Fig. 54. When the axis bisects the obeliscal area, and another straight line drawn from the apex represents the

Fig. 54.

axis of the pylonic area, we have what is commonly called the flail or whip of Osiris, an emblem of divinity, which he often holds in one hand, while in the other hand, crossed, he holds the crosier or curve of Osiris; sometimes the crux ansata, or sacred tau. So that this geometrical obeliscal representation of the laws of gravity becomes, in place of the whip, one of the most exalted emblems that the genius of man can assign to a divinity.

The obelisk was called "the finger of God." It now appears that the obelisk indicates the laws by which the universe is governed, and the granitic durability of this monolith is typical of the eternity of these laws and the monolith of unity. As such a symbol it was held in the greatest veneration, and placed within and at the entrance of the temples.

Nebuchadnezzar, who invaded and ravaged Egypt, erected in the plain of Dura a golden image, which he commanded

the people to worship.

From its dimensions, height 60 cubits, and breadth 6, it might have been an obelisk covered with gilded plates of metal.

In the Hippodrome at Constantinople there is a structure, or kind of obelisk, built with pieces of stone, said to be 94 feet high, "which was formerly covered with plates of copper, as we learn from the Greek inscription on its base." The pieces of copper were fastened together by iron pins, which were secured by lead; the holes in the stone are still visible. This obelisk, according to Bellonius, had the copper plates gilded so as to appear of gold.

Herodotus informs us that Pheron, after recovering his sight, erected, as an offering in the temple of the Sun, two obelisks, the height of each monolith being 100 cubits and breadth 8.

The golden thigh of Pythagoras was probably a small circular obelisk, by means of which he acquired a knowledge of the true solar system of the ancients; but Europe was not sufficiently enlightened in the age of Pythagoras to admit its truth, which he revealed only to a few of his select disciples.

The Chinese pagoda and Mahomedan minaret are varied, but false, forms of the obelisk, being devoid of the true principle of construction. Both these imitations of the obelisk continue to be dedicated to religion in the East. Probably some of the most ancient Chinese pagodas may be found to be true obelisks.

This sacred type of the eternal laws appears to have become more and more obscure as the days of science declined, till ultimately it ceased to be intelligible; when, instead of this spiritual symbol, a physical one, palpable to the senses and adapted to the capacity of the unlearned, was substituted, and so the Phallic worship became embodied and revered in the religious rites of Egypt, India, Greece, and Rome.

Squire concludes, from the American monuments, that this form of worship extended over that vast territory.

When the sacred tau, the symbolical generator of time,

velocity, and distance, ceased to be understood as a spiritual type, it was also adopted as a physical emblem.

It would seem that these types were properly understood, and most probably first associated with religion, by the Sabæans.

In the latest Assyrian palaces are frequent representations of the fire-altar in bas-reliefs and on cylinders, so that Layard thinks there is reason to believe that a fire-worship had succeeded the purer forms of Sabæanism.

The worship of planets formed a remarkable feature in the early religion of Egypt, but in process of time it fell into desuetude (Jablonski.)

To form the series of hyperbolic parallelograms,1,1,

.1 of 9.

(Fig. 40.) Series of inscribed parallelograms is

1, 1, 1, 1, 1, 1, 4, 1, 1

of 9.

Difference 1, 1, 12, 20, 30, 42, 36, 72

of 9.

Hence the series of inscribed rectangled parallelograms at right angles to 1, 2, 3, &c., will be, twice, three times. 1, &c. For 1st superficial rectangled parallelogram=1 of 9

129

In this hyperbolic series,,, .......

greatest parallelogram 9 is placed the last.

1 of 9, the

But in the series 1,,, &c. of 9, the greatest parallelogram is placed the first.

This last series of parallelograms overlap each other from

M to I L.

Also, as in fig. 37., when the first of the series is a square, the last will be a rectangled parallelogram.

But, as in fig. 38., when the first is a rectangled parallelogram, the last of the series will be a square.

By taking the difference of the series of rectangled parallelograms, 1,,, &c. in one square, we have the series of rectangled parallelograms, 4, 4, 1, &c. formed in the other

square.

The sum of the series ++ 1/2

terms will by construction=9-1x1=8.

So when 1,,, &c. of n is continued to n terms, the sum of the differential series +++, &c. of n to n-1 terms will n−1.

1st term by the 2nd, the 2nd by the 3rd, the 3rd by the 4th,

The sum of this series to n

1

n- -1 n

terms will = n−1.

By construction, it will be seen that the differential series

to 8 terms x by

=

1, 2, 3, 4, 5, 6, 7, 8

1, 1, 1, 1, 1, 4, 1, 1 of 9.

Thus the sum of this series to 8 terms + 9 for the 9th term = the hyperbolic series of rectangled parallelograms. The sum of the direct series

0+2+6+12+20, &c.,

which is formed from n2-n, will be seen to =

1

n

1

n

-1

of n2-n, from which the direct series 0, 2, 6, 12, 20,

1

&c. is formed. The last term of the series =

The more the radius of the quadrant is subdivided the nearer will the hyperbolic reciprocal curve approach its axis and the quadrantal arc, but still the axis of the curve will twice radius = twice the axis of the hyperbolic series of rectangled parallelograms within the square.

=

The hyperbolic area will also continually diminish as the area of the curve approaches to the area of the quadrant. For suppose the radius of the quadrant to be divided into 900 instead of 9 equal parts, then the axis of the hyperbola 900, and the area of the central or angular square 900 = 302.

will

LN

So the side of the central square will be to the axis of the hyperbola or radius of the quadrant, as 30: 900 :: 1:30.

But when the axis of the hyperbola = 9 = radius, the side of the central square, 3 axis of the hyperbola :: 3 : 9 :: 1 : 3.

2

When 6 hyperbolic parallelograms are inscribed in the square axis of curve = 62 = 36, the area of the series = 14.7. When 36 parallelograms are inscribed in the same axis, now = 362, the area of the series = 150.3.

2

2

.. area of 6 parallelograms: axis :: 147: 36

2

area of 36 parallelograms: axis :: 1503 362 :: 4.17 36

Thus 6 inscribed parallelograms will=14·7 of 62, or axis And 36 inscribed parallelograms will only 4.17 of the same square. First parallelogram in the series 36 will

=