frsas SB s^ GIFT OF BOHEMIAN CLUB THE LOST SOLAR SYSTEM OF THE ANCIENTS DISCOVERED. VOL. I. ' ty LONDON : Printed by SPOTTIS \^->I>K & Co., New -street-Square. THE LOST SOLAR SYSTEM OF THE ANCIENTS DISCOVERED. BY JOHN WILSON, IN TWO VOLUMES. VOL. I. LONDON: LONGMAN, BROWN, GPEF.N, LONGMANS, & ROBERTS. 1856. >(??.? / CONTENTS THE FIRST VOLUME. PART I. PAGE Gravitation near the Earth's Surface. Construction of the Obe- lisk. Variation of Time, Velocity, and Distance represented by the Ordinates and Axis of the Obelisk. The obeliscal and parabolic Areas compared. Construction and Summation of obeliscal Series of Numbers, Squares, and Cubes. Series of Obelisks and Pyramids compared and summed. Series to the second, fourth, and sixth Powers. Series of Cubes circumscribed by Squares. The obeliscal Star or Cross. Complementary obeliscal Series. Pylonic Curve generated by the Ordinate which varies inversely as the Ordinate of the Obelisk. The Horn of Jupiter Ammon formed by the spiral Obelisk - 1 PART II. Hyperbolic Series. Series of 1, -, &c., !,_,_,_, &c., A O A A A 1, , &c. Hyperbolic reciprocal Curve from which is generated the Pyramid and hyperbolic Solid, the Ordi- nates of which vary inversely as each other, that of the Pyramid varies as D 2 , that of the hyperbolic Solid varies as L._ Series I 2 , 2 2 , 3 2 , &c., and 1, -^ L,&c. The hyper- bolic Solid will represent Force of Gravity varying as or D 2 Velocity varying as Time t which varies as D 2 will be D- represented by the Ordinate of Pyramid, or by the solid Obelisk. Gravity represented symbolically in Hieroglyphics by the hyperbolic Solid. The Obelisk represents the 701077 yi CONTENTS OF f PAGE planetary Distances, Velocities, periodic Times, Areas de- scribed in equal Times, Times of describing equal Areas and equal Distances in different Orbits having the common Centre in the Apex of the Obelisk. The Attributes of Osiris sym- bolise Eternity - 74- PART III. Tower of Belus. Description by Herodotus. Content ^ Circumference of the Earth. Cube of Side or Enclosure equal to the Circumference of the Earth. The Equivalent of the Stade, Orgye, Cubit, Foot, and Palm of Herodotus in Terms of the Earth's Circumference and the Stature of Man The French Measurement of the Earth's Circum- ference. The Circuit of Lake Mceris, sixty Schaenes, com- pared with the Mediterranean Coast of Egypt ; with Indian Tanks and Cingalese artificial Lakes. Herodotus' Measure- ment of the Euxine from the Bosphorus to Phasis ; of existing Obelisks. Diodorus' Dimensions of the cedar Ship of Sesostris compared with modern Ships and Steam Vessels. The Canal of Sesostris from the Mediterranean to the Red Sea. The Egyptian Obelisks at Rome, Paris, Alexandria, Heliopolis, Fioum, Thebes. Colossal Statues at Memphis and Heliopolis. Monoliths at Butos, Sais, Memphis, Thmouis, Mahabalipuram. Celtic Monuments in Brittany - 151 PART IV. Pyramid of Cheops. Its various Measurements. Content equal the Semi-circumference of Earth. Cube of Side of Base equal ^ Distance of Moon, Number of Steps. En- trance. Content of cased Pyramid equal T J Distance of Moon. King's Chamber. Winged Globe denotes the third Power or Cube. Three Winged Globes the Power of 3 times 3, the 9th Power, or the Cube cubed. Sarcophagus. Cause- way. Height of Plane on which the Pyramids stand. First Pyramids erected by the Sabseans and consecrated to Religion. Mythology. Age of the Pyramid. Its sup- posed Architect. Sabaeanism of the Assyrians and Persians. All Science centred in the Hierarchy. Traditions about the Pyramids. They were formerly worshipped, and still continue to be worshipped, by the Calmucs. Were regarded as Symbols of the Deity. Relative Magnitude of the Sun_, Moon, and Planets. How the Steps of the Pyramid were made to diminish in Height from the Base to the Apex. Duplication of the Cube. Cube of Hypothenuse in Terms of the Cubes of the two Sides. Difference between two Cubes. Squares described on two Sides of Triangles having a THE FIRST VOLUME. Vll PACK common Hypothenuse. Pear-like Curve. Shields of Kings of Egypt traced back to the fourth Manethonic Dynasty. Early Writing. Librarians of Ramses Miamum, 1400 B. c. Division of Time. Sources of the Nile - 217 PART V. Pyramid of Cephrenes. Content equal to T 5 ^ Circumference, Cube equal to ^ Distance of Moon. The Quadrangle in which the Pyramid stands. Sphere equal to Circumference. Cube of Entrance Passage is the Reciprocal of the Pyramid. The Pyramids of Egypt, Teocallis of Mexico, and Burmese Pagodas were Temples symbolical of the Laws of Gravitation, and dedicated to the Creator. External Pyramid of Mycerinus equal to -^ Circumference equal to 1 9 Degrees, and is the Reciprocal of itself. Cube equal to ^ Circumference. Internal Pyramid equal to Jj- Circumference. Cube equal to ^ Circumference. The six small Pyramids. The Pyramid of the Daughter of Cheops equal to y^j Cir- cumference equal to 2 Degrees, and is the Reciprocal of the Pyramid of Cheops. The Pyramid of Mycerinus is a mean Proportional between the Pyramid of Cheops and the Pyramid of the Daughter. Different Pyramids compared. Pyramids were both Temples and Tombs. One of the Dashour Pyra- mids equal to ^ Circumference, Cube equal to twice Circum- ference. One of the Saccarah Pyramids equal to -f% Circum- ference. Cube equal to ^ Distance of Moon. Great Dashour Pyramid equal to | Circumference. Cube equal to ^ Dis- tance of Moon. How the Pyramids were built. Nubian Pyramids. Number of Egyptian and Nubian Pyramids. General Application of the Babylonian Standard - - 298 PART VI. American Teocallis. Mythology of Mexico before the Ar- rival of the Spaniards. Teocallis of Cholula, Sun, Moon, Mexitli. Their Magnitudes compared with the Teocallis of Pachacamac, Belus, Cheops, the Pyramids of Mycerinus and Cheops' Daughter, and Silbury Hill, the conical Hill at Avebury. The internal and external Pyramids of the Tower of Belus. Hill of Xochicalco. Teocalli of Pacha- camac in Peru. Ruins of an Aztec City. The Babylonian Broad Arrow. The Mexican formed like the Egyptian Arch. Druidical Remains in England. Those in Cumber- land, at Carrock Fell, Salkeld, Black-Comb. Those in Wiltshire, at West Kennet, Avebury, Stonehenge. Ex- ternal and Internal Cone of Silbury Hill. Mount Barkal in Upper Nubia. Assyrian Mound of Koyunjik at Nineveh. vill CONTENTS OF THE FIRST VOLUME. PAGK Rectangular Enclosure at Medinet-Abou, Thebes. The Circles at Avebury. Conical Hill at Quito, in Peru. Tomb of Alyattes, in Lydia. Conical Hill at Sardis. Stonehenge Circles and Avenue, conical Barrows. Old Sarum in Wiltshire, conical Hill. The Circle of Stones called Arbe Lowes in Derbyshire. Circle at Hathersage, at Graned Tor, at Castle Ring, at Stan ton Moor, at Banbury, in Berk- shire. Hill of Tara. Kist-Vaen. Stones held sacred - 352 THE LOST SOLAR SYSTEM OP THE ANCIENTS DISCOVERED. PART I. GRAVITATION NEAR THE EARTH'S SURFACE. CONSTRUCTION OP THE OBELISK. VARIATION OF TIME, VELOCITY, AND DISTANCE REPRESENTED BY THE ORDINATES AND AXIS OF THE OBELISK. THE OBELISCAL AND PARABOLIC AREAS COMPARED. CON- STRUCTION AND SUMMATION OF OBELISCAL SERIES OF NUMBERS, SQUARES, AND CUBES. SERIES OF OBELISKS AND PYRAMIDS COMPARED AND SUMMED. SERIES TO THE SECOND, FOURTH, AND SIXTH POWERS. SERIES OF CUBES CIRCUMSCRIBED BY SQUARES. THE OBELISCAL STAR OR CROSS. COMPLEMENTARY OBELISCAL SERIES. PYLONIC CURVE GENERATED BY THE OR- DINATE WHICH VARIES INVERSELY AS THE ORDINATE OF THE OBELISK. THE HORN OF JUPITER AMMON FORMED BY THE SPIRAL OBELISK. The Laws of Gravitation expounded by the Geometrical Pro- perties of the Obelisk. IT was found by Galileo that a heavy body, when allowed to fall freely from a state of rest towards the earth, described distances proportionate to the square of the times elapsed during the descent; or proportionate to the square of the velocities acquired at the end of the descent. That is, at the end of the 1st second the body had de- scribed a distance of IBy 1 ^ feet English, which call 1 P. VOL. I. B 2' THE LOST . SOLAR SYSTEM DISCOVERED. At the end of the 2nd second, from the beginning of motion, the body had described a distance of 4 p. At the end of the 3rd second, a distance of 9 P. At the end of the 4th second, a distance of 16 P. Thus the distances described at the end of 1, 2, 3, 4 seconds are I 2 , 2 2 , 3 2 , 4 2 , or 1st series 1, 4, 9, 16 P 1, 4, 9 2nd series 1, 3, 1, 5, 3, 7 difference 5 3rd series 1, 2, 2, 2 difference. Here 1, 4, 9, 16 P are the series of distances described in 1, 2, 3, 4, seconds. 1,3, 5, 7, the series of distances described in each second. 1, 2, 2, 2, the series of incremental distances described in each second more than was described in the preceding second. During the first second the distance described = 1 P. If the velocity had been uniform the distance would have been described in 1 second with the mean velocity = half the extreme velocities = |-(0 + 2)=lp. So that at the end of the 1st second the acquired velocity would = 2 p. The velocity acquired at the end of the 2nd second would = twice the mean velocity with which the whole distance 4 P was described in two seconds. The mean velocity will = i(0 + 4) = 2p; therefore the velocity at the end of the 2nd second will = 4 p ; at the end of the 3rd second = 6 P ; at the end of the 4th second = 8 P. The velocity acquired at the end of the 1st second, if con- tinued uniform during the 2nd second, would, of itself, have carried the body 2 P ; but during the 2nd second the body received an additional accelerating velocity from gravity equal to that which caused it to describe 1 P in the 1st second. So that during the 2nd second the distance described will = 2 + 1 = 3 = 1 + 2 p. In like manner, during the 3rd second, the distance described will =4 + 1 = 5 = 3 + 2 p. In the 4th second 6 + 1 = 7 = 5 + 2 p, will be described. CONSTRUCTION OF THE OBELISK. 3 The distances described in the successive seconds will be 1, 3, 5, 7 P. The velocities at the beginning of the 1st, 2nd, 3rd, and 4th seconds will be 0, 2, 4, 6r, at the end 2, 4, 6, 8 P. The mean of the extreme velocities in the successive seconds are 1, 3, 5, 7, For Generally, the distance (2 n 1) P, described in the n ih second with an accelerated velocity, will be uniformly de- scribed with the mean of the velocities at the beginning and end of the n ih second ; which mean velocity will = 1(2 n 2 + 2 ) = (2 n - 1) P. The whole distance described during n seconds will be proportionate to the square of the time, and = n 2 p. At the end of the descent the acquired velocity will be proportionate to the whole time elapsed, and = 2 n P in a second. During the descent equal increments of velocity 2 p are generated during each second. Hence the effect produced by gravity may be regarded as constant for so small a distance as the body describes while falling freely near the earth's surface. To construct the Obelisk. When a body falls from a state of rest, near the earth's surface, by the action of gravity, the time elapsed and the velocity acquired at the end of the descent will vary as the square root of the distance described. A body falling from rest will describe a straight line. B 2 THE LOST SOLAR SYSTEM DISCOVERED. Let the point whence the body begins to fall be the apex of the obelisk, and the distance described be along the axis. (Jig. 1.) If at the end of the descent a straight line be drawn perpendicular to the axis, and made = the square root of the axis, this line will be an ordinate, and equal the square root of the axis. * Since the ordinate varies as axis Fig. 1. and time varies as distance 2) the ordinate will represent the variation of the time of descent, and the axis that of the distance described. So that, when the body has de- scended 1 P along the axis, let an or- dinate be drawn at the distance of unity from the apex and made = VI, or 1 ; this ordinate will represent 1 second, the time of describing 1 p along the axis. Again when the body has fallen from the apex to a distance of 4 p, there draw an ordinate = >v/4 = 2, which will represent the time 2 seconds, during which the body fell from rest to a distance of 4 p. When the body has fallen from the apex to a distance of 9 P, there draw an ordinate = A/ 9 = 3, which will represent 3 seconds, the time of falling 9 p. Thus any number of ordinates may be drawn, and each made = the axis 2 . When the extremities of these ordinates are joined by straight lines, the area included by these lines, the axis and the last ordinate will be an obeliscal area. The ordinate of an obeliscal area will = in units the number of seconds elapsed during the descent from the apex to the ordinate ; and the axis will = in units the number of p's described during the descent from the apex to the ordinate. As the time and velocity both vary as the square root of the distance, and at the end of 1, 2, 3, 4 seconds 2, 4, 6, 8 P, are the acquired velocities, VARIATION OF TIME AND VELOCITY. 5 Then since ordinates made equal the square root of the axes represent the times, or number of seconds elapsed during the descent ; it follows, that double ordinates, or ordinates twice the length of the corresponding time ordinates, will represent the velocity acquired in the descent from the apex to these ordinates. As the n ih velocity ordinate will equal 2 n, or twice the corresponding time ordinate, so an additional ordinate like the time ordinate may be drawn on the other side of the axis ; these together will represent the velocity ordinate. So that during n seconds the distance described will =n 2 P, and the velocity acquired at the end of the descent will = 2 n P in a second. When the ordinates (fig. 6.) 1, 2, 3, 4, &c. are bisected and joined at the extremities by straight lines, an obeliscal area is formed equal to that of fig. 1. An obeliscal sectional axis is the part of the axis inter- cepted by two consecutive ordinates, and are as 1, 3, 5, 7. An obeliscal sectional area is the area included between two consecutive ordinates. Sum of n sectional axes = whole axis. or Sum of n ordinates=l+2 + 3-f-4 = l?z + l . n Difference = \ n 1 . n. Hence the difference between the sum of the sectional axes, or whole axis of the obelisk, and the sum of the cor- responding ordinates will equal \ axis \ ordinate ^tf ^n. Figs. 2. and 3. will represent 1st series, 1, 2, 3, 4 time ordinates. 2nd 2, 4, 6, 8 velocity ordinates. 3rd 1, 4, 9, 16 axes, or D. 4th 1, 3, 5, 7 sectional axes, or d. 5th 1, 2, 2, 2 sectional increments. The 1st series represents the time ordinates. The 2nd series the velocity ordinates. The 3rd series their correspond- ing axes, or distances D, described from the apex to the time or velocity ordinates. The 4th series the sectional axes, or dis- B 3 6 THE LOST SOLAB, SYSTEM DISCOVERED. tances d, described during successive seconds. The 5th series is formed by taking from each term of the 4th series Fig. 2. Fig. 3. the term immediately preceding. Similarly, the 4th series is formed from the 3rd series ; the 5th series form the incre- ments of the 4th series; for 4 terms of the 5th series = the 4th term of the 4th series. So the terms of the 4th series form the increments of the 3rd series ; since 4 terms of the 4th series = the 4th term of the 3rd series. Generally n terms of the 5th series = the n ih term of the 4th series ; or n terms of the 4th series = the n ih term of the 3rd series. The sum of 4 terms of the 5th series, the increments of the 4th series, described with accelerating velocities, will = the sectional axis 7, described with an accelerating velocity during one second. Also the mean of the extreme velocities with which the sectional axis 7 would be described in the 4th second =-J(6 + 8) = 7. Also n terms of the 4th series, the sectional axes, or distances d, described in n successive seconds, will = the n ih term of the 3rd series, or whole axis, or distance D, described in n seconds. TIME, VELOCITY, AND DISTANCE. 7 The mean velocity with which the axis or whole distance D, (w 2 p) would be uniformly described in n seconds = I the extreme velocities = ^(o + 2 . TZP) = HP in a second. The mean velocity with which the sectional axis 2nl . P, described in the n ih second would be uniformly described in one second = ^ the extreme velocities = \ (ni x 2p + n x 2p) = 2n I . P. Or, let the distance described = 100 . P = axis. The time ordinate will= ^100=10 seconds, and the velocity acquired at the end of 10 seconds, or of the descent, will = twice the time ordinate = 2 A/H)0 = 20 . p. If this acquired velocity were continued uniform during another 10 seconds, the distance described would = 10 x 20p = 200. P= twice the distance described, when the body fell from rest till the acquired velocity equalled 20 . P a second. The velocities acquired and the distances described at the end of 1, 2, 3, 4 seconds, are 2, 4, 6, 8 P velocities, and 1, 4, 9, 16 P distances. The distance described in 4 seconds with an accelerating velocity will = the distance described uniformly in 4 seconds with the mean velocity As the body had no velocity at the beginning of the descent, the mean velocity will = half the last acquired velocity. Hence with half the velocity acquired at the end of 4 seconds, if continued uniform during 4 seconds, the distance described would = the distance described in 4 seconds with an accelerating velocity. Thus the axis of the obelisk represents the distance de- scribed. The single ordinate, made = the square root of the distance or axis, will represent the time elapsed during the descent, and the double ordinate will represent the velocity acquired at the end of the time, or descent. B 4 8 THE LOST SOLAR SYSTEM DISCOVERED. The different distances intercepted by the ordinates, or the sectional axes, will represent the distances 1, 3, 5, 7p, described during the 1st, 2nd, 3rd, 4th seconds. The dis- tances 1, 3, 5, 7 also correspond with the mean velocities, or with the mean of the velocity ordinates at the beginning and end of each second. The axis and ordinates are multiples of the same unity, that of the obelisk, Unity in the axis = 1 . P Unity in the velocity ordinates = 1 . P but unity in the time ordinates = 1 second. The variation of velocity and distance described during each of six successive seconds will be seen below, where s, denotes seconds ; v, velocity at the beginning of each second ; g, the additional effect of gravity during each second ; d, distance described in each second ; v' 9 velocity acquired at the end of each second ; D, the whole distance described at the end of the several seconds. s. v. g. d. v f . D. 1st. 0+1= 1 . . . . = 2 .... = 1 2nd. 2+1 3 . . 4 .... = 4 3rd. 4+1= 5 . . . . = 6 . . . . = 9 4th. 6+1= 7 . . . . = 8 . . . . = 16 5th. 8+1= 9 . . . . = 10 = 25 6th. 10+1 = 11 . . . . = 12 = 36 30" 36 42 Half the sum ofv + half the sum oft/ = ^30 + 42 = 36. Or the mean of the sum of the velocities at the beginning and end of each of the six seconds = 36 = sum of the distances d, described during six seconds = whole axis = ordinate 2 = 6 2 . The 36 described with an uniform velocity during six Q (* seconds will =-=6 during each second. The mean of the velocities at the beginning and end of six seconds =^0 + 2x6 = 6. THE OBELISCAL AREA. 9 Let z. d denote the increment of d in a second, then during s. i. d. d. 1st. =1 . . . . = 1 2nd. 2 . . . . 3 . . 3rd. 2 .... 5 . . . . 4th. 2 .... 7 . . ~7 16 = 16 The sum of i. d = T = d, described in the fourth second. The sum of c?=16 = D, described during the four seconds. Let i. v denote the increments of velocity at the beginning and end of each of the four seconds. Then at the beginning of the s. 1st. 2nd. 3rd. 4th. . v. =0 . =2 =2 . v. =0 =2 =4 =6 At the end of the s. i. v. 1st. =2 . 2nd. =2 3rd. =2 4th. =2 v. =2 =4 =6 =8 The sum of.z. v at the beginning of the fourth second = 6; at the end =8. Also the acquired velocities at the beginning and end of the fourth second are 6 and 8. The mean =^6 + 8 = 7= the distance described in the fourth second. The Obeliscal Area. An obeliscal area =^ the area ofyzV/. 3, or the whole of Jig. 1 . or 6., and is composed of sectional areas intercepted by the 10 THE LOST SOLAR SYSTEM DISCOVERED. ordinates 1, 2, 3, 4, 5, 6, or defined by the sectional axes, 1,3,5,7,9, 11. Fig. 3. 1st sectional area = -J- or \ of 1 or I 2 2nd = 4^ 9 3 2 3rd =12i 25 5 2 4th =24^ 49 7 2 5th =40i 81 9 2 6th =60| 121 II 2 The area from the apex to the 1st ordinate =^, and f the circumscribing parallelogram = a parabolic area = f axis x ordinate=fl x l=-f. Difference = -j=-J- unity. Area from the apex to the 2nd ordinate =-- + 4^- =5. | axis x ordinate =|4x2 = |8 = 5^. Difference = 5J - 5 = J = -f . Area from the apex to the 3rd ordinate =5 + 12-J-= 17-Jv f axis x ordinate = f 9 x 3 =--27 = 18. Difference =18-17== Thus the curvilinear or parabolic areas will exceed the obeliscal areas contained by straight lines by -J-, f, -f-, -|, f, -J-, corresponding to the ordinates 1, 2, 3, 4, 5, 6. So that the difference between the curvilinear area and the area included by straight lines, or the parabolic and obeliscal areas at the 6th ordinate will be six times greater than the difference between these two areas at the 1st ordi- nate. The difference between the two areas at the 1st and nth ordinate will be as -^1 : ^n. Thus as n increases the two areas will continually approach to equality; since -fw 3 %n will continually approach to -f?i 3 . For parabolic area =- axis x ordinate. = f ft 2 x n = | n 3 . and obeliscal area=f/z 3 -J-w. Figs. 4, 5. The sum of the series Axis =1 + 3 + 5 + 7 + 9+ Il=w 2 = 36. SECTIONAL AREAS. 11 Ordinate = 2/z. f axis x ordinate =|*= 6 3 = 288, and 288 - 286 = 2 = 16 = \n. . 4. JFty. 5 a. Hence the sum of the series =^n 3 ^n, and J the sum of the series = frc 3 -g-w = the single obeliscal area. .Fzgr. 3. The sectional areas along the sectional axes, I, 3, 5, 7, 9, 11, &c., and between the ordinates and 1, 1 and 2, 2 and 3, &c., are 1 3 5 7 9 11 0x1+1= 1= = 41 = Ix I 3x1^ 5x21 6x4 + 1 = 24^ = 7x31 8x5 + 1 = 401= 9x41 .0x6 + 1 = 60^ = 11 x5J 143 = i of I 2 3 2 9 2 II 2 12 THE LOST SOLAR SYSTEM DISCOVERED. circumscribing parallelogram = axis x ordinate = 36x6 = 216. Parabolic area = f 2 1 6 = 144. 144-143 = 1 Obeliscal area = fra 3 - \n = |6 3 - 16 = 143. Though the actual difference between every two corre- sponding obeliscal and parabolic sectional areas equals ^ unity; yet the relative difference between two such areas will be greater nearer the apex, and less as the ordinates re- cede from the apex. Generally the corresponding areas of the n ili section will be as . 2nl 2 : 2n\* + n When w = 6, the areas will be as 60^ When n= 12, the areas will be as 264| : 266 J. The sum of the two ordinates = the axis of an obeliscal sectional area. As the successive sectional axes, or distance between the two ordinates, are continually increasing by 2, while the difference between the two ordinates, unity, remains the same, it fol- lows that the opposite sides of the single obelisk (j^.6.), will continually approach to parallelism, but which they can never attain ; for how great so- ever the sectional axes, or the sum of the two ordi- nates may be, still their difference will equal unity, so the sides of a sectional obeliscal area can never become parallel to the axis. The two sides of an obeliscal sectional area are always equal, and the two ordinates are always parallel. If the two ordinates were also equal, then the four sides would form a rectangular parallelo- gram, the opposite sides of which would be parallel Fig. 6. to each other, as are the ordinates. An ordinate equal the mean ordinate of any obeliscal sec- tional area will always correspond to an axis equal to the distance from the apex to the point of bisection of that sec- tional axis, less ^ unity, a constant quantity. For the sectional axis intercepted by the n 1 and n ih ordinates = 2w 1, the half of which = n \ = the mean of the two ordinates n 1 and n. SUMMATION OF OBELISCAL SERIES. 13 So the whole axis from the apex to the point of bisection of the sectional axis will = ri* (n i)=7z 2 7Z + |. But the axis corresponding to the ordinate n \ will = n \ = ft 2 TZ + ^ , which is less than ri 1 n + by ^. Hence the mean ordinate of the 1st sectional area, which = , will be at the distance from the apex = = unity; so that an or- dinate drawn at ^ from the apex, and made = J unity, will be an ordinate to the parabola. The parabolic area of the 1st section will be to the corre- sponding obeliscal area : : : J : : 4 : 3. The ordinates of the parabolic and obeliscal area are equal at the beginning and end of each section, but the intermediate ordinates of the parabola are greater than the corresponding intermediate ordinates of the obeliscal area. This difference of the ordinates makes a sectional area of the parabola exceed the corresponding sectional obeliscal area by unity. If the double ordinates, like the velocity ordinates, were made ordinates of an obeliscal area ; then the successive sec- tional areas would equal I 2 , 3 2 , 5 2 , 7 2 (Figs. 3, 4, 5), or equal twice the single obeliscal series of sectional areas of Figs. 1. or 6. Then each parabolic sectional area will exceed the corresponding obeliscal sectional by of 1. The Construction and Summation of Obeliscal Series. The sum of the series 1 + 2 + 3 +4, &c. = 1 ~n+\ . n. Fig. 7 2. The number of squares of unity = 1 + 2 + 3 + 4 + 5 + 6 = J the area of the triangle + 1 6 = | 6x6 + i 6 = n x n+ n = n+l . n. Fig. 7. The sum of the series ! 2 + 2 2 + 3 2 + 4 2 , &c. = 14 THE LOST SOLAR SYSTEM DISCOVERED. | axis x ordinate = f the circumscribing parallelogram, or n-\- 1 . n . n + J. | ' / I/H| f * _: 1 3 ! US d i: 4 | i ^ CO tfl Fig. 7. 7. 72. Foraxis = sum of the series 1 + 2 + 3+4, &c. = | n+ 1 . ft ; but here the ordinate = ^ of unity more than the number of terms, or side of the last square. Or ordinate = n + -% Sum of the series = axis x ordinate = fofi n+l . n . = \ n+l . TZ . H+-J The circumscribing parallelogram will = \ = axis x ordi- nate. Also by construction the sum of the areas limited by the ordinates will equal the sum of the corresponding squares = J n+l . n . w + i. For the straight line joining the two ordinates 8|- and 7|- cuts off a triangle from the square of 8 = the triangle added to the same square ; consequently the area contained by this SUMMATION OF OBELISCAL SERIES. 15 straight line, the sectional axis 8, and the two ordinates will = the square of 8. These series of areas would form an obeliscal area = the sum of the corresponding squares. Fig. 7. The ordinate of the series of squares = TZ + -J, the square of which = n+I . n + ^ twice the axis of the squares + \. In order to construct a parabolic area, the axis should vary as the square of the ordinate. If n + -J-, the ordinate of the series of squares, be made the ordinate of a parabolic area, the corresponding axis should = \(n + \ . w + i) = -i-w+1 . ft + -i- or = \ ordinate 2 of the parabolic area = the axis of the squares + -|. Hence the parabolic area will have an axis greater than the series of squares by i unity ; or equal % ordinate 2 = This parabolic area will = f axis x ordinate = f of -i- ordinate 2 x ordinate = ordinate = The apex of the parabola will be in the produced axis of the squares at the distance of \ above the first square. The n th ordinate of the squares, which = w + i, will be common to both areas ; but the parabolic area being curvilinear, the ordinate will continually vary as axis * from the apex to the n ih ordinate, which parabolic area so generated will be to the corresponding series of n squares, Difference = Fig. 7. The difference between the two areas at the 8 th ordinate, which are as 204-708 : 204, will = -708, or =JLj = .JL. + . 5 i r and -^ + 7^=:^ + ^. When 7z = 24 the two areas are as 4902-041 ! 4900. 16 THE LOST SOLAR SYSTEM DISCOVERED. Difference = 2-041 = 2 -j L 2 n + -z^i L 2 f n s q uar es of unity + -% of 1 square of unit, .Fzy. 7. The parabolic area corresponding to the series of squares has the apex 1, above the single obeliscal or para- bolic area on the other side of the axis. In order to com- pare the two parabolic areas having a common axis, let the two apices coincide. The parabolic area corresponding to the obeliscal area., will be to the parabolic area corresponding to the series of squares, as f axis x ordinate6 : f axis x ordinate V 72, f axis x axis 5 I axis x 2 axis 3 i : 2* or as side to diagonal of a square. Hence the first double Darabolic area will be to the parabolic area of the squares, as 2 : 2* 2* : i or as diagonal to side of a square. The difference between the 1st parabolic area and the 1st square, or the difference between the two areas to the 1st ordinate, will = ^n + -^ = iV + *V = i- The difference between the two areas to the 2nd ordinate will = iV* + -h = A + A from which take T ^ + -^ 9 the 1 st difference, and -^ will = the difference to be added to the 2nd square to equal the corresponding parabolic sectional area. So the difference between every two corresponding sec- tional areas in succession will = -^ unity. As n increases, the area of the series of squares \ n + 1 . n . n + ^-, will continually approach to equality with \ n + the corresponding parabolic area; though their difference -^n + 2^ will continually increase. Also, whatever be the increase of n, the last, or ra th square SUMMATION OF OBELISCAL SERIES. 17 will be less by ^ unity than the corresponding parabolic sectional area. The parabolic area may also be represented in terms of the axis. For parabolic area = f axis x ordinate, = axis x 2 axis *> = ^ 2 axis x 2 axis = -i- 2 axis i The series of squares may be so arranged that the axis shall divide the series into two equal parts. Fig. 8. The sum of the series 2 2 + 4 2 -f 6 2 , &C., will = f n+l . n . 2n 1, or = f w+1 . ra . Since the sum of 1+ 2 + 3, &c. = n+l . n, . . the sum of 2 + 4 + 6, &c. = n + l . n, = the axis of the series 2 2 + 4 2 + 6 2 , &c., and the ordi- nate will=2ra-f 1, or 2 . n + \ f axis x ordinate, or f n+l . n . 2 n + , or | ?Tn . ?z . w + i, will = the sum of the series 2 2 -(-4 2 +6 2 . VOL. i. c 18 THE LOST SOLAR SYSTEM DISCOVERED. Thus the axis and ordinate of n terms of the series 2 2 + 4 2 + 6 2 will be double the axis and ordinate of n terms of the series ! 2 + 2 2 + 3 2 , and their areas will be as their rectangles, or as 4 I 1. The parabolic area corresponding to the series of squares 2 2 + 4 2 + 6 2 will have an axis = the axis of the squares + J, in order that the parabolic axis may vary as ordinate 2 of the squares, or vary as (2 . n + 1-) axis of squares = n + 1 . n ordinate = 2 n+\ ordinate = 4 (ra+-J-) 2 = 4 (n+1 . rc + i) ^ ordinate 2 = + 1 . w + -J- = axis of squares + \ = axis of parabolic area. Hence parabolic area will = axis x ordinate = f of -J- ordinate 2 x ordinate $ = % ordinate 3 = (2 . n + i) or axis x ordinate = -- axis x 4 axis ui __ = -| axis x 2 . axis = axs 2 . The parabolic area will be to the corresponding series of n squares as -J. (2 . ?rHL) 3 : | M~l n . ^+i The difference = |(i w z^. 4, 5. The sum of the series I 2 4-3 2 -f5 2 + 7 2 , will = A n*-n. It has been shown that the single obeliscal area = (l a + 3 a + 5 2 + 7 2 ) = f w 3 -^7z., (^. 3.); consequently the double obeliscal area, or the sum of ! 2 + 3 2 + 5 2 + 7 2 , will = The single parabolic area = |^TZ S ,*. the double parabolic area will 4 ?i 3 . SUMMATION OF OBELISCAL SERIES. 19 The single parabolic area exceeds the single obeliscal area by i unity in each, corresponding sectional area. .*. the double parabolic area will exceed the double obeliscal area ^ unity in each sectional area. 1st S. 1+4 + 9 + 16 + 25 + 36 = 91 2nd & 4 +16 +36 = 56 3rd S. I +9 +25 =35 Sura of the 1st series to n terms = 91 when n = 6. Sum of \n terms of the 2nd series = 4 times the sum of ^n terms of the 1st series ; as 1+ 4+ 9 = 14 and 4 + 16 + 36 = 56 = 4x14; or n terms of the 2nd series = 4 times n terms of the 1st series, and n terms of the 3rd series = n 3 ^n. Hence sum of n terms of 1st series = \ n+\ . n . n terms of 2nd series = n+l.n.n n terms of 3rd series = -f- n?n. 1st series = ! 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 = 91 2nd series = 2 2 + 4 2 + 6 2 + 8 2 + 10 2 + 12 2 = 364 3rd series = ! 2 + 3 2 + 5 2 + 7 2 + 9 2 +ll 2 = 286 when n = 6. The difference between the 2nd and 3rd series will equal 3 + 7 + 11 + 15 + 19 + 23 = 78, or 2n* + n=S. 1st S. = ! 2 + 2 2 + 3' 2 + 4 2 + 5 2 + 6 2 = 91 2nd S. = 2 2 +4 2 +6 2 = 56 3rd S. = I 2 +3 2 +5 2 =35 S. 1st = \ . n+ 1 . n . 7Z+^=91 when n = 6 S. 2nd = n+l. n.n + \ =56 when n 3 S. 3rd = difference =35^ = ^n 3 \n =35 when n = 3. c 2 20 THE LOST SOLAR SYSTEM DISCOVERED. Sum of 1 2 + 2 2 + 3 2 = 14 and 4 x 14 = 56 = sum of 2nd series. Hence the sum of the 2nd series = 4 times the sum of -L n terms of the 1st series = 4 x 14 = 56. The difference between the two series = the sum of \n terms of the 3rd series. To sum the series 1+3 + 5, &c. sum of 1+2 + 3 + 4 + 5 + 6 = 21 2 + 4 + 6 = 12 1 + 3 + 5 =9 1 + 2 + 3 = 6, and 2x6 = 12 = sum of second series, which subtract from the first series = 21 12 = 9 = sum of 3rd series. Or, S. of -|-?z terms of the 1st series x by 2 = S. of \ n terms of the 2nd series, which subtracted from n terms of the 1st series = S. of -J- n terms of the 3rd series. The Formation of Increasing Series from a Series in which all the Terms are equal, excepting the first. By reversing the order of the three series, the least will be placed the first, from which the other two increasing series will be formed thus : 1, 2, 2, 2, 2, 2 Sum = 2w-l. and forms 1, 3, 5, 7, 9, 11 = ra 2 and forms 1, 4, 9, 16, 25, 36 = w+l..n + | and forms 1, 5, 14, 30, 55, 91 The first series represents the incremental distances de- scribed in each second more than was described in the pre- ceding second. The second series represents the distances described in each of the n seconds. So that the distance described in the ra th second will = the sum of the incremental distances described during n seconds. The third series represents the whole distances described during the several descents from the apex to the different ordinates ; as the whole distance described during n seconds SUMMATION OF OBELISCAL SERIES. 21 from the apex to the rc th ordinate will = the sum of the dis- tances described during each of the n seconds. The formation of these series may be further illustrated by the triangle, fig. 7 2., where the first horizontal line =1+2+2+2+2+2 = 6 times 2 less 1 = 2x6 1 = 11 = n times 2 less 1=2^1. Again, 2^ 1 forms the columnar series 1, 3, 5, 7, 9, 11, the sum of which series = the area of the triangle when each square =2, and each \ square =1. 1st series, 1+2 + 2 + 2 + 2 + 2 Sum = 2ra 1. 1 +2+2+2+2 1+2+2+2 1+2 + 2 1+2 1 2nd, formed from 2n 1 = 1+3 + 5 + 7 + 9+11. Sum = w 2 . The area of the triangle = 1 + 2 + 3 + 4 + 5 + 6 squares, each = 2 in area, less 6 half squares, = ^ n+ I . n less ^n = 21 3 = 18 squares. Or the triangle will contain 36 half squares = sum of Next, 1+3 + 5 + 7+ 9 + 11 1 + 3 + 5+ 7+ 9 1 + 3+ 5+ 7 1+ 3+ 5 1+ 3 1 1+4 + 9 + 16 + 25 + 36 1 + 4 + 9+16 + 25 1 + 4+ 9+16 1+ 4+ 9 1+ 4 1 = n+l . 1+5+14 + 30 + 55 + 91, which is formed from \n+\ .n.n +i. c 3 22 THE LOST SOLAR SYSTEM DISCOVERED. 5 4- 3 a 1 The series 1+4 + 9, or ! 2 + 2 2 + 3 2 is represented by the complementary area of the obeliscal series,./?^. 7.. These series and others may be formed from the column of units, and line of twos, by adding toge- ther two numbers in a diagonal line to form a third; the third with its diagonal number will form a fourth, and so the numbers may be in- creased to any extent. As2+l=3 2+3= 5 3+1=4 5 + 4= 9 4+1=5 9+5=14 1 1, 2, 2, 2, 2, 2 1, 3, 5, 7, 9, 11 1, 4, 9, 16, 25, 36 1, 5, 14, 30, 55, 91 I, 6, 20, 50, 105, 196 1, 7, 27, 77, 182, 378 1, 8, 35, 112, 294, 672 1, 9, 44, 156, 450, 1122. Fig. 7. a. The sum of any line of numbers = the number below the last term = the sum of the preceding column : as the line = 1+5 + 14 =20, and the column =2 + 3 + 4 + 5 + 6 =20. The two series cross each other at 5, the last term but one in both series. The last terms of the two series together =14 + 6 = 20, the sum of either series. Again, 36, the axis corresponding to the 6th ordinate of the obeliscal area, or distance described in 6 seconds, = the sum of 6 sectional areas, or = the line of series 1 + 3 + 5 + 7 + 9 + 11 = 36. 25 is the distance described in 5 seconds, 9 " " in the 5 th , and 9 + 2 = 11 " " 6 th . The distance described in 6 seconds = 25 + 11 =36= the column 2 + 9 + 25. SUMMATION OF OBELISCAL SERIES. 23 Thus the line of series = the column of series = the sum of the last terms of both series = the sum of either series. Fig. 7. The sum of the series of cubes of 1, 2, 3, 4, 5, 6, Forl +2+3 +4+5 +647 +8= n+l . n = axis = 36, and l 3 as =1296, as before call the ordinate n + J, then the last or 8 th ordinate = 8 '5, axis x ordinate =36 x 8*5 9 2 \ axis x ordinate = i 36 x 72-25 =1300-5, but the series of cubes = 1296 difference = 4-5 _ s_e 8 The first ordinate will =1*5, then \ axis x ordinate = |-1 x 1/5 =1*125, 1st cube= 1 _ difference= -125 = The 2nd ordinate = 2-5, 2 axis x ordinate = J3 x 2J-5 =9'375, The 2 cubes = I 3 and 2 3 =9 difference = -375 = I-. So -J- axis x ordinate exceeds the 1st cube by -J- cube of 1, the 1st and 2nd cubes, or I 3 + 2 3 " f, f, I 3 So the sectional solids having ordinate =(ra + ^) 2 exceed I 3 , 2 3 , 3 3 , 4 3 , 5 3 , 6 3 , 7 3 , 8 3 by i, f, -I, V } , V V 8 > V less i. JL. JH U> L5 2_J 28 1COP 8 , 8 , 8 , 8 8 > '8 > "8 or JL A A A JL 8 8 8' 8' 8' 8' 8 8' 8"> the sum of which = 3 ^ 6 , or 4-J- cubes of unity for the series of 8 cubes = - axis. c 4 24 THE LOST SOLAR SYSTEM DISCOVERED. Or sum of the 8 sectional solids \ axis x ordinate 2 . = axisx8-5 2 . " " =i axis x 72 -25. " " = (i axis x 2 axis) + \ axis. " " = axis 2 + ^ axis. Instead of taking the ordinate = (n + 1) 2 , let the ordinate 2 = ('ft + i x ) 2 1 axis x ordinate 2 = i ft + 1 . nx ft + 1 . w ? axis x ordinate = ^ (ft + 1 . ft) 2 content = Q ?/+ 1 . w) 2 = axis when ft = 8 = (J9x8) 2 = 36 2 =1296 = the content of the 8 cubes. Thus the series of n cubes of 1, 2, 3 &c., will = (-J- n + 1 . rc) 2 _.... jr _ J _. o o = axis = as many cubes of 1 as the axis contains squares of 1. Since the ordinate 2 a, axis, or ordinate a axis^, the solid will be of the parabolic form, and the content = the sum of the series of cubes, both having equal axes. Fig. 8. Sum 2 3 + 4 3 + 6 3 + 8 3 + 10 3 -1800 axis = 2 x J n-t I . n 2 x axis 1 + 2 + 3 = 2 + 4 + 6 &c. =n+l . n. Let ordinate = 2n + 1 axis x ordinate 2 = n + 1 . n . (2ra + l) 2 . Here the sectional solids having ordinate 2 c=(2w + I) 2 will exceed 2 3 , 4 3 , 6 3 , 8 3 , 10 3 by 1, 3, 6, 10, 15 less 1, 3, 6, 10 or 1, 2, 3, 4, 5, the sum = Jw+ 1 . n= 15, or 15 cubes of 1 for the series of 5 cubes, or 36 cubes of 1 for the series of 8 cubes, which = axis. SUMMATION OF OBELISCAL SERIES. 25 Let the ordinate 2 = (2w + I) 2 1 . w) axis x ordinate 2 = ?z + 1 . n . 4(n + 1 . n) | axis x ordinate 2 = 2(w + 1 . nf content = 2 axis 2 = 2 x (6 x 5) 2 = 1800, when n =5. _ i Here the ordinate a axis , so the solid will be parabolic, and the content = the series of cubes, both having equal axes. The content of n terms of 2 3 + 4 3 + 6 3 =8 times that of n terms of ! 3 -f 2 3 -t-3 3 . Thus the series of n cubes of 2, 4, 6 will = 2(n+l . ra) 2 = 2 xaxis 2 * Figs. 4, 5. Sum 1 3 + 3 3 -f 5 3 + 7 3 + 9 3 + 11 3 = 2556 axis = 1 + 3 + 5 &c. =n 2 . Let ordiuate =2/z, then ordinate will a axis 2 , and a para- bolic solid will be generated by the ordinate 2 , or 2n ---- n - .,_._._-_ -,r. 9 _ .O \ axis x ordinate = |-w 2 x 2n when ?z = l=ll 2 x2xl 2 = 2difference = 2 1=1 6 cubes =2556 difference = 36 When n = 1 difference = 2 1=1 n = 2 = 32- 28= 4 n = 3 = 162- 153= 9 n = 6 =2592-2556 = 36. The parabolic sectional solids when ordinate 2 =2w 2 will exceed I 3 , 3 3 , 5 3 , 7 3 , 9 3 , II 3 by ], 4, 9, 16, 25, 36 less 1, 4, 9, 16, 25 or 1, 3, 5, 7, 9, 11 26 THE LOST SOLAR SYSTEM DISCOVERED. Sum = ra 2 = 6 2 = 36, or 36 cubes of 1 for the series of 6 cubes = axis. Let ordinate 2 =(2ra) 2 -2 i axis x ordmate 2 =n 2 x(2n 2) = 2n* n 2 when w = l = |! 2 x(2 2 -2)=1 tt = 2 = 2 2 x4 2 -2 = 28 or sum of 6 terms of ! 3 + 3 3 + 5 3 &c. 2556. Thus the parabolic solid = sum of the series of cubes + axis, both having a common axis, = 2rc 4 -rc 2 + /z 2 = 2rc 4 = 2 axis 2 ; therefore sum of ! 3 + 3 3 + 5 3 &c. =2 axis 2 axis = 2rc 4 ra 2 . The axis of the series of 8 cubes of 1, 2, 3, 4, 5, 6, 7, 8 (.fy* ^'} = '2 n + l.w = 36, and the content of the series =axis 2 . The axis of the series of 6 cubes of 1, 3, 5, 7, 9, 11, (figs. 4, 5.) = ra 2 = 36, and the corresponding parabolic solid, having the same axis, = 2 axis 2 . .-. the series of 8 cubes of fig. 7. = J the content of the parabolic solid corresponding to the series of 6 cubes of figs. 4, 5. I 3 , 2 3 , 3 3 , 4 3 , 5 3 , 6 3 = 1, 8, 27, 64, 125, 216 1, 9, 36, 100, 225, 441 = 1 3 , ! 3 + 2 3 , ! 3 + 2 8 + 3 3 = 1 2 , 3 2 , 6 2 , 10 2 , 15 2 , 21 2 . The series 1, 3, 6 is formed of 1, 1 + 2, 1 + 2 + 3. The n ih term of 1 + 2 + 3 &c. =^7+1 . w = i( a + w). Thus the sum of n terms of the cubes of 1, 2, 3, &c., = the square of the sum of n terms of 1 + 2 + 3, &c., =(|w+l . nf = Let 7z = 6, s = (Jx = 441 cubes of unity = a stratum of the depth of unity and area = 21 2 = 441. If the squares 1, 2, 3, 4, 5, 6 represent the 6 cubes, then a square stratum having the side = the sum of the ordinates SUMMATION OF OBELISCAL SERIES. 27 = w ill = the sum of 6 cubes = of 6 2 + 6) 2 = 21 2 = 441 cubes of unity. If the sum of the cubes were ! 3 + 2 3 + 3 3 = 36, then a square stratum having the side = 6 would contain 36 cubes of unity, the sum of the cubes of 1, 2, 3 ; or the square of the sum of the sides of the cubes of 1, 2, 3 will = the sum of the cubes of 1, 2, 3. For ! 3 + 2 3 + 3 3 = 36 = 6x6 cubes of unity; if 6 cubes of unity be placed along the side of fig. 7. b. the squares of 6, then the square would contain 6 times 6 units, or 6 columns of 6 units each. Or the number of cubes of unity in ! 3 + 2 3 + 3 3 = the number of square units in (1+2 + 3) squared = 6x6 = 6 times 6 columns of single squares = one column of 36 squares = the length of 36 linear units. Thus 36 cubits of unity placed side by side in a straight line will extend to the same distance as 36 squares of unity placed in a straight line, equal to a straight line of 36 linear units. Thus the length of a side of a cube of unity = the length of a side of a square of unity = the length of a linear unit. So that, in measuring distances, a cube of unity, a square of unity, and a linear unit are all equal in length. Fig. 7. Sum the obeliscal series of axis =1+2 ..................... +8 =i- 64 + 8 = 36, ordinate at the end of the 8th cube, = 72-25-'25 = 72 i axis x ordinate 2 = \ 36 x 72 = 1296. But the sum of the series of 8 cubes = the square of the sum of the sides of the 8 cubes 28 THE LOST SOLAR SYSTEM DISCOVERED. which = axis of the obeliscal series = J axis x ordinate . belisc ordinate 2 = 2 axis. In this obeliscal series of cubes the axis = ^ ordinate , or The ordinate at the end of every cube of the series will = the square of (the side of the cube-f-*5) less -25; which will = 2 axis. Or the axis being known, the ordinate will = 2 axis. Let a straight line = 6 linear units. Then 6x1 will form a rectangled parallelogram = 6 square units, and 6 rectangled parallelograms will form a square = 6 x 6 = 36 square units, the square of 6. The square of 6 x by 1 will form a square stratum =6x6 = 36 cubes of unity, and 6 square strata will =6 x 36 = 216 cubes of unity = 6 x 6 x 6= the cube of 6. Otherwise. Let a straight line = a linear unit = 1. Then 6 x 1 = 6, a line of 6 units in length. Next 1x1 = square of 1, 6 x square of 1 = rectangled parallelogram of 6 squares of unity, 6 x rectangled parallelogram = 6 x 6 = 36 square units, = the square of 6. Thirdly. 1x1x1= cube of 1, 6 x cube of 1 = parallelopipedon of 6 cubes of unity, 6 x parallelopipedon = 6 x 6= square stratum of 36 cubes of unity, 6 x stratum =6x6x6 = 216 cubes of unity, = the cube of 6. Thus 1 = a linear unit, 6 x 1 = a line of 6 units in length, 1x1 = square of 1, 6x6= square of 6, 1x1x1 = cube of 1, 6 x6x6=cube of 6. OBELISKS AND PYRAMIDS COMPARED. 29 SOLID OBELISKS AND PYRAMIDS. To compare a Series of Obelisks with a corresponding Series of Pyramids. Figs. 9, 10. The first section of the obelisk is a pyramid, the 1st in the series of pyramids, and therefore the apices of both will coincide. Fig. 9. Fig. 10. All the other sections of the solid obelisk are frustums or frusta of pyramids having their several apices beyond the 30 THE LOST SOLAR SYSTEM DISCOVERED. apex of the obelisk, in the produced axis. Their several bases will = the square ordinates of the obelisk. The 2nd pyramid has the side of the base = 2, the side of the 2nd square ordinate of the obelisk. The height or axis of the pyramid is bisected by the 1st ordinate, which = 1=^ tne second ordinate 2; and 3 is the distance between, or the sectional axis of the 1st and 2nd ordinates, . . axis of pyramid will =2x3 = 6, = ordinate x sectional axis, = n times the sectional axis, = nx 2n I, or = 2/z 2 -ra = twice the axis less the ordinate; or = 2 ordinate less ordinate of obelisk. Axis of the 2nd pyramid beyond the apex of the obelisk = 6 4 = 2 = w 2 n = ordinate less ordinate = axis of obelisk less ordinate. So the axis of the 1st pyramid will =2n 2 1=2 1 = 1, when n = l. The axis of the 1st pyramid beyond the apex of the obelisk wil\ = n 2 nVl=0 9 when n = l. Hence the axis of the pyramid, having the n th ordinate obelisk for the base will n. 2nl = n times the sectional axis of the obelisk, or = 2ra 2 n = twice the axis less the ordinate. The distance of the apex of the pyramid from the apex of the obelisk will n L n-= axis of obelisk less ordinate. The whole axis of pyramid = axis of obelisk + produced axis ; and produced axis = axis obelisk - axis obelisk, 1st axis = 1, produced axis = 1 \^ 0, 2nd = 4, = 4- I 4= 2, 3rd =9, = 9- J9= 6, 4th =16, =16-116 = 12, 5th =25, =25-125 = 20, 6th =36, =36-136 = 30. OBELISKS AND PYRAMIDS COMPARED. 31 Thus series of whole axes will be 1st = 1+ 0= 1, or 1, 6, 15, 28, 45, 66, 2nd = 4 + 2= 6, D.I, 5, 9, 13, 17, 21, 3rd= 9+ 6 = 15, D.I, 4, 4, 4, 4, 4, 4th = 16 + 12 = 28, 5th =25 {-20 = 45, 6th =36 +30 = 66. Produced axes will be 0, 2, 6, 12, 20, 30, D.O, 2, 4, 6, 8, 10, D.O, 2, 2, 2, 2, 2. Axis of pyramid = 2 axis obelisk axis obelisk, = 2 axis obelisk ordinate, = 2 axis obelisk axis^, = sectional axis obelisk x ordinate, = 2 ordinate 2 obelisk ordinate. The several distances of the apices of 6 pyramids from the apex of the obelisk will be 0, 2, 6, 12, 20, 30. If from the end of the 6th ordinate a straight line be drawn to a distance from the apex of the obelisk along the pro- duced axis =n? w = 6 2 6 = 30, that line will represent the side of a triangle or pyramid ; and the frustum of that pyra- mid, between the ordinates 5 and 6, will be the 6th sectional solid of the obelisk. The axis of a pyramid = ra 2 + ri* n = 2n 2 n content = axis x ordinate From which take the section having the area of its base = nl. The axis of this pyramid = 2n I . n 2n l = 2n*-n-2n+I content =^(2n' 2 - r 3n + 1) . n 1 hence the frustum will equal 32 THE LOST SOLAR SYSTEM DISCOVERED. or rc-w which =^(6n* 9n? + 5n 1), when rc = = i(6x216-9x36 + 30-l) = similarly the 5th frustum =183 4th = 86^ 3rd = 31| 2nd = 7 1st = content of the obelisk = 642 The cubes of the sectional axes 1357 9 11 are 1 27 125 343 729 , 1331 =2556 |= J 6| 31i 85| 182 332|= 639 obeliscal series = i 7 31| 86^ 183 333|=642 diiFerence = T V A A A A H=A= 3 Thus the solid obelisk will exceed ^ the series of cubes, or of I 3 , 3 3 , 5 3 , 7 3 , 9 3 , II 3 , by T V the cube of 1 for every unit of the axis; or by 1 cube for every 12 units of the axis, or by as many cubes of 1 as would extend T T ^ the axis ; or by a stratum of cubes of 1 that would cover -^ the square ordi- nate. rn ^ _ o Sum of the cubes of 1, 3,5 = 2 axis axis, whenw = 6> = 2x36 2 -36 = 2556 sum of the cubes = 639 _ ^ sum of cubes =^ (2 axis axis) = - axis = 648-9 = 639. Content obelisk =^ sum of the cubes + ^ axis = axis ^ axis + -jL. axis Conten t parabolic obelisk = ^~ axis x ordinate 2 = axs OBELISKS AND PYRAMIDS COMPARED. If the quadruple parabolic obelisk generated by the double, 2 or velocity ordinate squared = 2n , descending along the axis and varying as axis = \ circumscribing parallelepiped. 2 Sum of the cubes = 2 axis axis, = 2x36 2 -36 = 2556. Content of quadruple obelisk = sum of the cubes + i axis, 2 . = 2 axis axis + J axis, = 2 axis f axis, = 2x36 2 -24, = 259224 = 2568. Content of quadruple para-7 = \ circumscribing parallelo- bolic obelisk 3 piped, = -i- axis x ordinate, 2 = \ axis x 2n , = \tf x 4w 2 , = 2rc 2 x rc 2 , 2 = 2 axis , = 2 x 36 2 = 2592. The content of the obelisk exceeds \ the sum of the series of ! 3 + 3 3 + 5 3 , &c., by ^ the axis; or by ^ of a cube for every unit of axis of the obelisk. The content of the obelisk is less than the content of the parabolic solid by j? the axis, or ^ of a cube for every unit of the axis. Hence the content of obelisk will lie between ^ the sum of the cubes of 1, 3, 5, &c., and parabolic content. But the sum of the series of cubes of 1, 2, 3, &c., = axis > and parabolic content = \ axis 2 ; therefore content of obe- lisk will lie between ^ the sum of the cubes of 1, 3, 5, &c., and | the sum of the cubes of 1, 2, 3, &c., the axes being equal. Again, the sum of the cubes of 2, 4, 6, &c., = 2 axis ' therefore content of obelisk will lie between ^ the sum of the cubes of 1, 3, 5, &c., and \ the sum of the cubes of 2, 4, 6, &c. VOL. I. D 34 THE LOST SOLAR SYSTEM DISCOVERED. Or the content of the quadruple obelisk, generated by the (double ordinate) 2 , will lie between the sum of the cubes of 1, 3, 5) &c., and the sum of the cubes of 2, 4, 6, &c. If, on the produced axis of the obelisk (jft?. 10.), squares be drawn having their sides equal 2, 4, 6, &c., the differences between the values of n 2 n, where ra = l, 2, 3, &c., these squares will represent the cubes of 2, 4, 6, &c. ; and the squares having their sides = the sectional axes 1, 3, 5, &c., will represent the cubes of 1, 3, 5, &c. Thus the content of the single obelisk will lie between ^ the sum of the 1st and ^ the sum of the 2nd series of cubes, if the axes were equal. 7+ 9 + ll = rc 2 = 6 2 =36, and + 2 + 4+ 6+ 8 + 10 = tt 2 n =30; therefore 1+5 + 9 + 13 + 17 + 21 = 2rc 2 -w = 66. Here v? = axis of obelisk, ri*n axis produced, 2 ra 2 n= axis of pyramid. Area of the triangle corresponding to the pyramid having base = w, and axis = 2n 2 n = -^(2n' 2 ri) . n=zn 3 ^ri 2 . f area of triangle = f of (2n 2 n) . n _ 2 M 3 1-2 ^ n ? n Area of obelisk = f w 3 ^n, and | the circumscribing parallelogram =fra 3 . Hence the area of obelisk, which =|?z 3 --^, will lie be- tween w 3 , which = the circumscribing parallelogram, or = the parabolic area, and fw 3 je? 2 , which = f the tri- angular area formed by the vertical section of the pyramid. When the series + 2 + 4 + 6, &c., begins with (and is reckoned a term), the sum of n terms = n 2 n. When the series begins with 2 (and n is reckoned from 2), the series 2+4 + 6 + 8 + 10 = rc 2 + ra = 30, when n 5, which is the same as 6 terms of + 2 + 4 + 6 + 8 + 10, where n 2 n = 30. OBELISKS AND PYRAMIDS COMPARED. 35 Since series 1+ 3+ 5+ 7 = ft 2 and series 2+ 4+ 6+ 8 = rc 2 + ?z. Therefore series 3 + 7 + and series 1 + 5 + Therefore series 2 = 4 x axis , or = 2 axis of obelisk. Ordinate and sectional axis obelisk = axis of pyramid. 1st. Ix 1= 1 2nd. 2x 3= 6 3rd. 3x 5 = 15 4th. 4x 7 = 28 5th. 5x 9 = 45 6th. 6x11 = 66. 1st. 1, 6, 15, 28, 45, 66 axes of pyramids. 2nd. 1, 5, 9, 13, 17, 21 difference of axes. 3rd. 1, 4, 4, 4, 4, 4 difference between the last dis- tances. Hence the sum of the series of the axes of pyramids will, if formed by squares of unity = the sum of the areas formed by each sectional axis and its ordinates, which area will exceed the area of the obelisk by half the number of squares of unity of the series 1, 3, 5, 7, 9, 11, or ^ the squares of unity along the whole axis of obelisk or ordinate For 1+6 + 15 + 28 + 45 + 66 = 161 and area of obelisk = n* n. when 71=6 = 1441 = 143 area of obelisk + ordinate 2 =143 + 18 = 161. Thus each number of the series of axes is formed by its sectional axis x ordinate, and the sum of this series = area of _ 2 obelisk + -J- ordinate Or sum of series =area obelisk + ^ ordinate The sum of any number of terms in the second series will = the number itself in the first series immediately above the last of these terms, which sum will also = the sectional axis D 2 36 THE LOST SOLAR SYSTEM DISCOVERED. x ordinate. Thus the sum of 6 terms of the second series, or 1+5 + 9 + 13 + 17 + 21 = 66,, the number above 21, or will = the 6th ordinate x its sectional axis = 6 x 1 1 = 66, or, generally, = n x 2 n 1. The sum of the 3rd series will = 1+4 + 4 + 4 + 4 + 4 = 21 the 6th term of the second series, or, generally, = 1+71-1 . 4 or = 4 n 3 Each of the sectional axes of the obelisk 1, 3, 5, 7, &c. equals the sum of the two ordinates, or the difference of their squares ; for n + n l = 2/z--l Subtracting the less from the next greater axis of the series of pyramids gives the series 1, 5, 9, 13 for the differences between the axes of the pyramids. To sum of 1 + 3 + 5 + 7 + 9 + 11, &c. = ra 2 add + 2 + 4 + 6 + 8 + 10, &c. = rc 1 . n then S. of 1+5 + 9 + 13 + 17 + 21, &c. will = axis of the n ih pyramid. Or by making the 3d the 1st series and the 1st the 3rd, it will be seen that the sum of n terms of the 1st series will form each of the n terms of the 2nd series, and the sum of the 2nd series will form each of the n terms of the 3rd series, and the sum of the 3rd series will form each of the n terms of a 4th series. 1, 4, 4, 4, 4, 4, Sum=4?z-3 and forms 1, 5, 9, 13, 17, 21, 2nl . n and forms 1, 6, 15, 28, 45, 66, =fw 3 + |-?z 2 n. and forms 1, 7, 22, 50, 95, 161, Thus the axis of obelisk + the rectangle by the two ordi- OBELISKS AND PYRAMIDS COMPARED. 37 nates of the last section will = the axis of pyramid having the same base as the obelisk. The axis of pyramid =n 2 below, and rf n above, the apex of the obelisk, or whole axis of pyramid =2n 1 n . 5 and 2n 1, forms the series 1, 3, 5,:7, the sectional axes, or series of the differences between the series of the whole axes of obelisk. If to this series there be added the series 0, 2, 4, 6, 8, &c., formed from 2n 2, the distances between the several apices of pyramids, the sum of which series = + 2 + 4 + 6, &c. = n 1 . n = the rectangle by the two ordinates of the last section of obelisk = the portion of the axis of each of these several pyramids beyond the apex of the obelisk. Then will 2nl + 2n 2=4ra 3, the difference between the entire axes of pyramids, form the series 1, 5, 9, 13, &c. ; the sum of which =1 + 5 + 9 + 13, &c. = n 2 +n 1 . n 2n l . n = the whole axis of the n th pyramid. The series of the axes of pyramids will be 1, 6, 15, 28, &c., each term being formed by 2n 1 . n, or by sectional axis x ordinate of obelisk, which equals the whole axis of a py- ramid = n x n ih sectional axis. Again, n 2 n, the distance of the apex of the pyramid from the apex of the obelisk, forms the series 0, 2, 6, 12, &c., the sum of which series =0 + 2 + 6+12+20 + 30, &c. = ^ 3 -\n. n? forms the series of the whole axes of obelisks, 1, 4, 9, 16, &c. which = the series of the parts of the axes of pyramids below the apex of obelisk ; the sum of which series = 1+4 + 9 + 16, &c. = i(rc+i . n. Then n 2 n-\-n 2 = 2n 1 . n will form the series 1, 6, 15, 18, &c. ; the sum of which = 1 + 6 + 15 + 18, &c. = n 3 + i n 2 -jl?z, the sum of the series of entire axes of pyramids. The formation and sum of each of these three series will be 1. S. ofn*-n =0 + 2+ 6 + 12 &c. = ^ 3 - J. n 2. S.ofn 2 = 1+4 + 9 + 16&c. = j r ?z 3 + ^rc 2 + 1 lrc 3. I. Sum of axes of pyramids beyond the apex of obelisk D 3 38 THE LOST SOLAR SYSTEM DISCOVERED. 2. Sum of axes of pyramids below the apex of obelisk, which is also the sum of the axes of obelisk. 3. Sum of the entire axes of pyramids. Since the axes of the pyramids are as 2n 1 . n, the areas of the triangular vertical sections of the pyramids will be as J axis x ordinate, or (2n 1 . ri 2 ), or n 3 %n 2 . The areas of the triangles will be expressed by the differ- ence between the series of ri 3 and -J-ra 2 . 8. n* =1 + 8 + 27 +64 + 125 +216 = 441 S. n* =1+2 + 4J-+ 8+ 12-|-+ 18= 45-*- 8. of difference = +6 + 22-J- + 56 + 112-J-+198 = 395i The sum of the cubes of 1 + 2 + 3 = Q- n+l . nf, and the sum of their squares = J n+l . n . So half the sum = \n+ 1 . n . Hence the sum of their difference, or the sum of the tri- angular areas will . n-n+l . n. . To Sum the Series of Pyramids. Content of pyramid = ^ axis x ordinate. Here ordinate = n 2 , and axis = 2n=l . n. Pyramid =(2n 1 . n . n 2 ). - Sum of series f w 4 4 + 3 4 &c.) ==f of(+l . n . Sum of series w 3 =i (1 3 + 2 3 + 3 3 &c.) When w = 6, Sum of series fw 4 = 151 6f i^ 3 = 147 Sum of their difference = 136 9f, = content of the series of pyramids. SERIES OF PYRAMIDS SUMMED. 39 These series, when n G, will be f rc 4 =f+10f + 54 + 170f + 416|- + 864 = 1516f, n 3 =+ 2f + 9+ 21^ + 41f + 72= 147, dif. =1 + 8 + 45 + 149^ + 375 + 792 = 1369f, or content of series of pyramids = 1369f ; or 4 )=1516f, = 147, dif. = 1516f-147 = 1369f. The series n* x by 2n will form the series of f ra 4 ; n*=, 2-f, 9, 2H, 41f, 72, 2w=2, 4, 6, 8, 10, 12, ffi 4 = f, 10f, 54, 170f, 416f, 864. Hence the w th term in the series f ra 4 will = the n ih term in the series n 3 x by 2n ; as the 6th in fra 4 = 864 = 72 x by 6x2. Also the series ^n* multiplied by 2n l will form the series of f n*= %n* =i, 2f, 9, 2H, 41f, 72, 2?z~l =1, 3, 5, 7, 9, 11, fra 4 -- i-w 3 =i-, 8, 45, 149|, 375, 792. Thus the series n 3 is a pyramidal series, each term being = to a pyramid, ^ n 3 , which, multiplied by twice the ordinate, or 2n, will form the first series f n 4 . The third series, the difference between the series n 4 and -^w 3 , will be formed by multiplying the pyramidal series ^/z 3 successively by 1, 3, 5, 7, or the corresponding sectional axes 2n 1. Thus each pyramid, the frustum of which forms a section of the obelisk, will = ^rc 3 , or ^ ord 3 obelisk multiplied by the sectional axis of that ordinate, or = ?i 3 x 2n l= Hence the pyramid having its axis = 2n 1 . n, and base = rc 2 , the base of the obelisk, may be compared with the cor- responding obelisk having its axis = n 2 . J> 4 40 THE LOST SOLAR SYSTEM DISCOVERED. Content pyramid content obelisk. The sum of the series of cubes of 1, 2, 3, 4 = (n + 1 . /0 s axis 2 . For l+2 + 3 + 4 + 5-f 6=+i . rc = 21=axis. or sum of the cubes = (n -f- 1 . w) 2 = axis The sum of the series of cubes of 2, 4, 6, 8 = 2(w + 1 . rif = 2 axis For the axis of n terms of this series will = twice the axis of n terms of the series 1, 2, 3, 4, and each term in the 1st series = 8 times the corresponding term in the last series. Therefore S. of 2 3 + 4 3 + 6 3 + 8 3 will = 8 times the S. of = 2 (+ w) 2 9 = 2 axis The sum of the series of cubes of 1, 3, 5, 7, = 2 w 4 7Z 2 = 2 axis axis. From I 3 , 2 3 , 3 8 , 4 3 , 5 3 , 6 3 , take 2 3 , 4 3 , 6 3 , difference "T 3 ^ 3^ o^T" Let w = the number of terms in each of the two last series^ then 2n will equal the number of terms in the 1st series. S. of 1st, which =(iwTl . w) 2 , will now =(2a + l . 2^) 2 . 2w ) 2 ^. of 2nd series =2w SERIES OF PYRAMIDS SUMMED. 41 S. of 3rd series = difference - 2 = 2?z 4 w 2 = 2axis axis. _____ _ For axis = 1 + 3 + 5 = n 2 ; 2 axis axis = 2 strata, each stratum having an area = axis , and a depth of unity, less a line of cubes of unity = the length of the axis. In the 3rd series of I 3 , 3 3 , 5 3 , H = 3, axis = 1 + 3 + 5 Sum of I 3 + 3 3 + 5 3 = 2 axis 2 - axis = 2x3 4 -3 2 = 2x81-9 = 153. Or in the series I 3 , 3 3 , 5 3 , Sum = 2 axis axis = 2x9 2 -9 = 153. Otherwise, ! 3 + 2 3 + 3 3 + 4 3 + 5 3 + 6 3 = 441 23 +43 +6 3 = 288 I 3 +3 3 +5 3 =153. Sum of the 2nd series = the sum of 8 x n terms of the 1st series. The difference of the two series = the sum of \ n terms of the 3rd series. As sum of ! 3 + 2 3 f 3 4 = 36, and 8x36 = 288 = sum of 2nd series, which, subtracted from the sum of the 1st series 441, leaves 153, the sum of the 3rd series, M , ~^- f> Sum of series of ! 3 + 2 3 + 3 3 + 4 3 = axis . Sum of series of 2 3 + 4 3 + 6 3 + 8 3 = 2 axis 2 . These axes become equal at the 20th term of the 1st series, and the 14th term of the second series. Sum of 1st series =(iw+ 1 . ft) 2 = = (J-21x20) 2 42 THE LOST SOLAK SYSTEM DISCOVERED. Sum of 2nd series =2(/z+ 1 . ra) 2 = 2axis = 2(15 x!4) 2 = 2x210 2 = 88200. Or when the two series have a common axis, their contents will be as 1 : 2. When both series have the same number of terms, their contents will be as 1 : 8. Let each of the series 1, 2, 3, 4, 5, 6, 2 9 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, form an axis,^*. 7, 8. 5., then the series of squares described on the side of the axis will represent both squares and cubes, or areas and solids of an obeliscal form. The axis of the 1st series = n + 1 . n 9 2nd = n+\ . n, 3rd = n 2 , The ordinate of the 1st 2nd =2( 3rd = 2n. Or sum of = axs = n + = axis = rc 2 . The sum of their squares, or . n .n + =% axis x ordinate, 2 2 + 4 2 + 6 2 = area = |-rc + 1 . n . n + =$ axis x ordinate, 5 2 =area=|^ 3 -i?z=f axis -i ordinate. The sum of their cubes, or 2 3 + 4 3 + 6 3 = solid = 2(raTT . nf = Also the axis of the obeliscal area = n 2 = ordinate , area= n?n = - axis ^ordinate, SERIES OF PYRAMIDS SUMMED. 43 solid = !?i 4 -ira 2 , = |-axis -i-axis; and \ the sum of I 3 + 3 3 + 5 3 = -J(2 axis axis), _ 2 solid obelisk = -J- axis ^axis. .*. the solid obelisk is greater than -J- the sum of ! 3 + 3 3 + 5 3 by -J- axis -J- axis, or T Vaxis, or -j^w 2 . The obeliscal area= n B n=n* w= the sum of The axis 1 + 3 + 5 = n 2 is common to both the obeliscal solid and to this obeliscal series of cubes. The corresponding parabolic area=f 3 , solid = axis 2 . Let each of the squares in the series l 2 4-3 2 -|-5 2 -f 7 2 + 9 2 + II 2 represent a cube of unity. Fig. 5 a. Then these square strata, each having a depth of unity, will form a terraced pyramid, the content of which will = f n 3 \n in cubes of 1. The content of the rectilineal-sided pyramid having a height n, and side of base = 2/z, will=f n 3 , which will ex- ceed the content of the stratified pyramid by n cubes of 1. Next compare their sectional triangular areas, made by dividing each pyramid vertically into two equal parts. Height of the triangle = n =6. Side of the base = 2n =12. . . Triangular area, = ^2ra . n=I2 x 6. = n 2 =6 2 = 36. Stratified area = 1 + 3 + 5 + 7 + 94- 11 = ^ 2 =6 2 = 36. Thus the triangular and stratified areas are equal. But the stratified pyramid is less than the triangular pyramid by in. Since the double obeliscal area,^. 5. = l 2 + 3 2 + 5 2 44 THE LOST SOLAR SYSTEM DISCOVERED. 9 2 +ll 2 , it follows that if each of these squares were con- verted into a stratum having a depth of 1, together they would form a stratified or terraced pyramid, Jig. 5 a., con- taining as many cubes of 1 as the double obeliscal area con- tains squares of 1, or =n 3 %n=% 6 3 -^6, = 288-2 = 286. Also content of rectilineal pyramid = f n 3 , " " stratified pyramid =fra 3 ^n. Double parabolic area=4w 3 , " obeliscal area=|^ 3 n. When the squares of 1 are arranged in the order 1, 2, 3, as in Jig. 7 2, the whole area will = \ n + 1 . n, which will equal the area of a triangle having its height = n, and base = ra+l, or = When the squares of 1, 2, 3 become strata of the depth of 1, and formed into a terraced pyramid, the content of the pyramid will = ^ra + l .n.n + %, which will = the content of a rectilineal pyramid having the sides of the base = n + 1 . by W+-J-, and height =n. These obeliscal series of solids are expressed in terms of 2 the axes, as sum of I 3 + 2 3 + 3 3 = axis . = axis axs. For when the obeliscal solid of the 1st series = axis, the _ __2 content is represented by a stratum, or by an area = axis , where for each square of 1, a cube of 1 is substituted, so that a stratum having an area = axis , and thickness that of unity, will form an obeliscal series of cubes having a content _ 2 = axis , = the sum of ! 3 + 2 3 + 3 3 . _ ! + , _ >$> = 4 6 i (7 x~6" 7 x 6~ + i 7 x 6) when n = 6, = | 6^- (74088 -1764 +14) = 67171 cubes of 1. = i n + i (ordinate 3 ordinate 2 + ^ ordinate.) Sum of 4th series = 4 times sum of 1st series. 5th = 4 2 2nd 6th =4 3 3rd 1st series = ! 2 + 2 2 + 3 2 = axis. 2nd = l 4 H-2 4 H-3 4 = faxis x ordinate $ axis. 3rd = I 6 -f 2 6 -f 3 G = | axis x ordinate 2 f axis x ordi- nate + \ axis ; or 1st = 2nd = 3rd =\ 4th =- u ). . 5th = n -+ (n+l . n i i . n n + l. , + w+1 . w). When ordinate oc axis 1 , area = j- circumscribing parallel- ogram, or ordinate 3 x axis 2 Axis 2 oc (n+l . n . n + -J-) Ordinate 3 oc (72 -f 1 . w) 3 . (n + I . 7z) 3 is less than (n + I . n . n by irc 4 + iw 3 -K?z 2 which =441, when w = 6. VOL. i. E 50 THE LOST SOLAR SYSTEM DISCOVERED. (n+1 . nf = 42 3 = 74088 add 441 74529 (n+l . n . 7z so that the ordinate 3 should = (42-08 &c.) when (n + l . n . n + ) 2 = 273 2 . Then ordinate 3 would !<%. 91 = 22f. From i, li, 3, 7, 13-|, 22f, take i, It, 3-1-, 7-J-, 13f, difference -J-, 1 , 2-J-, 4 , 6, 9. Thus we have to be added to the 1st obeliscal sectional area to make it a curvilinear area, 1 to the 2nd sectional axis, 2-J- to the 3rd, &c. The whole addition to the series will = 36 one-fourth squares of 1, or 9 squares of 1, equal to 1 axis* i, 1, 2, 4, 6, 9, = of 1, 4, 9, 16, 25, 36, = i of I 2 , 2 2 , 3 2 , 4 2 , 5 2 , 6 2 . The sum of ! 2 + 3 2 + 5 2 has been found. From I 4 , 2 4 , 3 4 , 4 4 , 5 4 , 6 4 , take 2 4 , 4 4 , 6 4 . Difference I 4 , 3 4 , 5 4 . The sums of the 1st and 2nd series are known; therefore the sum of I 4 , 3 4 , 5 4 , their difference, may be found, as the sum of the series I 3 , 3 3 , 5 3 was determined. E 2 52 THE LOST SOLAR SYSTEM DISCOVERED. Again from take I 6 , 2 6 , 3 6 , 4 6 , 2 6 , 4 6 Difference 1 G 3 6 . Since the sums of the 1 st and 2nd series are known, the sum of I 6 , 3 6 , 5 6 may be found. Or the sum of 6 terms of the series and 2 4 + 4 4 + 6 4 =1568 _|_ 3 4 The sum of ^6, or 3 terms of 1st series = ! 4 + 2 4 + 3 4 = 98, and 16 x 98 = 1568 = sum of 3 terms of the 2nd series, which subtracted from the sum of 6 terms of the 1st series = 707 = the sum of 3 terms of the 3rd series. Hence the sum of \n terms of the 1st series x by 16 = the sum of ^n terms of the 2nd series, which subtracted from the sum of n terms of the 1st series = the sum of \n terms of the 3rd series. When the series 1 + 2 + 3, &c. is squared, as ! 2 + 2 2 + 3 2 , &c., the sum of \n terms of this, the 1st series, x by 4, or 2 2 , = the sum of \n terms of the 2nd series, 2 2 + 4 2 + 6 2 . When cubed, as ! 3 + 2 3 + 3 3 , the sum of \n terms x by 8, or 2 3 = the sum of \n terms of the 2nd series, 2 3 + 4 3 + 6 3 . In the series ! 4 + 2 4 + 3 4 , the sum of \n terms x by 16, or 2 4 = the sum of - terms of the 2nd series, 2 4 + 4 4 + 6 4 . Fig. 25. SERIES SUMMED. 53 Thus from the sum of the series 14- 2 + 3 to the power of 2, 3, or 4, the sum of the series 1+3 + 5, to the power of 2, 3, or 4 may be found. Also in the series 1 + 2 + 3, &c. \n terms of the 1st series xby 2, or 2', = ^ terms of the 2nd series, 2 + 4 + 6. Fig. 25. Four series of the cubes of 1, 2, 3, 4, 5 are arranged star-like, radiating from a common centre, their axes being at right angles to each other. 2 As each series = axis , = -J- the circumscribing square, = i- 2 axis', = the circumscribing triangle, 2 .*. the 4 series of cubes will =2 axis =the circumscribing square stratum of the depth of unity. Fig. 26. When the axes of two series of cubes of 2, 4, 6, 8, Fig. 26. are in the same straight line, the sum of each series will 2 2 2 = 2 axis , and the sum of both series = 4 axis = 2 axis = the E 3 THE LOST SOLAR SYSTEM DISCOVERED. circumscribing square stratum having each side = twice the axis. Fig. 27. Let the two series of cubes of 2, 4, 6, 8, be each divided into 2 equal parts, then they will form 4 solid radia- tions from a common centre. thtr- Fig. 27. The content of the 4 radiations will = the content of two series of cubes of 2, 4, 6, 8 = 2 axis = the circumscribing square stratum having a depth of unity ; and the side = 2 axis. axis* Fig. 28. SERIES SUMMED. 55 Fig. 28. If two series of the cubes of 1, 3, 5, 7, have their 2 axes in the same straight line ; then as each series = 2axis 2 axis, the two series will =2 axis 2 axis. Let one side of the circumscribing rectangular stratum = 2 axis, and the other side = 2 axis 1, then the area of the rectangle will = 2 axis 2 axis = the content of the two series of cubes of 1, 3, 5, 7. In the fig. one side of the rectangle =2x16, and the other =2x 15-J. Fig. 29. represents 4 radiations, each formed of two single Fig. 29. obelisks, so that each ray represents 2 obelisks, or each ray represents the breadth of 2 and the depth of 1 obelisk. , 2 Content of a single obelisk = -J- axis ^axis, 2 . * . 8 obelisks = 4 axis -f- axis, 2 = 2 axis 4 axis, = 2 axis |-2axis. The side of the circumscribing square of the v 4 radiations = 2 axis. Let this square form a stratum of the depth of unity, __ o Then 2 axis | 2 axis = square stratum less a line of E 4 56 THE LOST SOLAR SYSTEM DISCOVERED. single cubes of unity extending -|2 axis, or f side of square: as when axis = 9, 2 axis = 18, and 18 2 f 18 = 324 12 = 312 cubes of unity. When the 4 solid obeliscal series of radiations become 4 solid parabolic series. Then each parabolic solid will =^ axis, and 8 = 4 axis 2 = 2 axis = the circumscribing square stratum having its side = 2 axis. Let m = 2 axis, the side of the square stratum circum- scribing the series of cubes, obeliscal and parabolic solids. Then content of 2 series of cubes = m xml = m 2 m. Content of the 4 obeliscal radiations = m xm ^ = m' 2 ~m. Content of the 4 parabolic radiations =mxm = m' 2 . Fig. 30. In the common multiplication table, called the Pythagorean, the compartments are squares. X Fig. 30. The numbers 1, 2, 3, along the top represent the ordi- nates corresponding to the axes 1, 4, 9 along the side, which _ 2 = ordmate . The numbers 1,8,27, at the extremities of the ordinates 1, 2, 3, represent the ordinate ; and 1, 16, 81, along the diagonal, represent the ordinate . Sum of Fig. 31. equals 7 2 = 49 squares of unity, which square of 7 is composed of the series 7 2 -5 2 , 5 2 -3 2 , 3 2 -l 2 , 1 2 -0, or 24 , 16 , 8,1, . 2 2 and sum of 1 +8 + 16 + 24 = 2ra 1 =8 1 =7 2 = 49. SERIES SUMMED. 57 Sum of the series 4 + \ / \ / \ / X / \ / \ 7. \ 72 Fig. 31. Fig. 32. Fig. 32. equals 8 2 = 64 squares of unity, which square of 8 is composed of the series 8 2 -6 2 , 6 2 -4 2 , 4 2 -2 2 , 2 2 -0, or 28 20 , 12 and sum of 4 + 12 + 20 + 28 = 2w 2 = 8 2 = 64, which also equals the sum of the series 4 (1+3 + 5 + 7), Draw the axis and ordinates of fig. 7. . like those of figs. 1. or 7. Then draw the ordinate at the apex =6, the greatest ordinate at the base. By joining this ordinate with the ordinates 1, 2, 3, 4, 5, 6 by lines parallel to the axis, another series of ordinates will be formed, between which will be included the areas 1, 4, 9, 16, 25, 36, or I 2 , 2 2 , 3 2 , 4 2 , 5 2 , 6 2 , sum of the series = ^rc + 1 . TZ . ?z + -J-. The series of areas along the axis will equal 0, 3, 10, 21, 36, 55, which are formed by rectangles of the sectional axes and ordinates. As term 1 = 0, 2= 3x1= 3, 3= 5x2 = 10, 4= 7x3 = 21, 5= 9x4 = 36, 6 = 11 x5 = 55. The circumscribing rectangled parallelogram including both series will = axis x ordinate = ordinate =6 3 = 216. 58 THE LOST SOLAR SYSTEM DISCOVERED. The rectangled parallelogram ft 3 , less the sum of the series 1 . n. n + , will n terms of the series 0, 3, 10. When w = 6 .ft.ra + -L wiU=-- 8 J-w 2 == 216-91 = 125 = 6 terms of the series + 3 + 10 + 21 + 36 + 55 = 125. jFY<7. 7. The complementary area of the obeliscal series of squares of 1, 2, 3, 4, 5, 6, 7, 8, formed by rectangles parallel to theaxis=l + 3 + 6 + 10 + 15 + 21+28, or 1= 1 1 + 2= 3 3 + 3= 6 6 + 4 = 10 10 + 5 = 15 15 + 6 = 21 21 + 7 = 28 84 = squares of unity. Here the number of squares = 8, and complementary rectangles = 7. The axis=|-ft + 1 . n, here n = S, = 19x8 = 36, and ordinate= 8, the side of 8th square. .-. the circumscribing rectangled parallelogram . n . ft, = 36x8 = 288, and area of the series of 8 squares . n. n + ^ = 19x8x8-5 = 204, .-. complementary area 1 . n . ft (-J-7Z + 1 .n. ft = 288 204 = 84. Fig. 8. The complementary area of the obeliscal series of squares of 2, 4, 6, 8, 10, formed by rectangles parallel to the axis = 4 + 12 + 24 + 40. SERIES SUMMED. 59 As 4 4 + 8 = 12 12 + 12 = 24 24 + 16 = 40 80. Here the number of squares = 5, and rectangles = 4. The axis =2 + 4 + 6 + 8 + 10, = n + 1 . n, here n 5, = 6x5 =30, and ordinate = 2ra = 2x5 = 10. .-. the circumscribing rectangled parallelogram = 7i+l . n. 2n, = 30x10 = 300. And area of the series of squares 2 +10 2 , = fw+1 . n . 2w+l = f 6 x 5 x 1 1 = 220. .-. the complementary area = n+l .n. 2n (f n+I . n. = 300 - 220 = 80. The complementary area = 4 + 12 + 24 + 40, = 4(1+ 3+ 6 + 10). Fig. 7. The complementary area of the obeliscal series of squares of 1, 2, 3, 4, 5, 6, 7, 8, formed by rectangles parallel to the ordinates equals 1x7= 7 2x6 = 12 3x5 = 15 4x4=16 5x3 = 15 6x2 = 12 7x1= 7 84. 60 THE LOST SOLAR SYSTEM DISCOVERED. Here the number of squares are 8, and the sides of the 7 rectangles parallel to the axis increase by 1, while the other sides parallel to the ordinates decrease by 1. The differences of the series 7, 12, 15, 16, 15, 12, 7, are 5, 3, 1, 1, 3, 5. The complementary area of the obeliscal series of squares of 1, 2, 3, to 12, will be 1x11 = 11 2x10 = 20 3x 9 = 27 4x 8 = 32 5x 7 = 35 6x 6 = 36 7x 5 = 35 8x 4 = 32 9x 3 = 27 lOx 2 = 20 llx 1 = 11 286. The differences between the terms of the series 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, are 9, 7, 5, 3, 1, 1, 3, 5, 7, 9. Hence, when the first term of the complementary series, which=tt 1, is an odd number, the series of differences de- creases by the odd numbers from n 3 to unity, and then recommences from unity and increases to n 3. The area of such a complementary increasing and de- creasing series will = n+l .n .n (n+l.n. n = 13 x 12 x 12 - |13 x 12 x 12-5, = 936 - 650 =286, = axis x ordinate series of squares. Let n, the number of squares, =11. Then n 1 = 10, an even number, SERIES SUMMED. 61 and 1x10=10 2x 9 = 18 3x 8 = 24 4x 7 = 28 5x 6 = 30 6x 5=30 7x 4 = 28 8x 3 = 24 9x 2 = 18 lOx 1 = 10 220 The differences between the terms of the series 10, 18, 24, 28, 30, 30, 28, 24, 18, 10 are 8, 6, 4, 2, 0, 2, 4, 6, 8. Here the complementary series of rectangles =n 1 = 10, an even number, and all the terms are even. The series of differences begins with n 3 = 8, an even number, and all the terms are even, each in succession de- creasing by 2 to 0, and then increasing by 2 to n 3, or 8. The sums of the second series of differences of the odd and even differential numbers are equal ; as 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 2 . difference =2 + 2 + 2 + 2 + + 2 + 2 + 2 + 2 = 16, and 8, 6, 4, 2, 0, 2, 4, 6, 8, 2. difference =2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16. When the number of rectangles are odd and =11, then 6x6 = 36 is equidistant from both extremes, being the middle term. When the number of rectangles are even and = 10, then 5 x 6 = 30 and 6x5 = 30 are the two nearest the middle, and equidistant, one from one extreme and the other from the other extreme. Sum of the 11 squares of 1, 2,3 = ^/z+ I .n. n + . Circumscribing rectangled parallelogram = axis x ordinate . n. 62 THE LOST SOLAR SYSTEM DISCOVERED. Complementary series of 10 rectangles = 1-12 x H 2 -lll x 11 x 11-5 = 220. The complementary area of the obeliscal series of squares of 2, 4, 6, 8, 10, formed by the rectangles parallel to the ordinatesj^. 