GRAMMAR 
 
 INFINITE FORMS. 
 
 
 
EDINBURGH, 
 PRINTED BY OLIVER & BO I'D. 
 TWEEDDALE-COURT. 
 
GRAMMAR 
 
 INFINITE FORMS; 
 
 THE MATHEMATICAL ELEMENTS OF ANCIENT 
 PHILOSOPHY AND MYTHOLOGY. 
 
 By WILLIAM HOWISON. 
 
 ramos annosaque brachia pandit 
 
 Ulmus opaca, ingens : quam sedem somnia vulgo 
 tenere ferunt, foliisque sub omnibus hserent. 
 
 EDINBURGH : 
 
 PUBLISHED BY 
 
 OLIVER & BOYD, TWEEDDALE-COURT ; 
 
 SOLD ALSO BY 
 G. & W. B. WHITTAKER, LONDON. 
 
 1823. 
 
 e*-~ -? 
 
I 
 
 
 I 
 
PREFACE. 
 
 The purpose of this treatise is, in the first place, 
 to recall from oblivion the knowledge of those 
 elementary powers of mathematical form which 
 were the basis of all the fables belonging to that 
 ancient system of philosophy and poetry which is 
 now called mythology. These are found as twelve 
 successive steps in one continued deduction, which 
 takes in all the simple or uncompounded powers 
 of mathematical form ; and, although the know- 
 ledge of their deductive order has been lost or for- 
 gotten during the intervening ages, it is shewn, by 
 many existing traces, to have been well understood 
 at one time by the Greeks and other cotempo- 
 rary nations. It was the chief source of the truth 
 of their conceptions and the correctness of their 
 
VI 
 
 taste, which always approached as near as possible 
 to abstract rule. 
 
 The present treatise is called a Grammar, be- 
 cause it shews the order and syntax of these powers, 
 and defines the qualities peculiar to each. But 
 the second part relates to the modes of composition 
 which are derived from the union of powers, and 
 which modes of composition are also found as steps 
 in a regular series. 
 
 The author of the following pages, in pursuing 
 these abstractions, may seem to be exerting an 
 useless ingenuity, and spending labour in vain, for * 
 the purpose of resuscitating antiquated and scarce- 
 ly amusing fables. He is of opinion, however, 
 that none of these were originally contrived at 
 random, or for commemorating particular events, 
 but were originally intended for a totally different 
 purpose, and are valuable memorials of the study 
 of abstract truths, which cannot even now be ex- 
 pressed by a more convenient set of symbols. 
 But, in this treatise, the compositions, and the 
 
Vll 
 
 modes of being which result from them, are fre- 
 quently illustrated by references to modern na- 
 tions, characters, events, or works of genius, in 
 which the natures belonging to them seem to have 
 been exemplified. And, although the coincidences 
 pointed out may perhaps sometimes awaken a 
 sense of the ludicrous, the author of this treatise 
 professes to have no intention of insulting any 
 department by the illustrations. It cannot but 
 serve a good end to exhibit the severe forms of 
 abstract truth without any accommodation to the 
 feelings of mankind ; who may be accused of what, 
 in the present condition of society, must be con- 
 sidered as an excessive respect for human nature. 
 

PREFACE. 
 
 The purpose of this Treatise is to help to deter- 
 mine an important question which occurs in phi- 
 losophy, and which is, Whether the number of 
 hinds or modes of being, exemplified in nature, be 
 limited or not ? It is evident that each kind may 
 have subordinate classes, but these cannot exist 
 apart from it. This Treatise is intended to shew 
 that every kind, which is really apart from others, 
 must be founded on some simple power, existing 
 in abstract idea, that is to say, having a mathema- 
 tical existence. Now, the number of simple ma- 
 thematical powers (which are capable of flux and 
 progression) will be found to be limited ; and, 
 therefore, the number of kinds, or modes or be- 
 ing, which are founded on them, must also be li- 
 mited. 
 
VI 
 
 The author of the following pages has been led 
 to suppose that the whole system of Grecian my- 
 thology had a mathematical origin, and that the 
 beautiful differences of character, in the chief di- 
 vinities, were the same as those of the simple ma- 
 thematical powers which are capable of flux and 
 progression. Thus the attributes of each divinity 
 became entirely distinct from those of the rest ; 
 and each inhabitant of Olympus assumed a cha- 
 racter so clearly and strongly discriminated as 
 never afterwards to be forgotten, either among 
 barbarians, or more civilized generations. 
 
 Part First of the following Treatise relates to the 
 simple mathematical powers which are capable of 
 flux ; and these are found deductively as steps in 
 a series, wherein each power presupposes that 
 which is immediately anterior. In this mathe- 
 matical theorem, the first steps are of the most 
 obvious nature ; but the rest are such as not to 
 occur so readily to the mind. The deductive 
 order of these powers, however, must undoubtedly 
 
 J 
 
Vll 
 
 have been studied and perceived by the ancients, 
 although it has been overlooked or allowed to fall 
 into oblivion during the intervening ages. 
 
 Part Second of the following Treatise is an in- 
 quiry into the mathematical flow or progression of 
 compositions which, in number and difference of 
 kind, correspond with the simple powers. In the 
 latter, there is merely the flux of quantity and 
 proportion which takes place in the same form 
 prolonged and continued. But composition pro- 
 duces a new beauty, which is the flux of kind, 
 or the progressive exemplification of the same ge- 
 neric power as it changes through an endless series 
 of particular instances or acts. This in fact is the 
 same as the nature of animated beings, and corre- 
 sponds with mental life or activity. In this de- 
 partment, ancient fable and imagination chiefly 
 delighted to luxuriate; and mythology, besides 
 particularizing the attributes of the chief divinities, 
 as elementary powers, farther abounded with the 
 characters of heroes, demigods, and local powers, 
 
Vlll 
 
 all intended to express the kinds of intellectual life 
 or mental sensation. These were, at the same 
 time, really exemplified in the nations and races 
 of mankind, to which, as well as to works of 
 genius, the author of the following Treatise has 
 frequently referred for illustration. 
 
CONTENTS. 
 
 PAUT I. 
 
 ON THE MATHEMATICAL ELEMENTS. 
 
 PAGE 
 
 Chap. I. On Contingency or External Relation . , 1 
 II. On Fixed Position, Form, and Measure- 
 ment . . . . . 5 
 
 III. On Continuity and Abstract Rule . . 9 
 
 IV. On Contrast, or Difference of Rule . 13 
 V. On Definition, or the Limitation of Ab- 
 stract Rules . . . . 16 
 
 VI. On Indefinite Volume or Diffusion . 20 
 VII. On Terminated Volume . . 24 
 
 VIII. On the Prospective Relation of Solid 
 
 Figure to Motion ... 28 
 
 IX. On acquired Motion ... 31 
 X. On Growing, or the Assumption of Ex- 
 traneous Forces ... 36 
 XI. On Composition ... .40 
 XII. On Comprehension and Imagination . 46 
 
PART II. 
 
 ON THE MODES OP COMPOSITION. 
 
 PAGE 
 
 Chap. I. On the Powers of Gradation, or the Com- 
 parison of different Finite Series. 
 Orion or Bacchus ... 49 
 II. On the Power of Searching out the differ- 
 ent Forms of the Saturnian Hyperbola. 
 Hercules and the Muses . . 60 
 
 III. On the Application of Straight Lines to 
 the Series of Hyperbolic Branches. Pan 
 or Msculapius .... 67 
 IV. On the Retrogressions of the Hyperbola. 
 Section I. On the Infinitude of a single 
 
 Curve. Pelops or Proserpina 72 
 II. On a Retrogressive Series of 
 Branches, or the Power of 
 Search into Antiquities. 
 Pluto or Adonis . 81 
 
 III. On the Evolution of Diminish- 
 ing Hyperbolic Branches. 
 Perseus ... 88 
 
 IV. On the Retrogression of the 
 Hyperbola through a Series 
 of Inconsecutive Forms. 
 Vertumnus or the Fates 94 
 
 
XI 
 
 PAGE 
 
 Chap. V. On the Interminable Forms of Trans- 
 verse Progression. 
 Section I. On the Powers of Continu- 
 ity and Syntax. Erich- 
 thonius or Hyperion . 100 
 II. On the Deduction of 
 Composite Hyperbolic 
 Branches. Theseus . 107 
 Chap. VI. On the Progression of Double Series 
 of Hyperbolic Branches. 
 Section I. On the Powers of Fluctua- 
 tion. Pollux or Eridanus 117 
 II. On the continued Relation 
 of two Simple Curves. 
 Castor or Bellerophon 123 
 Chap. VII. On the Powers of Collocation, or the 
 Distribution of Finite Parts of the 
 Hyperbola. Geryon or Silvanus . 131 
 
PART I. 
 
 ON THE MATHEMATICAL ELEMENTS. 
 
 CHAP. I. 
 
 ON CONTINGENCY, OR EXTERNAL RELATION. 
 
 From the entire and absolute sameness of a single 
 position no other mathematical idea is generated ; 
 and, therefore, the difference of positions must be 
 ranked as the first creative power, since it gener- 
 ates something else, namely, relation passing to and 
 fro between the positions. This cannot be fixed 
 or made perceptible to the senses ; but its exist- 
 ence is recognised by intellect. The contrariety 
 of points is the origin of adventitiousness ; because 
 external relation is not implied in the nature of 
 
either point taken separately. To the god Uranus 
 or Caslus, among the eldest deities, was attributed 
 the power of connecting all things ; and his char- 
 acter signified merely relation. To the same 
 power must also be ascribed the simplest and most 
 original idea of motion, as the transition between 
 different points ; and a moving body partakes of 
 the character of Caelus, by producing a connexion 
 between those places which it leaves and those 
 into which it comes. From motion comes fortune, 
 which belongs to existences capable of changing 
 their places, and creating novelty of circumstances. 
 Among the twelve chief heathen divinities, this 
 power belonged to Juno ; and, among the tribes 
 of Israel, the same character seems to have belong- 
 ed to the tribe descended from Reuben, the eldest 
 son of Jacob. The power of Juno, originating 
 from the difference of units, becomes also the same 
 as accession, and may be extended to the increase 
 of numbers by continued accession from without ; 
 because her power, being that of adventitiousness, 
 can never be supposed to stand still, or be limited 
 to any particular number. The arithmetical series 
 follows out the differences of units or equal num- 
 
bers, and assumes a new proportion at each step. 
 To Juno must also be ascribed the power of fluc- 
 tuation, as arising from the alteration of distances ; 
 because the relation of points or units continues 
 to subsist, however much their distances may be 
 changed, or into whatever order they be thrown. 
 But any form, having a certain order and arrange- 
 ment of parts, cannot be thus shaken or dis- 
 composed without losing its identity. The kind of 
 feeling, therefore, which must be ascribed to Juno 
 is that of temporary states and fluctuations. Thus, 
 the dancing of Salome procured from Herod a 
 boon which occasioned the decapitation of St John 
 the Baptist. To Juno, besides the fluctuations of 
 the feelings, may also be ascribed pilgrimage in re- 
 gard to the meeting of external novelty. Actuated 
 by a strong feeling of the changeableness of the 
 world, and its incessant agitation, the ancient 
 astrologers betook themselves to watch the revolu- 
 tions of the planets, and observe their conjunctions, 
 of which they endeavoured to find the connexion 
 with the fortunes of mankind in the world beneath. 
 To the same class with Caelus and Juno may be 
 referred the muse Urania. Juno was also my- 
 
thologically identified with the air, as the element 
 most capable of mutability and agitation. From 
 this power, however, the deduction of mathemati- 
 cal elements must begin. 
 
CHAP. II. 
 
 ON FIXED POSITION, FORM, AND MEASUREMENT. 
 
