nearly as follows, viz_:--

    Fig. 8 Mycerinus.         _Cheops     33 to 21 or 330 to 210_
                              _Cephren    32 to 21 or 320 to 210_
                              _Mycerinus  32 to 20 or 336 to 210_

The ratios of base to altitude are, Cheops, 33 to 21, Cephren, 32 to 21,
and Mycerinus, 32 to 20: not exactly, but near enough for all practical
purposes. For the sake of comparison, it will be well to call these
ratios 330 to 210, 320 to 210, and 336 to 210, respectively.

       *       *       *       *       *

    Fig. 9. Cheops.      Fig. 10. Cheops.

Figures 9 and 10 are meridional and diagonal sections, showing ratios
of Cheops, viz., half base to apothem, 21 to 34 _exactly_; half base to
altitude, 5½ to 7 nearly, and 183 to 233, nearer still (being the
ratio of Piazzi Smyth). The ratio of Sir F. James, half diagonal 10 to
altitude 9 is also very nearly correct.

My altitude for Cheops is 484·887 British feet, and the half base 380·81
British feet.

The ratio of 7 to 5½ gives 484·66, and the ratio of 233 to 183 gives
484·85 for the altitude.

My half diagonal is 538·5465, and ratio 10 to 9, gives 484·69 British
feet for the altitude.

I have mentioned the above to show how very nearly these ratios agree
with my exact ratio of 21 to 34 half base to apothem.

       *       *       *       *       *

    Fig. 11. Cephren.      Fig. 12. Cephren.

Figures 11 and 12 show the ratios of Cephren, viz., half base to
apothem, _20 to 33 exactly_, and half base, altitude, and apothem
respectively, as 80, 105, and 132, very nearly.

Also half diagonal, altitude, and edge, practically as 431, 400, and
588.

       *       *       *       *       *

    Fig. 13. Mycerinus.      Fig. 14. Mycerinus.

Figures 13 and 14 show the ratios of Mycerinus, viz., half base to
apothem, 20 _to_ 32 _exactly_, and half base, altitude, and apothem
respectively, as 20, 25, and 32 very nearly.

Also full diagonal to edge as 297 to 198, nearly. A peculiarity of this
pyramid is, that base is to altitude as apothem is to half base. Thus,
40 : 25 :: 32 : 20; that is, half base is a fourth proportional to base,
apothem, and altitude.


§ 5. THE EXACT DIMENSIONS OF THE PYRAMIDS.

                                 R.B. Cub.  Brit. Ft.
    Fig. 15. Cheops.              452      = 761·62
                                  287·767  = 484·887
    Fig. 16. Cheops.              365·9047 = 616·549
                                  430·058  = 724·647
                                  639·2244 = 1077·093

Figures 15 to 20 inclusive, show the linear dimensions of the three
pyramids, also their angles. The base angles are, Cheops, 51° 51′ 20";
Cephren, 52° 41′ 41″; and Mycerinus, 51° 19′ 4″.

                                R.B. Cub.  Brit. Ft.
    Fig. 17. Cephren.            420      = 707·70
                                 275·61   = 464·40
    Fig. 18. Cephren.            346·50   = 583·85
                                 405·16   = 682·69
                                 593·97   = 1000·84

                                R.B. Cub.  Brit. Ft.

    Fig. 19. Mycerinus.          210      = 353.85
                                 168      = 283.08
    Fig. 20. Mycerinus:          131.14   = 220.97
                                 198.10   = 333·7985
                                 296·9848 = 500·42

In Cheops, my dimensions agree with Piazzi Smyth--in the base of
Cephren, with Vyse and Perring--in the height of Cephren, with Sir
Gardner Wilkinson, nearly--in the base of Mycerinus, they agree with the
usually accepted measures, and in the height of Mycerinus, they exceed
Jas. J. Wild's measure, by not quite one of my cubits.

In my angles I agree very nearly with Piazzi Smyth, for Cheops, and with
Agnew, for Cephren, differing about half a degree from Agnew, for
Mycerinus, who took this pyramid to represent the same relation of [Pi]
that P. Smyth ascribes to Cheops (viz.: 51° 51′ 14·″3), while he gave
Cheops about the same angle which I ascribe to Mycerinus.

I shall now show how I make Cephren and Cheops of equal bases of 420
R.B. cubits at the same level, viz.--that of Cephren's base.

John James Wild made the bases of Cheops, Cephren, and Mycerinus,
respectively, 80, 100, and 104·90 cubits above some point that he called
Nile Level.

His cubit was, I believe, the Memphis, or Nilometric cubit--but at any
rate, he made the base of Cephren 412 of them.

I therefore divided the recognized base of Cephren--viz., 707·75 British
feet--by 412, and got a result of 1·7178 British feet for his cubit.
Therefore, his measures multiplied by 1·7178 and divided by 1·685 will
turn his cubits into R.B. cubits.

I thus make Cheops, Cephren, and Mycerinus, respectively, 81·56,
101·93, and 106·93 R.B. cubits above the datum that J. J. Wild calls
Nile Level. According to Bonwick's "Facts and Fancies," p. 31, high
water Nile would be 138½ ft. below base of Cheops (or 82·19 R.B.
cubits).

Piazzi Smyth makes the pavement of Cheops 1752 British inches (or 86·64
R.B. cubits) above _average Nile Level_, but, by scaling his map, his
_high Nile Level_ appears to agree nearly with Wild.

It is the _relative levels_ of the Pyramids, however, that I require, no
matter how much above Nile Level.

Cephren's base of 420 cubits being 101·93 cubits, and Cheops' base of
452 cubits being 81·56 cubits above Wild's datum, the difference in
level of their bases is, 20·37 cubits.

The ratio of base to altitude of Cheops being 330 to 210, therefore
20·37 cubits divided by 210 and multiplied by 330 equals 32 cubits; and
452 cubits minus 32 cubits, equals 420.

Similarly, the base of Mycerinus is 5 cubits _above_ the base of
Cephren, and the ratio of base to altitude 32 to 20; therefore, 5 cubits
divided by 20 and multiplied by 32 equals 8 cubits to be _added_ to the
210 cubit base of Mycerinus, making it 218 cubits in breadth at the
level of Cephren's base.

Thus, a horizontal section or plan at the level of Cephren's base would
meet the <DW72>s of the Pyramids so that they would on plan appear as
squares with sides equal to 218, 420, and 420 R.B. cubits, for
Mycerinus, Cephren, and Cheops, respectively.

    Fig. 21.
                                       R.B. Cub.
    Apex of Cephren above Base Cheops    295·98
    Apex of Cheops above Base Cheops     287·77
    Apex of Mycerinus above Base Cheops  156·51
    Base Cephren above base of Cheops     20·37
    Base Mycerinus above base of Cheops   25·37

Piazzi Smyth makes the top of the tenth course of Cheops 414 pyramid
inches above the pavement; and 414 divided by 20·2006 equals 20·49 R.B.
cubits.

But I have already proved that Cheops' 420 cubit base measure occurs at
a level of 20·37 cubits above pavement; therefore is this level the
level of the top of the tenth course, for the difference is only 0·12
R.B. cubits, or 2½ inches.

       *       *       *       *       *

I wish here to note as a matter of interest, but not as affecting my
theory, the following measures of Piazzi Smyth, turned into R.B. cubits,
viz.:--

                                        PYR. INCHES.  R.B. CUBITS.
    King's Chamber floor, above pavement    1702·    =  84·25
    Cheops' Base, as before stated          9131·05  = 452·01
    King's Chamber, "True Length,"           412·132 =  20·40
      "       "     "True First Height,"     230·389 =  11·40
      "       "     "True Breadth,"          206·066 =  10·20

He makes the present summit platform of Cheops 5445 pyramid inches above
pavement. My calculation of 269·80 R.B. cub. (See Fig. 21) is equal to
5450 pyramid inches--this is about 18 cubits below the theoretical apex.

       *       *       *       *       *

Figure 21 represents the comparative levels and dimensions of Mycerinus,
Cephren, and Cheops.

The following peculiarities are noticeable:--That Cheops and Cephren are
of equal bases at the level of Cephren's base;--that, at the level of
Cheops' base, the latter is only half a cubit larger;--that, from the
level of Mycerinus' base, Cheops is just double the height of
Mycerinus;--and that from the level of Cephren's base, Cephren is just
double the height of Mycerinus; measuring in the latter case, however,
only up to the level platform at the summit of Cephren, which is said to
be about eight feet wide.

The present summit of Cephren is 23·07 cubits above the present summit
of Cheops, and the completed apex of Cephren would be 8·21 cubits above
the completed apex of Cheops.

In the summit platforms I have been guided by P. Smyth's estimate of
_height deficient_, 363 pyr. inches, for Cheops, and I have taken 8 feet
base for Cephren's summit platform.


§ 6. GEOMETRICAL PECULIARITIES OF THE PYRAMIDS.

_In any pyramid, the apothem is to half the base as the area of the four
sides is to the area of the base._

    Thus--Ratio apothem to half base Mycerinus   32 to 20
            "     "         "    "   Cephren     33 to 20
            "     "         "    "   Cheops      34 to 21

              AREA OF THE FOUR SIDES.  AREA OF THE BASE.

    Mycerinus      70560·                   44100
    Cephren       291060·                  176400
    Cheops        330777·90                204304

              All in R.B. cubits.

    Therefore--32  :  20  : :   70560·    :  44100
               33  :  20  : :  291060·    : 176400
               34  :  21  : :  330777·90  : 204304

[2]Herodotus states that "_the area of each of the four faces of Cheops
was equal to the area of a square whose base was the altitude of a
Pyramid;_" or, in other words, that altitude was a mean proportional to
apothem and half base; thus--area of one face equals the fourth of
330777·90 or 82694·475 R.B. cubits, and the square root of 82694·475 is
287·56. But the correct altitude is 287·77, so the error is 0·21, or
4¼ British inches. I have therefore the authority of Herodotus to
support the theory which I shall subsequently set forth, that this
pyramid was the exponent of lines divided in mean and extreme ratio.