8. are 2x8 = 16 4x6 = 24 6x4 = 24 8x2 = 16 80 Here the number of squares = 5, and rectangles = 4. The complementary area = ?z + l .71. 2n (fra+1 . n. 2ra+l) when n = 5, = 300 -220 = 80. When ?z=10, the number of squares, the last term of the series 2, 4, 6 will be 20, and 9 the number of rectangles that form the complementary area, as 2x18= 36 4x16= 64 6x14= 84 8x12= 96 10x10 = 100 12 x 8= 96 14 x 6= 84 16 x 4= 64 18 x 2= 36 660 Thus the series of rectangles are formed by each being made equal to the two numbers equally distant from the extremes, or the mean of the series 2, 4, 6, 8, 10, 12, 14, 16, 18. When n, the number of squares, =11, the last term of the SERIES SUMMED. 63 series 2, 4, 6, &c. will be 22, and 10 the number of rect- angles that form the complementary area, as 2x20= 40 4x18= 72 6x16= 96 8x14 = 112 10x12 = 120 12x10=120 14x 8 = 112 16 x 6= 96 18 x 4= 72 20 x 2= 40 880 Sum of 11 squares of 2, 4, 6 = f n+ 1 .n.2n + l. Circumscribing rectangled parallelogram = axis x ordinate = n + l . n. 2n. Complementary series of 10 rectangles = TZ+ 1 . n. 2n -f-w-f- 1 . n . 2/z + l = 12xllx22-fl2xllx23 = 2904-2024 = 880. Or generally the series will be 2x(2?z-2), 4 x O-4), 6x(2w-6), 8 x (2/1 8), &c. ; and the sum = n+ 1 . n . 2w (frc-f 1 . n. 2ra + l), where n = the number of squares of 2, 4, 6, &c. that form the obeliscal series, and n 1 the number of rectangles that form the complementary area. Fig. 22. If the obeliscal series were formed of I 4 , 2 4 , 3 4 , 4 4 , 5 4 , 6 4 , the axis would = I 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 , and the area of the series of squares = (/l+l.W .W + -1 J-TZ+l.W.TZ+i), the circumscribing rectangled parallelogram would = axis x ordinate ; here ordinate = 6 2 = n 2 . 64 THE LOST SOLAR SYSTEM DISCOVERED. Therefore the complementary area would be known, which, if formed by a series of 5 rectangles between the ordinates, would be I 2 x(6 2 -l 2 ) or 1 x 35= 35 2 2 x(6 2 -2 2 ) 4x32 = 128 3 2 x(6 2 -3 2 ) 9x27=243 4 2 x(6 2 -4 2 ) 16x20 = 320 5 2 x(6 2 -5 2 ) 25x11 = 245 or generally I 2 x(rc 2 -! 2 ) 2 2 x(> 2 -2 2 ) 3 2 x( a -3 a ) 4 2 x(> 2 -4 2 ) 5 2 x(> 2 -5 2 ), &c. where n the number of squares that form the obeliscal series, and n 1 the number of rectangles that form the complementary obeliscal area. If the obeliscal series of squares were 2 4 , 4 4 , 6 4 , 8 4 , the axis would =2 2 + 4 2 + 6 2 + 8 2 , the area of the series of squares would . n. 2n + 1 LW+ 1 . n . 2n + 1), and circumscribing rectangled parallelogram = axis x ordi- nate ; consequently the complementary area would be known, which may be formed by a series of rectangles between the ordinates equal to 2 2 x 8 2 2 2 or generally 2 2 x (2w 2 2 2 ) 42 x8 2_ 4 2 4 2 x(2w 2 -4 2 ) 62 x s a -6 2 6 2 x (2rc 2 - 6 2 ), &c. where ft = the number of squares forming the obeliscal series, and n 1 the number of rectangles that form the comple- 2 mentary area. The ordinate will = 2n . Fig. 5. The complementary area of the obeliscal series of squares of 1, 3, 5, 7, 9, 11, to 6 terms, formed by 5 rect- angles parallel to the axis = SERIES SUMMED. 65 for 1x2= 2 = l 2 x2, 4x2= 8 = 2 2 x2, 9x2 = 18 = 3 2 x2, 16x2 = 32 = 4 2 x2, 25x2 = 50=5 2 x2. The axis = rc 2 , and ordinate=2ra 1, therefore circumscribing rectangled parallelogram =7i 2 . 2/z 1, here ra = 6, = 36x11 = 396, and area of the series of 6 squares, or Therefore the complementary area = 396 -286 = 110. Or the area of the series of 6 squares The complementary area=2(! 2 + 2 2 + 3 2 + 4 2 + 5 2 )=110. Therefore the area of the circumscribing rectangled paral- lelogram = I 2 + 3 2 + 5 2 + 7 2 + 9 2 + 1 1 2 , + 2 ( Fig. 5. The complementary area of the obeliscal series of squares of 1, 3, 5, 7, 9, 11, when formed by a series of rec- tangles parallel or between the ordinates are 1x10=10 3x 8 = 24 5x 6 = 30 7x 4 = 28 9x 2 = 18 110 1, 3, 5, 7, 9, 11, being the sides of the 6 squares parallel to the axis; 2, 4, 6, 8, 10, numbers between them; and 10, 8 6, 4, 2, the sides of the 5 rectangles parallel to the ordi- nates. VOL. I. F 66 THE LOST SOLAR SYSTEM DISCOVEKED. Here each rectangle is formed by an odd and even number ; the number of squares = n, and number of rec- tangles = n 1 . Sum of the series = n 2 (2n I ) (frc 3 %ri) = rectangled parallelogram less series of squares, = 6 2 xll-(|6 2 -i6), = 396-286 = 110. The duplicate ratio. 2, 3, 4, 5, 6. 4, 9, 16, 25, 36. 8, 27, 64, 125, 216, 1st power 1, 2nd " 1, 3rd " 1, 4th, &c. In each of the powers The first term : the 4th in the duplicate ratio of the first to the second. As 1 1 1 64:: 1 2 2 , also 1st 2nd:: 2nd 4th, 4 2 , as 1 2 ::2 4, 8 2 . 1 4 ::4 16, 1 8 ::8 64. The 1st I the 9th in the duplicate ratio of the 1st .' 3rd. The 1st : the 16th in the duplicate ratio of the 1st : 4th. In the geometrical progression of 1, 2, 4, 8, 16. 1st I last :: first 2 mean 2 . 1st: 2nd:: 2nd 3rd. 1st : 3rd ::3rd 5th. 1st : 4th ::4th 8th. The 1st I 3rd in the duplicate ratio of the 1st : 2nd. The 1st : 5th in the duplicate ratio of the 1st : 3rd. The 1st .' 8th in the duplicate ratio of the 1st I 4th. To construct the Pylonic Curve that shall have its Ordinate varying inversely as D*, from the Apex of the Obelisk, and the same Axis common to the Curve and the Obelisk, Fig. 34. Let the common axis of the obelisk and the curve = 81 ; then the last ordinate of the obelisk will = 9. PYLONIC CURVE. 67 Make the first ordinate of the curve at the apex of the obe- lisk =9, which will represent the mean time in which the first unit, or sectional axis 1, is described in the first second, so the Fiy. 34. Fig. 34. a. axis 1 will represent the velocity of the first second, then v x t = lx9 = 9 = a rectangled parallelogram having an area = 9. As the sectional axes of the obelisk are as 1, 3, 5, 7, &c., the distances described in each successive second, those axes will denote the velocities during those seconds, since VQC D* oc ordinate obelisk, and each of these axes being = the two ordinates by which it is bounded, = twice the mean ordinate of each section, = the mean velocity of each second, or the distance described in each successive second when a body falls freely near the earth's surface. oc ordinate of curve, As t GC OC OC D* V D* ordinate obelisk v x t will always equal a constant quantity = 9, the area of the first rectangled parallelogram. Hence the ordinates of the F 2 68 THE LOST SOLAR SYSTEM DISCOVERED. curve corresponding to the sectional axes 1, 3, 5, 7, 9, &c., will be as 9, f , f- , -f, f, &c. So that these ordinates of the curve will oc inversely as the sectional axes 1, 3, 5, &c. During the descent, the velocity with which unity is de- scribed along the axis 1, will be to the velocity with which unity is described along the 5th axis = 9, as 1 19. So that the velocity through axis 9 will be 9 times greater than the velocity through axis 1. The time t corresponding to these velocities will oc in- versely as the velocities, or as 9 I 1. So that the time of describing unity along the axis 1 will be 9 times greater than the time of describing unity with the mean velocity of the 5th second along the axis 9. The central unit of each sectional axis 1, 3, 5, 7, &c. will be described with the mean velocity of the corresponding second, and the time of describing any central unit will be the mean of the times in which the units along that sectional axis are described. Since time t oc and T the time of descent oc v .-.TOC-I Or t the time of describing unity at any distance oc in- versely as T, the time of descent to that distance. If an ordinate t, at the 1st axis 1, be made = 9 to re- present the time t in which unity is described in the 1st section 1, an ordinate t = 1 will represent the time t of de- scribing one of the nine units in the 5th sectional axis 9 with the mean velocity of that section. The 1st ordinate t=9 and V = 1 5th =1 andv=9 In the 1st section t x v=9 x 1 = 9. 5th t t x v=lx9 = 9. Or time t of describing unity in the 1st section .' time t of PYLONIC CURVE. 69 describing unity in the 5th section :: 9 I 1 ; and velocity with which unity is described in the 1st section : velocity with which unity is described in the 5th section :: 1 19. When, as in this Fig. 34, the 1st ordinate t = the last ordinate of obelisk = 9, the sectional axes 1, 3, 5, 7, &c. will _______ 2 = 9 in number, and axis of obelisk = ordinate = 9 2 =81. The mean time t in describing unity in any sectional axis will = 9 divided by that axis. When the 1st time t ordinate = the ra th ordinate of the obelisk = n, the time t of describing unity in the 1st sectional axis will be to the time t of describing unity in the last sec- tional, or 74 th axis, as * : -_ as 2n I : 1. The times t and corresponding velocities will be repre- sented by a series of equal rectangled parallelograms de- scribed along the sectional axes, so that each of the sectional axes 1, 3, 5, 7, &c., will represent the velocity, and the cor- responding t ordinates the mean time t in which unity is described in a section, and tx v will always = 9. In 1st sectional axis v=l and ordinate t= & =9 2 " =3 " = =3 3 " =5 " = f =1-8 4 " =7 " = =1-285 5 " =9 " = f =1 6 " =11 =r ! r= * 818 7 =13 " =T 9 3= * 69 8 " =15 " =1%-= '6 9 =17 =rV= "53 Since velocity cc D^, the sectional axes 1, 3, 5, &c., are described in equal times; hence the mean ordinate t, which. oc inversely as the sectional axes, will describe equal areas, or equal rectangled parallelograms in equal times. At the 9th ordinate the series of rectangled parallelograms described will =9, and the area of the whole = 9 x 9 = 9 2 = 81 F 3 70 THE LOST SOLAR SYSTEM DISCOVERED. = the square of the 9th ordinate of obelisk, or 1st ordinate t= L the circumscribing rectangled parallelogram which equals axis x ordinate = 9 2 x 9 = 9 3 . The series of rectangled parallelograms, when placed one above another, will form an Egyptian or Cyclopian door, gateway, vaulted roof, or arch, and each rectangled paral- lelogram will extend beyond the one below by a distance = 2. By making the first ordinate t=n, a variety of such arches may be formed. Since the t ordinate oc inversely as the sectional axes 1, 3, 5, &c., and each sectional axis = twice the mean ordinate of the obelisk. Therefore t ordinate will vary inversely as the mean ordi- nate, axis", or D*. If each rectangled parallelogram along the sectional axes be supposed to be described uniformly, each unit of a sec- tional axis would be described in equal times, corresponding to the mean t ordinate of the section. But the t ordinate at the beginning of each section, reckoning from the apex of the obelisk, will be greater than the m t ordinate, and at the end of the section the t ordinate will be less than the m t ordinate, since velocity continually increases. During the descent by the action of gravity, the T ordi- nate, or ordinate of the obelisk will oc D* and describe a cur- vilinear obeliscal or parabolic area. So the t ordinate, which 1> > 4-5 1-> -g- of 6, and the corresponding axes 1, 2, 3, 4, 5, 6, will be equal, and the ordinates will 0 = numerator + denominator of the sum of the series = ~^r - The sum of all the terms after the n ih term will never equal the n ih term. The sum of the series 1 _1 2_ J_ l l 1 1 + 3 + ~9 +27 + 81+243 f ^29 + 2187 1 1111 JL 1_ 3 3 2 3** 3 4 Q5 ~t~ Q6 "i" Q7 will never = 1, or the sum of all the terms after the first will never = , since 1 j_ 4 12^ 3 + 9 9 : 27 12 J_ 13 39 27 " 27 : = 27 : : 81 39 1 40 120 81 + 81 ~ 81 ~ 243 120 1 121 363 VOL. I. 243 + 243 ~"~ 243 ~ 729 82 THE LOST SOLAR, SYSTEM DISCOVERED. 363 1 364 1092 729 T 729 ~ 729 ~2187 1092 _1 __ 1093 3279 2187" l ~2187~~2187~6561 3279 1 3280 656T+6561 ^6561* whlch 1S less than Thus sum of all terms after the n ih term will never = -*- the n ih term. 1 1 -L W J- Sum of n terms of the series a + n &c., will = a n ., The reciprocal curve of contrary flexure is determined by the reciprocals of the sines of the quadrant, and the hyper- bolic series of, parallelograms is formed by the sines and their reciprocals. Fig. 40, Draw parallel and equidistant lines. At any radius, 9, describe a quadrant; then, where the arc inter- sects the 8th line, through that point, A, draw a straight line from the centre c, cutting the 9th line in B. Draw DAE parallel to c 9, then by similar triangles, AE : AC : I AD .* AB or 8 : 9 :: 1 : AB AB = I = 1 of 9 and AE x AB = AC x AD 9 or 8 Xo=9xl = 9, o or sine AE multiplied by its reciprocal AB = 9. Similarly F G = ? = I of 9. 7 7 9 and FC x FH = - x 7 = 9. So the remaining reciprocals, radiating from the centre C, multiplied by their respective sines 6, 5, 4, &c. will each = 9. SERIEF. Fig. 40. o 2 84 THE LOST SOLAR SYSTEM DISCOVERED. The extremities of these reciprocal sines will trace a curve of contrary flexure, beginning at 9 -f 1, or 10, and terminating at CK = ci = CL + Li = 9 + 9 = 18, or twice the radius, and K will be in the second line. With radii c 10, CB, CG, &c., describe circular arcs which will cut LI, = 9, at the dis- tances from L of g, 7, g, g, 4, 3, -, 1 of 9 or LI. Let LM be drawn parallel and = c 9, and similarly divided. From the points of division draw lines parallel to Li, which will cut at right angles the straight lines drawn from the points, at -, -, --, &c. of Li, the terminations of y o the circular arcs ; these lines will be respectively as 9, 8, 7, 6, 5, 4, 3, 2, 1, and will form with the lines drawn from LM a series of rectangular parallelograms which will form a hyperbolic area of parallelograms included by the two asymp- totes LI, LM, each of which = 9, for the greatest ordinate and greatest axis become asymptotes. The least ordinate at M cr I = 1, and the greatest ordinate at N, for this double hyperbolic area, will be 3, the side of the central or angular square ; then 1 : 3 : : 3 : 9, or least : greatest .: greatest ordinate : asymptote. The hyperbolic curve will be determined by the series of equal parallelograms inscribed between the curve and the asymptotes. Since the area of each of the 9 parallelograms in the series = 9, their whole area will = 9 x 9 = 9 2 = the area of the square that circumscribes the series of parallelograms arranged in hyperbolic order. But when so arranged the parallelograms overlap, or partially cover each other, so that the parallelogram along one asymptote, or side of the square, which = 1 x 9, or 9, has only -J- of 9, or 1 square of unity exposed, -|- being concealed below the next parallelogram, and this parallelogram is again partially covered by the next, and so on in succession, the last only being entirely exposed, so that the sum of those exposed, or superficial areas = the area of the hyperbolic series of parallelograms. Thus a series of parallelograms having each an equal area, and the area of the whole series being equal the square of SERIES. 85 the asymptote, can be so arranged that the superficial area of the series shall form an hyperbolic area, having the side of the circumscribing square equal the asymptote of the hyperbola. The area of such a series of parallelograms will = 1 + -J- + -J- + i, &c. of 9. Fig. 40. Radius' 2 sine 2 cosine 2 9 2 - 8 2 = 17 9 2 - 7 2 = 32 9 2 - 6 2 = 45 9 2 - 5 2 = 56 9 2 - 4 2 = 65 9 2 - 3 2 = 72 9 2 - 2 2 = 77 9 2 - I 2 = 80 9 2 - O 2 = 81 Cosine 2 81, 80, 77, 72, 65, 56, 45, 32, 17 Difference 1, 3, 5, 7, 9, 11, 13, 15 The cosines 2 decrease from the arc towards the centre, while their differences increase as the odd numbers 1, 3, 5, &c. If the 9th ordinate of the obelisk represent radius, the remaining 8 ordinates will represent the sines, and the difference between their squares will =81, 80, 77, &c., = the axes between the ordinates 1, 2, 3, &c. and ordinate 9. Again the difference between the terms of the last series will = the sectional axes 1, 3, 5, &c. The reciprocal of the sine also = (l + \ sine For radius = sine x reciprocal, radius 2 = sine 2 x reciprocal 2 , and radius 2 = sine 2 + cosine 2 , sine 2 -f cosine 2 . . reciprocal 2 = sine 2 _ cosine 2 - -I i ' n~ reciprocal^ (l + 22*52^ * V sine 2 /; G 3 86 THE LOST SOLAR SYSTEM DISCOVERED. so that when the 9th ordinate of the obelisk is made the radius of the quadrant, the other ordinates, 8, 7, 6, &c., will be as the sines. Fig.k\. The 9 rectangled parallelograms having their lengths = the cosines, or = the square root of 17, 32, 45, &c., and the breadth of each = unity, will circumscribe the quadrantal arc, and the first 8 of the series of the 9 rectangled paral- lelograms will be inscribed within the quadrantal arc. The quadrantal area will = the sum of the series of such inscribed rectangled parallelograms -!- radius 2 . For the difference between the eight inscribed parallelograms and the nine parallelograms that circumscribe the quadrant = the nine parallelograms along the arc = the last parallel- ogram c 1 =9x1 = 9 = - radius 9 Let the radius be divided into ninety equal parts, then the 1 difference will = radius ; when the radius is divided into 90 900 equal parts, the difference of the two series of rectangled parallelograms will = radius . Generally, the difference 1 2 will = radius For parallelogram c 1 will radius x radius = radius 2 . Hence, as n increases, the dif- n 9 ferential series of rectangled parallelograms will become evanescent, and the series of inscribed rectangled parallelo- grams will approach nearer and nearer to equality with the quadrantal area. Since the quadrantal area the series of inscribed parallelograms -f only half the evanescent series of parallelograms. For the diagonals of the differential paral- lelograms may ultimately be regarded as portions of the quadrantal arc. Thus, a Cyclopian arch may be constructed so that the semicircle shall touch the angular projections of the arch. (Fig. 41.) SERIES. 87 By varying the value of n in the hyperbolic series of rect- angled parallelograms, different Egyptian or Cyclopian hy- perbolic arches may be constructed. N Fig. 41. IT Fig. 42. Fig. 43. Fig. 42. is formed from the lower section of an hyperbolic series of rectangled parallelograms. Fig. 43. is formed by the hyperbolic series of rectangled parallelograms ; the first in the series is a square. G 4 88 THE LOST SOLAR SYSTEM DISCOVERED. Figs. 42. and 43. form hyperbolic galleries. That of 42. corresponds with the view of a gallery in the interior of the Pyramid of Cheops given by the French writers. The sides of the hyperbolic series of rectangled parallelo- grams formed within the square IM (Jig. 40.) have their sides 1, 2 } 3, &c., along the axis LM parallel and equal to the sines ; and the other sides of the rectangled parallelograms, which are at right angles to the sines, are equal to the reciprocals of the sines, and form the reciprocals of the sides 1, 2, 3, &c., of the rectangled parallelograms. Hence, as the reciprocals of the sines, which determine the curve 10 BGK, form the reciprocals of the hyperbolic series of rectangled parallelograms, we may call this curve the hyper- bolic reciprocal curve of contrary flexure. Having shown that the obelisk represents the laws of motion when a body falls near the earth's surface, or when a planet revolves in its orbit, we shall next attempt, by means of the pyramidal and hyperbolic temples, to interpret the ancient theory of the laws of gravitation when a body is supposed to fall from a planetary distance to a centre of force. With this view the velocity will first be supposed to at a , but afterwards in a greater inverse ratio. The pyramidal may not accord with the Newtonian theory of gravitation. We may not have interpreted the pyramid correctly ; but now we are unable to revise what has been done. The pyramid, like the obelisk, still points to the heavens as an enduring record of the laws of gravitation, though it has ceased to be intelligible for countless ages. If velocity 2 x hyperbolic ordinate will always equal a constant quan- 2 tity = 9 2 = axis, = the circumscribing square CM, or the square IM that contains the hyperbolic series of rectangled parallelograms. The axis being divided in 9 = parts, let the sphere of attraction have the centre of force in L, and the semi-dia- meter = 1, one of the 9 equal parts of the axis. Then if a body descending to the centre of force L, with a velocity oc % from L, should, at the distance of 9, or the 9th ordinate from the centre, have a velocity represented as I 2 , and that velocity should be continued uniformly through a semi- diameter = 1, along the axis from the 9th to the 8th ordinate, the solid thus generated by the velocity ordinate = I 2 would be represented by I 2 x 1, I 3 , or a cube of unity. 90 THE LOST SOLAR SYSTEM DISCOVERED. Since velocity oc 2 , the corresponding t ordinate, the reciprocal of the velocity ordinate will oc D 2 . Hence the square stratum generated by the corresponding t ordi- nate oc D 2 on the other side of the axis, will = 9 2 x 1 = a stratum having an area = 9 2 and a depth of 1, = 81 cubes of unity. In the descent through each successive semi- diameter, or 1, the rectangle by the velocity ordinate and the t ordinate will = aids =9 2 = 81, and 81 x 1 = 81 cubes of 1. At the distance of 1 from the centre of force the velocity ordinate will be represented by 9 2 , and the corresponding time t ordinate by I 2 . If these two ordinates descended to the centre of force with the acquired velocity, continued uni- form, then the respective strata so generated would be 8 1 and 1 cube of unity ; but the body cannot descend beyond the surface, or circumference of the spheres, at the distance of 1 from the centre. The area of the series of rectangular parallelograms, 1, 2, 3, &c., = -J-wH- 1 . n ^n 2 + ^n, as n increases by subdivision of the same axis or radius, the series will approach to rc 2 , the area of the triangle. Or the value of n varies inversely as the number of parts into which the same axis or radius is divided; but -J-w 2 still _ -2 _ __ 2 1 _ 2 -Jaxis = ^-radius , and -J-w = - area of axis ; which becomes evanescent as n increases numerically, and vanishes when ordinate of triangle continually oc axis or distance from the apex. Or \ n = ^ n squares of unity (Fig. 7-2.). Hence as the series of rectangular parallelograms approaches to a triangular area, so will the hyperbolic series of parallelo- grams approach to an hyperbolic area. In the same manner the series of strata generated by the ordinates of pyramid and hyperbolic solid will approach to a rectilinear pyramid and curvilinear hyperbolic solid. When the ordinate of triangle oc D, and ordinate of hyperbola cc -, each of their rectangles, or ordinate of SERIES. 91 triangle x by ordinate of hyperbola = area of the corre- sponding parallelogram inscribed along the axis = 9. The sum of the areas of the series of parallelograms =9 x 9 2 ___ 2 . 2 = axis ; and triangle generated by ordinate oc D = ^ axis . When the ordinate of pyramid oc i> 2 , and ordinate of hyperbolic solid oc , their product = the circumscribing square = 9 2 = axis , and as each of the 9 square strata has a depth of unity, the sum of the series of square strata will = 9 2 x 9 = 9 3 = axis 3 ; and -^ axis = pyramid generated by t ordinate oc D 2 from the apex. The content of the stratified pyramid = I 2 + 2 2 + 3 2 + 4 2 , &c. = iir+l. n. ^+i = l^ 3 + i^ + ^ = pyramid + triangle + -J-axis. For -J- n s = content of the rectilinear pyramid. J tt 2 = content of the triangular stratum of the depth of 1. n = a line or column of cubes of 1 = axis in length. If the same axis be continually divided, or n continually increased, the triangular stratum will become thinner, and so will the line of cubes = 1 axis. Thus they will ulti- mately become evanescent as the content of the stratified pyramid approaches to equality with the rectilinear pyramid, ^w 3 , or -i-axis 3 , and vanish when the ordinate continually oc 2 as axis . The solid = the axis = n 3 will always remain the same how much soever the axis be subdivided. A pyramid having the sides of the rectangular base as n -+ 1 by ft + -J-, and axis = ft, will = the stratified pyramid . n .~n + i, = ! 2 + 2 2 + 3 2 + 4 2 , & c ., each stratum having the depth of 1. (Fig. 43. a.) The two triangles are similar, equal and in- variable, each having the axis divided into 9 = parts ; the distance between the apices = 1. The circumscribing tri- angle includes 8 parallelograms ; the sum of which THE LOST SOLAR SYSTEM DISCOVERED. area triangle = = .i-9 2 =40-5 therefore 40-536 = 4-5, the area of the 2x8 triangles cut off from the series of 8 parallelograms by the lower triangle. Thus the tri- angular area exceeds the series of 8 parallelograms by 4-5. Or triangular area I difference of areas :: : 4-5 :: 40-5 : 4'5 i 9' 43> a ' When the axis of each triangle is divided into 81 equal parts, the distance between the apices = -^ axis. _ _ The series of 80 parallelograms =%n+l ,n ^80x81 = 3240. Area triangle =^axis 2 = i8 1 2 = 3280-5, therefore 3280-5- 3240 = 40-5. Or triangular area .* difference of areas :: -J-8 1 2 .' 40-5 :: 3280-5 I 40-5:: 81 : 1. The two triangles being always invariable and each = \ When axis = 9, difference of areas = 9 2 =81 = 9 3 = 729 =9 4 = 6561 On 99 & Si = - triangle When axis = 9, distance between apices = - axis y = 9 SERIES. 93 The distance between the bases of the 2 triangles = the distance between their apices. Next let a series of 9 instead of 8 parallelograms be de- scribed, then the area of the series will exceed that of the triangle. For area of 9 parallelograms =^n+ 1 . 7Z=|10 x 9=45 Area of triangle =9 a =40-5 therefore 45 40*5=4-5, the area of the 2x9 triangles, the excess of the 9 parallelograms above the triangle = axis 2 . So the excess of the parallelograms over the invariable tri- angle will be - triangle when axis = 9 y Hence the more the axis is subdivided the less will be the difference between the parallelograms and triangle, and the apices of the two triangles will approach each other, as will their bases, so that their coincidence will be the limiting ratio of the two series of parallelograms to equality with the invariable triangle = -J- axis 2 , or to the triangle generated by ordinate 2 . Fig. 45. m Fig. 45. Let 1, 2, 3, 4, the distances along the axis common to the 96 THE LOST SOLAR SYSTEM DISCOVERED. velocity and t ordinates, be equal the ordinates 1, 2, 3, 4, of the obelisk, and let 1, 4, 9, 16, the corresponding axes of the obelisk, be the t ordinates of this common axis. So that the ordinates of the obelisk will represent the axes or dis- tances, and the axes of the obelisk will represent the t or- _^. T ^ J . _^- JT .. ______ -Q dinates which will oc as the distance * The ordinate 1 6 will represent the t ordinate corresponding to the distance 4 ; 9 the t ordinate corresponding to the dis- tance 3, and 4, 1, to the distances 2, 1. Supposing the velocity acquired at the beginning of each distance were continued uniform through the distance of unity, then the corresponding t ordinate will describe the series of rectangled parallelograms 16, 9, 4, 1, and the whole area described will equal ! 2 + 2 2 -f 3 2 + 4 2 , or= -J-w+1 . n . n + generally. But as the time t 9 which GO , is continually varying during the descent, the area described by the t ordinate will be less than the sum of the rectangled parallelograms l 2 + 2 2 -f 3 2 + 4 2 , by the 4 triangles, or by half the sum of 1+ 3 + 5 4- 7, or half the axis x 1, which =^ra 2 . To reduce the area to the complementary obeliscal area, a further reduction of n must be made to form the complementary parabolic area, which will be described when the velocity continually cc y As series of rectangled parallelograms ! 2 + 2 2 + 3 2 + 4 =-J-wTl. n from which take then the complementary parabolic area will =n* or -J- the circumscribing parallelogram. Hence the whole area described by the t ordinate when it continually varies will = the complementary parabolic area. The whole time T of descent will oc the whole area de- scribed iV of 16. The inscribed velocity rectangled parallelograms having their sides along the axis =1, 2, 3, 4, will equal 1 x 16, 2 x i of 16, 3 x i of 16, 4 x ^ of 16, or equal 1, 1, , % of 16. The last series of rectangled parallelograms 1, -*-, -J-, ^ of 16, when they partially cover each other, form the series ^22 J 32* 42' of 16> as the series of equal rectangled parallelograms, when they partially cover each other, form the hyperbolic series of rectangled parallelograms 1, -J-, -*-, -, VOL. I. H 98 THE LOST SOLAR SYSTEM DISCOVERED. The greatest time ordinate =16, and the least =1, at the distance of 1 from the apex or centre of force. The least velocity ordinate = 1 , and the great- est, at the distance of 1 from the apex =16. Fig. 46. The time ordinates form the obelisk like half a canoe, perhaps the sacred boat. The velocity ordinates form an outline like the section of an architrave and column. The greatest rectangle, or rectangled parallelo- gram of one series = the greatest rectangle, or rectangled parallelogram of the other series = ri 1 . The least rectangle, or square of unity, in one series = the least rectangle, or square of unity in the other series. The first and greatest rectangled parallelogram in the t series becomes the last in the velocity series. The last and least rectangle, the square of unity, in the t series becomes the first in the velocity series. The circumscribing rectangled parallelograms of both series are equal. If instead of the lineal t ordinate, which a D 2 , the t ordinate were a rectangle having the length or ordinate would = the square root of the axis or of the cube inclosing the pyramid. Fig. 49. As the obelisk or parabolic solid oc axis ~ 2 , &c., or velocity at distance n + 1 I velocity at 1, the sun 1 p;: I 2 : n+l H 3 102 THE LOST SOLAR SYSTEM DISCOVERED. The velocity within the earth's orbit will be as - - --, ("- 1 ) 2 Let one asymptote divide the other asymptote at right angles into 2 equal parts. (Fig- 48.) So the ordinate in the descent will also be divided equally by the axis or asymptote ; then the solid generated will resemble the outline of a Bur- mese pagoda with its square terraced base, the sides of the terraces being as 1, -J-, -J-, of the side of the lowest terrace. The curve begins at the 3rd or 4th terrace, and is continued to the summit of the spire or tee. These pagodas are solid structures like the pyramids. So that when the velocity oc -5 the pyramid represents the varia- tion of the time, and the pagoda the variation of the velocity. Hence both the pyramidal and hyperbolic solid temples have originally been constructed as symbolical of the laws of gravitation. About one thousand five hundred and ninetieth part of the pyramid of Cheops is occupied by chambers and passages, while all the rest is solid masonry. Fig. 49. illustrates the velocity oc ^ in the descent of a body to the centre of force. The apices of the pyramid and obelisk are both in the centre of force. The ordinate of pyramid, and the solid obe- lisk itself, both of which vary as D 2 from the centre of force, will both oc time t ; so that the horizontal section or ordinate of pyramid at any point of descent, and the corresponding section of the obelisk intercepted between that point and the apex will both oc D 2 oc time t. The corresponding ordinate of the hyperbolic solid will a oo v corresponding to the time t at the given point. The hyperbolic solid, a horizontal section of which shows the variation of the velocity, has its base = the base of the 9 pyramid = 100 , passing through the centre, and its leas ordinate = the square of unity, is in a line with the bases of the pyramid and obelisk. The horizontal section of the pyra- TIME AND VELOCITY. 103 mid at the orb's surface also equals the square of unity. The axis, common to the obelisk, pyramid, and hyperbolic solid = 100; the side of the base of the pyramid and hyperbolic solid = the common axis = the side of the circumscribing square = 100. The rectangle of the t ordinate and velocity ordinate at any _ 2 distance = 100 . At the distance 10 from the centre of force _ 2 the t and velocity ordinates are equal, each = 10 , and their _ 2 rectangle =100 = the area of the circumscribing square. At the beginning of the descent the velocity ordinate x t ordinate = I 2 x 100* = UK) 2 . At the surface of the orb, the _ 2 end of the descent, velocity ordinate x t ordinate = 100 x I 2 = 100 2 . At 50 from the centre, or half the descent, velocity ordinate x t ordinate = 2 2 x50 2 = The side of the base, or greatest ordinate of obelisk, = axis 2 = 10 = side of the central or angular square of the -i hyperbolic area = asymptote 2 = axis = side of the circum- scribing square =10. The axis x ordinate obelisk = ordinate = 10 3 = the circumscribing rectangled parallelogram of the obelisk = -fa axis , or -fa the circumscribing square. Compare the area of the sections in fig. 49. made by a plane, which being at right angles to the sides of the base of the pyramid and obelisk, divides each into two equal parts by passing through their apices. 3, __ 2 Area of obeliscal parabola : area triangle : : f axis 2 : \ axis 1 *-* :: f . \ axis :: 4 : 3 axis' Fig. 47. The ordinate of the hyperbolic area at the dis- tance of 20 from the apex of the obelisk or centre of force = 5. At the distance of 25 from the apex the ordinate of the H 4 104 THE LOST SOLAR SYSTEM DISCOVERED. obelisk = 5. Hence the hyperbolic and obeliscal ordinates will become equal between the distances 20 and 25, where the hyperbolic curve will cut the obeliscal or parabolic curve. Fig. 49. Fig. 49. The two hyperbolic curves are continually ap- proaching each other and the common axis ; but as the last or din ate of the hyperbolic area = of n; therefore, how great soever the axis n may be supposed, still the ordinate of n will be a definite quantity, and although the curves are con- tinually approaching the axis, and to parallelism with each other, yet they can never meet, nor become parallel. On the contrary, the sides of the obelisk are continually diverging from each other and the common axis; yet they are continually approaching to parallelism with each other ENIGMAS. 105 and the axis n ; but they never can become parallel, because how far soever they may be extended, still the ordinate n 1 will exceed the ordinate n\ of the obelisk. It follows that although the hyperbolic curves are con- tinually approaching each other, and the sides of the obelisk continually diverging from each other, still the curve and side of the obelisk are continually approaching to parallelism with each other, although they are continually diverging from each other. The series, if continued beyond n, will become n+l 9 n + 2 to or - , which may again be continued to , and 2n ' so on to - or -., and then again, , JL, & c . Still 1 - - will be greater than 0. ft 4 This figure unveils three great enigmas ; the obelisk, the pyramid, and hyperbolic solid; temples around which the race who erected them, before history commenced, knelt and looked through Nature up to Nature's God. The Saba3ans worshipped these symbols of the laws of gravitation which govern the glorious orb of day, the planetary and astral systems the grandest and most sublime of the visible works of the Creator. The knowledge of these laws, and of the magnitude, distance, and motion of the heavenly bodies, inspired man with the most exalted feelings of reverence towards the Great First Cause. The sacred Tau is again represented in fig. 49. by the obelisk and hyperbolic solid, as the generators of time, velocity, and distance. Typhon, the son of Juno, conceived by her without a father, was of a magnitude so vast that he touched the East with one hand and the West with the other, and the heavens with the crown of his head. If a body be supposed to fall from the earth to the sun, the apex of the obelisk or pyramid would be in the centre of 106 THE LOST SOLAR SYSTEM DISCOVERED. the sun, and the base of the hyperbolic solid, like two arms, would extend from east to west. The following hieroglyphics, with the translation, is given by Gliddon in his " Ancient Egypt." KHNUM, (one of the forms of AMON, the creator) the creator (the idea denoted by a man building the walls of a city) of all Mankind, (literally men and women.) Ill " May thy soul attain to Khnum, the Creator of all mankind." Here we find the Creator represented as forming the laws of gravitation, and appears to be in the act of completing a counteracting force, similar, equal, and opposite to the one already made, so that where the central line bisects the distance between the two equal and opposite forces a body would gravitate to neither. If velocity oc , then velocity will x ordinate of the hy- perbolic column. If force of gravity oc , then force will oc ordinate of the hyperbolic column. The effect produced by the action of gravity on a body GRAVITATION. 107 that begins to fall freely at a distance near the earth's surface is that equal increments of velocity are generated in equal times. But the effect produced by gravity when a heavy body is freely acted upon by the earth at the distance of the moon from the earth will be different, as unequal increments of velocity will be generated in equal times. According to Newton the force of gravity varies inversely as the distance squared generally. Having deduced the properties of the obelisk from the effects produced by gravity acting on a body during its fall near the surface of the earth, let us now endeavour to illus- trate the effect produced by gravity generally. When the body falls from the apex of the obelisk, the distance described is reckoned from the apex, and the velocity acquired, as well as the time, T, elapsed, both vary as D* from the apex. But the time, t, in describing a small definite distance at any point in the descent x inversely as the velocity at that point, or t x v Or t ordinate of the curve x inversely as the ordinate of the obelisk. Also VXD*, and t x D* Newton found that the versed sine of the arc described by the moon in one minute was equal to the distance through which a heavy body at the earth's surface would fall in one second. Therefore the distance through which the latter would fall in one minute would be 3600 times greater than that through which the moon would fall in the same time. Or, according to Newton, the accelerating force of gravity x 2 ; that is, if the circular motion of the moon were de- stroyed and the moon descended as a heavy body towards the earth, it would in 1 second describe -00443 of afoot; a heavy 108 THE LOST SOLAR SYSTEM DISCOVERED. body falling from a state of rest near the earth's surface will describe 16*14 feet in a second. Now -00443 x 3600 = 15,948 feet, so that the force of gravity would, at the distance of 60 semi-diameters of the earth from its centre, cause a body to move from a state of rest and describe '00443 of a foot in one second ; while in the same time a body would descend from a state of rest and describe 16 '12 feet by the force of gravity at the earth's surface. Thus gravity is an accelerating force, and is 3600 times greater at the earth's surface than at the distance of the moon. So that if the -J- diameter of the earth be made = unity, this accelerating force will oc ^ Hence the figure that represents the velocity at different distances, from the centre of force to the moon's orbit, will also correspond to the force of gravity at the same distances. According to Newton, the times wherein any bodies would fall to the centre from different distances are between them- selves in the sesquialteral proportion of their distances di- rectly. Or time to centre oc D . But if instead of the accelerating force of gravity varying , the velocity be supposed to oo , then the time to cen- tre will oc D 3 . Since the force of gravity at the moon : the force of gravity at the surface of the earth :: 1 : 3600, if a body be supposed to fall from a state of rest at the moon and at the earth's surface ; the distance (unity) described in 1 second at the moon by the force of gravity : the distance described in 1 second at the surface of the earth by the force of gravity :: 1 .' 3600 :: velocity produced by the force of gravity at the distance of the moon : the velocity produced by the force of gravity at the earth's surface. The time t of describing unity at the distance of the moon : the time t of describing unity at the earth's surface :: 3600 : 1, for *xv=3600 oc 1. Hence if, at any point of the descent, sections of the hyper- GRAVITATION. 109 bolic solid and pyramid be made perpendicular to the axes, the area of the section of the hyperbolic solid will be propor- tional to the force of gravity at that point, and to the distance the force at that point would cause the body to fall from a state of rest in 1 second, which will be proportional to the velocity produced from rest, or to the distance described in 1 second by the force of gravity at that point. The section of the pyramid will be proportional to the time t of describing unity at that point. tx v will =3600, and ex 1. But supposing the force of gravity be such as to produce a velocity oc , time t will x D 2 , then we shall be enabled to illustrate these variations by the hyperbolic and pyramidal temples of the ancients. So calling the distance of the moon from the earth = 60 semi-diameters of the earth, we shall have velocity at moon : velocity at the earth's surface :: 2 : ~ :: I 2 : 60 2 :: 1 : 3600, or velocity acquired at the end of the descent will be 3600 times greater than the velocity at the beginning. In making some experiments we found that we could, without contact or external agency, attract and repel va- rious substances with a velocity that evidently varied in some inverse ratio of the distance ; and, as far as the eye could judge, the velocity seemed to vary inversely as the distance squared. The effects were produced by the finger touching the water on which the substances floated. This caused us to reflect on the laws of gravitation. So the experiments were abandoned, and our attention directed to other subjects mentioned in this work. Having shown by the obelisk that the time t in describing unity x inversely as the velocity at that point, or that t x v = a constant quantity, This relation of t to v will be the same whatever the law of velocity may be, or t x v will always equal a constant quantity. 110 THE LOST SOLAR SYSTEM DISCOVERED. Since the velocity at the earth is 3600 times greater than the velocity at the moon, it follows, that the time t in de- scribing a small definite distance at the moon will be 3600 times greater than the time t in describing the same distance at the earth, or t x v at the moon = 3600x1 = 3600, and t x v at the earth = 1x3600 = 3600. Similarly t x v at the intermediate distances will 3600. Since t oc v and v oc t oc D 2 . Fig. 49. When the obelisk is placed along with the pyramid, the bases of both being at the moon and their apices at the centre of the earth ; then as the ordinate of the pyramid descended as the time T elapsed from the beginning of the descent, the ordinate of the obelisk will correspond with the ordinate of the pyramid. The frustum of the pyramid above the ordinate will denote the time T elapsed during part of descent, and the remaining or lower part of the obelisk included between the descending ordinate and apex will oc D 2 oc time t. At the end of the descent the whole time T elapsed will be represented by the whole pyramid, and time t will vanish with the obelisk. Or the time T elapsed will increase as the frustum of the pyramid increases, while the time t will decrease as the obelisk decreases, so that at the end of the descent the pyramid will be completed and the obelisk will have vanished, excepting the small portions of the pyramid and obelisk each having an axis=l, since the descent of the body would cease at the earth's surface. Thus great T may be said to have consumed little t, or MYTHOLOGY. 1 1 1 Kronos to have devoured his offspring. But supposing the body to be repelled from the centre or apex, then during the ascent the obelisk, which was consumed at the end of the descent, will increase from the apex, so that at the end of the ascent the obelisk will be completed, or the offspring may be said to have attained the heavens. Again, the time of descent from the beginning to any point of the axis oc D 3 e? 3 , D being the whole axis described in the time T, and d the distance remaining to be described from the point in the axis to the apex of the pyramid. At the end of the descent the whole time T will oc D 3 , for d? will have vanished. If a body be repelled from the apex, time will oc d 3 ; at the end of the ascent the whole time T will a axis 3 oc d 3 oc D 3 . Here during the descent little d is consumed by great D, or Saturn devours his children. But during the ascent little d replaces great D, or Jupiter deposes his father Saturn, or Typhon destroys his brother Osiris. The Titans were brothers of Saturn/ one of whom was Typhseus or Typhon. They strove to depose Jupiter from the possession of heaven, but they were beaten and cast down into hell. _ . oc - oc ; . oc D oc axis oc pyramid deprived of its Jupiter t D 2 generating ordinate. Thus Kronos, when divided by his son Jupiter, may be said to be emasculated, as Caelum was by Saturn, and as Osiris by Typhon. Jupiter Ammon is represented with the horns of a ram. The ram's horn is symbolical of the spiral obelisk. The content of the obelisk oc D 2 . Kronos and Jupiter may be said to be divided against each other, when Jupiter wars against his father. Jupiter castrated Saturn or Kronos, as Saturn had cas- trated his father Caslum before with a sickle. The sickle may be symbolical of the curved obelisk. Saturn, like Time, has his scythe. Should the scythe represent the area of the obelisk, then the scythe of Saturn would be typical of the periodic time of the revolution of planets round the Sun. 112 THE LOST SOLAB SYSTEM DISCOVERED. Saturn holds in his hand a serpent with the tail in its mouth, forming a circle. The circular serpent is symbolical of the circular obelisk. The obelisk is typical of infinity or eternity, and the circle the orbit of a planet. So the circular serpent denotes that planets revolve in circular orbits, having their p . T oo area obelisk and velocity oc - : - , and that they will revolve ordmate in their orbits to eternity. The proud Neith says ( ( I am all that has been all that shall be and none among mortals has raised my veil." Neith is gravitation, by which the planets are preserved in their orbits, and supposed to continue their revolutions round the sun to all eternity. But what is gravitation, that causes planets to revolve in orbits having their p . T \? but t X D OC 1)3 X D X D* X P T. In the descents from different orbits to the same centre but T c oc D 3 , and t oc x D v t X D oc D 2 X DX D 3 X T C. Thus in the three laws of motion t x D will vary as T, p T, and T C. T Hence D x _ x PT D x TC In any of the three laws of motion, if the variation of v, T, or t be given, the other variations may be determined. Generally, T x v x D, and t <* - D D When v x D*, T x x ~\ x D I 4 120 THE LOST SOLAR SYSTEM DISCOVERED. D i - X i X D 2 V X T T D5 1 1 x - x x : x D D 3)3 V V X , P T. Or T X-XD X D^ X D* D* V D D 1 V cc - x oc - T D* D* T D^ i 1 t X X X D 2 X D D V V X , T C, OF T ~ X D X D 2 X D 3 D D 1 Vac- ^- 3 oc- 2 T D 3 1 t QC x xD^x D D \ Generally T x D x t T tec- When v x ^ P T 3 x D x axis obelisk D 5 p T3 x D5 x area obelisk p T3 x D 2 x content obelisk x orbicular area p T 2 x D 3 x content pyramid. In the orbicular velocities t, the time of describing unity, oc - x ordinate obelisk or t x ordinate obelisk VAKIATIONS. 121 axs P cc axis 2 oc area obelisk x P T. PT 2 oc axis x pyramid or -J- axis tf 3 x ordinate x pyramid or ordinate PT x ordinate x pyramid or -J- ordinate The Jig. 50. represents the pylonic area composed of a series of 6 equal parallelograms along the sectional axes I, 3, 5, 7, 9, 11, so that each sectional axis multiplied by its mean Fig. 50. ordinate will = 6, which equals the area of the first paral- lelogram or 6 x 1, 6 being the last ordinate of the obelisk corresponding to its axis 36, and the first ordinate of the pylonic area. The mean ordinates of 1, 3, 5, 7, 9, 11, the sectional axes, will correspond to the mean ordinates of the obelisk, which will lie between the ordinates 1, 2, 3, 4, 5, 6. 122 THE LOST SOLAR SYSTEM DISCOVERED. Hence the mean ordinate of obelisk multiplied by the mean ordinate of the py Ionic area will = 3, the half of 6,, the area of each parallelogram when one side = a sectional axis, or two ordinates of obelisk, and the other side = mean pylonic ordinate. as JL 2 H -f- = = 3 or i- = x - = f = 3 f =3 * = 3 x f = 3 H These mean ordinates, , f , &c. of the pylonic area will oc inversely as the mean ordinates *-, 1, 2^, 8cc. of the obelisk. So that if the orbits passed these two series of ordinates, the rectangle of each two corresponding ordinates would = 3. Fig. 51. Another series of parallelograms may be inscribed Fig. 51. between the axis and the curve by making the pylonic ordi- nates = ^ , 3^, f , &c. for one side of each parallelogram, and -the corresponding axis 36, 25, 1 6, &c. for the other sides. The VARIATIONS. 1 23 areas of these series of parallelograms along the axis of the curve will be as 18, 15, 12, 9, 6, 3, for the axis multiplied by its corresponding ordinate will be as 36 x ^ = 18, 25 x -f w = 15, 16 xf =12, and the areas 18, 15, 12, c., will ) for the wth parallelogram will = ordinate x axis = x n So as n increases, the last parallelogram will continually ap- proach to 5(3rt-3)=15x*-l. When n=l, 3ra 3 = 0, and first rectangled parallelogram = 5(3^-3 + -") =5 x-=5x -=1. V n/ n 5 So is incomparably greater than 3ra 3, being as 1 : 0; but as n continually increases, 3n 3 becomes vastly great com- pared with -. Since varies inversely as TZ, or as n increases n n - decreases, much more will 3n increase while - diminishes. n n If the ordinate oc r, and rc=5 = the 1st ordinate of the D* series of 5 parallelograms, the several axes will be I 4 , 2 4 , 3 4 , 4 4 , 5 4 , and ordinates 1 2 3 4 5 VOL. I. K 130 THE LOST SOLAR SYSTEM DISCOVERED, Sum of the series of parallelograms will be 5 = 5xl5 2 = 1125, axis of the series = 5 4 = 625. The series of 5 parallelograms along the sectional axes will be 5, 37-5, 108-33, 218-75, 369, or 5 (1, 7-5, 21-66, 43-75, 73-8). Let the series be continued to n terms while the 1st ordinate remains =5. The series will be 5(1, 7-5, 21-66 since the nth parallelogram will = ordinate x axis JL -2 When velocity cc i, p T will oc D 3 . Since the orbits of planets are supposed to be circular, and the velocity in each orbit uniform, the distance described will oc as the time, and the whole time T, or periodic time of a revolution, will oc directly as the orbit, or whole distance described, and inversely as the velocity, orbit i i 3 or P T oc i r- oc radius x D 2 oc D x D 2 oc D 2 velocity but area obelisk oc axis 2 oc D^, consequently p T will oc area obelisk oc D 5 _ 2 or p T oc D 3 . Hence, knowing the variation of the p T in different orbits round the same centre, the areas described in equal times by the radius vector in different orbits may be found. Areas described in equal times by the radius vector in VARIATIONS. 131 different orbits will oc directly as the orbicular area, and in- versely as the PT oc radius PT 00 oc D radius axis PT axis i oc axis oc ordinate of obe- Or area described oc lisk. Hence the area of obelisk from the apex to the ordinate, corresponding to any axis, radius, or distance will represent the P T of a body revolving in the orbit of that radius; and the ordinates themselves, corresponding to the different distances or axes, will represent the variation of the areas described in equal times in different orbits. (Figs. 50, 51.) By the tables the distance of Mercury from the Sun = 36,841,488 miles; that of Saturn = 907,956,130. The periodic time of Mercury = 88 days nearly. The periodic time of Saturn = 10,766 days. Taking 36 and 900, in round numbers, as the distances of Mercury and Saturn from the Sun, the corresponding ordi- nates will be 36^ and 900 or 6 and 30. So the area of Mercury's orbit will be to the area of Saturn's orbit ::36 2 : 900 2 :: 1296 : 810,000. Then areas described in equal times by Mercury and Saturn will be as orbicular area as 36 2 6 3 900 ::6 : 30, PT ordinate 3 6 3 30 3 which is the ratio of their ordinates and the inverse ratio of their velocities. The times of describing equal areas in different orbits oc PT D 5 : OC - OC orbicular area D 2 ordinate So times of describing equal areas in the orbits of Mer- , , . , ordinate 3 6 3 . 30 3 9 cury and Saturn will be as as . ^^ ..60 . b .. D 2 36 2 900 5:1, which is inversely as their ordinates and directly as their velocities. Thus in equal times the area described by Saturn with a K 2 132 THE LOST SOLAR SYSTEM DISCOVERED. velocity 1 will be to the area described by Mercury with velocity 5 :: 5 I 1. So that Mercury may describe an area equal to what Sa- turn describes in a given time as 1 second; the time required by Mercury will be 5 times greater than the time required by Saturn ; though Mercury moves with a velocity 5 times greater than that of Saturn. Or area described by Saturn in 1 second = D x velocity = \ 900 x 1 = 900 ; area described by Mercury in 1 second = ^Dx velocity =|-36x5 = 1 180; in 5 seconds =^180x5 = i. 900 = area described by Saturn in 1 second. As the areas described in circular orbits in a small portion of time, 1 second, oc radius x velocity x D x velocity ; the areas described in a greater portion of time will oc D x v ; for the latter areas will be equal multiples of the small areas. Or, as velocity is the distance described in a given time, it may be represented by a straight line, or the arc of a circle. For the area of circle = the rectangle of the radius x circumference. According to Archimedes a circle is equal to a right-angled triangle having one of the sides equal to the radius, and the other equal to the circumference of the circle. So the area described in a circular orbit can be repre- sented by a rectangle -J- D x velocity. Otherwise, calling the distance of Mercury and Saturn 36 and 900, since p T oc ordinate 3 , p T of Mercury .* p T of Saturn :: 6 3 : 30 3 . The orbicular area of Mercury : orbi- cular area of Saturn :: 36 2 I 90Q 2 . 36 2 I 2 1 Therefore orbicular area of Mercury = - = -^r, the ^O orbicular area of Saturn. So the time of describing an area in Saturn's orbit = the p T 30 3 area of Mercury's orbit will be as - = - - =43'2. Hence the times of describing equal areas in the orbits of Mercury and Saturn will be as VARIATIONS. 133 6 3 : 43 2:: 216 43-2 :: 5 1 :: 30 6, which are inversely as their ordinates, or directly as their velocities. Since the velocity in each orbit is uniform, the distances described in equal times in different orbits will oc velocities ordinate Also as time t of describing unity, oc - 12* L 2 0* 30* 42' 56' 72 Hence the series of inscribed rectangled parallelograms at right angles to 1, -J-, -J-, &c., will be |-, twice -J-, three times -Jj, &c. For 1st superficial rectangled parallelogram = of 9 9rrl 9 V 1 ^nu ,, & A 6 3rd ,, = 3 x -rV JL 3 4th 5th 6th 7th 8th 9th 4. ~ * i ~~ 7 JL = 9x1 r> i i. = 1. 1 of 9, the In this hyperbolic series, greatest parallelogram = 9 is placed the last. But in the series 1, -J-, , &c. of 9, the greatest parallelo- gram is placed the first. This last series of parallelograms overlap each other from M to i L. The series , $ &c., overlap one another from i to L M. 138 THE LOST SOLAR ED. 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 paral- lelogram, the last of the series will be a square. By taking the difference of the series of rectangled paral- lelograms, 1, |-, ^, &c. in one square, we have the series of rectangled parallelograms, \, -J-, ^ &c. formed in the other square. The sum of the series i + i+iV .... +-^ of 9 to 8 terms will by construction = 91x1 = 8, So when 1, -J-, -J-, &c. of n is continued to n terms, the sum of the differential series i + i+iV* & c< ^ n ^ w ~~l terms will = ft 1. The series ^, ^ -^ 9 &c. to ^. of n may also be formed n I n from the series 1, -J-, ^, , . . . . - of n, by multiplying the 1st term by the 2nd, the 2nd by the 3rd, the 3rd by the 4th, and the n I by n, as JL 1 2 1 a 1 . n The sum of this series to n I terms will = n 1. By construction, it will be seen that the differential series 2~0* 42' 56' 72 of 9 to 8 terms x by 1, 2, 3, 4, 5, 6, 7, 8 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 which is formed from ?z 2 n, will be seen to = -J- n 3 n SERIES. 139 .'.0 + ^ + + V.... +- of n to n terms will= n 1 n ~ of n 2 n, from which the direct series 0, 2, 6, 12, 20, &c. is formed. The last term of the series = === - n 1 . n' _ \ hence the sum of the series, n 1, will =- of the denomina- tor of the last term. 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 will = 900, and the area of the central or angular square L N = 900 = 30 2 . 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 I axis of the hyperbola : : 3 I 9 : : 1 : 3. When 6 hyperbolic parallelograms are inscribed in the _ 2 square = axis of curve = 6 2 = 36, the area of the series = 14 '7. When 36 parallelograms are inscribed in the same axis , now = 36 2 , the area of the series = 150-3. 2 .. area of 6 parallelograms ." axis :: 14-7 \ 36 area of 36 parallelograms : axis :: 150-3 : 36 2 ::4*17 : 36 Thus 6 inscribed parallelograms will= 14 '7 of 6 2 , or axis * And 36 inscribed parallelograms will only =4'17 of the same square. First parallelogram in the series 36 will = of 140 THE LOST SOLAR SYSTEM DISCOVERED. the first parallelogram in the series 6. And first 6 paral- lelograms in series 36 will =-J- of the first 6 parallelograms in series 6, = -J- 14-7 = 2-45, but whole series of 36 paral- lelograms == 4' 17 of axis , or of 6 2 . .-. 4-17 2*45 = 1-72 for the area of the remainder of the 36 parallelograms. Hence when the radius of the quadrant is divided into 6 equal parts, the area of the 6 hyperbolic parallelograms described in the square = axis , will = 14*7. When the same radius is divided into 36 equal parts, the area of the 36 hyperbolic parallelograms described in the 2 same square will = 4*17 of the axis , or of 6 2 . As n increases the more the radius is subdivided, the more will the angle B c 9 of the first or primitive triangle decrease, and the sine of the triangle will approach to equality with the hypothenuse, or radius, and the curvilinear area to that of the quadrant. The difference between the hypothenuse or radius C B and the sine that subtends the angle at c of the primitive triangle will always equal unity in the series 1, 2, 3, to n terms. Hence the radius will be to this sine as n : n 1; also twice the hypothenuse of the triangle = diameter of the circle = the axis of the reciprocal curve = the two asymptotes of the hyperbola. Thus the hypothenuse of the primitive triangle deter- mines the radius of the quadrant : the angle at c determines unity in the radius. These also determine the reciprocal curve, and the series of hyperbolic parallelograms as well as the series of parallelograms which form the triangular area. The outline of a dome is formed by the hyperbolic reci- procal curve, or the dome itself is formed by the revolution of the curve on its axis. Hodges thus describes his visit to the mosque of Moun- heyr, twenty miles distant from Patna, the capital of the province of Bahar. This edifice is not large, but very beau- tiful. A majestic dome rises in the centre, the line of whose curve is not broken, but is continued by a reverse curve till it terminates in a crescent. This appears to our author in- THE CAP OR HELMET. 141 finitely more beautiful than the European system of crowning the dome by some object making an angle with it. Area of quadrant =9 2 x *7854 = 63-6174 which x by 2 =127-2348. Hence, the radius continuing the same, as n increases, the curvilinear reciprocal area will continually approach to equality with that of the quadrant. The hyperbolic area, as n increases, will also continually decrease, when the same quadrant has its radius continually subdivided into equal parts for determining the reciprocals of the sines, which determine the hyperbolic area. The high cap having the hyperbolic reciprocal curve for the outline is one of the insignia of divinity or royalty (for kings shared the attributes of gods). Such a cap is some- times seen on the head of Osiris, and on the colossal statues at the entrance of the Luxor. Sometimes the top of the cap or helmet, like the hyperbolic area, terminates in a point ; such are found in Egypt, at Nimroud, and at Babylon. Also, in the Nimroud sculptures two archers have caps or helmets truncated at the top, like that in the constructed curvilinear area. The more truncated the top, the less will the radius be divided. The more pointed the top, the more will the same radius be subdivided. The two arches that have the truncated- like caps have both curled beards of the obeliscal form, like the Egyptians. The sphent may represent the hyperbolic area. The beards, or their casings, as seen in the Egyptian statues, are of the obeliscal form, typical of infinity. Similar beards are seen in the Assyrian sculptures. The hair of the head is frequently arranged in parabolic curved lines ; the focus being placed lower down than the crown of the head, over that part called by phrenologists the love of offspring. This parabolic arrangement of the hair is also symbolical of infinity. The focus may be supposed to be the sun, and 142 THE LOST SOLAR SYSTEM DISCOVERED. the parabolic curves the paths of the comets. Or they may together be supposed to represent a comet itself, or Stella crinita. The impression of Buddha's foot is like this parabolic or cometary system ; but with the addition of circular orbs placed round the focus, or sun, indicative of the planetary orbits. So that the foot-mark of Buddha represents both the cometary and planetary systems : the sun being placed in the centre of the heel, having concentric planetary circles ; the cometary parabolic paths extend to the toes, having the sun in the focus. The lower part of one form of Egyptian cap, as it rises from the head, is sometimes curved outwards, probably in- tended to denote the hyperbolic curve ; from this lower part rises the crown, of an egg-like shape. Such a combination is on the head of a colossal statue of polished red granite in the British Museum. The whole height of the statue is supposed to have been about 26 feet English, which would equal 37 Babylonian cubits. The egg-shaped part of the cap may represent the parabolic or hyperbolic conoid, both being typical of eternity. Or, if an ellipse revolve on its less axis, an oblate spheroid will be generated, like the figure of the earth. If the same ellipse revolve on its greater axis, an oblong spheroid will be generated, like the mundane egg. But the oblate spheroid, being the greater, would contain the oblong spheroid. So the world might be said to contain the mundane egg. We have since met with the reciprocal hyperbolic cap on a figure, supposed that of a priest, sculptured on stone, which Rich found at Hillah. He also informs us, " that among the gardens a few hundred yards to the west of the Husseinia gate, is the Mesjid-ess-hems, a mosque built on the spot where popular tradition says a miracle similar to that of the prophet Joshua was wrought in favour of Ali ; and from this the mosque derives its appellation. It is a small building, having instead of a minaret an obelisk, or rather hollow cone, THE HORN AS HEAD-DRESS. 143 fretted on the outside like a pine-apple, placed on an octagonal base. This form, which is a very curious one, I have ob- served in several very old structures ; particularly the tomb of Zobeide, the wife of Haroun-al-Raschid, at Bagdad ; and I am informed it cannot now be imitated. On the top of the cone is a mud cap, elevated on a pole, resembling the cap of liberty. This, they say, revolves with the sun ; a miracle I had not the curiosity to verify." The exaltation of the horn, an expression so frequent in scripture, is explained by the practice still existing in the East, of employing the horn in the head-dress. This is particularly the case among the Druses of Lebanon, where the horn is a tin or silver conical tube, about twelve inches long, and the size of a common post horn. The wife of an emir is distinguished by a gold horn enriched with pre- cious stones. This ornament of female attire is worn on the head in various positions, distinguishing their several con- ditions. A married woman has it affixed to the right side of the head, a widow to the left, and a virgin is pointed out by its being placed on the very crown : over this silver projec- tion the long veil is thrown, with which they so completely conceal their faces as rarely to have more than one eye visible. A similar horn is in use among the Christian women at Tyre; and ornaments of this kind are worn in some parts of the Russian territories. In Abyssinia Bruce found these horns worn by men : they attracted his par- ticular attention in a cavalcade, when he observed that the governors of provinces were distinguished by this head-dress. It consists of a broad fillet tied behind, from the centre of which projects a horn or conical piece of silver-gilt, about four inches long, and very much in its general appearance resembling a candle-extinguisher. It is called kirn (as in Hebrew), and is worn after a victory or on great public occasions. The hyperbolic reciprocal curve formed by the 4 quadrants will resemble a winged circle, which may be the origin of the winged globe or planet urged forward in its orbit by its reciprocal wings typical of positive and negative electricity. 144 THE LOST SOLAR SYSTEM DISCOVERED. The semicircle and reciprocal wings may represent the outline of Mercury's cap, which is hemispherical with wings attached to the sides. To his ankles the winged sandals, or talaria, are attached. The winged caduceus that he holds in his hand is entwined by two serpents in opposite directions, which may also denote positive and negative electricity. The Egyptians painted his face partly black and dark, and partly clear and bright, because he is supposed to converse sometimes with the celestial, and sometimes with the infernal gods. Or he may be regarded as flying by the aid of elec- trical wings, and so like an electrical telegraph communicating with heaven and earth. The positive and negative electric powers may have been indicated by his face being partly dark and partly bright. Nared, the son of Brahma, was, like Hermes or Mercury, a messenger of the gods. The wings of Mercury being hyperbolic and electrical, they denote that planetary distances would be traversed with the speed of electricity. The velocity of Mercury, which is nearest the sun, is greater than that of any other planet. But we suppose the wings of the globe to be symbolical of the obelisk, the exponent of the laws that urge a planet onwards with a velocity x , and p T co area obelisk. ordmate The motive power of the two wings by which the planet is propelled forward and preserved in its orbit may be positive and negative magnetism, galvanism, or electricity ; all of which have recently been discovered to be modifications of the same law of nature. By this agency the planet, like a bird, is supposed to fly with two electrical wings, which urge it forward and prevent its falling to the earth. Two serpents belong to the winged globe. The serpent is typical of the circular obelisk, or infinity. But the large expanded wings of the globe resemble the outline of an obeliscal or parabolic area, which denotes the periodic time of a planet. The serpent when formed into a circle with the tail in its PILASTER AT MEDINET-ABOU. 145 mouth, denotes the orbit in which the planet will revolve to eternity. Or if the serpent be supposed to eat its tail, the orbit will diminish so that the planet would ultimately fall to the centre of force, the sun. A caryatid pilaster, at Medinet-abou, 24 feet high, in- cluding the high cap, has the hands at the lower part of the chest resting upon a support rising from between the feet, VOL. I. L 146 THE LOST SOLAR SYSTEM DISCOVERED. and presenting in front a parallelogram having a breadth about ^ length. If the parallelogram were divided into -, ^ 3 , &c., the sum of all the parts, how far soever continued, would never equal the parallelogram itself; so the parallelogram would be symbolical of infinity or eternity. One hand holds the whip, or outline of the obelisk, the other the crosier or curve of Osiris. The beard is obeliscal. Above the cap is the globe and serpents with obeliscal or parabolic wings. The high cap itself is bounded on each side at the lower part by two serpents, and with feathered-like appendages of uniform breadth and of contrary flexure, extending the whole length of the sides. Fig. 55 OSIRIS. 147 Fig. 55. If the focal distance AS of the parabola =Aa = SB = i BC = latus rectum = 36. So AB = BC=latus rectum = 6 x 6 = 36. Parabolic area = 6 times area of obelisk. The ordinates pp SP at the different sections, ap will be a curve of contrary flexure traced by p. SP 2 = SQ 2 + PQ 2 = (AQ As) 2 -|- pq 2 = (axis \ L,) 2 + ordinate 2 = axis 2 L x axis -f -J-L 2 -H ordinate 2 = axis 2 + % L 2 .-. SP and Pp will always be greater than the axis, and the curve of contrary flexure ap will continually approach to, but can never touch the axis Aq. Hence the curve ap will be infinite, and the high cap of Osiris will be symbolical of eternity. The two feathered-like appendages along the curved side of the cap denote that the breadth of the cap will increase as the focal distance A s increases. If the focal distance were increased, the feathered-like appendages would become more like the curve which Osiris holds in his left hand. Thus the curve of Osiris will be typical of the parabolic curve of contrary flexure, or of infinity. When s P is above s, s p 2 = ordinate 2 + ( L axis) 2 When s P is below s, s p 2 = ordinate 2 + (axis L) 2 . The top of the cap and feathers being rounded off may denote their infinite extension. The serpents on the sides of the cap are typical of the obelisk or of infinity. The serpent here represented is perhaps the most common of all the Egyptian hieroglyphics. It is known by its erect position, swollen neck, and the entwining folds of the lower part of the body. Denon has given a sketch of this serpent I, 2 148 THE LOST SOLAR SYSTEM DISCOVERED. in the same attitude as we see it on the sculptured stone. It is the Naia Haje, a most venomous snake, which the ancient Egyptians assumed as the emblem of Cneph or the Good Deity. It is also a mark of regal dignity, and is seen on the fore part of the tiara of almost all Egyptian statues of deities and kings. This serpent in the erect position with its swollen neck resembles the parabolic curve of contrary flexure, the same as that of the cap, and the curve in one hand of Osiris. The Ibis, like the Naia Haje, may have been held sacred from its head and long beak having a resemblance to the parabolic curve of contrary flexure. In the other hand Osiris holds the obeliscal whip, by means of which he urges the heavenly bodies onwards in their orbits. Hence the myth of Phaeton driving the chariot of his father Sol. The Sun was worshipped by the Egyptians under the name of Osiris. The sun is the centre of force round which the planets re- volve with velocity oo =r- , and p T oo area obelisk, that ordmate is, the planets are urged onwards in their orbits by laws indicated by the obelisk ; or, metaphorically, they are driven by Sol or Osiris with the obeliscal whip. As the focal distance increases, the parabola increases, which is denoted by the feathered-like side of the cap ; for the short lines made by a series of increasing parabolas will be more inclined as they recede from the axis of the parabola, and thus give the outside of curve of contrary flexure a feathered appearance. The axis of the curve oc ordinate of parabola, and ordinate of curve oc s P axis of parabola. The revolution of the curve on its axis would generate a solid like the cap. The obeliscal beard typifies eternity. If equal parabolas, having their axes in the same straight line and their apices coinciding in A, but on opposite sides of Ap 9 then the parabolas described on one side of Ap will feather the curve generated by the parabolas on the opposite side of A/?. ATTITUDES OF THE GODS. 149 Again, if the apex of each parabola passed through the focus of the other, the sun would be in the axis of the curve, like the globe over the forehead of the figure ; then the two parabolas would represent the paths of two comets describing parabolas or ellipses round the sun as the common focus. The other globe on the top of the cap might denote a fixed star, or another sun placed beyond any definite distance from the sun. The Egyptian deities, when in a state of repose, are seated on hyperbolic steps, which decrease as 1, -J-, -J-, &c. So that the legs and thighs form a right angle, like the side and top of the seat ; the thighs and trunk form another right angle, like the top and back of the seat ; the arms also form a right angle, like the back and top of the seat. This hyperbolic attitude, which is typical of infinity, gives them a constrained appearance. Buddha, in the attitude of sitting cross-legged, assumes the form of the hyperbolic solid ; the Virginian Okee also assumes the same form ; so that by their constrained posi- tions they may be said to represent infinity or eternity. Wilkinson remarks that the same veneration for ancient usage, and the stern regulations of the priesthood, which for- bade any alteration in the form of the human figure, parti- cularly in subjects connected with religion, fettered the genius of the Egyptian artists, and prevented its develop- ment. The same formal outline, the attitudes and postures of the body, the same conventional mode of representing the different parts, were adhered to, at the latest as at the earliest periods: no improvements resulting from experience and observation were admitted in the mode of drawing the figure ; no attempt was made to copy nature, or to give proper action to the limbs. Certain rules, certain models, had been established by law, and the faulty conceptions of early times were copied and perpetuated by every successive artist. For, as Plato and Synesius inform us, sculptors were not suffered to attempt anything contrary to the regu- lations laid down regarding the figures of the gods; they L3 150 THE LOST SOLAE SYSTEM DISCOVERED. were forbidden to introduce any change, or to invent new subjects and habits ; and thus the art, and the rules which bound it, always remained the same. Some of the drawings of the Irish round towers represent them expanding towards the base, like a section of the hyper- bolic solid. 