 It is evident that more than two points, viewed 
 together, may either be considered as capable of 
 freely changing their order and distances, or they 
 may be fixed to certain places, by having their 
 distances compared, and a certain proportion esta- 
 blished among these distances. From the com- 
 parison of distances, therefore, comes fixed posi- 
 tion. One point may revolve round another and 
 preserve always the same distance; but a third 
 point cannot revolve round one of the two first, 
 without changing the proportion of the distances 
 between it and the other two. The comparison 
 and proportion of distances is also the origin of 
 form ; although the different measurements which 
 meet in one form may be innumerable. Among 
 the twelve chief heathen divinities, form and fixed 
 position was signified in the character of Jupiter ; 
 
6 
 
 who, in reference to fixed position, was supposed 
 to let down from his throne a great golden chain, 
 on which all things depended. In another point 
 of view, Jupiter, being only the offspring of Saturn 
 and Rhea, was of more recent birth than many 
 other divinities known in mythology; but, from 
 his character relating to a centre, he was hypo- 
 thetically called the father of gods and men. 
 Among the twelve tribes of Judea, this power 
 must have belonged to Simeon. In this power 
 also the number of points employed must be sup- 
 posed to be limited; because the characteristic 
 of Jupiter is stability, which rejects the change 
 produced by incessant external accession when 
 continued, as in the power of Juno. And the 
 only kind of flow or progression, which is appro- 
 priate to a system of fixed proportions, is that 
 which takes place in time ; as when a certain form 
 of rhythmus or musical measure, having been gone 
 through, is again begun, and repeated for ever in 
 a circle. Thus, in poetry, each stanza, or some- 
 times each verse, begins the same measure anew ; 
 and, in this kind of progression, no new form is 
 discovered, except in the increase of the whole 
 
series. The simple inhabitants of a pastoral coun- 
 try are averse to think that there can be any ad- 
 vantage not implied in the mode of existence which 
 they already enjoy; and thus they are induced 
 to harden themselves in a sort of stupidity, and to 
 reject all new affections. The power of Jupiter 
 must ever be opposed to accident and fluctuation, 
 and to every sort of feeling not resulting from 
 what is already possessed ; and, on the contrary, 
 it rejoices in seeing the same cycle of events re- 
 peated. To the same class with Jupiter must also 
 be referred the three original muses, who were the 
 daughters of Cselus and Terra, and, according to 
 ancient authors, were known before the nine 
 daughters of Jupiter and Mnemosyne. The three 
 earlier muses may have signified the three generic 
 powers either of musical rhythmus or verse ; name- 
 ly, first, the proportion of quantity, such as that 
 of a long to a short ; secondly, their order or dis- 
 tribution ; and, thirdly, their number, from whence 
 the extent of the whole form or measure to which 
 they belong. To Jupiter must also be ascribed 
 the power of acceleration and of producing the 
 same form on a different scale ; for it is evident 
 
8 
 
 that a system of points, having established dis- 
 tances, is capable of revolution round any one of 
 the points assumed as a centre. Those points, 
 which have unequal distances from it, will revolve 
 in concentric circles, and so trace the same form 
 on different scales ; and the power of acceleration 
 results from the more distant points being forced 
 to revolve with the greater speed, to preserve their 
 position or distance in regard to those nearer the 
 centre. To the power of Jupiter, therefore, may 
 be referred the low T er kinds of mental vehemence 
 and impetuosity. Among tribes of barbarians, 
 the feeling of rotatory motion is used for every 
 purpose, whether of religion or of war ; and, being 
 the lowest sort of mental enjoyment, it is given as 
 a free inheritance to all created beings. To the 
 same class with Jupiter may also be referred the 
 goddess Themis, that is justice, which is 'equality, 
 the simplest kind of proportion found by the com- 
 parison of quantities. Revenge is counted the 
 chief virtue among natures which are fixed, and 
 incapable of progressive change or expansion. 
 
CHAP. III. 
 
 ON CONTINUITY AND ABSTRACT RULE. 
 
 The third mathematical element is continuity, by 
 which is meant not relation passing to and fro be- 
 tween different points, but a line, whose parts 
 are fixed and remain in their places, and whose 
 continuity may be prolonged to any extent. A 
 line, however, can never be composed of points, 
 for these either must be separated by some dis- 
 tance, or they must become merged in one, with- 
 out extension. A line is that which has fixed in- 
 termediate parts extending through all the dis- 
 tance between any two points contained in it ; but, 
 at the same time, a line has in itself a continued 
 succession of points which remain in their places ; 
 and hence it becomes like a list or catalogue of 
 points bound together by perfect continuity, every 
 intermediate part of the line being as stable as the 
 rest. Nevertheless, the nature of continuity is a 
 
 a2 
 
10 
 
 mystery which can never be understood. But it 
 is evident that the extension or production of a 
 line must begin from a fixed point, which is first 
 found by the power of Jupiter. The first form of 
 continuity must be a straight line ; of which the 
 definition is, that it has equal relations on both sides. 
 Supposing two positions (A 
 and B) to be assumed equally ^^ B 
 
 distant from the fixed point \\ NsjC ^'/* 
 (C) from whence the line be- \ \ 
 
 gins ; and, if the line, as it is \ ] J) / 
 
 drawn out, preserves always 
 an equality in its two distances 
 (AD and BD) from these two lateral points 
 (A and B) then it will be a straight line, and may 
 be prolonged as such to any extent. But this 
 also gives the first and simplest idea of an abstract 
 rule prolonged and carried into successive in- 
 stances; because the two lateral distances, though 
 continually increasing and changing as the line ex- 
 tends, retain always the same proportion to each 
 other ; but their proportion to the extent of the 
 line is continually diminishing. Among the 
 twelve chief heathen divinities, continuity, or ab- 
 
11 
 
 stract rule, was represented by Apollo, whose 
 chariot was drawn through the sky by celestial 
 steeds ; for the mystery of continuity relates to 
 the yoking of different points together ; and the 
 point which advances is bound to that which is 
 thrown behind, in the same manner that the horses 
 are bound to the car which they draw. This 
 power may also be considered in another point of 
 view, for each side of a line has relation to any 
 point situated out of the line, and lying on that 
 side. This generates the first idea of relation 
 spreading into breadth, like that of a plane, be- 
 cause the relations of the extraneous point (B) to 
 the whole of the points situated in the line, are 
 blended into one Maya, or ideal extension having 
 breadth. Maya is the word used by the Indians 
 to signify mental creation or illusion. This Maya 
 was probably signified in Latona, whose character- 
 istic, according to Hesiod, was smoothness and 
 placidity. The lateral extension changes its 
 boundaries as the line extends, and exhibits form 
 in a state of transition through successive instances 
 all belonging to the same rule. This elementary 
 power must have belonged, among the tribes of 
 
12 
 
 Israel, to the sons of Levi. Continuity is also 
 the same as faithful tradition, which transmits 
 what has been heard, without departing from the 
 method which has been begun. Thus, the hard 
 and accurate style used by the Egyptian carvers, 
 in tracing the forms of animals, shews a rigid ad- 
 herence to tradition ; and such is the strength and 
 certainty which exists in the joints of the crocodile 
 of the Nile. The name Phoebus, in Greek sig- 
 nifying pure, applies well to the nature of abstract 
 rule, which, in ideal generation, admits of no ex- 
 ceptions ; for either the same principle is carried 
 on throughout, or that rule is abandoned, and 
 some other substituted in its place. 
 
13 
 
 CHAP. IV. 
 
 ON CONTRAST OR DIFFERENCE OF RULE. 
 
 The fourth mathematical element is an angle 
 which causes the line to assume a new direction, 
 by which it comes to have a lateral relation to the 
 previous part of the 
 line (CF) ; and, there- 
 fore, draws forth a 
 maya as to each dif- D 
 ferent point (C,D, E ? ) 
 situated in that part. 
 In each of thesemayas, 
 the lateral distance 
 (EG and DG and CG) increases according to a 
 different rate, and exemplifies a different rule ; 
 and the power of angle, therefore, becomes the 
 same as the contrast of different rules or methods 
 carried on at the same time, the reflected line (FG) 
 being capable of prolongation any extent. Among 
 
 E 
 
 
 a 
 
 ~r>^\ . 
 
14 
 
 the twelve chief heathen divinities, the power of 
 angle was represented by Diana, the goddess of the 
 moon, which sends back reflected light; and, among 
 the tribes of Israel, the same power must have been 
 represented by Judah, the tribe to which belonged 
 superiority and dominion over all the rest. The 
 characteristic of this elementary power is, that, 
 without the aid of any extraneous point not con- 
 tained in its own lines, it is capable of drawing 
 forth an infinite number of mayas, or different 
 forms of flowing extension within itself; because 
 these are always produced between the two parts 
 of the line ; but only the second part of the line is 
 capable of prolongation, the prior part standing 
 still, and affording fixed points, from whence the 
 different mayas or rules are drawn forth to infini- 
 tude. These pursue each other, from whence 
 perhaps the idea of the chase attributed to Diana. 
 But the flowing extension altogether, which is in- 
 cluded in the form of the angle, was perhaps signi- 
 fied in the character of Maia, and the Pleiades, 
 the daughters of Atlas, who, being changed into 
 stars, were made to shed a sweet and beneficial 
 influence. Maia was called bountiful, and her 
 
15 
 
 name signifies also a midwife or nurse, which ap- 
 plies well to the origin and increase of subject ex- 
 istence in the angle, one side of which is extending. 
 It is evident, however, that one of the mayas which 
 are drawn forth must be outermost and must out- 
 strip all the rest. For this reason, the elementary 
 power of Diana Corresponds with the nature of 
 pride, and the wish to be highest, or to achieve 
 the most. Such was the scriptural character of 
 Lucifer, the son of the morning ; and, as will 
 afterwards be shewn, such was likewise the mytho- 
 logical character of Hercules, as belonging to the 
 mode of composition which has most likeness to 
 this elementary power. From possessing in itself 
 the continuance of innumerable rules, which are 
 all different, the elementary power of Diana be- 
 comes the same as power over the different modes 
 of action, to choose among them, or contrast them. 
 
16 
 
 CHAP. V. 
 
 ON DEFINITION, OR THE LIMITATION OF 
 ABSTRACT RULES. 
 
 The fifth elementary power is the continuation of 
 the same line through successive angles, by which 
 the unlimited continuation of the same rules or 
 mayas is prevented. By a second angle there is 
 intercepted an intermediate portion of line which 
 can extend its own mayas no farther. The second 
 angle, to shew the power of alternation which be- 
 longs to it, must be supposed to be formed on the 
 other side of the line ; and the succession of angles 
 so alternating may be continued to infinitude. 
 Among the twelve chief heathen divinities this 
 power belonged to Vulcan, and is the same as the 
 power of definition, because it relates to what is 
 
17 
 
 situated on both sides of a line or boundary. 
 Thus, in a series of alternating _ 
 angles, any portion of line (EF) 
 draws forth a maya of its own, 
 in relation to the point (C) at 
 the beginning of all the line ; 
 and this maya, crossing any in- 
 termediate portion of line (DE) 
 is found on both sides of it, and 
 so becomes like the power of be- 
 ing present on both sides of a 
 boundary, and so perceiving both what is contain- 
 ed within it and also what is situated beyond it, 
 and every successive portion of line draws forth 
 its own peculiar may as or rules, but the whole 
 series of them is comprehended in the general 
 power of Vulcan. Among the twelve tribes of 
 Israel this power must have belonged to Zabulon. 
 By two or more angles the continuation of a 
 straight line is also capable of returning to a 
 former point in its own course, and so including a 
 space or forming a terminated figure, which, in 
 the simplest case, would be a triangle. The 
 
18 
 
 power of Jupiter also marks off a space or figure, 
 but only by relations alternating between distant 
 points. The kind of boundary which belongs to 
 Vulcan consists of line or continuous extension 
 having everywhere a series of fixed points contain- 
 ed in it. The series of alternating angles belong- 
 ing to Vulcan is the form which appears in light- 
 ning ; and he was also the god of fire, and the fa- 
 bricator of thunderbolts. A second angle pre- 
 vents the continuation of the flowing mayas which 
 were generated in the first angle, and which de- 
 pended upon the prolongation of the second part 
 of the line, from whence the relation of Vulcan to 
 halting or stopping, and shewing deficiency, which 
 is the source of the buffoonery ascribed to him. A 
 triangular figure encloses the flowing mayas alto- 
 gether within certain boundaries ; and the same is 
 the case with any other terminated figure, what- 
 ever be its form or the number of its sides. The 
 power of Vulcan is also the same as the end of 
 line or of continuity. Without such limitation, 
 there could be no particular continuity apart from 
 other things; therefore, to Vulcan belongs the 
 
19 
 
 power of separating those qualities which are con- 
 tained in a definition from those qualities which 
 are left out or denied by its terms, 
 
20 
 
 CHAP VI. 
 
 ON INDEFINITE VOLUME OR DIFFUSION. 
 