By taking the dimensions of the Pyramid from what I may call its
_working level_, that is, the level of the base of Cephren, this
peculiarity shows more clearly, as also others to which I shall refer.
Thus--base of Cheops at working level, 420 cubits, and apothem 340
cubits; base area is, therefore, 176400 cubits, and area of one face is
(420 cubits, multiplied by half apothem, or 170 cubits) 71400 cubits.
Now the square root of 71400 would give altitude, or side of square
equal to altitude, 267·207784 cubits: but the real altitude is
√(340²-210²) = √71500 = 267·394839. So that the error of Herodotus's
proposition is the difference between √714 and √715.

    Footnote 2: Proctor is responsible for this statement, as I am
    quoting from an essay of his in the _Gentleman's Magazine_. R. B.

This leads to a consideration of the properties of the angle formed by
the ratio _apothem_ 34 to _half base_ 21, peculiar to the pyramid
Cheops. (_See Figure 22._)

Fig. 22. Diagram illustrating relations of ratios of the pyramid Cheops.

Calling apothem 34, _radius_; and half base 21, _sine_--I find that--

    Radius is the square root of   1156
    Sine                            441
    Co-sine                         715
    Tangent                         713
    Secant                         1869
    and Co-versed-sine              169

So it follows that the area of one of the faces, 714, is a mean between
the square of the altitude or co-sine, 715, and the square of the
tangent, 713.

Thus the reader will notice that the peculiarities of the Pyramid
Cheops lie in the regular relations of the _squares_ of its various
lines; while the peculiarities of the other two pyramids lie in the
relations of the lines themselves.

Mycerinus and Cephren, born, as one may say, of those two noble
triangles 3, 4, 5, and 20, 21, 29, exhibit in their lineal developments
ratios so nearly perfect that, for all practical purposes, they may be
called correct.

    Thus--Mycerinus, [3]20² +  25² =   1025, and  32² =   1024.
    and Cephren,     [4]80² + 105² =  17425, and 132² =  17424.
    or              [5]400² + 431² = 345761, and 588² = 345744.

See diagrams, Figures 11 to 14 inclusive.

In the Pyramid Cheops, altitude is _very nearly_ a mean proportional
between apothem and half base. Apothem being 34, and half base 21, then
altitude would be √(34²-21²) = √715 = 26·7394839, and--

    21 : 26·7394839 :: 26·7394839 : 34, nearly.

Here, of course, the same difference comes in as occurred in
considering the assumption of Herodotus, viz., the difference
between √715 and √714; because if the altitude were √714, then would
it be _exactly_ a mean proportional between the half base and the
apothem; (thus, 21 : 26·72077 :: 26·72077 :: 34.)

    Footnote 3: Half base to altitude.

    Footnote 4: Half base to altitude.

    Footnote 5: Half diagonal of base to altitude.

In Cheops, the ratios of apothem, half base and edge are, 34, 21, and
40, very nearly, thus, 34² + 21² = 1597, and 40² = 1600.

The dimensions of Cheops (from the level of the base of Cephren) to be
what Piazzi Smyth calls a [Pi] pyramid, would be--

    Half base 210              R.B. cubits.
    Altitude  267·380304, &c.      "
    Apothem   339·988573, &c.      "

Altitude being to perimeter of base, as radius of a circle to
circumference.

My dimensions of the pyramid therefore in which--

    Half base = 210             R.B. cubits.
    Altitude  = 267·394839 &c.      "
    Apothem   = 340                 "

come about as near to the ratio of [Pi] as it is possible to come, and
provide simple lines and templates to the workmen in constructing the
building; and I entertain no doubt that on the simple lines and
templates that my ratios provide, were these three pyramids built.


§ 6A. THE CASING STONES OF THE PYRAMIDS.

Figures 23, 24, and 25, represent ordinary casing stones of the three
pyramids, and Figures 26, 27, and 28, represent angle or quoin casing
stones of the same.

The casing stone of Cheops, found by Colonel Vyse, is represented in
Bonwick's "Pyramid Facts and Fancies," page 16, as measuring four feet
three inches at the top, eight feet three inches at the bottom, four
feet eleven inches at the back, and six feet three inches at the
front. Taking four feet eleven inches as _Radius_, and six feet three
inches as _Secant_, then the _Tangent_ is three feet ten inches and
three tenths.

    Fig. 23. Cheops Casing Stone.

    Fig. 24. Cephren Casing Stone.

    Fig. 25. Mycerinus Casing Stone.

    Fig. 26. Cheops Angle or Quoin Stone.

    Fig. 27. Cephren Angle or Quoin Stone.

    Fig. 28. Mycerinus Angle or Quoin Stone.

Thus, in inches (√(75²-59²)) = 46·30 inches; therefore the inclination
of the stone must have been--slant height 75 inches to 46·30 inches
horizontal. Now, 46·30 is to 75, as 21 is to 34. Therefore, Col. Vyse's
casing stone agrees exactly with my ratio for the Pyramid Cheops, viz.,
21 to 34. (_See Figure 29._)

    Fig. 29. =Col. Vyse's Casing Stone.= 75 : 46·3 :: 34 : 21

This stone must have been out of plumb at the back an inch and seven
tenths; perhaps to give room for grouting the back joint of the marble
casing stone to the limestone body of the work: or, because, as it is
not a necessity in good masonry that the back of a stone should be
exactly plumb, so long as the error is on the right side, the builders
might not have been particular in that respect.

    Fig. 59. (Temple of Cheops, standing at angle of wall.)

Figure 59 represents such a template as the masons would have used in
building Cheops, both for dressing and setting the stones. (The courses
are drawn out of proportion to the template.) The other pyramids must
have been built by the aid of similar templates.

Such large blocks of stone as were used in the casing of these pyramids
could not have been completely dressed before setting; the back and
ends, and the top and bottom beds were probably dressed off truly, and
the face roughly scabbled off; but the true <DW72> angle could not have
been dressed off until the stone had been truly set and bedded,
otherwise there would have been great danger to the sharp arises.

       *       *       *       *       *

I shall now record the peculiarities of the 3, 4, 5 or Pythagorean
triangle, and the right-angled triangle 20, 21, 29.


§ 7. PECULIARITIES OF THE TRIANGLES 3, 4, 5, AND 20, 21, 29.

    Fig. 30 to 35. PECULIARITIES OF THE TRIANGLES

The 3, 4, 5 triangle contains 36° 52′ 11·65″ and the complement or
greater angle 53° 7′ 48·35″

    Radius          5   =  60  whole numbers.[6]
    Co-sine         4   =  48"
    Sine            3   =  36"
    Versed sine     1   =  12"
    Co-versed sine  2   =  24"
    Tangent         3¾  =  45"
    Secant          6¼  =  75"
    Co-tangent      6⅔  =  80"
    Co-secant       8⅓  = 100"

    Tangent + Secant = Diameter or 2 Radius
    Co-tan + Co-sec = 3 Radius
    Sine : Versed-sine :: 3 : 1
    Co-sine : Co-versed sine :: 2 : 1

Figure 30 illustrates the preceding description. Figure 31 shows the 3·1
triangle, and the 2·1 triangle built up on the sine and co-sine of the
3, 4, 5 triangle.

The 3·1 triangle contains 18° 26′ 5·82″ and the 2·1 triangle
26° 33′ 54·19″; the latter has been frequently noticed as a
pyramid angle in the gallery inclinations.

Figure 32 shows these two triangles combined with the 3, 4, 5 triangle,
on the circumference of a circle.

    Footnote 6: 60 = 3 × 4 × 5

The 20, 21, 29 triangle contains 43° 36′ 10·15″ and the
complement, 46° 23′ 49·85″.

Expressed in whole numbers--

    Radius         29  =  12180[7]
    Sine           20  =   8400
    Co-sine        21  =   8820
    Versed sine     8  =   3360
    Co-versed sine  9  =   3780
    Tangent            =  11600
    Co-tangent         =  12789
    Secant             =  16820
    Co-sec             =  17661

    Tangent + Secant = 2⅓ radius
    Co-tan  + Co-sec = 2½ radius
    Sine : Versed sine :: 5 : 2
    Co-sine : Co-versed sine :: 7 : 3

    Footnote 7: 12180 = 20 × 21 × 29

It is noticeable that while the multiplier required to bring radius 5
and the rest into whole numbers, for the 3, 4, 5 triangle is twelve, in
the 20, 21, 29 triangle it is 420, the key measure for the bases of the
two main pyramids in R.B. cubits.[8]

    Footnote 8: 12 = 3 × 4, and 420 = 20 × 21

I am led to believe from study of the plan, and consideration of the
whole numbers in this 20, 21, 29 triangle, that the R.B. cubit, like the
Memphis cubit, was divided into 280 parts.

The whole numbers of radius, sine, and co-sine divided by 280, give a
very pretty measure and series in R.B. cubits, viz., 43½, 30, and
31½, or 87, 60, and 63, or 174, 120 and 126;--all exceedingly useful
in right-angled measurements. Notice that the right-angled triangle 174,
120, 126, in the sum of its sides _amounts to_ 420.

Figure 33 illustrates the 20, 21, 29 triangle. Figure 34 shows the 5·2
and 7·3 triangles built up on the sine and co-sine of the 20, 21, 29
triangle.

The 5·2 triangle contains 21° 48′ 5·08″ and the 7·3 triangle
23° 11′ 54·98″.

Figure 35 shows how these two triangles are combined with the 20, 21, 29
triangle on the circumference, and Figure 36 gives a general view and
identification of these six triangles which occupied an important
position in the trigonometry of a people who did all their work by right
angles and proportional lines.

    Fig. 36. Ratios of Leading Triangles.


§ 8. GENERAL OBSERVATIONS.