151 PAKT III. TOWER OF BELUS. DESCRIPTION BY HERODOTUS. CONTENT -fa CIRCUMFERENCE OF THE EARTH. CUBE OF SIDE OF ENCLOSURE EQUAL TO THE CIRCUMFERENCE OF THE EARTH. THE EQUI- VALENT OF THE STADE, ORGYE, CUBIT, FOOT, AND PALM OF HERODOTUS IN TERMS OF THE EARTH'S CIRCUMFERENCE AND THE STATURE OF MAN. THE FRENCH MEASUREMENT OF THE EARTH'S CIRCUMFERENCE. THE CIRCUIT OF LAKE MCERIS, SIXTY SCHJENES, COMPARED WITH THE MEDITERRANEAN COAST OF EGYPT ; WITH INDIAN TANKS AND CINGALESE ARTIFICIAL LAKES. HERODOTUS' MEASUREMENT OF THE EUXINE FROM THE BOS- PHORUS TO PHASIS ; OF EXISTING OBELISKS. DIODORUS* DIMENSIONS OF THE CEDAR SHIP OF SESOSTRIS COMPARED WITH MODERN SHIPS AND STEAM VESSELS. THE CANAL OF SESOS- TRIS FROM THE MEDITERRANEAN TO THE RED SEA. THE EGYPTIAN OBELISKS AT ROME, PARIS, ALEXANDRIA, HELIOPOLIS, FIOUM, THEBES. COLOSSAL STATUES AT MEMPHIS AND HELIO- POLIS. MONOLITHS AT BUTOS, SAIS, MEMPHIS, THMOUIS, MAHA- BALIPURAM. CELTIC MONUMENTS IN BRITTANY. Tower of Belus. RICH, along with Rennell and Porter, concurs in the opinion that the temple of Belus was built upon the site of the tower of Babel, but is at variance as to which of the two ruins, the Mujelibe or Birs Niinroud, is best entitled to the distinction : Rennell decides in favour of the Mujelibe, Rich and Porter incline to the Birs. The brief notice of the extraordinary event which we find in Genesis serves little other purpose than to assure us of its actual occurrence. The first act of society that we find re- corded subsequently to the destruction of the whole human race, except the family of Noah, was an attempt to rally its forces round a common centre, and to organise and cement the new L 4 152 THE LOST SOLAR SYSTEM DISCOVERED. community by some bond of union, indispensable not only to the progress of civilisation, but to the existence of society. We are informed that the place selected for this great experi- ment was the plain of Shinaar, and that there men proceeded to found a city, with a tower, whose top, in the language of scripture, " should reach to heaven." The real intentions of the founders of this gigantic structure have been the subject of much controversy, which has not hitherto led to any very satisfactory solution, Herodotus, in describing the tower of Belus as he saw it, says, the Euphrates divides Babylon into two parts ; in one part is a square enclosure, with brazen gates, the wall on each side being two stadii, and consecrated to Jupiter Belus. In the middle of this holy place is a solid tower, having the length and depth of a stadium ; upon which there is another tower placed, and upon that another, and thus successively to the number of eight. On the outside of these towers are steps winding about, by which they go up to each tower. In the middle of this staircase is a lodge and seats, where those who mount up may rest themselves. In the last tower is a chapel, in the chapel an elegant bed, and near the bed a golden table. Herodotus does not state the height of the tower ; but Strabo says that the tomb of Belus was a pyramid, one stadium in height, by a stadium in length and breadth at its base. Fig. 56, A, Taking the 8 terraces to equal f of a stade in \ V V Fig. 56. height, the height of each terrace will equal ^ of a stade ; TOWER OP BELU8. 153 and as the side of the base, or the lowest platform on which the lowest tower stands, equals 1 stade = the height from the base to the apex of the teocalli or tower ; thus the height of the apex will = -J- stade above the highest platform, or the 8th tower. ___ Q Q Let 684 = circumference of the earth in stades; then 684 x 243 will = circumference in units. These formula are obtained by transposing the Babylonian numbers 243, so that the last 3 when placed the first, and the first 2 last, make 342, which multiplied by 2 =684, and 684 raised to the power of 2 = 684 =46 7 8 56 = circumference in stades, and ~684 2 x 243 = 113689008 = circumference in units. Next let us ascertain the value of the stade and unit in terms of English measurement. Since 24899 miles, or 131466720 feet, equal the equa- torial circumference of the earth (Herschel), then 131466720 -7-467856 = 280-99825, &c. feet = 1 stade. Hence a Babylonian stade, which = 243 units, may be said to equal 281 feet English ; then a Babylonian unit will =| or 1-156378, &c. of an English foot, or =13-876, &c. ^4o inches. The content of the tower, if made equal to ^ 243 , would exceed -^ of the earth's circumference, if the cubes of unity were placed in one continuous line. _ 3 So would 486 , or the cube of the side of the square inclo- sure, exceed in cubes of unity the whole circumference of the earth. The circumference, measured by cubes of unity, would lie between 484 and 485 ; and the content of the tower, to equal -^ the circumference in cubes of unity, would lie be- tween i 242 3 and i^43 . The way of correcting these differences will be seen in the construction of the Egyptian pyramids. 154 THE LOST SOLAR SYSTEM DISCOVERED. The sides of the 8 square terraces will be -J-, J-, -|, -|, %, &, % 9 f of a stade, so that the top of each of the 8 terraces will touch the sides of the circumscribing triangle, having the base = the height = 1 stade. Thus the content of the teocalli or terraced pyramid will = -^ of a cubic stade, = 243 cubic units, by taking the stade to equal 243 units, or 3 5 . So the height of each terrace will =3 3 =27 units; the height of the 8 terraces will = 8 x 3 3 . The sides of the ter- races will = 1 x 3 3 , 2 x 3 3 , 3 x 3 3 , 4 x 3 3 , 5 x 3 3 , 6 x 3 3 , 7 x 3 3 , 8 x 3 3 . The base of the circumscribing triangle = the height = 9 x 3 3 =3 2 x 3 3 =3 5 =243 units. 2 The base of the pyramid will = 243 height = 243 3 and content = 243 . This we suppose to have been the construction of the tower of Belus, for reasons which will be seen when we come to the formation and measurement of the teocallis, or trun- cated pyramids of America. Perhaps the lowest platform on which the lowest terrace stood might have been raised ; for what is called the great pagoda at Tangore is built of hewn stone, in the form of a truncated pyramid, and consists of 12 perpendicular stories or terraces, the lowest being built on huge blocks of stone, forming the pedestal, rising by 4 steps from the ground. On the top is a temple or chapel. The content of the 8 terraces will be to the content of the pyramid having the side of base and height equal the base and height of the circumscribing triangle, ::1 (! 2 + 2 2 + 3 2 + 4 2 + 5 2 + 6 2 + 7 2 + 8 2 ) : -J9x9 2 ::204 : 243. For (I 2 + 2 2 + 3^... + 8 2 )=-J-wTl . n. n + =9x8x8'5 = 204. But we only want the content of the complete pyramid having the height and side of base=l stade. So hereafter we shall only ascertain the content of the pyramid having the height and side of base equal the height and base of the triangle that circumscribes the sides and base of the teocalli. TOWER OF BELUS. 155 In this calculation of the terraced tower of Belus the sides of the terraces are supposed to be perpendicular; but possibly this was not the case, for all the American teocallis, as far as we know, have the sides of the terraces inclined, excepting one in Peru, where the sides are perpendicular. But whether the sides were perpendicular or inclined does not affect the content of the pyramid made by this cal- culation, that the base of the circumscribing triangle equalled the height, equalled 1 stade. The content of the 9 terraces, B, fig. 56., will be to the content of the rectilinear pyramid having the side of the base and height = the base and height of the inscribed triangle :: \n-\-\ . n . n + : n* :: 285 : 243, and | (204 + 285)= 244 -5 244-5- 243 = 1 = the difference between the mean of the 8 inscribed and the 9 circumscribing terraces and the rectilinear pyramid ^ 9 3 . Next double the height and side of the base of this pyramid. Then such a rectilinear pyramid will=^ 18 3 = 1944. The 17 inscribed terraces will 17x17^=1785. The 18 circumscribin terraces will 18x18^=2109 i(!785 + 2109)=1947, and 1947 1944 = 3 = the difference between the mean of the two stratified pyramids of 17 and 18 terraces and the rectilinear pyramid ^ 18 3 . Thus the rectilinear pyramid -- 18 3 is less than the mean of the content of 17 and 18 terraces by 3. When the content of the rectilinear pyramid ^ 9 3 , the rectilinear pyramid was less than the mean of the two strati- fied pyramids of 8 and 9 terraces by 1^-. Thus the rectilinear pyramid having the height and side of the base = n, will be less than the mean of the content of the two stratified pyramids, the one being within and the other without the triangle = \ ri 2 , by a cubic unit in every 6 ter- races, or of a cubic unit in every terrace. 156 THE LOST SOLAR SYSTEM DISCOVERED. The internal pyramid has n 1 terraces. The external pyramid has n terraces. All the terraces are rectangular. The side of the base of the tower = 1 stade, and the side of the square enclosure in which the tower stood = 2 stades. Therefore a cube having its side = that of the enclosed area, will nearly = the equatorial circumference of the earth, or 113689008 cubes of unity, which in extent will = 113689008 lines of unity. To recollect the number of English feet and Babylonian units that make a stade, say 3 2 x3 2 = 81. The index 2 placed before 81 makes 281, the number of feet in a stade. And 81 multiplied by the root 3, equals 81 x 3 = 243, the number of units in a stade. Also 3 4 or 9 2 = 81, and 3 5 =243. The Babylonian numbers 243 are derived from 3 5 . The pyramid of Belus =-J- cube of 1 stade = -^ cir- cumference, and height = side of base. So 24 pyramids = 8 cubes = circumference. Pyramid I circumference :: pyra- mid '. 8 cubes :: 1 I 24; height I twice the 12 edges of the cube : : height .' 6 times the perimeter of base. Cube of the side of the square enclosure = circumference of earth. Cube of side of base of pyramid : cube of side of en- closure :: -J- .' 1 circumference. The spire steeple of the church at Grantham, in Lincoln- shire, is said to be 280 feet high. The tower of the church at Boston is about 280 feet high. This tower has 365 steps, and the church fifty-two windows and twelve pillars. The knowledge of the properties of the tower, like all science, was confined to the sacred institutions, and not made known to the people. Bulwer relates that the art of printing was explained to a savage king, the Napoleon of his tribe. " A magnificent conception !" said he, after a pause ; "but it can never be in- troduced into my dominions. It would make knowledge equal, and I should fall. How can I govern my subjects BABYLONIAN NUMBERS. 157 except by being wiser than they?" Profound reflection, which contains the germ of all legislative control ! When knowledge was confined to the cloister, the monks were the most powerful part of the community. Alexander upbraided his tutor Aristotle for having pub- lished those branches of knowledge hitherto not to be acquired except from oral instruction : " In what shall I excel others if the more profound knowledge I gained from you be com- municated to all ? " Babylon had gone to decay since the extinction of the empire and the conquest of Cyrus. The citizens, like those of Egypt, received Alexander with joy ; and he aimed at gaining their attachment by treating them with confidence, giving back the vast revenues of the priesthood, and restoring the sacred buildings, especially the pyramidal temple of Belus, which he ordered to be rebuilt in its original magnificence. This project was never completed. Herodotus, describing Lake Mceris, says: "This great and wonderful lake extends from north to south in its length. The part the most profound has a depth of fifty orgyes. But what shows that it has been excavated by the hand of man is, that there is near the middle two pyramids, raised 50 orgyes above the water, and they are as much concealed below as they are exposed above. One sees on each a statue of stone, seated upon a throne. Each of them has 100 orgyes from the foot to the summit; and 100 orgyes make a stade of 600 feet. The orgye is a measure of 6 feet, or 4 cubits ; the foot is a measure of 4 palms, and the cubit is a measure of 6." Here the height of a pyramid is called a stade ; and the height of the tower of Babel has been called a stade, equal to 281 feet English. Now suppose the stade in both in- stances to have been the same; then the height of one of these pyramids = a stade = 281 feet, or, 100 orgyes = 281 feet, = 3372 inches. orgye = = 33-72 158 THE LOST SOLAR SYSTEM DISCOVERED. 1 cubit = 3 i^ = 8-43 1 foot = = 5-62 6 1 palm = = 1-405 If an orgye be called b, then a cubit = 4 a foot = - o b b a palm = - = 4x6 24 and a stade = 100 b. English money is subdivided in the same relative pro- portion : Let b = a silver two-shilling piece, then T = a silver sixpence, = a silver four pence, and 100 b = a ten-pound note, or ten gold sovereigns. To express in a popular way the proximate value of the terms in the table of Herodotus, in proportions of a man about 5 feet 7 inches or 2 orgyes in height. When the hand is placed flat, the fingers straight and touching each other ; then the breadth across the four fingers, in a straight line from the top of the nail of the last or least finger to a little above the nail joint of the first or fore finger, will =2*81 inches, the half of which will = 1/405 inches = a palm. Twice the breadth of the four fingers will = 5 '62 inches = 4 palms = 1 foot ; TABLE OF HERODOTUS. 159 And three such breadths will = 8*43 inches = 6 palms = 1 cubit. If a line be held between the thumb and fore finger of both hands, and the arms stretched horizontally to their full extent, the span, or length of the line, so intercepted, will = 67*44 inches = 2 orgyes. If the distance so spanned by the arms be called two arms' length, then half the distance may be called one arm's length. Thus half a span, or an arm's length, will =^67*44 = 3 3-7 2 inches =1 orgye. And a span, or two arms' length, will= 67*44 inches, or 5 feet 7-^tfo inches = 2 orgyes = the height of a man. Hence 100 arms, or the extended arms of 50 men, will = 1 stade. And the height of 50 men will= 1 stade. Also 100 orgyes =6 plethrons=l stade. By comparing the table of measurement of Herodotus with the corresponding value of each measure expressed in English feet and inches, and then by representing each por- tion of a stade by a part of the stature of man as its proxi- mate equivalent, we shall have E. Inches. 1 2 3 4 5 1-405 Palm, iraXcuarrTi. Foot, 7TOUS. Cubit, TTTJXVy. Orgye, opyvia. 5-62 4 8-43 6 H 33-72 24 6 4 67-44 48 12 8 2 Man's height. J =half the breadth of the four fingers = 1 palm. 2 = twice the breadth =1 foot. 3 = thrice the breadth =1 cubit. 4 = the length of an arm = l orgye. 5 = the height of a man =2 orgyes. 160 THE LOST SOLAR SYSTEM DISCOVERED. Possibly such a division of the stade into portions of the stature of man might originally have been given by the hierarchy to the people, as it would greatly assist the memory, and might have aided in establishing the Babylonian stade as the universal standard, since it combines the stature of man and the circumference of the earth. Better method : The distance from the first joint of the thumb to the end of the nail = a palm = 1*405 inches. When the first and little fingers are spanned, the nearest distance between the ends of the nails = a foot = 5 -62 inches. When the thumb and little finger are spanned, the distance between the ends of the nails = a cubit = 8 -43 inches. The measurement of seventeen mummies has been given by Pettigrew, from which it appears that the Egyptians were short in stature, as the average height of the male is 5 feet 3 inches, and of the female 5 feet. But the mummies which have been examined seem all to belong to the more modern times of Egyptian history, when the Egyptians were no longer an unmixed Coptic race, as they had been conquered successively by the Arabs of Ethiopia, by the Persians, and by the Greeks. Thus 100 arms would reach the height of the tower= 1 stade. The pyramidal tower, which represents the law of gravita- tion, is supposed to reach from earth to heaven. Hence the probable origin of the giant Briareus, with his 100 arms, who strove with heaven and made war against Jove. So fifty men of the stature of 2 x 33*72 inches, or 5 '62 feet, would equal a stade. The giants also warred against heaven. The heroes or kings of the Assyrians and Egyptians are represented as gigantic in stature when engaged in battle. Thus we find how the giants of antiquity might have been figuratively great, without supposing their stature to have exceeded that of an ordinary man. The present Moorish race, inhabiting the vast archipelago of oases in the great Sahara describe a depth as equal to the height of 100 men. In several of the oases in the Sahara of Algeria, and WEIGHTS AND MEASURES. 161 especially among the Rouara, according to Dumas, the whole irrigation is artificial, and all the water is derived from artesian wells, which have existed time out of mind in those remote regions. The Marabouts relate that an immense sub- terranean lake lies under the whole tract of the Sahara, at a depth of 25 to 200 fathoms ; and the Arabs all declare that, in many of the villages, these artesian wells are 100 men's height in depth. They are square, and supported by beams of the palm tree. When the workman taps the spring below, the water sometimes rushes up with such force as to throw him senseless to the surface of the earth. The public use of these waters is regulated by strict principles of equity, and an injury done to a well is the greatest of crimes. The Sheikh of each village is the recognised protector of the source. Richard I. caused several standard yards to be made in 1197 ; and it is said that the term yard was first applied to a measure exactly equalling in length the arm of a preceding monarch, Henry I. It appears that a wheat-corn was the first standard of weight in England; and it is supposed that the metallic weight called a grain became used as a representative of the wheat-corn, and that the modern troy grain is nearly the same. After a time the pennyweight or " sterling " was re- duced from 32 to 24 grains ; 20 pennyweights made an ounce, and 12 ounces one pound: this was called the troy pound, and became the standard of English weight, consist- ing of 5760 grains. But still the legislature could not en- sure uniformity in the weights ; for there was the moneyer's pound of 5400 grains, the avoirdupois pound of 7000 grains, and the old commercial pound of 7600 grains. The French weights and measures, until the last sixty years, were in principle but little better. Soon after the Revolution, the French mathematicians turned their atten- tion to the introduction of a decimal system of notation on as extensive a scale as might be practicable. It was pro- posed to introduce the decimal mode of division into weights and measures, but it was deemed expedient first to obtain a VOL. I. M 162 THE LOST SOLAK SYSTEM DISCOVERED. rigorous standard of weight, of length, and of bulk, in lieu of the imperfect ones then in use. For this purpose they sought for a standard among the unchangeable works of nature, as being of more constant application than any of the productions of man. The circumference of the globe was fixed upon ; for we have no reason to believe that this cir- cumference increases or diminishes. The distance of either pole from the equator is mathe- matically equal to one quarter of the circumference passing through both poles, and is, therefore, called a quadrant ; and it was determined to make the ten-millionth part of this quadrant a standard of measure from which a standard of weight might be deduced. The next point, therefore, was to determine the exact number of toises (or any other known measure of length) equal to a quadrant of the earth's circumference. This was a very delicate opera- tion, requiring the resources of the astronomer and the mathematician. The result arrived at was, that the distance from north pole to the equator was equal to 5,130,470 French toises, or 10,936,578 English yards. The ten- millionth part of this quantity was taken as the standard of length, and called a metre, being equal to about 39*371 English inches. From this standard were obtained not only other measures of length, but also measures of weight and of ca- pacity, the decimal mode of subdivision being employed throughout. Compare the measurements given by Herodotus with the Babylonian standard. The circuit of the lake Moeris, says Herodotus, equals 3600 stades, or 60 schasnes, which is equal to the length of the sea coast of Egypt. In describing the three mouths of the Nile, he remarks that " one on the east opens to the sea at Pelusium, another on the west at Can opus ; the third runs straight through the Delta to the sea." Then he mentions the canals supplied with water from these branches, and proceeds : f ( Besides the opinion I have of Egypt is confirmed by the testimony of an oracle, which was delivered by Jupiter Ammon, and LAKE MCERIS. 163 which I did not hear till after I was persuaded of what I be- lieve of Egypt. It appears that the inhabitants of the cities of Mosreotis and Apis, which are on the frontiers of Egypt, towards Libya, imagined that they were Libyans and not Egyptians, and as they began to be more negligent of their ceremonies, they would no longer abstain from sacrificing cows, and sent to the temple of Jupiter Ammon, asserting that they had nothing in common with the Egyptians ; that they dwelled beyond the province of the Delta ; that they spoke not the same language, and, therefore, they pretended that it was allowable for them to eat of everything. But the god would not grant the permission they asked, and answered them that Egypt included all the country that was watered by the Nile ; and that all who drank of these waters below the city of Elephantis were Egyptians," The distance between Lake Mosreotis and Pelusium equals about three degrees of longitude, corresponding to the sea coast of Egypt ; so that a degree will equal about 60 miles : then 3 x 60=180 miles for the distance between Lake Mce- reotis and Pelusium in a straight line ; but the curved coast of the Delta will exceed 180 miles. Again, 18-79 stades = 1 mile; 3600-4-18-79 = 191 miles, for the circuit of Lake Moeris and the extent of the sea coast of Egypt. In another place Herodotus says the Egyptian coast, ex- tending from the bay of Plinthene to the lake Selbonis, under Mount Casius, is sixty schaenes in length. This would ex- ceed the distance from Lake Mcerotis to Pelusium. The distance from the sea to Heliopolis (Herod.) equals 1500 stades. By the map the distance is about 80 miles ; then 80 x 18*79 = 1500 stades. The distance of Thebes from the sea is 6120 stades; and 6120-r-18-79 = 325 miles. By the map the distance is about 360 miles. The distances by the map are measured in straight lines, and not by the road or river. Herodotus calls the distance from the sea to Heliopolis M 2 164 THE LOST SOLAR SYSTEM DISCOVERED. 1500 stades; from Heliopolis to Thebes 4860 stades ; to- gether they equal 6360 stades. But he states the distance from the sea to Thebes 6120 stades. A parasange equals 30 stades. Volney remarks, that the description Herodotus gives of the soil, climate, and of all the physical state of Egypt is such that our most learned travellers have found as little to add as so criticise in it. Malte Brun thinks that the famous canal Joseph served to conduct the water of the Nile to the lake Moeris. It is probable that this canal called Joseph, like many other me- morable objects, was excavated by order of the king Mreris : the water would then fill the basin of the lake Birket-el- Karoun, to which they might have given the name of the prince, who had caused such a great alteration. Thus may be reconciled the different situations given to the lake Moeris by Herodotus, Diodorus, and Strabo ; and why the ancients said that the lake had been formed by the hand of man, since Birket-el-Karoun has no appearance of such a labour. The canal Joseph, which is partly filled with sand in some places, is about forty leagues in length, and from fifty to three hundred feet in breadth. The number of the principal canals in all Egypt is about ninety. Mallet, who has included in his calculation all the small canals of derivation, reckons six thousand for Upper Egypt alone. The Birket-el-Karoun is now only 7 or 8 leagues long, 2 or 3 broad, and 30 in circuit. Diodorus appears more correct than Herodotus, when he says that Moeris made the lake available for irrigation, not that he dug it. The following extract is from the popular geographies: f< Westward to Benisuef is the entrance to the fertile valley of Faioum. The chain of mountains that bounds the Libyan side of the Valley of the Nile elsewhere continuous here have a narrow opening, which, with a great artificial cut that continues it, admit the waters of the river into the valley. VALLEY OF FAIOUM. 165 This tract was, it is thought, the basin of an immense lake, called by the ancients Indris, which formed the grand sluice of the country, that drew off the waters when they were superabundant, and supplied them to the land when deficient. Some considerable dykes, used alternately for retaining and letting off the waters, indicate an extent of human labour only to be credited in the land of the Pyramids. The whole of the plain is about forty miles from east to west, and thirty from north to south ; but the lake is at present contracted in breadth to five miles, though it still runs the whole length of the valley ; and we are assured, after a close examination of the surrounding land by Jomard and Martin, that the present lake merely occupies a portion of the bed of the former one. In fact, the whole surrounding country bears every evidence of having been abandoned by the waters." " The entire valley is surrounded by hills, and forms the most compact province in Egypt, rivalling even the Delta both in soil and productions. The eye contemplates with delight its smiling fields, watered by a thousand canals, whose streams, besides giving fertility to the soil, add a picturesque freshness to the landscape. Plantations of roses, celebrated all over the East for their superior perfume, trees bearing the finest fruits, with fields of rice and flax, combine to give a charming diversity to the scene." This plain, having an extent of 40 miles by 30, will have a circuit of 2 x 40 + 2 x 30= 140 miles, which is less than the length of the sea-coast of Egypt. The circuit of Lake Mreris equalled 60 schaenes=3600 stades = 191 miles. The lake was oblong, extending from north to south. At Symbrumacum, a small town in the Carnatic, is a re- markable large tank, about eight miles in length by three in breadth, which has not been formed by excavation, like those in Bengal, but by shutting up with an artificial bank an opening between two natural ridges of ground. In the dry season the water is let out in small streams for cultivation, and it is said to be sufficient to supply the lands of thirty-two H 3 166 THE LOST SOLAR SYSTEM DISCOVERED. villages (should the rain fail), in which 5000 persons are employed in agricultural pursuits. Bopal, a town in the province of Malwah, is extensive, and surrounded with a stone wall. Outside of the town is a fort called Futteghghur, built on a solid rock. It has a stone wall with square towers, but no ditch. Under the walls of the fort is a very extensive tank or pond, formed by an embankment at the confluence of five streams issuing from the neighbouring hills. The tank is about six miles in length. (East India Gazetteer.) Like the pyramids rising out of the middle of Lake Moaris, we find a monument rising from the centre of an Indian lake or tank. Shere Khan, the Afghan, who expelled the emperor Humayoon (the father of Acbar) from Hindostan, was buried at Saseram, in the province of Bahar, in a magnificent mauso- leum rising from the centre of a large square lake, which is about a mile in circuit and bounded on each side by masonry, the descent to the water being by a flight of steps, now in ruins. The dome and the rest of the building are of a fine grey stone, at present greatly discoloured by age and neglect. " The Candelay Lake, about thirty miles from Trincomalee in Ceylon," says De Butt, "is situate in an extensive and broad valley, around which the ground gradually ascends towards the distant hills that envelope it. In the centre of the valley, a causeway, two miles long, principally made of masses of rock, has been constructed to retain the waters that from every side pour into the space enclosed within the circumjacent hills and the artificial dam thus formed. During the rainy season, when the lake attains its greatest elevation, the area of ground over which the inundation extends may be computed at fifteen square miles. This work of art, and others of equally gigantic proportions in the island, sufficiently indicate that at some remote period Ceylon was a densely- populated country, and under a government sufficiently en- lightened to appreciate and firm to enforce the execution of an undertaking which, to men ignorant of mechanical powers, must have been an Herculean labour ; for such is the ca- CANDELAY LAKE. 167 pricious nature of the mountain streams in this tropical island, where heavy rain frequently falls without inter- mission for several successive days, that no common barrier would suffice to resist the great and sudden pressure that must be sustained on such occasions. Aware of this peculi- arity in the character of their rivers, the Cingalese built the retaining wall that supports the waters of the Lake of Can- delay with such solidity and massiveness as to defy the utmost fury of the mountain torrents. Nearly the whole of its extent is formed with vast masses of hewn rock, to move which by sheer physical force must have required the united labour of thousands. The Cingalese have, from the earliest periods, been attentive to the formation of artificial reservoirs, wherever they could be advantageously constructed ; and the Lakes of Candelay, Minere, Bawaly, and many others of less note, attest the energy and perseverance of the ancient islanders in such constructions." " In Ceylon," observes Campbell, " there are many traces of an early civilisation, remains which show a great advance- ment in the arts, and that the country was well cultivated and thickly inhabited. There are extensive tracks of ruined canals, one of which was in some parts 15 feet deep and 100 wide. There are stone bridges ; in one the stones are from 8 to 14 feet long, jointed into one another, the upright pillars being grooved into the rocks below. The tanks are of an immense extent, with gigantic embankments, and the remains of a canal are seen, which brought the water from one of these tanks sixty miles to Anarajahpoora, the ancient capital. This city was surrounded by a wall sixteen miles square ; and there are the ruins of some great pagodas there, two of them 270 feet high, of solid brick- work, and which has been covered over with chunam, a lime cement which takes a polish like marble." No monuments of antiquity in the island of Ceylon are calculated to impress the traveller with such a conception of the former power and civilisation of the island, as the gigantic ruins of the tanks and reservoirs, in which the M 4 168 THE LOST SOLAR SYSTEM DISCOVERED. water, during the rains, was collected and preserved for the irrigation of their rice lands. " The number of these structures throughout vast districts now comparatively solitary is quite incredible," says Ten- nant, and their individual extent far surpasses any works of the kind with which he was acquainted elsewhere. Some of these enormous reservoirs constructed across the gorges of valleys, in order to throw back the streams that thence issue from the hills, cover an area equal to fifteen miles in length by four or five in breadth, and there are hundreds of a minor construction. These are mostly in ruins. A visit to one is described : it was that of Pathariecaloru, in the Wanny, about seventy miles to the north of Trincomalee, and about twenty-five miles distant from the sea. It is a prodigious work, nearly seven miles in length, at least 300 feet broad at the base, upwards of 60 feet high, and faced throughout its whole extent by layers of square stone. About the centre of the great embankment advantage has been taken of a rock about 200 feet high, which has been built on to give strength to the work. Some wild buffaloes and a deer came to drink from the water-course ; these were the only living animals to be seen in any direction. The embankment, estimated at the length of six miles, height 60 feet, breadth at base 200 feet, tapering to 20 at the top, would contain 7,744,000 cubic yards, and at Is. 6d. a yard, with the addition of one- half that sum for facing it with stone, and constructing the sluices and other works, it would cost 870,0007. sterling to construct the front embankment alone, according to the estimate of the government engineer. The existing sluice is a very remarkable work, not merely from its dimensions, but from its ingenuity and excellent workmanship. It is built of layers of hewn stones, varying from 6 to 12 feet in length, and still exhibiting a sharp edge, and every mark of the chisel. The ends of the retaining stones are carved with elephants' heads and other devices, like the extremities of Gothic corbels. As to human habitation, the nearest was the village, where RESERVOIRS IN CEYLON. 169 we had passed the preceding night ; but we were told that a troop of unsettled Veddahs had lately sown some rice on the verge of the reservoir, and taken their departure after securing their little crop. And this is now the only use to which this gigantic undertaking is subservient ; it feeds a few wandering outcasts ; and yet, such is its prodigious capa- bilities, that it might be made to fertilise a district equal in extent to an English county. Some thirty others, of nearly similar magnitude, are still in existence, but more or less in ruin, throughout a district of 150 miles in length from north to south, and about 90 from sea to sea. It is said that some one of the sacred books of Ceylon re- cords the name of the king who built this reservoir. It may be remarked that the length of this embankment = 6 miles = one side of the square that enclosed Babylon. The height of the embankment = 60 feet. of the walls of Baby Ion = 70 feet. The distance from the mouth of the Euxine Sea to the river Phasis is estimated by Herodotus at 11,100 stades. Taking Phasis as the extreme eastern part of the Euxine, as laid down by D'Anville, the latitude of Phasis is 42 north, and a degree of longitude corresponding to latitude 42 = 51-42 miles English, and 18-79 stades =1 mile. So that 11,100 stades will=ll|- degrees of longitude corresponding to latitude 42. The parallel of longitude between Phasis and the west side of the Euxine includes 13 by the map; but the distance from Phasis to the Bos- phorus will be somewhat less than 13. So that 11,100 stades will very nearly correspond to the distance from the Bosphorus to Phasis, according to modern geography ; and this is the distance assigned by Herodotus for the length of the Euxine. Herodotus makes his calculation by taking the average sailing of a vessel by day and by night, and the time oc- cupied in sailing from the Bosphorus to Phasis he calls nine days and eight nights. Next, try how this cubit of 8-43 inches English accords 170 THE LOST SOLAR SYSTEM DISCOVERED. with the measurement of any monument, still existing, given by Herodotus in cubits. Now Herodotus states that se Phe- ron, having recovered his sight, presented to all the temples magnificent offerings ; but he made especially to the temple of the Sun what are certainly remarkable and worthy the admiration of man ; there he erected two obelisks, each of a single stone, in height 100 cubits, in breadth 8." The temple of the Sun stood at Heliopolis. Now it appears, according to Ammianus Marcellinus, that three of the Roman obelisks were brought from Heliopolis, two by Augustus, and one conjointly by Constantine and Con- stantius. The latter is the great Lateran obelisk that for- merly stood in the Circus Maximus. It appears that one of the two brought by Augustus was first placed in the Campus Martius ; afterwards it was removed to where it now stands on the Monte Citorio. The whole height of the Citorio obelisk from the base to the apex measures 7 1 feet 5^ inches. Base ordinate = 8 feet y-oVo inch. Top ordinate =5 feet 1 1 1* inch. The other base ordinate is defective. Now compare the dimensions of this with one of the two obelisks erected at the Temple of the Sun. Taking the height given by He- rodotus at 100 cubits, then 843 x 100-f- 12 = 70-25 feet, and the whole height of the Citorio obelisk = 71 ft. 5-J- inches. Herodotus gives the breadth at 8 cubits. Now 8 '4 3 x 8 -r-12 = 5*62 feet only, a little more than the top ordinate. Diodorus informs us that Sesoosis erected two obelisks of very hard stone 120 cubits high. It appears from the inscriptions that the two obelisks which stood in front of the Luxor were erected by Eamses III. One of the Ramses was the Sesoosis of Diodorus and the Sesostris of Herodotus. One of these obelisks has been removed to Paris, which measures 74 French feet, or nearly 81 English feet, in height. The remaining obelisk is 3 French feet higher, which will make the height nearly equal 84-3 English feet. 120 cubits = 120 x 8*43 inches = 84-3 English feet. OBELISKS. 171 Ramses II., or the Great (says Sharpe), added to the temple of the Luxor, and set up two obelisks in front of it, one of which is now in Paris. Ramses III., who is said in the legends chiselled on the face of one of these obelisks, (f made these works (the propyla of the palace of the Luxor) for his father, Amun-Ra, and that he had erected these two great obelisks in hard stone before the Ramsesseion of the city of Amun." Rosellini attributes the rock-cut temple of Abousambel to Ramses III., whom he calls the Great. Wilkinson attri- butes the same temple to Ramses II., whom he calls Ramses the Great. In Rosellini's chronology the death of Ramses III. dates 1499 B. c. Several sovereigns were named Ramses, all belonging to the brilliant era when the great monuments were erected. The name of Ramses is inscribed at Ipsambul, and on nume- rous monuments of Nubia; on the two obelisks at Alex- andria ; on three lying on the ground at San, the ancient city of Tanis, the Zoan of Scriptures. The name is perpetuated on durable stone from the northern extremity of Egypt to the southern of Nubia. Sesoosis, the seventh from Moeris, was greater than any of his predecessors. According to Diodorus, he conquered Arabia and Libya. His army consisted of 600,000 foot, 24,000 horse, 28,000 chariots. He afterwards conquered Ethiopia, India beyond the Ganges, Scythia, and Thrace, and fixed the yearly tribute which the conquered nations should pay. He made two obelisks of hard stone, each 120 cubits high, on which he described the greatness of the king- dom, and the tributes of the subject states. Sesoosis II., his son, assumed the name of Sesoosis. The son was struck blind, but recovered his sight. Diodorus mentions that the wall erected by Sesoosis, be- tween Pelusium and Heliopolis, to prevent the plundering excursions of the Arabs, was 1500 stades long, which is the number of stades assigned by Herodotus for the distance from the sea to Heliopolis. 172 THE LOST SOLAE SYSTEM DISCOVERED, We make the distance from Heliopolis to the nearest coast less than the distance from Heliopolis to Pelusium. We have also a bulletin of Rameses III. or IV., almost as successful a conqueror as his great ancestor Sesostris. Be- neath a painting which depicts his return to Egypt, the fol- lowing address to his troops is put in his mouth t 6 ' Give yourselves up to joy ; let it rise to heaven ; the strangers are overthrown. The terror of my name is come over them, and has petrified their hearts. Like a lion I have opposed them, pursued them like a hawk, and have annihilated their guilty souls. I have passed over their rivers, and burned down their fortresses. I am a wall of brass for Egypt. Thou, my father, Ammon Ra, hast so commanded me, and I have pursued the barbarians ; I have passed victoriously through all parts of the earth, till at length the world itself withdrew from my steps. My arm subdued the kings of the earth, and my foot trampled on the nations." This reminds one of another affiliated child of Ammon, who, after having subdued the kings of the earth and trampled on the nations, cried for more worlds to conquer. The oriental bulletin of Buonaparte reminded his troops that the ages of 4000 years were regarding them from the summit of the great pyramid. The ages at different periods had also looked down from those pyramids on the armies of Rameses, Cambyses, Alex- ander, and many other triumphant kings, fluttering in the sunshine of glory. The obelisk in front of St. Peter's at Rome formerly stood in the Vatican Circus. Pliny says it was cut by Nunco- reus, the son of Sesostris, who corresponds to the Pheros of Herodotus. It seems to have been broken, and to have lost part of its length; yet it is still 83 feet 2 inches, or 120 cubits high. Diodorus mentions that Sesoosis placed in the temple of Vulcan his own and his wife's statue, 30 cubits in height. Herodotus states that Sesostris erected several statues at the entrance of Vulcan's temple. Two of these, representing OBELISKS. 173 himself and wife, are 30 cubits in height; and four other statues, representing his four sons, are 20 cubits each. So it appears that Diodorus and Herodotus made use of the same cubit in measuring these statues ; hence we may infer that they used the same cubit, that of Babylon, 8 '43 inches, in their measurements of obelisks. If the obelisk at St, Peter's be 120 cubits high, it cannot be one of the two obelisks erected by Pheros. Neither can the Lateran obelisk, which is said to have been brought from Heliopolis, have been one of Pheros' obelisks ; for this is said to be the largest obelisk in the world, measuring from the base to the apex 105 feet 7 inches, or 150 cubits. The sole remaining obelisk at Heliopolis is 67|- feet high, according to Pocock ; so this may be one of Pheros' obelisks, the com- panion to the Citorio obelisk. If so, one of the obelisks of Pheros, erected at Heliopolis, will be 100 cubits high, and the other rather less in height. So will one of the obelisks erected by Sesostris at the Luxor equal 120 cubits in height, and the other rather less. If the cubit of Diodorus be considered equal to the cubit of Herodotus, or of Babylon, we can measure the length of the ship of cedar wood built by Sesostris. Diodorus informs us that Sesostris having constructed a ship of cedar- wood, 280 cubits long, lined the inside with silver, and the outside with gold, made an offering of it to the god whom they adore at Thebes. 280x8-43 inches= 1967 feet English for the length of Sesostris' ship. Now the Gipsy Queen, an iron steamer built on the banks of the Thames, measures in length from the figure-head to the taffrail, 197 feet 6 inches, and between the perpen- diculars 175 feet. Breadth between the paddle-boxes, 24 feet. Burden 496 tons. Engines 240 horse power. What is generally considered as constituting a horse power is a power to raise 1.30 pounds 100 feet in one minute. The priests told Herodotus that Sesostris was the first king who, passing through the Arabian Gulf with a fleet 174 THE LOST SOLAB SYSTEM DISCOVERED. of long ships, subdued those nations that inhabit the Red Sea. The materials for ships were formerly transported overland from Gaza to the Eed Sea, having been originally brought from Mount Lebanon. This is a common occurrence at the present day on the shores of the Red Sea, where no tree grows. Laborde mentions that scarcely a year elapses in which the timbers of vessels may not be seen passing in single pieces, through the streets of Suez, on their way to the shore, in order to be put together and launched. In this manner, the cedar ship of Sesostris might have been built on the shores of the Red Sea with the cedars of Lebanon. Necus, the son of Psammitichus, was the first, according to Herodotus, who attempted to dig a canal from the Nile to the Red Sea, which was afterwards completed by Darius, the Persian; so broad that two vessels could easily sail on it together. It extended from a little above Bubastis, not far from the modern Grand Cairo, on the Nile, to Patumos, a city of Arabia on the Red Sea, near the present Suez, about four days' sail. Strabo says this canal was first cut by Sesostris, before the Trojan war, and that it terminated at the city Arsinoe, or Cleopatris. He makes it 100 cubits broad. Pliny makes it 100 feet broad, and 30 deep. Both these authors say that Darius was prevented from finishing the canal, from an apprehension that the Red Sea, being higher than the land of Egypt, if let in would inundate the country and spoil the waters of the Nile. This canal was finished or renewed by the Ptolemies. It was cleaned by Trajan, and afterwards restored by the Arabs in the time of Omar. It is now choked up ; and the trade between Cairo and Suez is carried on by caravans. Herodotus says 120,000 men perished in digging this canal under Necus. The king being hindered from finishing it by an oracle, built a number of ships, partly on the Mediter- ranean, which Herodotus calls the North Sea, and partly on the Arabian Gulf. Some of these he ordered to sail round Africa, which voyage they performed. CANAL OF SESOSTRIS. 175 Napoleon, accompanied by the French engineers in 1799, made a survey of the Suez canal. He was the first to dis- cover the undoubted traces of the canal of Sesostris, which he followed from the northern point of the Gulf of Suez for several leagues, and found that they were lost in the dry basin of the Bitter Lakes. This ancient work extends in a direct line north, through the trough or valley, for 13^- English miles. The walls of the canal are of solid masonry, from 6 to 16 feet deep, and the space between them is 146 English feet. Strabo states it at 150 feet. The breadth at the bottom of the canal, according to the plan, is not given ; but as the banks are inclined, this breadth may have been about half a stade, 200 cubits, or 140^ English feet. The bed of the canal has been raised by sand and earth, washed into it by the torrents ; and a new and higher bed has been curiously consolidated by natural means from the effect of calcareous filtrations. The French engineers dug through the fictitious bed, and found the real bed four or five feet beneath it They then detected the artificial com- position employed by the ancient engineers for retaining the waters of the canal, which they found to consist of moist saline sand, earthy clay, and gypsum. The French line, resulting from Jacotin's survey, passes through the bed of the Bitter Lakes, the lake El Timseh, thence to the marshy grounds of El Karesh (nearly on a level with the Red Sea), thence to Dar El Casseh, after- wards to El Do wade ; thence the line follows the traces of the old canal, and the ruins of the wall of defence of Sesos- tris, in a direct line, the ground being sandy, and lower than the Red Sea ; hence to the occasionally flooded strip of land by Lake Menzaleh, where the excavation of the ancient canal reappears in a sandy valley ; thence to the entrance of Tineh, passing between Faramah and Pelusium, where the land (having gradually declined, unobstructedly, the whole way from El Karesh) is 29 feet 6 inches lower than the Red Sea. The length of this line is 85 miles (being prolonged to save expense). 176 THE LOST SOLAR SYSTEM DISCOVERED. Linant, an engineer who surveyed the Isthmus in 1841-2, confirms the report and survey of Jacotin and the French engineers of 1799; and recommends the same line, both on account of its practicability and economy. An iron steam yacht for the Pacha of Egypt was launched from the banks of the Thames at Black wall, in 1851. Bur- then 2200 tons. Dimensions, length between the perpen- diculars 282 feet; length of keel for tonnage 258 feet; breadth for tonnage 40 feet; depth in hold 39 feet; draught of water 18 feet. Machinery 800 horse-power. She is pierced for the following number of guns : Spar deck, twelve 10-inch 84-pounders broadside, 56 cwt. ; spar deck, twelve 10-inch 84-pounders pivot guns, 85 cwt. ; main deck, fourteen 10-inch 32-pounders broadside, 56 cwt. Con- structed ostensibly for a yacht, she can be turned into the most powerful steamer afloat for war purposes. Length between perpendiculars = 282 feet, 281 feet = 1 stade = 400 cubits. Length of the cedar ship of Sesostris = 280 cubits. The Great Britain steam-ship is built entirely of iron, with the exception of the flooring of her decks and the floor- ing and ornamental parts of her cabins. She is 322 feet in length from figure-head to taffrail, and 50 feet 6 inches in breadth. She is registered at 3500 tons, so that her bulk was at the time she was launched nearly equal to any two steamers in the world. She has four decks, the lowest of which is of iron. The upper deck is flush from stem to stern, measuring 308 feet. She has four engines of 250 horse power each, and is fitted with the Archimedian screw propeller. The American ocean steam-ship Arctic is 3000 tons mea- surement ; length of keel 275 feet, of main deck 284 feet. Draught on her trial trip 18 feet, when fully loaded 19. The diameter of the wheels 35f feet. The engilies weigh 750 tons; their boilers contain 250 tons of water, of which they evaporate 8000 gallons an hour, with a consumption of 24 tons of anthracite coal in the same time. It takes ten CHINESE JUNKS. 177 engineers and assistants, 24 firemen, and 24 coal heavers, working in three gangs, with relays of 8 hours each, to di- rect, feed, and operate them. The length of the main deck exceeds 1 stade by 3 feet. The diameters of the wheels exceed stade by f of a foot. o / o The Himalaya, built of iron, at Black wall, on the banks of the Thames, is the largest ocean steam-ship in the world. She is 3550 tons register, equal to 4000 tons burden, and is of the extraordinary length of 372 feet 9 inches. The length of the keel is 311 feet; breadth for tonnage 46 feet 2 inches; depth of hold 24 feet 9 inches. These proportions, when contrasted with the dimensions of other ships, give a great advantage, particularly in length, to the Himalaya ; for ex- ample, the Duke of Wellington, a screw line of battle ship, of 131 guns, although of a greater beam and depth, is in- ferior in length by 92 feet to the Himalaya. The iron screw steamer Great Britain is 40 feet shorter than the Himalaya, while the American clipper ship Great Republic, recently destroyed by fire in New York, was 47 feet less in length than the Himalaya. Although the Himalaya exceeds in so large a degree the length of the Duke of Wellington, yet she is inferior in tonnage to that ship by 209 tons. The spar deck of the Himalaya is flush from stem to stern. An uninterrupted promenade of 375 feet, or 125 yards, is here provided. To walk round the spar deck pre- cisely one-seventh of a mile has to be traversed. The engines are 700 horse power. The saloon, nearly 100 feet in length, will dine 170 persons. The bed cabins are the largest ever yet appropriated to marine travellers. The Chinese Junk, lately arrived in London from China by the Cape of Good Hope, measures in length 165 feet; height of stern, 40 feet ; burthen about 700 tons. This is the first Chinese junk that has been seen in England ; hitherto it has been supposed that Chinese vessels were un- able to make extensive voyages, and therefore precluded from making discoveries. It is now proved that they are capable of circumnavigating the globe. This junk sailed from Canton, rounded the Cape of Good VOL. T. N 178 THE LOST SOLAR SYSTEM DISCOVERED. Hope, anchored at St. Helena, thence visited New York, North America, and ultimately arrived at London. The largest Chinese junks are about 1000 tons burden. The Chinese rarely make long voyages, for though they have been for many centuries acquainted with the use of the com- pass, they seldom lose sight of the coast. In their trading to Singapore, Batavia, and New Holland, they employ a foreign master, who is generally a Portuguese. The Chinese think that the magnetic attraction is to the south, and there- fore have that end of the needle coloured red. They have only twenty-four points in their compass. On the bows are placed two large eyes. There is, neither in the building nor in the rigging and fitting up of a Chinese junk, one single thing which is similar to what we see on board a European vessel. From her peculiar form, her measurement has not been ascertained, but it is supposed that she may measure about 400 tons, and carry 700. The figure of a cock is one of the zodiacal constellations of the Chinese. It is repre- sented on the stern with expanded wings. Athenasus thus describes a ship given to Philopater by Hiero, King of Syracuse. It was built under the care of Archimedes, and its timbers would have made sixty tri- remes. Besides baths and rooms for pleasures of all kinds, it had a library, and astronomical instruments, not for naviga- tion, as in modern ships, but for study, as in an observatory. It was a ship of war, and had eight towers, from each of which stones were thrown at the enemy by six men. Its machines, like modern cannons, could throw stones of 300 Iba. weight, and arrows of 18 feet in length. It had four anchors of wood and eight of iron. It was called the ship of Syra- cuse, but after it had been given to Philopater, it was known by the name of the ship of Alexandria. The royal barge, in which the king and court moved on the quiet waters of the Nile, was 330 feet long, and 45 feet wide. It was fitted up with state rooms and private rooms, and was nearly 60 feet high to the top of the royal awning. According to Plutarch, Ptolemy Philopater built a vessel of forty benches of oars, which was 420 feet long, and 72 FLOATING PALACE OF TRAJAN. 179 from the keel to the top of the poop, and carried 400 sailors, besides 4000 rowers, and near 3000 soldiers. Pliny says that it had fifty benches ; and he mentions another of Ptolemy Philadelphia with forty. Trajan selected Lake Aricinus (now the Lake of Nemi) as the scene of his retreat from the care of government. This lake is at the distance of about fifteen miles from Rome, in the vicinity of the Appian Way, and is surrounded with hills covered with trees, and always verdant. The atmosphere is salubrious and temperate, the soil fertile, and the scenery most beautiful, boasting, among other attractions, of the grotto and fountain of Egeria, so celebrated in the time of JSTuma Pompilius. The lake itself is very deep, and the water clear as crystal. It was here Trajan caused to be constructed a ship or bark of an immense size, composed of the most durable and expensive timber, on which a palace, decorated and adorned in a magnificent manner, was erected. The roof was supported and ornamented with massive beams of brass ; the pavement was inland with stones of the most varied and beautiful colours ; and the Egerian water was conducted by leaden pipes into the vessel, where it formed a refreshing fountain. The shores of the lake were laid out in gardens, planted with a diversity of trees and shrubs, and intersected with serpentine walks. Everything that imagina- tion could suggest was effected to improve and assist the natural beauties of the place. The bark was moored in the centre of the lake, and was built with the greatest strength and solidity ; the planks were of extraordinary thickness, and fastened not only with nails, of which great quantities were used, but also by smaller planks inserted in grooves, and secured in the most effectual manner. The outside was sheathed with plates of lead of a double thickness where ex- posed to the action of the water, and between the planks and sheathing were placed woollen cloths saturated with oil and pitch, in order to preserve the timbers from the water. The whole structure was most magnificent, and well fitted for the retirement of a prince. It was, however, in succeeding ages, and during the tyranny and misgovernment, the wars N 2 180 THE LOST SOLAR SYSTEM DISCOVERED. and troubles, the barbarian inroads, and the factious dissensions that ravaged Italy and the tributary states, and which caused the fall of the Roman Empire, neglected and suffered to fall into decay. Time and storms gradually reduced it to ruins, and it eventually sunk to the bottom of the lake, where it still remains imbedded and almost forgotten. Marchi, in his account of his descent in a diving-machine, states that it was then (A. D. 1535) 1340 years or more since the bark was submersed at the spot where it then remained sunk, at a great depth, by the eastern edge of the lake. He contrived to measure the bark, which he found to be, in English measure, about 500 feet in length by 270 in breadth, and 60 in depth. If we compare these dimensions with a British man-of-war, we shall have some idea of the immense size of the floating vessel, and of the importance of the build- ing erected on it. The length of a first-rate ship of war of 120 guns is about 205 feet (or two fifths of that of Trajan's floating palace), and the breadth 53 feet, being less than one-fifth the dimensions of the bark. This floating palace has recently been raised up; the timbers, which were of cypress and larch, were found sound after 1400 years' immersion. Ordinates of the Obelisk. Fig. 57 C. Let A B c D represent the four sides of an obelisk, having the two greater sides AD, BC, equal, and the two less jJJ. 13 Fig. 57. sides, AB, DC, also equal. The greater square equals the OBELISKS. 181 square of the greater side, and the less square the square of the less side. The following calculations are made for two square obelisks, one having the square of the greater side greater than the base of the obelisk ; the other having the square of the less side less than the base of the obelisk ABCD. The difference of the squares = -J- the perimeters of the 2 squares x -J- their difference, = i the perimeters x their difference = the rectangle by the sum of the two sides of the squares and their difference. The rectangle of the sum and difference of the sides of two squares, or the rectangle of the sum and difference of the two ordinates, =. the difference of their squares, or sectional axis of the obelisk. Fig. D. AB, AC, are two squares, rectangled parallelogram FH + rectangled paral- lelogram HE = their difference, FG or DE the difference of their sides, FGXFB+DEXEC=FH+DC Or FB+ECXFG orDE = FH + DC. Let AF = 6, AG = 4, then AF 2 AG 2 = 6 2 4 2 = area HF + HE = 6 x 2 + 4 x 2=12 + 8 = 20 square units. Thus the ordinates 6 2 4 2 = a line of 20 square units = a line of the length of 20 linear units = the axis intercepted by the two ordinates 6 and 4. Fig. E. Let the sides of the square ordinates be 18 and 12 ; 18 2 -12 2 = 324-144 = 180, and 6 x 18 + 12 = 180 = axis intercepted by the two ordinates. When the difference of the two ordinates = 1, the sum of the two ordinates = the difference of their squares : Asl0 2 =100 23 2 =529 9 2 = 81 22 2 = 484 19 =19 45 = 45 When the difference of the two ordinates = 2, twice the sum of the two ordinates = the difference of their squares. N 3 182 THE LOST SOLAR SYSTEM DISCOVERED. As, 10 2 = 100 23 2 = 529 82 = 64 21 2 2x18 = 36 2x44 = 88. When the difference of the two ordinates = 3, three times the sum of the two ordinates = the difference of their squares : As, 23 2 = 529 232 2 = 53824 2Q 2 = 40Q 3x43 =129 3x461 = Ib83. When the difference of the two ordinates = n, then n times the sum of the two ordinates = the difference of their squares ; or the difference of the two ordinates = 2 n times the greater ordinate less n 2 , or = 2n times the less ordinate -f ft 2 : As, 23 2 = 6x40 =240=12x23-6 2 =12xl7 23 2 = 529 8x38 =304=16 x 23-8 2 = 16 x 15 + 8 2 . Obelisks. We shall quote from the " Library of Entertaining Know- ledge" some extracts and the dimensions of the Egyptian obelisks now at Rome. " Of all the works of Egyptian art," says the writer, (( which, by the simplicity of their form, their colossal size and unity, and the beauty of their sculptured decorations, excite our wonder and admiration, none can be put in comparison with the obelisks. As lasting records of those ancient monarchs, whose names and titles are sculptured on them, they possess a high historical value, which is increased by the fact that some of the most remarkable of these venerable monuments now adorn the Roman capital. The Caesars seem to have vied with one another in transporting these enormous blocks from their native soil ; and since the revival of the study of antiquities in Rome, the most enlightened of her pontiffs LATERAN OBELISK. 183 have erected those which had fallen down and were lying on the ground in fragments. " An obelisk is a single block of granite, cut into a quadrila- teral form. The horizontal width of each side diminishes gradually, but almost imperceptibly, from the base to the top of the shaft, which is crowned by a small pyramid. Most obelisks, of which any accurate dimensions have been given, have only the opposite pairs of sides equal ; one pair often exceeding the other in the horizontal breadth by 6 or 7 inches, or even more than a foot. As an obelisk rises from its base in one continuous unbroken line, the eye, as it measures its height by following the clearly defined edges, meets with no interruption, while, the absence of all small lines of division allows the mind to be fully impressed with the colossal unity of the mass. " It would appear, from the inspection of the great gateway of the Luxor, from the remains of Heliopolis, and the two obelisks at Alexandria, that they were principally used in pairs, and placed on each side of the propyla, or great entrance of a temple. But they were also placed occasionally within the interior of the temples, but still in front of gateways, as at Carnak ; just as small obelisks are said to be found within the rock-cut temples of Ellora. " Of the two obelisks at Alexandria only one is standing. But they must have been both standing when Abd-el-Latif wrote, about the close of the twelfth century ; for he says he saw two obelisks near the sea, without making any mention of one of them being on the ground ; though when he speaks of the two obelisks of Heliopolis he takes care to say that one of them had fallen. " The Lateran obelisk now stands before the north portico of the Lateran church at Home, where it was placed in 1588 A. D. This is the largest of all the Roman obelisks, and perhaps the largest in the world. It is the same which the Emperor Constantius erected in the Circus Maximus. Mercati, who carefully measured it when lying on the ground, says it was broken into three pieces. The whole length of the three parts was 148 Roman palms ; but the base of the N 4 184 THE LOST SOLAR SYSTEM DISCOVERED. lowest part was so much damaged that it was necessary to take off four palms before it could be safely set on its pedestal. This reduces the length of the shaft to 144 palms, or 105 feet 7 inches English. The whole height, with the pedestal and ornaments at the top, is about 150 feet. The sides of the obelisk are not all of equal breadth. The width of the north and south sides (as they now stand) at the base is 9 feet 8J- inches ; the width of the same sides below the pyramidal top is 6 feet 9^ inches. The two other sides at the base and top are respectively 9 feet and 5 feet 8 inches. The obelisk is of Syene granite. The whole surface from the base to the very pointed top is covered with exquisite sculptures, superior to those of the other obelisks at Rome." Let us find a unit such that the difference between the squares of the base and top ordinates shall equal the height of the shaft intercepted by these ordinates. Such a unit for the Lateran obelisk will = 6 inches English. Elsewhere it is said the pyramidal top of the Lateran obelisk surpasses the width of the base by about one-third. F. I. So, from the entire height 105 7 Deduct for the pyramid, say . . . 8 1 Then the height of the shaft = 97 6 F. I. 1st base ordinate = 9 8J- = 19 '4 3 units 1st top ordinate = 6 9-J- = 13-55 and 19-43 =377 13-55 2 =183 2)194 Height =~97 feet Measured height = 97 1 feet. The ordinates on the two other sides are, F. I. 2nd base ordinate = 9 = 18 units, 2nd top ordinate = 5 8 = 11 -3 3 LATERAN OBELISK. 185 and 18 = 324 11-33 =128 2)196 Height = 98 feet Measured height = 97 \ feet. The whole height of the obelisk at present = 105 feet 7 inches =144 Roman palms, and 105f feet = 150 cubits = -f stade. The whole height, with pedestal and ornaments, = 150 feet. The addition of 4 palms, for the part cut off, would make the original height of this obelisk =154 cubits. The unit = 6 inches, or nearly so, for the two different sides of the obelisk. The height of the apex of the obelisk above the top of the shaft, or base of the pyramid, corresponding to the two greater sides will = 183 units. The height of the apex for the other two sides above the shaft will =128 units. Dif- ference = 55 units, or 27-J- feet. Pliny, speaking of the two large obelisks in his time, one of which stood in the Campus Martius, and the other in the Circus Maximus, the latter being the Lateran obelisk, says, " The inscriptions on them contain the interpretation of the laws of nature, the results of the philosophy of the Egyptians." On first beholding these obelisks, with their unbroken out- lines, their forms appeared as mysterious to us as their hiero- glyphics still continue to be. So we must leave others to ascertain whether any of the inscriptions admit of the inter- pretation mentioned by Pliny. But should that not be the case, still the obelisk itself, without any inscription, contains the interpretation of the laws of nature. Champollion re- marks that the Lateran obelisk belongs to Thouthmosis. If the height, from the base of the obelisk to the apex of the pyramid on the top, be made the height of a pyramid, similar to the top pyramid, then the content of the supposed pyramid may be found. Thus the supposed pyramid will be similar to the pyramid on the top of the obelisk, and their contents will bo as the cube of their heights. The part cut off the truncated obelisk is wanting ; but the truncated part is seen. 186 THE LOST SOLAR SYSTEM DISCOVERED. The part cut off the truncated pyramid is seen ; but the truncated part is wanting. The second obelisk in size is that which C. Caesar erected in the Vatican circus ; it was removed in the time of Sextus V. to its present position in front of St. Peter's, and was the first of the four which this pontiff restored. There are no hieroglyphics upon it. Pliny says it was cut by Nuncoreus, the son of Sesostris, who corresponds to the Pheros of Hero- dotus. It seems to have been broken, and to have lost part of its length ; yet it is still 83 feet 2 inches high (without the modern ornament at the top), of which six feet belong to the pyramidal apex. Each side is said to be of equal width, being at the base 8 feet 10 inches, and under the pyramid about 5 feet 11 inches. The height of the shaft will = 83 feet 2 inches, less 6 feet = 77 feet 2 inches. Let the unit = 6-66 inches, then base ordinate =8 feet 10 inches= 15-92 units top ordinate =5 feet 11 inches= 10'66 and 15-92 2 =253 10-66 2 =113-6 height = 139'4 units = 77-3 feet measured height =77 feet 2 inches. It appears, however, that there are great discrepancies about the dimensions of this obelisk, which induced Zoega to conclude that a more exact measurement was necessary, in order to determine if this were one of the obelisks of Pheros or not. It is, however, not easy to measure the obelisk at present. The whole height, with the pedestal and cross at the summit, is about 132 feet. The two obelisks of Pheros each equalled 100 cubits in height. .-.100 x 8-43 inches = 70-J- feet English, which is less than the obelisk at St. Peter's. St. Peter's obelisk is said to have lost part of its length, yet its present height is 83 feet 2 inches. 120 cubits = 84-3 feet. FLAMINIAN OBELISK. 187 Richardson says, near the centre of the great temple of Carnac there are three noble obelisks, about 70 feet high, and 9 square at the base ; a fourth obelisk is lying on the ground, cut into two pieces. In the vicinity of Syene, now Assouan, are those exten- sive quarries which furnished the ancient Egyptians with materials for their colossal statues and obelisks. Here is still to be seen a half-formed obelisk, between 70 and 80 feet long. The Flaminian obelisk (Flaminio del Popolo) is the next in size to the Vatican. This was one of the two obelisks that Augustus transported to Rome and erected in the Great Circus. It consists of three parts, which altogether, accord- ing to Mercati's measurements, made up 110 Roman palms ; but three palms were cut off from the lower part before it was put up in its present position, which will reduce the height to about 78 feet 5 inches. The sides are of unequal width ; those on the north and south, which correspond, are 7 feet 10 inches at the base and 4 feet 10 inches at the top. The other two, at the same positions respectively, are, at the base, 6 feet 11 inches and 4 feet 1 inch. The northern face of this obelisk shows marks of damage from fire, but the other sides are uninjured. No mention is made of the pyramidal top. In an engraving of this obelisk the height of the pyramid exceeds the side of the base. Call the height 5 feet 5 inches : Then the height of the shaft will =78 feet 5 inches less 5 feet 5 inches = 73 feet. Let the 1st unit = 6 '164 inches: Then, 1 . base ordinate = 7 feet 10 inches = 15*25 units, 1 . top ordinate = 4 feet 10 inches = 9 '408 and 15-25 = 232-5 2 = 88-5 height = 144 units, = 73-96 feet. Measured height = 73 feet. 188 THE LOST SOLAR SYSTEM DISCOVERED. The unit for the less side of this obelisk will = f the unit of the greater side = - 6-164 = 5-12 inches. Then 2nd base ordinate = 6 feet 11 inches = 16-21 units, 2nd top ordinate = 4 feet 1 inch = 9*55 ,, and 16-21 2 = 262-5 9-55 = 91 height = 171-5 units = 73 feet. Measured height = 73 feet. Or, let the unit of the greater sides =6-2 inches: Then, 1 . base ordinate = 7 feet 10 inches = 15-15 units, 1 . top ordinate = 4 feet 10 inches = 9-35 and 15-15 = 229-5 1T35 2 = 87-5 axis or height = 142 units, = 73-3 feet. Measured height = 73 feet. Let the unit of less sides = 5-12 inches. Then, 2nd base ordinate = 6 feet 11 inches = 16-12 units, 2nd top ordinate = 4 feet 1 inch = 9-55 and 16-12 = 262-5 9-55 2 = 91 axis or height = 171 '5 units, = 73 feet. Measured height = 73 feet. The mean of the two different units = (6-2 + 5-12) = 5-66 inches, a Babylonian foot = 5 -62 Height from base to apex = 78 feet 5 inches, 110 cubits = 110x8-43 inches =77-27 feet, 110 Koman palms was the original height. CITORIO OBELISK. 189 If to the present height, 78 feet 5 inches, there be added 2 feet 5 inches for the part cut off, we shall have for the original height of the obelisk, from the base to the pyramidal top, 80 feet 6 inches, which = 115 cubits. The Citorio obelisk is the fourth in size. Augustus placed this obelisk in the Campus Martius as a sun-dial. It was erected on the Monte Citorio in 1792 by Pius VI. It is about 7 1 feet 5 inches English in length. The height of the pyramidal top is 5 feet -^gfe inch. The south and north bases of the pyramid measure respectively 4 feet 11 finches; the east and west, 5 feet 1-jJkfe inch. The eastern and western sides of the base of the shaft measure each 8 feet Y^Q-Q inch. The bases on the north and south sides could not be measured, on account of the corrosion of the granite. The whole height of this obelisk, with its pedestal, is about 110 feet. This obelisk of the Campus was found broken in four pieces, the lowest of which was so injured by fire that it was necessary to substitute in its place another block of the same size ; the sculptures are also damaged on the re- maining parts. F. I. Height of the obelisk =71 5-578, pyramid = 5 5*578, say Height of shaft = 66 feet. Let the unit of the Citorio = -J- a Babylonian unit = x f-f-i. of a foot = 6-9382, &c. inches. F. I. Base ordinate = 8 j-^-o = 13-83 units, Top ordinate = 5 lyffo =8-8 .2 and 13-83 = 191 JF8* = 77 Height = 114 units = 57 Babylonian units, = 66 feet. Measured height = 66 feet. The height of the Citorio obelisk, from its base to the apex 190 THE LOST SOLAR SYSTEM DISCOVERED. of the pyramid = 71| feet, which corresponds with the height of one of the obelisks of Pheros = 100 cubits = 70-^ feet. Pliny says this obelisk came from Heliopolis, and was the work of King Sesostris. The measure of the ordinates of the four largest obelisks only are given ; but, including the false obelisks, there are altogether twelve at Rome. There are two obelisks at Alexandria; but only one of them is standing, which is called Cleopatra's Needle. Its dimensions are : F. I. Width of one base - ---82 Width of same face of the obelisk at the base of the pyramidal top -5 If Width of the adjacent base (the two opposite ones, as usual, being equal) - - 7 8^ Width of base of pyramidal top -4 S~ Height of obelisk from base of shaft to base of o pyramidal top - - 57 6f Height of pyramidal top - - 6 6|- Whole height of obelisk - - 64 1 These dimensions of the base are not taken quite at the bottom of the shaft, but on one side 3 feet and inch above the bottom, and on the other side somewhat less. Let the unit of the greater sides = 8 '43 inches = a Babylonian cubit. F. I. 1st base ordinate = 8 2 =11-62 units, 1st top ordinate = 5 If = 7-35 and 11-62 = 135 T 7 ^ 2 = 54 Height = 81 units, = 57 feet. Measured height = 57J feet. Let the unit for the less sides = 1 cubit = 8 inches. 