 The sixth mathematical element is the power of 
 rising through the difference of planes, from 
 whence comes the generation of volume. If the 
 power of Vulcan, or terminated figure, be assum- 
 ed for the basis ; and if, from one of its angles, as 
 a fixed point, a new line be made to rise, leaving 
 the plane below, it will draw forth two new may as 
 or surfaces, beginning from the two sides of the 
 angle which is left beneath. These two surfaces 
 follow the extension of the rising line, and, be- 
 tween them, will be generated the first idea of 
 bulk or volume, having the original terminated 
 figure for its basis, and having one of the rising 
 planes for its boundary, on each side ; but its in- 
 crease altogether accompanies the prolongation of 
 the rising line. Among the twelve chief heathen 
 
21 
 
 divinities this power was represented by Neptune ; 
 and, among the tribes of Israel, the same character 
 must have belonged to Issachar. The birth of 
 volume is like nature, or the first diffusion of the 
 waters of chaos, which were subject material, and, 
 at first, had nothing to prevent them from spread- 
 ing indefinitely. Volume is only a secondary kind 
 of maya, and differs from the first in its degree of 
 affinity to lines, in which affinity it is altogether 
 inferior to the first. The maya of Neptune is not 
 produced under lines, but only under planes, 
 which are themselves a kind of ideal creation; 
 and volume is therefore a subject existence pro- 
 duced under subject existence. According to the 
 philosophy of the Indians, that which constitutes 
 the nature of material substance is altogether an 
 illusion, and has no real existence. In a plane, all 
 the relations are clear and certain, because they re- 
 fer to fixed points, situated in lines, and having an 
 order or series ; but, in the maya of t Neptune 'or 
 volume, the internal relations are of a different na- 
 ture, because no fixed points can be found in the 
 planes between which they are generated. Hence 
 
the character of Proteus, as a sort of illusive and 
 unintelligible nature. Nevertheless, to the utmost 
 height of the line of Neptune, there must be found 
 a succession of fixed points, each of which has its 
 own maya of volume in relation to the plane be- 
 low ; but all these forms of volume are blended 
 into one, which is continually increased as the line 
 ascends. This power is the ideal source of me- 
 lancholy, and obscuration from the intermediate 
 parts of volume, as in the depths of the ocean ; 
 but Neptune was also reckoned the god of intel- 
 lect. A certain character of darkness and native 
 sadness must be ascribed to all the marine powers, 
 such as Oceanus, Tethys, Doris and her nymphs 
 accustomed to twilight under the waves, and re- 
 joicing in the obscurity of clouds and showers. 
 Nevertheless there must always be a highest or 
 outermost part of volume not blended with any of 
 the mayas beneath ; and thus Neptune was con- 
 ceived as generally driving his chariot, in day- 
 light, over the surface of the brine, while the me- 
 lancholy of the ocean was confined to the depths 
 beneath. The power of Neptune is also the origin 
 
23 
 
 of the form given to spires and pinnacles which 
 are placed on the highest parts of buildings, and 
 meant to appear as if rising into the sky, and ca- 
 pable of unlimited ascent. 
 
24 
 
 CHAP. VII. 
 
 ON TERMINATED VOLUME. 
 
 The seventh mathematical element is the power 
 of separating the different forms of volume ; for, 
 if the line of Neptune be made to form an angle, 
 the next point in it will produce a maya of volume, 
 which no longer entirely absorbs those beneath, 
 but, having its apex situated in a new line, leaves 
 the previous form of volume partially uncovered. 
 This is like the appearance of dry land ; and every 
 successive point, in the new line, becomes the apex 
 of another form of volume, partially separated 
 from the rest. Thus, by the second part of the 
 line, there is produced a ridge of pyramidal mayas, 
 having their forms partially separated from each 
 other, and no longer capable of being lost in the 
 general increase of volume. And, if the rising 
 line be made to pass through other angles, it will 
 only produce farther exemplifications of the same 
 
 1 
 
25 
 
 principle. But, if the line be made to descend 
 and return to the original plane of Vulcan, and be 
 fastened to some point in the lines which form its 
 limits, then there will be produced a terminated 
 solid, incapable of farther increase, and having, on 
 all sides, planes which enclose the mayas of volume 
 within. This power of solidity belonged to Vesta, 
 among the twelve chief heathen divinities; and, 
 among the tribes of Israel, the same character 
 must have belonged to Dan. The line of Nep- 
 tune, after forming an angle, which should be on 
 the other side of the line, to shew its power of al- 
 ternation, may be called the line of Vesta ; and in 
 relation to the previous part of the same line, it 
 draws forth a plane of its own, at the sides of which 
 the two other planes belonging to each apex are 
 joined. 
 
 These are left by twos, as relating to a fixed point 
 for their apex ; but the middle plane produced by 
 the line if Vesta continues to extend, till another 
 angle be formed at which that ridge terminates, 
 and another begins. Farther, the sides of these 
 ridges will not be uniform surfaces, but will consist 
 of two different planes diagonally joined. The ge- 
 
26 
 
 neration of this figure is regular and beautiful, but 
 cannot easily be expressed by language ; nor could 
 it well be represented by lines drawn on a plane. 
 
 Another characteristic by which the power of 
 Vesta is distinguished from that of Neptune is, that 
 every point in the line of Vesta has two planes op- 
 posite to it, in relation to each of which it produces 
 a maya of volume ; so this power becomes the same 
 as the coincidence of different mayas of volume in 
 regard to the same point as the apex ; and this is an 
 union which can never happen by the power of Nep- 
 tune. The line of Ops or Vesta is the power of se- 
 parating the forms of volume, and finding out new- 
 examples of terminated solids or individual na- 
 tures, the varieties of which it may explore to in- 
 finitude. 
 
 It is evident that solids, when complete in them- 
 selves, and externally separated from others, are 
 capable of relative motion, and of freely changing 
 their places. The name of Rhea, in Greek ~Pux, 
 signifying " flowing," may refer to the flowing of 
 multitudes which are capable of freely changing 
 their places ; as is exemplified in a shoal of fishes 
 moving through the waters. The external rela- 
 
27 
 
 tion of solids to each other also gives birth to a 
 new kind of maya, alternating between them, and 
 having relation to their figures and sizes. This 
 may be called the maya of Faunus, a rural power 
 belonging to the earth. 
 
2S 
 
 CHAP. VIII. 
 
 ON THE PROSPECTIVE RELATION OF SOLID 
 FIGURE TO MOTION. 
 
 The eighth mathematical element is the relation 
 of the form of a solid to the various courses 
 through which it might move ; and this is the 
 source of the infinite extension of parallels ; be- 
 cause whenever a solid begins to move, it must 
 trace parallel lines, which are the sides of its course, 
 if it continue to present the same front in moving. 
 The idea of parallel lines situated in a plane is first 
 found belonging to the power of Vulcan. But, 
 when that which moves is a solid, the parallels which 
 it draws are not situated in one plane, but in every 
 plane; and the course which the solid traces is 
 therefore also solid. But this power assumes an- 
 other form; because, even when the solid is at 
 rest, it has what may be called an anticipation, or 
 prospective view, of all the solid courses through 
 
29 
 
 which it might move; but, from being solid, they 
 are blended together at their origin, and still con- 
 tinue partially to retain their union as they diverge, 
 altogether forming one maya, which is that of the 
 courses through which the solid might move. 
 Among the twelve chief heathen divinities this 
 power was represented by Mars ; and among the 
 tribes of Israel it must have belonged to Gad. In 
 every solid, although the number of sides be limit- 
 ed, there is an infinite multiplicity of aspects, to 
 each of which belongs a solid course, in the form 
 of parallels extending to infinitude ; but it is evi- 
 dent that every course must have a different solid 
 form, according to the aspect from whence it pro- 
 ceeds. To the power of Mars belongs a sentiment 
 of freedom, might, and power, extending far be- 
 yond the local existence of the individual ; but this 
 is a feeling which might also hypothetically belong 
 to every particle among the sands of the sea. The 
 union of the different courses must be most dense 
 at first ; and, as they diverge and separate, the 
 general maya, which they form, becomes rarer, 
 like an aerial haze departing from the solid. 
 The elementary power of Mars is the beginning 
 
30 
 
 of a new series of powers all belonging to indivi- 
 dual being; because all of them are produced 
 from the nature of courses which originate from 
 solidity, or the power of Vesta. The maya of 
 Mars, however, is distinguished from any former 
 maya by the characteristic of not being produced 
 under lines or planes already extended ; for it is 
 only prospective or potential. 
 
31 
 
 CHAP IX. 
 
 ON ACQUIRED MOTION. 
 
 The ninth mathematical element is the generation 
 of the parabola from the bending of parallel courses, 
 since these cannot be prolonged without encounter- 
 ing other solids ; as every solid has relation not 
 merely to all the courses through which it might 
 move, but also to all the opposition which it would 
 meet with in them ; and from hence, in abstract 
 idea, is the beginning of gravitation. For the sake 
 of illustration, let it be considered that, if two pa- 
 rallel lines be required to bend, and yet to retain 
 the same distance between them, it is evident they 
 must turn otherwise than by an angle, in which 
 there will always be an oblique distance between 
 the lines. Therefore it becomes necessary that 
 the lines assume the form of two parallel curves, 
 one of which, being the outer, must be the larger. 
 For this purpose one of the lines must borrow and 
 
32 
 
 gain from the other a continued increase of exten- 
 sion ; but, if the outer line, at first, were to begin 
 suddenly b) r borrowing some certain quantity, the 
 curvature would begin suddenly, without any 
 previous gradations between it and the original 
 straightness of the lines, or their equality of pro- 
 portion ; and if the borrowing were continued at 
 the same rate, the two lines would bend together 
 as the circumferences of a larger and smaller circle, 
 and consequently would be finite. But this mode 
 of beginning and continuing the curve is hypothe- 
 tically impossible, because the first beginning of 
 change of proportion between the straight lines 
 must always have been infinitely remote, and in 
 the least possible degree. 
 
 The outer line, therefore, must have been gain- 
 ing extension from the other, at a rate always in- 
 creasing ; and the outer line is supposed in these 
 two curves to be found borrowing according to 
 the same proportion as exists in the decreasing 
 series of numbers, or their approach to unity by 
 continued diminution as in the series five, four, 
 three, two, one ; and this is the kind of propor- 
 tion which is expressed in the parabola. In this 
 
33 
 
 mode of exhaustion, the unit which is taken away, 
 at each step, becomess always a greater quan- 
 tity in proportion to the number which remains to 
 be exhausted. But if this principle be carried back 
 towards the beginning of the curve, it would make 
 the first change of proportion infinitely remote, and 
 incapable of being found. Thus, if the outer line, 
 in a certain portion of the course, had been gaining 
 from the inner line one-seventeenth part, and had 
 become to it as eighteen or sixteen ; and if, in the 
 next preceding part, (in which the length of the 
 outer and inner lines taken together had the same 
 total amount), the outer line had been borrowing 
 only one-eighteenth part, and had become to the 
 inner line as nineteen to seventeen ; then, in the 
 part still anterior to that, the outer line would have 
 borrowed only one-nineteenth part from the inner, 
 and would have become to it as twenty to eighteen. 
 Thus the difference becomes always less ; and this 
 progressive refinement or change of proportion is 
 capable of being pursued for ever, and is express- 
 ed in the proportions assumed by the parabola, 
 when retraced towards its origin, which can never 
 
 b2 
 
36 
 
 CHAP. X. 
 
 ON GROWING, OR THE ASSUMPTION OF 
 EXTRANEOUS FORCES. 
 
 The tenth mathematical element is the generation 
 of the hyperbola from the assumption of an ex- 
 traneous force which, being united or blended with 
 those already operating, gives birth to a new form. 
 If, at any part in the united courses of the para- 
 bola, another straight course, or form of Mars, be 
 applied, and made to take a share of curvature from 
 the two courses which are already bending in the 
 parabola, so as to bend along with them, and enable 
 all the three to proceed together as parallels united 
 in one curve ; in this case the newly-added course 
 must communicate part of its straightness to the 
 two former, and must receive from them part of 
 their curvature. This produces an evolving form 
 which is the hyperbola, and which is the original 
 form of support, or nourishment, or vegetation, 
 
37 
 
 which is from the assumption of extraneous forces. 
 That which characterizes this elementary power is 
 the sharing of the curvature among the courses 
 which are united as parallels. Any quantity, con- 
 tinually lessened by taking away the same part 
 from the amount which remains, will give a series 
 of proportionals in continued diminution ; and this 
 is the rule or proportion which is expressed in the 
 decreasing curvature of the hyperbola. Thus if, 
 to the two parallel courses of the parabola, an- 
 other single course be added and made to extend 
 along with them, assuming a portion of their cur- 
 vature, its share will be always one-third of the 
 curvature which remains to be exhausted. There- 
 fore, after each farther extension of the courses 
 through another equal length, the quantity of cur- 
 vature left will be only two-thirds of what it was 
 before. It would first be as two-thirds of the ori- 
 ginal quantity, then as two-ninths, and then as two 
 twenty-sevenths, and so on in continued propor- 
 tion. The hyperbola also in expanding repeats 
 the same form of curvature in lengths which in- 
 crease in continued proportion. Thus, if a certain 
 form were found in a length which was as three ; 
 
38 
 
 and if the same form were next found, on a larger 
 scale, in a length which was to the former as nine 
 to three ; then the same form would next be found, 
 on a still larger scale, in a length which was, to 
 the preceding, as twenty-seven to nine ; and so on 
 in continued proportion. The hyperbola, there- 
 fore, corresponds with the geometrical series of 
 numbers. Among the twelve chief heathen divi- 
 nities this power was represented by Ceres, the 
 goddess of vegetation; and among the tribes of 
 Israel the same character must have belonged to 
 Naphthali. When the decrease of curvature is 
 slow, the hyperbola evolves as a spiral ; but, in 
 other instances, it takes place so rapidly that the 
 curve forms no circumvolutions, but is capable of 
 extending beside an infinite straight line placed be- 
 side it as an asymptote. This is the form of the 
 hyperbola found by that section of the cone. It may 
 easily be shewn that, besides the parabola and hyper- 
 bola, there can be no other simple and continuous 
 forms of curves expressing change of quantity ; be- 
 cause quantity can increase or diminish only in two 
 continuous manners ; namely, first according to the 
 sam e proportion which is expressed in an arithme- 
 
39 
 
 tical series, and which belongs to the parabola ; 
 and, secondly, according to the proportion which 
 is expressed in a geometrical series, and which 
 belongs to the hyperbola. To Ceres must be 
 ascribed elasticity and expansion ; and the powers 
 of recoil, which belong to elementary particles, al- 
 ways operate according to a series of proportionals. 
 The hyperbola is the last continuous line found in 
 the mathematical deduction of forms ; and there 
 is no end to its extension; for, if there remains 
 any curvature at all in a line, it can never be ex- 
 hausted, however long it may continue to be shared 
 by another line extending along with it as a paral- 
 lel, and adopting a form, in which their qualities 
 unite. But the outer and inner lines or courses, 
 in the hyperbola, are different curves, otherwise 
 they could not be parallel ; and this is proved by 
 the difference of the forms of parallel ellipses, 
 traced within each other. The same hyperbola 
 traced on a larger scale would diverge. 
 