It must be admitted that in the details of the building of the Pyramids
of Gïzeh there are traces of other measures than R. B. cubits, but that
the original cubit of the plan was 1·685 British feet I feel no doubt.
It is a perfect and beautiful measure, fit for such a noble design, and,
representing as it does the sixtieth part of a second of the Earth's
polar circumference, it is and was a measure for all time.

It may be objected that these ancient geometricians could not have been
aware of the measure of the Earth's circumference; and wisely so, were
it not for two distinct answers that arise. The first being, that since
I think I have shown that Pythagoras never discovered the Pythagorean
triangle, but that it must have been known and practically employed
thousands of years before his era, in the Egyptian Colleges where he
obtained his M.A. degree, so in the same way it is probable that
Eratosthenes, when he went to work to prove that the earth's
circumference was fifty times the distance from Syene to Alexandria, may
have obtained the idea from his ready access to the ill-fated
Alexandrian Library, in which perhaps some record of the learning of the
builders of the Pyramids was stored. And therefore I claim that there is
no reason why the pyramid builders should not have known as much about
the circumference of the earth as the modern world that has calmly stood
by in its ignorance and permitted those magnificent and, as I shall
prove, useful edifices to be stripped of their beautiful garments of
polished marble.

My second answer is that the correct cubit measure may have been got by
its inventors in a variety of other ways; for instance, by observations
of shadows of heavenly bodies, without any knowledge even that the earth
was round; or it may have been evolved like the British inch, which Sir
John Herschel tells us is within a thousandth part of being one five
hundred millionth of the earth's polar axis. I doubt if the
circumference of the earth was considered by the inventor of the British
inch.

It was a peculiarity of the Hindoo mathematicians that they tried to
make out that all they knew was _very old_. Modern savants appear to
take the opposite stand for any little information they happen to
possess.

The cubit which is called the Royal Babylonian cubit and stated to
measure O·5131 metre, differs so slightly from my cubit, only the
six-hundredth part of a foot, that it may fairly be said to be the same
cubit, and it will be for antiquaries to trace the connection, as this
may throw some light on the identity of the builders of the Pyramids of
Gïzeh. Few good English two-foot rules agree better than these two
cubits do.

While I was groping about in the dark searching for this bright needle,
I tried on the plan many likely ancient measures.

For a long time I worked in Memphis or Nilometric cubits, which I made
1·7126 British feet; they seem to vary from 1·70 to 1·72, and although I
made good use of them in identifying other people's measures, still they
were evidently not in accordance with the design; but the R.B. cubit of
1·685 British feet works as truly into the plan of the Pyramids _without
fractions_ as it does into the circumference of the earth.

Here I might, to prevent others from falling into one of my errors,
point out a rock on which I was aground for a long time. I took the base
of the Pyramid Cheops, determined by Piazzi Smyth, from Bonwick's
"Pyramid Facts and Fancies" (a valuable little reference book), as
763.81 British feet, and the altitude as 486.2567; and then from Piazzi
Smyth's "Inheritance," page 27, I confirmed these figures, and so worked
on them for a long time, but found always a great flaw in my work, and
at last adopted a fresh base for Cheops, feeling sure that Mr. Smyth's
base was wrong: for I was absolutely grounded in my conviction that at a
certain level, Cheops' and Cephren's measures bore certain relations to
each other. I subsequently found in another part of Mr. Smyth's book,
that the correct measures were 761.65 and 484.91 British feet for base
and altitude, which were exactly what I wanted, and enabled me to be in
accordance with him in that pyramid which he appears to have made his
particular study.

For the information of those who may wish to compare my measures, which
are the results of an even or regular circumference without fractions,
with Mr. Smyth's measures, which are the results of an even or regular
diameter without fractions, it may be well to state that there are just
about 99 R.B. cubits in 80 of Piazzi Smyth's cubits of 25 pyramid inches
each.


§ 9. THE PYRAMIDS OF EGYPT, THE THEODOLITES OF THE EGYPTIAN LAND
SURVEYORS.

About twenty-three years ago, on my road to Australia, I was crossing
from Alexandria to Cairo, and saw the pyramids of Gïzeh.

I watched them carefully as the train passed along, noticed their clear
cut lines against the sky, and their constantly changing relative
position.

I then felt a strong conviction that they were built for at least one
useful purpose, and that purpose was the survey of the country. I said,
"Here be the Theodolites of the Egyptians."

Built by scientific men, well versed in geometry, but unacquainted with
the use of glass lenses, these great stone monuments are so suited in
shape for the purposes of land surveying, that the practical engineer or
surveyor must, after consideration, admit that they may have been built
mainly for that purpose.

Not only might the country have been surveyed by these great
instruments, and the land allotted at periodical times to the people;
but they, remaining always in one position, were there to correct and
readjust boundaries destroyed or confused by the annual inundations of
the Nile.

The Pyramids of Egypt may be considered as a great system of landmarks
for the establishment and easy readjustment at any time of the
boundaries of the holdings of the people.

The Pyramids of Gïzeh appear to have been main marks; and those of
Abousir, Sakkârah, Dashow, Lisht, Meydoun, &c., with the great pyramids
in Lake Moeris, subordinate marks, in this system, which was probably
extended from Chaldea through Egypt into Ethiopia.

The pyramid builders may perhaps have made the entombment of their Kings
one of their exoteric objects, playing on the morbid vanity of their
rulers to induce them to the work, but in the minds of the builders
before ever they built must have been planted the intention to make use
of the structures for the purposes of land surveying.

The land of Egypt was valuable and maintained a dense population; every
year it was mostly submerged, and the boundaries destroyed or confused.
Every soldier had six to twelve acres of land; the priests had their
slice of the land too; after every war a reallotment of the lands must
have taken place, perhaps every year.

While the water was lying on the land, it so softened the ground that
the stone boundary marks must have required frequent readjustment, as
they would have been likely to fall on one side.

By the aid of their great stone theodolites, the surveyors, who
belonged to the priestly order, were able to readjust the boundaries
with great precision. That all science was comprised in their secret
mysteries may be one reason why no hieroglyphic record of the scientific
uses of the pyramids remains. It is possible that at the time of
Diodorus and Herodotus, (and even when Pythagoras visited Egypt,)
theology may have so smothered science, that the uses of the pyramids
may have been forgotten by the very priests to whom in former times the
knowledge belonged; but "a respectful reticence" which has been noticed
in some of these old writers on pyramid and other priestly matters would
rather lead us to believe that an initiation into the mysteries may have
sealed their lips on subjects about which they might otherwise have been
more explicit.

The "_closing_" of one pyramid over another in bringing any of their
many lines into true order, must even now be very perfect;--but now we
can only imagine the beauties of these great instrumental wonders of the
world when the casing stones were on them. We can picture the rosy
lights of one, and the bright white lights of others; their clear cut
lines against the sky, true as the hairs of a theodolite; and the sombre
darkness of the contrasting shades, bringing out their angles with
startling distinctness. Under the influence of the Eastern sun, the
faces must have been a very blaze of light, and could have been seen at
enormous distances like great mirrors.

I declare that the pyramids of Gïzeh in all their polished glory,
before the destroyer stripped them of their beautiful garments, were in
every respect adapted to flash around clearly defined lines of sight,
upon which the lands of the nation could be accurately threaded. The
very thought of these mighty theodolites of the old Egyptians fills me
with wonder and reverence. What perfect and beautiful instruments they
were! never out of adjustment, always correct, always ready; no magnetic
deviation to allow for. No wonder they took the trouble they did to
build them so correctly in their so marvellously suitable positions.

If Astronomers agree that observations of a pole star could have been
accurately made by peering up a small gallery on the north side of one
of the pyramids only a few hundred feet in length, I feel that I shall
have little difficulty in satisfying them that accurate measurements to
points only _miles_ away could have been made from angular observations
of the whole group.


§ 10. HOW THE PYRAMIDS WERE MADE USE OF.

It appears from what I have already set forth that the plan of the
Pyramids under consideration is geometrically exact, a perfect set of
measures.

I shall now show how these edifices were applied to a thoroughly
geometrical purpose in the true meaning of the word--to measure the
Earth.

I shall show how true straight lines could be extended from the Pyramids
in given directions useful in right-angled trigonometry, by direct
observation of the buildings, and without the aid of other instruments.

And I shall show how by the aid of a simple instrument angles could be
exactly observed from any point.

This Survey theory does not stand or fall on the merits of my theory of
the Gïzeh plan. Let it be proved that this group is not built on the
exact system of triangulation set forth by me, it is still a fact that
its plan is in a similar shape, and any such shape would enable a
surveyor acquainted with the plan to lay down accurate surveys by
observations of the group even should it not occupy the precise lines
assumed by me.

And here I must state that although the lines of the plan as laid down
herein agree nearly with the lines as laid down in Piazzi Smyth's book,
in the Penny Cyclopædia, and in an essay of Proctor's in the
_Gentleman's Magazine_, still I find that they do not agree at all
satisfactorily with a map of the Pyramids in Sharp's "Egypt," said to be
copied from Wilkinson's map.

We will, however, for the time, and to explain my survey theory, suppose
the plan theory to be correct, as I firmly believe it is.

And then, supposing it may be proved that the respective positions of
the pyramids are slightly different to those that I have allotted to
them on my plan, it will only make a similar slight difference to the
lines and angles which I shall here show could be laid out by their aid.

Let us in the first place comprehend clearly the shape of the land of
Egypt.

A sector or fan, with a long handle--the fan or sector, the delta; and
the handle of the fan, the Nile Valley, running nearly due south.

The Pyramids of Gïzeh are situate at the angle of the sector, on a rocky
eminence whence they can all be seen for many miles. The summits of the
two high ones can be seen from the delta, and from the Nile Valley to a
very great distance; how far, I am unable to say; but I should think
that while the group could be made general use of for a radius of
fifteen miles, the summits of Cephren and Cheops could be made use of
for a distance of thirty miles; taking into consideration the general
fall of the country.