191 F. I. 2nd base ordinate = 7 8-7 = 11'6 units, 2nd top ordinate = 4 8-5 = 7 '06 and fT~6 2 = 134-56 T^OG 2 = 49-84 Height = 84-72 units, = 56-48 feet. Measured height = 57-J feet. The whole height of the obelisk, from the base to the pyramidal top = 64 feet 1 j- inch, and 63 feet = 90 cubits, F. I. Height of the pedestal on which the obelisk rests 611 Respective height of the three plinths on which the base stands, 1 foot 7 inches, 1 foot 9^ inches, 2 feet If inch, making altogether - - 5 5^-J- Whole height of the obelisk and its supports - 76 6^- 108 cubits = 76-9 feet. The whole height from the base of the pedestal to the pyramidal top of the obelisk - - 71 04- 100 cubits = 70|- feet = -J- stade = the height of one of the obelisks of Pheros. The standing obelisk contains three different cartouches ; two of which are titles, and the third is the name of Ramses. That which lies on the ground contains five different car- touches ; three of which, with some slight variations, are the same as on the other obelisk. The name of Ramses is found here also, together with another name. In these calculations we have only made use of the height of the shaft ; but the height of the obelisk may be regarded as the height of the single block of granite, which includes the shaft and pyramidal top. So the height of Cleopatra's Needle = 64 feet 1 inch ; and 63-22 feet = 90 cubits ; height of pedestal =10 cubits. Denon makes the heiht of the cubical kind of base = 6 feet 6 inches, French. Taking the Paris foot =1-[V Eng- 192 THE LOST SOLAR SYSTEM DISCOVERED. lish feet, the side of the cube will = 7 feet English =10 cubits. The cubical base is no part of the obelisk, being a sepa- rate block, like the base of the obelisk which Belzoni removed from Philse. While the French army was at Alex- andria, the earth was removed from the base of Cleopatra's Needle, and it was laid bare to the lowest foundation stone, when the French measures were obtained, which are some- what different from those given on English authority. Not having Denon's nor Belzoni's works to refer to, we cannot say what may be the precise meaning of the cubical kind of base. Nor do we know the dimensions of the cubical base of the Philre obelisk. The length was 22 feet, and width at the base 2 feet. 30 cubits = 21-075 feet. 3 = 2-1075. Pliny states that Ptolema3us Philadelphus erected at Alex- andria an obelisk 80 cubits high, which King Nectanebus had cut out ; but it took much more labour to take the stone to its destination and set it up than it did to cut it out. This obelisk, being inconvenient to the naval station, was brought to the Forum at Rome by a certain Maxim us, a prefect of Egypt, who cut off the top, intending to add a gilded one ; but this was never done. We do not know the measure of Pliny's cubit ; but 80 cubits of 8-43 inches each = 56 feet. The obelisk now standing in the Piazza Navona (at Rome), called the Pamphilian obelisk, is said to be 54 feet high; but it is ranked among the pseudo-obelisks at Home. Besides the obelisks now standing at Rome, others which cannot be found are mentioned by writers of the 16th and 17th centuries; while various fragments which still exist, or lately existed, in different parts of the city, attest the number of works of this kind which once adorned the imperial capital, and the devastations of barbarians, both foreign and domestic. The only obelisk now standing at Heliopolis is supposed to be one of the most venerable monuments of antiquity that the land of Mizraim possesses; but one about which there OBELISK AT HELIOPOLIS. 193 is considerable discrepancy in the accounts of travellers. Pococke states that he found by the quadrant it was 67^ feet high. This obelisk is 6 feet wide to the north and south, and 6 feet 4 inches to the east and west ; and it is discoloured by the water (the annual inundation) to the height of nearly 7 feet. It is well preserved ; except that on the west side it is scaled away for about 15 feet high. The pedestal on which this obelisk stands is said by some writers to be entirely covered with earth. If so, the whole height would exceed that taken by the quadrant. If the height were 70| instead of 67| feet, it would = 100 cubits. " It was during the reign of Osirtasen," remarks Wilkinson, " that the temple of Heliopolis was either founded or received additions, and one of the obelisks bearing his name attests the skill to which they had attained in the difficult art of sculp- turing granite. Another, of the same materials, indicates the existence of a temple erected or embellished by this monarch in the province of Crocodilopolis. The accession of the first Osirtasen, I conceive to date about the year 1740 B. c." Rawlinson, in his " Assyrian Researches," says that the city of Ra-bek, in the land of Misr, or Egypt, which was always spoken of as the chief place in the country, was the Biblical " On " and the Greek Heliopolis ; the name being formed from <{ Ra," the sun, and " bek " (Coptic baki\ a city. " Nothing remains of the celebrated city of Heliopolis," says Lepsius, " which prided itself of possessing the most learned priesthood next to Thebes, but the walls, which re- semble great banks of earth, and an obelisk standing upright, and perhaps in its proper position. This obelisk possesses the peculiar charm of being by far the most ancient of all known obelisks ; for it was erected during the old empire by King Sesurtesen I., about 2300 B. c., the broken obelisk in the Faium near Crocodilopolis, bearing the name of the same king, being rather an obelisk-like long-drawn stele. Boghos Bey has obtained the ground on which the obelisk stands as a present, and has made a garden round it. The flowers of the garden have attracted a quantity of bees, and these could find no more commodious lodging than in the deep and VOL. I. O 194 THE LOST SOLAR SYSTEM DISCOVERED. sharply-cut hieroglyphics of the obelisk. Within the year they have so covered the inscriptions of the four sides that a great part has become quite illegible. It had, however, already been published ; and our comparison presented few difficulties, as three sides bear the same inscription, and the fourth is only slightly varied." Afterwards Lepsius found, standing in its original place in a grave of the beginning of the seventh dynasty, an obelisk, of but a few feet in height, but well preserved, and bearing the name of the person to whom the tomb was erected. " This form of monument," remarks Lepsius, " which plays so conspicuous a part in the New Empire, is thus thrown some dynasties farther back into the Old Empire than even the obelisk at Heliopolis." Abd-al-Latif spent some years in Egypt, and saw two obelisks at Ain-schems (Heliopolis), one standing and the other fallen. " Among the monuments of Egypt we must reckon those of Ain-schems (the Fountain of the Sun), a small town which was surrounded by a wall, now easily recognised, though in ruins. These ruins belong to a temple, where we see sur- prising colossal figures cut in stone, which are more than 30 cubits in height, with all their limbs in proportion. Of these figures some were standing on pedestals, others seated in different positions in perfect regularity. In this town are the two famous obelisks called Pharaoh's Needles. They have a square base, each side of which is 10 cubits long, and about as much in height, fixed on a solid foundation in the earth. On this base stands a quadrangular column of py- ramidal form, 100 cubits high, which has a side of about 5 cubits at the base, and terminates in a point. The top is covered with a kind of copper cap, of a funnel shape, which descends to the distance of 3 cubits from the top. This copper, through the rain and length of time, has grown rusty and assumed a green colour, part of which has run down along the shaft of the obelisk. I saw one of these obelisks that had fallen, and was broken in two, owing to the enormity of the weight. The copper which had covered its OBELISK AT HELIOPOLIS. 195 head was taken away. Around these obelisks are many others, too numerous to count, which are more than a third or one-half as high as the large ones." The breadth of the base is here said to be 5 cubits only which is evidently too small to be proportionate to the height Pocock's measurement is 6 feet 4 inches, which = 9 cubits for the greater sides. Herodotus tells us that Pheros erected two obelisks in the temple of the Sun, each of a single stone, 100 cubits in height and 8 cubits in breadth. Hence it would appear that the two obelisks called Pha- raoh's Needles, at Heliopolis (the City of the Sun), were the two which Pheros erected at the temple of the Sun on the recovery of his sight. The Citorio obelisk, pronounced to be one of the most beautiful of all now existing at Rome, both for the proportion of its parts and the colour of the material, corresponds in height to one of Pharaoh's Needles and to one of Pheros' obelisks. On the pedestal of the Citorio obelisk is the following inscription : " This obelisk of King Sesostris, once erected as a sun-dial in the Campus by C. Caesar Augustus, after suffering much, both from time and the action of fire, was taken out of the rubbish by Pope Benedict XIV. Pius VI., after repairing and beautifying the obelisk, removed it from the place where Benedict had left it, and again placed it on a pedestal, in the year 1792, and the eighteenth of his pon- tificate." The son of Sesostris corresponds to the Pheros of Hero- dotus. Of the obelisk at Heliopolis Hasselquist says, "At Matarie (Heliopolis) is an obelisk, the finest in Egypt. I could not have believed that natural history could be so useful in matters of antiquity as I found it here. An ornithologist can determine at the first glance to what genus those birds belong which the ancient Egyptians have sculptured." According to Norden, the hieroglyphics, though inferior o 2 196 THE LOST SOLAR SYSTEM DISCOVERED. to those of the obelisks of Luxor, are still well executed. Hasselquist pronounces the sculptured birds to be so well cut that it is very easy to point out the originals in nature. He recognises the screech-owl, a kind of snipe, a duck or goose, and none more readily than the stork, in the very attitude in which he may now be seen on the plains of Egypt with upraised neck and drooping tail. The obelisk now standing a few miles from Medinet-el- Faioum is described by Pococke as being of red granite, and 43 feet high, measuring 4 feet 2 inches on the north side, and 6 feet 6 inches on the east. The hieroglyphics are di- vided by lines into three columns on each side. The obelisk is much decayed all round for 10 feet high; the whole is very foul, from the birds sitting on the top, so that it would have been difficult to have taken off the hieroglyphics. This obelisk has the top rounded in Burton's drawings. The height of this obelisk = 43 feet, and 60 cubits = 42-15 feet The less breadth = 4 feet 2 inches, = 6 cubits. The golden image erected by Nebuchadnezzar in the plains of Dura was 60 cubits high and 6 cubits in breadth. At Axuin in Abyssinia (lat. 14 6') there is an obelisk of a single block of granite. The height has been stated to equal 80 feet ; it has also been called equal to 60 feet. Several other obelisks lie broken on the ground, one of which is of still larger dimensions. Among other antiquities discovered at Nimroud by Layard is an obelisk in basalt, six feet high, in a perfect state of preservation, and ornamented with twenty-four bassi-relievos, representing battles, camels of Bactriana, and monkeys ; which, it is said, involuntarily recalls to mind the expedition of Semiramis to India. Pliny records an incident which strikingly illustrates the importance the ancients attached to obelisks. An obelisk being hewn and brought to its destination, was about to be erected : so anxious was the monarch that it should meet with no accident in this difficult operation, that, to oblige his RUINS AT MEMPHIS. 197 engineers to exert all their prudence and skill, he bound his own son to the apex. " The far Syene " was renowned for its granite quarries, and the well into which the sun is said to shine without a shadow, though the town is in fact north of the tropic. It stands immediately before the cataract opposite to the isle of Elephantine. The chisel-marks in the quarries of Syene are still sharp. In one place is seen an obelisk half severed from the rock, but broken and abandoned. That Abd-al-Latif made use of the same cubit as Herodo- tus would appear probable from the dimensions both give of the colossal statues at Memphis. Abd-al-Latif describes what Memphis was, even in the twelfth century. He says, " Its ruins offer to the spectator a union of things which confound him, and which the most eloquent man in vain would attempt to describe. As to the figures of idols found among these ruins, whether we consider their number or their prodigious size, the thing is beyond description. But the accuracy of their forms, the justness of their proportions, and their resemblance to nature, are most worthy of admiration. I measured one which, without its pedestal, was more than thirty cubits, its breadth from right to left about ten cubits, and from front to back it was thick in proportion. This statue was formed of a single block of red granite, and was covered with a red varnish, to which its antiquity seemed only to give a new freshness." Both Herodotus and Diodorus mention the height of each of the statues of Sesostris and his wife at the temple of Vulcan to be thirty cubits. Lying among the ruins of Memphis there is a noble speci- men of Egyptian sculpture, said (in the " Athenasum ") to be a colossal statue of Ramses the Second, the Sesostris of the Greeks one of the two statues mentioned by Herodotus as having been in front of the temple of Vulcan. This statue is almost entire, wanting only the top of the royal head- dress and the lower part of the legs ; and in its present state it measures 36 feet 6 inches in length. o 3 198 THE LOST 80LAB SYSTEM DISCOVEKED. 30 cubits of Herodotus = 21 feet English ; but the height of the discovered statue = 36^ feet. By reference again to that authority, we find it mentioned that among the many magnificent donations which Amasis presented in the most famous temples, he caused a colossus, lying with the face upwards, 75 feet in length, to be placed before the temple of Vulcan at Memphis ; and on the same basis erected two statues, of 20 feet each, wrought out of the same stone, and standing on each side of the colossus. Like to this another is seen at Sais, lying in the same posture, cut in stone, and of equal dimensions. Now 75 feet of Herodotus = 35 feet English ; for 600 feet = 1 stade = 281 English feet, and 75 feet = -J- stade = 50 cubits. An obelisk stands in the public place at Aries in France, where it was erected in 1676, having been found in some gardens near the Rhone. There is no record of the time when it was brought to France, but it would appear a probable conjecture that it had lain up to 1676 just in the position in which it was landed from the ship. It consists of a single piece of granite : the height is 52 feet French ; the base has 7 feet diameter. Taking the Paris foot to = 1^ of an English foot, the 52 French feet will be between 56 and 57 feet English, and 56-2 feet English = 80 cubits. Pliny mentioning the obelisk, 80 cubits high, which was brought from Alexandria to Rome, states that six such obelisks were cut out of the same mountain, and the architect received a present of fifty talents. The obelisk sent to Rome is said to have been clean cut out. Should that be under- stood as having been cut and left without sculptures ? If the obelisk at Aries be a true one, which can now be determined, since the geometrical construction of ancient obelisks is known, it may possibly have been one of the six mentioned by Pliny, as it is 80 cubits high, and has no hieroglyphics inscribed upon it. Bouchaz says " The obelisk at Aries came from Egypt, like those at Rome. There are no hieroglyphics upon it, and probably the Romans brought it from Egypt, intending to erect it in honour of some of their emperors." POMPEY'S PILLAR. 199 Pompey's Pillar stands on a small eminence between the walls of Alexandria and the shores of Lake Maraeotis, about three quarters of a mile from either, and quite detached from any other building. It is of red granite ; but the shaft, which is highly polished, appears to be of earlier date than the capital or pedestal, which have been made to correspond. It is of the Corinthian order. The column consists only of three pieces the capital, the shaft, and the base and is poised on a centre stone of breccia, with hieroglyphics on it, less than a fourth of the dimensions of the pedestal of the column, and with the smaller end downwards; from which circumstance the Arabs believe it to have been placed there by God. The earth about the foundation has been examined, probably in the hopes of finding treasures. It is owing, probably, to this disturbance that the pillar has an inclination of about seven inches to the north-west. The centre part of the cap-stone has been hollowed out, forming a basin on the top; and pieces of iron still remaining in four holes prove that this pillar was once ornamented with a figure, or some other trophy. Various dimensions of Pompey's Pillar have been given ; the following, however, were taken by one of the party who assisted in making the ascent by means of a rope-ladder : Top of the capital to the astragal (one stone) Astragal to first plinth (one stone) Plinth to the ground Whole height Measured by a line from the top It is to be remembered, however, that the pedestal of the column does not rest on the ground, its elevation being - - The height of the column itself is therefore Diagonal of the capital Circumference of the shaft (upper part) (lower part) - Length of side of the pedestal - 10 - 99 4 16 11 24 2 27 2 16 6 o 4 200 THE LOST SOLAR SYSTEM DISCOVERED. Here the height of the shaft = the assigned height of the obelisk at Heliopolis, according to Pococke's measurement : according to another measurement by British officers, who found the Greek inscription dedicating the pillar to the Roman Emperor Diocletian, the height of the shaft = 64 feet, which = the height of Cleopatra's Needle from the base of the shaft to the pyramidal top. Neither Strabo nor Diodorus make mention of this pillar. Denon supposes it to have been erected about the time of the Greek emperors or of the caliphs of Egypt. With regard to the inscription, some have remarked that it might have been added after the erection of the column. Few monuments of antiquity have afforded so wide a field for conjecture and speculation as Pompey's Pillar. Its erection has been as- signed to Pompey, Vespasian, Hadrian, and Diocletian. As Alexandria was embellished by the Ptolemies with works of art collected from the ancient cities of Egypt, the shaft may have originally been a circular obelisk, which, on being removed to Alexandria, was placed on a pedestal and crowned with a capital. When the difference of 2 ordinates = 5, then as both ordi- nates increase by ^, the difference of their squares will in- crease by 1, and the difference of the two ordinates will always = 5. Or when each of the ordinates has increased by 1, as from 15 and 10 to 16 and 11, the difference of the squares of the last set of ordinates will exceed the difference of the squares of the first set by 10 : since 16 2 -11 2 =135 and 15 2 -10 2 =125 Difference = 10 The height of the shaft of an obelisk = the sum x differ- ence of the two ordinates = the difference of their squares. If an obelisk have the lowest ordinate = 6, and highest ordinate = 1, the height of the shaft = 6 2 - 1 2 = 35 ; then, by adding ^ to each of these two ordinates, they become ORDINATES. 201 6-1 and 1*1; the difference of their squares will = 36, and so on. OED. SQUARE, DIFF. 6 36 1 1 35 6-1 37-21 1-1 1-21 36 6-2 38-44 1-2 1-44 6-3 1-3 6-4 1-4 6-5 1-5 39-69 1-69 40-96 1-96 42-25 2-25 37 38 39 40 Thus the difference of the squares of 6 and 1 =35 15 and 10 =125 6-1 and 1-1 = 36 15-1 and 10-1 = 126 6*2 and 1-2 = 37 15-2 and 10-2 = 127 6-3 and 1-3 = 38 15-3 and 10-3 = 128 6-4 and 1-4 = 39 15-4 and 10-4=129 6-5 and 1-5=40 15-5 and 10-5 = 130 7 and 2 =45 16 and 11 =135 8 and 3 =55 17 and 12 =145 9 and 4 65 18 and 13 =155 10 and 5 =75 19 and 14 =165 11 and 6 =85 20 and 15 =175 Diff. of sq. 6-1 and 1*1 exceeds diff. of sq. 6 and 1 by 1 6'5 and 1*5 5 7 and 2 - 10 11 and 6 - 50 202 THE LOST SOLAR SYSTEM DISCOVERED. Diff. of sq. 15-1 and 1(H exceeds diff. of sq. 15 and 10 by 1 15-5 and 10-5 5 16 and 11 - 10 20 and 15 - 50 25 and 20 - 100 30 and 25 - 150 Thus 15 exceeds 10 by 5, and 30 exceeds 25 by 5 30 15 15, 25 10 15, 30 2 -25 2 exceeds 15 2 -10 2 by 150. When ! is added both to 15 and 10, their sum is increased by *2. The increase of their sum x their difference = *2 x 5 _ -I When *5 is added to both, increase x difference = 1x5 -=5. When 1 is added, increase x difference = 2 x 5= 10. When 5 is added, increase x difference = 10 x 5= 50. When 10 is added, increase x difference = 20 x 5 = 100. If the difference between two ordinates = 6, then, when both ordinates are increased by 1, the difference of their squares will be increased by 2 x 6, or 12. ORD. SQUARE. DIFF. 7 49 1 1 48 8 64 2 4 60 9 81 3 9 72 Hence, when the difference between two ordinates = n, then, as each ordinate increases by 1, the difference of their squares will increase by 2n. Or, when the difference between two ordinates = n, then when both ordinates are increased by m, the difference of their squares will be increased by 2mn. DRUIDICAL STONES. 203 The rude Druidical quadrilateral, monolithic, obeliscal monuments descend many feet below the surface. The three monolithic obelisks called the " Devil's Arrows," near Boroughbridge, in Yorkshire, are nearly a]l of the same height. The base of the central one has been traced to 6 feet below the surface ; its height above the surface is 22^- feet. At Rudston, in the same county, stands a similar obelisk, upwards of 29 feet high; its depth in the ground has been traced to 12 feet, without coming to the bottom. It stands 40 miles from any quarry where the same sort of stone is found ; and, like all similar monuments, it remains without either historical or traditional record. The men-hir or stone- long, in Brittany, is 40 French feet high above the surface ; and not less than 10 feet of the same obeliscal monolith is supposed to descend below the surface. Along the coast of Carnac (Morbiham), a bay in Brittany, rude Druidical stones, ranged in many lines over a surface of half a league, may be counted by hundreds ; they present the appearance of an army in battle. An obelisk, now fallen and broken, measuring 64 feet English in length, and computed to weigh upwards of 300 tons, is described among the remarkable monuments, usually called Druidical, at the Bourg of Carnac, in the Department of Morbiham (the country of the ancient Veneti), on the south coast of Brittany. Jablonski's Lexicon gives a derivation of the word Osiris, which he deduces from Osh Iri, that is, he who makes time. Osiris holds in one hand a kind of key, with a circular handle, which from its having some resemblance to the letter T, is often called the Sacred Tau, or crux ansata. The serpent of the* Egyptians may have been held sacred from its form resembling the circular obelisk, the emblem of eternity. Burton found sculptured, on the obelisk at San, the crux ansata, or tau, with the circle attached to the top, suspended from the middle part of the serpent. The tau formed by the double ordinate, and the sectional axis of the obelisk, may be regarded as symbolical of time, 204 THE LOST SOLAR SYSTEM DISCOVERED. velocity, and distance, or the generator of lines, areas, and solids. This sacred tau or key, as represented in the hand of Osiris, unfolds to view the long concealed type of the law of gravitation embodied by the geometrical and mechanical skill of the ancients in a single block of granite. Some obelisks remain perfect after having endured the revolution of three or four thousand years. The Egyptian obelisk being truncated, the part wanting above the top might be supposed to denote the legendary period elapsed before history commenced. The visible part of the obelisk from the truncated top to the surface of the earth might indicate the historic period. A future indefinite period might be symbolised by the supposed continued descent of the obelisk below the earth's surface. The Jains say that time has neither beginning nor end. " The temple of Latona, at Butos, near the mouth of the Nile, where oracles are given, is a magnificent structure adorned with a portico 10 orgyes in height. But of all things I saw there, nothing astonished me so much as a quadrangular chapel in this temple, cut out of one single stone, and containing a square of 40 cubits on every side, entirely covered with a roof of one stone, having a border 4 cubits thick. This chapel, I confess, appeared to me the most prodigious thing I saw in that place. (Herodotus.) The height of the portico =10 orgyes = 40 cubits = ^ stade. The exterior of the stone chapel is a cube of 40 cubits. 40 cubits = T V stade = 28-1 feet Eng. Taking the thickness of the sides of the cubic chapel = the thickness of the stone that formed the roof = 4 cubits. Then 40 - 8 = 32 40 3 = 64000 32 3 = 32768 = 40 3 . Thus the external cube is double the internal cube. The content of the walls = the internal cube = ~ the ex- ternal cube = 32000 cubic cubits. The height of the granite pedestal on which is placed the STONE CHAPEL AT BUTOS. 205 equestrian bronze statue of Wellington, in the front of the London Exchange = 14 feet, and the height of the statue = 14 feet ; together they = 28 feet = -JL. stade = the height of the cubic stone chapel at Butos. The content of the Lateran obelisk may be compared with the content of the chapel formed out of one stone. Taking the shaft of the obelisk = 200, and height from base to apex = 400 units ; here unity = 6 inches. The height x ordinates of the greater side = 400 x 20 2 = 160000 and 200 x 14'4 2 = 40000 "Difference = 120000 half = 60000 = content in units = 7500 = in feet Thus the content of obelisk when estimated by the greater ordinates = 60000 cubic units. When estimated by the lesser ordinates = 54200 ; the mean = (60000 + 54200) = 57100 cubic units = 7137 cubic feet = 525 tons by taking a cubic foot of granite to equal 165 pounds avoirdupois. Wood makes by measurement a stone at Balbec = 14128 cubic feet, which will equal twice the content of the Lateran obelisk. The thickness of the sides of the chapel = 4 cubits = 33-72 inches. then 2 x 33'72 = 5-62 feet, is to be deducted from 28 feet, the side of the external cube. 28 5*62 = 22-38 feet for the side of the internal cube. 28 3 = 21972 cubic feet 22-2 3 ? &c. = 10986 = -J- 28 3 = the internal cube = the content of the walls of the chapel, from which deduct the top part, or roof, = 28 2 x 2*81 = 2203 cubic feet, and 10986 2203 = 8783 cubic feet for the content of the 5 sides of the cubic chapel that would have to be transported to Butos in one piece. The weight, if granite, would be about 646 tons. Thus the content of the 206 THE LOST SOLAR SYSTEM DISCOVERED. Lateran obelisk : the content of the cubic chapel :: 7137 I 8783 in cubic feet. Or weights as 525 : 646 tons. Herodotus says that Psammitichus, having sent to Butos to consult the oracle of Latona, which is the truest of all oracles in Egypt, was answered that he would be avenged by men of copper coming from the sea. The same oracle announced that Mycerinus would live only 6 years, and die in the 7th. It was at Butos the oracle answered Cambyses : "It is destined that Cambyses, the son of Cyrus, shall end his days at Ecbatan." Probably the oracle might be given in this cubic chapel. When the Athenians were afflicted with the plague, an oracle ordered the cubic altar of Apollo to be doubled. There were also temples at Butos dedicated to Apollo and Diana. Stonehenge, on Salisbury Plain, is supposed by Davis to have been the round temple dedicated to Apollo, according to this substantive description given by Diodorus : ' ' Among the writers of antiquity, Hecateus and some others relate that there is an island in the ocean, opposite to Celtic Gaul, and not inferior in size to Sicily, lying towards the north, and inhabited by Hyperborei, who are so called because they live more remote from the north wind. The soil is excellent and fertile, and the harvest is made twice in the same year. Tradition says that Latona was born there, and therefore Apollo is worshipped before any other deity ; to him is dedi- cated a remarkable temple of a round form." Latona, the daughter of Titan, had an oracular temple at Butos, formed of one gigantic stone. These oracles were celebrated for their truth, and for the decisive answers given. The oracles at the temple of her son Apollo, at Delphi, delivered by the priestess Pythia, were celebrated in every country. It is said Neptune, moved with compassion towards Latona, when driven from heaven and wandering from place to place, because Terra, influenced by Juno, refused to give her a MYTHOLOGY. 207 place where she might rest and bring forth, struck with his trident Delos, one of the Cyclades, and so made immoveable that island, which before wandered in the ^Egean, and appeared sometimes above and sometimes below the sea. There Apollo was born, to whom the island became sacred. One of the altars consecrated to Apollo at Delos was reckoned among the seven wonders of the world. Here we find a striking similarity between the temple of Latona in Egypt, and those of her son Apollo in Greece. The temple at Delos stood on a once floating island. The temple at Butos stood near the great lake, on which floated the island of Chemmis. The oracles delivered at the temples of Latona and Apollo were greatly celebrated. The altars at both were reckoned among the wonders of the world, and at both were cubic altars ; at one the external cube was double the central cube, at the other, the cubic altar was required to be doubled. No wonder then that Herodotus recognised in Egypt the gods of his country, as the Sepoys in the British army that came from India during the Egyptian campaign recog- nised the gods of their country, and worshipped them in the colossal temples of Egypt. Burckhardt says the excavated temples of Nubia, from their strong resemblance, recalled to his mind those of India. Here are the links of the mythological chain, like those of learning and science, connecting Asia, Africa, and Europe. The following extract, descriptive of a visit to the Temple of Dendera, is from " Scenes and Impressions in Egypt." The author traverses Egypt in the Overland route from India. " To one who has just quitted a country where the priest still officiates, and the worshipper bows down and prostrates himself in the temples of idolatry, who is familiar with the aspect, the habits and customs, the rites and ceremonies of the Hindoo, this temple is an object of no common interest ; for here the Indian soldier fancied he recognised the very gods he worshipped, and with sadness and indignation complained to his officers, that the sanctuary of his god was 208 THE LOST SOLAR SYSTEM DISCOVERED. neglected and profaned. He saw a square and massive building, a colossal head on the capitals of huge columns ; on the walls, the serpent ; the lingam, in the priapus ; the bull of Iswara, in the form of Apis ; Garuda, in Arueris ; Hanu- man, in the round headed cynocephalus ; a crown very similar to that of Siva, on the head of Osiris ; and in the swelling bosom of Isis, that of the goddess Parvati: while on the staircase, the priest and the sacred ark must have reminded him, and strongly, of the Brahmins, and the palanquin litter of his native country. Many, many forms he must have missed, many too have observed, to which he was an entire stranger." Again, speaking of the low tombs near the great pyramid, two of which have their walls covered with paintings. " There is the birth and story of Apis, the cow calving ; there are sacrifices, feasting, dancing; there is an antelope in a small wood ; and there is a figure (though a mere trifle) called and fixed my attention, a man carrying two square boxes across the shoulder on a broad flat bending piece of wood ; exactly similar to this is the manner in which burdens are borne in India, by what we there call bangy-coolies. It suggests to me, what I had forgotten before to remark, the peculiar way in which you see, in paintings at Thebes, the end of the girdle or loin cloth gathered, plaited, as it were, and hanging down before their middles ; this is exactly Indian ; nor in my eye is either the complexion or feature, either in the paintings or statues, very different from some tribes of Brahmin." For the following mythological details history is indebted to Herodotus: "The Pelasgians, the most ancient people of Greece, honoured their gods without knowing them, and even without giving them names. They were called gods, and regarded masters of all things. It was not till a period far distant from their origin that they knew the names of their gods came from Egypt. Then they went to consult the oracle of Dodona, the most ancient in Greece, and in- quired if they ought to receive the names of the gods given by barbarians. Upon the oracle answering that they ought STONE CHAPEL AT SAIS. 209 to receive them, they sacrificed to the gods, and invoked them by names. It was from the Pelasgians the Greeks re- ceived these names. One remains still ignorant whence each god came, if he had always existed, what was his form ? For myself, I believe they came from Egypt ; .and if I should be told that the Egyptians knew not Neptune, Castor, Yesta, Themis, the Graces and Nereids, I should answer, that the Pelasgians learned these names from the Samothracians with whom they associated. As to all the other gods, their names came from Egypt." Thus it appears that at a remote period an intercourse had been established from India to the west of Asia ; thence to Egypt and the Mediterranean, through the agency of com- merce, migratory masons, wandering philosophers, or magi. So that India had long been enlightened before the first ray of science had pierced the last European darkness. Though India may appear to stand the first, and Europe the last in the scale of antiquity of science and learning, yet perhaps China may contend with India, and America with Europe for priority. These remote epochs call to mind the exclamation which Plato, in the " Timaeus," puts into the mouth of the priests of Sais " O Solon, O Solon ! ye Greeks still remain ever children ; nowhere in Hellas is there an aged man. Your souls are ever youthful. Ye have no knowledge of antiquity, no ancient belief, no wisdom grown venerable by age." Herodotus, describing Sais, says, " What I admire above all other things is a house made out of one stone, which was brought by Amasis from Elephantis. Two thousand men were employed during three whole years in transporting this house, which has in front 21 cubits, in depth 14, and 8 in height ; this is the measure of the outside. "The inside is 18 cubits in length, 12 in depth, and 5 in height. This wonderful edifice is placed by the entrance of the temple of Minerva." External measurement =21, 14, 8 cubits Internal = 18, 12, 5 Difference =3, 2, 3 VOL. i. p 210 THE LOST SOLAR SYSTEM DISCOVERED, Let the common difference = 2 '5 ; then 21, 14, 8 less 2-5, 2-5, 2-5 equals 18'5, 11/5, 5'5 for internal sides, and 18'5 x 11*5 x 5'5 = 1170 internal content, 2x1170 = 2340; but 21 x 14 x 8 = 2352 external content. Thus the external content = double the internal content. The chamber, according to this calculation, would not ex- ceed a 15^- feet sectional length of a London sewer. By placing the chamber on one side, a man might walk upright on a floor about 15^ feet by 3 feet 10 inches. These dimen- sions are too insignificant for a monolith which took 2000 men, for three whole years, to transport from Elephantis. If the dimensions had originally been written orgyes in- stead of cubits, then, by this supposition, the orgye being = 4 cubits, the content of the mass to be moved, which = the sides of the chamber = \ the external content, will = about 52266, or 26133 cubic feet = 1925 tons, if the stone were granite. The content would = nearly twice the content of the Balbec stone, and the Balbec stone = twice the content of the Lateran obelisk. The granite block which composes the pedestal of the bronze equestrian statue of Peter the Great, at St. Peters- burgh, was estimated at the weight of 1500 tons. Then, according to the preceding calculation, the weight of the monolithic temple transported from Elephantis to Sais would be to the weight of the monolithic block of granite transported from the Gulf of Finland to St. Petersburg!! ::1925 : 1500. The St. Petersburgh block formed the remnant of a huge rock which lay in a morass about four miles from the shore of the Gulf of Finland, and at the distance of about fourteen miles by water from St. Petersburgh. The means adopted in conveying this block, both by land and water, are also stated. " I found the rock," says the engineer employed, " covered STATUE OF PETER THE GREAT. 211 with moss. Its length was 42 feet, its breadth 27, and its height 21 feet." " The expense and difficulties of transporting it," says Coxe, (f were no obstacles to Catherine the Second. The morass was drained, the forest cleared, and a road formed to the Gulf of Finland. It was set in motion on huge friction- balls and grooves of metal by means of pulleys and windlasses, worked by 500 men. In this manner it was conveyed, with 40 men seated on the top, 1200 feet a day, to the shore ; then embarked on a nautical machine, transported by water to St. Petersburgh, and landed near the spot where it is now erected. Six months were consumed in this undertaking, which was certainly laborious in the extreme; for the rock weighed 1500 tons. In its natural state the stone would have been a magnificent support for the statue ; but the artist, in his attempts to improve it, deprived it of half its grandeur." The height of the figure of the emperor is 1 1 feet ; that of the horse, 17 feet. The weight of both together is 36,636 pounds English. Since 500 Russians conveyed a monolith weighing 1500 tons, in six months, to St. Petersburgh, the conveying a monolith, weighing 1873 tons, by water, in three years, by 2000 Egyptians, from Elephantis to Sais, does not seem an impossibility. Wilkinson thus describes the broken statue in the Memno- nium, which was formerly in a sitting attitude : " To say that this is the largest statue in Egypt will convey no idea of the gigantic size or enormous weight of a mass which, from an approximate calculation, exceeded, when entire, nearly three times the solid contents of the great obelisk at Karnak, and weighed about 887 tons. The obelisk weighs about 297 tons, allowing 2650 ounces to a cubic foot." The smaller of two Luxor obelisks, lately removed to Paris, was calculated by Lebas to weigh 246 tons English. Montverrand, a French architect, has raised a granite column at St. Petersburgh, which is a single block, about p 2 212 THE LOST SOLAE SYSTEM DISCOVERED. 96 feet high, and weighs three times as much as the obelisk of Luxor. The monolithic granite temple, called the " Green Taber- nacle, or Chamber," at Memphis, was, according to Arab writers, formed of one single stone, 9 cubits high, 8 long, and 7 broad. In the middle of the stone a niche or hole is hol- lowed out, which leaves 2 cubits of thickness for the sides, as well as for the top and bottom. Exterior 9, 8, 7 cubits Deduct 4, 4, 4 Interior 5, 4, 3 Then 9 x 8 x 7 = 504 exterior content. and 5x4x3= 60 interior content. 8 x 60 =480. In order that the interior content should = ^ exterior con- tent, the internal dimensions should = the external dimen- sions ; or internal = 4*5 x 4 x 3'5 = 63, and 8x63 = 504: so 8 times the internal content will = the external content. Makrizi, speaking of the same monolith, adds, te There was at Memphis a house (chamber) of that hard granite which iron cannot cut. It was formed of a single stone, and on it there was sculpture and writing. On the front, over the entrance, there were figures of serpents presenting their breasts. This stone was of such a weight, that several thousand men together could not move it. The Emir S. S. Omari, broke this green chamber about the year 750 of the Hegira (A. D. 1349), and you may see pieces of it in the jamy (mosque) which he caused to be built in the quarter of the Sabasans, outside of Cairo." A monolith at Tel e' Tmai, the ancient Thmouis, in the Delta, still remains ; it is of polished granite, and rectangular. According to Burton, it is 21 feet 9 inches high, 13 feet broad, and 1 1 feet 7 inches deep ; the thickness of the walls being about 2-J- feet. This will make the height, breadth, and depth of the chamber, each 5 feet Jess than the external height, breadth, TEMPLES AT MAHABALIPURAM. 213 and depth. If instead of 5 feet, 5*2 feet be deducted from each external measure, this will give the interior content, ^ of the exterior content. Exterior 21-8 13 11-6 deduct 5-2 5-2 5-2 Interior 16-6 Ts 64 then 21-8 x 13 x 11-6 = 3287 and 16-6 x 7*8 x 6-4 = 3287 or the interior content = J the exterior content. The height, 21 feet 9 inches, will be about 31 cubits. The Butos monolith being a cube of 40 cubits, or the height of 40 cubits. In the vicinity of Mahabalipuram, on the sea-coast of the Carnatic, are the celebrated ruins of ancient Hindoo temples, dedicated to Vishnu. Facing the sea there is a pagoda of one single stone, about 16 or 18 feet high, which seems to have been cut on the spot out of a detached rock. On the outside surface of the rock are bas-relief sculptures, repre- senting the most remarkable persons whose actions are celebrated in the Mahabharat. Another part of the rockjs hollowed out into a spacious room. On ascending the hill, there is a temple cut out of the solid rock, with some figures of idols in alto relievo upon the walls, very well finished : at another part of the hill, there is a gigantic figure of Vishnu, asleep on a bed, with a huge snake wound round in many coils as a pillow, which figures are all of one piece, hewn out of the rock. A mile and a half to the southward of the hill are two pagodas, about 30 feet long by 20 wide, and the same in height, cut out of the solid rock, and each consisting originally of one single stone. Near to these is the figure of an elephant, as large as life, and a lion much larger than the natural size ; but otherwise a just representation of a real lion, which is, however, an animal unknown in this neighbourhood, or in the south of India. The whole of these sculptures appear to have been rent by some convulsion of nature, before they were finished. The great rock above described is about 100 yards from the sea; but on the rocks washed by the sea are sculptures P 3 214 THE LOST SOLAR SYSTEM DISCOVERED. indicating that thej- once were cut out of it. East of the village, and washed by the sea, is a pagoda of stone, contain- ing the Lingam, and dedicated to Mahadeva. The surf here breaks far out, and (as the Brahmins assert) over the ruins of the city of Mahabalipuram, which was once large and magni- ficent ; and there is reason to believe, from the traditional records of the natives, that the sea, on this part of the Coro- mandel coast, has been encroaching on the land. All the most ancient buildings and monuments at this place are con- secrated to Vishnu, whose worship appears to have predo- minated on this coast ; while, on the opposite coast, in the neighbourhood of Bombay, that of Mahadeva, or Siva, pre- vailed to a greater extent. (East India Gazetteer.) We do not know that any such rectangular Druidical monolith monuments exist; but we find a description of a large dolmen formed by 17 or 18 blocks of stone. The finest Celtic monument, the largest and most regular, within the limits of Brittany or Anjou, is seen near the village of Bagneux, about a mile from Saumur. This monument is a dolmen of a rectangular form, raised on the side of a hill, and composed of enormous blocks of sand- stone. It is 58 feet long, 21 wide, and about 7 feet high from the ground. The disposition of the stones is perfectly uniform, four at each side for the walls, four for the roof, one on the left side near the entrance, one at the west, closing up the dolmen at that end ; two smaller ones standing up near the entrance, and a single isolated block at the bottom, like a pillar, helping to sustain the weight of the roof. There are altogether seventeen of these immense blocks, and from some rough masonry, which may be seen supplying a vacancy on the right of the entrance, it is inferred that there were originally eighteen. Scattered about in disorder outside the entrance are some flat stones, which it is conjectured may have once stood upright in continuation of the northern wall. The great blocks which form this singular structure are all unhewn, yet of such equal dimensions that, with a single ex- ception, the result apparently of an accident, they lie almost CELTIC MONUMENT. 215 as closely together as if they had been carefully smoothed for the places they occupy. They vary in thickness from 18 inches to 2\ feet, and are all of extraordinary magnitude ; the largest, that which closes the west end, presenting a square surface of twenty-one feet to the side. It is said, that upon digging round the monument, the walls are found to be buried nearly 9 feet in the earth, which would give the upright blocks a height of almost 16 feet. The fact is remarkable, as Celtic stones in general are seldom sunk to such a depth. But in this instance there appears to have been a necessity for it, as the blocks, instead of being vertical in the usual way, incline so far towards the centre, that a plummet dropped from the top would fall more than a foot from the base. It is impossible to visit these prodigious masses of stone without renewed astonishment at the marvel- lous mechanical power by which they were raised from their quarries, transported to their destination, and arranged in symmetrical order. In the vineyards, about 40 or 50 yards distant, is a solitary peulven, about 6 or 7 feet high, out of the line of the dolmen, and apparently having no connection with it ; and on the top of a hill not far from the neighbour- ing village of Riau is a smaller dolmen, consisting of six great stones, also set towards the east, equally regular in form, but considerably dilapidated by the action of the weather. This dolmen presents the additional peculiarity of a flooring of flag stones. The blocks of which these monu- ments are built are composed of sandstone, found in the environs of Saumur ; but at such a distance from the place selected for the mystical purposes to which the Celts applied them, that they must have been carried at least half a league over a difficult country, intersected with ravines and valleys. The work of cutting these prodigious blocks out of the quarry, and raising them from their beds, is intelligible to a people who understand the use of the wedge and the lever ; but the mechanical power by which they were conveyed across rivers and hills, and placed in this regular order of walling and roofing, is utterly incomprehensible. A glance into the dolmen of Bagneux, this vague damp p 4 216 THE LOST SOLAR SYSTEM DISCOVERED. hall, fills the mind with a sort of dreary wonder not very easy to describe. What could have been the object of this rude, stony temple, mausoleum, or whatever else it was? The twilight within is by no means impressive, except in the same way, but with a sort of palpable horror in it, as a great sub- terranean sepulchre can be felt to be impressive. When you creep in, rather shudderingly, you have an instinctive conviction of the tremendous solidity of the masses of stone around and above you, which have stood there for centuries heaped upon centuries; yet it is of so dismal a kind, that you can hardly overcome a certain sense of terror, lest the whole mass should fall and crush you to atoms. It is probably the consciousness of your own weakness and insignificance in the presence of so ponderous a mystery that produces this feeling. Formerly the neighbourhood of Saumur was scattered over with Celtic ruins, of which few are now remaining, and of these which are still described in the local books some have already disappeared. They have been broken up for ma- terials to mend the roads. The sides of this dolmen would seem from the description to resemble the sides of an Egyptian propylon, the sides of both being inclined, and both structures colossal. Perhaps rectangular structures formed of several large stones to resemble a rectangular monolith may be found among Druidical remains. In Gaul, the power of the Druid priesthood was so directly inimical to the Roman domination, that, as Gibbon remarks, under the specious pretext of abolishing human sacrifices, the emperors Tiberius and Claudius suppressed the dangerous power of the Druids ; next the priests themselves ; their gods , and their altars subsisted in peaceful obscurity until thejinaljlestruction of paganism. 217 PAET IV. PYRAMID OF CHEOPS. ITS VARIOUS MEASUREMENTS. CONTENT EQUAL THE SEMI-CIRCUMFERENCE OF EARTH. CUBE OF SIDE OF BASE EQUAL \ DISTANCE OF MOON. NUMBER OF STEPS. EN- TRANCE. CONTENT OF CASED PYRAMID EQUAL y L DISTANCE OF MOON. KING'S CHAMBER. WINGED GLOBE DENOTES THE THIRD POWER OR CUBE. THREE WINGED GLOBES THE POWER OF 3 TIMES 3, THE 9TH POWER, OR THE CUBE CUBED. SARCOPHAGUS. CAUSEWAY. HEIGHT OF PLANE ON WHICH THE PYRAMIDS STAND. FIRST PYRAMIDS ERECTED BY THE SABJEANS AND CON- SECRATED TO RELIGION. MYTHOLOGY. AGE OF THE PYRAMID. ITS SUPPOSED ARCHITECT. SAB^ANISM OF THE ASSYRIANS AND PERSIANS. ALL SCIENCE CENTRED IN THE HIERARCHY. TRADITIONS ABOUT THE PYRAMIDS. THEY WERE FORMERLY WORSHIPPED, AND STILL CONTINUE TO BE WORSHIPPED, BY THE CALMUCS. WERE REGARDED AS SYMBOLS OF THE DEITY. RE- LATIVE MAGNITUDE OF THE SUN, MOON, AND PLANETS. HOW THE STEPS OF THE PYRAMID WERE MADE TO DIMINISH IN HEIGHT FROM THE BASE TO THE APEX. DUPLICATION OF THE CUBE. CUBE OF HYPOTHENUSE IN TERMS OF THE CUBES OF THE TWO SIDES. DIFFERENCE BETWEEN TWO CUBES. SQUARES DE- SCRIBED ON TWO SIDES OF TRIANGLES HAVING A COMMON HYPO- THENUSE. PEAR-LIKE CURVE. SHIELDS OF KINGS OF EGYPT TRACED BACK TO THE FOURTH MANETHONIC DYNASTY. EARLY WRITING. LIBRARIANS OF RAMSES MIAMUM, 1400 B.C. DIVISION OF TIME. SOURCES OF THE NILE. Pyramid of Cheops. HAVING made repeated attempts, and as many failures, to ascertain the magnitude of the Pyramid of Cheops from stated measurements whicfrdiffered so greatly from each other, we at last abandoned all hopes of arriving at any satisfactory conclusion. Herodotus only says, " The Pyramid of Cheops is quadri- 218 THE LOST SOLAR SYSTEM DISCOVERED. lateral ; each side being 8 plethrons in length, and height the same." These statements we found to be inaccurate ; for we had already ascertained the value of the plethron of Hero- dotus. Savary gives the dimensions of the Great Pyramid from the following authors : HEIGHT. BA.SE. Feet. Feet. Herodotus - - 800 800 Strabo - - 625 600 Diodorus - 600 700 Pliny - 708 Le Brun - 616 704 Prosper Alpinus - - 625 750 Thevenot - 520 682 Niebuhr - 440 710 Greaves - 444 648 To these might be added a list more numerous, with dis- crepancies not less. The number of sides of the pyramid - = 4 Suppose each side - - = 4' 2 (linear plethra) then the perimeter will - - = 4 3 and the area of the base - - = 4 4 (square plethra). Let the sum of the indices of 4, or 1 + 2 + 3+4 = 10, be the height in plethra : Since 1 phlethrum - =40*5 units, 10 plethra will - - = 405 = 468^ feet. The side of the base will = 16 x 40-5 = 648 units, = 749^ feet. By the addition of somewhat more than unity to the height, we have Content of the pyramid = J height x base area, = |- 406, &c. x "648 > = I 113689008 units, = \ circumference of the earth PYRAMID OF CHEOPS. 219 which may also be expressed by -J- (324 x 2 243 x 324 x 2 ) ; 324 being the Babylonian numbers 243 transposed. Height : base :: 10 : 16 Height = -J-S- or -| base. Herodotus makes the height the same as the base : Height = 405 units, Base = 648 units, from which take 243, or 1 stade, and there will be left 405 units for the height, which makes the height = the side of the base, less 1 stade. The cube of the side of the base =648 3 = 272097792 4 cubes = 1088390065 units. The distance of the moon from the earth = 60 semi- diameters of the earth = 9*55 circumference, say = 9 '5 7 circumference, then 9-57 x 113689008 = 1088003806 units, and 9-55 circumference = 1085730026 Hence the distance of the moon from the earth = 4 times the cube of Cheops = the cubes of the four sides. Diameter of the earth = 7926, and circumference = 24,899 miles. Distance of Mercury from the Sun = about 150 times the distance of the moon from the earth. Distance of moon = 4 cubes, .*. distance of Mercury = 4 x 150 = 600 cubes, = 10 x 60 cubes of Cheops. Distance of the moon = 9*57 x circumference = 4 cubes, = 9*57 x 24899, = 238283-43 miles, .-. 150 x 4 = 600 cubes = 150 x 238283-43 or, distance of Mercury = 35742514 miles. By the tables, the distance of Mercury = about 36 or 37 millions of miles. So the distance of Mercury from the sun will somewhat exceed 150 times the distance of the moon from the earth, or 600 cubes of Cheops. The distance of the moon from the earth, by the tables, = 60 and 61 semi-diameters of the earth. 220 THE LOST SOLAR SYSTEM DISCOVERED. According to Herschel, the mean distance of the centre of the moon from that of the earth is 59*9643 of the earth's equatorial radii, or about 237,000 miles. The mean distance of Mercury from the Sun is about 36,000,000 miles. Thus 152 x 237000 miles = 36,024,000 miles for the distance of Mercury, which is nearly 150 times the distance of the moon. It will be seen hereafter, that the distance of Mercury I distance of Belus :: 1 : 150 nearly, and distance of Mercury = 150 x distance of moon = 150 x 4 cubes. Hence the distance of Belus will = 150 2 x distance of moon = 150 2 x 4 cubes = 22500 x 4 = 90000 = 300 2 = (5 x 60) 2 cubes of Cheops. The distance of Saturn = 25 times the distance of Mer- cury = 25 x 150 x 4 = 15000 = f 100 2 cubes or Mercury = 600 Saturn = 15000 Belus = 90000. Cube of side of base = J distance of moon 2 sides = 2 4 =16 The cube of twice the side = (2 x 64 8) 3 = twice the dis- tance of the moon. Distance of Mercury =150 times the distance of moon =75 times the cube of twice the side. Distance of Belus 150 x 75 times the cube of twice the side. 2 pyramids = circumference. Twice pyramid : cube of perimeter of base circumference I 16 distance of moon 1 : 152 distance moon from earth : distance Mercury from sun. The side of the base does not = 8 plethrons, but it = Q- 8) 2 = i the square of 8 plethrons, = i8 2 =16= twice 8 pleth- rons, or 1600 feet of Herodotus, and the height, 10 plethrons, will equal 1000 feet. PYRAMID OF CHEOPS. 221 The perimeter will =8 2 = 64 =70 -6 =70 plethrons less 1 stade, and side = -J- 64 = 16 plethrons. Height = 10 = 16 6 plethrons = side of base less 1 stade. Thus the side = twice 8 plethrons, and height equals the side less 1 stade. Herodotus says the side equals 8 plethrons, and the height equals the side. Hence dimensions of the pyramid of Cheops, which re- presents the | circumference of the earth, might easily be impressed on the memory by saying the perimeter of the base equals 70 plethrons less one stade, and the height equals the side of the base less 1 stade. The number 1600, which indicates the side of the base in feet of Herodotus, corresponds with the number of talents of silver which the interpreter told Herodotus was inscribed on the side of the pyramid as having been expended in furnishing the workmen with radishes, onions, and garlic. A pyramid having the same base as that of Cheops, and height = side of base would = -J- the cube of Cheops, = of ^ = iV distance of moon. The contents of all the pyramids were assigned without reference to the cube of the sides of the bases, for we did not discover that these cubes were measures of the distance of the moon and planets till after the estimates of the pyramids were made. Since pyramid of Cheops = \ circumference and cube of Cheops = i distance of moon. Height I side of base of pyramid of Cheops :: 406 : 648 :: pyramid of Cheops I a pyramid having same base and height = side of base. Distance of moon =4 cubes = 12 pyramids each having height = side of base. So 406 : 648 x 12 :: circumference : 9*57 circumference. Thus distance of moon = 9 '5 7 circumference. The cube has 12 edges, each = height or side of base of cube. So -J- circumference : distance of moon : : height of pyra- mid : the 12 edges of the cube :: height of pyramid .' 3 times the perimeter of the base. The pyramid of Belus = ^ cube of 1 stade = -% circurn-r ference and height = side of base. 222 THE LOST SOLAR SYSTEM DISCOVERED. So 24 pyramids = 8 cubes = circumference ; pyramid : circumference :: 1 I 24 :: height : twice 12 edges of the cube :: height : 6 times the perimeter of base. The mean distance of the moon from the earth = 237000 miles 400 x 237000 = 94800000 miles and 95000000 miles = the distance of the earth from the sun. Hence the distance of the earth from the sun = 400 times the distance of the moon from the earth = 400 x 4 = 1600 cubes = 1600 times the cube of the side of the base of the pyramid of Cheops. But the side of the base of the pyramid of Cheops = 2f stades = 648 units = 1600 Babylonian feet. Hence the distance of the earth from the sun = as many cubes of the side of the base of the pyramid of Cheops as the side of the base contains Babylonian feet. The distance of the moon from the earth = as many cubes of Cheops as the pyramid has sides, or 3 5 = 243 transposed = 324 doubled = 648 four times the cube of 648, or half the cube of twice 648 = distance of moon, or the cube of twice the side of the base = twice the distance of the moon from earth, = diameter of the orbit of moon. The cube of the perimeter of the base =8x2=16 times the distance of moon. Distance of earth from sun = 400 times the distance of moon from earth = 4 -f/ = 25 times the cube of the perimeter of the base. Side of base : height :: 648 : 406, &c., and 648 3 : 407 3 , &c. : : -J I ~Q distance of moon. Thus cube of side of base = -J- distance of moon and cube of height = % cube of side of base, or cubes are as 1 : 4. Cube of twice side of base = (2 x 648 ) 3 = i x 2 3 = f = 2 distance of moon = diameter orbit of moon Cube of twice height = (2 x 407 &c.) 3 = -jig- x 2 3 = -f$ i distance of moon PYRAMID OF CHEOPS. 223 Cubes are as -J- I 2 :: 1 I 4 Cube of 4 times height = (4 x 407 &c.) 3 = I Lx4 3 = -f-f- = 4 distance of moon = 2 diameter orbit of moon. Thus cube of 4 times height = twice cube of twice side = twice diameter orbit of moon = twice 30 diameters of earth. If 30 radii divided the circumference of earth into 30 equal parts, then pyramid would = 15 of these parts, and cube of side of base =15 radii. The inclined side of pyramid will = 521 units, and 52 1 3 , &c. = -J circumference. Thus the cube of the inclined side of pyramid of Cheops will = height x area base of pyramid of Cephrenes = f circumference. Cube of twice inclined side of pyramid of Cheops = -f x 2 3 = 4J> = 10 circumference. Cube of twice side of base = twice distance of moon. Cube of perimeter = 16 distance of moon 2 = 128 4 = 1024 2 cubes of 4 times perimeter of base = 2048 distance of moon and 2045 = distance of Jupiter Side of base - height = 648 - 406 = 242 and 242 3 , c = circumference (2 x 242, &c.) 3 = i x 2 3 = f = 1 Cube of twice difference = circumference. The celestial distances are expressed in terms of the dis- tance of the moon from the earth, and in terms of the cir- cumference, which means the circumference of the earth. Twice side of base = 2 x 648 = 1296 units. Cylinder having height = diameter of base = 1296 units, will = 1296 3 x-7854 = 15 circumference, Inscribed sphere - = 10 cone - = 5 Perimeter of base - =4 x 648 Cylinder = 15 x 2 3 = 120 circumference, Sphere = f =80 Cone = 40 224 THE LOST SOLAR SYSTEM DISCOVERED. 12 cylinders = 120 x 12 = 1440 circumference, = distance of Mercury. Twice perimeter of base = 8 x 648 Cylinder = 120 x 2 3 = 960 circumference, Sphere = f = 640 Cone = I = 320 4 cylinders = 4 x 960 = 3840 = distance of earth. Distances in terms of the cube of side of base : Moon = 4 cubes, Mercury = 600 Earth - = 1600 Saturn - =15000 Belus - 90000 Distances in terms of the cube of twice side of base : Moon = \ cube, Mercury = 75 Earth - = 200 Saturn - = 1875 Belus - =11250 Distance of the earth = 1600 cubes of side = 200 cubes of two sides = 25 cubes of perimeter. Cylinder, diameter = twice perimeter base = 960 circumference 1 \ cylinder = distance of Mercury. 4 Earth. 75 = Uranus. 225 = Belus. Moon. For distance of moon = j-^ distance of Mercury = TToXli = Tio cylinder. So the distance of Belus from the sun will = 225 x 100 = 22,500 times the distance of the moon from the earth. Or the distance of Belus = 15 2 cylinders = 15 2 x 100 times distance of the moon. PYRAMID OF CHEOPS. 225 Thus 4 cubes of side of base = distance of the moon. 4 cylinders, diameter = 2 perimeter = distance of the earth. Or 1 cube of 2 sides = diameter of the orbit of the moon. 1 cylinder, diameter 4 perimeters = diameter of orbit of the earth. 2 cubes of 4 perimeters = distance of Jupiter. Pyramid : cube of side of base, :: -^ circumference I -J- distance of the moon, : : J circumference : - radii of the earth, :: J circumference : 15 :: arc of 12 degrees : radius of the earth. Cube of side of base = 648 3 =^ distance of the moon. Cube of twice side = 1296 3 = 2 = diameter of orbit of the moon ; but 1296 = 6 4 .-. (6 4 ) 3 =6 12 = diameter of the orbit of the moon. 3 5 = 243. Place the last numeral the first of the series, and 243 becomes 324 ; then 324 doubled and cubed = 648 3 = -J- dis- tance of the moon in units. Again : Transpose the first and last numerals, and 243 becomes 342 : then 342 doubled and squared = 684 2 = cir- cumference of the earth in stades. 243 x 684 2 = 3 5 x 684 2 = circumfer. of the earth in units. (2 x 648) 3 = diameter of the orbit of the moon in units, = 6 12 = (2x3) 4 * 3 = twice 3 to the power of 4 times 3. Cube of twice side of base = 6 12 = 1296 3 = diameter of the orbit of the moon. Cube of perimeter = 2 3 x 6 12 . 50 cubes = 50 x 2 3 x 6 12 = 400 x 6 12 = 400 times diameter of the orbit of the moon, = diameter of the orbit of the earth. Cube of twice side = diameter of the orbit of the moon. 150 cubes = Mercury. 150 2 cubes = Belus. VOL. I. Q 226 THE LOST SOLAR SYSTEM DISCOVERED. Cylinder having height = diameter of base s= 1296 = 6 4 will =15 circumference Sphere = 10 Cone =5 Sphere, diameter = side of base of pyramid = 648 will =. -i^- = -J circumference = height x area base of Cephrenes' pyramid. The dimensions of Cheops' pyramid will be, side of base = i ( 6 ) 4 =! ( 6 ) 4 > and height = | side of base. Or, height = side of base less 1 stade, = \ (6) 4 -243 units, = i(6) 4 -3*. 3 5 = 243 (3 x 342) 3 = distance of the moon ; (3 x 432) 3 = diameter of orbit of the moon. 2, 3, 4 are Babylonian numbers derived from 3 5 . 3 5 = 243 read backwards = 342 (3 x 342, &c.) 3 = 1028 3 = distance of the moon. 3 5 = 243 first figure placed last =432 (3 x 432) 3 = 1296 3 = 6 12 = diameter of orbit of the moon. 3 5 read backwards, tripled, and cubed = distance of the moon. 3 5 first figure being placed last, tripled, and cubed = dia- meter of the orbit of the moon. 3 5 = 243 (2 x 243) 3 = circumference of the earth. 3 5 doubled and cubed = circumference of the earth. In English measures we make the height of Cheops' pyra- mid 468 feet and side of base 749 Davidson makes the height 461 and side of base - 746. It appears that Davidson in 1763 took the height of this pyramid, first, by measuring the steps or ranges of stone, and subsequently with a theodolite, and both accounts PYRAMID OF CHEOPS. 227 agreed. He found the number of ranges to be 206, and the platform on the top composed of six stones. Colonel Coutelle was with the army of Napoleon in 1801, and officially employed with M. Le Pere, an architect, at the pyramids of Gizeh. By measuring the height of each step, and including the two ruined tiers at the top, they made the whole height to the platform = 139*117 metres, which =456*4 feet English. If stade =456*625 feet English. The trigonometrical survey agreed with this measured height. This will make the height to the platform = If stade, and height to apex = 10 plethrons = If stade. Side of base = 16 plethrons = 2-f stades. So height : side of base :: 10 I 16 :: 5 : 8. Height =f side of base. So we shall have 10 plethrons less If stade, or 468-33 feet less 456-625 feet =11*7 feet for the completion of the pyramid to its apex, which, according to Greaves, in 1638, wanted about 9 feet. Height to platform = If stade. Height to apex = f side of base. Height to platform of the teocalli of Cholula = f stade. Pyramid of Cheops = \ circumference. Teocalli of Cholula = 1 circumference. The pyramid of Cheops is terraced, and has a platform at the top like the Mexican teocalli. The estimate of the teocalli of Cholula has since been modified, and the external pyramid made = -^ distance of the moon. Herodotus says that all the stones composing the pyramid of Cheops are 30 feet long, well squared, and joined with the greatest exactness, rising on the outside by a gradual ascent, which some call stairs, others little altars. No mention is here made of the breadth or depth of these stones. Now if we take 30 feet as the average of the greatest perimeter of these squared stones, these 30 feet will Q 2 228 THE LOST SOLAR SYSTEM DISCOVERED. equal 14-05 feet English, which will allow 4 feet for the length of each of the two greatest sides, and 3 feet for each of the two less sides; since a Babylonian foot equals 5*62 inches, which is less than half a foot English. Again, if 30 feet be taken for half the greatest perimeter, or for the length of the two adjacent sides of the largest stones, this will allow 9 by 5 feet English for these two sides. Coutelle says the stones of the Great Pyramid and those of the second, belonging to the outer covering, rarely exceed 9 feet in length and 6^ in breadth. The height of the steps do not decrease regularly, as we ascend the pyramid, but steps of greater height are sometimes interposed between steps of less height ; but, he adds, the same level and the same perfectly horizontal lines appear in all the faces. The height of the steps decreases from the lowest to the highest ; the greatest height being 4'628 feet, and the least 1'686 feet. The mean width of the steps is a little more than 1 foot 9 inches, which is deduced from the length of the base, and the side of the platform at the top, which in its present state is 32 feet 8 inches. Greaves makes the side of the platform 13*28 feet, and says it is not covered with one or three massy stones, but with nine, besides two that are wanting at the angles. Pliny makes the breadth at the top to be 25 feet. Diodorus makes it but 9 feet. The measurement of one of the larger stones of the pyramid by Coutelle = 9 feet by 6|- feet. Herodotus makes the length of one of these stones =30 feet, which = 14 feet English. If that represented the length and breadth, and were written equal to 9 + 5 or 9 by 5 feet, then the dimen- sions of Herodotus and Coutelle would agree. For 18 by 12 feet of Herodotus would nearly =9 by 6|- of Coutelle's feet. The number of steps assigned to this pyramid by different authorities vary, according to Greaves, from 260 to 210; who says, that which by experience and by a diligent calcu- lation I and two others found is this, that the number of PYRAMID OF CHEOPS. 229 degrees from the bottom to the top is 207, though one of them in descending reckoned 208. The least and greatest distances of the sun from the earth has been estimated at 204 and 210 semi-diameters of the sun, the mean of which = 207 semi- diameters. The entrance to the pyramid of Cheops is on the north side, and said to be about 47 feet above the base, and on a level with the fifteenth step, reckoning from the foundation ; 1 plethron = 46f feet. Greaves says the entrance has ex- actly a breadth of S^^LSL. English feet. Entrance about 47 feet above the base =41 units (10x40-8, &c.) 3 = 408 3 , &c. = f circumference (10 x 10 x 40-8, &c.) 3 = sjLoo =60 4-J- cubes of 100 times height = 2700 circumference = distance of Venus 60 cubes = Saturn (2 x 10 x 10 x 40-8, &c.) 3 = 600 x 2 3 = 4800 circumference 15 cubes of 200 times height = 72000 = distance of Uranus 45 cubes = Belus or 30 cubes = diameter of the orbit of Uranus 90 cubes = Belus 9 cubes of 100 times height = diameter of the orbit of Venus. Height to entrance = 40 '8, &c, units Height to apex = 407. If height to entrance = -^ height to apex, then cube of 10 times height to entrance = cube of height to apex = -^ distance of the moon = -J- cube of side of base. Breadth of entrance = 3-463 feet = 2-993 = 3 units 3 5 = 243 242, &c. 3 = -J- circumference (2 x 242, &c.) 3 = 1 or (2 x 3 5 ) 3 = cube of (2 x 3 5 ) = circumference in units Q 3 230 THE LOST SOLAR SYSTEM DISCOVERED. Twice breadth =2x3 = 6 units 6 4 =1296 1296 3 = diameter of the orbit of the moon ( 6 4)3 = 6 i 2 = or cube of (2 x3) 4 = and cube of (2 x 3 5 ) = circumference of the earth. Again, 3 units x 243 = 3 stades 3 5 = 243 2 x3 5 = 2x 243 = 486 684 2 = circumference in stades or 3 5 doubled, transposed, and squared = 684 2 = circum- ference of the earth in stades 243 x 684 2 = circumference in units or 3 5 x (3 5 doubled, transposed, and squared) = circumference in units. Cube of 6 4 = diameter of the orbit of the Moon 150 cubes of 6 4 = Mercury 400 cubes of 6 4 = Earth 150 2 cubes of 6 4 = Belus Sphere diameter of 6 4 = 10 circumference. Writers since Greaves, in 1638, make the number of steps as follow : 1655. Thevenot - - 208 1692. Maillet - 208 1711. Pere Sicard - - 220 1743. Pococke - 212 1763. Davidson - 206 1799. Denon - 208 The Leaning Tower of Pisa is. inclined more than 14 feet from the perpendicular. It is built of marble and granite, and has 8 stories, formed by arches, supported by 207 pil- lars, and divided by cornices. The different stated heights are from 150 to 187 feet. Here arc associated the 8 stories of the tower of Babylon. The 207 pillars, the same number as the terraces of the Great Pyramid, and the height of a teocalli = -| stade =175 feet. PYRAMID OF CHEOPS. 231 This tower was built A. D. 1174: so these associations have only been preserved by repeated copies, like the minarets, which are only imperfect copies of the circular obelisk, be- cause they are devoid of the principle by which the obelisk is constructed. The number 1600, which represents the side of the base of the Great Pyramid in feet, is also associated with the number of pillars in a Ceylon temple, said to have had 9 stories none now exist but 1600 stone pillars, upon which the building was erected, remain. They form a perfect square, each side about 200 feet, containing 40 pillars; around which temple are immense solid domes, having alti- tudes equal to their greatest diameter. They are for the most part surmounted by spiral cones, that, in some mea- sure, relieve the vastness and massiveness of their gigantic proportions. Like the pyramids of Egypt, their simplicity and solidity of construction have defied the ravages of time. The solid content of the largest of them has been esti- mated to exceed 450,000 cubic yards. Its greatest diameter and altitude are equal, and measure 270 feet. From this description these large domes seem to corre- spond with the solid generated by the hyperbolic reciprocal curve of contrary flexure, which has an altitude equal the diameter of the base ; and the dome terminates in a spiral curve of contrary flexure to the body of the dome. 1 stade = 281 feet. Side of square = 200 feet =173 units 175 3 , &c. = yV distance of moon = -jL cube of Cephrenes (10 x 175, &c.) 3 = ^MHP- = 20 20 cubes of 10 times side =400 times distance of the moon = distance of the earth (2 x 10 x 175, &c.) 3 = 20 x 2 3 = 1600 Cube of 20 times side or of 5 times perimeter = 1600 times distance of the moon = twice the diameter of the orbit of the earth Q 4 232 THE LOST SOLAR SYSTEM DISCOVERED. 10 cubes of 10 times side = 200 distance of the moon. 20 cubes = 400 = distance of the earth. Vyse's Measurements of the Pyramid of Cheops. Feet. Former base - 764 Present base - 746 Present perpendicular height - 450-9 Present height inclined - 568*3 Former height inclined 611 Perpendicular height by casing stones - 480-9 Having calculated the terraced pyramid of Cheops = \ circumference, a plain pyramid having the sides cased, and side of base and height = the former base and height by Vyse's measurement will = ^ distance of moon. Former base = 764 feet = 660 units Former height = 480-9 =416 Then height x area base ~ 413, &c. x 662 2 = distance of moon Pyramid = i 1 8 Thus it appears that the pyramid of Cheops in its present state may be regarded as a teocalli or terraced pyramid having the content = \ circumference of the earth. But if the terraced pyramid were completely cased on all sides, the plain pyramid would = -^ distance of the moon. Vyse's former base = 764 feet former height = 480-9 present base = 746 present height = 450-9 . . former base .* present base : : former height : height to apex of present pyramid or 764 : 746 :: 480-9 : 469-6 feet 469-6 feet = 406 units PYRAMID OF CHEOPS. 233 According to our calculation height to apex = 406 units side of base = 648 So that the completely cased pyramid would be similar to the terraced pyramid if completed to the apex. Height of each pyramid will = -| side of base. Cube of height = 414 3 = f circumference Cube of 2 == 5 Cube of 4 40 Cube of 4 times height = 40 times circumference Cube of side of base = 662 3 = -f^ distance of the moon Cube of side of base of terraced pyramid = i Cubes will be as $ : |- :: 32 I 30 Former inclined side = 611 feet = 528 units 528 3 = -f^ distance of the moon (5 x 528) 3 = ^L. x 5 3 = ^0 = 20 20 cubes of 5 times inclined side = 400 times distance of the moon = distance of the earth (10 x 528) 3 = *& = .S.JJQL distance of the moon 5 cubes of 10 times inclined side = 800 times the distance of the moon = diameter of the orbit of the earth Area of base of cased pyramid = 662 2 units terraced . = 648 2 Vyse makes the area of the A. K. P. former base =131 22 present base = 12 3 3 Terraced pyramid height = 405 units side of base = 648 Cased pyramid height =414 side of base = 662 234 THE LOST SOLAR SYSTEM DISCOVERED. In both pyramids, height = f side of base. Their contents are as * circumference I vV distance of 2 moon. :: i circumference : -y^- circumference :: 18 : 19-1 Greaves, in describing the interior of the Great Pyramid, says, this gallery or corridor, or whatever else I may call it, is built of white and polished marble, which is very evenly cut in spacious squares or tables. Of such materials as is the pavement, such is the roof, and such are the side walls that flank it; the knitting of the joints is so close, that they are scarcely discernible to a curious eye ; and that which adds grace to the whole structure, though it makes the passage the more slippery and difficult, is the acclivity and rising to the ascent. The height of this gallery is 26 feet, the breadth is 6-ftf^Q feet ; of which 3 T 4 ^- feet are to be allowed for the way in the midst, which is set and bounded on both sides with two banks (like benches) of sleek and polished stone ; each of these hath 1 ? l 7 feet in breadth, and as much in depth. Breadth of gallery = 6 -8 7 feet =5-94 units if = 6 The way in the middle = -J- breadth of gallery = ^6 = 3 units, 3 5 =243, 243 transposed, doubled and squared, = 684 2 , 243 x 6 84 2 = circumference of earth. Thus 3 5 x (3 5 transposed, doubled and squared) = circum- ference. Or (2x3 5 ) 3 =(2x243) 3 , and (2 x 242, &c.) 3 = circumference. So 2 3 x 3 15 = circumference nearly. Breadth of gallery = 6 units. PYRAMID OF CHEOPS. 