40 
 
 CHAR XI. 
 
 ON COMPOSITION. 
 
 The eleventh mathematical element is that from 
 whence originates the power of composition ; be- 
 cause, the eleventh element is not a continuous 
 form generated according to one rule, but includes 
 in itself the juncture of two forms extended ac- 
 cording to different rules. If, at any part in the 
 first hyperbola, another straight course, or power 
 of Mars, be added, and made to proceed along 
 with the first three united courses, as an addition- 
 al power taking a share of the original curvature, 
 it is evident that, after this addition^ the hyper- 
 bola will not continue to extend according to the 
 same rule, but will start off in a new direction, 
 and diminish its curvature more rapidly than 
 before ; because, after the juncture, the power 
 which is sharing or diminishing the original cur- 
 vature has become twice as much as before. A 
 
41 
 
 second form of the hyperbola will therefore be ge- 
 nerated, departing, at a crisis or juncture, from 
 what would have been the continuation of the first 
 curve. This shift or change of direction will be 
 towards the outer side of the curve ; but the conca- 
 vities of both the first and second parts will still be 
 towards the same side. The share of curvature as- 
 sumed by the first-added parallel was one-third ; 
 but the first form of the curve being left, the pa^ 
 rallel courses are continued into another form, in 
 which the quantity of curvature assumed by the 
 added courses becomes one-half; and, therefore, 
 after each farther equal extension of the four pa- 
 rallel courses together, the curvature left is only 
 one-half of what it was before, and thus the quan- 
 tity diminishes according to continued proportion 
 in the second curve, as in the first, but according 
 to a different rate. In this form are found two 
 different qualities of curve, their partition being 
 marked by a curve, angle, or juncture, at the place 
 where the hyperbola branches off, and assumes a 
 new direction. Among the twelve chief heathen 
 divinities, this power was represented by Minerva ; 
 and among the tribes of Israel, the same power 
 
42 
 
 must have belonged to Joseph. From containing 
 in itself the juncture of forms essentially different, 
 this elementary power the origin of composition, 
 which is the freedom of prolonging the same con- 
 tinuity according to a rule different from that 
 which has been begun. The ellipsis is an ex- 
 ample of composition; because it is a figure not 
 found by the continuation of a single curve, 
 but composed of four similar parts joined to- 
 gether in four inverted positions ; and only one- 
 fourth part of it is continuous, being the same 
 as the hyperbola, in abstract principle. By 
 the power of Minerva, the prolongation of a line 
 is enabled to change, at once, from any quali- 
 ty of form to another ; and, in flowing lines, the 
 junctures or crises cannot be easily detected. The 
 elementary parts of form are capable of extension 
 only according to one rule ; and the minutest parts 
 of every line must be mathematical forms ; because 
 a line cannot consist entirely of junctures or tran- 
 sitions from one kind of extension to another : and 
 if there be any portion, however minute, in which 
 the mode of extension does not change, that por- 
 tion must be a mathematical form which is part 
 
43 
 
 either of a circle, a straight line, an angle, a pa- 
 rabola, or an hyperbola ; since every rule in a 
 plane must produce some one of these. To Mi- 
 nerva, therefore, must be ascribed the power of 
 elegance, or choice and freedom of transition from 
 one principle to another. Considered as an ele- 
 ment found by continued deduction, Minerva is a 
 second form of Ceres or growing, and is produced 
 from the assumption of a second extraneous force, 
 after the power of the first extraneous force had 
 been for some time exemplified. But the essence 
 of this element is the juncture which it includes, 
 and which necessarily implies that both the first 
 and second curves must appear in the power of 
 Minerva, when it is viewed as a single element, 
 apart from others. 
 
44 
 
 CHAP. XII. 
 
 ON COMPREHENSION AND IMAGINATION, 
 
 The twelfth and last element, found by continued 
 deduction, is a general curve, comprehending a 
 succession of different hyperbolic parts. For, if 
 to the second hyperbola, which belongs to Miner- 
 va, a third extraneous power or parallel course be 
 added, it will only have the effect of producing a 
 third hyperbola, which again starts off in a new 
 direction, and exhibits a new quality of curve, con- 
 stituting the third branch ; and, if farther addi- 
 tional forces be successively added, they will only 
 have the effect of producing a succession of hyper- 
 bolic forms, changing their quality in continued 
 gradation ; and, having the united courses of the 
 original hyperbola continued through the whole of 
 them ; but with the addition of one more parallel 
 course added at the beginning of each successive 
 
45 
 
 branch. And if the additions be made at such 
 points as to have always equal lengths of course 
 between them, then all the points or junctures, at 
 the inner side, will be situated in an imaginary 
 curve, which is also an evolving hyperbola, tend* 
 ing in the same general direction as the particular 
 branches, and having its convexity to the same 
 side as theirs. This curve is not actually traced, 
 but derives an ideal existence from the nature of 
 the comprehending form. This elementary power 
 was represented by Mercury, among the twelve 
 chief heathen divinities ; and, among the tribes of 
 Israel, the same character must have belonged to 
 Benjamin. This is the last elementary power 
 which is found by continued deduction ; since the 
 successive assumptions of extraneous forces will 
 produce no other form, although they be continued 
 to infinitude. To express the relation of Mercury 
 to number it was allegorically said that, while he 
 was yet an infant, Juno had been induced to 
 suckle him ; but, discovering that he was the son 
 of Maia, she put him away ; and, the milk flow- 
 ing from his lips through the celestial regions, 
 produced the milky-way. The share of curvature 
 
46 
 
 assumed in the third hyperbolic branch is three- 
 fifths ; in the fourth branch, four-sixths ; in the 
 fifth branch, five-sevenths ; and so on, the differ- 
 ence of quality in each successive branch becoming 
 always more refined. Mercury was called the in- 
 terpreter, because of the power of an intermediate 
 branch intervening between two others, and con- 
 necting them, although they are entirely different 
 in quality. To Mercury may also be ascribed the 
 form of the syllogism; in which the minor, or 
 second proposition, connects the first proposition 
 with the conclusion. Three successive branches 
 of the hyperbola are a mathematical syllogism, in 
 which the original parallel courses are carried 
 through an intermediate curve, and transferred 
 into a third curve, of which the quality is deduci- 
 ble from the first. Thus, by reasoning, the force 
 of an acknowledged proposition is carried into 
 other applications. But the characteristic of this 
 elementary power is the comprehension of particu- 
 lar forms in a general one ; which may be called 
 the Saturnian hyperbola ; because of its fulness of 
 parts, and because of the imputed character of 
 time, which is poetically said to swallow up all 
 
47 
 
 particular forms of existence. The Saturnian hy- 
 perbola, besides occupying a real place by the suc- 
 cessive forms contained in it, has also reference to 
 a series of points situated in a totally different 
 curve ; but the relations between these points 
 would be direct, and like straight lines connecting 
 them, and, therefore, would not in reality trace 
 the curve ; which, nevertheless, is traced by the 
 power of imagination ; and, therefore, to Mercury 
 belongs a new kind of maya or ideal creation, dif- 
 ferent from any other ; because it is not produced 
 by the direct relations between points existing in 
 any form really extended. Thus, the Saturnian 
 hyperbola teaches the mind to create relations 
 which are not there ; and this is the nature of 
 fancy or conception. The twelfth element, there- 
 fore, has two different forms, namely, that which 
 has real extension and place, and that which is 
 only imagined or conceived. Some of the ancients 
 ascribed to Mercury the invention of letters, and 
 said that he had learnt them from seeing: cranes 
 tracing figures in the air by their flight ; and it 
 is evident that these forms must have been dis- 
 
48 
 
 covered by conceiving the relation of the crane to 
 places in which it no longer was, and which were 
 beyond its real or local extension. 
 
PART II. 
 
 ON THE MODES OF COMPOSITION. 
 
 CHAP. I. 
 
 ON THE POWERS OF GRADATION, OR THE COM- 
 PARISON OF DIFFERENT FINITE SERIES. 
 
 Orion or Bacchus. 
 
 The last of the simple elements is the Saturnian 
 hyperbola, which comprehends a series of dif- 
 ferent branches all included in one general curve. 
 Beyond this no other elementary power is found. 
 But, by resorting to composition, a new series of 
 infinite forms is discovered. Composition must 
 begin from the application of the power of Juno, 
 which is the first. 
 
 If the power of Juno, or alternation, be applied 
 to the Saturnian hyperbola, it can have no other 
 
50 OIIION OR BACCHUS. 
 
 effect than to make it alternate, that is to say, to 
 make each branch cross itself, and so change its 
 direction to the other side. But this change of 
 direction is balanced or corrected by the succeed- 
 ing branch also crossing itself and again extend- 
 ing in the opposite direction ; so that the whole 
 series, instead of bending as a general curve, is 
 forced to stretch forward and assume a form some- 
 what like that of alternating angles. To this 
 form, however, there is a limit; because the 
 breadth or thickness of each branch is increased 
 by another parallel course added to it ; and the 
 thickness of a single branch must at last become 
 greater than will permit of its crossing. Thus, 
 after a certain number of branches have alternated, 
 the transverse progression comes to an end, and a 
 limited series is found. If the series of branches 
 be farther prolonged, they must resume the ori- 
 ginal form of the Saturnian hyperbola. 
 
 The power of Juno, when employed alone in 
 composition, could not produce any infinite pro- 
 gression ; and the second power, which is Jupiter, 
 must therefore be applied. Jupiter is the repeti- 
 tion of an identical form on a smaller scale, as in 
 concentric circles. If this power be applied to the 
 succession of hyperbolic branches, it will have the 
 effect of transposing the remainder of it into a 
 smaller scale, in which the same deduction is con- 
 tinued. For it is evident that, on whatever scale 
 
ORION OR BACCHUS. 51 
 
 an hyperbolic form is produced, it remains the 
 same as to quality, and has still the same proper- 
 ties as a particular curve. 
 
 Suppose the transposition to be into a scale 
 one-third of the former, then the breadth of a 
 single branch will have become only one-third of 
 what it was before ; but, if its length be drawn 
 out to the same extent as formerly (as it must be 
 to allow the investigation to proceed), then there 
 will be a renewed freedom of making each branch 
 cross and the series resume the transverse mode of 
 progression; because the thickness of a single 
 branch has now become less in proportion to its 
 length ; and the thickness will be more slowly in- 
 creased than formerly, because the same change of 
 scale must be applied to the parallel courses which 
 are added to the successive branches, and by which 
 their thickness is increased. Therefore the new 
 transverse progression will admit of a greater num- 
 ber of branches than before, but it will also come 
 to an end, and will constitute another finite series. 
 After which, a new transposition into a scale of 
 one-third must be again resorted to ; and, the 
 length of a single branch, being drawn out to the 
 same length as before, there will be a renewed 
 freedom of transverse progression, which will also 
 come to an end, and constitute another finite series 
 still more extensive than the second. After which 
 another transposition must be again resorted to ; 
 
52 OBION OR BACCHUS. 
 
 and so on to infinitude. In each new transverse 
 progression the number of branches will be greater 
 in continued proportion ; because the rate of the 
 increase made to the thickness of a single branch 
 is changed after each transposition, and is diminish- 
 ed in continued proportion ; so that the number 
 of steps becomes greater, and the exhaustion of 
 the whole series slower. 
 