It must be admitted that if meridian observations of the star Alpha of
the Dragon could be made with accuracy by peeping up a small hole in one
of the pyramids, then surely might the surveyors have carried true north
and south lines up the Nile Valley as far as the summit of Cheops was
visible, by "_plumbing in_" the star and the apex of the pyramid by the
aid of a string and a stone.

True east and west lines could have been made to intersect such north
and south lines from the various groups of pyramids along the river
banks, by whose aid also such lines would be prolonged.

Next, supposing that their astronomers had been aware of the latitude of
Cheops, and the annual northing and southing of the sun, straight lines
could have been laid out in various sectoral directions to the
north-eastward and north-westward of Cheops, across the delta, as far as
the extreme apex of the pyramid was visible, by observations of the sun,
rising or setting over his summit. (That the Dog-star was observed in
this manner from the north-west, I have little doubt.)

    Fig. 37. Sun above Pyramid and sinking behind Pyramid

For this purpose, surveyors would be stationed at suitable distances
apart with their strings and their stones, ready to catch the sun
simultaneously, and at the very moment he became transfixed upon the
apex of the pyramid, and was, as it were, "swallowed by it." (_See
Figure 37_.) The knowledge of the pyramid <DW72> angle from different
points of view would enable the surveyor to place himself in readiness
nearly on the line.

Surely such lines as these would be as true and as perfect as we could
lay out nowadays with all our modern instrumental appliances. A string
and a stone here, a clean-cut point of stone twenty miles away, and a
great ball of fire behind that point at a distance of ninety odd million
miles. The error in such a line would be very trifling.

Such observations as last mentioned would have been probably extended
from Cephren for long lines, as being the higher pyramid above the
earth's surface, and may have been made from the moon or stars.

In those days was the sun the intimate friend of man. The moon and stars
were his hand-maidens.

How many of us can point to the spot of the sun's rising or setting? We,
with our clocks, and our watches, and our compasses, rarely observe the
sun or stars. But in a land and an age when the sun was the only clock,
and the pyramid the only compass, the movements and positions of the
heavenly bodies were known to all. These people were _familiar_ with the
stars, and kept a watch upon their movements.

How many of our vaunted educated population could point out the Dog-star
in the heavens?--but the whole Egyptian nation hailed his rising as the
beginning of their year, and as the harbinger of their annual blessing,
the rising of the waters of the Nile.

    Fig. 38. From the North West Bearing 315° Sun in the West.

    Fig. 39. From the South East Bearing 135° Sun in the West.

    Fig. 40. From the North East Bearing 45° Sun in the East.

    Fig. 41. From the South West Bearing 225° Sun in the East.

It is possible therefore that the land surveyors of Egypt made full use
of the heavenly bodies in their surveys of the land; and while we are
pitifully laying out our new countries by the circumferenter and the
compass, we presume to speak slightingly of the supposed dark heathen
days, when the land of Egypt was surveyed by means of the sun and the
stars, and the theodolites were built of stone, with vertical limbs five
hundred feet in height, and horizontal limbs three thousand feet in
diameter.

Imagine half a dozen such instruments as this in a distance of about
sixty miles (for each group of pyramids was effectually such an
instrument), and we can form some conception of the perfection of the
surveys of an almost prehistoric nation.

The centre of Lake Moeris, in which Herodotus tells us two pyramids
stood 300 feet above the level of the lake, appears from the maps to be
about S. 28° W., or S. 29° W. from Gïzeh, distant about 57 miles, and
the Meidân group of pyramids appears to be about 33 miles due south of
Gïzeh.

Figures 38, 39, 40 and 41, show that north-west, south-east, north-east,
and south-west lines from the pyramids could be extended by simply
plumbing the angles. These lines would be run in sets of two's and
three's, according to the number of pyramids in the group; and their
known distances apart at that angle would check the correctness of the
work.

A splendid line was the line bearing 43° 36′ 10·15″, or 223° 36′
10·15″ from Cheops and Cephren, the pyramids covering each other, the
line of hypotenuse of the great 20, 21, 29 triangle of the plan. This I
call the 20, 21 line. _(See Figure_ 42.)

Figure 43 represents the 3, 4, 5 triangle line from the summits of
Mycerinus and Cheops in true line bearing 216° 52' 11·65". This I call
the south 4, west 3 line.

The next line is what I call the 2, 1 line, and is illustrated by figure
44. It is one of the most perfect of the series, and bears S. 26° 33'
54·9" W. from the apex of Cephren. This line demonstrates clearly why
Mycerinus was cased with red granite.

Not in memory of the beautiful and rosy-cheeked Nitocris, as some of the
tomb theory people say, but for a less romantic but more useful object;
simply because, from this quarter, and round about, the lines of the
pyramids would have been confused if Mycerinus had not been of a
different color. The 2, 1 line is a line in which Mycerinus would have
been absolutely lost in the <DW72>s of Cephren but for his red color.
There is not a fact that more clearly establishes my theory, and the
wisdom and forethought of those who planned the Gïzeh pyramids, than
this red pyramid Mycerinus, and the 2, 1 line.

Hekeyan Bey, speaks of this pyramid as of a "_ruddy complexion_;" John
Greaves quotes from the Arabic book, Morat Alzeman, "_and the lesser
which is coloured_;" and an Arabic writer who dates the Pyramids three
hundred years before the Flood, and cannot find among the learned men of
Egypt "_any certain relation concerning them_" nor any "_memory of them
amongst men_," also expatiates upon the beauties of the "_coloured
satin_" covering of this one particular pyramid.

    Fig. 42. South 21. West 20. Bearing 223°.36'.10·15".

    Fig. 43. South 4. West 3. Bearing 216°.52'.11·65".

    Fig. 44. South 2. West 1. Bearing 206°.33'.54·18".

    Fig. 45. South 96. West 55. Bearing 209°.48'.32·81".

    Fig. 46. South 3. West 1. Bearing 198°.26'.5·82".

    Fig. 47. South 5. West 2. Bearing 201°.48'.5".

    Fig. 48. South 7. West 3. Bearing 203°.11'.55".

Figure 45 represents the line south 96, west 55, from Cephren, bearing
209° 48' 32·81"; the apex of Cephren is immediately above the apex of
Mycerinus.

Figure 46 is the S. 3 W. 1 line, bearing 198° 26' 5.82"; here the dark
<DW72> angle of the pyramids with the sun to the eastward occupies half
of the apparent half base.

Figure 47 is the S. 5, W. 2 line, bearing 201° 48' 5"; here Cephren and
Mycerinus are in outside <DW72> line.

Figure 48 is the S. 7 W. 3 line, bearing 203° 11' 55"; here the inside
<DW72> of Cephren springs from the centre of the apparent base of
Mycerinus.

I must content myself with the preceding examples of a few pyramid
lines, but must have said enough to show that from every point of the
compass their appearance was distinctly marked and definitely to be
determined by surveyors acquainted with the plan.


§ 11. DESCRIPTION OF THE ANCIENT PORTABLE SURVEY INSTRUMENT.

I must now commence with a single pyramid, show how approximate
observations could be made from it, and then extend the theory to a
group with the observations thereby rendered more perfect and delicate.

We will suppose the surveyor to be standing looking at the pyramid
Cephren; he knows that its base is 420 cubits, and its apothem 346½
cubits. He has provided himself with a model in wood, or stone, or
metal, and one thousandth of its size--therefore his model will be O.42
cubit base, and O.3465 cubit apothem--or, in round numbers, eight and
half inches base, and seven inches apothem.

This model is fixed on the centre of a card or disc, graduated from the
centre to the circumference, like a compass card, to the various points
of the compass, or divisions of a circle.

The model pyramid is fastened due north and south on the lines of this
card or disc, so that when the north point of the card points north, the
north face of the model pyramid faces to the north.

The surveyor also has a table, which, with a pair of plumb lines or
mason's levels, he can erect quite level: this table is also graduated
from the centre with divisions of a circle, or points of the compass,
and it is larger than the card or disc attached to the model.

This table is made so that it can revolve upon its stand, and can be
clamped. We will call it the _lower limb_. There is a pin in the centre
of the lower limb, and a hole in the centre of the disc bearing the
model, which can be thus placed upon the centre of the table, and
becomes the _upper limb_. The upper limb can be clamped to the lower
limb.

The first process will be to clamp both upper and lower limbs together,
with the north and south lines of both in unison, then revolve both
limbs on the stand till the north and south line points straight for
the pyramid in the distance, which is done by the aid of sights erected
at the north and south points of the perimeter of the lower limb. When
this is adjusted, clamp the lower limb and release the upper limb; now
revolve the upper limb until the model pyramid exactly covers the
pyramid in the distance, and shows just the same shade on one side and
light on the other, when viewed from the sights of the clamped lower
limb--and the lines, angles, and shades of the model coincide with the
lines, angles, and shades of the pyramid observed;--now clamp the upper
limb. Now does the model stand really due north and south, the same as
the pyramid in the distance; it throws the same shades, and exhibits the
same angles when seen from the same point of view; just as much of it is
in shade and as much of it is in light as the pyramid under observation;
therefore it must be standing due north and south, because Cephren
himself is standing due north and south, and the upper limb reads off on
the lower limb the angle or bearing observed.

So far we possess an instrument equal to the modern circumferenter, and
yet we have only brought one pyramid into work.

If I have shown that such an operation as the above is practically
feasible, if I have shown that angles can be taken with moderate
accuracy by observing one pyramid of 420 cubits base, how much more
accurate will the observation be when the surveyor's plane table bears
a group of pyramids which occupy a representative space of about 1400
cubits when viewed from the south or north, and about 1760 cubits when
viewed from the east or west. If situated a mile or two south of the
Gïzeh group our surveyor could also tie in and perfect his work by
sights to the Sâkkarah group with Sâkkarah models; and so on, up the
Nile Valley, he would find every few miles groups of pyramids by aid of
which he would be enabled to tie his work together.