235 Breadth to the power of 4 times 3 = (6 4 ) 3 = 6 12 = diameter of the orbit of moon, or (2 x 3) 12 = twice distance of moon. Height of gallery = 26 feet = 22-41 units, (10 x 22-4, &c.) 3 = 224 3 , &G. = ^ circumference (10 x 10 x 22-4, &c.) 3 = JL ^ Q -= 100. Cube of 100 times height = 100 times circumference. Sphere diameter 6 4 =10 circumference. This gallery is of the hyperbolic order. See^. 42, hyper- bolic areas. Greaves, describing what is now commonly called the King's Chamber, containing a granite sarcophagus, says the length of this chamber on the south side, most accurately taken at the joint where the first and second row of stones meet, is 34-^fo English feet. The breadth of the west side, at the joint or line where the first and second row of stones meet, is 17-J^- feet. The height is 19^ feet. " These proportions of the chamber, and those of the length and breadth of the hollow part of the tomb, were taken by me with as much exactness as it was possible to do ; which I did so much the more diligently, as judging this to be the fittest place for fixing the measures for posterity ; a thing which hath been much desired by learned men, but the manner how it might be exactly done hath been thought of by none." Chamber. Length 34-38 feet =29*72 units. Breadth 17-19 =14-86 Height 19-5 =16-85 (10 x 29-6, &c.) 3 = 296 3 , &c.= T ff distance of moon (5 x 10 x 29-6, &c.) 3 = roo~o x 53 =finro == 3 (5 x 5 x 10 x 29-6, &c.) 3 = 3 x 5 3 = 375 236 THE LOST SOLAR SYSTEM DISCOVERED. 10 cubes of 250 times length = 3750 distance of Moon = distance of Saturn 20 cubes = Uranus 60 cubes =Belus. Breadth = 14-86 units. (10 x 14-8, &c.) 3 = 148 3 , &c. = T -oVo distance of moon (10 x 10 x 14-8, &c.) 3 = ii^L=3 Height= 16-85 units. (10 x 16-5, &c.) 3 = 165 3 , &c.= T fc circumference (10 x 10 x 16-5, &c.) 3 = -*fl$- =40 (2 x 10 x 10 x 16-5, &c.) 3 = 40 x 2 3 = 320 12 cubes of 200 times height = 320 x 12 =3840 circum- ference = distance of Earth. Cube of 100 times breadth = 3 distance of Moon. Pyramid =i cube = distance of Moon. 50 cubes =150 distance of Moon. = distance of Mercury. Content = 29-6, &c. x 14-8, &c. x 16'5, &c. = 7290, 4 times content = 4 x 7290=29160, and distance of Belus = about 29160 3 = the cube of 4 times content of chamber = the cube of Babylon = the cube of 120 stades. In the " Library of Entertaining Knowledge," the Height of this chamber = 19-214 feet Length on south side =34-348 Width on west side =17-056 Cube of 4 times content I cube of 5 times content :: 4 3 : 5 3 ::64 : 125 :: 1 : 2 nearly. Thus cube of 4 times content = distance of Belus = cube of Babylon ; cube of 5 times content = twice distance of Belus = twice cube of Babylon = distance of Ninus = cube of Nineveh. PYRAMID OF CHEOPS. 237 Length -f breadth + height = 29-6, &c. + 14-8, &c. + 16-5, &c. = 61 units (600 x 61) 3 = 36600 3 = diameter of orbit of Belus, cube of 600 times (sum of 2 sides + height) = 36600 3 = diameter of orbit of Belus = distance of Ninus = cube of Nineveh. There is a very small temple at Philse, by some supposed to be Grecian. There is only a single chamber in it, about 11^ feet long by 8 wide, with a doorway at each end, opposite to one another. 11-5 by 8 feet = 10 by 7 units. 10-2 3 = 12 seconds. 7'1 3 = 4 Cubes of the sides are as 1 I 3. Hamilton found at Gau Kebir, at the furthest extremity of the temple, a monolith chamber of the same character. It had a pyramidal top, and measured 12 feet in height and 9 in width at the base. Within were sculptured hawks and foxes, with priests presenting offerings to them, and the same orna- ments on the doorway as are seen on the entrances of the great temples. 12 feet =10-3 units, 9 = 7-7 If the base be a square, content will = 7*7 2 x 10*3 = 610 units, and 610 3 = twice circumference. 7 seconds = 613-9 units, I" = 87-7 =101-4 feet English, V" = 1-461,, = 1-69 5'" = 5x1-69 = 8-45 12 cubits = 8-43 113689008 units = circumference = 360 degrees. 315802 =1 degree. 5263 =1 minute. 87*7 =1 second. 1-461 = V" 238 THE LOST SOLAE SYSTEM DISCOVERED. Denon found granite monoliths of small dimensions at Philas, both of them in the great temple, and placed respec- tively at the extremity of the two adjoining sanctuaries. The dimensions of one of them are 6 feet 9 inches in height, 2 feet 8 inches in width, and 2 feet 5 inches deep, French measure. Not knowing the exact proportion between the French and English foot, but taking the French to exceed the English b 7 TO P art > Dimensions in English feet : = 6-75 2-66 2-41 = 5-84 2-33 2-1 units A= _^9 jll J_ = 6-13 2-44 2-2. Content = 6-13 x 2-44 x 2-2 = 33, and about 3 3 '2 9 = diameter of the orbit of Belus = distance of Ninus. 30-7 3 , &c. = 29160 units=120 stades, = side of Babylon. 33-2 3 , &c. = 36450 units=150 stades, = side of Nineveh. Thus content raised to the power of 3 times 3 = cube of Nineveh = distance of Ninus. Three winged globes, one above another, decorate the architrave of the doorway. The frieze and cornice are orna- mented with a series of serpents erect. The holes in which the hinges of the door were fastened are still visible. The winged globe, flanked on each side by the erect serpent, usually ornaments the frieze of the doorway of an Egyptian temple. The cube of the dimensions of these temples denote celestial distances. Hence the winged globe denotes the third power. Three winged globes denote three times the third, or the ninth power. " From the top to the bottom of this chamber (of Cheops) are six ranges of stone, all of which being respectively sized to PYRAMID OF CHEOPS. 239 Ian equal height, very gracefully in one and the same altitude run round the room. The stones which cover this place are of a strange and stupendous length, like so many huge beams lying flat and traversing the room, and withal supporting that infinite mass and weight of the pyramid above. Of these there are nine, which cover the roof; two of them are less by half in breadth than the rest ; the one at the east, the other at the west." "Within this glorious room," says Greaves, "as within some consecrated oratory, stands the monument of Cheops or Chemmis, of one piece of marble, hollow within and uncovered at the top, and sounding like a bell. This tomb is cut smooth and plain, without any sculpture or engraving. The exterior superficies of it contains in length 7 feet 3^ inches ; in depth it is 3 feet 3f inches, and the same in breadth. The hollow part within is in length, on the west side, 6 ^o 8 ^ feet. In breadth, at the north end, 2-f^ feet. The depth is 2^^- feet." Sarcophagus outside : length 7 ft. 3 in. =6-3 units depth 3 ft. 3| in. = 2-863 breadth =2-863 Sum =12-026 units Content = 51-53 units (10 x 51-4) 3 = 514 3 =! distance of the Moon . (2xlOx51-4) 3 = -f=l Cube of 20 times content = 150 cubes = Mercury 150 2 cubes Belus depth = breadth = 2-863 units 10x2-86 = 28-6 Distance of Neptune = 28 -6 9 or 10 times breadth to the power of 3 times 3 = distance of Neptune. Length = 6-3 units (100 x 6-32) 3 = 632 3 = 2 distance of the Moon (3 x 100 x 6'32) 3 = V x3 3 = 60 240 THE LOST SOLAR SYSTEM DISCOVERED. 5 cubes of 300 times length = 3.00 times distance of the Moon = diameter of the orbit of Mercury. 6'4 2 , &c. = 41, &c. i(6-4, &c,) 2 = 20-5, &c. and 20-5 9 , &c. = distance of Mars (1(6-4, &c.) 2 ) 9 = 20-5 9 , &c. = ^ square of length to the power of 3 times 3 = distance of Mars. Depth = 2-863 units 2-87 3 = 23-5, &c. and 23'5 9 , &c. = distance of Jupiter (2-87 3 ) 9 = 2-87 27 = 2-87 3x3x3 = 23 5 9 , &c. Depth to the power of 3 times 3 = 23-5 9 , &c. = distance of Jupiter. Sum of length, depth, and breadth = 12-026 units (\ 12) 12 = 6 12 = diameter of the orbit of the Moon. Depth x breadth =2-86 x 2-86 = 8-17, &c. 100x8-17 = 817 and 816 3 = i distance of the Moon Sarcophagus inside : length 6-488 feet = 5*61 units breadth 2-218 feet = 1-917 depth 2-86 feet = 2-473 content =26-595 units (10 x 26-7) 3 = 267 3 = ^-o distance of Moon (10 x 2 x 10 x 26-7) 3 =^HHr>-= 14 2 cubes of 200 times content = 280 distance of Moon = distance of Venus Length + breadth + depth = 5'61 + 1-917 + 2-473 = 10 units length = 5-61 (100 x 5-65, &c.) 3 = 565 3 , &c.=-J- distance of Moon breadth = 1-917 10x1-917 = 19-17 and 19 9 = distance of Venus SARCOPHAGUS. 241 lOOx 1-917 = 191-7 and 189 3 , &c. = -/$ circumference, depth = 2-473 100x2-473= 247-3 and 24 7 3 , &c. = -^ circumference. Cube of 10 times external content : cube of 10 times in- ternal content :: ' -j-J-g- distance of Moon :: 400 : 56 :: 50 : 7. Depth = 2-473 units 2-48 3 , &c. = 15-35 2x(2-48, &c.) 3 = 30-7 and 30 -7 9 = distance of Belus (2 x (2-48, &c.) 3 ) 9 = 30-7 9 . Twice cube of depth to the power of 3 times 3 = 30 -7 9 = distance of Belus. The measurements of the sarcophagus made by Greaves differ from those lately made by Vyse. The latter makes the external length 7 ft. 6 in. = 6-51 units breadth 3 ft. 3 in. =2-81 height 3 ft. 5 in. = 2-95 Internal length 6ft. 6 in. = 5-62 units breadth 2 ft. 2-J- in. = 1-908 depth 2 ft. 10| in - = 2 *48 external length = 6-51 3x6-51 = 19-53 distance of earth = 19-5 9 , &c. 3 times length to the power of 3 times 3 = distance of earth. External content =6-51 x 2-81 x 2-95 = 53-96 i=26-98 distance of Uranus =26'9 9 , &c. Half content to the power of 3 times 3 = distance of Uranus. Davison has since discovered a chamber immediately over VOL. I. R 242 THE LOST SOLAR SYSTEM DISCOVERED. the king's chamber, which is now called Davison's chamber. It is reached by mounting, with the help of a ladder, to a hole at the top of the upper part of the high ascending gallery. The stones which form the ceiling of the king's chamber form also the floor of the upper chamber, but the room is four feet longer than that below. More recently Caviglia has discovered a large chamber cut in the rock, and under the centre of the pyramid. The dimensions are not minutely given. The chamber is stated to be about 66 feet by 27, with a flat root and very irregular floor. 27 feet =23-34 units 66 feet =57- 23 -5 9 , &c. = distance of Jupiter -1-57 = 28-65 and 28-6 9 = distance of Neptune. Wilkinson observes, no doubt it was by the causeways that stories were carried on sledges to the pyramids ; that of the Great Pyramid is described by Herodotus as 5 stades long, 10 orgyes broad, 8 orgyes high, of polished stones, adorned with figures of animals (hieroglyphics), and it took no less than ten years to complete it. Though the size of the stade is uncertain, we may take an average of 610 feet, which will require this causeway to have been 3050 feet in length (a measurement agreeing very well with the 1000 yards of Pococke, though we can now no longer trace it for more than 1424 feet, the rest being buried by the alluvial deposite of the inundation). Its present breadth is only 32 feet, the outer faces having fallen ; but the height, 85, exceeds that given by Herodotus, and it is evident, from the actual height of the hill, from 80 to 85 feet, to whose surface the causeway actually reached, and from his allowing 100 feet from the plain to the top of the hill, that the expression 8 orgyes (48 feet) is an oversight either of the historian or his copyist. It was repaired by the caliphs and Memlook kings, who made use of the same causeway to carry back to the CAUSEWAY. 243 " Arabian shore " those blocks that had before cost so much time and labour to transport from the mountains; and several of the finest buildings of the capital were constructed with the stones of this quarried pyramid. The length of the causeway of Herodotus = 5 stades =1405 feet The breadth *=10orgyes = 28-1 The height = 8 = 22-48 The length of the causeway of Wilkinson = 1424 feet Breadth = 32 Height = 85 or 30 orgyes. The causeway, which formed the wonderful approach to the pyramidal temples, was 5 stades in length (the line of measure so frequently associated with the sacred structures in the four quarters of the world). As 5 stades is so frequently mentioned, it may be as well to give an instance of a granite structure of nearly that length. Waterloo bridge, over the Thames, has nine arches, is built entirely of granite, and is 1280 feet in length. The breadth of the carriage road or causeway is 28 feet. The parapet, or foot walk on each side of the carriage road, is 7 feet in breadth. 5 stades =1215 units i = 607-5 60 1 3 = cube of Cephrenes=|- distance of Moon 12023 ^ = s 610 3 = 2 circumference 1220 3 =16 5 stades =1215 units 1424 feet=1231 123 3 , &c. =- 1 o circumference (10 x 123, &c.) 3 = ^^ = -ijyi (6x 10 x 123, &c.) 3 = ^4p-x6 3 R 2 244 THE LOST SOLAR SYSTEM DISCOVERED. 10 cubes of 6 times length = 36000 circumference = distance of Saturn 20 cubes = Uranus 60 cubes = Belus. Should the length have equalled originally 1296 units = 5^ stades. Then cube of length=1296 3 = 6 12 = diameter of orbit of Moon. Sphere, diameter 1296=10 circumference. 1 stade = 243 units and 242 3 , &c.=| circumference (2x242,&c.) 3 =l (4x242, &c.) 3 = 8 5 stades =12 15 units 5 stades + 5 units =1220 units. Cube of (5 stades + 5 units)=1220 3 = 16 circumference. Cube of 5 times (5 stades + 5 units) = 16 x 5 3 = 2000 circumference 3 x 5 x (5 stades + 5 units) = 2000 x 3 3 = 54000 circumference 4 cubes of 15 times (5 stades + 5 units) = 216000 circumference = distance of Belus 2 x 3 x 5 x (5 stades + 5 units) = 54000 x 2 3 = 432000 circumference = diameter of orbit of Belus. Cube of 30 times (5 stades -f 5 units) = diameter of orbit of Belus = distance of Ninus. According to Ctesias, the bridge over the Euphrates at Babylon was 5 stades in length. Strabo says the Euphrates at Babylon was a stade in breadth. It is stated in the "Athenaeum " that the blocks of which the pyramid of Cheops is composed are roughly squared, but built in regular courses, varying from 2 feet 2 inches to 4 feet 10 inches in thickness, the joints being properly CONSTRUCTION OF THE PYRAMID. 245 broken throughout. The stone used for casing the exterior, and for the lining of the chambers and passages, were ob- tained from the Gebel Mokattam, on the Arabian side of the valley of the Nile ; it is a compact limestone, called by geo- logists swine-stone, or stink-stone, from emitting, when struck, a fetid odour, whereas the rocks on the Libyan side of the valley, where the pyramids stand, are of a loose granu- lated texture, abounding with marine fossils, and, conse- quently, unfit for fine work, and liable to decay. The mortar used for the casing and for lining the passages was composed entirely of lime ; but that in the body of the pyramid was compounded of ground red brick, gravel, Nile-earth, and crushed granite, or of calcareous stone and lime, and in some places a grout, or liquid mortar, of desert sand and gravel only has been used. It is worthy of especial notice that the joints of the casing-stones, which were discovered at the base of the northern front, as also in the passages, are so fine as scarcely to be perceptible. The casing-stones, roughly cut out to the required angle, were built "in horizontal layers, corresponding with the courses of the pyramid itself, and afterwards finished, as to their outer surface, according to the usual practice of the ancients. In order to insure the stability of the superstructure, the rock was levelled to a flat bed, and part of the rock was stopped up in horizontal beds, agreeing in thickness with the courses of the artificial work. The plain on which the pyramids at Gizeh stand is a dry, barren, irregular surface. According to Jomard the ele- vation of the base of the foundation stone, let into the solid rock, at the north-east end of the Great Pyramid, is 140 feet above the superior cubit of the Nilometer at Rouda; nearly 130 feet above the valley, and the mean ele- vation of the floods (from the year 1798 to 1801); and nearly 164 above the mean level of the low state of the Nile for the same period. 140-5 feet English = \ stade = 300 feet of Herodotus, who states that the pyramids of Cheops and Cephrenes are of equal height, and stand on the same hill, which is about 100 feet high. R 3 246 THE LOST SOLAR SYSTEM DISCOVERED. On this platform of rock stand the massive pyramids, monuments of the skill of man and the antiquity of science, temples of a remote epoch, where man adored the visible symbol of nature's universal law, and through that the invi- sible God of creation. Here the pyramid of Cheops indicates the -J- circumference of the earth, and the \ diameter of the earth's orbit. Its towering summit may be supposed to reach the heavens, and the pyramid itself to represent the law of the time of a body gravitating from the earth to the sun. The solid hyperbolic temple the Shoemadoo at Pegu represents the law of velocity corresponding to this law of the time. These two symbols of the laws of gravitation that pervade the universe resemble the close alliance of Osiris and Isis, husband and wife, brother and sister, the two ancient deities said to comprehend all nature. On the statue of the goddess were inscribed these words : " I am all that has been, that shall be, and none among mortals has raised my veil." The Brahmins say the gods are merely the reflecting mirrors of the divine powers, and finally of God himself. The pyramid may be supposed to reach the heavens. So it was by building pyramids that the giants of old were said, figuratively, to have scaled the heavens. L'Abbe de Binos (1777), in his letters addressed to Ma- dame Elizabeth of France, mentions that the pyramids of Egypt are supposed by some to be the tombs of the ancient kings ; that they are called by others the mountains of Pha- raoh; that the poets have described them as rocks heaped one upon the other by the Titans, in order to scale Olympus. The Abbe ascended the Great Pyramid, and found the top of it about twelve feet square ; and upon it he observed six large stones, arranged in the form of an L, which he was told signified a hieroglyphic. The pyramids may.be regarded as scientific and religious monuments. The great pyramid of Cheops may have been both a temple and fortress, like the teocalli of Mexitli, or, like BURMESE TEMPLES. 247 the great teocalli of Teotihuacan, a temple of the Sun, before which the glorious orb of day may have been worshipped as an emblem of God, when he rose above the eastern range of hills between the Nile and the Red Sea, then passing to the west till he set beyond the Libyan desert, a region of deso- lation and aridity, extending from the pyramids, through the Sahara, to the f6 Sea of Darkness," the distant Atlantic Ocean. De Sacy has endeavoured to trace the origin of the word pyramid, not in the Greek language, but in the primitive Egyptian language. The radical term signifies something sacred, the approach to which is forbidden to the vulgar. The worship of the planets, says Jablonski, formed a re- markable feature in the early religion of Egypt, but in pro- cess of time it fell into desuetude. The Burmese hyperbolic temples, like the Egyptian and Mexican pyramidal temples, were most probably originally dedicated to the worship of the heavenly bodies. The Persian poet, Firdausi, represents them as "pure in faith, who, while worshipping one supreme God, contemplate in sacred flame the symbol of divine light." The fire- worship- pers abhorred alike the use of images and the worship of temples ; they regarded fire as the symbol of God. The Sabseans regarded the pyramidal and hyperbolic tem- ples and the obelisk as the symbols of divinity. Thus, a simple quadrilateral monument, without a cypher, has transmitted to the present age a proof of the scientific acquirements of an epoch that long preceded the earliest dawn of European civilisation. The pyramidal, like the obeliscal records of science, monuments combining the phy- sical and intellectual power of man, have endured ages after all traditional and written records have perished. The laws formed by the Creator for the government of the celestial bodies had become, by the uniformity of their action, known to man, after a lengthened series of astrono- mical observations. These laws, when symbolised in geome- trical forms, became objects of reverence, and the invisible R 4 248 THE LOST SOLAR SYSTEM DISCOVERED. Creator was worshipped through the visible type of his laws. Such appears to have been the origin and mode of worship of the ancient Sabseans. Yet, however remote the period of its origin might have been, and however generally it might have been adopted at an early epoch, at the present time it embraces very few votaries in comparison with those it formerly numbered. The obelisk and pyramid are symbolical of the laws that govern the heavens. Religion taught the people to kneel before these sublime monuments, to look with reverential awe on heaven's law, and worship heaven's God. The Egyptians of a later period also believed in the unity of the Deity ; but when they spoke of his attributes they personified them separately, and, in process of time, fell into the natural course of idolatry. They mingled truth with error, and, as is usually the case, truth was obscured and error prevailed. The Egyptians believed in the immortality of the soul. They adopted the doctrine of the transmigration of the souls of the wicked, through various animals, for a period of 3000 years, or " the circle of necessity," to expiate the sins of the flesh : whereas the souls of the just were absorbed into the Deity ; they became part of Osiris, and their mummies were invested with the emblems of the gods, to signify that their soul had become a part of the divine essence. Champollion Figeac thus expresses his views of the Egyptian theocracy : "A theocracy, or a government of priests, was the first known to the Egyptians; and it is necessary to give this word priests the acceptation that it bore in remote times, when the ministers of religion were also the ministers of science and knowledge ; so that they united in their own persons two of the noblest missions with which man can be invested, the worship of the Deity, and the cultivation of intelligence." " This theocracy was necessarily despotic. On the other hand, with regard to despotism (we add these reflections to reassure our readers, too ready to take alarm at the social MYTHOLOGY. 249 condition of the early Egyptians), there are so many dif- ferent kinds of despotism, that the Egyptians had to accept one of them, as an unavoidable condition. In fact, there is in a theocratic government the chance of religious despotism ; in an aristocracy, or oligarchy, the chance of a feudal des- potism ; in a republic, the chance of a democratic despotism everywhere a chance of oppression. The relative good will be where these several chances are most limited. And, with respect to the form of government best adapted to the social happiness of man, opinions are as varied as are the countries and human races on the earth. That institution which is admirably suited to Europeans may be odious and deleterious to Orientals." The early mythology of -the ancient nations would appear to have centred in the divine attributes and operations, which created, animated, and preserved the celestial and ter- restrial systems, this mythology being represented under an embodied form, which, not being generally understood, led eventually to the introduction of idolatrous practices. Thus superstition and darkness spread over these countries. The purity of the original faith being sullied, the whole mythology was misunderstood, and its tenets and symbols misrepresented and perverted. The primeval theology peculiar to those early ages may be deemed the spiritual. The less refined system prevalent in later times, and from which most of the writers, both ancient and modern, have drawn their inferences, may be termed the physical. The spiritual, which may be regarded as arcanic, comprised the more abstruse stores of ancient wisdom, and was revealed to the initiated only. The phy- sical, being rendered palpable to the senses, was adapted to the capacity of the unlearned and unreflecting. Herodotus attributes the building of the three pyramids at Gizeh to Cheops, Cephrenes, and Mycerinus. He says, " They informed me that Cephrenes reigned 56 years, and that the Egyptians, having been oppressed by building the pyramids, and all manner of calamities, for 160 years, during all which time the temples were never opened, had con- 250 THE LOST SOLAR SYSTEM DISCOVERED. ceived so great an aversion to the memory of the two kings, that no Egyptian will mention their names, but they always attribute their pyramids to one Philition, a shepherd who kept his cattle in those parts. They said, also, that after the death of Gephrenes, Mycerinus, the son of Cheops, became king; and, disapproving the conduct of his father, opened the temples, and permitted the people, who were reduced to the last extremities, to apply themselves to their own affairs, and to sacrifice as in preceding times." Since Mycerinus permitted the people to sacrifice as in pre- ceding times, it follows that sacrifice was not practised during the two preceding reigns at least, since the Egyptians had been oppressed for 160 years. Cheops reigned 50 years. The religious rites of Boodha are performed at this day before the solid hyperbolic temples, where sacrifice is never practised. We shall not stop to inquire whether Cheops and Cephrenes were, as sovereign pontiffs, innovators or reformers of the national religion; or whether they wished by closing the temples to compel the people to worship before the pyramids, teocallis and obelisks; or whether these pyramids were built by Cheops and Cephrenes, for, according to Manetho, that of Cheops was built by Suphis, 1000 years before their reigns. The builders, however, adopted the same Babylonian standard of unity in the construction of these pyramids as that used in the sacred structures of the Brahmins, Boodhists, Chaldeans, Druids, Mexicans, and Peruvians. Suphis was arrogant towards the gods ; but, when penitent he wrote the sacred book which the Egyptians value so highly. From this account of Suphis he appears also to have been a reformer of the Egyptian religion. These inquiries may be left to those conversant with hiero- glyphics and Egyptian researches, whose recent labours have thrown so much light on the manners and customs of ancient Egyptians, that, it is said, Lepsius intends to write the Court Journal of the Fourth Memphite dynasty. Wilkinson thinks that the oldest monuments of Egypt, and probably of the world, are the pyramids to the north of Mem- ARCHITECT OF THE PTRAMID. 251 -phis ; but the absence of hieroglyphics and of every trace of sculpture precludes the possibility of ascertaining the exact period of erection, or the names of their founders. " From all that can be collected on this head it appears that Suphis and his brother Sensuphis erected them about the year 2120 B. c.; and the tombs in their vicinity have been built, or cut in the rock, shortly after their completion. These present the names of very ancient kings, whom we are still unable to refer to any certain epoch, or to place in the series of dynasties." Sayuti and other Arabic writers conceive that the pyramids were erected before the Deluge, or more correct accounts of them would have existed. Jomard says that the tradition that the pyramids were antediluvian buildings only proves their great antiquity, and that nothing certain was known about them. They have been attributed to Yenephes, the fourth king of the first dynasty ; and to Sensuphis, the second king of the fourth Memphite race. According to Lepsius, the pyramids of Gizeh were built under the fourth dynasty of Manetho, 4000 B. c. Vyse found Shoopho, whom the Greeks called Suphis the First, in the quarriers' marks in the new chamber of the Great Pyramid, scored in red ochre, in hieroglyphics, on the rough stones. (( The tombs around the pyramids," remarks Gliddon, " afford us abundance of sculptural and pictorial illustrations of manners and customs, and attest the height to which civilisation had attained in the reign of Shoopho ; while, in one of them, a hieroglyphical legend tells us that this is e the sepulchre of Eimei, great priest of the habitations of King Shoopho.' This is probably that of the architect, according to whose plans and directions the mighty edifice near the foot of which he once reposed the largest, best-constructed, most ancient, and most durable of mausolea in the world, was built, and which, for 4000 to 5000 years after his decease, still stands an imperishable record of his skill." Shoopho's name is also found in the Thebaid as the date of a tomb at Chenaboscion. In the peninsula of Mount Sinai his name and tablets show that the copper mines of the Arabian 252 THE LOST SOLAR SYSTEM DISCOVERED. district were worked by him. Above his name the titles " Pure King and Sacred Priest " are in strict accordance with Asiatic institutions, wherein the chief generally combines in his person the attributes of temporal and spiritual dominion. His royal golden signet has recently been discovered. The sculptures of the Memphite necropolis inform us that Mem- phis once had a palace called " the abode of Shoopho." Lepsius thinks the tomb to be that of Prince Merhet, who, as he was a priest of Chufu (Cheops), named one of his sons ts Chufu-mer-nuteru," and possessed eight villages, the names of which are compounded with that of Chufu. And the position of the grave on the west side of the pyramid of Chufu, as well as the perfect identity of style in the sculptures, render it more than probable that Merhet was the son of Chufu* by which the whole representations are rendered more interesting. This prince was also " Superintendent- General of the Royal Buildings," and thus had the rank of high court architect, a great and important post in these times of mag- nificent architecture, and which we have often found under the direction of princes and members of the royal family. It is therefore to be conjectured that he also overlooked the building of the Great Pyramid. If the pyramid be regarded as typical of Osiris, " he who makes time," and the hyperbolic solid symbolic of Isis, velocity oc inversely as time then the Egyptian pyra- mid and Burmese hyperbolic temple, both being typical of gravity, may be supposed to represent Osiris and Isis, hus- band and wife, brother and sister, both of divine origin. In the great hyperbolic temple, the Shoemadoo of Pegu, is a statue of Mahasumdera, the protectress of the world ; but, when the time of general dissolution arrives, by her hand the world is to be destroyed. The obelisk, combined in the same figure (49.) with the pyramid and hyperbolic solid, is symbolical of both time and velocity at a small distance only from the surface of the earth. So the obelisk may have been regarded as Horus, the son of Osiris and Isis. Typhon assumed the form of a crocodile to avoid the ven- MYTHOLOGY. 253 geance of Horus. The crocodile we suppose to have been held sacred, from its round and tapering body resembling a circular obelisk. As the pyramid is generated from the base to the apex, the obelisk, which oc D 2 from the apex, decreases from the base to the apex ; so that at the end of the descent the pyramid is completed and the obelisk consumed. So Osiris may be said to devour his own child. For Osiris substitute Saturn, who was also Kronos or time, and we have the myth of Saturn, the son of Crelum and Terra, or Vesta devouring his own offspring. Ccelum married his own daughter Terra. Saturn succeeded Ccelum, and married his own sister Ops, Rhea, or Cybele. The ancients dedicated the cube to Cybele. The brothers of Saturn and Cybele were the Titans, Cen- timani, or hundred-handed giants with fifty heads. The height of the tower of Belus equalled the height of 50 men, or the length of 100 arms. But the pyramidal tower, like the hyperbolic solid, would represent any sup- posed distance in the heavens. So the giants may be said to have scaled the heavens. The tower contained as many cubes of unity as equalled in extent ~ of the earth's circumference. After Saturn was deposed by Jupiter, he ordained laws and civilised the people of Latium, as Osiris did the Egyp- tians. Both instructed the people in agriculture. The curve of Osiris resembles a crosier or sickle. Saturn received from his mother a scythe or sickle. The hour-glass, formed of two hollow cones or circular pyramids, is a symbol of Saturn. The marriage of Creluin and Terra is figurative of the laws of gravitation by which the earth and the heavenly bodies are mutually influenced, and the harmony of the solar system preserved during 08) = 600. If the side of square base of pyramid = 601 units, and sight x base = -f- circumference, then 5 times the cube of side of the base = 5 x 601 3 = 1085409005 units. Distance of moon = 9-55 circumference = 1085730026 mits. Hence 5 times the cube of the side of the base of the pyra- iid of Cephrenes will = 9 '55 circumference = distance of le moon from the earth. The cube of Cheops will be to the cube of Cephrenes as : 4. Distance of Mercury from the sun will =150 times the listance of the moon from the earth, =150x5 = 750 cubes )f Cephrenes, = 150 x 4 = 600 cubes of Cheops. Should one side of the base of the pyramid = 610 units, id the other side r=592 units, the cube of the greater side 610 3 = 2 circumference; the mean of the 2 sides =i ;610 + 592) = 601. The cube of the mean will = 60 1 3 = distance of the loon. A sphere having a diameter = 601 units will = circum- 'ence. If the base of pyramid be a square having a side =601 mits, and height = 392, &c., then height x base = 392 &c. x 601 2 = f circumference. Pyramid = -f^ circumference. Cube of height I cube of side of base :: 392 3 , &c. : 601 3 :: 4- distance of moon :: 5 I 18. 302 THE LOST SOLAR SYSTEM DISCOVERED. Cube of height = T 5 ^ cube of side of base. Cube of side of base = distance of moon = 6 / = 12 radii of the earth. If 12 radii divided the circumference of the earth into 12 equal parts, then pyramid would = 5 of these parts, and the cube of the side of the base would = the 12 radii. The inclined side of the pyramid will = 494 &c. units, and 494 3 &c. = distance of the moon. So cube of height I cube of inclined side :: 392 3 , &c. : 494 3 , &e. ::-&:* :: 1 I 2. Cube of inclined side I cube of side of base :: I dis- tance of the moon :: 5 I 9. Cube of inclined side = -f- cube of side of base. Pyramid = 5 times 30 degrees. Cube of side of base = -J- of 60 radii = distance of the moon. Cube of perimeter of base = 6 y 4 distance of the moon O _ 5 1 2 " 5 y i) A 4096 5> J5 5 55 ?) 5 cubes of 4 times perimeter = 4096 2 cubes =2048 and distance of Jupiter = 2045 Thus 2-|- cubes of 4 times perimeter of pyramid of Ce- phrenes = distance of Jupiter = 2 cubes of 4 times peri- meter of pyramid of Cheops. Sphere having diameter = 601 units = side of base of Cephrenes = 60 1 3 x '5236 = circumference, or sphere of Cephrenes = circumference = twice the pyramid of Cheops. Cube of Cephrenes = 60 1 3 = -J- distance of the moon Cylinder = circumference Sphere = f Cone =1 Cone of Cephrenes = pyramid of Cheops Cylinder = f circumference = height x area of the base of Cheops. PYRAMID OP CEPHRENES. 303 Cube of side of base = J- distance of the moon 5 cubes distance Cube of 4 times perimeter = 4 -*L2J 5 cubes = diameter of the orbit of Jupiter. (5 x 601) 3 = x 5 2 = 25 distance of the moon 6 cubes of 5 times side of base = 150 times distance of the moon = distance of Mercury 16 cubes = 400 times distance of the moon = distance of the earth. (10 x 601) 3 = u*f* = 200 distance of the moon 2 cubes of 10 times side of base = 400 times distance of the moon = distance of the earth 3 cubes = 600 distance of the moon distance of Mars = 604 Sphere of Cephrenes = circumference = pyramid of Cholula Cone = -i = pyramid of Cheops. Sphere, diameter 2 x 601 =8 circumference Sphere, diameter = perimeter of base = 4 x 601 = 64 cir- cumference. Cube of Cephrenes = distance of the moon Cube of Cheops = -J- Pyramid of Cephrenes = y 5 ^- circumference Pyramid of Cheops =TJ * Cylinder having height = diameter of base = 601 will = 60 1 3 x *7854 = f circumference Sphere = f Cone = -J- Cylinder having height = diameter of base = 2 x 601 will =12 circumference Sphere =8 Cone =4 Cylinder having height = diameter of base = 4 x 601 = perimeter of base will =96 circumference Sphere = 64 Cone = 32 304 THE LOST SOLAR SYSTEM DISCOVERED. 15 cylinders = 1440 circumference = distance of Mercury 40 = 3840 = distance of the earth ^ cylinder = 9'6 = distance of the moon. Distance of moon I distance of Mercury :: distance of Mer- cury : distance of Belus 1 : 150:: 150 : 150 2 1 : 15:: 15 : 15 2 xlO. distance of Mercury =15 cylinders distance of Belus = 15 2 x 10 5 cylinders having height = diameter of base = twice peri- meter of base will = 3840 circumference = distance of the earth. Height x area of base of pyramid = 393, &c. x 601 2 = f circumference, Pyramid = of -f = -JL Yyse makes the former height = 454 '3 feet = 392-8 units, former base = 707 '9 =612 If height x area base = 400, &c. x 615 2 |= ^circumference, Pyramid = ^ of -J = A The heights and sides of bases of the two pyramids will be proportionate to each other. So that if the first pyramid were completely cased, the cased pyramid might be = to the latter, supposing the bases were square, which seems doubtful. The first pyramid would be to the latter pyramid as A :: 45 :: 15 circumference, 48 16 When reference is made to the pyramid of Cephrenes, the content is supposed = circumference, and cube of one side, or of the mean of two sides of base = distance of the moon. Wilkinson makes the sides 684 by 695 feet = 591-4 by 610-9 units say 592 by 610 610 3 = 2 circumference mean = 601 60 1 3 = f distance of the moon. PYRAMID OF CErilKENES. 305 Height x area of base = 393, &c. x 592 x 610 f circumference Pyramid = -J- of f = -ft It will probably be found that the sides of the base of some of the pyramids are unequal. From the two great pyramids we learn that the quadrant was divided into 3 equal parts ; or the circumference into 12, the zodiacal division. The pyramid of Cheops = -ft- circumference, Cephrenes = -ft- Or the parallelopipedon of Cheops = f = 6 quadrants. Cephrenes = -f- = 5 The pyramid of Cheops = \ circumference =180 degrees 3 , Belus = ^ a sign =15 which are as 12 : 1. The pyramid of Belus : pyramid of Cephrenes::^ -ft ::1 : 10. The pyramid of Cheops : pyramid of Cephrenes : : -ft- I -f$ : : 6 : 5 signs, if the equator be supposed to be divided into 12 equal parts or signs. The distance of the moon from the earth = 5 cubes of Cephrenes. The distance of the earth from the sun =400 times the distance of the moon from the earth = 400 x 5 = 2000 cubes of Cephrenes. In Vyse's measurements of the interior of the pyramid of Cephrenes, the length of the entrance passage from the first covering stone to the horizontal passage = 104 feet 10 inches. Total length of the entrance passage to the bottom of the incline = 104 feet. 105-2 feet = 91-2 units 10 x 91-2 = 912 and 912 3 = 2 -f circumference 3= 91'2 3 = 3 g Q circumference = 2f degrees = Y^-Q- circumference = ^ degree. Pyramid of Cephrenes = -ft- circumference = 150 degrees. VOL. i. x 306 THE LOST SOLAR SYSTEM DISCOVERED. . . the cube of the length of the entrance passage is the reciprocal of the content of the pyramid of Cephrenes. The Birman solid hyperbolic temples are symbolical of the law of the velocity described by a body gravitating to the centre of force. The Egyptian solid pyramidal temples are typical of the law of the time corresponding to that velocity. On each side of the hyperbolic temple, as the Shoemadoo at Pegu, are dwellings for the priests, who still officiate at the altar; but the former science of the priesthood has departed. Along the sides of the quadrangular area in which stands the pyramid of Cephrenes are dwellings for the priests, excavated out of the solid rock ; but the hierarchy exists no longer, and the knowledge accumulated for ages, and held sacred by the priesthood, has perished. The Birman pagodas are solid structures, without any opening. Vyse computes the space occupied by the chambers in the pyramid of Cheops at 15 1 90 of the whole. The teocallis of Mexico are solid pyramidal temples. Montezuma was emperor and high-priest. The temple of Mexitli had five terraces. It was on the platform of this teocalli that the Spaniards, the day preceding the "noehe triste," or " melancholy night," attacked the Mexicans, and, after a dreadful carnage, became masters of the temple. It stood within a great square, surrounded by a wall of hewn stone. " Close to the side of the wall," says De Solis, " were habitations for the priests, and of those who, under them, attended the service of the temple ; with some offices, which altogether took up the whole circumference, without re- trenching so much from that vast square but that eight or ten thousand persons had sufficient room to dance in it upon their solemn festivals. In the centre of the square stood a pile of stone, which in the open air exalted its lofty head, overlooking the towers of the city, and gradually diminishing till it formed half a pyramid. Three of its sides were smooth, the fourth had stairs wrought in the stone, a sumptuous building, and extremely well proportioned. It was so high that the stair- PYRAMID OF MYCERINUS. 307 case contained 120 steps; and of so large a compass, that on the top it terminated in a flat 40 feet square. Pyramid of Mycerinus. Herodotus states that Mycerinus left a pyramid less than that of his father, wanting on all sides, for it is quadran- gular, 20 feet ; it is 3 plethrons on every side, and one half is made of Ethiopian stone. Instead of the side of the base being 3 plethrons, suppose the perimeter of the base to equal 30 plethrons, or 5 stades. Then each side will equal 7 plethrons, or f stade, or 351-21 feet, or 304 units, which is 20 units less than half the side of Cheops' pyramid. For the side of the base of Cheops' pyramid = 648 units, and ^-648 = 324 units, from which take 20 units, and we have 304 units left for the side of the base of Mycerinus' pyramid. These 20 units may have been called feet by Herodotus. If the priesthood in his time knew the value of the Babylonian unit, it appears they never made him acquainted with it, for in his tables neither this measure nor its equivalent is ever mentioned, though this unit formed the basis of his table of measures. Its value may probably have been unknown to all, except the elect of the sacred colleges of a philosophical priesthood. At whatever period, remarks Maurice, the Egyptian hieroglyphics were first invented, their original meaning was scarcely known, even to the priests themselves, at the asra of the invasion of Cambyses. And at the time when the Macedonian invader erected Alexandria, probably out of the ruins of Memphis, the knowledge of them was totally ob- literated from their minds. The difference between the sides of these two pyramids may be expressed by saying The perimeter of the base of the pyramid of Mycerinus equals half the perimeter of the base of the pyramid of Cheops, less 80 units, or less 20 units on every side. The perimeter of Cheops = 64 plethrons = 2592 units i=1296 x 2 308 THE LOST SOLAR SYSTEM DISCOVERED. .-. The perimeter of Mycerinus = 1296-80=1216 units and side of base = 304 5 stades = 30 plethrons = 1215 units = perimeter of the base. Jomard's dimensions of this pyramid are, Base, measured on the north side, 100'7 metres =330 feet English. Height 53 metres = 173*84 feet; but height not determined with great accuracy. Angle made by the plane of the face with the plane of the base, about 45. Vyse makes the former base =354*6 feet present height =203 former height =218 Wilkinson's present base =333 present height =203 '7 by calculation with the angle of 51 given by Vyse. Pliny makes the distance between the angles, or side of the base -363 feet. By Arbuthnot's table a Roman foot = 11 -604 inches English. 363 x 11-604 inches=351-02 feet English and f stade =351-25 If the angle of inclination of the side =45, according 'to Jomard, the height will = half the side of the base. Assuming the height = f stade = 175-625 feet=152 units, side of base will = f- stade = 351-25 feet = 304 units, e\ and 152, &c. x 305 = -J circumference of earth pyramid = -J- of = -^ circumference. Such a pyramid would combine the height of the teocalli, -| stade, with the content of the tower of Belus, -^ circum- ference, and the height f stade would be f that of the tower. PYEAMID OF MYCERINUS. 309 These supposed dimensions of this pyramid, height= 175-6 feet, base =351-25, accord nearly with Jomard's height, 173-84 feet, and with Vyse's former base, 354*6 feet, as well as with Pliny's base, 351-02 feet, and also with the 304 units, or 351-25 feet, obtained by comparing the side of this pyramid with that of Cheops. But the difference between the heights of Yyse and Jomard=55 feet, and the difference between their bases 24 feet. Cube of side of base = 305 3 = -J circumference Cube of twice side = 610 3 =2 Cube of height = 152 3 , &c, =^\. If this pyramid had formerly been a teocalli having the height to the side' of base as f : -f stade, or 152, &c. .' 305 units, or I75f : 351^ feet. Supposing such a teocalli to have had 4 terraces of equal heights, and the height of the 4 to equal f stade, and the height to the apex to equal the height of 5 terraces, or 175| + i 175| feet = 220 feet. Then the content of this pyramid, having the same base, would exceed that of the pyramid having the height to side of base as f : -J stade by ^. Or content = -^ + -J- of -^ circumference == !MF or iV circumference nearly, and -^g circumference = 19 degrees nearly, for 19x19 = 361. Such a pyramid would accord with the base and former height by Vyse's measurement. By this supposition the mode we have adopted for mea- suring the content of a teocalli is practically illustrated. Here the circumscribing triangle of the pyramid and teocalli are equal, and so are their contents, for the content of the pyramid or teocalli = -J- (the square of the base of the circumscribing triangle x the height). The bulk of the pyramid has been more carefully and compactly built than the two larger ones, and the stones x 3 310 THE LOST SOLAR SYSTEM DISCOVERED. have been better finished, and are of a greater size. It has been carried up in steps or stages, diminishing towards the top like those in the fourth and fifth pyramids ; and the angular spaces have been filled up so as to complete the pyramidal form. ( Vyse^) The dimensions of such a pyramid will be Height to apex = f + -J- of f stade = 175'6 + 43-9 = 219-5 feet = 190 units. Side of base = -J- stade = 35 1 -25 feet = 304 units ; then height x base Q = 190, &c. x 305 2 = r circumference 360* pyramid =- r circumference = 360* degrees. Yyse's height to apex = 218 feet =188-5 units side of base =354-6 feet = 306 -5 square of platform at the top about 9. According to Yyse's dimensions the content of the pyramid of Mycerinus will = -, circumference = 360* degrees. 360* The perimeter of the base will =fx 4 = 5 stades = 30 plethrons. 5 1-5 25 5 2 , , Height =- + i of - = _=_ stade. Also a pyramid having the height to apex=| stade.' Side of base = twice the height = f stade; Or perimeter of base = 5 stades = 30 plethrons will = -^5- circumference = 15 degrees = the content of the tower of Belus. Both these formulas will require a small correction, the addition of a unit to a stade, as will be seen afterwards. Side of base of pyramid = 305 units, and 305 3 = cir- cumference. So 4 cubes = circumference. Or if a cube be described on each of the 4 sides the sum of the cubes will = circumference (2 x 305) 3 = f = 2 circumference, PYRAMID OF MYCEIUNUS. 311 or cube of sum of 2 sides = 2 circumference. 189 3 &c. = -j-fo- circumference (10 x 189 &c.) 3 = VW = 60 cube of 10 times height = 60 circumference. Inclined side will = 242 &c. units 242 3 &c. = J|- circumference (2 x 242 &c.) 3 = 1 Cube of twice inclined side = circumference. Cube of side of base '. cube of inclined side :: 1 * 4- circum- ference ::2 : 1. Pyramid = - r circumference = (360) degrees. (360) 2 So the pyramid of Mycerinus will be the reciprocal of itself. Cube of perimeter of base =(4 x 305) 3 = 16 circumference. Cube of perimeter of base of Cheops =16 distance of the moon. Cubes of perimeters are as circumference : distance of the moon. Pyramid of Mycerinus : pyramid of Cheops : ^ circumference 19 9-5 distance of the moon. The pyramid of Mycerinus will be similar to the pyramid of Cheops, so the height will = ^ side of base. 4 cubes of Mycerinus = circumference 4 Cheops = distance of the moon. Taking Jomard's base as that of the internal pyramid, side of base = 330 feet = 285 units and 283 3 &c. = % circumference. Content of external : content of internal pyramid :: I ' iV A circumference. Thus we shall have the content of the external pyramid circumference 1 9 X 4 312 THE LOST SOLAR SYSTEM DISCOVERED. Cube of side of base = J circumference Content of internal pyramid = -^ Cube of side of base =5- s> The external and internal pyramids will be similar, having height = -| side of base. The content of the internal pyramid will = that of the tower of Belus = -^ circumference. Content of external pyramid = -^ circumference = 19 degrees, or = circumference 2 " = 360 2 = 19 degrees. Herodotus says the pyramid of Mycerinus was built up to the middle with Ethiopian stone. The casing has been taken away at different times : some of it was removed a few years ago to assist in the construction of the arsenal at Alexandria. The lower part of the casing consisted' of polished granite, as the ancient historians have described ; but the eleven or twelve courses towards the bottom are not worked smooth, but form a sort of rusticated base, inclining like the rest of the pyramid. The style of building of the pyramid of Cephrenes is said to be inferior to that of Cheops, the stones used in its con- struction being less carefully selected, though united with nearly the same kind of cement. Nor, says Wilkinson, was all the stone of either pyramid brought from the quarries of the Arabian mountains, but the outer tier or casing was com- posed of blocks hewn from their compact strata. This casing, part of which still remains on the pyramid of Cephrenes, is, in fact, merely formed by levelling or planing down the upper angle of the projecting steps, and was consequently commenced from the summit. The pyramid of Mycerinus is described as being built in almost perpendicular degrees, to which a sloping face has afterwards been added. The outer layers, many of which still remain, were of red granite, of which material the lowest row of the pyramid of Cephrenes was also composed, as is evident by the block and fragments which lie scattered about its base. In measuring the content of the teocalli, this sloping face, which included the outer layers of the pyramid, has been in- SMALL PYRAMIDS. 313 eluded ; since the inclining side of the teocalli, according to estimation, is that straight line which touches all the exterior angles of the terraces, or degrees, and terminates at the apex and ground base of the teocalli. The following measurements of the small pyramids at Gizeh are those made by Col. Yyse, who, in his description of the pyramids, has given the measurements of the interior chambers and passages of all the pyramids. The fourth central and sixth western pyramids south of the third pyramid, that of Mycerinus, are both built of large square blocks put together in the manner of Cyclopian walling, and are at present in steps or degrees. These two pyramids are of equal dimensions and similar in construction, each having four terraces, like a teocalli. Both are in a di- lapidated state. Height to the top platform, 6 9 '6 feet, i- stade = 70-1- feet. Side of the base of the lowest terrace = 102 '5 feet. Suppose the height to the apex = the height of 5 terraces = 70 + 70 = 87-5 feet = 75 &c. units = JL stade. Let the base of the circumscribing triangle =128 feet = 111 units = 2f plethrons, then height x base = 75 &c x 111 2 = 3 degrees. Pyramid -^3 = 1 degree, or -j^- circumference. Thus the fourth and sixth pyramids or teocallis, &c., each = 1 degree. The fourth pyramid is much dilapidated on the northern front ; but the masonry on the other sides is very fine, and the stones exceeding large and apparently of great antiquity. Like the sixth pyramid it has been built in regular stages. Side of base =111 units Height = 75 110 3 &c. = 4( / 00 distance of the moon (2 x 10 x 110 &c.) 3 = | x 2 3 = \ = 10. Cube of 20 times side, or of 5 times perimeter = 10 times distance of the moon 314 THE LOST SOLAR SYSTEM DISCOVERED. 15 cubes =150 times distance of the moon = distance of Mercury (2 x 2 x 10 x 110 &c.) 3 = 10 x 2 3 = 80 distance of moon 5 cubes of 40 times side = 400 times distance of the moon = distance of the earth. 75 3 &c. = g^-o circumference (10x75&c.) 3 = VW = V (2 x 10 x 75 &c.) 3 = 3 -gx2 3 = 30. Cube of 20 times height = 30 times circumference. (4x2x10x75 &c.) 3 = 30 x 4 3 = 1920. 2 cubes of 80 times height = 3840 circumference, = distance of the earth. The fifth pyramid is to the south-east of the third. Height to apex = 93-3 feet = 80 units. Side of base = 145-9 feet = 125 units. Height x base = 80 &c. x 125' 2 =-^- circumference = 4 degrees. Pyramid = -f- of a degree = -j-i circumference, and height to side of base :: 80 .* 125 :: 5 : 8 nearly. Or height = f side of base nearly. Perimeter of base = 500 units, and height =80 50 1 3 &c. = l -f circumference (3 x 501 ) 3 = LO x 33 _ 30 c i rcum ference ; or cube of 3 times perimeter = 30 times circumference. The fifth pyramid had at the time of Richardson a flat top, which was covered with a single stone. The two pyramids to the west of this, but in the same line, consist each of four receding platforms, like the Mexican teocallis. The several divisions of these pyramids are ascended by high narrow steps to the summit, which is a platform. The third pyramid, that of Mycerinus, appears also to have been originally a teocalli, and that at a later period the SMALL PYRAMIDS. 315 terraces of the ancient teocalli had been built up so as to form a plain-sided pyramid. We know of no pyramid of which the fifth pyramid will be the reciprocal. Such a pyramid should = -f circumference, = 270 degrees. The seventh, eighth, and ninth pyramids are situated to the eastward of the great pyramid. The seventh (northern) and eighth (central) pyramids are both in very ruined condition. The dimensions of both are supposed by Vyse to be equal. Height to apex =111 feet, and side of base = 172*5 feet. If the height be supposed = 105 feet = 92 units, and side of base =164 feet = 142 units, Then height x base = 92 &c. x 142 2 = -fa circumference, = 6 degrees. So each pyramid will = -J- of -fa = -^ circumference, = 2 degrees. Thus the side of base of each pyramid will = 3|- plethrons, = 141-75 units. Height will = 2| plethrons = 91-25 units. 14 1 3 &c. = -^Q circumference. Cube of side = -fa circumference. 9 1 3 &c. = j-J-Q- circumference. Cube of height = y^- circumference. Several of the casing stones of the central pyramid had been roughly chiselled into the proper angle, and then worked down to a polished surface after they had been built ; and in many places the operation had not been entirely performed. They were as firmly laid as the blocks in the Great Pyramid, and the masonry of the buildings had a great resemblance. It is to be remembered that tradition assigns the building of this pyramid to the daughter of Cheops. The pyramid of Cheops = circumf. = 180 degrees. The pyramid of his daughter = y^- circumf. zz 2 degrees. The height of the pyramid of Cheops = If stade. The side of base = 2f stades. The height of the pyramid of his daughter = 2 plethrons. The side of base = 3^ plethrons. 316 THE LOST SOLAR SYSTEM DISCOVERED. The height : side of base of Cheops' pyramid : : 5 I 8 ; of his daughter :: 5 17. The side of the base of the great pyramid = 16 plethrons. The perimeter of the base of the small pyramid = 3|- x 4 = 14 plethrons. Having since found that the pyramid of Mycerinus is a mean proportional between the pyramid of Cheops and the pyramid of his daughter. So that if all the three pyramids be similar, we can determine the height and side of base of the pyramid of Cheops' daughter. The three pyramids are Cheops : Mycerinus : : Mycerinus .* Daughter 180: 360*:: 360* : 2 degrees. The three pyramids being similar, the cubes of the sides of bases will be as their contents. Cube of Cheops I cube of Mycerinus :: cube of Mycerinus : cube of Daughter 648 3 : 305 3 ::305 3 : 144 3 ^distance of moon .* -J- circumference :: -\ circumference .' -J- y _.._. T ^_. T ^^^_. _ distance of moon J- 9*5 circumference \ -J- circumference :: -J- circumference I -J. x circumference, y *o Hence the cube of the side of base of pyramid of Cheops' daughter will =144 3 = - x circumference = circum- 4 y *o GO ference. Since the three pyramids are similar, and height of each =-! side of base o .*. height of pyramid of Cheops' daughter = -f 144 = 90 units. Height x area base = 90 x 144 3 , c. =- g 1 o circumference. Pyramid = i of -^Q=^^O circumference = 2 degrees. . . Height will be 90 and side base 144 instead of 92 142 SMALL PYRAMIDS. 317 The pyramid of the Daughter is the reciprocal of the pyramid of Cheops. The pyramid of Mycerinus is the reciprocal of itself. The pyramid of Mycerinus is a mean proportional be- tween the pyramid of Cheops and the pyramid of his daughter. The three pyramids are all similar. The height of each = ^- side of base. Hence knowing the side of the base of pyramid of Cheops, the dimensions of all the three pyramids can be deter- mined. Wilkinson mentions " that on the east side of the great pyramid stand three smaller ones, built in degrees or stages, somewhat larger than the three on the south side of the pyramid of Mycerinus. The centre one is stated by Herodotus to have been erected by the daughter of Cheops. It is 122 feet square, which is less than the measurement given by the historian of 1^- plethron, or about 150 feet; but the difference may be accounted for by its ruined con- dition." Wilkinson makes the side of the base of the pyramid of Cheops' daughter to equal 1 22 feet. Vyse makes the side of the base to equal 172*5 feet. If side of base = 122 feet= 105-5 units 104 3 , &c. =^-5-0 circumference (10 x 104, &c.) 3 = ^1^-= 10 circumference, or cube of 10 times side of this base = 10 times circumference. Side of base of pyramid = 143 units ; 143 3 =^o circumference. The dimensions of the seventh pyramid are, Height to the apex (supposed) 111 feet. Side of the base (supposed) 172 -5 feet. Height 111 feet = 96 units. Side of base 172-5 feet = 149 units. Let the height = 98 units, and side of base = 152 units =4- stade. 318 THE LOST SOLAR SYSTEM DISCOVERED. Then height x base = 98, &c. x 152 2 = giQ circumference = 7*2 degrees. Pyramid = y^- circumference = 2 -4 degrees = Ti~o circumference =^- degrees; 152 < 5 3 = 3 1 J circumference and 150 3 , &c. = 3-^ distance of moon. The second pyramid, that of Cephrenes, =yV circumference = 150 degrees. Thus the seventh pyramid will be the reciprocal of the second, that of Cephrenes, as the eighth is the reciprocal of the first, that of Cheops. Perimeter of the eighth pyramid = -f x 4 = 2-J- stade = one of the sides of the base of the pyramid of Cephrenes. The fourth and sixth pyramids are both teocallis, each having four terraces, and the content of each = -g-J-o circum- ference =1 degree. The teocalli of Cholula has four terraces, and the content = 1 circumference = 360 degrees. Hence the fourth and sixth pyramids are the reciprocals of the teocalli of Cholula. The third pyramid, that of Mycerinus = i T circum- 360* ference = 360 degrees, and is .'. the reciprocal of itself. Height : side of base of Cephrenes :: 392 : 601 :: 5 : 7-64. Height to side of base of seventh pyramid :: 98 : 152 :: 5 : 775. Height to side of base of Mycerinus as 5 .' 8. If the three pyramids were similar, and height of each = |- side of base, then cubes of sides of base will be as 601 3 : 305 3 ::305 3 : 155 3 , &c. - distance of moon : - circumference :: - circumference 5 1 30-4 circumference ; LARGE PYRAMIDS. 319 - circumference : - circumference :: - circumference 544 1 circumference. 30-4 Thus side of base of seventh =155, &c. height =f 155, &c. = 97. Height x area base = 97 x 155 2 , &c. = 5 1 Q circumference Pyramid = ^- of 5V == T5~o circumference = 2 -4 degrees. Vyse makes the former height of Cephrenes somewhat more than f side of base. So it would seem that the pyramid of Cephrenes is dis- similar to Mycerinus, though it may have been similar to the seventh. If so the cubes of the sides of the bases of the three pyramids will not be as their contents. Thus the pyramid of Cephrenes and the seventh will be reciprocals, and may be similar to each other, though dis- similar from the pyramid of Mycerinus ; still the pyramid of Mycerinus will be a mean proportional between Cephrenes and the seventh. Cephrenes : Mycerinus : : Mycerinus I seventh. 150 : 360* :: 360* : 2-4 degrees. Cube of side of base of Cheops = 648 3 = -J- distance of the moon. f cube = f x pyramid = ^ .47-75 96 = 1 circumference. Cube of side of base of Cephrenes = 601 3 = the moon. distance of pyramid = -J- of -1-=^" 9*55 = ^4 x 9*55 = circumference ; 320 THE LOST SOLAR SYSTEM DISCOVERED. but pyramid =^-=|-J circumference, 9*55 which is greater than . Thus the pyramid of Cephrenes exceeds -^ of -f- cube of side of base, and, therefore, is dissimilar to the pyramid of Cheops or Mycerinus. The angle of inclination of the side of Cheops is less than the angle of inclination of the side of Cephrenes. Should the side of base of a pyramid = 601 units, and height = % side = 375 &c. units ; then height x area of base = ^ distance of the moon ; pyramid = -J- of |- = -^ distance of the moon ; tower of Belus = -^ circumference. Thus a pyramid having side of base = that of Cephrenes, and height -J- side of base, will = ^4 distance of the moon. Pyramid of Cheops has height = |- side of base, and content = \ circumference. These two pyramids will be similar, and as circumference 3-1416 5 distance of the moon TT 2 4 circumference |-J radii of the earth 5 , ,, 2-5 4 nearly. Cubes of sides of bases are as -J- : $ distance of the moon 5:4 Such a pyramid would be to the tower of Belus as -J^ distance of the moon : -^ circumference as '. circumference. Pyramid of Cephrenes .' tower of Belus as -f^ '. -^ circum- ference :: 10 I 1. Having found that a pyramid has two dimensions, one in- ternal, the other external, let us try how nearly two such pyramids of Cephrenes may be made to accord with the measurements of Vyse. Former height 454 '3 feet = 392 units Present height 447*6 = 376 LARGE PYRAMIDS. 321 Former base Present base 707-9 feet = 612 units 690-9 = 579 Internal Pyramid. Let height x base =376 x 601 2 = distance of the moon pyramid = of = -^ cube of side of base = 601 3 = j- distance of the moon. External Pyramid. Let height x base = 381 &c. x 610 2 = *g circumference pyramid = -J- of *g = -|-J = I ^- cube of side of base = 610 3 = 2 circumference. The internal and external pyramids will be similar, having height = -J side of base, which is the proportion of the in- ternal and external pyramids of Cheops ; therefore the two pyramids of Cephrenes are similar to the two pyramids of Cheops, or the four pyramids are all similar. The two pyramids of Cephrenes will be external '. in- ternal : : circumference .' -^ distance of the moon :: 10 : distance External pyramid of Cephrenes .' tower of Belus ::\% : -^ circumference:: 10 .' 1. Internal pyramid of Cheops C tower of Belus :: I -% cir- cumference :: 12 : 1. External pyramid of Cephrenes '. internal pyramid of Cheops '. : yV A circumference :: 5 : 6. Internal pyramid of Cephrenes I external pyramid of Cheops : : -^ '. -^ distance of the moon. :: 3 I 4. In all the four pyramids cube of height '. cube of side of base::5 3 : 8 3 :: 125 : 512 :: 1 : 4 nearly. External cube of Cephrenes .' internal pyramid of Cheops :: 2 : -J- circumference :: 4 : 1. Internal cube of side of base of Cephrenes : external cube of side of base of Cheops :: .' -fa distance of the moon ::3 : 4. VOL. I. Y 322 THE LOST SOLAR SYSTEM DISCOVERED. The ninth southern pyramid is in much better preservation than the seventh and eighth. The height to apex = 101 '8 feet, and side of base =160 ft. 101-8 feet = 88 &c. units; 160 feet =138 units. Height x base 88 x 139 2 &c. = -^Q circumference. Pyramid = ^-J-^ circumference = ~ degree. The great pyramidal teocalli at Dashour = -- circumference = 200 degrees, the reciprocal of which will be the ninth pyramid. 139 3 &c. = -4-J-Q distance of the moon, or = -g^- circumf. and distance of the moon = -^L- distance of the earth, 139 ' &c " = = distance f the eartL (10 x 139 &c.) 3 = VW = i Distance of the moon. (10 x 10 x 139 &c.) 3 = 5 -^- = 2500. 3 cubes of 100 times side = 7500 distance of the moon, = distance of Uranus. 9 cubes = distance of Belus. Height = 88 units. 88 3 = 5-g-Q- circumference (10 x 88) 3 = V pyramid = -/ cube of side of base. 10 times cube of side of base = distance of the moon 20 times cube of side =5 48 circumference = 5 3x4 2 = 5 Side of base of Cheops' pyramid = 648 units = 2f =| stade. Height =f side of base = 405 units =A of -|=4 stade. o o o Height x area base =-f x (-|) 2 =-| x -y~= s / 7 cubic stades = _JL2_Q_ x 243 3 cubic units. Pyramid = content = circumference, very nearly. PYRAMIDS. 329 The addition of part of a unit to height and side of base will be required to make pyramid =-J- circumference. Thus side of base =-f stade, height =|- side of base =f stade, content of pyramid =-^- x 243 3 = 320 x but circumference will lie between OAQ3 320and321x-^L, o or 320 and 321 x3 n for __ n "^""I* 3*"" 320 x3 n = 56687040 units and tvx6 n = 56687040 J- circumference = 56844504 6 n = 3 n x320x V = 3 11 x 32 2 x V Q = 3 n x 32 2 x2 = 3 u x(2 5 ) 2 x2 = 3 n x2 n = 6 11 6 12 = diameter of orbit of moon in units = (2 x 648) 3 = cube of twice side of Cheops' base ; 3 5 transposed, doubled and squared, = 684 2 = circumference of earth in stades; 3 5 x 6 84 2 = circumference of earth in units. Pyramid of Cheops = of x (f) 2 02 = 5 x -= cubic stades = 5 x -5 x 243 3 cubic units 9 330 THE LOST SOLAR SYSTEM DISCOVERED. = 5x8 2 x3 n = 5x2 6 x3 u or =5 x4 3 x3 u = 56687040, when corrected = |- circumference = 56844504 Pyramid = ^ of cube of side of base 4 3 = 5 x cubic stades 3 43 = 5 x x 243 3 cubic units o = 5x4 3 x3 12 Pyramid = *-= 5 x 4 3 x 3 H Cube of side = (f ) 3 stade = -J- distance of moon 16 3 Cube of 2 side = stade =2 distance of moon o Ifi 3 = i- x 243 3 units = 16 3 x3 12 = 2 12 x3 12 = 6 12 Cube of 2 side of base = diameter of orbit of moon = 6 12 = 19'2 circumference Cylinder =15 Sphere =10 Cone = 5 Pyramid of Cheops : Cone .' Sphere 1 1 I Cylinder : : 1 10 20 30 PYRAMID OF CHEOPS. 331 Cube of side of base = -J- distance of moon = 2 4 circumference Cylinder = if Sphere = ^ Cone - Pyramid of Cheops .' Cone :: =| : Sphere ::8 : Cylinder:: 8 10 20 30 Cone having height = diameter of base = side of base of Cheops' pyramid will = f pyramid =A x =f circumference. Cone having & height will 5 5 25 5' 5 2 1 5 2 5 2 . f = 2* x 2 = 2"6 = g2 circumference. Height of cone = ^ diameter of base Content =(f) 2 circumference. Pyramid of Cheops I cone having diameter = side of base of pyramid of Cheops, and height = height of Cheops :: 32 : 25::2 5 : 5 2 . Cone having same height as the last and diameter of base = diagonal of base of Cheops' pyramid will 5 2 1 5 2 . - = x -= circumference. 2 4 2 2 5 Diameters being as 1 .* 2*, Cones are as 1 : 2, Heights being equal. Cone having height and diameter of base = diagonal of base of pyramid will =x(2) 3 = 5x=l pyramid = X = circumference> 332 THE LOST SOLAR SYSTEM DISCOVERED. Cone having height and diameter of base = twice side of pyramid will =x (2*)3 = 5 x?= 10 pyramid 4 = 10x^=5 circumference. Pyramid having height and side of base = twice side of base of Cheops' pyramid will = twice distance of moon = -L 19 '2 = 6*4 circumference. Cone : pyramid :: 5 I 6 -4 :: 25 : 32 :: 5 2 : 2 5 :: 5 to second power : 2 to the fifth. Sphere : pyramid :: 2 x 5 2 2 5 Cylinder '. pyramid : : 3 x 5 5 Cube I pyramid : : 3 The proportions are only proximate and will require cor- rection. Since cone : pyramid : : 25 : 31 '82, &c. The pyramids of Saccarah are numerous and of irregular formation, some towering aloft, others greatly decayed, some constructed of brick, and some of stone. Champollion con- siders the brick pyramids of more ancient date than those of stone. There are several large pyramids at Saccarah and Dashour. The largest one at Saccarah is about 350 feet high, and has only four retreating steps or terraces. The teocalli of Cholula has four terraces. The pyramids of Djizeh, like those of Abousir, Saccarah, and Dashour, are placed at various distances from each other. The multitude of pyramids scattered over the district of Saccarah, observes Denon, prove that this territory was the necropolis (city of the dead) to the south of Memphis, and that the village opposite to this, in which the pyramids of Djizeh are situated, was another necropolis, which formed MAUSOLEA. 333 the northern extremity of Memphis. The extent of the an- cient city may thus be measured. The remains of some of the kings of Egypt, who were sovereigns and pontiffs, may have been deposited within a pyramidal temple ; as the remains of the popes, who were sovereigns and pontiffs, are still interred within the temple of St. Peter's at Home. Doubtless both in the old and new world, tumuli, which are but rude imitations of pyramids, have been raised as sacred memorials over the ashes of kings and chiefs. The custom of depositing the remains of man in or near some sacred place is not confined to any country. Some Mahomedans carry a corpse a journey of many months to be deposited near a sacred shrine. The Hindoos carry their dead and dying great distances to the sacred Ganges. The Moslem emperors have erected many splendid mau- solea as monuments to their posthumous fame ; as the Burra- Gombooz at Bejapore, which exceeds the dome of St. Paul's at London in diameter, and is only inferior to that of St. Peter's at Rome. It was constructed in the lifetime of the monarch, Mahomed Shah, and under his own auspices. So the pyramids, like the modern cathedrals, may have been erected as temples, and used as mausolea. They were used as temples of worship and places of sacrifice when the Spaniards arrived in America ; and remains of the dead have been found in some Mexican teocallis. The hyperbolic temple still continues to be used in the Burmese empire as a place of worship, but not of sacrifice. Bohlen mentions that the Burmese priests are embalmed exactly in the Egyptian fashion. The intestines are taken out of the body, the cavity of which is filled with spices, and the whole is protected from the external air by a covering of wax. The arms are then placed on the breast, the body is swathed in bandages varnished with gum, covered with gold leaf, and at the expiration of one year it is burned ; the re- mains are then placed in a pyramidal-formed building. The sepulchres of the Egyptian kings were not always in a remote and sequestered place, like the valley of Bidan-el- 334 THE LOST SOLAK SYSTEM DISCOVERED. Molouk, but even within the precincts of the temple. Thus all the Saite kings were buried near the temple of 'Athenaea, and within its enclosing wall. Here also was a tomb of Osiris. When a Scythian king dies, says Herodotus, they smear his body all over with wax, after having opened it and taken out the intestines. The cavity is filled with chopped cypress, pounded aromatics, parsley, and aniseed, and then the inci- sion is sewn up. One of the pyramids at Dashour, according to Davidson, has a base, each side of which is 700 feet, a perpendicular height of 343 feet, and 154 steps. There is an entrance into the north side, which leads down by a long sloping pas- sage, and then by a horizontal base to a large room, the upper part of which is constructed of stones of polished granite, each projecting six inches beyond that below, and thus forming in appearance pretty nearly a pointed arch. Height = 343 feet, side of base = 700 feet. Let the height to apex = \ the side of the base, and height to apex : side of base :: -J I | stade :: 351-25 feet : 702-5 feet :: 303-25 units ! 607 -5 units :: 305 I 610 when corrected ; then height x base = 305 &c. x 610 2 = circumference. Pyramid = circumference =120 degrees. The reciprocal pyramid should = y^ circumference = 3 degrees. A pyramid representing -J- circumference of the earth will have for the side of the base 705-39 feet, and height 352-69 feet. The perimeter of the base will =|-x4 = 10 stades = 60 plethrons, or = 60 plethrons+ 10 units, when corrected. The height will = -J- perimeter. \ x Cl) 2 stade = -J- circumference pyramid = -^ i x (|) 2 stade = pyramid = DASHOUK PYRAMID. 335 These formulas will require to be corrected by the addi- tion of unity to a stade. Here we find the solution of the frequent recurrence of 5 stades and % stade in the measurements of the sacred monu- ments in both hemispheres. Five stades being a whole number was not so mysterious as the fraction |- stade, which has so often and unexpectedly crossed our path of inquiry. Taking the measurements with the small correction, we have 5 x 2 = 10 stades = 60, or 3 score plethrons 5 stades = 30, or -J- of 3 score plethrons. Thus a square base having a perimeter of 3 score plethrons or 10 stades, and height = the side will = circumference of the earth in units, and pyramid = - circumference. A square base having a perimeter of of 3 score plethrons or 5 stades, and height = the side, will = -J- circumference, and pyramid = -^ circumference. The cube of the side of base of pyramid = 610 3 = twice circumference. If perimeter of a square = SOpleths., side= 7^=303*75 units, and 305 3 = -J-circum. = 60 =15 =607-5 610 3 = 2 = 120 . =30=1215 1220 3 =16 Hence the cubes of 305, 610, 1220 units, which respec- tively = -J-, 2, 16 circumference, will have the perimeters of their bases somewhat greater than 30, 60, 120 plethrons. The tower of Belus has the height = the side of the base = 1 stade. 