 The crossing of each branch within itself is the 
 cause of measuring and intersecting the curve by 
 other parallel lines going through it. These are 
 the prolongation of the same curve, but being a 
 different part, their form is different from that of 
 the lines which they cross. The multiplicity of 
 forms in the intersection is increased in each new 
 branch by another added parallel. This mode of 
 composition is the same as internal sensation or 
 the consciousness of form ; and among the an- 
 cients it was signified in the character of Bacchus ; 
 because the effect of the vine is to produce an in- 
 creased power of mental sensation, which is the 
 same as the knowledge of form. In another point 
 of view, this mode of composition was probably 
 signified in Orion, the hero of the chase, or the 
 power of exhausting intermediate distances. But 
 this power, in effect, becomes the same as the com- 
 parison of different modes of gradation, or the 
 sense of beauty in finite portions of continuity ; 
 because the transverse mode of progression is the 
 
ORION OR BACCHUS. 53 
 
 same as the sense of continuity which binds con- 
 trary points together and blends them into one 
 form ; as each branch, in crossing, returns to the 
 greater curvature in the preceding part of the 
 branch, but is, at the same time, diminishing its 
 curvature ; and this produces the blending of con- 
 trary forms and directions, which resembles the 
 binding of contrary points together in the con- 
 tinuity of a line. Therefore Bacchus, like 
 Apollo, was represented as drawn in a car. Con- 
 tinuity is also the belt of Orion. But, as every 
 transverse progression is finite, Bacchus can only 
 be the same finite continuity ; or, rather, this 
 mode of composition is like that comparison of in- 
 dividual forms which belongs to the elementary 
 power of Vesta, which draws forth successive ridges. 
 It is therefore said in the Scripture, that the tribe 
 of Dan should become as a judge among the rest. 
 The comparison of different kinds of gradation pro- 
 bably constituted also the characters of the Graces, 
 Aglaia, Euphrosyne, and Thalia ; who, by some 
 authorities, were called the daughters of Bacchus, 
 by others of Jupiter and Venus. These minute 
 goddesses were judges of refinement, and had the 
 power of reproving coarseness or inelegance. The 
 exhaustion of successive finite series, which in- 
 crease in continued proportion, also extends to a 
 sense of the unlimited ; because, although each 
 particular series comes to an end, it is evident, 
 
54 ORION OR BACCHUS. 
 
 from comparing those series which have been al- 
 ready completed, that the number of steps in the 
 future ranges to be found must pass beyond the 
 limits of all calculation. This is probably ex- 
 pressed in the Scripture in Jacob's ladder. Tak- 
 ing a stone (that is Vesta) for his pillow, he fell 
 asleep, and saw a scale extending up to heaven. 
 Some poets, such as Pope, have applied the same 
 conception to the gradations of being in the uni- 
 verse, likening it to a great chain of which no link 
 or intermediate degree could be wanted. This 
 notion, however, is in a great measure erroneous, 
 since the order of the universe probably depends 
 more upon differences of kind than of degree, 
 which latter is left open for progressive change, 
 except among the brutes. 
 
 To this mode of composition may also be re- 
 ferred lyrical poetry, in which there is the com- 
 pletion of limited forms or stanzas. This charac- 
 ter belonged to the people of Mytelene, among 
 whom lyrical poetry received its origin from 
 Alcaeus. The transition into another series was 
 also expressed in the action of Sappho, who leapt 
 from the Leucadian rock, to escape from the bond- 
 age of her passion. Theophrastus, a native of the 
 same island, satirized the improprieties of man- 
 ners, or the want of refinement and taste. But, 
 mythologically, the exhaustion of a particular se- 
 ries was expressed in the fable of Meleager the 
 
ORION OR BACCHUS. 
 
 55 
 
 iEtolian, the duration of whose life depended on 
 the consumption of a brand snatched from the 
 hearth. The ancient Romans, whose character 
 corresponded with this mode of composition, 
 thought that the highest glory and virtue was in 
 the progressive enlargement of their prospects, or 
 increase of range, as exemplified in their conquests ; 
 for they were not contented to have their views 
 terminated by any impassable barrier. These 
 characteristics led them to approve of suicide. 
 Among the modern inhabitants of Italy, the same 
 taste appears in their love of the fine arts which, in 
 regard to sensation, are essentially a trial of the 
 various modes of gradation, or a comparison of 
 different limited series. To the power of Orion 
 belongs also emulation, which will not endure to be 
 exceeded or surpassed. The character of the an- 
 cient Romans is accounted to have expired when 
 there was placed over their heads an emperor, to 
 whom all the other officers of state were virtually 
 accountable and subordinate ; for then the glory 
 of their separate duties and powers appeared to be 
 lost. Such is the nature of this mode of composi- 
 tion as to republicanism ; but in the fine arts, 
 where it is more useful, and can be more happily 
 exercised, it tends to the multiplication of different 
 departments, each having a manner and prospect 
 of its own. Thus, republican virtue is converted 
 into what is called vertu, or the sense of beauty. 
 
56 ORION OR BACCHUS. 
 
 This mode of composition has also a certain re- 
 lation to the nature of the parabola, which must 
 always reach a termination, and which, in ap- 
 proaching to that termination, traces the inter- 
 mediate gradations of form according to the na- 
 ture of a decreasing arithmetical range, in which 
 the same quantity is taken away at each step. 
 Now, in the exhaustion of any limited series, the 
 same form is virtually expressed by the continual- 
 ly changing proportion of a single step to the 
 quantity which remains to be exhausted. There- 
 fore, Orion is virtually the same as the power of 
 searching out the different forms of the parabola, 
 always passing from a more rapid form, in which 
 the gradations are coarser, to a slower form in 
 which the gradations are more refined, and, by 
 their greater number, extend the range or com- 
 pass of the form to which they belong. Suppos- 
 ing the first series had thirty steps, the next 
 transposition into a scale of one-third would begin 
 a series of ninety ; which, by another transposi- 
 tion, would pass into a series of two hundred and 
 seventy, and so on ; each of these series having a 
 form of the parabola which corresponds with it. 
 One of the mythological representations of the 
 finite series was Hebe, the cup-bearer of the gods ; 
 and her character has relation either to the com- 
 pletion or exhaustion of a finite quantity. This 
 is also the character of Priapus, god of lakes, as 
 
ORION Oil BACCHUS. 57 
 
 determinate quantities ; and, in another point of 
 view, of Pomona, goddess of orchards, because 
 the fruits are finite quantities or completions. 
 From each branch crossing within itself, and 
 so changing its direction, this mode of composi- 
 tion is also like a search into the differences of 
 direction ; and, therefore, it has a certain relation 
 to the elementary power of Mars ; from whence 
 one of his names, Gradivus. This, however, is 
 but a partial resemblance, and must always come 
 to an end ; since the changes of direction in each 
 series are finite in their number ; although each 
 new transposition produces a more extended range. 
 Therefore, having only a partial resemblance to 
 Mars, this mode of composition has most relation 
 to the power of Vesta; and the character of the 
 inhabitants of Italy is that which comes nearest to 
 it. The fable of Saturn hiding himself in Latium, 
 signified that the form of the Saturnian hyperbola 
 is lost or hidden in the transverse mode of pro- 
 gression, which is the same as Latium. This is 
 the dominion of the Pope, whose character and 
 place virtually coincides with that of Orion. 
 Among the Homeric heroes, in the Grecian troops, 
 the same power was perhaps represented by 
 Diomed, whose native country was ^Etolia, the 
 birth-place of Meleager. In the Trojan battles, 
 the son of Tydeus was distinguished for his auda- 
 
 c2 
 
58 ORION OR BACCHUS. 
 
 city, which, like that of Orion, passed beyond all 
 limits, and prompted him to lift his spear even 
 against divinities. The same refusal to acknow- 
 ledge any superior power was signified also in the 
 fable of the giants, sons of the earth, or pow r ers of 
 Vesta, who, wishing to scale the heavens, placed 
 Pelion on Ossa, and, upon these, piled other 
 mountains. 
 
 The kings of Latium were descended from 
 Faunus, a rustic power attached to the earth, or, 
 in other words, belonging to Vesta. To the 
 Fauns, however, there was ascribed a complete 
 human figure, with the addition of small horns in 
 the forehead, and a short tail at the termination 
 of the spine. The Fauns may have had relation 
 to the sense of form, which corresponds with a 
 single branch crossing within itself and producing 
 the intersections of parallel curves. Something 
 like this is exhibited in the patterns of windows 
 used in Gothic architecture, where parallel forms 
 of the hyperbola are crossed by others, which, 
 however, are generally exactly similar, and are 
 not farther parts of the same curves, as must be 
 the case when a single branch crosses by changing 
 its direction. Closely connected with the sense of 
 form is the love of the grotesque, which may also 
 be ascribed to the Fauns, and which is not incapa- 
 ble of being mingled with the sublime. These 
 
ORION OR BACCHUS. 
 
 59 
 
 forms, along with the Satyrs, accompanied Bacchus, 
 a divinity ever young, and only reaching the 
 end of any series to pass into another more exten- 
 sive. But the chief characteristic of this power is 
 the comparison of the different kinds of gradation ; 
 and, from this comparison, the sense of unlimited 
 range is generated. 
 
60 
 
 CHAP. II. 
 
 ON THE POWER OF SEARCHING OUT THE DIF- 
 FERENT FORMS OF THE SATURNIAN HYPER- 
 BOLA. 
 
 Hercules and the Muses. 
 
 The exhaustion of successive finite series, found 
 by transposition into a smaller scale, may also be 
 considered in another point of view, in which it 
 becomes the power of searching out the different 
 forms of the Saturnian hyperbola ; for, at the 
 end of each transverse progression, if the same 
 series of branches be continued without crossing, 
 they will give a new form of the Saturnian hyper- 
 bola, that is, a new and different example of the 
 comprehending curve. It is evident that the ra- 
 pidity of any single form of that curve depends 
 upon the number of branches or accelerations it 
 contains within a certain extent of course ; or, in 
 other words, it depends upon the shortness of a 
 single branch. Now, at each transposition into a 
 smaller scale, the length of a single branch being 
 virtually increased, it follows that, after the trans- 
 
HERCULES AND THE MUSES. 61 
 
 verse progression has been completed, the farther 
 continuation of the same series of branches, with- 
 out crossing, must give birth to a new Saturnian 
 form, slower than the preceding. This form, 
 however, must not be considered as capable of 
 infinite prolongation, but only as a short specimen, 
 which is tried before the series of branches is again 
 transposed into a smaller scale. The comparison 
 of these successive specimens is the power belong- 
 ing to the second mode of composition ; for the 
 power of Bacchus does not necessarily imply the 
 continuation of a series of branches after they have 
 ceased to cross. Among the ancients, this second 
 mode of composition was represented by Hercules, 
 the god of achievement and acquisition ; and those 
 persons who acquired wealth were said to have 
 been befriended by Hercules ; because all the Sa- 
 turnian forms are forms of comprehension. Each 
 transverse progression terminates in the discovery 
 of a new Saturnian form, and corresponds with 
 the spirit of investigation and adventure. Thus 
 Hercules, in his wanderings, pierced into the 
 stupendous scenes of rocks in the Hyperborean 
 regions. Contemplating the wild shapes of ca- 
 verns which suddenly divided light from darkness, 
 he heard the cries of the wolf or eagle, and saw 
 the distant mountains covered with snow. Pur- 
 suing his way through the labyrinth of rugged 
 cliffs, he saw, beneath him, the concave edges of 
 
62 HERCULES AND THE MUSES. 
 
 cliffs, as it were set with teeth, while at every step 
 his powers of scrutiny were sharpened, and the 
 keenness of his vision increased. Descending to 
 a milder climate, he found the Pierian maids al- 
 ways ready to welcome their associate, and to hail 
 him by the name of Hercules Musagetes, or the 
 leader of the muses. The ancients were accustom- 
 ed to build temples to Hercules and the muses, as 
 powers associated and joined together. The rea- 
 son of this seems to be, that the muses are the 
 powers of inventing and conceiving ; and imagina- 
 tion, or the power of comprehension, which be- 
 longs to Saturn, must also be attributed to them, 
 who are rather the varied powers of comprehen- 
 sion, or its different modes in the different arts 
 and sciences to which fancy and invention are 
 applied. Hence their various attributes of music, 
 painting, lyrical poetry, tragedy and comedy, and 
 the rest. But, in English, to muse signifies to 
 fancy or conceive. 
 
 This mode of composition, although its practical 
 power is to search out the differences of Saturnian 
 forms, has most relation to the elementary power 
 of Diana or contrast, because the differences which 
 it examines are those of forms of the same kind, 
 namely, forms of Mercury. Now contrast is the 
 comparison not of forms without similitude or rela- 
 tion, but rather of forms which are allied by prin- 
 ciple, so as to induce comparison of their courses. 
 