If the Gïzeh group of pyramids is placed and shaped in the manner I have
described, it must be clear that an exact model and plan, say a
thousandth of the size, could be very easily made--the plan being at the
level of the base of Cephren where the bases of the two main pyramids
are even;--and if they are not exactly so placed and shaped, it may be
admitted that their position and dimensions were known to the surveyors
or priests, so that such models could be constructed. It is probable,
therefore, that the instrument used in conjunction with these pyramids,
was a machine constructed in a similar manner to the simple machine I
have described, only instead of there being but one model pyramid on the
disc or upper limb, it bore the whole group; and the smaller pyramids
were what we may call vernier points in this great circle, enabling the
surveyor to mark off known angles with great accuracy by noticing how,
as he worked round the group of pyramids, one or other of the smaller
ones was covered by its neighbours.[9]

    Footnote 9: See general plan of Gïzeh Group op. page 1.

The immensity of the main pyramids would require the smaller ones to be
used for surveys in the immediate neighbourhood, as the surveyor might
easily be too close to get accurate observations from the main pyramids.

The upper limb, then, was a disc or circular plate bearing the model of
the group.

Cheops would be situated in the centre of the circle, and observations
would be taken by bringing the whole model group into even line and even
light and shade with the Gïzeh group.

I believe that with a reasonable-sized model occupying a circle of six
or seven feet diameter, such as a couple of men could carry, very
accurate bearings could have been taken, and probably were taken.

The pyramid shape is the very shape of all others to employ for such
purposes. A cone would be useless, because the lights and shades would
be softened off and its angles from all points would be the same. Other
solids with perpendicular angles would be useless, because although they
would vary in width from different points of view they would not present
that ever changing angle that a pyramid does when viewed from different
directions.

After familiarity with the models which I have made use of in
prosecuting these investigations, I find that I can judge with great
accuracy _from their appearance only_ the bearing of the group from any
point at which I stand. I make bold to say that the pocket compass of
the Egyptian surveyor was a little model of the group of pyramids in his
district, and he had only to hold it up on his hand and turn it round in
the sun till its shades and angles corresponded with the appearance of
the group, to tell as well as we could tell by our compasses, perhaps
better, his bearing from the landmarks that governed his surveys.

The Great Circle of Gold described by Diodorus (_Diod. Sic. lib. X.,
part 2, cap. 1_) as having been employed by the Egyptians, and on which
was marked amongst other things, the position of the rising and setting
of the stars, and stated by him to have been carried off by Cambysses
when Egypt was conquered by the Persians, is supposed by Cassini to have
been also employed for finding the meridian by observation of the rising
and setting of the sun. This instrument and others described by writers
on Egypt would have been in practice very similar to the instrument
which I have described as having been probably employed for terrestrial
observations.

The table or disc comprising the lower limb of the instrument, might
have been supported upon a small stand with a circular hole in the
centre, so arranged that the instrument could be either set up alone and
supported by its own tripod, or rested fairly on the top of any of those
curious stone boundary marks which were made use of, not only to mark
the corners of the different holdings, but to show the level of the Nile
inundations. (_See Figure 49, copied from Sharpe's Egypt,_ _vol. I.,
p_. 6.) The peculiar shape of the top of these stone landmarks, or
"sacred boundary stones," appears suitable for such purposes, and it
would have been a great convenience to the surveyor, and conducive to
accuracy, that it should be so arranged that the instrument should be
fixed immediately over the mark, as appears probable from the shape of
the stone.

    Fig 49. Sacred boundary stone.

A noticeable point in this theory is, that it is not in the least
essential that the apex of a pyramid should be complete. If their
summits were left permanently flat, they would work in for survey
purposes quite as well, and I think better, than if carried to a point,
and they would be more useful with a flat top for defined shadows when
used as sun dials.

In the Gïzeh group, the summit of Cheops appears to me to have been left
incomplete the better to get the range with Cephren for lines down the
delta.

In this system of surveying, there is always a beautiful connection
between the horizontal bearings and the apparent or observed angles
presented by the <DW72>s and edges of the pyramid. Thus, in pyramids like
those of Gïzeh, which stand north and south, and whose meridional
sections contain less, and whose diagonal sections contain more than a
right angle, the vertex being the point at which the angle is
measured--this law holds:-- That the smallest interior angle at the
vertex, contained between the inside edge and the outside edge, will
exhibit the same angle as the bearing of the observer's eye from the
apex of the pyramid _when the angle at the apex contained by the outside
edges appears to be a right angle_.

Figures 50 to 55 inclusive illustrate this beautiful law from which it
will be seen that the Gïzeh surveyors possessed, in this manner alone,
eight distinctly defined bearings from each pyramid.

    Fig. 50. Cheops from points bearing

              °  ′   ″
           S 19.12.22 W
           W 19.12.22 N
           N 19.12.22 E
           E 19.12.22 S

    Fig. 51. Cheops from points bearing

              °  ′   ″
           S 19.12.22 E
           W 19.12.22 S
           N 19.12.22 W
           E 19.12.22 N

    Fig. 52. Cephren from points bearing

              ° ′   ″
           S 23.7.50·24 W
           W 23.7.50·24 N
           N 23.7.50·24 E
           E 23.7.50·24 N

    Fig. 53. Cephren from points bearing

              ° ′   ″
           S 23.7.50·24 E
           W 23.7.50·24 S
           N 23.7.50·24 W
           E 23.7.50·24 N

    Fig. 54. Mycerinus from points bearing

              ° ′   ″
           S 17.1.40·4 W
           W 17.1.40·4 N
           N 17.1.40·4 E
           E 17.1.40·4 S

    Fig. 55. Mycerinus from points bearing

              °- ′ ″
           S 17.1.40·4 E
           W 17.1.40·4 S
           N 17.1.40·4 W
           E 17.1.40·4 N

       *       *       *       *       *

    Fig. 56. Cheops model.

    Fig. 57. Cephren model

    Fig. 58. Mycerinus model

       *       *       *       *       *

I recommend any one desirous to thoroughly comprehend these matters, to
make a plan from my diagram, _Figure_ 5, using R.B. cubits for measures,
and to a suitable scale, on a piece of card-board. Then to cut out of
the card-board the squares of the bases of the pyramids at the level of
Cephren, viz., 420, 420 and 218 cubits respectively, for the three main
pyramids. One hundred cubits to the inch is a convenient scale and
within the limits of a sheet of Bath board.

By striking out the models on card-board in the manner shown by diagrams
(_see Figures_ 56, 57, and 58) they can be cut out with a
penknife--cutting only _half through_ where the lines are
_dotted_--bent up together, and pasted along the edges with strips of
writing paper about half an inch wide.

These models can be dropped into the squares cut out of the card-board
plan, thus correcting the error caused by the thickness of the
card-board base, and if placed in the sun, or at night by the light of
_one_ lamp or candle properly placed to represent the sun in the
eastward or westward, the clear cut lines and clear contrasting shades
will be manifest, and the lines illustrated by my figures can be
identified.

When inspecting the model, it is well to bear in mind that the eye must
be kept very nearly level with the table, or the pyramids will appear as
if viewed from a balloon.

       *       *       *       *       *

I believe that the stones were got up to the building by way of the
north side of each pyramid. The casing on the south, east, and west, was
probably built up as the work proceeded, and the whole of these three
faces were probably thus finished and completed while there was not a
single casing stone set on the north side. Then the work would be closed
up until there remained nothing but a great gap or notch, wide at the
bottom, and narrowing to the apex. The work on the north side would then
be closed from the sides and top, and the bottom casing stone about the
centre of the north side, would be the last stone set on the building.
These old builders were too expert not to have thus made use of all the
shade which their own building would thus afford to a majority of the
workmen.

       *       *       *       *       *

Many of the obelisks were probably marks on pyramid lines of survey.

The pyramid indeed may have been a development of the obelisk for this
purpose.

Their slanting sides might correspond with some of the nearly upright
slant angles of the pyramids, in positions opposite certain lines.
Reference to several of my figures will show how well this would come
in.

Herodotus speaks of two obelisks at Heliopolis, and Bonwick tells us
that Abd al Latif saw two there which he called Pharaoh's Needles. An
Arab traveller, in 1190, saw a pyramid of copper on the summit of the
one that remained, but it is now wanting. Pharaoh's Needles appear to
have been situated about 20 miles NE. of the Gïzeh group, and their
<DW72> angles might have coincided with the apparent <DW72> angles of
Cephren or Cheops on the edge nearest the obelisk.

The ancient method of describing the meridian by means of the shadow of
a ball placed on the summit of an obelisk points to a reasonable
interpretation for the peculiar construction of the two pillars, Jachin
and Boaz, which are said to have been situated in front of the Hebrew
Temple at Jerusalem, and about which so much mysterious speculation has
occurred.

They were no doubt used as sun-dials for the morning and afternoon sun
by the shadow of the balls or "chapiters" thrown upon the pavement.

Without presuming to dispute the objects assigned by others for the
galleries and passages which have been discovered in the pyramid Cheops,
I venture to opine that they were employed to carry water to the
builders. They are connected with a well, and the well with the Nile or
canal. Whether the water was slided up the smooth galleries in boxes, or
whether the cochlea, or water screw, was worked in them, their angles
being suitable, it is impossible to conjecture; either plan would have
been convenient and feasible.

These singular chambers and passages may indeed possibly have had to do
with some hydraulic machinery of great power which modern science knows
nothing about. The section of the pyramid, showing these galleries, in
the pyramid books, has a most hydraulic appearance.

The tremendous strength and regularity of the cavities called the King's
and Queen's chambers, the regularity and the _smallness_ of most of the
passages or massive stone connecting pipes, favor the idea that the
chambers might have been reservoirs, their curious roofs, air chambers,
and the galleries or passages, connecting pipes for working water under
pressure. Water raised through the passages of this one pyramid nearest
to the canal, might have been carried by troughs to the other pyramids,
which were in all probability in course of construction at the same
period of time. A profane friend of mine thinks that the sarcophagus or
"sacred coffer" in the King's chamber may have been used by the chief
architect and leading men of the works as a _bath_, and that the King's
chamber was nothing more or less than a delightful bath room.

