Part 4

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 slope angles might have coincided with the apparent slope 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 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.