HERCULES AND THE MUSES. 63 
 
 The most perfect contrast is in forms which are 
 different specimens belonging to the same general 
 rule ; because, between these, there is always a re- 
 lation which bends them together and forces 
 comparison, while they diverge and pursue dis- 
 similar courses. Contrast, therefore, may be 
 ascribed to Hercules and the muses ; and this 
 mode of composition has most resemblance to the 
 elementary power of Diana, and has relation to 
 wealth and the aggregation of powers. To the 
 same class with the muses may be referred Mor- 
 pheus, a deity who had in his power all the diver- 
 sity of dreams, or all the forms of the Saturnian 
 hyperbola. 
 
 The kind of mental feeling which belongs to this 
 mode of composition may easily be known by its 
 characteristics; for in every thing it desires the 
 result. It also finds pleasure in the knowledge of 
 particular examples of every power ; from whence, 
 perhaps, the character of Eros, or the celestial 
 Cupid, as the power of fixing on particular speci- 
 mens. But from the love of contrast springs also 
 allegory, which is some abstract truth presented 
 as if disguised or hidden in a particular form or 
 conception. Spenser is perhaps the poet near- 
 est to English genius. Among the ancient poets, 
 the most disposition to professed allegory is shewn 
 in Anacreon, whom Love, in the form of a child, 
 
64 HERCULES AND THE MUSES. 
 
 frequently calls forth to battle, or invites to run a 
 race with him. 
 
 To Hercules and the muses must be ascribed 
 freedom and generosity of mind, and productive 
 power ; because every particular specimen sug- 
 gests the possibility of others. This characteristic 
 leads again to the power of achievement and ac- 
 quisition. It has been said that, in England, the 
 power of science is always seen in its result ; and 
 the character of the English nation is that which 
 comes nearest to this mode of composition ; both 
 as to achievement and the search after results, 
 shewn in the doctrines of the chief English philoso- 
 pher, Lord Bacon, and also as to the national 
 disposition to acknowledge the freedom of indivi- 
 duals, or to suffer each to pursue apart his own 
 course, as one more added to the number of dis- 
 similar modes of conception, which this power 
 must always wish to see multiplied. The succes- 
 sive transpositions into smaller scales have relation 
 to the sense of the picturesque, from the boldness 
 and suddenness of the results ; corresponding with 
 which is the spirit of adventure advancing precipi- 
 tately, while yet unable to guess the conclusion. 
 The picturesque, in another point of view, seems 
 to result from the composition of forms which are 
 themselves composite ; in the same manner that 
 the power of Hercules searches out different Sa- 
 
HERCULES AND THE MUSES. 65 
 
 turnian curves, which are themselves forms of 
 comprehension, including hyperbolic parts. Thus 
 the picturesque may be produced from the rugged 
 compilation of cliffs which exemplify different kinds 
 of ruggedness. 
 
 To the elementary power of Mercury belongs 
 
 conviction. Now, this mode of composition being 
 
 the means of searching out the various forms of 
 
 the Saturnian hyperbola, is the same as the power 
 
 of comparing the various modes of conviction, or 
 
 of weighing and considering arguments together, 
 
 Minos, a form of this power, was appointed to be 
 
 one of the infernal judges. Ariadne, the power of 
 
 continuous deduction, the daughter of Minos, gave 
 
 to Theseus the thread or principle by which he 
 
 was guided through that maze which resembled 
 
 the perplexities and difficulties to be encountered 
 
 in reconciling the different aspects assumed by the 
 
 same truth ; for all the different Saturnian forms 
 
 discovered are examples of the same curve. To 
 
 Hercules or Minos, therefore, may also be ascribed 
 
 the comparison of arguments, as a judge listens to 
 
 what is said on both sides, till he finds out the 
 
 truth, through the disguises of different statements. 
 
 Diana was known under the name of Dictynna, or 
 
 the Cretan, as if to signify the application of this 
 
 character to her. Others said that the original 
 
 Dictynna was a nymph beloved by Minos, and 
 
 celebrated for having first invented hunting nets, 
 
66 HERCULES AND THE MUSES. 
 
 and using them on Dicte, a mountain in Crete. 
 This fable corresponds with the power of acquisi- 
 tion. The voracity ascribed to Hercules was often 
 a source of ridicule ; but in the Muses it would 
 appear as the desire to be always acquiring or 
 comprehending something farther. Thus the la- 
 borious Aristotle, having explored all the different 
 forms of syllogism, turned his powers of research 
 into various other departments of science ; or thus 
 St Paul would never have an end of adding one 
 more to the number of those he had converted to 
 the Christian faith. Among the apostles, the 
 place of Minos or Diana seems at first to have be- 
 longed to Judas Iscariot ; but afterwards the 
 same character was shewn in a totally different 
 point of view in St Paul, who, of all the sacred 
 writers, shewed the most inclination for the exer- 
 cise of reason, or for that opposition of arguments 
 which belongs to Minos, the comparer of evi- 
 dence. 
 
67 
 
 CHAP. III. 
 
 ON THE APPLICATION OF STRAIGHT LINES TO 
 THE SERIES OF HYPERBOLIC BRANCHES. 
 
 Pan or JEsculapms. 
 
 The third mode of composition is from the appli- 
 cation of the third power, which is that of Apollo, 
 or straight lines, to the forms of the hyperbolic 
 branches. The straight lines, proceeding from the 
 beginning of the whole as from a centre, diverge 
 and become the power of marking off successive 
 portions of any particular curve, by which it is 
 shewn to have one nature throughout, or to con- 
 tinue producing similar forms. Or, when applied 
 to successive branches, the straight lines are still 
 capable of finding their harmony and relation of 
 these branches, and of giving so much for so much, 
 in two forms measured at once. Hence the balance 
 may be ascribed to Pan, although it belongs truly 
 to Pollux or Eridanus, as will afterwards be shewn. 
 This mode of composition is like the concurrence 
 of abstract rule with nature, to help it, support it, 
 and bear it out; and it corresponds with the 
 
68 PAN OR ^SCULAPIUS. 
 
 medica, or strengthening power of iEsculapius; 
 for it sooths and flatters nature by going along 
 with it. This mode of composition contains no 
 power of creative progression ; but the straight 
 lines, diverging from and moving round a centre, 
 can only continue to measure or try those curves 
 along which they spread. This mode of com- 
 position was expressed in the character of Pan, 
 the inspirer of alarms ; but he was also a wood- 
 land power, loving to sit tranquilly amidst the 
 forest recesses, and to diffuse the sound of his 
 reeds through the trembling air ; and his pastoral 
 attribute is another characteristic of the power of 
 superintendence proceeding from one centre. This 
 mode of composition has most relation to the ele- 
 mentary principle of Jupiter, or revolution, and 
 the completion of cycles. But since the parallel 
 courses which extend together in any hyperbolic 
 form must always be different curves, it follows 
 that a straight line moving over them must al- 
 ways measure different forms at the same time ; 
 and must give or mark off corresponding portions 
 in the curves which bend together as parallels. 
 The characteristics of this mode of composition 
 seem to belong to the Scotch nation, or more par- 
 ticularly to the Scotch Highlanders. The Low- 
 landers of Scotland may perhaps rather be taken 
 for forms of Pluto, a power which will after- 
 wards be explained, and which was also mythologi- 
 
PAN OR JESCULAPIUS. 69 
 
 cally exemplified in Epirnetheus, deriving know- 
 ledge from events after they were past, but never 
 able to foresee them. But the character of Pro- 
 metheus, who was distinguished for wisdom or 
 foresight, in some respects corresponded with 
 iEsculapius, as the power of animating and sup- 
 porting. iEsculapius was the son of the nymph 
 Coronis, who was afterwards changed into a raven, 
 the type of foresight and vaticination. To the 
 same mode of composition with Pan may be also re- 
 ferred the Pythian Apollo, whose oracles were ut- 
 tered at Delphi, in Phocis. The name of one of 
 the places in that region was Anemoreia, that is, 
 " the rushing of blasts," which is like some of the 
 names in the poems of Ossian. In the Trojan 
 war, the Phocians were led by Schedios and Epis- 
 trophos ; but this power was not represented by 
 any distinguished character among the Grecian 
 troops. Of all the modes of composition, this, 
 which coincides with the elementary power of Ju- 
 piter, is the coarsest and the most vulgar, but, at 
 the same time, it has a certain beauty of its own, 
 which arises from its universality and simplicity. 
 To St Andrew may be ascribed the pastoral flute ; 
 and the essential characteristic of Pan is the incli- 
 nation to superintend and interfere. Thus the 
 Delphic oracle, by its responses, found means to 
 direct the affairs not only of Greece, but of many 
 surrounding countries. But Pan is likewise the 
 
70 PAN OR .ESCULAPIUS. 
 
 power of all taken together ; and a system of mu- 
 tual control and interference or usurpation, if it 
 were perfectly equal, in a large community, might 
 he compatible with the simplicity and kindness of 
 the golden age, or the good-nature of mankind 
 exercised in prompting and admonishing each 
 other. This is the beautiful ideal of democracy. 
 But it would preclude all difference of manners or 
 habits ; for the characteristic of Pan is the inca- 
 pability of progressive change ; and this is the 
 true meaning of his form being partly similar to 
 that of beasts. The conviction, that the same 
 things which now are must continue for ever, con- 
 stituted the sunshine of the golden age, and was 
 exemplified in the unchangeable character of shep- 
 herds engaged in superintending their flocks, or in 
 sounding their vocal reeds. But Pan was also 
 much reverenced by the ancients for the wonders 
 he accomplished by inspiring panics in war, al- 
 though the exercise of this power was generally of 
 short duration. To this class may perhaps be re- 
 ferred Napoleon Buonaparte, who is now for ever 
 laid asleep in the island of St Helena. It was the 
 opinion of Napoleon, as of many others, that every 
 thing in the world ought to depend upon the mu- 
 tual consent and concurrence of numbers. This, 
 at first, has the appearance of being liberal and 
 just, but it would soon preclude all the differences 
 of manners, and make an end of individual free- 
 
PAN OR /ESCULAPIUS. 71 
 
 clom; since the multitude leagued together will 
 always be inclined to persecute and insult those 
 who refuse to combine with them. The power of 
 universal consent would knit mankind [together in 
 a world of vulgar falsehood of their own establish- 
 ing, and would keep them, like tribes of savages, 
 ignorant of every thing beyond. There can be 
 no difference of manners or opinions among the 
 brute creation ; but, in proportion as men rise in 
 the scale of intellectual life, it becomes more ne- 
 cessary that individuals should have the liberty of 
 keeping their motives and feelings apart from those 
 of others. 
 
72 
 
 CHAP. IV. 
 
 9 
 
 ON THE RETROGRESSIONS OF THE HYPERBOLA. 
 
 SECT. I. 
 
 ON THE INFINITUDE OF A SINGLE CURVE. 
 
 Pelops or Proserpina. 
 
 The fourth mode of composition is from the ap- 
 plication of the elementary power of Diana, or 
 angle, to the hyperbola. This is the means of 
 shewing that the curve is infinite, in which ever 
 direction it is traced ; or, in other words, that it 
 has no real centre, or that its course may be trac- 
 ed back for ever into the depths of space. For if 
 successive portions of straight line be placed along 
 the inside of the curve, touching it, and always 
 joined to each other at the same angle, it fol- 
 
PELOPS OR PROSERPINA. 73 
 
 lows, from the nature of the curve, that these por- 
 tions of line will be a series of proportionals di- 
 minishing towards the origin of the curve ; because 
 they are joined together alw r ays at the same angle, 
 and because the successive angles which they form 
 all touch the inside of the hyperbola, and their 
 series participates of its quality, which is founded 
 on continued proportion. Now, when the origin 
 of the hyperbola is arrived at, and when the curve 
 extends no farther in that direction, as a guide, 
 then the series of angles may still be continued, 
 according to that proportion which is already be- 
 gun, and they will continue to trace back the curve 
 for ever, as it retires by diminution into the depths 
 of space, and disappears from the reach of the 
 senses. But the parallel courses of the original 
 hyperbola must also be supposed to be prolonged 
 in the retrograde direction, and to follow the series 
 of angles. The breadth of the parallel courses 
 w r ould at last become an obstacle to their farther 
 retrogression ; but these also, by successive falls or 
 contractions, greater in continued proportion, may 
 be supposed to approach the innermost line, and 
 so to permit a farther extension of the curve by a 
 number of angles greater in continued proportion, 
 after each fall. These will draw out successive 
 forms of the same curve, always more extended ; 
 these portions being marked off by the successive 
 
 D 
 
74 PELOPS OR PROSERPINA. 
 
 falls. Supposing the first portion had ten angles, 
 and the next portion had twenty, then the second 
 portion would contain the first form twice ; the 
 third portion would have forty angles, and would 
 contain the second form twice, and so on to in- 
 finitude; each successive portion being a more 
 extended form or reach of the innermost curve. 
 But since, after each fall or contraction, the breadth 
 of the outer parallel courses is made less, their forms 
 must become different, while the innermost curve 
 touching the angles continues always the same. 
 Therefore, after each fall or contraction, the outer 
 courses of the hyperbola are made to produce new 
 hyperbolic forms, nearer to the innermost curve, 
 to which they are always parallel ; and, the power 
 of Proserpina, although apparently one curve, has 
 an infinite series of different hyperbolic forms, all 
 fitted to the outer side of the simple and continued 
 curve which touches the angles. 
 