       *       *       *       *       *

The following quotation from the writing of an Arabian author (Ibn Abd
Alkokm), is extracted from Bonwick's "Pyramid Facts and Fancies," page
72:--"The Coptites mention in their books that upon them (the Pyramids)
is an inscription engraven; the exposition of it in Arabicke is
this:--'I, Saurid the King built the Pyramids (in such and such a time),
and finished them in six years; he that comes after me, and says he is
equal to me, let him destroy them in six hundred years; and yet it is
known that it is easier to pluck down than to build; _and when I had
finished them, I covered them with sattin, and let him cover them with
slats._'"

The italics are my own. The builder seems to have entertained the idea
that his work would be partially destroyed, and afterwards temporarily
repaired or rebuilt. The first part has unfortunately come true, and it
is possible that the last part of the idea of King Saurid may be carried
out, because it would not be so very expensive an undertaking for any
civilized nation in the interest of science to re-case the pyramids of
Gïzeh, so that they might be once more applied to land-surveying
purposes in the ancient manner.

It would not be absolutely necessary to case the whole of the pyramid
faces, so long as sufficient casing was put on to define the angles. The
"_slats_" used might be a light wooden framework covered with thin
metal. The metal should be painted white, except in the case of
Mycerinus, which should be of a reddish color.


§ 12. PRIMARY TRIANGLES AND THEIR SATELLITES;--OR THE ANCIENT SYSTEM OF
RIGHT-ANGLED TRIGONOMETRY UNFOLDED BY A STUDY OF THE PLAN OF THE
PYRAMIDS OF GIZEH.

=Main Triangular Dimensions of Plan are Represented by the Following
Eight Right-angled Triangles.=

TABLE TO EXPLAIN FIGURE 60.

    +--------------------------++-------------------------------------+
    | AB  28}  × { 84} × { 672 || DG   3}  × { 72} × {576             |
    | BJ  45}  3 {135} 8 {1080 || GE   4} 24 { 96} 8 {768             |
    | JA  53}    {159}   {1272 || ED   5}    {120}   {960             |
    +--------------------------++-------------------------------------+
    | DC   3} ×  {135} × {1080 || FW  48}  × { 48} × {384             |
    | CA   4} 45 {180} 8 {1440 || WV  55}  1 { 55} 8 {440             |
    | AD   5}    {225}   {1800 || VF  73}    { 73}   {584             |
    +--------------------------++-------------------------------------+
    | EB   3} ×  { 63} ×  {504 || FB  20}  × { 80} × {640             |
    | BA   4} 21 { 84} 8  {672 || BA  21}  4 { 84} 8 {672             |
    | AE   5}    {105}    {840 || AF  29}    {116}   {928             |
    +--------------------------++=====================================+
    | FH   3} ×  { 96} × { 768 ||                                     |
    | HN   4} 32 {128} 8 {1024 || Note.--In the above table the first |
    | XF   5}    {160}   {1280 || column _is the Ratio_, the second   |
    +--------------------------++ _the connected Natural Numbers_, and|
    | AY   3} ×  { 36} × { 288 || the third column represents _the_   |
    | YZ   4} 12 { 48} 8 { 384 || _length each line in R.B. cubits_.  |
    | ZA   5}    { 60}   { 480 ||                                     |
    +--------------------------++-------------------------------------+

    Fig. 60.

Reference to _Fig. 60_ and the preceding table, will show that the main
triangular dimensions of this plan (imperfect as it is from the lack of
eleven pyramids) are represented by four main triangles, viz:--

                  Ratio.

    C A D C .. .. 3, 4, 5

    F B A F .. .. 20, 21, 29

    A B J A .. .. 28, 45, 53

    F W V F .. .. 48, 55, 73

Figures 30 to 36 illustrate the two former, and _Figures_ 61 and 62
illustrate the two latter. I will call triangles of this class "primary
triangles," as the most suitable term, although it is applied to the
main triangles of geodetic surveys.

We have only to select a number of such triangles and a system of
trigonometry ensues, in which base, perpendicular, and hypotenuse of
every triangle is a whole measure without fractions, and in which the
nomenclature for every angle is clear and simple.

An angle of 43° 36′ 10·15″ will be called a 20, 21 angle, and an
angle of 36° 52′ 11·65″ will be called a 3, 4 angle, and so
on.

In the existing system whole angles, such as 40, 45, or 50 degrees, are
surrounded by lines, most of which can only be described in numbers by
interminable fractions.

In the ancient system, lines are only dealt with, and every angle in the
table is surrounded by lines measuring whole units, and described by the
use of a couple of simple numbers.

Connecting this with our present system of trigonometry would effect a
saving in calculation, and general use of certain peculiar angles by
means of which all the simplicity and beauty of the work of the ancients
would be combined with the excellences of our modern instrumental
appliances. Surveyors should appreciate the advantages to be derived
from laying out traverses on the hypotenuses of "primary" triangles, by
the saving of calculation and facility of plotting to be obtained from
the practice.

The key to these old tables is the fact, that in "primary" triangles the
right-angled triangle formed by the sine and versed sine, also by the
co-sine and co-versed-sine, is one in which base and perpendicular are
measured by numbers without fractions. These I will call "satellite"
triangles.

Thus, to the "primary" triangle 20, 21, 29, the ratios of the co-sinal
and sinal satellites are respectively 7 to 3, and 2 to 5. (_See Figure
35._) To the 48, 55, 73 triangle the satellites are 11, 5 and 8, 3
(_Fig. 62_); to the 3, 4, 5 triangle they are 2, 1 and 3, 1 (_Fig. 30_);
and to the 28, 45, 53 triangle, they are 9, 5 and 7, 2 (_Fig. 61_). The
primary triangle, 7, 24, 25, possesses as satellites the "primary"
triangle, 3, 4, 5, and the ordinary triangle, 4, 1; and the primary
triangle 41, 840, 841, is attended by the 20, 21, 29 triangle, as a
satellite with the ordinary triangle 41, 1, and so on.

    Fig. 61. The 28-45-53 Triangle.

    Fig. 62. The 48-55-73 Triangle.

Since any ratio, however, whose terms, one or both, are represented by
fractions, can be transformed into whole numbers, it evidently follows
that every conceivable relative measure of two lines which we may
decide to call co-sine and co-versed-sine, becomes a satellite to a
corresponding "primary" triangle.

Now, since the angle of the satellite on the circumference must be
_half_ the angle of the adjacent primary triangle at the centre, it
follows that in constructing a list of satellites and their angles, the
angles of the corresponding primary triangles can be found. For
instance--

    Satellite 8, 3, contains 20° 33′ 21·76″
    Satellite 2, 7, contains 15° 56′ 43·425″

Each of these angles doubled, gives the angle of a "primary" triangle as
follows, viz.:--

    The 48, 55, 73 triangle = 41°  6′ 43·52″
    The 28, 45, 53 triangle = 31° 53′ 26·85″

The angles of the satellites together must always be 45°, because the
angle at the circumference of a quadrant must always be 135°.

From the Gïzeh plan, as far as I have developed it, the following order
of satellites begins to appear, which may be a guide to the complete
Gïzeh plan ratio, and to those "primary" triangles in use by the pyramid
surveyors in their ordinary work.

    +-------+-------+-------+------+-------+------+------+------+
    | 1,  2 |  2, 3 | 3,  4 | 4, 5 | 5,  6 | 6, 7 | 7, 8 | 8, 9 |
    |       |       |       |      |       |      |      |      |
    | 1,  3 |  2, 5 | 3,  5 | 4, 7 | 5,  7 |      | 7, 9 |      |
    |       |       |       |      |       |      |      |      |
    | 1,  4 |  2, 7 | 3,  7 | 4, 9 | 5,  8 |      |      |      |
    |       |       |       |      |       |      |      |      |
    | 1,  5 |  2, 9 | 3,  8 |      | 5,  9 |      | 7, 1 |      |
    |       |       |       |      |       |      |      |      |
    | 1,  6 |       |       |      | 5, 11 |      |      |      |
    |       |       |       |      |       |      |      |      |
    | 1,  7 |       | 3, 11 |      | 5, 13 |      |      |      |
    |       |       |       |      |       |      |      |      |
    | 1,  8 |       | 3, 13 |      |       |      |      |      |
    |       |       |       |      |       |      |      |      |
    | 1,  9 |       |       |      |       |      |      |      |
    |       |       |       |      |       |      |      |      |
    | 1, 11 |       |       |      |       |      |      |      |
    |       |       |       |      |       |      |      |      |
    | 1, 13 |       |       |      |       |      |      |      |
    |       |       |       |      |       |      |      |      |
    | 1, 15 |       |       |      |       |      |      |      |
    |       |       |       |      |       |      |      |      |
    | 1, 17 |       |       |      |       |      |      |      |
    +-------+-------+-------+------+-------+------+------+------+

Primary triangles may be found from the _angle of the satellite_, but it
is an exceedingly round-about way. I will, however, give an example.

Let us construct a primary triangle from the satellite 4, 9.

    Rad. × 4
    --------  = ·4444444 = Tangt. < 23° 57′ 45·041″
        9

                 ∠ 23° 57′ 45·041″ × 2 = 47° 55′ 30·083″.

  therefore the angles of the "primary" are 47° 55′ 30·083″.

                                        and 42°  4′ 29·917″.

  The natural sine of 42° 4′ 29·917″ = ·6701025.

  The natural co-sine 42° 4′ 29·917″ = ·7422684.

The greatest common measure of these numbers is about 102717,
therefore--

    Radius 10000000 ÷ 102717 = 97
    Co-sine 7422684 ÷ 102717 = 72
    Sine    6701025 ÷ 102717 = 65

and 65, 72, 97 is the primary triangle to which the satellites are 4, 9,
and 5, 13. (_See Fig_. 63.) The figures in the calculation do not
balance exactly, in consequence of the insufficient delicacy of the
tables or calculations.