1 X I 2 stade = -I- circumference o pyramid = -^ Here unity must be subtracted from a. stade, for 243 x 243 2 exceeds -J- circumference but 242 &c. x 242 2 = i and pyramid = A So the formula for the tower will be the side of the base = 1 stade less unity, and height = the side of the base. 336 THE LOST SOLAR SYSTEM DISCOVERED. Thus a square base having a perimeter of ^ of 3 score plethrons Jess one stade, and height = the side, will = -J- circumference, and pyramid = -^ circumference. Perimeter = 4 stades =4x6 = 24 = 30 6 plethrons = \ of 3 score plethrons less 1 stade. The Dashour pyramid has 154 steps. The distance of the earth from the sun = 220 semi- diameters of the sun. Distance of the earth = 400 distance of the moon Venus = 281 400 : 281 ::220 : 154, or distance of Venus = 154 semi-diameters of the sun. There is a pyramid at Saccarah, the sides of which, on an average, are said to be about 656 feet, and the height 339 feet. This is the pyramid which contains hieroglyphics in relief round the doorway of a small chamber. Height = 339 feet, and side of base =656 feet. If the height = 338 feet = 292 units and side of base = 654-5 feet = 567 units = 2-J- stade = 14 plethrons ; then height x base = 292 x 567 2 =f circumference = 300 degrees, pyramid = -J-of-f- =-& =100 Perimeter of base =4 x 14 = 56 plethrons height =7 +8^ units. The reciprocal of this pyramid will equal y^- circumference = l -f degrees. This pyramid : the pyramid of Cephrenes : : ^ : - circum- ference :: 100 I 150 degrees :: 2 : 3. The Dashour pyramid \ Cheops' pyramid : : -J- * -J- circum- ference:: 120 I 180 degrees:: 2 I 3. Six times the cube of the side of the base of the pyramid at Saccarah =6 x 566 3 &c. = 9'55 circumference = distance of the moon from the earth. Thus 6 cubes will = distance of the moon 150 x 6 = Mercury 150 2 x 6 = Belus from the sun. SACCARAH PYRAMID. 337 6 cubes of 566 = distance of the moon 3x6 or 18 cylinders, diameter 4 x 566 = dist. of Mercury. |- or cube of 2 x 566 = distance of the moon o * j = diameter of orbit f- or f cylinder, diameter 16 x 566 = distance of the earth -f- = diameter of orbit. cube of side of base = ^ distance of the moon cube of perimeter = sg cube of 4 perimeters = ^^ 6 cubes = 4096 3 cubes = 2048 and distance of Jupiter = 2045. Thus 3 cubes of 4 times perimeter of base = distance of Jupiter. 566 3 = -|- distance of moon (6 x 566) 3 = i x 6 3 = 36. 25 cubes of 6 times side of base = 800 times distance of the moon = diameter of the orbit of the earth. Height x area of the base of the cased pyramid of Cheops = Jr distance of the moon = cube of side of base of Saccarah pyramid. There is another pyramid at Dashour that has a base line of 600 feet : at the height of 184 feet the plane of the side is changed, and a new plane of inclination completes the pyra- mid with a height of 250 feet more. The platform is 30 feet square. The entrance passage, which is on the north face, cuts the side of the pyramid at right angles ; and as the in- clination of the passage is 20 degrees, according to Jomard, it follows that the side of the pyramid makes an angle of 70 degrees with base. In its present state the pyramid consists of 198 steps, 68 large steps from the ground to the angle, and 130 smaller ones from the angle to the top. Fig. 66. A. The platform at the top is 30 feet square, so the height from the platform to the apex will be 15 feet, or very nearly. VOL. I. Z 338 THE LOST SOLAR SYSTEM DISCOVERED. Stated height to platform =184 + 250 = 434 feet. Therefore height from base to apex will =434 + 15 = 449 feet. D E, the bsse of the teocalli, = 600 feet. The circumscribing triangle ABC having the height r A = 449 feet, and the sides AB, AC, drawn from the apex A Fig. 66. A. touching the sides of the teocalli DG, EH, inclined 70 degrees at the height Fi = 189 feet, will have a base BC = 820 feet. The stated height FL to platform =1844 250 = 434 feet. Let us take 5 from 250, and add 5 to 184, then 189 + 245 = 434 feet or FI+ IL =FL. Thus the whole height F L to the platform will remain = 434 feet. The height to apex will = F I + I L +L A = F A = 189 + 245 + 15 =449 = F I + AI =F A = 189 + 260 =449. AF : AI :: BC : GH 449 : 260:: 820 : 474 feet. AF = 449 feet = 380 units = (f) 2 stade AI = 260 =225 BC = 820 = 708-75 = 17-J- plethrons =410. GREAT DASHOUR PYRAMID. 339 Height AFX base B c = 380 x 706 2 = f circumference = 600 degrees pyramid ABC=ioff = f = 200 379 3 = -Q distance of the moon. Height A I x base G H = 225 &c. x 410 2 = -L circumference = 120 degrees pyramid AGH = of = -J- =40 pyramid ABC pyramid A G H = frustum G H B c | i = -| circumference or 200 - 40 =160 degrees. Pyramid A B c + frustum G H B c = 200 + 160 =360 degrees = twice the pyramid of Cheops. Pyramid A B C + pyramid A G H = 200 + 40 = 240 degrees = twice the other pyramid at Dashour. Pyramid A B c = 200 degrees = twice the pyramid at Saccarah. The frustum G H B c, if completed, would be 10 degrees greater than the pyramid of Cephrenes, and 20 degrees less than the pyramid of Cheops. The height to apex 2|x| = f|=(|) 2 stade = 380 units. Perimeter of base =4 x 17-J-==70 plethrons. This pyramid will be to that of Cheops as 200 : 180 de- grees:: 10 I 9 ; and to that of Cephrenes as 200 .* 150 de- grees :: -- : -fj circumference :: 4 .* 3. It may be remarked that the number of steps at present are 198, and the pyramid = 200 degrees. The pyramid is built of a hard white stone, which contains fossils. Its sides face the cardinal points. This structure, which is partly a pyramid and partly a teocalli, will explain how the pyramids were built. The height to the apex = 2-J- x -| or 5 x -f^ stade. Suppose F K, the height from the base to the platform of a second terrace, = 2 x f or stade, then the height from a second platform to the apex will = -J- of f = T V stade. Or the height I A might be divided into any convenient 340 THE LOST SOLAR SYSTEM DISCOVERED. number of terraces, and the stones raised from terrace to ter- race till the teocalli was completed by building upwards. Then to give the structure the pyramidal form, the builders would begin at the highest platform and build up to the apex ; then from the next platform in the descent they would build up the angular spaces so that the pyramidal part would be completed from the apex down to the second terrace, and so in succession till they had finished the pyramid from the apex to the base G H, by building downwards as they had completed the teocalli by building upwards. At GH, when ~ the pyramid was finished, the building ceased, and the remaining J- was left incomplete, which might probably have been the original intention, for the structure combines the teocalli with the pyramid and different proportions of the earth's circumference. This mode of building is in accordance with the method described by Herodotus, who says, " the pyramid of Cheops was first built in form of steps or little altars. When they had finished the first range they carried stones up thither by a machine ; from thence the stones were moved by another machine to the second range, where there was another to receive them, for there were as many machines as ranges or steps." Others say they transferred the same machine to each range. Both accounts have been related to us. The upper part of the pyramid was first finished, then the next part, and last of all the part nearest the ground. These are all the Egyptian pyramids of which we have found any stated measurements. If the side of base BC = 713 units, then 3 x 713 3 = 9'55 circumference = distance of the moon. Thus 3 times the cube of the base BC will = distance of the moon from the earth. 150 times 3 cubes will = distance of Mercury. 150 2 times 3 cubes will = distance of Belus from the sun. Should the side of the base B c = 713 units, it will exceed 706 by 7 units ; so that, if the side of the base be increased, the height must be diminished in order thai; the pyramid ABC may equal circumference, or 200 degrees. GREAT DASHOUR PYRAMID. 341 Four times the cube of the side of the base of the pyramid of Cheops = the distance of the moon. Hence the great Dashour cube will be to the Cheops' cube as 4 : 3. The cube of the side of this great Dashour pyramid = -J- distance of the moon. The other Dashour pyramid, which has 154 steps, = circumference of the earth. So 3 cubes = distance of the moon, and 3 pyramids = circumference of the earth. Side of base DE = 600 feet = 519 units 52 1 3 = -J- circumference 5 1 4 3 = -J- distance of the moon. Cube of 3 times side of base BC = (3 x 713) 3 = 3 3 x \ = 9 times distance of the moon. Let BC = 713 units. (Fig. 66. B.) GH = 414 AF = 374 DE == 521 z 3 342 THE LOST SOLAR SYSTEM DISCOVERED. Height AF x base BC = 374 x 713* = ^ circumference. Pyramid ABC = -J-of|- = |- circumference = 200 degrees. BC 3 = 71 3 3 = ^ distance of the moon. So 3 times the cube of B c = distance of the moon. The cube of the side GH = 414 3 &c. = -f- circumference. A pyramid = ^ of |- = -^ circumference = 5 times the pyramid of Belus. The cube of the side DE = 52 1 3 &c. = -J- circumference. A pyramid = i of -J- = -^ = pyramid of Cephrenes. Thus the cube of the side DE is double the cube of the side GH. % The cube of twice the side GH = (2 x 414) 3 = 5 circumf. The cube of perim. of base GH = 40 The cube of twice the side DE = (2 x 52 1) 3 = 10 The cube of perim. of base DE = =80 In order that GH = 414 may be within the circumscribing triangle ABC, the height n will be somewhat less than 158 units, since the height to the apex AF is less than 380 units by 6. Let A I = 221 units ; Height A I x base GH = 221 x 414 2 = -J- circumference. Pyramid AGH = 1 of ^ = circumference = 40 degrees. FI = AF Ai = 374 221 = 153 units. This great Dashour pyramid = -J circumference. A pyramid having the same base and height = side of base = j- the cube = -J- of 3 cubes = * distance of the moon. The pyramid AGH = -J- circumference. 3 times the cube of the side of base B c = distance of the moon from the earth. 150 times the distance of the moon = distance of Mercury. 150 times the distance of Mercury = distance of Belus. This great pyramid of Dashour : 3 times the cube of the side of the base BC :: - circumference I distance of moon :: -J circumference .* 9'55 circumference. Pyramid AGH : pyramid = -J- cube of BC :: J circumference -i- distance of the moon. GREAT DASHOUR PYRAMID. 343 If the sides DG, EH be produced till they meet at M, it will be found that MF = 838 units. Height MF x base DE = 838 x 521 2 = 2 circumference. Pyramid MDE = ^of2=f circumference. Thus 3 times the pyramid MDE = 2 circumference, and 3 times the cube of B c = distance of the moon. Pyramid ABC + pyramid AGH = -J + -i- = -| = -| circumference ; .'. pyramid ABC + pyramid AGH = pyramid MDE. The 2* is a quantity impossible to express in numbers ; but all the ordinates, as GH, DE, cc MF, the distance from M, and continually increase from at M to 414 &c., or 1 at GH to 521 &c., or 2*, which is represented by the line DE. 1-26 3 = 2-000376. So the cube root of 2 will be less than 1*26, or DE will be to GH in a less proportion than 1*26 I 1. GH 3 : DE 3 :: 1 2 (2 GH) 3 : 1 : 4 (2 GH) 3 : 1 : 8 (2 DE) 3 : 1 : 16 (4 GH) 3 : 1 : 32 (4 DE) S : 1 : 64 DE 3 = f circumference. The cube of perimeter = (4 DE) 3 = ^ x 64 = 80 circumf. 18 x 80 = 1440 circumference = distance of Mercury. The 2 sides of the bases of the pyramids ABC, AGH, are BC, GH. Their sum = BC + GH = 713 + 414 = 1127 units. Cylinder having height = diameter of base 1131 = 1131 3 x -7854 = 10 circumference 1130 3 &c. = -J distance of the moon. The difference of the sides = BC GH = 713 -414= 299 units ; and 300*5 3 = -fa distance of the moon. For the cube of Cephrenes = 601 3 = distance of moon. DE 3 GH 3 GH 3 DE 3 DE 3 344 THE LOST SOLAR SYSTEM DISCOVERED. So the cube of 300-5 = -J- 601 3 = of = TO distance of the moon. The cube of the side DE + the cube of the side GH ~ i + I circumference = 52 1 3 &c. + 414 3 &c. Sum of the two sides = DE + GH = 521 + 414 &c. = 935 ; 934 3 = f distance of the moon. Cylinder having height = diameter of base 2 BC, 2x713 units, will = 20 circumference = double the cylinder diameter BC + GH, 1131 units. The cube, sphere, and cone, diameter 2 BC will be double the cube, sphere, and cone, diameter BC + GH. 72 cylinders diam. 2 BC = 1440 circumf. = dist. of Mercury 72 x 150 Belus 9 cylinders diam. 4 B c = Mercury 9 x 150 Belus Hence 3 cubes of B c - = distance of the moon and 9 cylinders, diameter 4 BC = Mercury 5 cubes of side of base of Cephrenes = moon and 3x5 = 15 cylinders, diam. = perimeter of base = distance of Mercury 4 cubes of side of Cheops' - = moon and 3 x4 = 12 cylinders, diam.= perimeter will = distance of Mercury 3 cubes of the side of the great Dashour pyramid = distance of moon and 3x3 = 9 cylinders, diam. = perimeter will = distance of Mercury. Since 3 cubes of BC = distance of moon, so 3 x 3 or 9 cylinders, diameter 4 BC = distance of Mercury. Also |- cylinder, diameter 8 B c = V = 3 = earth. Thus 3 cubes of BC = moon 3 cylinders, diameter SBC = ,, earth 3x3 or 9 - 4 BC = Mercury So f cube of 2 B c moon J- cube - = 2 = diameter of orbit of the moon. GREAT DASHOUR PYRAMID. 345 -f cylinder, cliam. 16 BC - = distance of the earth f cylinder - - = 2 = diameter of orbit of the earth. - = distance of moon Cube of side of base - Cube of perimeter - Cube of 4 perimeters - 3 cubes of 4 times perimeter = 4096 = 2 x 2048 and 2 x 2045 = 2 distance of Jupiter. Thus 3 cubes = diameter of the orbit of Jupiter. If the cubes of the sides of the base of the Saccarah and Dashour pyramids be 566 3 : 713 3 as ^ : -J- distance of moon i : 2. Then if a = side of base, and b = height of Dashour and c= d= a 3 will = 2 c 3 a = 2*c if b* = b = then a 2 x ^ = (2^) 2 x = x . . pyramid 2 x b =2 pyramid but b = Saccarah Hence when the cubes of the sides of base are as 1 : 2, and contents as 1 : 2, the cubes of their heights will be as 1:2, and cubes of hypothenuses as 1 12. For hy pothenuse 2 = (a 2 + # 2 ) hypothenuse =( hypothenuse 3 = (a 2 -f or hypothenuse 3 = (sum of squares of 2 sides)3 346 THE LOST SOLAR SYSTEM DISCOVERED. Thus cube of hypothenuse of the greater triangle = twice cube of hypothenuse of the less triangle. Hence it appears, by similar triangles, that when the sides of a right-angled triangle are double the sides of another, each to each, then the hypothenuse of the greater triangle will be double the hypothenuse of the less triangle. When the squares of the sides are double, each to each, then the square of the hypothenuse of the greater triangle will be double the square of the hypothenuse of the less triangle. When the cubes of the sides are double, each to each, then the cube of the hypothenuse of the greater triangle will be double the cube of the hypothenuse of the less triangle. Hence height x base of Saccarah pyramid = 293 x 566 2 , &c.:=-f- circumference Pyramid =-^3. Height x base of Dashour pyramid = 370 = 713 2 =f circumference Pyramid =f. Cubes of heights are as 293 3 : 270 3 : : -J : I circumference : : 1 : 2. Cubes of sides of bases are as 566 3 : 713 3 : : i : | distance of moon : : 1 I 2. Contents as ^ : -J circumference : : 1 : 2. Cubes of hypothenuses are as 637 3 I 802 3 : : 1 I 2. The Nubian pyramids are said to be about 80 in number, but generally of small dimensions. Some have propyla in front of one side. One portico is sculptured, and has an arched roof constructed with a keystone ; the whole curve consists of five stones. There is an arched portico, similarly constructed, at Jebel Barkal, near the Nile, where there are NUBIAN PYRAMIDS. 347 also pyramids with propyla in front of them. There are also pyramids at Nouvri, a few miles north of Jebel Barkal. Waddington describes the largest as containing within it another pyramid of a different date, stone, and architecture. The inner is seen from a part of the outer one having fallen off. The base line of this pyramid is 159 feet (48*5 metres), according to Cailliaud, or 152 feet, according to Waddington, who states the height at 103 feet 7 inches. Taking Caillaud's base and Waddington's height, we have height =103 -6 feet, say ^105 feet = 91 units =f stade, side of base = 159 feet =137*5 units. Then -- height x area base = content pyramid =^- 91 x 137 , &c. circumference of earth. Or the content of such a pyramid will = i~o~o f * nat f Cheops. -| stade is associated negatively with the height of this pyramid which = 1 -J=-| stade. 8 o We have not met with the dimensions of any other Nubian pyramid. Pyramid =^oT circumference = degrees. Great Dashour pyramid =f circumference = 200 degrees. Consequently the pyramid at Nouvri is the reciprocal of the Great Dashour pyramid. Most of the Nubian pyramids have not their sides placed opposite to the four cardinal points. None of them appears to have been entered. The following description of the Nubian pyramids is ex- tracted from " Egypt and Mehemet AIL" " Two groups of pyramids stand near Djebel-Birkel, in Nubia ; one contains only a few pyramids, but the other has twice as many in good condition. Among the former is one that has almost entirely fallen in, which is larger and dif- 348 THE LOST SOLAR SYSTEM DISCOVERED. ferent in form from the others; and it appears to be of a more remote age. The others, 17 in number, vary con- siderably in style from the Egyptian pyramids, but they are certainly not older, nor, indeed, are they very old. In fact, they look as smooth and uninjured as if they had been but just completed. I ascended one of them, which may be done without difficulty, because each layer of stones forms a convenient step, and only the four corners, from top to bottom, are covered with a polished, rounded stone mould- ing, and found on the summit a square wooden beam fixed in the wall, which had come to light by the falling of a stone, and, though thereby exposed to the wind and weather, was still as sound as if new. " None of these pyramids are above 80 feet high, and they are comparatively smaller at the base than the Egyptian pyramids, and more tapering. " Only a few of these pyramids had sculptures, which~were softer and more voluptuous than the Egyptian style admits ; one of these high reliefs represented a queen seated on] a throne, the pedestal of which consisted of lions, with a rich covering thrown over them. "I consider the majority of the pyramids of Nour to be the most ancient of all the Ethiopian monuments now extant. They are not so taper as the pyramids of Birkel, and conse- quently more nearly resembling the Egyptian ; neither has any of them the peculiar projecting entrance of those at Birkel, nor do the layers of stone form steps by which to ascend them. On the whole the remains of rather more^than forty may be distinguished, but only sixteen of them are in tolerable preservation, and even these are much injured by the weather, and in a dilapidated state. They are built en- tirely of rough-hewn sandstone and a kind of ferruginous pudding-stone, cemented with earth, and many of them appear to have been tumuli of mould, afterwards covered with stones. The nature of the circumjacent ground af- fords reason to conjecture that not only all these pyramids were encompassed by a canal communicating with the Nile, but even that several others traversed the place on which NUBIAN PYRAMIDS. 349 they stand. One of these monuments exceeds all its com- panions in extent, and its outer sides are so broken and shat- tered that we had no difficulty in ascending its summit. The form of this singular structure differs entirely from those that surround it ; and it appears to have consisted of several stories, of various degrees of steepness. The entire height of this truncated pyramid, as it now stands, is nearly 100 feet, and its circumference about four times that extent." A pyramid having height = side of base = 113 feet = 98 &c. units would = 1 degree, or -^fao circumference ; 2\ plethrons less 2 units = 101-5-2-5 = 98-75 units. Hence a pyramid having the height = side of base = f __ 3 plethron less f unit, will = a degree nearly, or = \ 98 &c. units. If the height to the platform of this teocalli were 100 feet, then 13 feet would be the height of the apex of this hypo- thetical pyramid above the platform of the teocalli. As we know of no complete dimensions of any of the small pyramids, we have given this, as an example among other similar cases, to show the general application of the method of calculation for ascertaining the contents of such pyramids in terms of the cubic unit, or circumference of the earth. We only suppose this method of calculation to be appli- cable to some of the small pyramids for pyramids and obe- lisks continued to "be erected ages after their geometrical principles of construction were lost. Again, if the height 113 feet were divided into 9 equal parts, like the tower of Belus, then the height of each ter- race would = 113 -f- 9 = 12*55 feet. So that the height of the apex above the platform would = 12 '5 5 feet = 11 units. The contents of the two pyramids will be as 113 3 I 28 1 3 :: 1 I 15 degrees :: TFO ^V circumference. The height of the tower of Belus = side base = 1 stade = 281 feet. 350 THE LOST SOLAR SYSTEM DISCOVERED. Lepsius reckons 69 Egyptian pyramids in the vicinity of Memphis, all within a line of 56 miles, and 139 at and near Meroe, in Upper Nubia. SIDE OF BASE. 80 pyramids at Meroe sandstone, 60 to 20 feet 42 Noori 100 to 20 17 Gabel Birkel 80 to 23 The arch, both round and pointed, is coeval with the era of these last pyramids. Gliddon remarks that the style of Egyptian architecture was grand and chaste, while the column now termed Doric, and attributed to the Greeks, was in common use in the reign of Osortasen, which precedes the Dorians by 1000 years. The arch, both round and pointed, with its perfect key- stone, in brick and in stone, was well known to the Egypt- ians long before this period ; so that the untenable assertion, that the most ancient arch is that of the Cloaca Maxima at Rome falls to the ground. In architecture, as in everything else, the Greeks and Romans obtained their knowledge from their original sources in Egypt, where still existing ruins attest priority of inven- tion 1000 years before Greece, and 1500 years before Rome. These topics are now beyond dispute, and may be found in the pages of the Champollion school. Until the last few years they were utterly unknown to history. It is by these chronicles, or " foolish things," as Josephus calls the enduring pyramids, that the scientific claims of the ancients have been transmitted to posterity, ages after every other record had perished. These monumental records of science and skill have been found in all parts of the world, constructed by colonies, com- bining a priesthood with the learning and science of an early age. These colonies may have been founded by some of the great Cyclopian family, known by the various designations of Shepherd Kings of Egypt, the Anakim of Syria, the Os- cans of Etruria, and the Pelasgians of Greece. Such were the wandering masons, who appear to have been both archi- NUMBER OF PYRAMIDS. 351 tects and civil engineers, to have travelled round the world, building cities, erecting temples for worship, and constructing canals for irrigation and commerce ; thus making the barren land fruitful, and, at the same time, facilitating the transport of the productions of the soil, and so promoting the temporal welfare of man ; while the priests or magi administered to his spiritual wants, and controlled him by laws which they made and enforced. We have applied the Babylonian standard to the measure- ments of the numerous passages, chambers, and sarcophagi within the pyramids of Gizeh, made by Colonel Yyse, where the cubes represent planetary distances. But the numerous instances already given may be sufficient to show the mode of application, and the importance of accurate measurements of ancient monuments, designed by the builders as permanent records of the astronomical knowledge of a race unknown when history began. 352 PAET VI. AMERICAN TEOCALLIS. MYTHOLOGY OP MEXICO BEFORE THE AR- RIVAL OF THE SPANIARDS. TEOCALLIS OF CHOLULA, SUN, MOON, MEXITLI. THEIR MAGNITUDES COMPARED WITH THE TEOCALLIS OF PACHACAMAC, BELUS, CHEOPS, THE PYRAMIDS OF MYCER1NUS AND CHEOPS' DAUGHTER, AND SILBURY HILL, THE CONICAL HILL AT AVEBURY. THE INTERNAL AND EXTERNAL PYRAMIDS OF THE TOWER OF BELUS. HILL OF XOCHICALCO. TEOCALLI OF PACHA- CAMAC IN PERU. RUINS OF AN AZTEC CITY. THE BABYLONIAN BROAD ARROW. THE MEXICAN FORMED LIKE THE EGYPTIAN ARCH. DRUIDICAL REMAINS IN ENGLAND. THOSE IN CUMBER- LAND, AT CARROCK FELL, SALKELD, BLACK-COMB. THOSE IN WILTSHIRE, AT WEST KENNET, AVEBURY, STONEHENGE. EX- TERNAL AND INTERNAL CONE OF SILBURY HILL. MOUNT BARKAL IN UPPER NUBIA. ASSYRIAN MOUND OF KOYUNJ1K AT NINEVEH. RECTANGULAR ENCLOSURE AT MEDINET-ABOU, THEBES. THE CIRCLES AT AVEBURY. CONICAL HILL AT QUITO, IN PERU. TOMB OF ALYATTES, IN LYDIA. CONICAL HILL AT SARDIS. STONEHENGE CIRCLES AND AVENUE, CONICAL BARROWS. OLD SARUM IN WILTSHIRE, CONICAL HILL. THE CIRCLE OF STONES CALLED ARBE LOWES IN DERBYSHIRE. CIRCLE AT HATHERSAGE, AT GRANED TOR, AT CASTLE RING, AT STANTON MOOR, AT BAN- BURY, IN BERKSHIRE. HILL OF TARA. KIST-VAEN. STONES HELD SACRED. AMERICAN TEOCALLIS. (Described by Humboldt.) ef AMONG the tribes of people who, from the seventh to the twelfth century of our era, appeared successively in the country of Mexico, five are enumerated, the Tolteques, Cicimeque, Acolhues, Tlascalteques, and Azteques, who, though politically divided, spoke the same language, ob- served the same worship, and constructed pyramidal edifices, which they regarded as the teocallis, or the houses of their gods. These edifices, though of dimensions very different, TEOCALLIS. 353 had all the same form ; they were pyramids of several stories, the sides of which were placed exactly in the direction of the meridian and parallel of the place. The teocalli rose from the middle of a vast square enclosure surrounded by a wall. This enclosure, which one may compare to the 7rspi(3o\os of the Greeks, contained gardens, fountains, habitations for the priests, and sometimes even magazines of arms ; for each house of a Mexican god, like the ancient temple of Baal Berith, burned by Abimelech, was a place of strength. A great staircase led to the top of the truncated pyramid. On the summit of this platform were one or two chapels in the form of towers, which contained colossal idols of the divinity to whom the teocalli was dedicated. This part of the edi- fice ought to be regarded as the most essential ; it was the vaos, or rather the O-TJKOS of Grecian temples. It was there that the priests kept up the sacred fire. By the peculiar arrangement of the edifice, as we have just shown, the sacri- ficer could be seen by a great mass of people at the same time. One saw from a distance the procession of the teo- pixqui, as it ascended or descended the staircase of the pyra- mid. The interior of the edifice served as a sepulchre for the kings and principal personages of Mexico. It is im- possible to read the descriptions which Herodotus and Dio- dorus Siculus have left of the temple of Jupiter Belus, without being struck with the features of resemblance which the Babylonian monument presents when compared with the teocallis of Anahuac. When the Mexicans, or Azteques, one of the seven tribes of the Anahuatlacs (bordering people), arrived in the year 1190, in the equinoctial region of New Spain, they found there the pyramidal monuments of Teotihuacan, Cholula or Cholollan, and Papantla, already erected. They attributed these great works to the Tolteques, a powerful and civilised nation that inhabited Mexico 500 years before. They made use of hieroglyphical writing, and had a year and a chrono- logy more accurate than most of the people of the ancient continent. The Azteques did not know for a certainty if >ther tribes had inhabited the country of Anahuac before VOL. I. A A 354 THE LOST SOLAR SYSTEM DISCOVEKED. the Tolteques. In regarding these houses of the god of the Teotihuacan and Cholollan as the work of the latter people, they assigned to them the highest antiquity of which they could form an idea. It might, however, be possible that they were erected before the invasion of the Tolteques, that is, about the year 648 of the common era. We should not be astonished that the history of any American people did not commence before the seventh century, and that the history of the Tolteques should be also as uncertain as that of the Pelasgians or Ausonians. The deeply read M. Schlrezer has proved almost to evidence that the history of the north of Europe does not ascend beyond the tenth century, an epoch when the Mexican plane already presented a civilisation much further advanced than that of Denmark, Sweden, or Russia. The teocalli of Mexico was dedicated to Tezcatlipoca, the first of the Azteque divinities after Teotl, who was the su- preme and invisible Being, and to Huitzilopochtli, the god of War. It was erected by the Azteques after the model of the pyramids of Teotihuacan, only six years before the dis- covery of America by Christopher Columbus. This trun- cated pyramid, called by Cortez the principal temple, had a base 97 metres long, and about 54 metres high. It is not surprising that a building of these dimensions should have been destroyed in so short a time after the siege of Mexico. In Egypt there remains scarcely any vestige of the enor- mous pyramids that rose from the middle of Lake Mceris, and which Herodotus says were ornamented with colossal statues. The pyramids of Porsenna, of which the descrip- tion appears somewhat fabulous, had statues^ according to Yarro, more than 80 metres high ; these also have disap- peared from Etruria. But if the European conquerors have overthrown the Azteque teocallis, they have not equally succeeded in de- stroying the more ancient monuments, those which are at- tributed to the Tolte"que nation. We shall now give a short description of these monuments, remarkable for their form and magnitude. TEOCALLIS. 355 The group of Teotihuacan pyramids stands in the valley of Mexico, eight leagues distant and north-east of the capital, on the plain called Micoatl, or path of the dead. One still observes two great pyramids dedicated to the sun (tonatiuh) and the moon (metzitli), and surrounded by some hundreds of small pyramids forming streets running exactly from north to south and from east to west. One of the two great teocallis has 55, the other 44 metres perpendicular eleva- tion. The base of the first is 208 metres long ; whence it re- sults that the Tonatiuh Yztaqual, from the measurements of M. Oteyza, made in 1803, is more elevated than Mycerinus, or the third of the great pyramids of Djizeh in Egypt, and the length of its base is nearly that of Cephrenes. The small pyramids that surround the great houses of the sun and moon have scarcely 9 or 10 metres of elevation. Ac- cording to the tradition of the natives, they served as sepulchres for the chiefs of the tribes. Around those of Cheops and Mycerinus in Egypt are also seen eight small pyramids placed symmetrically and parallel to the sides of the great ones. The two teocallis of Teotihuacan had four principal stories ; each of these was subdivided in small steps, of which the edges may still be distinguished. The middle is clay mixed with small stones ; it is covered with a thick wall of porous amygdaloid. This construction re- calls to mind one of the Egyptian pyramids at Saccarah, which has six stories, and which, according to Pococke, is a mass of stone and yellow mortar, covered externally with rough stones. At the top of the great Mexican teocallis were placed two colossal statues of the sun and moon. They were of stone and covered with plates of gold ; these plates were carried away by the soldiers of Cortez. While the Bishop Zumaraga, a Franciscan monk, under- took to destroy all that related to the religion, history, or antiquities of the indigenous people of America, he also broke the idols in the plain of Micoatl. There may still be seen the remains of a staircase, formed of large hewn stones, which formerly led to the platform of the teocalli. To the east of the group of pyramids of Teotihuacan, in A A 2 356 THE LOST SOLAR SYSTEM DISCOVERED. descending the Cordilleras near the Gulf of Mexico, in a thick forest called Tajin, rises the pyramid of Papantla. Its discovery was accidentally made by some Spanish hunters about thirty years ago; for the Indians contrive to con- ceal from the whites every object of ancient veneration. The form of this teocalli, which has six, or perhaps seven, stories, is more tapering than that of any of the other monu- ments of this kind. Its height is about 18 metres, while the length of its base is only 25 ; it is consequently lower by almost one half than the pyramid of Caius Cestius at Rome, which is 33 metres high. This little edifice is constructed of hewn stones of an extraordinary size, very finely and regu- larly cut. Three staircases lead to the top. The coating of these stories is ornamented with hieroglyphical sculpture, and small niches are symmetrically disposed. The number of these niches appear to allude to the 318 signs simple, and composed of the days of Cempohualilhuitl, or calendar civil of the Tolteques. The greatest, the most ancient, and most celebrated of all the pyramidal monuments of Anahuac is the teocalli of Cholula. At this day it is called the mountain made by the hands of man. When seen at a distance, one is tempted to take it for a natural hill covered with vegetation. Cortez described Cholula as being more beautiful than any- city in Spain, and well fortified. From a mosque (teocalli) he reckoned more than 400 towers. Humboldt reckoned the number of inhabitants, when he visited it, at 16,000. Since then Bullock has estimated them at 6000 only. The plane of Cholula is 2200 metres above the level of the sea. At a distance is seen the summit of the volcanic Orizaba covered with snow. This colossal mountain is 5285 metres in height, from the sea. The teocalli of Cholula has four platforms of equal height, and its sides appear to have been placed with great exactness opposite the cardinal points of the compass ; but as the angles are not very well defined, it is difficult to discover with correctness their exact original direction. This pyra- midal monument has a more extended base than any other TEOCALLIS. 357 edifice of the same description found in the old continent. I have measured it with care, and am satisfied that its perpendicular height is not more than fifty-four metres, and that each side of its base is 439 metres in length. Bernal Diaz del Castillo, a private soldier in the expedition of Cortez, amused himself in counting the number of steps in the staircases, which led to the platforms of the different teocallis ; he found 114 in the great temple of Tenochtitlam, 117 in that of Tescuco, and 120 at Cholula. The base of the pyramid at Cholula is twice as large as that of Cheops, in Egypt, but its height is very little greater than that of Mycerinus. In comparing the dimensions of the temple of the sun, at Teotihuacan, with those of the pyramid at Cholula, one sees that the people who constructed these remarkable monuments had the intention of making them all of the same height, but with bases of which the lengths should be in the pro- portion of one to two. As to the proportion between the base and height, one finds it very different in different monuments. In the three great pyramids of Djizeh, their heights are to their bases as 1 : 1*7; in the pyramid of Papantla, covered with hieroglyphics, this proportion is as 1 : 1-4; in the great pyramid of Teotihuacan, as 1 I 3 -7 ; and in that of Cholula as 1 I 7*8. This last monument is built with unburned bricks alternating with layers of clay. The Indians of Cholula assured me that the interior is hollow, and that while Cortez occupied their town, their ancestors had concealed within it a number of warriors, with the intention of making a sudden attack on the Spaniards ; but the materials of which the teocalli is constructed, and the silence of contemporary historians, render this assertion but little probable. However it cannot be doubted but that there were in the interior of this pyramid, as in other teocallis, considerable cavities which served for sepulchres; the discovery of them was owing to accident seven or eight years ago ; the route from Puebla to Mexico, which formerly passed by the north of the pyramid, was changed, and in forming the new road they cut through the first platform, so A A 3 358 THE LOST SOLAR SYSTEM DISCOVERED. that an eighth part of it remains isolated, like a heap of bricks. In making this cut they found in the interior a square house, formed of stones and supported by props of cypress ; it contained two bodies, idols formed of basalt, and a great number of vases skilfully painted and enamelled. No care was taken to preserve these objects ; but it is said .to have been carefully ascertained that this chamber had no outlet. In supposing this pyramid not to have been built by the Tolteques, the first inhabitants of Cholula, but by prisoners made by the Cholulains, one might believe that these were the bodies of unfortunate slaves that had been caused to perish intentionally in the interior of the teocalli. We examined the ruins of this subterraneous chamber, and observed a particular arrangement of bricks, tending to diminish the pressure on the roof. The natives being ig- norant of the arch, placed very large bricks horizontally, so that the upper course should pass beyond the lower ; hence resulted an assemblage of steps, which supplied in a measure the Gothic arch. Similar vestiges of this rude substitute for the arch have been found in several Egyptian edifices. It would be interesting to excavate a gallery through the centre of the teocalli of Cholula, to examine its internal construction ; and it is astonishing that the desire to discover hidden treasures has not already caused an attempt to be made. During my travels in Peru, in visiting the vast ruins of the city of Chimu, near Mansiche, I entered the interior of the famous Huaca of Toledo, the tomb of a Peruvian prince, in which Garci Gutierez of Toledo dis- covered, while digging a gallery, in 1576, more than the value of five millions of francs (about 208,3337. sterling), in solid gold ; this is proved by accounts preserved in the town- hall of Truxillo. The great teocalli of Cholula, called also the mountain of tmburned bricks (Tlalchihualtepec), had on its summit an altar dedicated to Quetzalcoatl, the god of the air. This Quetzalcoatl (a name signifying serpent covered with green feathers, from coatl, serpent, and quetzalli, green feather) is without doubt the being the most mysterious of all the MYTHOLOGY. 359 Mexican mythology : tins was a white man with a beard like the Bochica of the Muyscas, of whom we have already spoken : he was chief priest to Tula, the lawgiver, the chief of a religious sect who, like the Sonyasis and the Buddhists ofHindostan, imposed upon themselves penances the most cruel ; he introduced the custom of piercing the lips and ears, and wounding the rest of the body with thorns of the aloe, or the prickles of the cactus, and introduced reeds into the wounds to cause the blood to flow more freely. In a Mexican drawing, at the Vatican, I have seen a figure re- presenting Quetzalcoatl assuaging by his penitence the anger of the gods, when, 13,060 years after the creation of the world (I give the chronology very vaguely stated by Father Rios), there was a great famine in the province of Culan ; the saint retired towards Tlaxapuchicalco, near the volcanic Catcitepetl (talking mountain), where he marched with naked feet over the leaves of the aloe armed with thorns. This reminds one of the Rishi, hermits of the Ganges, the pious austerity of whom the Pouranas celebrate. The reign of Quetzalcoatl was the golden age of the people of Anahuca : then all the animals, and even men, lived in peace, the earth produced without culture the richest harvests, the air was filled with a multitude of birds admired for their songs and beauty of their plumage ; but this reign, like that of Saturn, and the happiness of the world, was not of long duration ; the great spirit Tezcatlipoca, the Brahma of the people of Anahuac, offered to Quetzalcoatl a draught, which, in rendering him immortal, inspired him with the desire to travel, and particularly with an irresistible wish to visit a remote country, which tradition called Tlapallan. The analogy of this name with that of Huehuetlapallan, the country of the Tolteques, appears not to have been acci- dental ; but how can one conceive that this white man, priest of Tula, should direct his course, as we shall soon see, to the south-east, towards the plains of Cholula, thence to the eastern coast of Mexico, to arrive at a northern country, whence his ancestors departed in the year 596 of our era. Quetzalcoatl, in traversing the territory of Cholula, acceded A A 4 360 THE LOST SOLAR SYSTEM DISCOVERED. to the entreaties of the inhabitants, who offered him the reins of government ; he remained during twenty years among them, taught them the fusion of metals, instituted great fasts of twenty-four days, and regulated the inter- calations of the Tolteque year ; he exhorted them to peace ; he desired they should make no other offerings to the divinity than the first fruits of the seasons. From Cholula, Quetzal- coatl passed to the mouth of the river Goasacoalco, whence he disappeared after having announced to the Cholulains that he should return hereafter to govern them again, and renew their happiness." The descendants of this saint the unfortunate Montezuma believed he recognised in the companions in arms of Cortez : (< We know by our books," said he, in his first interview with the Spanish general, "that myself and all those who inhabit this country are not the original inhabitants, but that we were strangers that came from a great distance. We know also that the chief who brought our ancestors returned for a time to his native country, and when he returned here to seek those who were established, he found them married with the women of this country, having a numerous posterity, and living in cities which they had built ; our people would not obey their ancient chief, and he returned alone. We have always believed that his descendants would come some day to take possession of this country. Considering that you come from that part where the sun was born, and that, as you assure me, you have known us for a long time, I can no longer doubt that the king who sent you is our natural chief." The marvellous account which the Abbe Clarvigero gives of the more than oriental pomp of the barbaric Sultan of Te- nochtitlan, his luxurious living, magnificent palaces, and extensive menageries may be compared with the following extract from the " Journal des Debats," which states that Layard's Assyrian discoveries confirm all that ancient authors tell us of the luxury indulged in by the most magnificent of the Asiatic sovereigns ; and if already we knew, by the tes- timony of Lucian, that a number of wild beasts were kept in the Assyrian temples, we now learn from Layard that the MYTHOLOGY. 361 great king furnished his menagerie with rare animals from different countries, either for utility or curiosity, such as the elephant, the rhinoceros, the camel with two humps, from Bactriana, the large kind of monkey called the sylvan, &c. Among the numerous varieties of the feathered race which enliven the forests of Guatimala, Juarros says the quetzal holds the first rank for its plumage, which is of an exquisite emerald green : the tail feathers, which are very long, are favourite ornaments with the natives, and were formerly sent as a valuable present to the Sultans of Tenochtitlan. Great care was taken not to kill the birds ; and they were released after being despoiled of their feathers. The birds, them- selves, adds Juarros, as if they knew the high estimation their feathers were held in, build their nests with two open- ings, that, by entering one, and quitting them by the other, their plumes may not be deranged. This most beautiful bird is peculiar to this kingdom. Manrique witnessed at Arracan a splendid ceremony of the idol Paragri, eleven palms high, made of silver, and trampling under foot a bronze serpent, covered with green scales. The Indian god of the visible heavens is called Indra, or the King, and Divespetir, Lord of the Sky. He has the character of the Roman Genius, or Chief of the Good Spirits. His weapon is Vajra, or the thunderbolt. He is the regent of winds and showers ; and though the east is peculiarly under his care, yet his Olympus is Meru, or the North Pole, allegorically represented as a mountain of gold and gems. He is the prince of the beneficent genii. (Jonet.) The Parsis historians in the Persian Chronicles, says Yolney, relate that the reign of Djem-Chid was glorious, when God, to punish him for exacting adoration, excited against him Zohak. Zohak overturned Djem-Chid, who disappeared and tra- velled 100 years over the whole earth. Zohak, when king, became a cruel tyrant; he invented various tortures, among others, that of crucifying and flaying alive : he had several surnames, among them one was Quas-lohoub, that is to say, the Quaisi of the glittering arms ; another name was Ajde- 362 THE LOST SOLAR SYSTEM DISCOVERED. hac and Mar, that is to say, serpent, because he had on his shoulders two serpents attached to two ulcers, which the devil had produced there by two kisses. We shall next quote the historical authority of an empire that has been from a remote period aristocratically exclusive, where we find the mythological antiquity of the serpent. It is stated in the Magasin Pittoresque, from manuscripts in the King's Library at Paris, that Fo-hi civilised China 3254 years before our era, and reigned 115 years. He had the body of a dragon, the head of an ox, according to some ; others say he had the body of a serpent and the head of Kilin. It is easy here to distinguish an Indian type. Again, others say he had a long head, fine eyes, irregular teeth, lips of the dragon, a white beard that reached to the earth ; his height was 9 feet 1 inch ; he belonged to heaven, and de- parted for the east. He was adorned with all the virtues, and he united whatever there was of the highest or lowest. Here we find half the body that of a dragon or serpent, the beard white, reaching to the ground, and Fo-hi's departure easterly. The name of Quetzalcoatl signified a serpent covered with green feathers ; he was a white man with a beard ; he also disappeared, and was thought to have gone northerly, though he departed from the east coast of Mexico. He promised to return. The reign of Quetzalcoatl was the golden age of Anachua. He taught them how to fuse the metals, and desired they would make no further offerings to the divinity than the first fruits of the seasons. " The pyramid of Belus was a temple and a tomb. In like manner, the tumulus of Calisto in Arcadia, described by Pausanias as a cone made by the hands of man, but covered with vegetation, had on its top a temple of Diana. The teo- callis were also both temples and tombs ; and the plain in which are built the houses of the sun and moon at Teoti- huacan is called the Path of the Dead. The group of pyra- mids at Djizeh and Saccarah in Egypt, the triangular pyra- mid of the queen of the Scythians, mentioned by Dioclorus, the fourteen Etruscan pyramids, which are said to have been enclosed in the labyrinth of king Porsenna at Clusium, the TEOCALLIS. 363 tumulus of Alyattes at Lydia, the sepulchres of the Scandi- navian king Gormus, and his queen Daneboda, the tumuli found in Virginia, Canada, and Peru, in which numerous galleries built with stone communicate with each other by shafts, and extend through the interior of these artificial hills, also the pagoda of Tanjore, although pyramidal, and formed of many stories, wants the temple on the top, and therefore, like all other pagodas in Hindostan, is said to have nothing in common with the Mexican temples. The platform of the pyramid of Cholula, upon which I made a great number of astronomical observations, measures 4200 square metres. A small chapel dedicated to Notre- Dame de los Remedios, and surrounded with cypress, has replaced the temple of the god of the Air, or the Mexican Indra : an ecclesiastic of Indian race daily celebrates mass on the summit of this ancient monument. At the time of Cortez, Cholula was regarded as a holy city ; nowhere was there to be found a greater number of teocallis, more priests and religious orders, more magnifi- cence in the worship, more austerity among the fasting and penitent. We have before noticed the striking analogy observable between the Mexican teocallis and the temple of Bel or Belus, at Babylon. This analogy had already occurred to M. Zoega, though he was only able to procure very incom- plete descriptions of the group of pyramids at Teotihuacan. According to Herodotus, who visited Babylon, and saw the temple of Belus, this pyramidal monument had eight stages : its height was a stade ; the length of its base equalled its height ; the area included by the exterior wall equalled four square stades. The pyramid was constructed with bricks and asphalt; at the top there was a temple (vaoi), and another near the base ; the first, according to Herodotus, was without statues; there was only a table of gold, and a bed, upon which reposed a woman chosen by the god Belus. Diodorus Siculus, on the contrary, asserts that this higher temple had an altar and three statues, to which he gave, after the idea imbibed from the Greek worship, the names 364 THE LOST SOLAR SYSTEM DISCOVERED. of Jupiter, Juno, and Rhea ; but these statues and monu- ments neither existed at the time of Diodorus nor Strabo. In the Mexican teocallis one distinguishes, as in the temple of Bel, the naos inferior to that which is found upon the platform of the pyramid ; this destination is clearly indicated in the letters of Cortez, in the " History of the Conquest," written by Bernal Diaz, who resided many months in the palace of the king Axajacatl, and, consequently, opposite the teocalli of Huitzilopochtli. No ancient author, neither Herodotus, Strabo, Diodorus, Pausanias, Arrian, nor Quintus Curtius intimated that the temple of Belus was placed according to the four cardinal points of the compass, as are the Egyptian and Mexi- can pyramids. Pliny merely observes that Belus was regarded as the inventor of astronomy. Diodorus reports that the temple at Babylon served the Chaldaeans as an ob- servatory : " One understands," says he, " that this erection was of an extraordinary height, and that the Chaldeans there made their observations of the stars, so that their risings and sittings could be very accurately noted from the elevation of the building." The Mexican priests also observed the position of the stars from the tops of the teocallis, and announced to the people, by the sound of the horn, the hours of the night. These teocallis have been erected in the in- terval between the epoch of Mahomet and the reign of Fer- dinand and Isabella ; and one cannot regard without astonish- ment that these American edifices, of which the form is almost identical with that of one of the most ancient monu- ments on the banks of the Euphrates, should belong to a period so near our own." Having quoted the descriptions and measurements of dif- ferent American teocallis, we shall state the results of our calculations in succession, and draw all the teocallis on the same scale, so that their relative magnitudes may be com- pared. The internal and external pyramids of each teocalli will be similar. The side of the base of the internal pyramid will equal the side of the base of the lowest terrace, and the apex will be in the centre of the top platform. The side of TEOCALLIS. 365 the base of the external pyramid will equal the base of the circumscribing triangle, and height to apex equal height of triangle. Each teocalli has two pyramids, and the number of ter- races represented. Belus and Cheops' pyramids are both teocallis, or terraced pyramids ; their internal and external pyramids are drawn on the same scale. The eight terraces of Belus are represented, but not those of Cheops, the number being about 208. The pyramids of Mycerinus and of Cheops' Daughter are similar to Cheops'. Fig. 69. Teocalli of Cholula. 70. Sun. ,,71. Moon. 72. Mexitli. 73. Pachacharnac. 74. Belus. 3 , 75. Cheops. 76. Pyramid of Mycerinus 77. Cheops' Daughter. 78. Silbury Hill. 366 THE LOST SOLAR SYSTEM DISCOVERED. Fig. 73. Fig. 74. Fig. 75. Fig. 76. Fig. 77. Fig. 78. TEOCALLI OF CHOLULA. 367 Side of base of lowest terrace of Cholula = 439 metres=1245 units. Height to platform = 54 metres = 153 units, | stade =3'75 plethrons=151'875 units, -f stade +-| unit =152, &c. units, 5 stades + 5 units =1220 units. Internal Pyramid. Height x area base = 152, &c. x 1220 2 , &c. = 2 circumference. Pyramid = f circumference = 240 degrees. External Pyramid. Height x area base = 172, &c. x 1374 3 -^ I ^ distance of moon. Pyramid = T V distance of moon. 1374 = 687=!- side of base, if =684=2x342, 342 being Babylonian numbers. Cube of side of base = (2 x 684) 3 = 22 6 circumference. Cube of perimeter =(8 x 684) 3 = 1446 circumference distance of Mercury = 1440. But, as has been stated, the distance assigned, 1440 cir- cumference, is less than the calculated distance. Thus the distance of Mercury expressed in Babylonian numbers, which are derived from 3 5 rr243, will be = (16x342) 3 = (8x684) 3 6 84 2 = circumference of earth in.stades, 684 2 x243= units. Both these pyramids will be similar. The apex of the less pyramid will be in the centre of the top platform. The apex of the greater pyramid will be 21 units above the top platform, if the teocalli or terraced pyramid were cased as 368 THE LOST SOLAR SYSTEM DISCOVERED. the pyramid of Cheops is said to have been, and the cube of perimeter of its base will = distance of Mercury. Both pyramids will have height to side of base as 1:8. The side of the top platform will accord with that of Humboldtj 184 units; but the side of base of the lowest terrace or side of base of less pyramid will 1220 units. Humboldt's =1245 units. The height to platform accords with that of Humboldt. Cube of height to platform = (! side of base) 3 = (^ 1220) 3 = 152'5 3 = x 16 =:- circumference ; 8 32 2 cubes r::^ circumference = 22*5 degrees. Twice cube of height to platform : cube of side of base of external pyramid : : 22*5 degrees I 22 -6 circumference : : degree I circumference. Cube of side of base of external pyramid =720 times cube of height to platform. Cube of perimeter = 720 x 4 3 = 46080 times cube of height to platform. Cube of 4 times height of external pyramid ==(4 x 172) 3 = 688 3 ^ distance of moon (10 x 688) 3 = ^f^=300. Cube of 40 times height = 300 distance of moon = diameter of orbit of Mercury. Cube of 40 times height ! cube of 4 times side of base : : diameter of orbit of Mercury .' distance of Mercury :: 2 I 1. All the terraces are of equal height ; J- side of lowest terrace = 6 10 units, ^610 = 152, &c. = height to platform, i 152, &c.= 38, &c.=height of a terrace. TEOCALLI OF CHOLULA. 369 610 3 = 2 circumference, 152 3 ,&c. = = = . 4 2 64 32' 38 3 ,&c. = -Lx = _!_. 32 4 2 2048 Side of base of lowest terrace = 2x610=1220. Cube of side = 1220 3 = 2 x8 = 16 circumference. Cube of perimeter = 16 X4 3 = 1024. Content of internal pyramid = -| circumference. Content of external pyramid = I 1 5 - distance of moon. Cube of perimeter of base = distance of Mercury. When the teocalli of Cholula is compared with other pyramids, it is made = circumference of earth; for this estimate was made before we knew that a teocalli repre- sented two pyramids. In all the Mexican teocallis we find the measurement of only one side of the base stated. Humboldt supposes that they were intended to have the sides as 2 : 1. First we calculated the teocallis having the sides as 2 I 1 ; but afterwards by making the base equal to the square of the given side. Should the sides of the base be as 2:1, the content of a teocalli will only equal half of what has been calculated. The cube of the perimeter of the base of the teocalli of Cholula we make = distance of Mercury. Though the sides should be as 2 .* 1, still the cube of 4 times the greater side will = distance of Mercury; and the content of a pyramid having base = square of that side will be what has been computed. Bullock remarks that at a distance the appearance which the teocalli of Cholula assumes is that of a natural conical hill, wooded and crowned with a small church ; but, as the traveller approaches it, its pyramidal form becomes distin- guishable, together with the four stories into which it is shaped, although covered with vegetation, the prickly pear, the nopal, and the cypress. VOL. I. B B 370 THE LOST SOLAR SYSTEM DISCOVERED. This descriptive view of the teocalli suggests the idea that the hanging gardens of Babylon might have been formed by planting trees and shrubs on the terraces of some old teocalli. The tumulus of Calisto, in Arcadia, described by Pausanias as a cone made by the hands of man, but covered with vege- tation, had on the top a temple of Diana. The teocalli of the Sun has four terraces, and the height to the top platform = 55 metres = 180 feet English = 156 units. The side of the base of lowest terrace =208 metres =682 feet = 5 90 units. Height x area base = 156 x 590 2 = ^V distance of moon. Pyramid = i of T V= TO or internal pyramid = -^ distance of moon. If height of external pyramid =181 units, and side of base =685, &c. Height x area base = 181 x 685 2 , &c. =! circumference, The two pyramids will be similar. The apex of the less will be in the centre of the top plat- form. The internal pyramid of Sun will =-J- the external pyramid of Cholula. The external pyramid of Sun will be to internal pyramid of Cholula : : -J- I -f- circumference, :: 3 : 8. 1 4 2 y The internal pyramid of Sun =- = = the external 6 24 24 pyramid of Cholula. 3 9 3 2 The external pyramid of Sun = - = = the internal 8 24 24 pyramid of Cholula. TEOCALLI OF THE SUN. 371 Cube of twice perimeter of base of external pyramid of Sun = (8 x 685) 3 = 5480 3 = distance of Mercury, and (8 x 684) 3 distance of Mercury. or (16 x 342) 3 = distance of Mercury, in Babylonian numbers. (4 x 181) 3 =724 3 = V circumference. 3 cubes of 4 times height of external pyramid = 10 cir- cumference. (3x724) 3 =V x3 3 = 270 10 cubes of 12 times height = 2700 circumference = distance of Yenus 100 cubes of 24 times height =r 216000 circumference = distance of Belus. Internal '. external pyramid : : -J- circumference '. -fa distance of moon : : |- circumference I radius of earth :: quadrantal arc I radius :: circumference : 2 diameters. Cube of height of internal pyramid = 156 3 = -3^- circumference =12 degrees 3 5 = 243 (3 x 342 &c.) 3 = 1028 3 = distance of moon (16 x 342) 3 = distance of Mercury Distance of moon .* distance of Mercury nearly as 3 3 : 16 3 ::1 : 151-7 (2 x 342) 2 x 243 = circumference of earth. Thus the circumference of earth, distance of moon, and distance of Mercury are expressed in Babylonian numbers. The teocalli of the moon is stated to be 11 metres = 36 feet = 31 units lower than the teocalli of the sun, and the base much smaller. .-. height will =15631 = 125 units. No measurement of the base or top platform is given. If the teocallis of the sun and moon were similar then 156 : 125:: 590 or 156 : 590:: 125 say as 123 B B 2 472 &c. 472 &c. 470. 372 THE LOST SOLAK SYSTEM DISCOVERED. Height x area of base of internal pyramid = 123 x 470 2 = -^-Q distance of moon pyramid = -^ Internal pyramid of sun = -fa Pyramids are as 1 : 2. Height x area base of external pyramid of moon will = 143 x 545 2 &c. = -| circumference External pyramid = i External pyramid of sun = -J- Pyramids are as 1 : 2. Cube of side of base of external pyramid of moon = 545 3 &c. (8 x 545 &c.) 3 = distance of Mercury, or cube of twice perimeter of base of external pyramid of sun = twice cube of twice perimeter of base of external pyramid of moon. The cubes of the similar sides of these two pyramids will be as 1 : 2. The cubes of the similar sides of the internal pyramids will be in the same ratio, and so will the pyramids them- selves. Cube of height of external pyramid of moon = 143 3 &c. - 38-4 circumference. Cube of height of external pyramid of sun = 18 1 3 = - circumference. 19-2 Cubes are as 1 : 2. - : 1 :: 1 I 19-2 circumference. 19-2 Cube of height of external pyramid of sun I circumference :: circumference I twice distance of moon. Cube of height of external pyramid of moon '. ^ circumference '. \ circumference I distance of moon. -Bullock, who visited these pyramids, says : " On de- TEOCALLI Or THE MOON. 373 scending the mountain, the pyramids are seen in a plain at about five or six miles distance. As we approached them the square and perfect form of the largest became at every step more and more visibly distinct, and the terraces could now be counted. We soon arrived at the foot of the largest pyramid, and began to ascend. It was less difficult than we expected, though, the whole way up, lime and cement are mixed with fallen stones. The terraces are perfectly visible, particularly the second, which is about 38 feet wide, covered with a coat of red cement eight or ten inches thick, com- posed of small pebble-stones and lime. In many places, as you ascend, the nopal trees have destroyed the regularity of the steps, but nowhere injured the general figure of the square, which is as perfect in this respect as the great pyra- mid of Egypt. On reaching the summit, we found a flat surface of considerable size, but which had been much broken and disturbed." The width of 38 feet for the terraces agrees with the width in the outline we have given of this teocalli with its two bases and two heights. On the summit of the teocalli of the moon are the remains of an ancient building, 47 feet long and 14 wide ; the walls are principally of unhewn stone, three feet thick and eight feet high. Forty-seven feet = 1 plethron. This pyramid is more dilapidated than the greater py- ramid. Sides of ancient building 47 by 14 feet = 40-63 12-1 units 10x40-7 = 407 407 3 &c. = -jig- distance of moon (2 x 407 &c.) 3 = i Cube of 20 times greater side = distance of moon 10x12-3 = 123 123 3 &c. = -^ circumference Cube of 10 times less side = $ circumference. BBS 374 THE LOST SOLAR SYSTEM DISCOVERED. The teocalli, or great temple of Mexitli, occupied the present site of the great cathedral of Mexico. Humboldt mentions its four sides as having corresponded exactly with the cardinal points of the compass ; its base was 97 metres, and height 37 metres ; the point, terminated by a cupola, was 54 metres in height from the base, and its having had five stories, like many of the pyramids of Saccarah, particu- larly like that of Meidoum. It formed a pyramid so trun- cated, that when viewed at a distance it appeared like an enormous cube, upon which were placed small altars with cupolas made of wood ; the point where these cupolas termi- nated was 54 metres above the base. The stair-case to the platform contained 120 steps. Teocalli of Mexitli. Side of base =97 metres =318 feet =275 units, say =279. Height to platform =37 metres =121-4 feet =105 units, say =108. Height x area base of internal pyramid =108 &c. x 279 2 = -f-Q circumference. Pyramid =i of -^ =&. Cube of height =108 3 = gV circumference. Cube of side of base = 27 9 3 = -gV distance of moon. If height of external pyramid =130 units, and side of base =336, then height x area base = 130x336 2 = T ^-o distance of moon, pyramid = -J- of -^ =T6o Cube of height =129 3 &c. = 3-^ distance of the moon. Cube of side of base =336 3 = -J- circumference. Cube of height of external pyramid : cube of side of base of internal pyramid : : -g-i~o : -^ distance of the moon, :: 1 I 10. TEOCALLI OF MEXITLI. 375 Cube of height of internal pyramid .' cube of side of base of external pyramid :: oV 1 circumference, :: 1 : 30. The two pyramids will be similar. If the height of the five terraces be equal, the height of each will =21*6 units = difference of height of the two py- ramids. Cube of side of base of external pyramid = 336 3 =J cir- cumference = 120 degrees. The number of steps to the platform were 120. The Mexitli teocalli is said to have been built after the model of the pyramids of Teotihuacan, only six years before Columbus discovered America, The Cathedral of Mexico stands on the site of the teocalli. This teocalli may be passed over as unimportant, if it were a modern structure, and as no traces of it remain. Tower of Belus. If the height = side of base of internal pyramid = 242 &c. units, Fig. 67. then cube of side of base = 242 3 &c. = circumference, pyramid = of -J = ^ If the height of external pyramid = side of base =262 &c. units, B B 4 376 THE LOST SOLAK SYSTEM DISCOVERED. Cube of side will = 262 3 &c. = -^Q distance of the moon. = radius of the earth. Pyramid = -f- of -^ = T ^ T distance of the moon. Cubes of the sides are as Radius : |- circumference of the earth. Cubes of twice the sides are as 8 Radii' : circumference. 4 diameters I circumference of the earth. The two pyramids are similar. 360 external pyramids = ff = 2 distance of the moon. = diameter orbit of the moon. 360 internal pyramids 3 -/f = 15 circumference. External pyramid of Belus : external pyramid of Cheops, : : y~ .' T *g distance of the moon, :: 1 : 10 Internal pyramid of Belus : internal pyramid of Cheops : : A : \ circumference, :: 1 : 12 If the internal pyramid of the tower of Belus = -^ cir- cumference, and the external pyramid = y^ distance of the moon, the sides of the terraces will be inclined as in Fig. 67., and not perpendicular as in Fig. 56. The top of the tower in Fig. 67. forms the outline of the Royal tent. Cube of Babylon = 120 3 stades, = 29160 3 units. Twice height of external pyramid of tower = 2 x 262 &c. = 524 &c. Section of cube to the height of 524 units 524 1 = = . cube. 29160 55-6 Cube of Babylon = distance of Belus. = 22500 distance of the moon, distance of the earth =400 TEOCALLI OF BELUS. 377 400 1 22500 5625 So distance of the earth will nearly equal a section of the cube of Babylon having the height = twice height of ex- ternal pyramid of tower. Distance of Mercury = -^ distance of Belus. = y^-Q cube of Babylon, and -^ of 29160 = 194. So distance of Mercury = section of cube having height of 194 units, = f of 242 &c. = A stade, = |- height of internal pyramid. Distance of Venus = -^ distance of Belus, = -gL cube of Babylon, and gV of 29160 = 364-25 units =f stade. So distance of Venus = section of cube having height = J- stade, = f- height of internal pyramid. The teocallis and pyramids are drawn on the same scale, so an estimate may be formed of their relative magnitudes. To form a conception of their real magnitudes, their dimen- sions may be compared with some of the public buildings in London. Waterloo Bridge over the Thames is built with granite ; the length = 1280 feet. The side of the base of the internal pyramid of Cholula = 1410 feet. Side of base of the external pyramid = 1589 feet. So that the side of the base of the internal pyramid would exceed the length of the bridge by 130 feet; and side of base of the external pyramid by 309 feet. The height of each pyramid = -J- side of base. The square area of Lincoln's Inn Fields = about that of the base of Cheops' pyramid, and height = f side of base. The dimensions of St. Paul's Cathedral from east to west, 378 THE LOST SOLAR SYSTEM DISCOVERED. within the walls, are stated at about 510 feet ; and the line from north to south, within the portico doors, at 282 feet. 281 feet = 1 stade = height = side of base of the tower of Belus. Humboldt says, " Another monument well worthy the attention of the traveller is the intrenched military station of Xochicalco. This is an isolated hill, 117 metres high, sur- rounded by fosses, and divided by the hand of man into 5 stories or terraces ; the sides of the terraces being formed of masonry. The whole forms a truncated pyramid, having the four sides placed exactly according to the four cardinal points. The platform of this extraordinary monument nearly equals 9000 square metres ; on the top is seen the ruins of a small square edifice, that served, no doubt, as the last resort of the besieged. "The terraces have about 20 metres of perpendicular eleva- tion. They contract towards the top, as in the teocallis or Aztec pyramids, the summit of which is ornamented with an altar. All the terraces are inclined towards the south-east ; probably to facilitate the flow of water during the rains, which are very abundant in this region. The hill is sur- rounded by a fosse pretty deep and very broad : the whole entrenchment has a circumference of about 4000 metres. The magnitude of these dimensions ought not to surprise us: on the ridge of the. Cordilleras of Peru, and on heights almost equal to that of the Peak of Teneriffe, M. Bonpland and my- self have seen monuments still more considerable. Lines of defence and entrenchments of extraordinary length are found in the plains of Canada. The whole of these American works resemble those that are daily discovered in the eastern part of Asia. Nations of the Mongol race, especially those that are more advanced in civilisation, have built walls that separate whole provinces. " The summit of the hill of Xochicalco presents an oblong platform, which from north to south has 72 metres, and from east to west 86 metres in length. This platform is surrounded by a wall of hewn stone, having a height exceeding 2 metres, that served as a defence to the attacked. TEOCALLI OF XOCIIICALCO. 379 ft In the centre of this spacious place of arms is found the remains of a pyramidal monument that had five terraces ; the form resembling that of a teocalli. The first terrace only has been preserved ; the proprietors of a neighbouring sugar manufactory having been barbarous enough to destroy this pyramid, by tearing away the stones to construct their furnaces. The Indians of Tetlama assert that the five terraces still existed in 1750 ; and from the dimensions of the first step or terrace (gradiii) it may be supposed that the whole edifice had an elevation of 20 metres. The sides are placed exactly according to the four cardinal points. The base of this edifice has a length of 20*7 metres, and a breadth of 17*4 metres. What is very remarkable, no vestige of a staircase leading to the top of the pyramid has been discovered, though it is asserted that a stone seat or chair (ximoilalli), ornamented with hieroglyphics, had been found. " Travellers who have examined this work of the native Americans have not been able sufficiently to admire the cutting and polishing of the stones, which are all of the form of parallelopipedons ; the care with which they have united them without any cement being interposed, and the execution of the reliefs with which the terraces or steps are adorned, each figure occupying many stones, and their forms not in- terrupted by the joints of the stones; so that one might suppose the reliefs had been sculptured after the edifice had been built. " Among the hieroglyphical ornaments of the pyramid of Xochicalco we distinguish the heads of crocodiles spouting water, and figures of men sitting cross-legged, according to the custom of several nations of Asia. " The fosse that surrounds the hill, the coating of the terraces, the great number of subterraneous apartments cut in the north side of the rock, the wall that defends the approach to the platform, all concur to give to the monument of Xochicalco the character of a military monument. The natives designate to this day the ruins of the pyramid that rises in the middle of the platform by a name equivalent to that of citadel. The great analogy in form remarkable 380 THE LOST SOLAR SYSTEM DISCOVERED. between this presumed citadel and the houses of the Aztec gods, the teocallis, makes me suppose that the hill of Xochi- calco was nothing else than a fortified temple. The pyramid of Mexitli, or the great temple of Tenachtitlan, contained also an arsenal in its enclosure, and served, during the siege, as a stronghold, sometimes to the Mexicans and sometimes to the Spaniards. The sacred writings of the Hebrews inform us that, from the highest antiquity, the temples of Asia, as, for instance, those of Baal-Berith, at Sichem in Canaan, were at the same time edifices consecrated to worship and entrenchments into which the inhabitants of the city might fly to shelter themselves against the attacks of the enemy. In short, nothing can be more natural to men than to fortify the places in which they preserved the tutelary deities of the country ; nothing more confiding, when public affairs were endangered, than to take refuge at the foot of their altars, and combat under their immediate protection. Among the people where the temples had preserved one of the forms the most ancient, that of the pyramid of Belus, the construction of the edifice might answer the double purpose of worship and defence. In the Greek temples the wall alone that formed the 7rspi/3o\os afforded an asylum to the besieged." Humboldt's description of the hill of Xochicalco, in his " Monuments de Peuple Indigenes de 1'Amerique," differs from that given in his " Essai Politique." One account makes the area of the platform 9000 square metres ; the other makes the sides 72 by 86 metres, which = 6192 square metres. No dimensions of the base are mentioned. "We may remark that 117 metres, the height to the plat- form, = 383-8 feet = 331-9 units, and 33 1 3 &c. = -^ distance of the moon, or cube of height = -^ distance of the moon, = diameter of the earth. Sides of platform are 72 by 86 metres = 203 by 243 units (1 stade) 204 3 &c. = -fe circumference 242 3 &c, = tV TEOCALLI OP XOCHICALCO. 381 Sum of cubes of 2 sides = A + A" == ~o = i circumference = 72 degrees. The height to the platform of the teocalli of Cholula = 152 3 &c. Cube of the heights of the two teocallis will be as 153 3 &c. : 331 3 :: 1 :: 10. It appears that Humboldt had never seen the hill of Xochi- calco, and has given the measurement of M. Alzate. If the teocallis of Cholula and Xochicalco were similar, their contents would be as 1 .' 10. The external pyramid of Cholula = -fa distance of moon. So the external pyramid of Xochicalco would = distance of the moon. The external pyramid of Cheops = ^ distance of the moon, = -fa part of the external pyramid of Xochicalco. The circumference of the fosse is about 4000 metres. The French measured -J- circumference of the earth passing through the poles. A ten-millionth part of this quadrant was made a standard of length and called a metre; being equal to 39*371 English inches. Circumference of fosse = 4000 metres, i = 1000 i circumference = T?r uuuL_. = _J , = one ten-thousandth part of the quadrant from the equator to the pole. .-. cirumference of the fosse will = one ten-thousandth part of the circumference of the earth passing through the poles. Hence the measurement of the earth's circumference made at a very remote period by an unknown race, who constructed the great teocalli of Xochicalco, accords with the measure- ment lately made by the French, if the circumference of the fosse = 4000 metres. Pyramidal Monument. Sides 20-7 by 17 -4 metres, = 58*7 by 49-26 units. 382 THE LOST SOLAR SYSTEM DISCOVERED. 