 Farther, by this mode of composition, it may 
 also be shewn that any hyperbola, although extend- 
 ed in the evolving direction along an asymptote, 
 will at last, in its retrogressive course, assume the 
 form of an involute ; because the repetition of any 
 angle, however obtuse, w T ill at last make the curve 
 turn in. The characteristic of this power, neverthe- 
 less, is that, being such as to withdraw from the reach 
 of the senses, it can at last only be made a subject of 
 
PELOPS OR PROSERPINA. 75 
 
 cogitation ; for the remainder of the form assum- 
 ed by the hyperbola in its retrogression can never 
 be seen, but can only be ruminated upon in silence. 
 This is the kind of refinement and beauty which 
 belongs to Proserpina, the queen of the shades, and 
 which has the characteristic of purity and firm- 
 ness in belonging all to one curve. But this curve 
 may also escape from its perfect uniformity, and 
 become susceptible of change of direction, by con- 
 tinuing always to cross itself in the same direction 
 towards the inner side of the curve. In this case 
 the general course of the curve would continue the 
 same as before, and every second portion of it 
 would return to the original course of the simple 
 form, but every intermediate portion being turned 
 in the opposite direction, so as to have its concave 
 side outwards, would give to the power of Proser- 
 pina a sense of continually changing direction. 
 Each crossing may be supposed to take place 
 after an equal number of angles, and the number 
 of crossings, after each fall, would therefore in- 
 crease in continued proportion. Such may be 
 imagined the meanders of the river Phlegethon. 
 The successive intersections of the curve would 
 also be connected with a consciousness of in- 
 creasing refinement in the successive parts of the 
 same form ; or the continued reproduction of the 
 same form on a smaller scale by the same curve 
 in its retrogressive course. 
 
76 PELOPS OR PROSERPINA. 
 
 This mode of composition has most relation to 
 the elementary power of Venus ; because the para- 
 bola is retraced towards its origin, which is in- 
 finitely remote, and can never be found. The 
 power of Proserpina is similar to this ; but it re- 
 tires within itself, and traces back the form of the 
 hyperbola for ever to its origin, by means of 
 cogitation or deep internal feeling. Thus Phle- 
 gethon continued to restrain its sound the more 
 in proportion as it descended into a deeper shade, 
 till it was lost in silence. But, in another point 
 of view to Proserpina may be ascribed the power 
 of fashion, which excludes or refuses to acknow- 
 ledge whatever is not found within a certain line, 
 or manner, or quality ; in the same way that 
 Proserpina refuses to acknowledge any form which 
 is not found in tracing that one curve which has 
 been begun. The refinements of social feeling 
 may also be compared to the connexion of succes- 
 sive forms, all having their places in one curve, 
 and constituting a whole by the mutual corre- 
 spondence of their quality or fashion which is con- 
 formed to the same ruling principle, although the 
 forms themselves must all have different places 
 and functions in the series. Among the ancients, 
 the country whose inhabitants seem to have had 
 most relation to this power was Elis, where a gene- 
 ral assembly was held of the Grecian nations at the 
 Olympic games; and where, probably, at every 
 
PKLOPS OR PROSERPINA. 77 
 
 celebration of the games, a sort of fashion or pre- 
 vailing mode was communicated, to retain its in- 
 fluence for the next five years. The inhabitants 
 of Elis were exempted from the hardships of mi- 
 litary service, and their region was veiwed as a 
 privileged field, and as sacred to the refinement 
 of all the nations. This was probably also the 
 character of Pelops, whose shoulder was said to 
 have been devoured by Ceres, and from whom 
 the Grecian peninsula took its ancient name. In 
 modern Europe, the same character is perhaps 
 exemplified in the Prussians, as to the variety of 
 particulars included in one fashion. To the same 
 class may be referred the leopard, the most beau- 
 tiful of the tiger species. This is perhaps also the 
 true character of Ariadne, the companion of Bac- 
 chus ; because of the increasing range and exten- 
 sion of the retrogressive forms which are traced 
 by the power of Proserpina ; in which respect it 
 has a correspondence with the character of Bac- 
 chus. But, in another point of view, the succes- 
 sive falls in the breadth of the curve have relation 
 to the condensation of substance, and its power of 
 involution, which is expressed in the proboscis of 
 the elephant, as condensation is in the tusks ; 
 from whence perhaps the ivory shoulder of 
 Pelops. 
 
 In the Italic region the power of Proserpina 
 seems to have belonged to the Etrurians, a nation 
 
78 PELOPS OR PROSERPINA. 
 
 always distinguished for taste, and also for a cer- 
 tain characteristic firmness and purity in these 
 forms which they designed. The Etrurians are 
 said by Herodotus to have been originally a colony 
 from Lydia in Asia Minor. To the school of Pro- 
 serpina may evidently be referred Michael Angelo, 
 and Dante, the explorer of the infernal regions, 
 who frequently endeavours to express what is be- 
 yond the reach of the senses, as when, at the end 
 of his poem, he endeavours to describe the visible 
 appearance of the Trinity. 
 
 This talent is of a nature exclusive, and often 
 severe, but at the same time capable of uniting 
 beauty with a stiff and peculiar manner. It is 
 also compatible with recluseness ; because the re- 
 cluse may be as a system of fashion to himself, by 
 rejecting and excluding all that is not within a cer- 
 tain line which he has chosen. To this mode of 
 composition may perhaps be referred one of the 
 modern English poets, namely, Wordsworth, who 
 has the power of finding out much within any 
 given limits. To the same power may perhaps be 
 referred topography, which finds out all the re- 
 sources and localities contained within a given re- 
 gion. Thus, Dante investigated the different cir- 
 cles of the shades. Galileo, applying the tele- 
 scope to portions of the heavens, found in them 
 clusters of minute stars, which had not before 
 been discovered. The power of Proserpina, being 
 
PELOPS OR PROSERPINA. 79 
 
 confined to one curve, has more relation to the dis- 
 covery of different places than of different manners 
 and kinds of extension. Nevertheless, though, for 
 this reason, connected with stiffness and simplicity, 
 it has a strong relation to the discovery of form ; 
 which it augmented by the curve being capable of 
 a sort of alternation, which changes the situation 
 of particular parts, while its general course as an 
 involute is preserved. 
 
 Elis was also the place in which the muses were 
 met by Thamyris, who offered to contend with 
 them in song, and whom they overcame and de- 
 prived of sight, because it was previously agreed 
 that the party vanquished should be at the disposal 
 of the victor. Tiresias was another blind prophet, 
 celebrated for his wonderful knowledge, and for a 
 fabled change of sex. That Proserpina was con- 
 sidered by the ancients as one of the most refined 
 powers, is shewn by the remark of Vitruvius, who 
 says that, in building temples to Proserpina, Flo- 
 ra, or Venus, the Corinthian order should be em- 
 ployed as the most conformable to the delicate 
 and feminine nature of these powers. There was 
 also a restorative influence ascribed to the daughter 
 of Ceres, and one of her titles was Proserpina the 
 Salutary; perhaps from the power of order and 
 regular distribution of parts within a certain space, 
 which was before explained. 
 
 In regard to feeling, Proserpina is the same as the 
 
80 PELOPS OR PROSERPINA. 
 
 elementary power of Venus or acquired motion ; 
 and this is also the character of Eros or love, in his 
 most universal form. As in the physical world 
 the gravitating power of a body relates to all the 
 solids in the universe, and compares their respec- 
 tive influences ; so the power of Proserpina is the 
 same as the feeling of all things taken together ; 
 from whence the character of the counsellor, Nes- 
 tor of Pylos, who weighed different considerations 
 with each other. Among the modern European 
 poets this character seems to belong to Goethe, 
 whose poetry exemplifies a sublime enthusiasm 
 and sensibility in regard to the powers of the whole 
 universe modifying each other. This is the same 
 as the love of variety which liberates the mind, and 
 enables it to feel an unconfined enthusiasm, in 
 yielding to the powers of all external affections to- 
 gether. This is also the character of Hesiod, 
 whose theogony is imbued with a feeling of the 
 influential powers of the whole universe, and of 
 Love, the most beautiful of the divinities. Hence 
 the image of jealousy spoken of in the scripture ; 
 for this is the love of created being more than the 
 love of God. Thus at the Olympic games, the 
 ancients could scarcely perform religious rites with 
 sufficient devotion for thinking of the vast assembly 
 which was present. 
 
81 
 
 SECT. II. 
 
 ON A RETROGRESSIVE SERIES OF BRANCHES, OR 
 THE POWER OF SEARCH INTO ANTIQUITIES. 
 
 Pluto or Adonis. 
 
 The application of a series of similar angles to the 
 interior of a curve is capable of other powers bed- 
 sides that which belongs to Proserpina, and gives 
 birth to other modes of composition. Throughout 
 the whole series, the angle itself must be supposed 
 to remain the same, but the proportion of its two 
 sides being capable of being changed, the series of 
 angles may be employed to retrace different hy- 
 perbolic curves. If, in the series of lines dimi- 
 nishing according to continued proportion, the 
 later or shorter line be increased to nearer the 
 same length with the preceding, then the nature 
 of the curve which is traced will be changed, 
 and it will be transposed, in the retrograde direc- 
 tion, into a slower form of the hyperbola ; which, 
 after a certain number of angles, may again be 
 changed into a slower, by another addition to the 
 length of the later line. But the series of added 
 
 d2 
 
82 PLUTO OR ADONIS, 
 
 quantities must diminish in continued proportion, 
 to prevent the later line from even becoming equal 
 to the anterior ; for a series of additions, diminish- 
 ing in continued proportion with sufficient rapidi- 
 ty, will never increase the later line beyond a cer- 
 tain extent. If this precaution be observed, the 
 retrogressive series of branches will be infinite; 
 and, if to each branch the same number of angles 
 be allowed, the lengths of the branches will dimi- 
 nish in continued proportion, the whole series as- 
 suming the form of a general retrogressive curve, 
 having hyperbolic angles like those of the Satur- 
 nian hyperbola. But, by increasing the number 
 of angles allowed to a branch, the lengths of the 
 successive branches may be drawn out according 
 to various rules, or even with perfect freedom, 
 while their regular deduction as to quality is pre- 
 served. 
 
 This new mode of composition was probaby the 
 one which belonged to the character of Pluto, 
 the ruler of the shades, to whom also may be 
 ascribed the power of exploring internal posses- 
 sions. To this mode of composition may also be 
 referred the power of search into antiquities, be- 
 cause in exploring a retrogressive series it re- 
 mounts to those earlier and slower forms which 
 must have been anterior, according to that rule. 
 Thus Pluto, contemplating his possessions in the 
 infernal regions, was enabled to remount to the 
 
PLUTO OR ADONIS. 83 
 
 earliest generations which had been upon the 
 earth. This power is also capable of a sort of al- 
 ternation by each branch crossing itself, and 
 changing its direction towards the inner side of the 
 general series, on which principle the whole series is 
 enabled to preserve its general course as an invo- 
 lute. The first branch, by crossing, will place the 
 concave side outwards ; the next branch, by cross- 
 ing, will again turn out the convex side ; and so 
 on alternately. Thus Acheron, the river of pain, 
 so named perhaps from drawing out particular por- 
 tions of form, continued its descent through the 
 infernal regions. This mode of composition has 
 most relation to the elementary power of Mars, 
 because of its perfect freedom of changing its 
 course in every direction ; not only because of the 
 general curve being an involute having an infinite 
 series of branches, each of which has a direction of 
 its own, but also because of the power of draw- 
 ing out the length of any particular branch, by 
 which the position of the branch that is to follow 
 may be totally altered, and by which there is a 
 perfect freedom of giving it any direction. This 
 is like the power of communicating ductility to 
 forms ; and, in the celebrated hymn in honour of 
 Proserpine, which is said to have been sung by 
 Pindar appearing after his death, there was ap- 
 plied to Pluto the ephithet of K^vrnms, the gold- 
 en. To this class may be referred some of the 
 
84 PLUTO OR ADONIS. 
 
 greatest painters who have excelled in gracefulness, 
 like Raphael; because the source of freedom of 
 design is the power of shortening some parts of 
 forms while others are increased or drawn out. To 
 the same class may be referred Rubens, who did 
 not excel in delineating beauty, but who had the 
 same perfect freedom of changing the course of his 
 lines, and making the figures appear as if turning. 
 Another form of this power was Adonis, the be- 
 loved of Venus. 
 
