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 slopes 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 slopes 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 slopes and measure his work at one and the same
time without the aid of a staff or level; the slope measures would
reveal the depth, and the slope measures and bottom width would be all
the measures required, while the top width would prove the correctness
of the slopes 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.