588 3 = - distance of the moon. Supposed height 20 metres. Height x area of base = 588 x 588 x 494 = f- circumfer. Pyramid = of f = 1 Pyramid of 10 times the dimensions of the internal pyramid of the small teocalli = circumference, = internal pyramid of Cheops. Cube of 10 times greater side = ^ distance of the moon. Cube of 10 times less side = i >, , Cube of 20 times greater side = -f-f- = J- Pyramid = of J- = -J- Or pyramid having height = side of base = 20 times the greater side of base of small pyramid = -J- distance of the moon. Should the fosse form a square, side would = 1000 metres. Side of base of teocalli of Cholula = 439 metres. Side of base of a similar teocalli having content =10 times content of the teocalli of Cholula will = 946 metres. So that a similar teocalli of 1 times the content of that of Cholula will have a square base less than the square formed by the fosse. Small pyramid = ^oVo circumference, = 5 times circumference of fosse, 5 x 4000 = 20000 metres. The teocalli has 5 terraces. The pyramid of Pachacamac in Peru is thus described in the recent narrative of the United States' exploring expe- dition : " The Temple of Pachacamac, or Castle, as it is called by the Indians, is on the summit of a hill, with three terraces ; the view of it from the north is somewhat like that of the pyramid of Cholula, given by Humboldt, except that the flanks were perpendicular. The whole height of the hill is 250 feet, that of the mason-work 80 ; the form is rectangular, TEOCALLI OF PACHACAMAC. 383 the base being 500 by 400 feet, At the south-eastern ex- tremity the three distinct terraces are not so perceptible, and the declivity is more gentle. The walls, where great strength was required to support the earth, were built of unhewn square blocks of rock ; these were cased with sun-dried bricks (adobes), which were covered with a coating of clay or plas- ter, and stained or painted of a reddish colour. A range of square brick pilasters projected from the uppermost wall, facing the sea, evidently belonging originally to the interior of a large apartment. These pilasters gave it the aspect of an Egyptian structure. In no other Peruvian antiquities have pilasters been seen by us. On one of the northern ter- races were also remains of apartments ; here the brick ap- peared more friable, owing to a greater proportion of sand ; where they retained their shape their dimensions were nine inches in width by six inches deep, varying in height from nine inches to two feet ; and they were laid so as to break joint, though not always in a workmanlike manner. The remains of the town occupy some undulating ground, of less elevation, a quarter of a mile to the northward. This also forms a rectangle, one-fifth by one-third of a mile in size : through the middle runs lengthwise a straight street, twenty feet in width. The walls of some of the ruins are thirty feet high, and cross each other at right angles. The buildings were apparently connected together, except where the streets intervened. The larger areas were again divided by thinner partitions, and one of them was observed to contain four rec- tangular pits, the plastering of which appeared quite fresh. No traces of doors or windows towards the streets could be discovered, nor indeed anywhere else. The walls were ex- clusively of sun-dried brick, and their direction north east and south-west, the same as those of the temple, which fronted the sea. Some graves were observed to the southward of the temple, but the principal burying-ground was between the temple and town. Some of the graves were rectangular pits, lined with a dry wall of stone, and covered with layers of reeds and canes, on which the earth was filled in to the depth of a foot or more, so as to be even with the surface. 384 THE LOST SOLAR SYSTEM DISCOVERED. The skulls brought from this place were of various charac- ters ; the majority of them presented the vertical elevation, or raised occiput, the usual characteristic of the ancient Pe- ruvians, while others had the forehead and top of the head depressed. Eight of these were obtained, and are now depo- sited at Washington. The bodies were found enveloped in cloth of various qualities, and a variety in its colours still existed. Various utensils and other articles were found, which seemed to denote the occupation of the individual : wooden needles and weaving utensils ; netting made in the usual style ; a sling ; cordage of different kinds ; a sort of coarse basket ; fragments of pottery, and plated stirrups. They also found various vegetable substances : husks of Indian corn, with ears of two varieties, one with the grain slightly pointed, the other, the short and black variety, which is still very commonly cultivated ; cotton-seeds ; small bunches of wool ; gourd-shells, with a square hole cut out, precisely as is done at present. These furnished evidence of the style of the articles manufactured before the arrival of the Spaniards, and of the cultivation of the vegetable pro- ducts ; when to these we add the native tuberous roots (among them the potato) cultivated in the mountains, and the animals found domesticated, viz., the llama, dog, and Guinea-pig, and the knowledge of at least one metal, we may judge what has since been acquired." Teocalli of Pachacamac. Height to platform 250 feet = 216 units. Sides of base of lowest terrace = 500 by 400 feet = 432 by 345-5 units. Height x area of base of internal pyramid =220x342 &c. x 432 = -j-2-Q distance of the moon pyramid = T External Pyramid. Height x area of base = 294 x 454 &c. x 574 =f circum- ference pyramid = -|. TEOCALLI OF PACHACAMAC. 385 Cube of sum of 2 sides = (454 + 574) 3 = 1028 3 = distance of the moon. Cube of perimeter = 8 times distance of moon. Internal Pyramid. Sides are 342 &c. by 432 _3 _3 1028 1296 Cube of 3 times less side = 1028 3 = distance of the moon. Cube of 3 times greater side = 1296 3 = 6 12 = diameter of the orbit of the moon. The 2 pyramids are similar, and the sides of the terraces will be perpendicular to base. Lowest terrace = ^ height x area of base of internal pyramid = internal pyramid. Height of the 3 terraces = 220 |= 73-3 293-3. The 3 terraces are as I 2 , 2 a , 3 2 , their heights being equal. The 3 pyramids rising from the base of the 1, 2, 3 ter- races will be as I 3 , 2 3 , 3 3 . 3rd pyramid =^1-0 = ^for distance of the moon 2nd =-A-of 1st = -fr of 3rd terrace = 3rd pyramid =^= 1 ^= 2nd = of 3rd terrace = 1st = 1st terrace = j 1 f difference = 1st pyramid =- 2nd terrace = 2nd pyramid = 3rd terrace = = 3rd pyramid =jffo5 1st pyramid = i of 1st terrace 2nd =fof2nd 3rd =fof3rd VOL. i. c c 386 THE LOST SOLAR SYSTEM DISCOVERED. The 3 sections of the internal pyramid, made at equal dis- tances, are as I 3 , 2 3 -l 3 , 3 3 -2 3 1 , 7 , 19 == 2 7 * 2700 > 2700* Sum of 3 sections = ir fi = whole pyramid. 3rd terrace 3rd section = 2nd -2nd 1st 1st Common difference = Sum of differences =2- And 3rd terrace 2nd terrace = -$%fa. Therefore 3rd terrace 2nd terrace = sum of 3 terraces sum of 3 sections internal pyramid 3rd terrace = sum of 1st and 2nd terrace = 2 *% distance of the moon. Sides of the base of lowest terrace are 342 by 432. These are Babylonian numbers derived from 3 5 243. (342 x 2) 2 x 243 = circumference of the earth (342 &c. x 3) 3 = distance of the moon (432 x 3) 3 = diameter of the orbit of the moon (342, &c. x 16) 3 = distance of Mercury (432 x 16) 3 = diameter of the orbit of Mercury. Cubes of the sides of the base of the lowest terrace are as 342 3 &c : 432 3 :: 1 : 2. (342 &c. x 2 4 ) 3 = distance of Mercury. (432 x 2 4 ) 3 = diameter of the orbit of Mercury. The cube of the sum of 3 5 when transposed by changing the places of the first and last numbers and multiplying by 2 4 = distance of Mercury. The cube of the sum of 3 8 when transposed by placing the first number the last and multiplying by 2 4 = diameter of the orbit of Mercury. TEOCALLI OF PACHACAMAC. 387 The cube of 3 times the first transposed numbers = distance of the moon. The cube of 3 times the last transposed numbers = diameter of the orbit of the moon. The square of twice the first transposed numbers multiplied by 3 5 = circumference of the earth. Circumference of the earth = (342 x 2) 2 x 243 Diameter of the orbit of Belus = 432000 circumference = 432 x 10 3 .-. (342 x 2) 2 x 243 x 432 x 10 3 = diameter of orbit of Belus; or, 4 x 342 2 x 243 x 432 x 10 3 = . so, 2 x 342 2 x 243 x 432 x 10 3 = distance of Belus. Cube of height to platform = 2 1 6 3 = -J- cube of less side of base = -J- (342) 3 = i cube of greater side of base = -J- (432) 3 Internal pyramid = y-^ distance of the moon. Height = 220 units, and 22 1 3 &c. =yj-o distance of the moon. The rectangular enclosure of the town = by j- of a mile. a mile =18*7 9 stades i= 6^26 j= 3-76 1 + 1=10-02 2 perimeter = 20-04. One side of the rectangular enclosure = 6 '2 6 stades=1521 units, say = 1538; then 1538 3 = 32 circumference = -Jg- distance of Mercury, or 45 cubes = 1440 circumference = dis- tance of Mercury from the Sun. The two sides of the rectangle = 10 stades = 2430 units, and one side = 1538, so the other side will = 2430 - 1538 = 892, and 898 3 = f distance of Moon, or 3 cubes =2 li =1- Thus the two sides will = 1538 + 898^= 2426, and 1 peri- meter = 10 stades = 2430 units. c c 2 388 THE LOST SOLAR SYSTEM DISCOVERED. Perimeter of the walls of Pachacamac = 20 stades = -J- 120 = -i- the side of the square enclosure of Babylon. Cube of 20 stades = ^ the cube of 120 = ^-J-Q- cube of Babylon = j|-g- distance of Bel us = -g-J-g- of 216000 circumference = 1000 circumference. Thus the cube of the perimeter of the walls of Pachacamac = -j-J-g the cube of Babylon, = -^-^ the distance of Belus, = 1000 times the circumference of the earth, = more than 100 times the distance of the moon from the earth. 1|- cube of 898 = distance of moon from earth. Distance of earth from sun = 400 times the distance of moon from earth, = 400 x 1 = 600 times 898 3 . Distance of Mercury from sun =150 times the distance of moon from earth, = 150 x I-L = 225 = 15 2 times 898 3 . or 153 3 &c. = 3-J--Q distance of the moon, (10 x 153, &c.y = VVV = V- Cube of side = ^ . 45 cubes =150 distance of the moon, = distance of Mercury, 120 cubes = distance of the earth, or 15 cubes of twice side = distance of the earth. We find in " Tschudi's Travels in Peru," that prior to the Spanish conquest, the valley of Lurin was one of the most populous parts of the coast of Peru. The whole of the broad valley was then called Pachacamac, because near the sea-shore and northward of the river, there was a temple sacred to the " Creator of the Earth." Pachacamac was the greatest deity of the Yuncas, who did not worship the sun till after their subjugation by the Incas. The temple of Pachacamac was then dedicated to the sun by the Incas, who destroyed the idols which the TEMPLE OF PACHACAMAC. 389 Yuncas had worshipped, and appointed to the service of the temple a certain number of virgins of royal descent. In the year 1534, Pizarro invaded the village of Lurin ; his troops destroyed the temple, and the Virgins of the Sun were dis- honoured and murdered. The ruins of the temple of Pachacaniac are among the most interesting objects on the coast of Peru. They are situ- ated on a hill about 558 feet high. The summit of the hill is overlaid with a solid mass of brickwork about thirty feet in height. On this artificial ridge stood the temple, enclosed by high walls, rising in the form of an amphitheatre. It is now a mass of ruins; all that remains of it being some niches, the walls of which present faint traces of red and yellow painting. At the foot and on the sides of the hill are scat- tered ruins, which were formerly the walls of habitations. The whole was encircled by a wall eight feet in breadth, and it was probably of considerable height, for some of the parts now standing are twelve feet high, though the average height does not exceed three or four feet. The mania for digging for treasures every year makes encroachments on these vestiges of a bygone age, whose monuments are well deserving of a more careful preservation." De la Vega adds that the name by which the Peruvians called the devil was Capay, which they never pronounced but they spit, and showed other signs of detestation. Their principal sacrifice to the sun were lambs, but they offered also all sorts of cattle, fowls, and corn, and even their best and finest clothes, all which they burned in the place of incense, rendering their thanks and praises to him, for having sustained and nourished all those things, for the use and support of mankind; they had, also, their drink-offerings, made from maize ; and when they first drank after their meals (for they never drank while they were eating), they dipped the tip of their finger into the cup, and lifting up their eyes with great reverence to heaven, gave the sun thanks for their liquor, before they presumed to take a draught of it ; and here he takes an opportunity to assure us, that the Incas always detested human sacrifices, and would c c 3 390 THE LOST SOLAK SYSTEM DISCOVERED. not suffer any such in the countries under their dominion, as they had heard that the Mexicans, and some other coun- tries did. He admits that the ancient Peruvians sacrificed men to their gods. The oracle at Rimac was consulted before the introduction of the worship of the sun by the Incas. " The valley of Rimac," says De la Yega, " lies four leagues to the northwards of Pachacamac, and received its name from a certain idol of the figure of a man, that spoke, and answered questions like the oracle of Apollo at Delphos. The idol was seated in a mag- nificent temple, to which the great lords of Peru either went in person, or inquired by their ambassadors, of all the im- portant affairs relating to their provinces; and the Incas themselves held this image in great veneration, and consulted it after they conquered that part of the country. De la Vega, who was descended from the Incas, makes a remarkable concession in relation to the Peruvians worship- ping Pachacamac, the almighty invisible God, before the Incas introduced the worship of the sun. The royal historian assures us the Peruvians acknowledged one almighty God, maker of heaven and earth, whom they called Pacha Camac, Pacha in their language signifying the universe, and Camac the soul. Pacha Camac, therefore, signified him who ani- mated the world. They worshipped him in their hearts as the unknown God. This doctrine was more ancient than the time of the Incas, and dispersed through all the kingdoms, both before and after the conquest. They believed that he was invisible, and therefore built no temples to him, except one in the valley of Pacha Camac, dedicated to the Unknown God, which was standing when the Spaniards arrived in Peru ; neither did they offer him any sacrifices, as they did to the sun, but showed, however, the profound veneration they had for him, by bowing their heads, lifting up their eyes, and by other outward gestures, whenever his sacred name was mentioned. Though he was seldom worshipped, because they knew so little of him, or in Avhat manner he ought to be adored. De la Vega describes the principal rites and ceremonies in RELIGION OF THE INCAS. 391 the religion of the Incas. He informs us they had four grand festivals annually, besides those they celebrated every moon. The first of their great feasts, called Raymi, was held in the month of June, immediately after the summer solstice, which they did not only keep in honour of the sun, that blessed all creatures with its heat and light, but in commemoration of their first Inca, Manca Capac, and Coya Mana Oclo, his wife and sister, whom the Inca looked upon as their first parents, descended immediately from the sun, and sent by him into the world to reform and polish man- kind. They fasted three days, as a preparative to the feast, eating nothing but unbaked maize and herbs and drinking water. The morning being come, the Inca, accompanied by his brethren and near relations, drew up in order, according to their seniority, went in procession at break of day to the market place, in Cusco, barefoot, where they remained looking attentively towards the east in expectation of the rising sun, which no sooner appeared than they fell down and adored the glorious luminary with the most profound veneration, acknowledging him to be their god and father. The king rising upon his feet (while the rest remained in a posture of devotion), took two great gold cups in his hands, filled with their common beverage made of Indian corn, and invited all the Incas, his relations, to partake with him and pledge him in that liquor. The Caracas and nobility drank of another cup of the same kind of liquor, prepared by the wives of the sun ; but this was not esteemed so sacred as that consecrated by the Inca. The Inca offered the vases or golden bowls, with which he performed the ceremony of drinking, and the rest of the royal family delivered theirs into the hands of the priests. Then the priests went out into the court and received from the Caracas and governors of the respective provinces their of- ferings, consisting of gold and silver vessels, and the figures of all sorts of animals cast of the same metals. These offerings being made, great droves of sheep and lambs were brought ; out of which the priests chose a black c c 4 392 THE LOST SOLAR SYSTEM DISCOVERED. lamb, and having killed and opened it, made their prognostics and divinations thereupon relating to peace or war, and other events, from the entrails of the beast ; always turning the head of the animal to the east when they killed it. After the first lamb, the rest of the cattle provided were sacrificed, and their hearts offered to the sun ; and their carcases were flayed and burnt, with fire lighted by the sun's rays, contracted by a piece of crystal, or something like a burning-glass. They never make use of common fire on these occasions, unless the sun was obscured. Some of the fire was carried to the temple of the sun, and to the cloister of the select virgins, to be preserved the following year with- out extinction. The sacrifice being over, they returned to the market- place, where the rest of the cattle and provisions were dressed and eaten by the guests ; the priests distributing them first to the Incas and then to the Caracas and their people in their order; and after they were done eating, great quan- tities of liquor were brought in. It should have been observed, that the people fell down on their knees and elbows when they adored the sun, covering their faces with their hands ; and it is remarkable that the Peruvians expressed their veneration for the temple, and other holy places, by putting off their shoes, as the Chinese, the people of the East Indies, and other Asiatics do, though at the greatest distance from them, and not by uncovering their heads, as the Europeans do at divine service. The nuns of Cusco were all of the whole blood of the Incas, dedicated to the sun, and called the wives of the sun. The select virgins in the other provinces were either taken out of such families as the Incas had adopted, and given the privilege to bear the name of Incas, or out of the families of the Caracas and nobility residing in the respective pro- vinces, or such as were eminent for their beauty and ac- complishments : these were dedicated to the Inca, and called his wives. As to the notions the Peruvians had of a future state, it is evident they believed the soul survived the body, by the ADOKATION OF THE SUN. 393 Incas constantly declaring they should go to rest, or into a state of happiness provided for them by their god and father the sun, when they left this world. Manco Capac not only taught all his subjects to adore his father (the sun), but instructed them also in the rules of morality and civility, directing them to lay aside their pre- judices to each other, and to do as they would be done by. He ordained that murder, adultery, and robbery should be punished with death ; that no man should have but one wife ; and that in marriage they should confine themselves to their respective tribes. Besides the worship of the sun, they paid some kind of adoration to the images of several animals and vegetables that had a place in their temples. These were the images brought from the conquered countries, where the people adored all manner of creatures, animate or inanimate; for whenever a province was subdued, their gods were imme- diately removed to the temple of the sun at Cusco, where the conquered people were permitted to pay their devotions to them, for some time at least, for which there might be several political reasons assigned. " The bodies of the Incas were embalmed and placed in the temple of the sun, where divine honours were paid them, but their hearts and bowels were solemnly interred in a country place of the Incas, about two or three leagues from Cusco, where magnificent tombs were erected, and great quantities of gold and silver plate and other treasures buried with them ; and at the death of the Incas and Caracas, or great lords, their principal wives, favourites, and servants either killed themselves, or made interest to be buried alive with them in the same tomb, that they might accompany them to the other world," says De la Vega, " and renew their immortal services in the other life, which, as their religion taught them, was a corporeal, and not a spiritual state." And here he corrects the errors of those historians who relate, that these people were killed or sacrificed by the successors of the deceased prince, which he seems to abhor ; and ob- serves further, that there was no manner of occasion for any 394 THE LOST SOLAR SYSTEM DISCOVERED. law or force to compel them to follow their benefactors or masters to the other world ; for when these were dead, they crowded after them so fast, that the magistrates were forced sometimes to interpose, and by persuasion, or by authority, to put a stop to such self-murders, representing that the deceased had no need of more attendants, or that it might be time enough to offer him their service when death should take them out of this world in a natural way. What the form or dimensions of the Temple of the Sun were, neither De la Yega nor any other writer pretend to describe ; but relate, that amongst all their buildings, none were comparable to this temple. It was enriched with the greatest treasures, every one of the Incas or emperors adding something to it, and perfecting what his predecessor had omitted. The image of the sun was of a round form, consisting of one plate of gold, twice as thick as the plates that covered the walls. On each side of this image were placed the several bodies of the deceased Incas, so embalmed, it is said, that they seemed to be alive. These were seated on thrones of gold, supported by pedestals of the same metal, all of them looking to the west, except the Inca Haana Capac, the eldest of the sun's children, who sat opposite to it. Besides the chapel that contained the sun, there were five others of a pyramidal form, the first being dedicated to the moon, deemed the sister and wife of the sun. The doors and walls thereof were covered with silver, and here was the image of the moon, of a round form, with a woman's face in the middle of it. She was called Mama Quilca, or Mother Moon, being esteemed the mother of their Incas : but no sacrifice was offered to her as to the sun. Next to this chapel was that of Venus, called Chasea, the Pleiades, and all the other stars. Venus was much esteemed as an attendant on the sun, and the rest were deemed maids of honour to the moon. This chapel had its walls and doors plated with silver, like that of the moon ; the ceiling repre- senting the sky, adorned with stars of different magni- tudes. ADORATION OF THE SUN. 395 The third chapel was dedicated to thunder and lightning, which they did not esteem as gods, but as servants of the sun ; and they were not represented by any image or pic- ture. This chapel, however, was sealed and wainscotted with gold plates like that of the sun. The fourth chapel was dedicated to Iris, or the rainbow, as owing its origin to the sun. This chapel was also covered with gold, and had a representation of the rainbow on one side of it. The fifth apartment was for the use of the high priest, and the rest of the priests, who were all of the royal blood ; not intended for eating or sleeping in, but was the place where they gave audience to the sun's votaries, and consulted concerning their sacrifices. This was also adorned with gold from the top to the bottom, like the chapel of the sun. Though there were no other image worshipped in this temple but that of the sun, yet they had the figures of men, women and children, and all manner of birds, beasts, and other animals of wrought gold, placed in it for ornament. The Indians not only adorned themselves, their houses, and temples with gold, but buried it with them when they died. They also buried and concealed gold from the Spa- niards ; but never purchased houses or lands with it, or esteemed it the sinews of war, as the Europeans do. It has been observed that the Burmese at the present day make use of their gold for ornamenting their temples, but employ none as a medium of circulation or commerce. Diodorus Siculus relates that the Egyptians worshipped the sun under the name of Osiris, as they did the moon by the name of the goddess Isis. Techo, the Jesuit, relates that the natives of La Plata, which is contiguous to Peru, worship the sun, moon, and stars; and in some part of the country the Jesuits relate that they worshipped trees, stones, rivers, and animals, and almost everything animate and inanimate. One of the objects of their adoration was a great serpent. The Hindoos record two races of their early monarch?, 396 THE LOST SOLAR SYSTEM DISCOVERED. and claim for them a supernatural descent one from Surya, the sun ; the other from Indu, the moon. These solar and lunar kings are said to have, between them, ruled the countries of India for, as Jones calculates, thirty-two generations. Another dynasty, sprung from the lunar branch, is said to have eclipsed them both. This was the line of the kings of Magadha, found by the Greeks in the provinces of the Ganges. Chandragupta, who is said to have usurped their power, is believed to be the Sandracottus who received the ambassadors of Seleucus, and whose seat of government was at Palibothra. Wilkes, in the United States exploring expedition, re- marks that at the Tonga Islands, though it is not known that any person is actually worshipped, as elsewhere, there are two high chiefs, whose official titles are Tuitonga and Veati, and a woman called Tarnaha, who are believed to be descended from the gods, and are treated with reverence on that account by all, not excepting the king, who regards them as his superiors in rank. In New Zealand the great warrior-chief, Hongi, claimed for himself the title of a god, and was so called by his followers. At the Society Islands, Tamatoa, the last heathen king of Raiatea, was worshipped as a divinity. At the Marquesas there are, on every island, several men who are termed atua, or gods, who receive the same adoration, and are believed to possess the same powers as other deities. In the Sandwich Islands the reverence shown to some of the chiefs borders on religious worship. At the Depeyster's group, the westernmost cluster of Poly- nesia, we were visited by a chief, who announced himself as the atua or god of the islands, and was acknowledged as such by the other natives. This singular feature in the religious system of the Poly- nesians, appearing at so many distant and unconnected points, must have originated in some ancient custom, or some tenet of their primitive creed, coeval, perhaps, with the formation of their present state of society. There is cer- tainly no improbability in the supposition that the law-giver, whose decrees have come down to us in the form of the PACIFIC ISLANDS. 397 tabu system, was a character of this sort a king, invested by his subjects with the attributes of divinity. It is worthy of remark, that in all cases in which we know of living men having been thus deified, they were chiefs of high rank, and not ordinary priests (tufuna), or persons performing the sacerdotal functions. But of all the qualities that distinguish this race, there is none which exerts a more powerful influence than their superstition; or, perhaps, it would be more just to say, their strong religious feeling. When we compare them with the natives of Australia, who, though not altogether without the idea of a god, hardly allow this idea to influence their con- duct, we are especially struck with the earnest devotional tendencies of this people, among whom the whole system of public polity, and the regulation of their daily actions, have reference to the supposed sanction of a supernatural power ; who not only have a pantheon surpassing, in the number of divinities and the variety of their attributes, those of India and Greece, but to whom every striking and natural pheno- menon, every appearance calculated to inspire wonder and fear, nay, often the most minute, harmless, and insignifi- cant objects, seem invested with supernatural attributes, and worthy of adoration. It is not the mere grossness of idolatry, for many of them have no images, and those who have look upon them simply as representations of their deities ; but it is a constant, profound, absorbing sense of the ever-present activity of divine agency, which constitutes the peculiarity of this element in the moral organisation of this people. Yet, this religious feeling is wholly independent of morality, to which the Polynesians lay no manner of claim. They expose their children, sacrifice them to idols, bury their parents alive, indulge in the grossest licentiousness, lie and steal beyond example, yet they are devout. Father Leander, of the order of bare-footed Carmelites, says, in describing Balbec, that by following the road by the cavern, to the extent of 50 paces, an ample area of a spherical figure presents itself, surrounded by majestic columns of granite, some of them of a single piece, and 398 THE LOST SOLAR SYSTEM DISCOVERED. others formed of two pieces, the whole of them of so large a dimension that two men can with difficulty girt them. They are of the Ionic order of architecture, and are placed on bases of the same stone, at such distances from each other that a coach and six might commodiously turn between them. They support a flat tower or roof, which projects a cornice with figures wrought with matchless workmanship ; these rise above the capitals with so nice a union, that the eye, however perfect it may be, cannot distinguish the part in which they are joined. At the present time the greater part of this colonnade is destroyed, the western part alone remaining perfect and upright. This fabric has an elevation of 500 feet, and is 400 feet in length : 500 feet =432 units 400 =345-5 (3 x 342, &c.) 3 =1028 3 = distance of moon (3 x432) 3 =1296 8 = 2 distance of moon (2 x 342) 3 x 243 = circumference. In the year 1773, two monks, Fathers Graces and Font, after a journey of nine days from the presidency of Horca- sitas, arrived at a fine open plain at the distance of a league from the south bank of the river Gila. There they found the ruins of an ancient Aztec city covering an extent of about a square league ; in the midst rose an edifice called Casa Grande. This great house accords exactly with the cardinal points, and has from north to south a length of 136 metres, and from east to west 84 metres. A wall with towers surrounds this edifice. Vestiges of an artificial canal for conducting the water from the Gila to the city were found : 136 metres = 446 feet=385 units 84 =275 =238 385 3 = \ circumference 238 3 = gV distance of moon. Cube of sum of 2 sides = (385 + 238) 3 = 623 3 =f distance of moon KUINS OF AN AZTEC CITY. 399 (3x623) 3 = f x3 3 = 6 (5 x 3 x 623) 3 = 6 x 5 3 = 750. 10 cubes of 15 times sum of 2 sides = 7500 distance of moon = distance of Saturn, 20 cubes = Uranus, 60 cubes = Belus. 9 cubes of sum of 2 sides = 2 distance of moon = diameter of orbit. Cube of perimeter = = - distance of moon. 9 3 Cube of less side : cube of 2 sides : : 238 3 8 1 1 623 3 -f- distance of moon 18 385 3 =|- circumference (60 x 385) 2 =| x 60 3 =|- 216000 circumference. 2 cubes of 60 times greater side = 216000 circumference = distance of Belus (9 x 238) 3 =^ L r x 9 3 = 9 distance of moon. Cube of 9 times less side = 9 times distance of moon. The ruins of the ancient city covered about a square league. Taking a league at three English miles, a side would = 3 x 18-79 = 56-37 stades. If side = 60 stades, then cube of side = cube of 60 stades = -J- cube of Babylon. Cube of twice the side, or of 2 x 60 stades would = cube of 120 stades = cube of Babylon = distance of Belus. " At Mai- Amir," says de Bode, " in the middle of the plain, rises an immense artificial mound, the dimensions of which are certainly not less imposing than those at Shush and Babylon. It is surrounded by broken and uneven 400 THE LOST SOLAR SYSTEM DISCOVERED. ground ; but a luxuriant carpet of green grass conceals its structure from the inquisitive eye. Its external form and appearance resembling the Susian and Babylonian mounds, and the circumstance of cuneatic inscriptions being found in its vicinity, bespeak the high antiquity of the place, and afford a strong argument in favour of the existence here, in former times, of a considerable fort, corroborating my im- pression that Mai-Amir is the site of the Uxian town besieged by Alexander." If the triangle and pyramid of Belus be divided like Fig. 68., then the several sections will represent the Baby- Fig. 68. Ionian broad arrow. The straight lines intercepted by the two apices of each double set of triangles are equal. The areas of all the single triangles are equal, and therefore of each double set. The content of each of the differential solid sections, the difference between the 1st and 2nd, the 2nd and 3rd pyramid, &c., are also equal. The two opposite triangles which form the arrow head are similar and equal. Each of these two triangles, though equal to each of the other triangles in every set, are similar only to each other. Or the arrow heads, though dissimilar to each other, have their breadths, areas and contents, equal. Thus when the heights of the triangles and pyramids vary as 1, 2, 3, &c, while their bases are equal or common, then DRUIDICAL REMAINS. 401 Height oc 1, 2, 3, 4, 5, 6, &c. Difference QC 1, 1, 1, 1, 1, 1, &c. Areas oc 1, 2, 3, 4, 5, 6, &c. Difference x 1, 1, 1, 1, 1, 1, &c. Solids oc 1, 2, 3, 4, 5, 6, &c. Difference x 1, 1, 1, 1, 1, 1, &c. No arches were found in the teocallis, but, as a substitute, large bricks were placed horizontally, so that the upper course passed beyond the lower, which supplied in a measure the gothic arch, like those found in several Egyptian edifices; whence it is inferred that the inhabitants of both countries were ignorant of the method of constructing arches. We should say that the arch was not found because it was not admissible in the obeliscal style of architecture, since the curve never appears in the construction of the obeliscal series of squares ; but the overlapping of bricks is said to have supplied in a measure the gothic arch. This is the obeliscal arch, if it may be so called. The framework or mould for such an arch would be the obeliscal series of squares, where the sections, as they increase from the apex, project laterally beyond each other. These projections have their sides bounded by ver- tical and horizontal straight lines, so that each layer of bricks would be necessarily placed horizontally, and the upper course project beyond the lower as the sides of the arch approached the apex. It has been seen that such obeliscal or parabolic and hy- perbolic arches, which symbolise the laws of gravitation, can be constructed in a variety of ways. Druidical Remains in England. Those in Cumberland. About three miles south-west of Castle Sowerby is a stupendous mountain, called Carrock I^ell, being 803 yards above the level of the sea, and 520 yards above the surrounding meadows. The whole of this mountain is a ridge of horrid precipices, abounding with chasms, not to be fathomed by the eye. Close under it, for nearly two miles, is a winding path, just wide enough for a VOL. I. D D 402 THE LOST SOLAR SYSTEM DISCOVERED. horse to pass singly, and everywhere intercepted by enor- mous stones, which have fallen from the summit of the mountain. In the year 1740, a cavern was discovered at the end of it, which has never been explored ; near which is a remarkable pool of water, called Black Hole, 150 yards in circumference, and in some places 65, and in others 45, fathoms deep. The eastern end of Carrock Fell, for upwards of a mile in length, is almost covered with masses of granite of various sizes, some of them not less than 300 tons in weight ; and on the highest part is a singular monument of antiquity, of which the following description is given in the history of Cumberland. The summit of this huge fell is of an oval form. Round its circumference is a range or enclosure of stones, which seem to be incontestably the work of men's hands. The stones of the sides of the enclosed area are about eight yards perpendicular below the ridge of the mountain, but at the ends not more than four. In some places, however, the height is six feet, in others three only, or even less ; this variation is probably owing to a practice continued from age to age of rolling some of the stones down the sides of the mountain for amusement, or rather from a desire of wit- nessing the effects of their increasing velocity. The stories are in general from one to two or three, and even four hun- dred-weight ; but many of them are considerably smaller. From the few stones that may be found within the area, it would seem that the whole range has been formed by the stones obtained in the enclosed space, which is nearly desti- tute of vegetation. The direction of the ridge of the top of the fell in its transverse diameter is nearly east and west ; and in this direction within the surrounding pile of stones it measures 252 yards: the conjugate diameter is 122 yards, and the content of the space enclosed is about three acres and a half. The entrances are four, one opposite each point of the com- pass ; those on the west and south sides are four yards in width ; that on the east appears to have been originally of the same dimensions, but is now about six yards wide ; the OLD CARROCK. 403 width of the northern entrance is eight yards. Besides these on the north-west quarter there is a large aperture or passage twelve yards in width ; which, if the nature of the ground is attended to, and the apparent want of stones in this part considered, seems never to have been completed. At the distance of 66 yards from the east end of this range, on the summit of the hill, stands an insulated pile of stones, appearing at a little distance like the frustum of a cone. Its base is about 11 yards in diameter, and its per- pendicular height 7 yards. On clambering to the top, the interior is found to be funnel-shaped ; the upper part or top of the funnel being five yards diameter ; but as the hollow gradually slopes downwards, the width at the bottom is little more than two feet : the largest stones appear to weigh about 1^ cwts. The crowned head of Old Carrock is by no means perfectly uniform, the end to the westward being about 15 yards higher than the middle of the oval. On the highest point is a fragment of rock projecting about three yards above the surface of the ground, having stones heaped up against two of its sides, and at a distance assuming the appearance of the one just described, though of twice its magnitude. Both these piles seem to be coeval with the surrounding range, but there are other smaller heaps that are evidently of modern contrivance, and appear to have been erected, speaking locally, as ornaments to the mountain. The name given to this monument by the country people is the Sunken Kirks. Transverse diameter = 252 yards = 756 feet = 653*6 units Conjugate diameter = 122 = 366 =317 20x653 = 13040 Distance of Jupiter = 13040 3 30x319 = 9570 Diameter of orbit of earth = 9560 3 Sum of diameters = 653 + 319 = 972 30x972 = 29160 Distance of Belus = 29160 8 D D 2 404 THE LOST SOLAR SYSTEM DISCOVERED. Cube of 20 times greater diameter = distance of Jupiter. Cube of 30 times less diameter = diameter of orbit of earth. Cube of 30 times sum of 2 diameters = distance of Belus. Mean of 2 diameters = (653 + 317) = 485 485 3 = circumference. Cube of mean = circumference (2 x 485) 3 = 8 (30x2x485) 3 = 8 x 30 3 = 216000 circumference. Cube of 30 times sum of 2 diameters = distance of Belus. These diameters are within the pile of stones : the breadth of the pile is not stated. Circumference of circle having diameter 655 &c. = 2056 units -J- = 1028 1028 3 = distance of moon (2 x 1028) 3 = 8 Cube of circumference = distance of moon Cube of circumference = 8 If diameter of a circle = 648 units diameter 3 = 648 3 = cube of Cheops = -J- distance of moon, circumference 3 will = y. But if diameter = 655 &c. units cube of circumference will = \ 2 = 8 distance of moon. If diameter of a circle = 1 circumference = 3'1415 &c. square of circumference = 9*869 &c. cube of circumference = 31*004 &c. Cube of diameter : cube of circumference : : 1 ; 31 Square of diameter '. square of circumference :: 1 : 9-869 &c. OLD CARROCK. 405 Circumference of earth : distance of moon ::1 : 9-55 &c. Internal diameters are 653 and 317 units. Cylinder having height = 321 and diameter of base = 657 will = 321 x657 2 x -7854 iV or ~fo distance of moon. Spheroid = -^ Cone = ^V Cylinder of 10 times dimension will = l ^o^ = 100 distance of moon. Cylinder of 20 times dimension will = 100 x 2 3 = 800 distance of moon. = diameter of orbit of earth. Less internal diameter = 317 units. Circumference of circle of diameter 317 = 996 996 3 = ig circumference (2 x 996) 3 = 70 Cube of twice circumference of circle = 70 times circumference of earth. Sum of 2 circumferences =2056 + 996 = 3052 units. 2x3052 = 6104 610 3 &c. = 2 circumference (10x610&c.) 3 = 2000 Cube of 2 sum of 2 circumferences (or of 4 times mean) = 2000 times circumference of earth. The frustum of the cone of stones has a diameter of 11 yards = 33 feet = 28 -5 8 units; circumference will=89'6 units. 898 5 = f distance of moon. 3 cubes of 10 times circumference = diameter of orbit of moon. If diameter = 28 *3 units. (10 x 28-3) 3 = i circumference. (10 x 10 x 28-3) 3 = i- /- = 200 circumference. D D 3 406 THE LOST SOLAR SYSTEM DISCOVERED. Cube of 100 times diameter = 200 circumference of earth; therefore, cube of 40 times circumference = 400. Cylinder having height = diameter of base = 27, &c., units, = A degree. = 3 minutes. Sphere = f = 2 Cone - - =1 = 1 Cone = 1 minute = 1 geographical mile. (10 x 89*8) 3 = f distance of moon. (3 x 10 x 89-8) 3 = f x 3 3 = 18. (5 x 3 x 10 x 89-8) 3 = 18 x 5 3 = 2250. 10 cubes of 150 times circumference = 22,500 distance of moon = distance of Belus. iV CUDe Mercury ; or, -jJg- cube of 150 times circumference =150 times distance of moon = distance of Mercury. Should diameter of cone =29*16 units, circumference will= 91*6. 1000x29*16 = 29160 Distance of Belus = 29160 3 60x91*6 5496 Distance of Mercury = 54 90 3 . Cube of 1000 times diameter = distance of Belus. Cube of 60 times circumference = distance of Mercury. Mean distance of Mercury may be between 5460 3 and 5490 3 . " There is a conical hill, called Tagsher, in Western Barbary ; near which, as I learned from the kaid, are some curious ruins. He described them as being those of a large castle, built of extraordinary materials, every stone of which being of such a size that no hundred men of modern times could move it; some of them, he said, were as much as twenty feet square, and about fifteen feet high. He described the entrance as having been blocked up by earth and sand, except in one place through which he entered LONG MEG CIRCLE. 407 and proceeded some distance under ground ; the passage be- coming at last so narrow that he could not advance further, although by light he perceived it was of yet greater extent. At a short distance from the building lay a flat stone, which he lifted up, and found beneath it a pit, that, by his descrip- tion, was of an inverted conical form : it was empty." -(Hay.) At the village of Salkeld, on the summit of a hill, is a large and perfect Druidical monument, called by the country people Long Meg and her Daughters. A circle of about eighty yards in diameter is formed by massy stones, most of which remain standing upright. These are sixty-seven in number, of various qualities, unhewn or untouched with any tool, and seem by their form to have been gathered from the surface of the earth. Some are of blue and gray limestone, some of granite, and some of flints. Many of such of them as are standing measure from twelve to fifteen feet in girt, and ten feet high ; others are of an inferior size. At the southern side of the circle, at the distance of eighty-five feet from its nearest member, is placed an upright stone, natu- rally of a square form, being of red freestone, with which the country about Penrith abounds. This stone, placed with one of its angles towards the circle, is nearly fifteen feet in girt, and eighteen feet high, each angle of its square answering to a cardinal point. In that part of the circle most contiguous to the column, four large stones are placed in a square form, as if they had constructed or supported the altar ; and towards the east, west, and north, two large stones are placed, at greater distances from each other than any of the rest, as if they had formed the entrances into this mystic round. What creates astonishment to the spectator is, that no such stones, nor any quarry or bed of stones, are to be found within a great distance of this place ; and how such massy bodies could be moved, in an age when we may suppose the mechanical powers were little known, is not easily to be determined. Diameter of circle = about 80 yards = 240 feet = 208 units. D D 4 408 THE LOST SOLAR SYSTEM DISCOVERED. Cylinder having height = diameter of base = 208 nuits will = 208 3 , 8rc. x -7854 = T V circumference. -J= Inscribed sphere = -^ ,j i= Inscribed cone = iV If diameter = 238, &c., feet, circumference = 749 feet = 648 units = side of base of pyramid of Cheops. Then cube of circumference = 648 3 = ^ distance of moon. 4 cubes = distance of moon from earth. If diameter = 208 units, 208 3 , &c. = Y!~O circumference. (10x208, &c.) 3 =8^ =80. (5 x 10 x 208, &c.) 3 =80 x 5 3 = 1000 circumference. Cube of 50 times diameter=1000 circumference. (6 x 5 x 10 x 208, &c.) 3 = 1000 x 6 3 = 216,000. Cube of 300 times diameter = 216,000 circumference. = distance of Belus. Pyramid - - = Uranus. Should diameter =209 units, Circumference will=657, &c. 657 3 , &c. = ^ circumference. (4 x 657) 3 = V x 4 3 = 160. 9 cubes of 4 times circumference of circle = 1440 circumference of earth = distance of Mercury. (3 x 4 x 657) 3 = 160 x 3 3 = 4320 circumference. 100 cubes of 12 times circumference of circle, = 432,000 circumference of earth = diameter of orbit of Belus. Circumference of circle, diameter 208 = 653, 20x653=13,060. Distance of Jupiter =13,040 3 . Cube of 20 times circumference = distance of Jupiter. Cube of 50 times diameter =^ distance of Jupiter. Cube of 1 diameter = ( 1 x 1 ) 3 = 1 3 Cube of 4 circumference = (4 x 3-1416) 3 = 12-5664 3 . BLACK-COMB CIRCLE. 409 10 3 =1000 12-5 3 &c. = 2000 Cubes are as 1 I 2. Cube of 10 diameter =\ cube of 4 circumference. Cube of diameter =i ^- Cube of circumference = 2 ^diameter. The cube of 10 times diameter being = -J- cube of 4 times circumference must only be regarded as an approximation. Neither is the cube denoting planetary distances to be other- wise regarded. We first used the cube for planetary distances in a rough manner only, not having tables for the higher numbers, and not then expecting to make so general a use of these expres- sions. When accurate measurements of ancient monuments have been made, the expressions of planetary distances ought also to be corrected. The same observations will apply to l -f diameter =2*5 -^ circumference = 1*2 5 6 &c. *g diameter : --fa :: 2 : 1. Cubes are as 8 : 1. Cube of 10 diameter =|- cube of 4 circumference Cube of -^Q circumference = LO diameter. Several Druidical circles, and other remains of antiquity, are to be seen in the neighbourhood of Black-comb; the most remarkable of which is the Druidical temple called Sunken Kirk, situated in the level part of a wet meadow, about a mile east from this mountain. It is a circle of large stones, and is thus described by Gough : " At the en- trance are four large stones, two placed on each side at the distance of 6 feet ; the largest, on the left-hand side, is 5 feet 6 inches in circumference. Through this you enter into a circular area, 29 yards by 30. The entrance is nearly south-east : on the north or right-hand side is a huge stone, of a conical form, its height nearly 9 feet. Opposite the entrance is another large stone, which has once been erect, but has now fallen within the area ; its length is 8 feet. To 410 THE LOST SOLAR SYSTEM DISCOVERED. the left hand, to the south-west, is one, in height 7 feet, in circumference 11 feet 9 inches. The altar probably stood in the middle, as there are some stones still to be seen, though sunk deep in the earth. The circle is nearly complete, ex- cept on the western side, where some stones are wanting ; the large stones are 31 or 32 in number. The outward part of the circle, upon the sloping ground, is surrounded with a buttress, or rude pavement of small stones, raised about half a yard from the surface of the earth. The situation and aspect of the Druidical temple near Keswick is, in every respect, similar to this, except the rectangular recess formed by ten large stones, which is peculiar to that at Keswick ; but upon the whole (I think), the preference will be given to this, as the stones appear much larger, and the circle more entire." If diameter = 30 yards = 90 feet, circumference = 281 feet = 1 stade = 243 units. Transpose 2 and 3, or read the figures backwards, and 342 is expressed, which, multiplied by 2, and raised to the power of 2 = 684 , and 684 x 243 = circumference of the earth. Thus by means of a circle, having a circumference of 1 stade, the Druids could show that the circumference of the earth equalled 684 stades, or 684 x 243 units. Or circumference of circle I circumference of the earth ::1 : 684 2 . Circumference = 1 stade = 243 units; 243 transposed, by placing 3 the first, = 324 ; and 324 x 2 = 648= side of base of pyramid of Cheops ; the cube of which = 648 3 = ^ distance of the moon. 4 cubes = 4 x 648 3 = distance of the moon from the earth. Cube of circumference of circle =243 3 = -L circumference of the earth. Cube of twice circumference of circle = circumference of the earth. Cube of 120 times circumference of circle = cube of 120 stades = cube of Babylon = distance of Belus. BLACK- COMB CIRCLE. 411 The circumference of circle at Black-comb = the height of the tower of Belus. Diameter 29 yards = 75*2 units. Cylinder having height = diameter of base = 74 units will = 1 degree = -g-J-g- circumference Sphere = f = f Cone = % J- Circumference of circle, diameter 75*2 units = 236 &c., should circumference = 239. 40x239 = 9560 diameter of the orbit of the earth = 9560 3 . Cube of 40 times circumference = diameter of the orbit of the earth. Cube of 3 times 40 times circumference of 243 units = distance of Belus Sphere = Neptune Pyramid = Uranus = diameter of the orbit of Saturn. Diameter of circle = 29 yards = 87 feet = 75 2 units circumference =236 235 3 &c. = 5--o distance of the moon (10 x 235 &c.) 3 = V7o = 12 (5 x 10 x 235 &c.) 3 = 12 x 5 3 = 1500. 5 cubes of 50 times circumference = 7500 distance of moon = distance of Uranus 15 cubes i9 = Belus. Diameter of circle = 30 yards = 90 feet = 7 7 '8 units circumference = 244 243 3 = circumference (2 x 243) 3 = 1 (10 x 2 x 243) 3 = 1000. Cube of 20 times circumference of circle = 1000 times circumference of the earth = Jg- distance of Saturn = -h Uranus Belus. 412 THE LOST SOLAR SYSTEM DISCOVERED. (30 x 2 x 243) 3 = 1 x 30 3 = 27000. Cube of 60 times circumference of circle = 27000 times circumference of the earth = 10 times distance of Venus. (60 x 2 x 243) 3 = 1 x 60 3 = 216000. Cube of 120 times circumference of circle = 216000 times circumference of the earth = distance of Belus. Cube of twice circumference of circle = circumference of the earth. Diameter of circle = 29 yards = 87 feet = 75-22 units if= 75-84 circumference =238*35. 100x75-84 = 7584 distance of the earth = 7584 3 . 40x238-5 = 9540 diameter of the orbit of the earth = 9540 3 . Cube of 100 times diameter = distance of the earth Cube of 40 times circumference = diameter of the orbit of the earth. The cubes are as 1 I 2. There is a Druidical circle on the summit of a bold and commanding eminence called Castle-Eigg, about a mile and a half on the old road, leading from Keswick, over the hills to Penrith. Castle-Rigg is the centre-point of three valleys that dart immediately under it from the eye, and whose mountains form part of an amphitheatre which is completed by those of Borrowdale on the west, and by the precipices of Skiddaw and Saddleback close on the north. Such seclusion and sublimity were indeed well suited to the dark and wild mysteries of the Druids. The circle at present consists of about forty stones, of different sizes, all, or most of them, of dark granite ; the highest about seven feet, several about four, and others con- siderably less. The form may with more propriety be called an oval, being 35 yards in one direction, and 33 yards in CASTLE-RIGG CIRCLE. 413 another, in which respect it assimilates exactly to that of Rollick, in Oxfordshire ; but what distinguishes this from all other Druidical remains of a similar kind is the rectangular enclosure on the eastward side of the circle, including a space of about eight feet by four. Diameters = 35 by 33 yards, = 105 by 99 feet, = 90-75 by 85-6 units. Circumference of circle, diameter 90 = 283 &c. units, )) 3) j) 85 = 267 ,, 283 3 &c. = -J circumference, 266 3 &c, = $ Cube of circumference of greater diameter = j- circumference of the earth. Cube of circumference of less diameter = circumference of the earth. Cube of 5 times greater circumference = ix5 3 = 25 = 5 2 circumference. Cube of 6 times less circumference = i x 6 3 = 36 = 6 2 circumference. Cube of 60 times greater circumference = x 60 3 =i 216000 circumfer. = distance of Belus. Cube of 60 times less circumference = i. x 60 3 =i 216000 circumfer. = -J- distance of Belus, = distance of Saturn. Sum of 2 circumferences = 283 + 266 = 549 units, mean = 279 &c. 279 3 &c. = -^Q distance of the moon, = |JL = J- radius of the earth. Cube of mean = f radius of the earth. Cube of 10 times mean, or of 5 times sum = (10 x 279 &c.) 3 = -Lj-jP- = 20 distance of the moon. 414 THE LOST SOLAR SYSTEM DISCOVERED. 20 cubes of 10 times mean, or of 5 times sum, = 20 x 20 = 400 distance of the moon, = distance of the earth, 30 cubes = distance of Mars. Sum of 2 diameters = 90 + 85 = 175 units. 175 3 &c. = -J^-Q distance of the moon. (10 x 175 &c.) 3 = VVV = 5. Cube of 10 times sum of 2 diameters = 5 times distance of the moon. Cube of 20 times sum of 2 diameters = 40 times distance of the moon, = -^ distance of the earth. Or, 10 cubes = 400 distance of moon = distance of earth. At West Kennet, in Wiltshire, there is a kind of walk about a mile long, which was once enclosed with large stones : on one side the enclosure is broken down in many places, and the stones taken away ; but the other side is almost entire. On the brow of the hill near this walk is a round trench, enclosing two circles of stones, one within another : the stones are about 5 feet in height ; the diameter of the outer circle 120 feet, and of the inner, 45 feet. At the distance of about 240 feet from this trench have been found great quantities of human bones, supposed to have been those of the Saxons and Danes who were slain at the battle of Kennet, in 1006. Diameter of the outer circle =120 feet, = 104 units, say = 106. Cylinder having height = diameter of base will = 106 3 &c. x '7854, = 3 degrees. Inscribed sphere = f of 3 = 2. Inscribed cone = . o f 3 = 1. Diameter of inner circle = 45 feet. Circumference = 141 feet = -^stade = 121-5 units. Twice circumference = 2 x 121-5 = 243 = 3 5 . 243 transposed = 342. WEST KENNET CIRCLE. 415 342 x 2 = 684. 684 2 = circumference in stades. 684 2 x 243 = circumference in units. Circumference of circle = ~ stade. 2 circumference 1 stade. = side of tower of Belus. Cube = JT circumference of the earth. o Cube of 4 times circumference = cube of 2 sides of tower = cube of side of square enclosure of the tower = circum- ference of the earth. Diameter = 45 feet = 38 % 9 units. Cylinder having height = diameter of base = 39 &c. units will = 34*00 circumference = 9 minutes. Sphere = f = mrVo = 6 Cone = J- = y^Vo = 3 . 1 minute = 1 geographical mile. Or, cylinder having height = diameter of base = 37 units will = -J- degree. Sphere = T V Cone = -^ Diameter of outer circle = 104 units. 104 3 &c. y-J-o circumference. (10 x 104 &c.) 3 = VV / = 10 or, 1044 3 =10 circumference, and 1028 3 = distance of the moon. The transverse and conjugate diameters of the Druidical circles are often stated as differing from each other. If one diameter of the outer circle of West Kennet = 104*4 units, cube of 10 times diameter = 10 times circum- ference of the earth. If the other diameter = 102-8 units, cube of 10 times this diameter = distance of the moon. Diameter of outer circle = 120 feet = 103*75 units. Circumference = 326 40x326 = 13040. Distance of Jupiter = 13040 3 416 THE LOST SOLAR SYSTEM DISCOVERED. Cube of 40 times circumference = distance of Jupiter. Cube of 100 times diameter = \ distance of Jupiter. Diameter of inner circle = 45 feet = 38*9 units. Circumference =122 300 x 122 = 36600. Diameter of orbit of Belus = 36600 3 . Cube of 300 times circumference = diameter of orbit of Belus. 122 3 = g-LQ- distance of moon. (10 x 122) 3 = LQJH) = u>. (6 x 10 x 122) 3 = LO x 6 3 = 360. Cube of 60 times circumference = 360 distance of moon. (5 x 6 x 10 x 122) 3 = 360 x 5 3 = 45000. Cube of 300 times circumference = 45000 dist. of moon, = diam. of orbit of Belus. Cube of circumference = -^^ distance of the moon, = ^L. = -JL. radius of the earth. Cube of 10 times circumference = ^g = 100 radius of the earth. Cube of ^ x 300 times diameter r= distance of Belus. Diameter of outer circle = 120 feet = 10375 units, Circumference = 326 If circumference of a circle = 324 &c< units, 324 3 &c. = -fa circumference, (1 x 324 &c.) 3 = -3^- = 300. Cube of 10 times circumference = 300 circumf. of earth. (4 x 10 x 324 &c.) 3 = 300 x 4 3 = 19200. Cube of 40 times circumference = 19200 circumference. Distance of Jupiter = 19636 If circumference of a circle = 324 units, 40x324= 12960 1296 3 = 6 12 = diameter of orbit of the moon. (10 x 1296) 3 = 1000 diameters = 2000 distance of the moon. Distance of Jupiter = 2045 SILBURY HILL. 417 Cube of 40 times circumference = 2000 distance of moon ; but if circumference = 326, the cube of 40 times circumference = distance of Jupiter ; .-. cube of 100 times diameter = \ distance of Jupiter. The walk is about a mile in length. 1 mile = 18-79 stades = 4566 units, if = 4770 Then, cube of length = 4770 3 = \ distance of earth. Cube of 2 length = 9540 3 = f = 2 = diameter of orbit of the earth. If = 4345, 2x4345 = 8690. Cube of twice length = 8690 3 = distance of Mars. The two great Druidical temples of Avebury and Stone- henge are both in Wiltshire. The mound at Avebury, according to Stuckeley, is in a situation that seems to leave no doubt that it was one of the component parts of the grand temple. This artificial mound of earth is conical, and called Sil- bury hill ; it is the largest tumulus in Europe, and one worthy of comparison with those mentioned by Homer, Herodotus, and other ancient writers. The circumference of the hill, as near the base as possible, measures 2027 feet; the sloping height, 316 feet ; the per- pendicular height, 170 feet ; and the diameter of the top 120 feet. This artificial hill covers the space of 5 acres and 34 perches. A proof that this wonderful work was raised before the Roman-British period is furnished by the Roman road from Bath to London, which is straight for some dis- tance, till it reaches the hill, where it diverges to the south to avoid it, and then again continues its direct course. Many barrows are found in the neighbourhood, one of which the Roman road just mentioned has cut through. Other Druidical remains are found around Avebury, including circles, cromlechs, and stones erect, confirming the impres- VOL. I. E E 418 THE LOST SOLAR SYSTEM DISCOVERED. sion that this place must have been the greatest and most important of the kind in Britain. No marks of tools are anywhere visible on the stones of Avebury ; they were set up in their rude natural grandeur. Two circles of stones, not concentric, are enclosed by a great circle of stones ; a very deep circular trench was dug without these stones. The inner slope of this bank measures 80 feet, and circumference at the top was 4442 feet ; the area thus enclosed was about 28 acres. Silbury Hill. (Fig. 79.) Fig. 79. Height to platform = 170 feet = 147 units. Circumference of base =2027 =1828 . Diameter ,, = 645 = 557 . Diameter of platform = 120 =104 . Say, originally =115 . Height to apex of external cone, according to these data, will = 1 86 units. Height x area base, = 186 &c. x 557 2 x -7854= f. circumference. External cone = J of f = T %- The internal cone will have the apex in the centre of the platform, height to apex = height to platform, and diameter of base = diameter of base of external cone, less diameter of platform =557 115 = 442 units. Height x area of base = 148 x 442 2 x '7854 = circumference. Internal cone \ of = -Jg- SILBURY HILL. 419 The two cones will be as iV : iV circumference. :: 1 : 2 and the cones will be similar. Cube of height to apex of external cone = 186 3 &c. = ! 6 o o distance of the moon. Cube of height to apex of internal cone = cube of height to platform = 148 3 = 10 3 00 distance of the moon. The cubes of the heights of the two cones are as 1 ? KH)o distance of the moon, as 1 I 2. External cone less internal cone = difference of cones, = the sides of the hollow cone, = the hollow cone, = the internal cone, = yV circumference = 24 degrees, which is the reciprocal of the tower of Belus. For the tower = -^circumference = 15 degrees. The tower = = internal cone. Inclined side of internal cone = 266 units. Cube of side = 266 3 &c. = -J- circumference. The inclined side of internal cone will equal sloping side of hill. Inclined side of internal cone =266 units = 308 feet. Sloping side by measurement =316 feet. Inclined side of external cone = sloping side of hill con- tinued to apex = 335 &c. units, and 335 3 &c. = -- circumference. Cubes of the inclined sides of the two cones are as -J- I -J circumference, as 1 : 2. The cubes of the diameters of their bases are in the same ratio. Cube of 3 times circumference of base of external cone = (3 x 1828) 3 = 5484 3 , = distance of Mercury. E E 2 420 THE LOST SOLAR SYSTEM DISCOVERED. Cube of 3 times circumference of base of internal cone = -J- distance of Mercury. Cube of 10 times height of internal cone = (10 x 148 &c.) 3 = 3 distance of moon. 50 cubes = 150 distance of moon. = distance of Mercury. Cube of 10 times height of external cone = (10 x 186 &c.) 3 = fg- = 6 distance of moon. 25 cubes = 150 distance of the moon. = distance of Mercury. A conical hill, having diameter of base = 2317 units, and height to apex = 773 units, will = distance of the moon. The internal similar cone, having diameter of base = 1091 units, and height to apex = 364 units, will = circumference of the earth. Fig. 80. Fig. 80. The apex of the internal cone will be in the centre of the platform of the truncated cone. As there are many large mounds, both in Asia and Ame- rica, with circular or rectangular bases, possibly one may be found to represent the distance of the moon combined with the circumference of the earth. If the mound be circular the diameter of base should = 2317 units =. 9-17 stades = i mile English nearly. Height to platform = 364 units = 421 feet English. Many circles, like the Druidical, are surrounded by slop- ing entrenchments, or raised embankments, probably to represent the frustum of a cone, which would require less labour than the construction of an artificial mound, though MOUNT BARKAL. 421 in this case advantage would likely be taken of a natural hill, by forming it into the required dimensions. If the height and sides of base were reduced to ^ the di- mensions of the supposed conical mound, the contents would be reduced to -fa the supposed contents ; but the proportion of the distance of the moon to the circumference of the earth would remain. At Mount Barkal, in Upper Nubia, lat. 18 25', there was once a city : the remains prove it to have been an ancient es- tablishment of priests, who possessed a kindred worship to that of Egypt. The temples lay between the mountain and the Nile. It is not said whether the sides of Mount Barkal are cir- cular or rectangular. The height corresponds to the height of the supposed mound, or truncated cone, the circumference of which would = \\ mile. The peculiar form of Mount Barkal, says Ruppel, must have fixed attention in all ages. From the wide plain there arises up perpendicularly on all sides a mass of sandstone, nearly 400 feet high, and about 25 minutes in circuit. The unusual shape of the mountain must have become still further an object of curiosity, from the phenomena with which it is connected. The clouds, attracted from all around to this isolated mass, descend in fruitful showers ; and hence we need hardly wonder if, in ancient times, it was believed that the gods here paid visits to man, and held communion with him. Temple rose after temple ; and who can say how far many a devotee came to ask advice of the oracle ? The circuit of 25 minutes would be about \\ mile. The sides of Mount Barkal are perpendicular. Height to platform x diameter of base of cone = 364 x 2317 2 , which will lie between 10 circumference and distance of moon 9*55 So by a slight reduction of base and height we shall have a solid square terrace, having a height of 421 feet = distance of moon. E E 3 422 THE LOST SOLAR SYSTEM DISCOVERED. Perpendicular height of Mount Barkal = about 400 feet. Fig. 81. If the square terrace = distance of the moon, and if, upon the platform, a cone be made similar and equal to the cone at the base, then we shall have a square terrace = distance of the moon, and cone on the platform = circum- ference of the earth. Fig. 81. The height of the cone will equal height of terrace. The Assyrian mound of Koyunjik, at Nineveh, is 2563 yards in length, nearly = 1-^ mile English. If a square mound or terrace had the side = 2563 yards = 6638 units and height = 12 &c. the content would = distance of the moon. The mound of Koyunjik is bounded by a ditch, which, like the rampart, encircles the whole ruins. Layard, in some remarks on his recent researches at Nine- veh, states, that the date of the ruins discovered was still a mystery, but there could be no doubt of their extreme an- tiquity. He would afford one proof of it ; the earliest buildings in Nineveh were buried, and the earth which had accumulated over them had been used as a burial-place by a nation who had lived 700 years before Christ. Probably the buildings dated from 1200 years before Christ. The rooms were lined with slabs of marble, covered with bas- reliefs, which were joined together by double dovetails of iron. The doorways were flanked by winged figures of greater height than the slabs ; on all these figures was the mark of blood, as if thrown against them, and allowed to trickle down. The walls were of sun-dried bricks, and where these showed above the sculptured slabs, up to the ceiling, they were covered with plaster and painted. The NINEVEH. 423 beams, where they remained, were found to be of mulberry. That the slabs should have been preserved so long puzzled many. In truth, however, the bricks being simply dried in the sun, in falling had returned to earth, and had thus buried the tablets and protected them. The buildings were pro- vided with a complete system of sewerage. Each room had a drain connected with a main sewer. In the midst of these ruins he discovered a small chamber formed of bricks regularly arched. The bas-reliefs sent to England by him were, in many cases, found in positions showing that they had been taken from other buildings and re-used the sculptured face of the slab being turned to the wall, and the back re- worked. The small chamber is perfectly vaulted with unburnt bricks, the diameter of the arch being 13 or 14 feet, and the form semicircular. Another curious fact mentioned was the existence of cramps of iron, of a dovetailed form at each end, which had been used to connect the slabs of the internal walls. The "Journal de Constantinople" publishes an extract from a letter written by Layard from Mousoul. " My exca- vation has so far succeeded," he says, "that I have penetrated to the interior of eight . chambers, and found four pairs of winged bulls of gigantic forms. These blocks of marble are covered with sculptures of perfect workmanship, but so in- jured by fire that it is impossible to take their impression. Among the bas-reliefs which have more particularly attracted my notice, is one that represents a mountainous country. Another has also mountains covered with pines and firs. In a third there are vines in a fourth a sea-horse. In one is seen the sea ploughed by many vessels in others cities, which, bathed by the waters of a river, and shadowed by palm-trees, represent, perhaps, the ancient Babylon. The palace brought to light appears to have occupied a consider- able extent of ground, and would require large sums of money for its due examination. An artist should be sent out to draw these bas-reliefs, which differ essentially in style and execution from those of Khorsabad. The palace where these E E 4 424 THE LOST SOLAR SYSTEM DISCOVERED. discoveries have been made is better known to travellers than Nimfoodj and would certainly interest them more. Major Rawlinson makes sensible progress in his reading of the cuneiform characters. It seems certain that the first palace explored at Nimroud was reared by Ninus ; that the obelisk records the exploits of that one of his sons who built the cen- tral palace ; and that thirty years of his reign were employed in the embellishment of these monuments. They treat of the conquest of India and other countries as also of the principal acts of certain other monarchs, ancestors of Ninus. At Nineveh, Botta has laid open fifteen rooms of what appears to have been a vast palace, some of which are 160 feet long, and the walls covered with sculpture and inscrip- tions, the latter historical, and the former illustrating sieges, naval combats, triumphs, &c. The characters employed re- semble those of Persepolis, at Ecbatana (Hamadan), and Van. The sculpture is admirably executed, original in design, and said to be much superior to the figures on the monu- ments of the Egyptians, and show a remarkable knowledge of anatomy and the human face, great intelligence, and har- mony of composition. The ornaments, robes, &c., are exe- cuted with extraordinary minuteness, and the objects, such as vases, drinking-cups, are extremely elegant; the bracelets, ear-rings, &c., show the most exquisite taste. Botta is in- clined to place the sculpture and inscriptions in the period when Nineveh was destroyed by Cyaxares. 160 feet =138 &c. units 139 3 &c.= T -J- 5 - dist. of moon (20xl39) 3 = 4^x203 = 20 cube of 20 times length = 20 dist. of moon 20 cubes of 20 times length =400 dist. of moon = 20 2 distance of moon =dist. of earth. There are curious traces of a large rectangular enclosure south of Medinet-Abou, Thebes, and bordering very near on the enclosure of the temples. " This rectangle, according to Heeren, is about 6,392 feet in length, and 3,196 in breadth, comprising an area of 2,269,870 square yards, MEDINET-ABOU. 425 which is about seven times as much as the Champ de Mars, at Paris, and consequently offered room "enough for the ex- ercises and manoeuvres of a large army. The whole had an enclosure, which is indicated by elevations of earth, between which may still be distinguished the entrances, which have been counted to the number of thirty-nine ; there may, however, have been as many as fifty or more. The prin- cipal entrance was on the east side, where a wider opening is seen. The whole enclosure shows distinctly that it was once adorned with the splendid architecture of triumphal monuments. Probably this extensive circus lay out of the city, but still close to it. A similar one of smaller dimen- sions is seen to the east side of the river, nearly opposite to this on the west, and we may therefore, with some degree of certainty, determine from this double evidence the southern limits of the city. It is highly probable that these spacious enclosures were not merely intended for games, such as chariot races, but also for the mustering and exercising of armies, which, under Sesostris and other conquerors, here began their military expeditions, and returned hither tri- umphant after victory." Sides of the rectangled enclosure are 6392 by 3196 feet = 5527 2763 units. Supposing the height of the enclosing walls, which are in- dicated by the elevation of the earth, to have originally been 12 units, then height x area base = 12x5527x2763 = ^ dist. of moon. Or the content might have equalled circumference of earth. 10 times height of walls = 10x 12 = 120 units = 120x5527x2763 = dist. moon, or 120 &c. x 5527 x 2763 = 10 circumference i stade=121-5 units sum of 2 sides= 5527 + 2763 = 8290 426 THE LOST SOLAR SYSTEM DISCOVERED. 829 3 =5 circumference 8290 3 =5000 (2x8290) 3 = 40000. Cube of sum of 2 sides =5000 circumference, Cube of perimeter = 40000 perimeter =16580 units dist. saturn = 15990 3 = 36000 circumference. Avebury Circle. If the circumference at the top of the mound of the Avebury circle = 4442 feet, diameter will = 1413 feet = 1222 units 1220 3 units=16 circumference. Maurice says the diam. of the Avebury circle =1400 feet, which will=1210 units. Suppose the diam. to equal 1202 units, area of circle will= 1202 2 x '7854. If the area be made a stratum of the depth of unity, the circular stratum will = -j-J-g- circumference. 5 stades = 30 plethrons=1215 units =1405 feet. Cube of 1202 =|- distance of moon, Inscribed cylinder =12 circumference sphere = 8 cone = 4 pyramid =-f- distance of moon 4 cube of 1202 = distance of moon o =2 distance = diam. of orbit of moon cylinder, diam. 8 x 1202, distance of earth f 2 distance of earth = diam. of orbit of earth, or (3 x 1220) 3 = 16 x 3 3 = 432 circumference (10x3xl220) 3 = 432000 Cube of 30 times diam. of circle = 432000 times circumference of earth = diam. of orbit of Belus. AVEBURY CIRCLE. 