The Solution of the Pyramid Problem Part 7

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 Gzeh 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 b.a.l.l.s or "chapiters" thrown upon the pavement.

Without presuming to dispute the objects a.s.signed by others for the galleries and pa.s.sages 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 ca.n.a.l. 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 pa.s.sages 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 pa.s.sages or ma.s.sive stone connecting pipes, favor the idea that the chambers might have been reservoirs, their curious roofs, air chambers, and the galleries or pa.s.sages, connecting pipes for working water under pressure. Water raised through the pa.s.sages of this one pyramid nearest to the ca.n.a.l, 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 Copt.i.tes 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 Gzeh, 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 ill.u.s.trate the two former, and _Figures_ 61 and 62 ill.u.s.trate the two latter. I will call triangles of this cla.s.s "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' 1015? will be called a 20, 21 angle, and an angle of 36 52' 1165? 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 circ.u.mference 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' 2176?

Satellite 2, 7, contains 15 56' 43425?

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

The 48, 55, 73 triangle = 41 6' 4352?

The 28, 45, 53 triangle = 31 53' 2685?

The angles of the satellites together must always be 45, because the angle at the circ.u.mference of a quadrant must always be 135.

From the Gzeh plan, as far as I have developed it, the following order of satellites begins to appear, which may be a guide to the complete Gzeh 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'="">

9

? 23 57' 45041? 2 = 47 55' 30083?.

therefore the angles of the "primary" are 47 55' 30083?.

and 42 4' 29917?.

The natural sine of 42 4' 29917? = 6701025.