    Fig. 63.

The connection between primaries and satellites is shown by figure 64.

    Fig. 64.

Let the triangle ADB be a satellite, 5, 2, which we will call BD 20, and
AD 8. Let C be centre of semi-circle ABE.

    AD : DB :: DB : DE      = 50 (_Euc. VI_. 8)

    AD + DE      =  AE      = 58 = diameter

    AE ÷ 2       =  AC = BC = 29 = radius

    AC-AD        =  DC      = 21 = co-sine

                     and DB = 20 = sine

From the preceding it is manifest that--

    sine²
    -----  + ver-s = dia.
    ver-s

The formula to find the "primary triangle" to any satellite is as
follows:--

Let the long ratio line of the satellite or sine be called _a_, and the
short ratio line or versed-sine be called _b_. Then--

  (1) a          = sine.

      a² + b²
  (2) -------   = radius.
        2b

      a² - b²
  (3) -------   = co-sine.
        2b

Therefore various primary triangles can be constructed on a side DB
(_Fig_. 64) as sine, by taking different measures for AD as versed-sine.
For example--

             }    5         = sine    =  5
             }
             }    5² + 1²
     From    }    -------   = radius  = 13
   Satellite }    2  × 1
     5, 1.   }
             }    5² - 1²
             }    -------   = co-s.   = 12
             }    2  × 1

       *       *       *       *       *


             }    5         = sine   = 5   }     {20
             }                             }     {
             }    5² + 2²                  }     {
      From   }    -------   = radius = 7¼  } × 4 {29
    Satellite}    2  × 2                   }     {
      5, 2.  }                             }     {
             }    5² + 2²                  }     {
             }    -------   = co-s.  = 5¼  }     {21
             }    2  × 2                   }     {


       *       *       *       *       *

Finally arises the following simple rule for the construction of
"primaries" to contain any angle--_Decide upon a satellite which shall
contain half the angle_--say, 5, 1. Call the first figure _a_, the
second _b_, then--

    a² + b² = hypotenuse.
    a² - b  = perpendicular.
    a  × 2b = base.

                              "PRIMARY" LOWEST RATIO.
    Thus--             | 5² + 1²   = 26 = 13
        Satellite 5,1  | 5² - 1²   = 24 = 12
                       | 5 × 2 × 1 = 10 =  5
       --------------- |----------------------------
    and--              | 5² + 2²   = 29 = 29
        Satellite 5,2  | 5² - 2²   = 21 = 21
                       | 5 × 2 × 2 = 20 = 20

Having found the lowest ratio of the three sides of a "primary"
triangle, the lowest whole numbers for tangent, secant, co-secant, and
co-tangent, if required, are obtained in the following manner.

Take for example the 20, 21, 29 triangle, now 20 × 21 = 420, and 29 ×
420 = 12180, a new radius instead of 29 from which with the sine 20, and
co-sine 21, increased in the same ratio, the whole canon of the 20, 21,
29 triangle will come out in whole numbers.

Similarly in the triangle 48, 55, 73, radius 73 × 13200 (the product of
48 × 55) makes radius in whole numbers 963600, for an even canon without
fractions. This is because sine and co-sine are the two denominators in
the fractional parts of the other lines when worked out at the lowest
ratio of sine, co-sine, and radius.

After I found that the plan of the Gïzeh group was a system of "primary"
triangles, I had to work out the rule for constructing them, for I had
never met with it in any book, but I came across it afterwards in the
"Penny Encyclopedia," and in Rankine's "Civil Engineering."

The practical utility of these triangles, however, does not appear to
have received sufficient consideration. I certainly never met with any
except the 3, 4, 5, in the practice of any surveyor of my acquaintance.

(For squaring off a line nothing could be more convenient than the 20,
21, 29 triangle; for instance, taking a base of 40 links, then using the
whole chain for the two remaining sides of 42 and 58 links.)


Table of Some Primary Triangles and their Satellites.

    ANGLE OF PRIMARY       PRIMARY         SATELLITE.    ANGLE OF SATELLITE
    DEG.  MIN.  SEC.     RAD. CO.-S. SINE.              DEG.  MIN.   SEC.
     2    47    39·70     841  840  41      41     1     1      23   49·85

     6    43    58·62     145  144  17      17     1     3      21   59·31

     8    47    50·69      85   84  13      13     1     4      23   55·34

    10    23    19·89      61   60  11      11     1     5      11   39·94

    12    40    49·37      41   40   9       9     1     6      20   24·68

    14    14    59·10      65   63  16       8     1     7       7   29·55

    16    15    36·73      25   24   7       7     1     8       7   48·36

    18    55    28·71      37   35  12       6     1     9      27   44·35

    22    37    11·51      13   12   5       5     1    11      18   35·75

    25     3    27·27      85   77  36       9     2    12      31   43·63

    25    59    21·22      89   80  39      13     3    12      59   40·61

    28     4    20·94      17   15   8       4     1    14       2   10·47

    30    30    36·49      65   56  33      11     3    15      15   18·24

    31    53    26·85      53   45  28       7     2    15      56   43·42

    36    52    11·65       5    4   3       3     1    18      26    5·82

    41     6    43·52      73   55  48       8     3    20      33   21·76

    42     4    30·08      97   72  65      13     5    21       2   15·04

    43    36    10·15      29   21  20       5     2    21      48    5·07

    46    23    49·85      29   20   21      7    3     23      11   54·92

    47    55    29·92      97   65   72      9    4     23      57   44·96

    48    53    16·48      73   48   55     11    5     24      26   38·24

    53     7    48·35       5    3    4      2    1     26      33   54·17­

    58     6    33·15      53   28   45      9    5     29       3   16·57

    59    29    23·51      65   33   56      7    4     29      44   41·75

    61    55    39·06      17    8   15      5    3     30      57   49·53

    64     0    38·78      89   39   80      8    5     32       0   19·39

    64    56    32·73      85   36   77     11    7     32      28   16·36

    67    22    48·49      13    5   12      3    2     33      41   24·24

    71     4    31·29      37   12   35      7    5     35      32   15·64

    73    44    23·27      25    7   24      4    3     36      52   11·63

    75    45     0·90      65   16   63      9    7     37      52   30·45

    77    19    10·63      41    9   40      5    4     38      39   35·31

    79    36    40·11      61   11   60      6    5     39      48   20·05

    81    12     9·31      85   13   84      7    6     40      36    4·65

    83    16     1·38     145   17  144      9    8     41      38    0·69

    87    12    20·30     841   41  840     21   20     43      36   10·15

Reference to the plan ratio table at the commencement, and to the tables
here introduced, will shew that most of the primary triangles mentioned
are indicated on the plan ratio table principally by the lines
corresponding to the ratios of the satellites. Thus--

    PRIMARY TRIANGLE                     INDICATED BY

    17, 144, 145.       Triangle FP, PA, AF on plan.
    13, 84, 85.         Plan ratio of SJ to SU, 7 to 6.
    11, 60, 61.         Plan ratio BC to FB, 6 to 5, and DN to NR, 61
                          to 60.
    12, 35, 37.         Plan ratio EO to AY, 37 to 12, and EA to AY,
                          35 to 12.
     5, 12, 13.         Plan ratio CY to BC, 3 to 2; JE to EX, 3 to 2;
                          CA to YA, 5 to 1; and NZ to ZA, 12 to 5.
     8, 15, 17.         Plan ratio FB to BY, 5 to 3, and AC to BC, 15
                          to 8.
    33, 56, 55.         Plan ratio YX to AY, 7 to 4; AB to BO, 7 to 4;
                          and EA to AZ, 7 to 4.
    28, 45, 53.         Exists on plan, AB, BJ, JA.
     3,  4,  5.         Pervades the plan, and is also indicated by plan
                          ratio GX to DG, 2 to 1; SU to SV, 2 to 1;
                          and CY to YZ, 3 to 1.
    48, 55, 73.         Exists on plan, FW, WV, VF--and is also indicated
                          by plan ratio FO to OZ, 8 to 3.
    65, 72, 97.         Plan ratio AC to CH, 9 to 4; MY to YZ, 9 to 4.
    20, 21, 29.         Exists on plan FB, BA, AF; and plan ratio, GU
                          to DG, 5 to 2.

It seems probable that could I add to my pyramid plan the lines and
triangles that the missing eleven pyramids would supply, it would
comprise a complete table on which would appear indications of all the
ratios and triangles made use of in right-angled trigonometry, a
"_ratiometer_" in fact.

I firmly believe that so far as I have gone it is correct--and it is
possible, therefore, with the start that I have made, for others to
continue the work, and add the eleven pyramids to the plan in their
correct geometrical position. By continuing the system of evolution by
which I defined the position of Cephren, and the little pyramid to the
south-east of Cheops, after I had obtained Cheops and Mycerinus, may be
rebuilt, at one and the same time, a skeleton of the trigonometrical
tables of a forgotten civilization, and the plan of those pyramids which
are its only link with the present age.


§ 13. THE SIZE AND SHAPE OF THE PYRAMIDS INDICATED BY THE PLAN.

I pursued my investigations into the <DW72>s and altitudes of the
pyramids without reference to the plan, after once deciding their exact
bases.

Now it will be interesting to note some of the ways in which the plan
hints at the shape and size of these pyramids, and corroborates my work.

The dimensions of _Cheops_ are indicated on the plan by the lines EA to
YA, measuring 840 and 288 R.B. cubits respectively, being the half
periphery of its horizontal section at the level of Cephren's base, and
its own altitude from its own base. (_See Fig_. 5.)

The line EA, in fact, represents in R.B. cubits the half periphery of
the bases of either Cheops or Cephren measured at the level which I
have set forth as the _plan level_, viz., base of Cephren.

The ratio of Cephren's base to Cephren's altitude is indicated on the
plan by the ratios of the lines BC to EB, or FO to OR, viz., 32 to 21.
(_See Fig._ 4.)

The altitude of Mycerinus above Cephren's base appears on plan in the
line EF, measuring 136 R.B. cubits.