427 A cylinder having the height = diameter of base =1202 units will = 1202x 1202 2 x -7854 = 1202 3 x '7854 = 12 circumference. Inscribed sphere = f circumscribing cylinder = f 12 = 8 circumference. Inscribed cone =4- 12 = 4 circumference. o Sphere diam. 1202 units = 8 circumference, 601 =1 300 =i 150 = Thus a sphere having a diameter that of the circle (of stones) will = 8 times circumference. A sphere having the diameter = radius of the circle of stones will = circumference. Suppose the diameter of the circular trench having sloping sides to have equalled, originally, 1202 + 72 = 1274 units, cylinder having height = diameter of base will = 1274 3 x 7854. Inscribed sphere will = f 1274 3 x -7854 = 9-55 circumference = distance of moon from earth. Call distance = 30 diameters earth = 30x7926 = 237780 miles circumference =24899 miles, and 9-55 x 24899 = 237780 miles. Thus a sphere having the diameter = the diameter of the circular trench will 9-55 circumference = distance of moon from earth. The circumference at the top of the mound = 4442 feet =3854 units 384 3 , &c.=- circumference Cube of circumference of circle =500 circumference of earth. 428 THE LOST SOLAK SYSTEM DISCOVERED. Cube of 2 circumference = 4000 circumference of earth. 2 circumference = 20 x 384, &c. = 7680 nearly. If circumference of a circle = 3790 units, the cube of twice circumference of circle would = 7580 3 = distance of earth. Cube of circumference would = distance of earth =4JLfi. = 50 distance of moon. 3 cubes of circumference = distance of Mercury, 75 = Saturn, 150 = Uranus, 450 = Belus. The inner slope of the bank of the trench = 80 feet. Should the circumference of the outer circle = 4335 units, Cube of circumference will = ^ distance of Mars. Cube of 2 circumference =1 Cube of 2 circumference at top of mound = 4000 circum- ference. Cube of 3 x 2 circumference = 4000 x 3 3 = 108000. Cube of 6 times circumference of circle =108000 circum- ference of earth =^ distance of Belus. Cube of 3 x 2 circumference = 3 distance of Saturn. Measured circumference = 4442 feet, diameter = 1413 feet = 1222 units, 1 220 3 = 16 circumference, (30 x 1220) 3 = 16 x 30 3 =432000 circumference, = diameter of orbit of Belus. Cube of 30 times diameter = diameter of orbit of Belus. Cube of 24 times diameter = distance of Belus. For 30 : 24 : : 5 : 4 5 3 : 4 3 : : 125 : 64 : : 2 : 1 nearly. If diameter of circle = 1202 units, 1 = 601= side of base of the pyramid of Cephrenes, the cube of which = 601 3 =i distance of moon. AVEBURY CIRCLE. 429 5 cubes = 5 x 601 3 = distance of moon. Cube of diameter = 1202 3 =f distance of moon (5xl202) 3 =f x5 3 = 200. 2 cubes of 5 times diameter =400 distance of moon = distance of earth. Near Avebury is a fallen cromlech ; and various barrows are visible in different parts of the neighbourhood. According to another description of Avebury, the remains originally consisted of one large circle of stones, 138 feet by 155, inclosing two smaller circles, and having two extensive avenues of upright stones. Diameters are 138 by 155 feet = 119 by 133 units say =117 by 131 Cylinder having height =117 and diameter of base = 131, will = 117xl31 2 x -7854= -fa circumference =6 degrees Spheroid =f =^ =4 Cone =-J. = 1^ =2 Cylinder having height = 131 and diameter of base = 117, will = 131 x 117 2 x -7854 = -J0- circumference = 4 '5 degrees Spheroid =A =T I_. =3 Cone =1=^ =1-5 Diameters are 138 by 155 feet = 119 by 133 units. If diameter = 119, circumference = 374 units 378 3 , &c. =-^0 distance of moon. Or cube of circumference = ^V distance of moon, cube of 10 circumference = i J^ Q - = 50. 3 cubes = 150 distance of Mercury, 8 cubes = 400 earth. Or 1 cube of 20 times circumference = distance of earth. 430 THE LOST SOLAR SYSTEM DISCOVERED. If diameter =133, circumference = 41 7 units 414 3 = circumference (2x414) 3 = 5, or cube of 2 circumference of circle = 5 circumference of earth. The diameters to these circumferences will be about 120 and 132 units, 121 3 , &c. = g-J--o distance of moon =iftro = iV radius of earth 131 3 , &c. =- 1 o circumference. If circumference of less circle = 379*2 units 20x379-2 = 7584 distance of earth = 7584 3 cube of 20 times circumference = distance of earth, and cube of 50 times diameter \ distance of earth. If circumference of greater circle = 42 1-2, &c. units 8x421-2, &c. = 3370 i distance of Venus = 3370 3 . Cube of 8 times circumference = 4- distance of Venus. o Cube of 16 times circumference = distance of Venus. Cube of 16 x-^-, or 40 times diameter \ distance of Venus. Sum of diameters = 121, &c. + 131, &c. = 253 units 253 3 , &c. = Y^O" distance of the moon (10 x 253, &c.) 3 = \ ^o = 15 10 cubes of 10 times sum =150 distance of the moon = distance of Mercury. De Ulloa states, that at about 50 toises north of the palace of the Incas of Quito, still called by the ancient name Callo, and fronting its entrance, is a mountain, the more singular as being in the midst of a plain ; its height is between 25 and 30 toises, and so exactly, on every side, formed with the conical roundness of a sugar-loaf, that it seems to owe its form to industry ; especially as the end of its slope on all sides forms exactly with the ground the same angle in every part. And what seems to confirm the opinion is, that guacas, or mausoleums, of prodigious magnitude, HILL AT CALLO. 431 were greatly affected by the Indians in those times. Hence the common opinion that it is artificial, and that the earth was taken out of the breach north of it, where a little river now runs, does not seem improbable. But this is no more than conjecture, not being founded on any evident proof. In all appearance this eminence, now called Panecillo de Callo, served as a watch-tower, commanding an uninter- rupted view of the country, in order to provide for the safety of the province on any sudden alarm of an invasion, of which they were under continual apprehensions, as appears from the account of their fortresses. Taking the toise as equal to 6*44 feet English, we have 27 toises=175 feet, which, if taken as the height of the conical hill at Callo, would make it nearly of the same height as the conical hill at Silbury, and also = the height of the teocalli of Cholula, or = ^ stade. Ulloa gives the proportion of the French to the English foot as 846 to 811, and 6 French feet make 1 toise; so that -f stade will 28-12 toises, 1 stade =45 There is in Lydia a tomb of Alyattes, the father of Croesus, which exceeds in magnitude, according to Herodotus, other monuments, with the exception of those of Egypt and Babylon. The base is formed of large stones, and the rest is terraced. There are five termini placed on the summit of the tomb, on which are inscribed letters indicating what portion of the work each party had accomplished, whence it appears from the measurements that the women had executed a larger portion than the men. The circuit of the tomb measures 6 stadia and 2 plethra, the length, thirteen plethra. Circuit = 6 stades and 2 plethrons, = 1539 units, i = 769 &c. 769 3 = 4 circumference. (2x769) 3 = 32. Cube of circuit =32 circumference, (5 x 2 x 769) 3 = 32 x 5 3 = 4000. 432 THE LOST SOLAE SYSTEM DISCOVERED. 9 cubes of 5 times circuit = 36000 circumference, = distance of Saturn, 18 = Uranus, 54 = Belus. 2 cubes of 15 times circuit = Belus. Length = 13 plethrons = 526 -5 units, 525 3 &c. = 4 distance of the moon. (3x525) 3 =^x3 3 =ig., (5 x 3 x 525) 3 =.fg- x 5 3 = 450. Cube of 15 times length = 450 distance of the moon. =3 Mercury. 50 cubes = 22500 the moon. = distance of Belus. (10 x 3 x 525) 3 = _3JLQjLo_ = 3600 dist. of moon. = 3750-150 Cube of 30 times length = distance between Saturn and Mercury. Circuit =38 plethrons. 2 length = 26 2 breadth =12 Breadth = 6 plethrons = 1 stade = 243 units. 243 3 = -i- circumference, (2 x 243) 3 = 1 circumference, or cube of twice breadth = circumference of the earth. Cube of 120 times breadth = cube of Babylon = distance of Belus. Cube of 12 times breadth = 1 * 00 . Cube of 20 times breadth = -jj-g-. In the environs of Sardis is a colossal tumulus, believed to be the tomb of Alyattes. It is a cone of earth 200 feet high. Leake regards it as one of the most remarkable antiquities in Asia. The base is now covered with earth, but the tomb still retains the conical form, and has the appearance of a natural hill. Newbold describes Sardis, the ancient capital of Croesus, as being now desolate, scarcely a house remaining. The me- STONEHENGE. 433 lancholy Gygaean lake, the swampy plain of Hermus, the thousand mounds forming the necropolis of the Lydian mo- narchs, among which rises conspicuous the famed tumulus of Alyattes, produce a. scene of gloomy solemnity. Massive ruins of buildings still remain, the walls of which are made of sculptured pieces of the Corinthian and Ionic columns that once formed portions of the ancient pagan temples. The Pactolus, famed for its golden sands, contains no gold ; but the sparkling grains of mica with which the sand abounds, have probably originated the epithet. Stonehenge stands in the middle of a flat area, near the summit of a hill. It is enclosed by a double circular bank and ditch, nearly thirty feet broad, after crossing which an ascent of nearly thirty yards leads to the work. The whole fabric was originally composed of two circles and two ovals. The outer circle is about 108 feet in diameter, consisting, when entire, of 60 stones, 30 uprights, and 30 imposts. 11 uprights have their 5 imposts on them by the grand entrance; these stones are from 13 to 20 feet high. The smaller circle is somewhat more than 8 feet from the in- side of the outer one, and consisting of 40 smaller stones, the highest measuring about 6 feet, 19 only of which now remain, and only 11 standing. The walk between these two circles is 300 feet in circumference. The " adytum," or cell, is an oval formed of 10 stones, from 16 to 22 feet high, in pairs, and with imposts above 30 feet high, rising in height as they go round, and each pair separate, and not connected as the outer pair; the highest 8 feet. Within these are 19 other smaller single stones, of which 6 only are standing. At the upper end of the adytum is the altar, a large slab of blue coarse marble, 20 inches thick, 16 feet long, and 4 feet broad; it is pressed down by the weight of the vast stones which have fallen upon it. The whole number of stones, uprights and imposts, comprehending the altar, is 140. Another account makes the circumference of the surround- ing ditch 369 yards. According to another description of Stonehenge, the whole VOL. I. F F 434 THE LOST SOLAR SYSTEM DISCOVERED. structure was composed of 140 stones, including those of the entrance, forming two circles and two ovals, respectively concentric. The whole is bounded by a circular ditch, origi- nally 50 feet broad, the inside verge of which is 100 feet distant, all round, from the greater extremity of the greater circle of stones. The circle is nearly 108 feet in diameter ; so that the diameter of the area wherein Stonehenge is situ- ated, is about 408 feet. The vallum is placed inwards, and forms a circular terrace, through which was the entrance to the north-east by an avenue of more than 1700 feet in a straight line, bounded by two ditches, parallel to each other, about 70 feet asunder. Avenue is more than 1700 feet, or 1470 units. (148 &c.) 3 = ToVo distance of the moon (10x148 &c.) 3 =1482 3 = fgJHi=3. Cube of length = 3 distance of the moon. (5 x 10 x 148 &c.) 3 =3 x 5 3 = 375. 10 cubes of 5 times length = 3750 distance of the moon = distance of Saturn 20 = Uranus 60 = Belus. Distance between the parallel ditches is about 70 feet, or 60 units. Sum of 2 sides = 1482 -f 60= 1542. 153 3 &c.= 3-J-Q distance of the moon (10 x 153 &c.) 3 = Vdr = V - Cube of sum of 2 sides = ^ distance of the moon. (3x10x153 &c.) 3 =Vx 33 = 90 (5x3x 10x153 c.) 3 = 90x5 3 = 11250. 2 cubes of 15 times sum of 2 sides = 22500 distance of the moon = distance of Belus. Cube of greater side I cube of sum of 2 sides:: 3 : ^>::9 : 10. or breadth = 60 units (10x60-l) 3 = 601 3 = |- distance of the moon (10 x 10 x 60-1) 3 = UULP = 200. STONEHEXGE. 435 2 cubes of 100 times breadth = 400 distance of the moon = distance of the earth. Diameter of circumscribing circle = 408 feet= 353 units circumference =1109 JT= 554 &c. 554 3 &c. = f circumference (2x554 &c.) 3 = V = 12 - Cube of circumference of circle =12 times circumference of the earth. Cube of twice circumference=12 x 8 = 96. 15 cubes j, = 1440 circumference = distance of Mercury 40 cubes =3840 circumference = distance of the earth or 5 cubes of 4 circumference = Cube of 10 times circumference = 12000 circumference of the earth = ^ distance of Saturn = -|- 3, Uranus = 3*3 Belus. Diameter of great circle of stones=108 feet = 93'37 units circumference = 2 93 &c. 293 3 &c. =f circumference (3x293 &c.) 3 =|x3 3 = 6. Cube of 3 times circumference of circle = 6 times circum- ference of the earth. Cube of 6 times circumference of circle =48 times circum- ference of the earth. 30 cubes =1440 circumference = distance of Mercury 80 cubes = 3840 the earth. Cube of 30 times circumference = 6 000 circumference of the earth = distance of Saturn = iV Uranus = T V Belus F F 2 436 THE LOST SOLAR SYSTEM DISCOVERED. Should circumference = 29 1-6 units, cube of 100 times cir- cumference would = 29 160 3 = distance of Belus = cube of Babylon. Circumference of ditch = 369 yards = 1107 feet = 957 units 954 3 = - distance of the moon Cube of 10 times circumference = 800 distance of the moon = diameter of the orbit of the earth. Cube of 10 x V * or f 25 diameter = distance of the earth nearly. Cube of 5 times circumference = 100 distance of the moon. 3 cubes = 300 distance of the moon = diameter of the orbit of Mercury 4 cubes = distance of the earth 75 = Uranus 225 = Belus. Breadth of ditch = 50 feet. So that the diameter of the circle on the inside verge will = 408-100 = 308 feet circumference = 967 feet = 836 units, and 828 3 = 5 circumference, or cube of circumference of circle = 5 times circumference of the earth. (4 x 828) 3 = 5 x 4 3 = 320 circumference. 12 cubes of 4 times circumference of the circle = 3840 earth = distance of the earth. Cube of 10 times circumference of the circle = 5000 cir- cumference of the earth. Cube of 10 x y, or of 25 diameter, = 2500 circumference of the earth. If circumference = 841 &c. units 8x841 &c. = 6730 distance of Venus = 67 30 3 . STONEHENGE. 437 Cube of 8 times circumference = distance of Venus. Diameter of inner circle of stones is somewhat more than 92 feet, or 79'54 units, circumference = 249 '8 if =254-4 100x254-4 = 25440 diameter of the orbit of Uranus = 25440 3 . Cube of 100 times circumference = diameter of the orbit of Uranus. Cube of 100 x V diameter, or of 250 diameter = distance of Uranus. Circumference of ditch = 957 units 10x956 = 9560 diameter of the orbit of the earth = 9560 3 . Cube of 10 times circumference = diameter of the orbit of the earth. Cube of 10 x y, or of 25 times diameter = distance of the earth. Twice circumference of inner circle of stones = 2 x 243 = 486 units. Cube of twice circumference of circle = 486 3 = circum- ference of the earth. The numerals 486 transposed and squared = 6 84 2 = circum- ference of the earth in stades. Sum of 2 diameters of circles of stones = 93-37 + 79-5 = 173 units circumference = 544 546 3 -/Q distance of the moon (20 x 546) 3 = T V x 20 3 = 1200 Cube of 20 times circumference = 1200 distance of the moon pyramid i = 400 = distance of the earth. (10x546) 3 = -3-f-|p=150 distance of the moon Cube of 1 times circumference = distance of Mercury = 150 distance of the moon 150 cubes = distance of Belus = 22500 distance of the moon. F F 3 438 THE LOST SOLAR SYSTEM DISCOVERED. Cube of circumference of greater circle of stones : cube of circumference of less :: f .* ^ :: 16 : 9 :: 4 2 I 3 2 . If sum of 2 diameters = 173 &c. units 50x173 &c. = 6890 distance of Mars = 6890 3 10 times circumference = 5460 distance of Mercury ~5460 3 . Cube of 50 times diameter = distance of Mars 10 circumference = Mercury 120 circumference of circle = 243 = Belus. The outer circle, when entire, consisted of 60 stones, 30 uprights, and 30 imposts; 17 of the uprights remain stand- ing, and 6 are lying on the ground, either whole or in pieces, and 1 leaning at the back of the temple, to the south-west, upon a stone of the inner circle ; these 24 .uprights and 8 imposts are all that remain of the outer circle. The up- right stones are from 18 to 20 feet high, from 6 to 7 broad, and about 3 feet in thickness ; and being placed at the dis- tance of 3^ feet from each other were joined at the top by mortise and tenon to the imposts, or stones laid across like architraves, uniting the whole outer range in one continued circular line at the top. The outsides of the imposts were rounded a little to favour the circle, but within they were straight, and originally formed a polygon of 30 sides. At the upper end of the adytum, or cell, is the altar, a large slab of blue coarse marble, 20 inches thick, 16 feet long, and 4 broad : it is pressed down by the weight of vast stones that have fallen upon it. At some distance round this famous monument are great numbers of sepulchres, or, as they are called, barrows, being covered with earth, and raised in a conical form. They extend to a considerable distance from the temple, but are so placed as to be all in view of it. Such as have been opened were found to contain either human skeletons or ashes of burnt bones, together with warlike instruments, and such things as the deceased used when alive. OLD SARUM. 439 From these sepulchres being within sight of the temple, as we have seen the small pyramids and sepulchral chambers erected near the great pyramidal temples, we may conclude that, like the Christians of the present age, the ancients thought it was most proper to bury their dead adjoining those places where they worshipped the Supreme Being. Indeed, all worship indicates a state of futurity, and they might reasonably imagine that no place was so proper for depositing the relics of their departed friends as the spot dedicated to the service of that Being with whom they hoped to live for ever. The sentiment is altogether natural ; no objection can be made to it, while the depositories of the dead are detached from populous towns or cities ; but no one can excuse the present mode of crowding corrupt bodies into vaults under churches, adjoining to the most public streets, where the noxious effluvia may be attended with the most fatal consequences to the living. Close to the village of E'Mozora, in Western Barbary, is the site of an heliacal temple, whereof, among numerous remains now prostrate, one stone, called vulgarly by the Moors Al Ootsed, or the peg, stands yet erect, and is of such large dimensions, that it would not discredit the stu- pendous structure on Salisbury Plain. (Hay.} The ancient Sorbiodunum, or Old Sarum, is about a mile north of Salisbury, and was one of the ten British cities ad- mitted to the privileges of the Latin law. Of this once flourishing and celebrated place nothing now remains but its ruins. It is to this place the present city owes its origin. The name is supposed to be derived from a British compound word, signifying a dry situation ; and the Saxons, who called this place Searysbyrze, seem to have a reference to the same circumstance ; searan, in the Saxon language, signifying " to dry." Leland supposes Sorbiodunum to have been a British post prior to the arrival of the Romans, with whom it afterwards became a principal station, or castra stativa. Besides the evidence of the Itineraries, and the several roads of that people which here concentrate, the great number of F F 4 440 THE LOST SOLAR SYSTEM DISCOVERED. Roman coins found within the limits of its walls prove its occupation as a place of consequence by the Romans. Ac- cording to the author of " Antiquitates Sarisburiensis," some of the Roman emperors actually resided at Old Sarum. Le- land mentions this place as having been very ancient and ex- ceedingly strong. It covers the summit of a high steep hill, which originally rose equally on all sides to a point. The area was nearly 2000 feet in diameter, surrounded by a fosse or ditch of great depth, and two ramparts, some remains of which are still to be seen. On the inner rampart, which was much the highest, stood a wall, nearly 12 feet thick, made of flint and chalk strongly cemented together, and cased with hewn stones, on the top of which was a parapet, with battlements quite round. Of this wall there are some re- mains still to be seen, particularly on the north-west side. In the centre of the whole rose the summit of the hill, on which stood a citadel or castle, surrounded with a deep en- trenchment and very high rampart. In the area under it stood the city, which was divided into equal parts, north and south, by a meridian line. Near the middle of each division was a gate, which were the two grand entrances ; these were directly opposite to each other, and each had a tower and a mole of great strength before it. Besides these, there were two other towers in every quarter, at equal distances, quite round the city ; and opposite to them, in a straight line with the castle, were built the principal streets, intersected in the middle by one grand circular street. In the north-west angle stood the cathedral and episcopal palace ; the former, according to Bishop Godwin, was consecrated in an evil hour ; for the very next day the steeple was set on fire by lightning. The foundations of these buildings are still to be traced, but the site of the whole city has been ploughed over. Leland adds to his account, that " without each of the gates of Old Sarum was a fair suburb, and in the east suburb a parish church of St. John, and thereon a chapel, yet standing. There had been houses in time out of mind in- habited in the east suburb ; but there is not one within or without the city. There was a parish church of the Holy OLD SARUM. 441 Rood, in Old Saresbyrie, and another over the gate, whereof some tokens remain." About the time the West Saxon kingdom was established, King Kenric, or Cynric, resided here, after having defeated the Britons. This prince, about four years after, incor- porated Wiltshire with Wessex. About the middle of the tenth century, in the reign of Edgar, a great council, or witenagemote, was summoned by that prince, when several laws were enacted for the better government of church and state. Soon afterwards (in the year 1003) it was plundered and burnt by S \veine, the Danish king, in revenge for the massacre committed by the English on his countrymen the preceding year. It was, however, rebuilt, and became so flourishing, that the bishop's see was removed thither from Sherborne, and the second of its bishops built a cathedral. William the Conqueror summoned all his states of the king- dom hither, to swear allegiance to him, and several of his successors often resided here. In 1095, William II. held a great council, which impeached William, Earl de Ou, of high treason, for conspiring to raise Stephen, Earl of Albemarfe, to the throne. His cruel punishment marks the barbarity of the age. Henry I. held his court here in 1100, and again in 1106. In 1116, he ordered all the bishops, abbots, and barons, to meet here, to do homage to his son William, as his successor to the throne. Here, in 1483, was executed Henry Stafford, Duke of Buckingham, who had exerted all his influence, and used every effort, to advance Richard III. to the throne. James I. frequently visited Salisbury, as did Charles I. On one oc- casion, when the latter was here, in 1632, a boy only fifteen years of age was hanged, drawn, and quartered, for saying he would buy a pistol to kill the king. We find the first prelude to its downfall was a quarrel that happened between King Stephen and Bishop Roger, the latter of whom espoused the cause of the Empress Maud, which enraged the king to such a degree, that he seized the castle, which belonged to the bishops, and placed a governor and garrison in it. 442 THE LOST SOLAR SYSTEM DISCOVERED. This was looked upon as a violation of the rights of the church, and occasioned frequent differences between the military and the monks and citizens, the issue of which was, that the bishop and canons determined to remove to some place where they might be less disturbed, having in vain ap- plied to the king for redress of their grievances. From the time that Stephen put a garrison into the castle, Old Sarum began to decay. The removal of the city was first projected by Bishop Herbert, in the reign of Richard II. ; but the king dying before it could be effected, and the turbulent reign of John ensuing, the plan could not be carried into execution until the reign of Henry III., when Bishop Richard Poore fixed upon the site of the present cathedral, and translated the episcopal see. The inhabitants of Old Sarum speedily fol- lowed, being intimidated by the insolence of the garrison, and at the same time suffering great inconvenience through the want of water. By degrees, Old Sarum was entirely de- serted, and at present there is but one building left within the precincts of the ancient city. However, it is still called the borough of Old Sarum, and sent two members to Par- liament, till the Reform Act of 1832, who were chosen by the proprietors of certain lands adjacent. The area of the base of the conical hill is nearly 2000 feet, or 1730 units in diameter. Diameter of external cone of Silbury will = 557 units, and content = -f^ circumference. Diameter of external cone at Sarum = 1738 units. If the conical hills at Silbury and Sarum were similar their contents would be as 1 : 30, Then content of the conical hill at Sarum would = -%-x 30 = ^-4=4 circumference 1 O 1 O = 8 times pyramid of Cheops. The hill is surrounded by a fosse and two ramparts. If the diameter of one of these circles should be about 1740 units, circumference would = 5466. Cube of circumference would = 5466 3 = distance of Mercury. OLD SARUM. 443 The height of the steep hill, which originally rose equally on all sides to a point, is not stated. The principal streets radiated from the castle and were intersected in the middle by one grand circular street. Sarum appears to have been the Rome of Britain, the residence of her pontifical Druids, whose altars were over- turned and religion extirpated by the Romans. The throne of the Caesars at Rome has since been sup- planted by the hierarchal chair of St. Peter, where the sovereign pontiff by his supreme temporal and spiritual authority rules the Eternal City and states ; as the Roman emperors previously ruled the destinies of kingdoms by military power. The glory of Sarum gradually became extinct : the last ray was when, reduced to only one house, she retained the power of returning two members to parliament; among those whom towards the last she sent to commence their political career was Chatham, the father of Pitt. At last Sarum, after having been the Rome of the Druids, and the Windsor of kings and emperors, who ruled by their will, was deprived of even a representative in the Commons of England, and is now forgotten. It would seem more than probable that the great teocallis were originally constructed for religious purposes, and also as places of defence in time of danger. The old ballad, alluding to Sarum, says " 'Twas a Roman town, of strength and renown, As its stately ruins show. Therein was a castle for men and arms, And a cloister for men of the gown." The cathedral of Salisbury, or New Sarum, is a Gothic structure. From the centre of the roof, which is 116 feet high, rises a beautiful spire of freestone, the altitude of which is 410 feet from the ground, and is esteemed the highest in the kingdom ; being nearly 70 feet higher than the top of St. Paul's, and just double the height of the Monument in London. 444 THE LOST SOLAR SYSTEM DISCOVERED. 1 stade = 281 feet = height of tower of Belus 1 = 421-5 = 410 = height of the spire. The singularity of there being in this cathedral 365 win- dows, &c. is explained in the following verses: " As many days as in one year there be, As many windows in this church you see ; As many marble pillars here appear As there are hours throughout the fleeting year ; As many gates as moons one here does view ; Strange tale to tell ! yet not more strange than true." Between Ashbourne and Buxton in Derbyshire is a circle of stones, or Druidical temple, called Arbe Lowes, 150 feet in diameter, surrounded by a large bank of earth, about 11 yards high in the slope, but higher towards the south or south-east, and formed by a large barrow ; the ditch within is four yards in width, with two entrances, east and west. Diameter = 150 feet = 129-6 units 10 x 129-6 = 1296 and 1296 3 = diameter of orbit of moon. Cube of 10 times diameter = diameter of orbit of moon. Cube of 4 times circumference of diameter 1296 = 2 diameter of orbit of moon. Circumference = 407 units. 90x407 = 36630 diameter of orbit of Belus = 36630 3 . Cube of 90 times circumference = diameter of orbit of Belus. Cube of 90 x V diameter, or of 225 diameter = 29160 3 = distance of Belus. At Hathersage, in Derbyshire, above the church, at a place called Champ Green, is a circular area, 144 feet in diameter, encompassed with a high and pretty large mound of earth, round which is a deep ditch. DERBYSHIRE CIRCLES. 445 Diameter 144 feet= 124*5 units. Circumference =391, &c. 60x126-4 = 7584. Distance of the earth =7584 3 . Cube of 60 times diameter = distance of the earth. If circumference = 399 units, 30x399 = 15960. Distance of Saturn = 15960 3 . Cube of 30 times circumference = distance of Saturn. If circumference = 3 92 units, 392 3 = I 1 g distance of the moon, (6 x 392)3=^x63= 12, (5 x 6 x 392) 3 = 12 x 5 3 = 1500. Cube of 6 times circumference = 12 distance of the moon. Cube of 30 =1500 = 10 times distance of Mercury, = I 3 g- distance of Belus. On Stanton Moor, a rocky, uncultivated waste, about two miles in length, and one and a half broad, are numerous remains of antiquity, as rocking-stones, barrows, rock- basons, circles of erect stones, &c., which have generally been supposed of Druidical origin. The following Druidical circles are also in Derbyshire. In a field north of Grand Tor, called Nine-stone Close, are the remains of a circle called Druidical, about 13 yards in diameter, now consisting of seven rude stones of various dimensions : one of them is about eight feet in height, and nine in circumference. Between seventy and eighty yards to the south are two other stones, of similar dimensions, standing erect. Diameter 13 yards = 39 feet = 33 '5 units. If diameter = 33'2 units, circumference = 104, &c. Diameter of circle to the power of 3 times 3 = 33'2 9 = diameter orbit of Belus. Cube of circumference =104 3 , &c. = T -J- 5 - circumference. Cube of 10 times circumference of circle = 1 j Q ^y ) = 10 = 10 times circumference of earth. 446 THE LOST SOLAR SYSTEM DISCOVERED. Diameter=13 yards = 33'5 units. If = 33-65. 100x33-65 = 3365, J- distance of Venus = 3365 3 , 200x33-65 = 6730, distance of Venus =6730 3 . Cube of 200 times diameter = distance of Venus. Cube of 200 x -fa or of 80 times circumference = diameter of orbit of Venus. Circumference =105, &c., if=106,&c. 90x106, &c,, =9540. Distance of the earth = 9540 3 . Cube of 90 times circumference = distance of the earth. About a quarter of a mile west of the little valley which separates Hartle Moor from Stanton Moor is an ancient work, called Castle Ring, supposed to have been a British encampment. Its form is elliptical; its shortest diameter, from south-east to south-west, is 165 feet; its length, from north-east to south-west, 243. It was encompassed by a deep ditch and double vallum, but part of the latter has been levelled by the plough. Greater diameter of Hartle Moor ellipse = 2 10 units = 243 feet. Less diameter=142 units=165 feet. 2 1 1 3 , &c. = -jL circumference. 141 3 , &c.= T V Circumference of circle diameter 210 = 659 units. 658 3 = 1 - f circumference. (2x658) 3 =\=20. Cube of twice circumference of circle = 20 circumference of the earth. Circumference of circle diameter 142 = 446 units. 449 3 , &c. = |- circumference. (5 x 449, &c.) 3 = x 5 3 = 100. (3 x 5 x 449, &c.) 3 = 100 x 3 3 = 2700. STANTON MOOR CIRCLE. 447 Thus cube of 5 times circumference =100 circumference of the earth. Cube of 15 times circumference of circle = 2700 circum- ference of the earth = distance of Venus. 80 cubes = distance of Belus, or 10 cubes of 30 times circumference = distance of Belus. Circumferences = 659 and 446, if =.652 434. 30x652 = 19560, Distance of Uranus =195 60 3 , 30x434, &c. = 13040. Distance of Jupiter = 13040 3 . About half a mile north-east from the Router rocks, on Stanton Moor, is a Druidical circle, eleven yards in diameter, called the Nine Ladies, composed of the same number of rude stones, from three to four feet in height, and of dif- ferent breadths. A single stone, named the King, stands at a distance of thirty-four yards. Diam. = 11 yards = 33 feet = 28-53 units. say = 27 &c. Cylinder having height = diam. of base will = 27 3 , &c. x -7854 = -/Q degree = 3 minutes Sphere = f = -ft- . =2 Cone =i = -gV =1 Near this circle are several cairns and barrows; most of which have been opened, and various remains of ancient customs discovered in them. Urns, with burnt bones, &c. have been found in these and some of the other barrows. Under one of the cairns human bones were found, together with a large blue glass bead. Cone = 1 minute = L^= _!__ circumference = 1 geographical mile. Cube of Babylon = 216000 circumference = 21600x10. 448 THE LOST SOLAR SYSTEM DISCOVERED. Circumference = 21600 cones of Stanton Moor. So cube of Babylon = 21600 2 x 10 cones = 21600 2 x 10 miles, or distance of Belus =10 times the square of the earth's circumference when' unity = 1 geographical mile. Diameter = ] 1 yards = 33 feet = 28-53 units. Circumference = 89-53. (10 x 89-8) 3 = f distance of the moon. (SxlOx 89-8) 3 = | x 3 3 = 18. (5 x 3 x 10 x 89-8) 3 = 18 x 5 3 x 2250. Cube of 150 times circumference = 2250 distance of the moon = -^Q distance of Belus. 3 cubes of 10 times circumference = 2 distance of the moon = diameter of the or- bit of the moon. Diameter = 28 '53 units. (10 x 28-4) 3 = ^ circumference. 10 cubes of 10 times the diameter of the circle = twice the circumference of the earth. 28 -6 9 = distance of Neptune. Should circumference = 91 units 60 x 91 = 5460 Distance of Mercury = 5460 3 . Cube of 60 times circumference = distance of Mercury. On the top of Banbury Hill, in Berkshire, is a supposed Danish camp of a circular form, 200 yards in diameter, with a ditch of 20 yards wide. Diameter 200 yards = 600 feet = 519 units. Circumference = 1630. i- = 815. 816 3 =1 distance of moon. (2x816) 3 = f = 4. Cube of circumference = 4 distance of the moon. BANBURT HILL CIRCLE. 449 100 cubes =400 dist. of moon = dist. of earth. 70 =280 = Venus. Cube of 10 times circumference = 4000 distance of moon, = 10 times the distance of the earth. Twice width of ditch = 40 yards = 120 feet = 103 units. Diameter of outer circle will = 519 + 103 = 622 units, and circumference = 1952. i- = 976. 96 8 3 , &c. = 8 circumference (2 x 968, &c.) 3 = 64 Cube of circumference of circle = 64 circumference of the earth (5 x 2 x 968, &c.) 3 = 64 x 5 3 = 8000. 9 cubes of 5 times circumference of the circle = 72000 circumference of the earth = distance of Saturn 18 39 Uranus 54 = Belus 60 cubes of circumference =64 x 60=38 40 circum- ference = distance of the earth. Should circumference of less circle = 1596 units 10 x 1596 = 15960 Distance of Saturn = 15960 3 . If circumference of greater circle =1956 10 x 1956 = 19560 Distance of Uranus = 1 9560 3 . Cube of 10 times less circumference = distance of Saturn Cube of 10 times greater circumference = Uranus. " Kath " is a Celtic word for " fort." It abounds in Scot- land, but usually with a variety of pronunciations. Such forts are usually mere earth-works, forming a circle, or set of con- centric circles, on plain ground, or cutting off the outer angles of a bank overhanging a rivulet. The enclosure is supposed to have contained temporary buildings for residence. VOL. I. G G 450 THE LOST SOLAR SYSTEM DISCOVERED. The celebrated hill of Tara, in the county of Meath, Ireland, is covered with a cluster of raths, and presents few other objects. From an indefinitely early period down to the sixth century it was a chief seat of the Irish kings, according to Wakeman. Shortly after the death of Dermot, the son of Fergus, in the year 563, the place was deserted, in consequence, as it is said, of a curse pronounced by St. Ruadan, or Rodanus, of Lorha, against that king and his palace. After thirteen centuries of ruin, the chief monu- ments for which the hill was at any time remarkable are distinctly to be traced. They consist for the most part of circular or oval enclosures and mounds, within or upon which the principal habitations of the ancient city undoubtedly stood. The rath called Rath Righ, or Cathair Crofinn, appears anciently to have been the most important work upon the hill, but it is now nearly levelled with the ground. It is of an oval form, measures in length from north to south about 850 feet, and appears in part to have been constructed of stone : within its enclosure are the ruins of the Forradh, and of Teach Cormac, or the house of Cormac. The mound of the Forradh is of considerable height, flat at the top, and encircled with two lines of earth, having a ditch between them. In its centre is a very remarkable pillar stone, which formerly stood upon, or rather by the side of a small mound, lying within the enclosure of Rath Righ, and called Dum- hana-n-Giall, or the mound of the Hostages, but which was removed to its present site to mark the grave of some men slain in an encounter with the king's troops during the rising of 1798. It has been suggested by Petrie, that it is extremely probable that this monument is no other than the celebrated Lia Fail, or Stone of Destiny, upon which, for many ages, the monarchs of Ireland were crowned, and which is generally supposed to have been removed from Ireland to Scotland for the coronation of Fergus Mac Eark, a prince of the blood royal of Ireland, there having been a prophecy that in whatever country this famous stone was preserved, a king of the Scotic race should reign. The Teach Cormac, lying on the south-east of the For- HILL OF TARA. 451 radh, with which is joined a common parapet, may be described as a double enclosure, the rings of which upon the western side became connected. Its diameter is about 140 feet Q- a stade = 140J- feet.) Diameter of ellipse = 850 feet = 734-6 units. 734 3 =^- distance of the moon. (3x734) 3 =J r x3 3 =9 Cube of diameter = Cube of 3 diameter = 9 Circumference of circle of diameter 734 = 2206 units. 22 1 3 &c. =7^-0 distance of the moon. (10x221 & C .) 3 =Y^ = 10. Cube of circumference = 10 distance of the moon. Diameter of Teach Cormac=i stade. Cube of diameter =^5- circumference. Cube of circumference = |i = i circumference nearly. Cube of 4 times diameter = circumference. Cube of 4 times circumference = 31. Cube of 20 times circumference = 31 x5 3 =3875 circum- ference. distance of the earth = 3840. This is the only measured Druidical monument in Ireland that we have met with. We find Druidical monuments in Denmark, Sweden, and Norway, quoting from the French " Idolatrous Nations." It appears that the Laplanders in Denmark, natives of Finland, and the proper Laplanders in former times all worshipped Jumela as the Supreme Being, and likewise the Sun and Moon. Storjunkare is represented under the form of a large unpolished stone, such as is met with in the mountains ; sometimes it is sculptured. This stone-god is frequently supplied with a numerous family ; one of them is his wife, others his sons and daughters, and the rest his domestics. Kein-deers are sacrificed to Thoron, but to the Sun only young female deers. They have tutors ^and academies for the particular study of the black art. They G G 2 452 THE LOST SOLAR SYSTEM DISCOVERED. stand in awe of their manes, or the souls of their dead, till they are actually transmigrated into new bodies ; whence it is manifest that their notion, with respect to souls, is the same as that received among the Tartars and Scythians, who borrowed it from the eastern nations. There is an ancient chapel, in ruins, situated between Revel and Nerva, where some devotees strip themselves naked and fall down on their knees, before a great stone, which stands in the middle of the chapel ; they also dance round it, and offer oblations of fruits and other provisions. This ceremony is a relic of that religious worship of the Goths which all the people in general of the north, the Ger- mans, Gauls, c., paid formerly to stones ; , and we are as- sured that this divine adoration of them was grounded on a notion, which was then established among all those idolaters, that some diminutive sprites, or imps of the devil, resided within those stones ; nay, they carried the point still further, and were fully persuaded that those stones were oracles. At this day the peasants in part of Brittany believe that at certain periods of the year, when the moon shines brightly, that hideous dwarfs, whom they call Cormandons, rise from their subterranean abodes, form an infernal ring about the dol-mens and men-hirs, and try to attract travellers by ring- ing gold upon the sacred stone. "We consider some of the single upright Druidical stones to be rough representations of the accurately proportioned and highly-finished obelisk of the Egyptians. Indeed some of them assume the rough, square, tapering, truncated form, as already noticed. Others are sculptured. In Scotland four or five ancient obelisks are still to be seen, called the Danish stones of Aberlemno, and are adorned with bas-reliefs of men on horseback, and many emblematical figures and hieroglyphics, not intelligible at this day. The stone near Forrest rises about 23 feet above the ground, and is supposed to be not less than 12 or 15 feet below, so that the whole height will be at least 35 feet, and its breadth is nearly 5 feet. A great variety of figures in relief are carved on it. Many Druidical monuments and temples are dis- SACKED STONES. 453 cernible in the northern parts of Scotland as well as in the isles. They are circular, and equally regular with those in England, but not on so large a scale. The cromlech at Plas Newydd, in Anglesea, is formed by a massive irregular-shaped stone, supported laterally by other stones, which incline inwards from the base to the stone that forms the roof; the whole structure resembles an Egyptian propylon. In the Druidical circle at Jersey, a large stone is repre- sented as forming a projecting roof, which is supported late- rally by other stones inclining inwards from the base to the top like Kit's Cotty house, in Kent ; one view of the last represents the external sides as but little inclined to the roof, which is nearly flat and projecting, like the top of a pro- pylon, or the roof of the monolithic chapel at Butos, already described by Herodotus as having a single flat stone pro- jecting over the sides of the chapel. The tomb of Cyrus seems to have been formed like the Butos stone chapel, with a projecting roof, according to the " Antiquities of Persia," Kit's Cotty is called a Kist- vaen, or stone chest, which not only accords with our views, but it will be seen that the use made of the Kist-vaen may throw some light on the monolithic chambers or chest of the Egyptians. Davies describes the probation of Taliesim, a Druidical noviciate. " I was first modelled in the form of a pure man in the hall of Ceridwin, who subjected me to penance. Though small within my ark, and modest in my deportment, I was great. A sanctuary carried me above the surface of the earth. Whilst I was enclosed within its ribs the sweet Awen rendered me complete." Whence Davies infers that the Kist-vaen is very probably the ark here referred to. The Kist-vaen, like other Druidical monuments, is found in different and remote parts of the world. There is one on the banks of the Jordan resembling Kit's Cotty. The opinion of Clemens Alexandrinus is that columns were worshipped as the images of God. Herodian says the Phre- nicians worshipped a great stone circular below, and ending QO 3 454 THE LOST SOLAR SYSTEM DISCOVERED. with a sharpness above in the figure of a cone, and of a black colour. They report it to have fallen from heaven, and to be the image of the sun. The vertical section, or plane of an Egyptian obelisk revolving on its axis, would generate a solid answering this description, like the pointed minaret. This conical stone was called Elseogabalis. M. Aurelius Antoninus, a Roman emperor, called Helio- gabalus, because he had been a priest of that divinity in Pho3nicia, obliged his subjects to pay adoration to the god Heliogabalus, which was no other than a large black stone, having the form of a cone, that he brought with him to Rome on his being elected emperor by the army ; he built a temple to the god, and continued priest himself, commanding the Vestal fire, the palladium, and consecrated bucklers to be transported thither. Mahomet destroyed other superstitions of the Arabs, but he was obliged to adopt their rooted veneration for the black stone, and transfer to Mecca the respect and reverence which he had designed for Jerusalem. (Pitts.) It appears that history can still trace among various other nations the worship of conical and pyramidal stones. The Paphian Venus was the celestial Venus of the Assy- rians, and represented, according to Tacitus, by a cone, but, according to Maximus Tyrius, by a white pyramid. The Paphian Venus, says Pausanias, was worshipped first by the Assyrians, afterwards by the Paphians and Phoenicians of Ascalon. The Cythereans acquired these rites from the Phrenicians. He also states that it was the custom of the Greeks, at an early period, to reverence the form of rude stones instead of statues ; and adds, that several such existed at his time. At Pherse there were thirty square stones, each called by the name of some deity. Mercury was frequently represented by a rude stone. The Apollo Carynus and Jupiter Milichius, in the forum of Sicyon, were worshipped under the form of small pyramids: the Diana Patroa, in the same place, under that of a column. The Hercules of Hyettus was a rude stone. The symbol of Cupid at Thespia was also a rude stone. According to Cle- SACKED STONES. 455 mens Alexandrinus, the Delphic Apollo was once a column. Lactantius mentions the worship of Terminus under the form of a rude stone. Hamilton describes one of the idols in the pagoda of Jug- gernaut as a huge black stone, of a pyramidal form, Mau- rice mentions a black stone, 50 cubits high, that stood before the gate of a temple erected to the sun by an ancient rajah. In the pagoda at Benares is an idol of a black stone. Boodh was represented by a huge column of black stone. Hercules, Neptune, Cupid, Jupiter, Juno, and Diana, says Legrew, were first worshipped under the symbols of cones and pyramids ; at a later period, these statues pre- sented forms of transcendent beauty. On the old coins of Apollonia, according to D'Anker- ville, Apollo was represented by an obelisk a little different from those of Egypt. On the medal of the Chalcidians is an ancient representation of Neptune, in the form of a pyra- mid. On a medal of Ceos, Jupiter and Juno appear in the form of pyramids ornamented with draperies. Damascius, in his life of Isidorus, states, that many consecrated stones were to be seen near Heliopolis, in Syria ; and adds, that they were dedicated to Gad, Jupiter, the Sun, and other deities. Seetzen having assumed the character of a Mahometan, took a passage in a vessel from Suez, where there were a number of other pilgrims destined for Mecca. Before reach- ing Jidda, they came to a village called Rabog, where the ceremony took place of putting on the ehhran the pilgrim's dress. Thus transformed into pilgrims, they began to cry aloud Lubbaik, Allahoumme Lubbaik, an ancient form of prayer which Seetzen suspects of being appropriated to Bac- chus. At Mecca he found the holy temple composing a most majestic square, 300 feet by 200, and surrounded with a triple or quadruple row of columns. The houses of the town rose above it, and the surrounding mountains high above them, so that he felt as in the arena of a magnificent theatre. He had an opportunity of seeing the Kaaba en- circled by more than a thousand pilgrims, Arabs from every province, Moors, Persians, Afghans, and natives of all the GG 4 456 THE LOST SOLAR SYSTEM DISCOVERED. countries of the East In their enthusiastic zeal to kiss the black stone, they rushed pell-mell in confused crowds, so as to cause an apprehension that some of them must have been suffocated. This religious tumult, with the multitude and various aspect of the groups, presented the most extra- ordinary spectacle he ever beheld. TABLES OF SQUARES, CUBES, AND POWERS, TABLE OF SQUARES, From 1 to 1000. Root or Numb. Square. Boot or Numb. Square. Boot or Numb. Square. Boot or Numb. Square. I I 37 1369 73 5329 109 11881 2 4 38 1 444 74 547 6 no I2IOO 3 9 39 1521 75 5625 III 12321 4 16 4 1600 76 5776 112 12544 5 2 5 4i 1681 77 5929 I! 3 12769 6 36 42 1764 78 6084 114 12996 7 49 43 1849 79 6241 "5 13225 8 64 44 1936 80 6400 116 I345 6 9 81 45 2025 8l 6561 117 13689 10 IOO 46 2116 82 6724 118 *39 2 4 ii 121 47 2209 83 6889 119 14161 12 144 48 2304 84 7056 120 14400 13 169 49 2401 85 7225 121 14641 H 196 5o 2500 86 739 6 122 14884 15 22| 5i 2601 87 7569 I2 3 15129 16 2 5 6 52 2704 88 7744 I2 4 15376 J 7 289 53 2809 89 7921 I2 5 15625 18 3H 54 2916 9 8100 126 15876 19 3 6l 55 3025 9 1 8281 127 16129 20 400 56 3136 9 2 8464 128 16384 21 441 57 3 2 49 93 8649 I2 9 16641 22 484 58 s 33 6 4 94 8836 I 3 16900 23 52 9 59 348i 95 9025 I3 1 17161 24 576 60 3600 96 9216 132 17424 25 625 61 3721 97 9409 133 17689 26 676 62 3844 98 9604 '34 17956 27 729 63 39 6 9 99 9801 '35 18225 28 784 64 4096 IOO IOOOO 136 18496 29 841 65 4225 IOI IO20I 137 18769 3 900 66 435 6 102 10404 138 19044 3i 961 67 4489 I0 3 10609 39 19321 32 IO24 68 4624 104 I08l6 140 19600 33 1089 69 4761 105 II025 141 19881 34 1156 70 4900 106 11236 142 20164 35 1225 7i 5041 107 II449 H3 20449 36 1296 72 5184 108 ! 11664 144 20736 460 TABLE OF SQUARES. Boot or Numb. Square. Koot or Numb. Square. Eoot or Numb Square. Koot or Numb Square. *45 21025 I 9 I 36481 237 56169 283 80089 146 21316 I 9 2 36864 238 56644 284 80656 147 21609 193 37249 239 57121 285 81225 148 21904 194 37636 240 57600 286 81796 149 222OI '95 38025 241 58081 287 82369 150 22500 196 38416 242 58564 288 82944 151 22801 197 38809 243 59049 289 83521 152 23104 198 39204 244 59536 290 84100 i53 23409 199 39601 245 60025 291 84681 i54 23716 200 40000 246 60516 292 85264 155 24025 201 40401 247 61009 2 93 85849 156 2 4336 202 40804 248 61504 294 86436 *57 24649 20 3 41209 249 62001 295 87025 158 24964 204 41616 250 62500 296 87616 '59 25281 20 5 42025 251 63001 297 88209 1 60 25600 2O6 42436 252 63504 298 88804 161 25921 207 42849 253 64009 299 89401 162 26244 208 43264 254 64516 300 90000 163 26569 209 43681 255 65025 301 90601 164 26896 210 44100 256 65536 302 91204 165 27225 211 44521 257 66049 303 91809 1 66 27556 212 44944 2 5 8 66564 304 92416 167 27889 213 45369 259 67081 305 93025 168 28224 214 45796 260 67600 306 93636 169 28561 2I| 46225 261 68121 307 94249 170 28900 216 46656 262 68644 308 94864 171 29241 217 47089 263 69169 309 95481 172 29584 218 47524 264 69696 310 96100 1 73 29929 219 47961 265 70225 3ii 96721 *74 30276 22O 48400 266 70756 312 97344 *75 30625 221 48841 267 71289 313 97969 176 30976 222 49284 268 71824 3H 9 8 59 6 177 31329 22 3 49729 269 72361 3i5 99225 178 31684 224 50176 270 72900 3i6 99856 179 32041 225 50625 271 73441 317 100489 1 80 32400 226 51076 272 739 8 4 318 101 124 181 32761 227 51529 273 74529 3*9 101761 182 33124 228 51984 274 75076 320 102400 '83 33489 229 52441 275 75625 321 103041 184 33856 230 52900 276 76176 322 103684 185 34225 231 5336i 277 76729 323 104329 1 86 34596 232 53824 278 77284 324 104976 187 34969 233 54289 279 77841 325 105625 188 35344 234 54756 280 78400 326 106276 189 35721 235 55225 281 78961 327 106929 190 36100 236 55696 282 79524 328 107584 TABLE OP SQUARES. 461 Root Root Root Root or Square. or Square. or Square. or Square. Numb. Numb. Numb. Numb. 329 108241 375 140625 421 177241 467 218089 330 108900 376 141376 422 178084 468 219024 331 109561 377 142129 423 178929 469 219961 332 110224 378 142884 424 .179776 470 220900 333 110889 379 143641 425 180625 471 221841 334 111556 380 144400 426 181476 472 222784 335 1 12225 38i 145161 427 182329 473 223729 336 112896 382 145924 428 183184 474 224676 337 113569 383 146689 429 184041 475 225625 338 114244 384 147456 430 184900 476 226576 339 114921 385 148225 431 185761 477 227529 34 i 15600 386 148996 432 186624 478 228484 34 1 116281 387 149769 433 187489 479 229441 342 116964 388 150544 434 188356 480 230400 343 117649 389 151321 435 189225 481 231361 344 118336 39 152100 436 190096 482 232324 345 119025 391 152881 437 190969 483 233289 346 119716 392 153664 438 191844 484 234256 347 1 20409 393 154449 439 192721 485 235225 348 121 104 394 155236 44 193600 486 236196 349 I2I80I 395 156025 441 194481 487 237169 35 I225OO 39 6 156816 442 "PSS^ 488 238144 35 1 I2320I 397 157609 443 196249 489 239121 352 123904 398 158404 444 197136 490 240100 353 124609 399 159201 445 198025 49 * 241081 354 125316 400 160000 446 198916 492 242064 355 126025 401 160801 447 199809 493 243049 356 126736 402 161604 448 200704 494 244036 357 127449 403 162409 449 201601 495 245025 358 128164 404 163216 450 202500 496 246016 359 I2888I 405 164025 45 * 203401 497 247009 360 129600 406 164836 452 204304 498 248004 361 I3032I 407 165649 453 205209 499 249001 362 131044 408 166464 454 2061 1 6 500 250000 363 131769 409 167281 455 207025 501 251001 364 132496 410 168100 456 207936 502 252004 365 133225 411 168921 457 208849 53 253009 366 133956 412 169744 458 209764 504 254016 367 134689 4i3 170569 459 210681 55 255025 368 J 354 2 4 414 171396 460 21 I6OO 506 256036 3 6 9 136161 4'5 172225 461 2I252I 57 257049 370 136900 416 173056 462 213444 508 258064 37 1 137641 4*7 173889 4 6 3 214369 509 259081 372 138384 418 174724 464 215296 510 260100 373 139129 419 175561 465 216225 5 11 261121 374 139876 420 176400 466 217156 512 262144 462 TABLE OF SQUARES. Root or Square. Koot or Square. Root or Square. Root or Square. Numb. Numb. Numb Numb. 5 J 3 263169 559 312481 605 366025 6?i 423801 5M- 264196 560 313600 606 367236 652 425104 5!5 265225 56i 314721 607 368449 653 426409 516 266256 562 3^844 608 369664 654 427716 5 X 7 267289 563 316969 609 370881 655 429025 518 268324 564 318096 610 372100 656 430336 S 1 9 269361 565 319225 611 373321 657 431649 520 270400 566 320356 612 374544 658 432964 521 271441 567 321489 613 375769 659 434281 522 272484 568 322624 614 376996 660 435600 5 2 3 273529 569 323761 615 378225 661 436921 524 274576 57o 324900 616 379456 662 438244 5 2 5 275625 57i 326041 617 380689 663 439569 526 276676 572 327184 618 381924 664 440896 5 2 7 277729 573 328329 619 383161 665 442225 528 278784 574 329476 620 384400 666 443556 529 279841 575 330625 621 385641 667 444889 53 280900 576 331776 622 386884 668 446224 53i 281961 577 332929 623 388129 669 447561 S3 2 283024 578 334084 624 389376 670 448900 533 284089 579 335 2 4I 625 390625 671 450241 534 285156 580 336400 626 391876 672 451584 535 286225 581 3375 61 627 393 I2 9 673 452929 53^ 287296 582 338724 628 394384 674 454276 537 288369 583 339889 629 395 6 4i 675 455625 538 289444 584 341056 630 396900 676 456976 539 290521 55 342225 631 398161 677 458329 54 291600 586 34339 6 632 3994 2 4 678 459684 54 1 292681 587 3445 6 9 633 400689 679 461041 542 293764 588 345744 634 401956 680 462400 543 294849 589 346921 635 403225 681 463761 544 295936 59 348100 636 404496 682 465124 545 297025 59 1 349281 637 405769 683 466489 546 298116 592 350464 638 407044 684 467856 547 299209 593 351649 639 408321 685 469225 548 300304 594 352836 640 409600 686 470596 549 301401 595 354025 641 410881 687 471969 55 302500 596 355216 642 412164 688 473344 55 1 303601 597 356409 643 4^3449 689 474721 55 2 304704 598 357604 644 4H736 690 476100 553 305809 599 358801 645 416025 691 477481 554 306916 600 360000 646 417316 692 478864 555 308025 60 1 361201 647 418609 693 480249 550 309136 602 362404 648 419904 6 94 481636 557 310249 603 363609 649 421201 695 483025 55* 311364 604 364816 650 422500 696 484416 TABLE OF SQUARES. 463 Koot or Numb. Square. Koot or. Numb. Square. Koot or Numb. Square. Koot or Numb. Square. 697 485809 743 552049 789 622521 835 697225 698 487204 744 553536 7 9 624100 836 698896 699 488601 745 555025 79 i 625681 837 700569 700 490000 746 556516 ?9 2 627264 838 702244 701 491401 747 558009 793 628849 839 703921 702 492804 748 55954 794 630436 840 705600 73 494209 749 561001 795 632025 841 707281 704 495616 75 562500 796 633616 842 708964 705 497025 75 1 564001 797 635209 843 710649 706 498436 75 2 565504 798 636804 844 712336 707 499849 753 56 7 oo9 7 99 638401 845 71402; 708 501264 754 568516 goo 640000 846 715716 709 502681 755 5 7 oo25 goi 641601 847 717409 710 504100 75 6 57 J 53 6 g02 643204 848 719104 711 505521 757 57349 83 644809 849 720801 712 506944 75 8 574564 804 646416 850 722500 7i3 508369 759 5 7 6o8i 805 648025 851 724201 7'4 509796 760 5 77 66o go6 649636 852 725904 7i5 511225 761 579 121 8Q 7 651249 853 727609 716 512656 762 580644 808 652864 854 729316 717 514089 7 63 582169 809 654481 855 731025 718 5 J 5524 7 64 583696 gio 656100 856 732736 719 516961 7 65 585225 8H 657721 857 734449 720 5 i 8400 766 586756 812 659344 8 5 8 736164 721 519841 7 67 588289 8^3 660969 859 737881 722 521284 768 589824 8H 662596 860 739600 723 522729 769 591361 8i5 664225 861 741321 7H 524176 77 592900 gi6 665856 862 74344 725 525625 77 i 594441 8i7 667489 863 744769 726 527076 77 2 595984 8i8 669124 864 746496 727 528529 773 5975 2 9 819 6 7 o 7 6i 865 748225 728 529984 774 5997 6 820 6 7 24oo 866 749956 729 53 H4 1 775 600625 821 674041 86 7 751689 73 532900 77 6 602 i 7 6 822 675684 868 753424 73i 53436i 777 6o3 7 29 823 6 7 7329 869 755161 732 535 82 4 77 8 605284 824 6 7 89 7 6 8 7 o 756900 733 537289 779 606841 825 680625 8 7 i 758641 734 53875 6 7 8o 608400 826 682276 8 7 2 760384 735 540225 7 8i 609961 82 7 683929 873 762129 736 541696 7 82 61 1524 828 685584 874 763876 737 543169 7 83 613089 829 687241 875 765625 738 544644 7 84 614656 830 688900 876 767376 739 546121 785 616225 831 690561 877 769129 74 547600 7 86 6i 779 6 832 692224 878 770884 74i 549081 7 87 619369 833 693889 879 772641 742 550564 7 88 620944 834 695556 880 774400 464 TABLE OF SQUARES. PvOOt or Numb. Square. Boot or Numb. Square. Root or Numb. Square. Root or Numb. Square. 881 776161 9 II 829921 941 885481 971 942841 882 777924 912 831744 942 887364 972 944784 883 779689 9'3 8335 6 9 943 889249 973 946729 884 781456 914 83539 6 944 891136 974 948676 885 783225 9 T 5 837225 945 893025 975 950625 886 784996 916 839056 946 894916 976 952576 887 786769 917 840889 947 896809 977 954529 888 788544 918 842724 948 898704 978 956484 889 790321 919 844561 949 900601 979 958441 890 792100 920 846400 95 902500 980 960400 891 793881 921 848241 95 1 90440 1 981 962361 892 795664 922 850084 952 906304 982 964324 893 797449 923 851929 953 908209 983 966289 894 799236 924 853776 954 910116 984 968256 895 801025 925 855625 955 912025 985 970225 896 802816 926 857476 956 913936 986 972196 897 804609 927 859329 957 915849 987 974169 898 806404 928 861184 958 917764 988 976144 899 808201 929 863041 959 919681 989 978121 900 810000 930 864900 960 921600 99 980100 901 811801 93i 866761 961 923521 991 982081 902 813604 93 2 868624 962 925444 992 984064 93 815409 933 870489 9 6 3 927369 993 986049 904 817216 934 872356 964 929296 994 988036 905 819025 935 874225 965 931225 995 990025 906 820836 93 6 876096 966 933156 996 992016 907 822649 937 877969 967 935089 997 994009 908 824464 938 879844 968 937024 998 996004 909 826281 939 881721 969 938961 999 998001 910 828100 94 883600 97 940900 1000 IOOOOOO TABLE OF CUBES, From 1 to 1000. Root or Number. Cube. Hoot or Number. Cube. Root or Number. Cube. , ! 37 5 6 53 73 389017 2 8 38 54872 74 405224 3 27 39 593*9 75 421875 4 64 4 64000 76 438976 5 125 4 1 68921 77 45 6 533 6 216 42 74088 78 47455 2 7 343 43 79507 79 49339 8 512 44 85184 80 512000 9 729 45 91125 81 53H4I 10 1000 46 97336 82 55n68 1 1 1331 47 103823 83 571787 12 1728 48 i 10592 84 592704 '3 2197 49 117649 85 614125 H 2744 5 125000 86 636056 15 3375 5 1 132651 87 658503 16 4096 5 2 140608 88 681472 1 7 49 J 3 53 148877 89 704969 18 5832 54 157464 90 729000 J 9 6859 55 166375 9' 753571 20 8000 56 175616 92 778688 21 9261 57 185193 93 804357 22 10648 58 195112 94 830584 2 3 12167 59 205379 95 857375 24 13824 60 216000 96 884736 2 5 15625 61 226981 97 912673 26 17576 62 238328 98 941192 27 19683 63 250047 99 970299 28 21952 64 262144 IOO IOOOOOO 29 24389 65 274625 IOI 1030301 30 27000 66 287496 IO2 1062208 31 29791 67 300763 10 3 1092727 32 32768 68 3H432 104 1124864 33 35937 69 328509 105 1157625 34 39304 70 343000 1 06 1 191016 35 42875 7i 3579 11 107 1225043 36 46656 72 373 2 48 108 1259712 VOL. I. H H 466 TABLE OF CUBES. Root or Number. Cube. Root or Number. Cube. Root or Number. Cube. 109 1295029 J 55 3723875 20 1 8120601 110 1331000 156 3796416 202 8242408 Ml 1367631 J 57 3869893 203 8365427 112 1404928 158 3 9443 l 2 204 8489664 "3 1442897 J 59 4019679 205 8615125 114 1481544 1 60 4096000 206 8741816 "5 1520875 161 4173281 207 8869743 116 1560896 162 4251528 208 8998912 117 1601613 163 4330747 209 9123329 118 1643032 164 4410944 2IO 9261000 119 1685159 165 4492125 21 I 939393 1 120 1728000 166 4574296 212 9528128 121 1771561 167 4 6 57463 2I 3 9663597 122 1815848 168 4741632 214 9800344 I2 3 1860867 169 4826809 215 9938375 I2 4 1906624 170 4913000 216 10077696 I2 5 1953125 171 5000211 2I 7 10218313 126 2000376 172 5088448 218 10360232 127 2048383 173 5177717 219 10503459 128 2097152 174 5268024 22O 10648000 I2 9 2146689 175 5359375 221 10793861 130 2197000 176 5451776 222 10941048 I 3 I 2248091 177 5545233 22 3 11089567 132 2299968 178 5639752 22 4 11239424 133 2352637 179 5735339 22 5 11390625 '34 2406104 1 80 5832000 226 11543176 135 2460375 181 5929741 22 7 11697083 136 2515456 182 6028568 228 11852352 137 2571353 183 6128487 22 9 12008989 138 2628072 184 6229504 230 12167000 139 2685619 185 6331625 2 3 I 12326391 140 2744000 186 6434856 232 12487168 141 2803221 187 6539203 233 12649337 142 2863288 188 6644672 2 34 12812904 H3 2924207 189 6751269 235 12977875 144 2985984 190 6859000 2 3 6 13144256 H5 3048625 191 6967871 237 13312053 146 3112136 192 7077888 238 13481272 H7 3176523 193 7189057 239 13651919 148 3241792 194 73 i384 240 13824000 149 337949 '9S 7414875 241 13997521 150 3375000 196 7529536 242 14172488 I 5 I 3442951 '97 7645373 243 14348907 152 3511808 198 7762392 244 14526784 153 35 8l 577 199 7880599 245 14706125 154 3652264 200 8000000 246 14886936 TABLE OF CUBES. 467 Root or "Number. Cube. Root or Number. Cube. Eoot or Number. Cube. 247 15069223 2 93 25*53757 339 38958219 248 15252992 294 25412184 340 39304000 249 15438249 295 25672375 34i 39651821 250 15625000 296 25934336 342 40001688 2 5 I 15813251 297 26198073 343 40353607 252 16003008 298 26463592 344 40707584 253 16194277 299 26730899 345 41063625 254 16387064 300 27000000 346 4H2I736 255 16581375 301 27270901 347 41781923 256 16777216 302 27543608 348 42144192 257 l6 974593 303 27818127 349 42508549 2 5 8 I7I735 12 304 28094464 35o 42875000 259 ! 7373979 35 28372625 35i 43243551 260 17576000 306 28652616 352 43614208 261 i777958i 37 28934443 353 43986977 262 17984728 308 29218112 354 44361864 263 18191447 309 29503629 355 44738875 264 18399744 310 29791000 356 45U8oi6 265 18609625 3" 30080231 357 45499 2 93 266 18821096 312 30371328 358 45882712 267 19034163 3i3 30664297 359 46268279 268 19248832 3H 3 959I44 360 46656000 269 19465109 3i5 3 1 255875 361 47045881 270 19683000 316 3I55449 6 362 47437928 271 19902511 3i7 31855013 363 47832147 272 20123648 3i8 32157432 364 48228544 273 20346417 319 32461759 365 48627125 274 20570824 320 32768000 366 49027896 275 20796875 321 33076161 367 49430863 2 7 6 21024576 322 ! 33386248 368 49836032 277 21253933 323 ! 33698267 369 50243409 2 7 8 21484952 324; 34012224 37 50653000 279 21717639 325 34328125 37 1 51064811 280 21952000 326 j 34645976 372 51478848 281 22188041 327! 349 6 5783 373 51895117 282 22425768 328 ! 35287552 374 52313624 283 22665187 329; 35611289 375 52734375 284 22906304 330 35937000 376 53157376 285 23149125 33i i 36264691 377 53582633 286 23393656 332 36594368 378 54010152 287 23639903 333 36926037 379 54439939 288 23887872 334 ' 37259704 380 54872000 289 24137569 335 37595375 381 553o634i 290 24389000 336 37933056 382 55742968 291 24642171 337 38272753 383 56181887 292 24897088 338 38614472 384 1 56623104 H II 2 468 TABLE OF CUBES. Koot Koot Koot or Cube. or Cube. or Cube. Number. Number. Number. 385 57066625 431 80062991 477 108531333 386 57512456 43 2 80621568 478 109215352 387 57960603 433 81182737 479 109902239 388 58411072 434 81746504 480 110592000 389 58863869 435 82312875 481 111284641 39 59319000 436 82881856 482 1 1 1980168 39 1 59776471 437 83453453 483 112678587 392 60236288 438 84027672 484 113379904 393 60698457 439 84604519 485 1 14084125 394 61 162984 44 85184000 486 114791256 395 61629875 441 85766121 487 115501303 39 6 62099136 442 86350888 488 116214272 397 62570773 443 86938307 489 116930169 398 63044792 444 87528384 490 117649000 399 63521199 445 88121125 491 118370771 400 64000000 446 88716536 492 119095488 401 64481201 447 89314623 493 119823157 402 64964808 448 89915392 494 120553784 403 65450827 449 90518849 495 121287375 404 65939264 45 91125000 496 122023936 405 66430125 45 * 9'73385i 497 122763473 406 66923416 452 92345408 49 8 123505992 407 67419143 453 92959677 499 124251499 408 67911312 454 93576664 500 125000000 409 68417929 455 94196375 501 125751501 410 68921000 456 94818816 502 126506008 411 69426531 457 95443993 503 127263527 412 69934528 458 96071912 504 128024064 413 70444997 459 96702579 505 128787625 414 70957944 460 97336000 506 129554216 4i5 7H73375 461 9797 2I 8i 507 130323843 416 71991296 462 98611128 508 131096512 4i7 72511713 463 99252847 509 131872229 418 73034632 464 99897344 510 132651000 419 73560059 465 100554625 5 11 133432831 420 74088000 466 101 194696 512 134217728 421 74618461 467 101847563 5 J 3 135005697 422 75I5H48 468 102503232 5H 135796744 423 75686967 469 103161709 5*5 136590875 424 76225024 470 103823000 516 137388096 425 76765625 47i 104487111 5i7 138188413 426 77308776 472 105154048 518 138991832 427 77854483 473 105823817 5*9 J 3979 8 359 428 78402752 474 106496424 520 140608000 429 78953589 475 107171875 521 141420761 430 79507000 476 107850176 522 142236648 TABLE OF CUBES. 469 Root or Nnmbor. Cube. Hoot or Number. Cube. Hoot or Number. Cube. 5 2 3 143055667 569 184220009 615 232608375 524 143877824 57 185193000 616 233744896 525 144703125 57 1 18616941 l 617 234885113 526 H553I576 57 2 187149248 618 236029032 5 2 7 H 6 3 6 3 l8 3 573 188132517 619 237176659 528 147197952 574 189119224 620 238328000 529 148035889 575 190109375 621 239483061 530 148877000 576 191102976 622 240641848 53i 149721291 577 192100033 623 241804367 532 150568763 578 193100552 624 242970624 533 I 5 I 4'9437 579 194104539 625 244140625 534 J 5 22 7334 580 195112000 626 2 453H37 6 535 i53i3 375 581 196122941 627 246491883 536 153990656 582 197137368 628 247673152 537 I54854I53 583 198155287 629 248858189 538 155720872 584 199176704 630 250047000 539 156590819 585 200201625 631 251239591 540 157464000 586 201230056 632 252435968 54' 158340421 587 202262003 633 253636137 542 159220088 588 203297472 634 254840104 543 160103007 589 204336469 635 256047875 544 160989184 59 205379000 636 257259456 545 161878625 59 1 206425071 637 258474853 546 162771336 592 207474688 638 259694072 547 163667323 593 208527857 639 260917119 548 164566592 594 209584584 640 262144000 549 165469149 595 210644875 641 263374721 55 166375000 596 21 1708736 642 264609288 55 1 167284151 597 212776173 643 265847707 55 2 168196608 598 213847192 644 267089984 553 169112377 599 214921799 645 268336125 554 170031464 600 216000000 646 269586136 55 I 170953875 60 1 217081801 647 270840023 55 6 171879616 602 218167208 648 272097792 557 172808693 603 219256227 649 273359449 55 8 173741112 604 220348864 650 274625000 559 174676879 605 221445125 6;, 275894451 560 175616000 606 222545016 652 277167808 561 176558481 607 223648543 653 278445077 562 177504328 608 224755712 654 279726264 563 '78453547 609 225866529 655 281011375 564 179406144 610 226981000 656 282300416 S6 J 180362125 611 228099131 657 28 3593393 566 181321496 612 229220928 658 284890312 5 67 182284263 613 2334 6 397 659 286191 179 568 183250432 614 2 3H75544 660 287496000 470 TABLE OF CUBES. Koot or Number. Cube. lioot or Number. Cbe. Root or Number. Cube. 66 1 288804781 707 353393243 753 426957777 662 290117528 708 354894912 754 428661064 663 291434247 709 356400829 755 430368875 664 292754944 710 357911000 756 432081216 665 294079625 711 35942543 1 757 433798093 666 295408296 712 360944128 758 4355 I 95 12 667 296740963 7 1 3 362467097 759 437245479 668 298077632 7*4 3 6 3994344 760 438976000 669 299418309 7 X 5 365525875 761 440711081 670 300763000 7,6 367061696 762 442450728 671 3021 1 171 1 7 1 7 368601813 763 444194947 672 303464448 718 370146232 764 445943744 673 304821217 719 371694959 765 447697125 674 306182024 720 373248000 766 449455096 675 307546875 721 374805361 767 451217663 676 308915776 722 376367048 768 452984832 677 310288733 7 2 3 377933067 769 454756609 678 311665752 724 379534 2 4 77 456533000 679 313046839 725 381078125 771 458314011 680 314432000 ' i } 726 382657176 77 2 460099648 681 315821241 727 384240583 773 461889917 682 317214568 728 385828352 774 463684824 683 31861 1987 729 387420489 775 465484375 684 320013504 73 389017000 776 467288576 685 32Hi9 l2 5 73i 390617891 777 469097433 686 322828856 73 2 392223168 778 470910952 687 324242703 733 393832837 779 472729139 688 325660672 734 395446904 780 474552000 689 327082769 735 397065375 781 47637954 1 690 328509000 736 398688256 782 478211768 691 329939371 737 400315553 783 480048687 692 331373888 738 401947272 784 481890304 693 332812557 739 403583419 785 483736625 694 334 2 5S3 8 4 740 405224000 786 485587656 695 335702375 74 1 406869021 787 487443403 696 337'53536 742 408518488 788 489303872 697 338608873 743 410172407 789 491 169069 698 340068392 744 411830784 79 493039000 699 341532099 745 413493625 791 494913671 700 343000000 746 415160936 79 2 496793088 701 344472101 747 416832723 793 498677257 702 345948008 748 418508992 794 500566184 73 347428927 749 420189749 795 502459875 704 348913664 750 421875000 79 6 50435833 6 705 350402625 751 42356475! 797 506261573 706 351895816 752 425259008 798 508169592 TABLE OF CUBES. 471 Root or Number. Cube. Root or Number. Cube. Root or Number. Cube. 799 510082399 845 603351125 891 707347971 800 5 i 2000000 846 605495736 892 709732288 80 1 513922401 847 607645423 893 712121957 802 515849608 848 609800192 894 714516984 803 517781627 849 6 i 1960049 895 716917375 804 519718464 850 614125000 896 719323136 805 521660125 851 616295051 897 721734273 806 523606616 852 618470208 898 724150792 807 525557943 853 620650477 899 726572699 808 527514112 8 5 4 622835864 900 729000000 809 529475129 855 625026375 901 731432701 810 531441000 856 627222016 902 733870808 811 53341173! 857 629422793 93 736314327 812 535387328 858 631628712 904 738763264 813 537366797 859 633839779 95 741217625 814 539353144 860 636056000 906 743677416 815 54"343375 861 638277381 97 746142643 816 54333849 6 862 640503928 908 748613312 817 5453385*3 863 642735647 909 751089429 818 547343432 864 644972544 910 753571000 819 549353259 865 647214625 911 756058031 820 551368000 866 649461896 912 758550528 821 553387661 867 651714363 913 761048497 822 823 824 825 826 827 828 555412248 55744*767 559476224 561515625 563559976 565609283 567663552 868 869 870 871 872 873 874 653972032 656234909 658503000 660776311 663054848 665338617 667627624 914 9'5 916 917 918 919 920 763551944 766060875 768575296 771095213 773620632 776151559 778688000 829 830 831 832 833 834 835 836 837 838 569722789 571787000 573856191 575930368 578009537 580093704 582182875 584277056 586376253 588480472 875 876 877 878 879 880 881 882 883 884 669921875 672221376 674526133 676836152 679151439 681472000 68379784 1 686128968 688465387 690807104 921 922 923 924 925 926 927 928 929 93 781229961 783777448 786330467 788889024 79H53I25 794022776 796597983 799178752 801765089 804357000 839 840 841 842 590589719 592704000 594823321 596947688 885 886 887 888 693154125 695506456 697864103 700227072 93 i 932 933 934 806954491 '809557568 812166237 814780504 8 43 844 599077107 60121 1584 889 890 702595369 704969000 935 936 817400375 820025856 II H 4 472 TABLE OF CUBES. Root or Number. Cube. Root or Number. Cube. Root or Number. Cube. 937 822656953 959 881974079 980 941 192000 938 825293672 960 884736000 981 944076141 939 827936019 961 887503681 982 946966168 940 830584000 962 890277128 983 949862087 941 833237621 9 6 3 893056347 984 952763904 942 835896888 964 895841344 985 955671625 943 838561807 965 898632125 986 958585256 944 841232384 966 901428696 987 961504803 945 843908625 967 904231063 988 964430272 946 846590536 968 907039232 989 967361669 947 849278123 969 909853209 99 970299000 948 851971392 970 912673000 991 973242271 949 854 6 7349 97 1 915498611 99 2 976191488 950 857375000 972 918330048 993 979146657 95i 860085351 973 921167317 994 982107784 952 862801408 974 924010424 995 985074875 953 865523177 975 926859375 996 988047936 954 868250664 976 929714176 997 991026973 955 870983875 977 93 2 574833 998 99401 1992 956 873722816 978 93544I35 2 999 997002999 957 876467493 979 938313739 IOOO 1000OOOOOO 958 879217912 TABLE OF POAVEKS. 473 ON 00 ON 1 N VO vO 3- N u-i 1 ON 00 vo ON OO oo OO vO 00 ON J ON OO to to OO VO to ON N 00 ON to 00 CO vO OO vO N ON vO 00 vO ON to ON 00 cT OO vO t-o VO ON OO VO to 1 N LO 1 vO VO oo N oo to ON to ON ON CO oo vO to vo ON oo VO GO OO oo to oo vo "1" ON to TJ- to oo oo 00 * N to OO to to ON co O O 00 vo vO 00 OO vo o vo O ON CO N vo to ON oo 00 to 3- ON OO OO vo ON oo N to CO vo" to ON ON vo VO vO to vO vO ON vO vo ON ON vO vO ON VO ON o vo vo vo vO ON N vO to vO to to OO VO vo o to vO VO to 00 r-- 00 ON oo o M M N vO u/~ to N 00 r-. 1 ON to N to ON N vO VO ON 00 N OO CO vo O N to O N VO LO to C LO 00 t-o LO to vO VO VO vO ON rj- OO vC vO vo vO OO O to ON vO vO vO CO CO o VO vO to oo vO N OO to to ON N 00 J ON oo N VO vO oo vo ON ON ON I to LO to N to ON ON vO ON OO ON 00 to rH OO 1 1 vo N I 1 ^- OO N vO N fH N 00 o" ! ON i ON 00 00 to vO 00 VO to g 1 1 PH u