The line EO on plan measures 888 cubits, which would be the length of a
line stretched from the apex of Cheops to the point E, at the level of
Cheops' base.

This merits consideration:--the lines EA and AY are connected on plan at
the centre of Cheops, and the lines EO and EA are connected on plan at
the point E.

Now the lines EO, EA and AY are sides of a "primary triangle," whose
ratio is 37, 35, 12, and whose measure in cubits is 888, 840, and 288;
and if we suppose the line EA to be stretched horizontally beneath the
pyramids at the level of the base of Cheops from E to A on plan, and the
line AY to be a plumb line hanging from the apex of Cheops to the level
of his base, then will the line EO just stretch from the point E to the
apex of Cheops, and the three lines will connect the two main pyramids
by a vertical triangle of which EA, AY and EO form the base,
perpendicular, and hypotenuse. Or, to explain it in another manner: let
the line EA be a _cord_ stretching horizontally from A at the centre of
the base of Cheops to the point E, both ends being at the same level;
let the line AY be a _rod_, lift it on the end A till it stands erect,
then is the end Y the apex of Cheops. Now, the line EO would just
stretch from the top of the rod AY to the point E first described.

It is a singular coincidence, and one that may be interesting to
students of the _interior_ of the Pyramids, that the side EP, of the
small 3, 4, 5 triangle, EP, PF, FE, in the centre of the plan, measures
81·60 R.B. cubits, which is very nearly eight times the "_true breadth_"
of the King's chamber in Cheops, according to Piazzi Smyth; for 81·60/8
= 10·20 R.B. cubits, or 206·046 pyramid inches (one R.B. cubit being
20·2006 pyramid inches). The sides of this little triangle measure
81·60, 108·80, and 136, R.B. cubits respectively, as can be easily
proved from the plan ratio table.


§ 14. A SIMPLE INSTRUMENT FOR LAYING OFF "PRIMARY TRIANGLES."

A simple instrument for laying off "primary triangles" upon the ground,
might have been made with three rods divided into a number of small
equal divisions, with holes through each division, which rods could be
pinned together triangularly, the rods working as arms on a flat table,
and the pins acting as pointers or sights.

One of the pins would be permanently fixed in the table through the
first hole of two of the rods or arms, and the two other pins would be
movable so as to fix the arms into the shape of the various "primary
triangles."

Thus with the two main arms pinned to the cross arm in the 21st and 29th
hole from the permanently pinned end, with the cross arm stretched to
twenty divisions, a 20, 21, 29 triangle would be the result, and so on.


§ 14_a_. GENERAL OBSERVATIONS.

I must be excused by geometricians for going so much in detail into the
simple truths connected with right-angled trigonometry. My object has
been to make it very clear to that portion of the public not versed in
geometry, that the Pyramids of Egypt must have been used for land
surveying by right-angled triangles with sides having whole numbers.

A re-examination of these pyramids on the ground with the ideas
suggested by the preceding pages in view, may lead to interesting
discoveries.

For instance, it is just possible that the very accurately and
beautifully worked stones in the walls of the King's chamber of Cheops,
may be found to indicate the ratios of the rectangles formed by the
bases and perpendiculars of the triangulations used by the old
surveyors--that on these walls may be found, in fact, corroboration of
the theory that I have set forth. I am led to believe also from the fact
that Gïzeh was a central and commanding locality, and that it was the
custom of those who preceded those Egyptians that history tells of, to
excavate mighty caverns in the earth--that, therefore, in the limestone
upon which the pyramids are built, and underneath the pyramids, may be
found vast excavations, chambers and galleries, that had entrance on the
face of the ridge at the level of High Nile. From this subterraneous
city, occupied by the priests and the surveyors of Memphis, access may
be found to every pyramid; and while to the outside world the pyramids
might have appeared sealed up as mausoleums to the Kings that it may
have seen publicly interred therein, this very sealing and closing of
the outer galleries may have only rendered their mysterious recesses
more private to the priests who entered from below, and who were,
perhaps, enabled to ascend by private passages to their very summits.
The recent discovery of a number of regal mummies stowed away in an out
of the way cave on the banks of the Nile, points to the unceremonious
manner in which the real rulers of Kings and people may have dealt with
their sovereigns, the pomp and circumstance of a public burial once
over. It is just possible that the chambers in the pyramids may have
been used in connection with their mysteries: and the small passages
called by some "ventilators" or "air passages," sealed as they were from
the chamber by a thin stone (and therefore no ventilators) may have been
_auditory passages_ along which sound might have been projected from
other chambers not yet opened by the moderns; sounds which were perhaps
a part of the "hanky panky" of the ancient ceremonial connected with
the "mysteries" or the "religion" of that period.

Down that "well" which exists in the interior of Cheops, and in the
limestone foundations of the pyramid, should I be disposed to look for
openings into the vast subterraneous chambers which I am convinced _do_
exist below the Pyramids of Gïzeh.

The priests of the Pyramids of Lake Moeris had their vast subterranean
residences. It appears to me more than probable that those of Gïzeh were
similarly provided. And I go further:--Out of these very caverns may
have been excavated the limestone of which the pyramids were built, thus
killing two birds with one stone--building the instruments and finding
cool quarters below for those who were to make use of them. In the
bowels of that limestone ridge on which the pyramids are built will yet
be found, I feel convinced, ample information as to their uses. A good
diamond drill with two or three hundred feet of rods is what is what is
wanted to test this, and the solidity of the pyramids at the same time.


§ 15. PRIMARY TRIANGULATION.

Primary triangulation would be useful to men of almost every trade and
profession in which tools or instruments are used. Any one might in a
short time construct a table for himself answering to every degree or so
in the circumference of a circle for which only forty or fifty triangles
are required.

It would be worth while for some one to print and publish a correct set
of these tables embracing a close division of the circle, in which set
there should be a column showing the angle in degrees, minutes, seconds
and decimals, and also a column for the satellite, thus--

    SATELLITE.  PRIMARY.     ANGLE.

     5  2      20  21  29   43° 36′ 10·15″

     7  3      21  20  29   46° 23′ 49·85″

and so on. Such a set of tables would be a boon to sailors, architects,
surveyors, engineers, and all handi-craftsmen: and I make bold to say,
would assist in the intricate investigations of the astronomer:--and the
rule for building the tables is so simple, that they could easily be
achieved. The architect from these tables might arrange the shape of his
chambers, passages or galleries, so that all measures, not only at right
angles on the walls, but from any corner of floor to ceiling should be
even feet. The pitch of his roofs might be more varied, and the monotony
of the buildings relieved, with rafters and tie-beams always in even
measures. The one solitary 3, 4, 5 of Vitruvius would cease to be his
standard for a staircase; and even in doors and sashes, and panels of
glass, would he be alive to the perfection of rectitude gained by
evenly-measured diagonals. By a slight modification of the compass card,
the navigator of blue water might steer his courses on the hypotenuses
of great primary triangles--such tables would be useful to all sailors
and surveyors who have to deal with latitude and departure. For
instance, familiarity with such tables would make ever present in the
mind of the surveyor or sailor his proportionate northing and easting,
no matter what course he was steering between north and east, "the
_primary_" embraces the _three ideas in one view_.

In designing trussed roofs or bridges, the "primaries" would be
invaluable to the engineer, strain-calculations on diagonal and upright
members would be simplified, and the builder would find the benefit of a
measure in even feet or inches from centre of one pin or connection to
another.

For earthwork <DW72>s 3, 4, 5; 20, 21, 29; 21, 20, 29; and 4, 3, 5 would
be found more convenient ratios than 1 _to_ 1, and 1½ _to_ 1, etc.
Templates and battering rules would be more perfect and correct, and the
engineer could prove his <DW72>s and measure his work at one and the same
time without the aid of a staff or level; the <DW72> measures would
reveal the depth, and the <DW72> measures and bottom width would be all
the measures required, while the top width would prove the correctness
of the <DW72>s and the measurements.

To the land surveyor, however, the primary triangle would be the most
useful, and more especially to those laying out new holdings, whether
small or large, in new countries.

Whether it be for a "squatter's run," or for a town allotment, the
advantages of a diagonal measure to every parallelogram in even _miles_,
_chains_, or _feet_, should be keenly felt and appreciated.

This was, I believe, _one_ of the secrets of the speedy and correct
replacement of boundary marks by the Egyptian land surveyors.

I have heard of a review in the "Contemporary," September, 1881,
referring to the translation of a papyrus in the British Museum, by Dr.
Eisenlohr--"_A handbook of practical arithmetic and geometry," etc.,
"such as we might suppose would be used by a scribe acting as clerk of
the works, or by an architect to shew the working out of the problems he
had to solve in his operations_." I should like to see a translation of
the book, from which it appears that "_the clumsiness of the Egyptian
method is very remarkable_." Perhaps this Egyptian "_Handbook_" may yet
shew that their operations were not so "_clumsy_," as they appear at
first sight to those accustomed to the practice of modern trigonometry.
I may not have got the exact "hang" of the Egyptian method of land
surveying--for I do not suppose that even their "clumsy" method is to be
got at intuitively; but I claim that I have shewn how the Pyramids could
be used for that purpose, and that the subsidiary instrument described
by me was practicable.

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 <DW72> ratios
should govern each building; before I conclude, I shall show how I got
at my <DW72> 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 <DW72> 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 <DW72> RATIOS OF THE PYRAMIDS WERE ARRIVED
AT.

The manner in which I arrived at the <DW72> 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 <DW72>s, 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 <DW72>s.

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 <DW72>s.

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 <DW72> ratios.

Reasoning thus, with the addition of the knowledge I possessed of the
angular estimates of these <DW72>s 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 <DW72>s 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 <DW72>s 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 <DW72>s of
the three pyramids is exactly as one might expect after examination of
Figure 3, 4 or 5.





End of the Project Gutenberg EBook of The Solution of the Pyramid Problem, by 
Robert Ballard

*** 