 Among the ancient nations, the character of 
 the Arcadians seems to have been that which came 
 nearest to this mode of composition. Living in an 
 inland country, they passed their time in a state of 
 rural freedom, enjoying the pleasure of contem- 
 plating their internal possessions of flocks and 
 herds, and easily changing their intentions as they 
 wandered from mountain to valley. A troop of 
 Arcadians was present at the siege of Troy ; and 
 they are mentioned by Homer as skilful in war. 
 But, like Pluto, they preferred nothing to their 
 own region, and those inland solitudes, which were 
 called by the poets nigri colles Arcadia?, " the 
 dark hills of Arcadia." Pan also was said to fre- 
 quent them ; but this was not because the charac- 
 ter of the Arcadians coincided with his own power, 
 but only because of a certain similitude between 
 them ; as the power of Pan, which is straight lines 
 diverging from a centre, has an obvious relation 
 
PLUTO OR ADONIS. 85 
 
 to change of direction, and so far coincides Math 
 the power of Mars. The regions frequented by 
 the ancient divinities were generally not peculiar- 
 ly their own, but merely had some partial coinci- 
 dence with the power which belonged to them, and 
 therefore, were supposed to be agreeable places of 
 resort for them. Thus Bacchus frequented Rho- 
 dope, and the cold mountains of Thrace belonging 
 to Mars. The graces dwelt at Orchomenus in 
 Bceotia; and Sicily was called sacred to Proser- 
 pina, although it had more relation, perhaps, to 
 the power of Ceres. To Pluto may be ascribed 
 also the power of acceleration ; because the effect 
 of the addition made to the later line in the series 
 of proportionals, is to carry out the curve to a 
 greater extent or circuit than the same number of 
 angles would have produced if the former propor- 
 tion of the lines had been continued. For al- 
 though each successive branch, to which the same 
 number of angles is allotted, is shorter than that 
 which preceded it, nevertheless each new branch, 
 by altering the course of the curve into a wider or 
 more circuitous form, has a real power of making 
 the extent of the curve have an increasing propor- 
 tion to the same number of the angles which trace 
 it. Thus Acheron may be conceived as rolling 
 its floods with increasing vehemence, but, at the 
 time, with perfect freedom and fluidity of parts. 
 The characteristic of this power is internal profu- 
 
86 PLULO OR ADONIS. 
 
 sion and affluence. Thus Rubens drew a great 
 picture representing the fall of the damned. To 
 the same class with Pluto may perhaps also be re- 
 ferred Pythagoras, one of the greatest among the 
 ancient philosophers, and particularly curious about 
 the powers of finite numbers. This power, from 
 being like a retrogressive form of the Saturnian 
 curve, corresponds also with the power of gloomy 
 imagination searching into the depths, and was 
 shewn in the poetry of Milton, who chose Satan 
 for his hero ; for Mars must always be rebellious, 
 and wish to have the free prospect of the whole 
 universe to himself. This seems also to have been, 
 the character of the ancient Persians, who, like 
 Milton, delighted in the conception of an evil prin- 
 ciple opposed to the good. 
 
 Supposing the additions made to the later line 
 in the series were too great, or according to too 
 slow a series of proportionals, the length of the 
 later line might ultimately become equal to that of 
 the preceding one ; and, in this case, the curve, be- 
 ing no longer conducted by lines of unequal pro- 
 portion, would cease to bend as a retrogressive hy- 
 perbola, and would become part of a circle. This 
 was perhaps signified in Styx, which was consider- 
 ed as only an aestuary of Acheron. Styx was 
 sworn by, because the angles, if continued, will 
 complete and fulfil the circle, and, unless by a 
 farther addition, there is no escape from it. This 
 
PLUTO OR ADONIS. 87 
 
 name was also given to a lake in Arcadia. If the 
 lengths of the two equal lines were continually di- 
 minished by the same quantity being taken from 
 both, it would give a type of the wearisome and 
 stationary power of Orcus, and the series of angles 
 would continue tracing portions of circles always 
 smaller, and descending to infinitude. Pirithous, 
 who attempted to carry away Proserpina, and who 
 afterwards was bound in the infernal regions, re- 
 presented this power. When he asked what was it 
 o^clock, the answer was, eternity. This was also 
 the power of Circe, who had the power of chang- 
 ing the human species into brutes, which signifies 
 making their nature stationary and unprogressive* 
 
88 
 
 SECT. III. 
 
 ON THE EVOLUTION OF DIMINISHING HYPERBOLIC 
 BRANCHES. 
 
 Perseus. 
 
 If, in the series of angles which trace the curve, 
 the later line had at last become equal to the pre* 
 ceding one, then, if length still continued to be 
 added to the later line, it would become the long- 
 er of the two, and the series of angles would be- 
 gin to trace an evolving curve, which would also 
 be the hyperbola. And, if always, after a certain 
 number of angles, another addition were made to 
 the later line, but, according to the same series of 
 quantities, diminishing in continued proportion as 
 formerly ; in this case there would be produced a 
 succession of evolving branches, for ever expanding 
 and increasing in length. This power is derived 
 immediately from that of Pluto, and is produced 
 by the continuation of the same series of added 
 quantities diminishing in continued proportion. It 
 was signified in Demogorgon, of which the Arca- 
 dians were afraid to pronounce even the name. 
 
PERSEUS. 
 
 89 
 
 The chief representation of it, however, was in the 
 characters of the furies Megaera, Alecto, and Ti- 
 siphone, the daughters of Acheron and night ; and 
 also, perhaps, it belonged to Cocytus, the river 
 of lamentation, because of its evolving form ; for 
 it was understood that Acheron flowed into Co- 
 cytus. Thus the power of Pluto may come to an 
 end and be lost in that of Cocytus ; from whence 
 the ceremony of weeping for Adonis. 
 
 But as this series would have no limit to the 
 increase of its size, it becomes necessary to sup- 
 press and restrain the expanding power, by trans- 
 posing the angles which trace each successive branch 
 into a smaller scale, for instance a fourth. This 
 will make the branches continually diminish their 
 length, with immense rapidity, though they con- 
 tinue evolving, and pursuing the same deduction 
 as before. This mode of composition was proba- 
 bly signified in the character of Perseus, who slew 
 Medusa, and placed on his shield the gorgon's 
 visage, surrounded with snaky hair, and still re- 
 tailing sufficient terrific powers to convert the 
 spectators into stone. But, this mode of compo- 
 sition, by the successive changes of scale, passes 
 into infinite refinement, and acquires another pe- 
 culiarity, which is, that the branches, by their suc- 
 cessive diminutions, turn in, and change the course 
 of the whole series so fast that it becomes an invo- 
 lute, while all the particular branches continue 
 
90 PERSEUS. 
 
 evolving. This is like those powers of magic by 
 which rivers were made to flow back towards their 
 sources. The characteristic of this mode of com- 
 position is, that it is like a vast extent gathered up 
 within narrow limit s, because each branch in the 
 diminishing series represents one which would have 
 been found at an immense distance, and situated in 
 a totally different curve, if the original series had 
 been followed out, without transposition into a 
 smaller scale. But that order is reversed ; and in 
 proportion as the branches would have been far 
 off and of great extent, they are represented by 
 branches more minute and delicate, and farther 
 withdrawn into the bosom of the curve. The cha- 
 racter of the Spartans was that which came nearest 
 to this mode of composition, because they were 
 distinguished for brevity of expression, for theft 
 and concealment, and for great courage and power, 
 which seemed ridiculously to defy what they were 
 totally disproportioned to. Perseus was called the 
 grandson of an Argive prince, but had no relation 
 to the Argive region, and may be assumed as the 
 type of the Lacedaemonian character. This mode 
 of composition has most relation to the elementary 
 power of Mercury ; because it partakes of the nature 
 of a series of evolving hyperbolic branches, such 
 as exists in the Saturnian curve. One of the attri- 
 butes of Mercury was speed in going to distant 
 places, and the winged sandals were also ascribed 
 
PERSEUS. 91 
 
 to Perseus. Mercury was also the protecting god 
 of thieves. To this mode of composition may also 
 be referred the character of Somnus, the brother 
 of death, with the power of imagination; since 
 the curve of Perseus resembles the Saturnian hy- 
 perbola, to which imagination originally belongs. 
 The muses, in searching out the different forms of 
 the Saturnian hyperbola, virtually explore the dif- 
 ferent modes of imagining, and contrast them ; but 
 the power of Perseus belongs all to one curve, and 
 is like the infinitude of one vast and endless dream. 
 This character may be ascribed to the ancient 
 Danes, among whom the mythological poems of 
 the north were composed. This mode of compo- 
 sition may also be made to alternate, by each 
 branch crossing itself and changing its course to- 
 wards the inner side of the general series, by which 
 means the whole together will preserve its course 
 as an involute. The first branch, by crossing, 
 will place the concave side outwards; the next 
 branch, by crossing, will again turn out the con- 
 vex side, and so on alternately. This becomes the 
 same as the consciousness of sleep, or a sublime 
 feeling of mental illusion, which some of the eastern 
 sages endeavoured to cultivate. Mercury was called 
 the son of Maia ; and he may be conceived sliding 
 through the region of the air, as if lost in dream. 
 To this mode of composition belongs a bold and 
 oriental mode of imagining, and a style of expres- 
 
92 PERSEUS. 
 
 sion passing into hyperbola and metaphor; since 
 metaphor is the substitution of another form for 
 that which was originally meant. This power is 
 not so much allied to the feelings of modern 
 Europeans, such as they have hitherto existed. 
 Among the English poets, Coleridge seems to 
 come nearest to it In another point of view, this 
 mode of composition is the power of bringing near 
 that which is remote, as if the vision of Perseus 
 extended to the summit of Olympus, or to the re- 
 motest of the glittering tents and pavilions of genii 
 spread in the air, and not appearing till after sun- 
 set. The true sage sees things afar off; he seems 
 as if he could lean the arm of his understanding 
 on the horns of the moon ; he has nine hundred 
 and ninety-nine ears for every sort of instruction ; 
 the thousandth is reserved for sleep, and, in the 
 daytime, for the habitual songs of the grasshopper, 
 or the murmur of a neighbouring brook. The want 
 of self-constraint allows troubles to expand, and 
 leads to a state like the whirl of dancing santons ; 
 but surely the virtuous man enjoys peace ; and the 
 only things which he covets are those which belong 
 to time and distance. In the fine arts, that which 
 seems most connected with this mode of feeling is 
 relievo, in which the dimensions of solid figures 
 are made to approach to the nature of a plane ; 
 because this is like the power of contraction and 
 compression which belongs to the Lacedaemonian 
 
PERSEUS. 93 
 
 character. If there were Spartan artists in ancient 
 times, it is probable that they would have a pecu- 
 liar talent for works in relievo. The nature of re- 
 lievo, however, would be coarse, and would have 
 nothing sufficiently peculiar to itself if the figures 
 were too prominent ; and its greatest delicacy is 
 when the figures seem almost to sink and vanish 
 in the plane; for, since its origin is from solid 
 figures, it is evident that the boldest relievo must 
 be, in reality, that which makes them approach 
 nearest to a plane, where they appear as if about 
 to vanish. This mode of composition was also the 
 peculiarity of Horace, in whose odes is shewn the 
 power of contraction bringing together materials 
 from great distances animoque rotundum percur- 
 risse polum, geographical references, which often 
 compress a vast extent of dominions within the 
 compass of a single ode. Jerusalem was built in 
 the region which belonged to the tribe of Benja- 
 min ; and, perhaps, the Jewish style of architec- 
 ture may have had some relation to this mode of 
 composition, in which the proportions are not of a 
 direct and simple meaning, but in proportion to 
 their decrease expressive of something farther off, 
 and much greater in size. Such was Nemesis, 
 who reversed the fortunes of the great and proud. 
 This was perhaps also the character of the ancient 
 Chaldeans, who delighted in observing the distant 
 phenomena of the heavens. 
 
94 
 
 SECT, IV. 
 
 ON THE RETROGRESSION OF THE HYPERBOLA 
 THROUGH A SERIES OF INCONSECUTIVE FORMS. 
 
 Vertumnus or the Fates. 
 
 But, differing from all these powers, and existing 
 entirely apart from them, a new retrogressive series 
 will be found, if in the succession of angles as they 
 existed at first in the power of Proserpina, the 
 later line instead of being added to, as in the power 
 of Pluto, at each new branch, be, on the contrary, 
 diminished at the beginning of each new branch, 
 according to a series of quantities diminishing in 
 continued proportion, so as never to exhaust the 
 extent of the later line. In this case, the general 
 series, instead of passing always into slower forms 
 of the hyperbola, will make its transitions into 
 more rapid forms. These exist together in an 
 order which is the reverse of that which belongs 
 to the original form of the Saturnian hyperbola, to 
 which the curve of Pluto has a certain resem- 
 blance ; because the curve of Pluto coincides with 
 the natural order or derivation of branches, each 
 
VERTUMNUS OR THE FATES. 95 
 
 leading to a slower form which had preceded it. 
 But, when the later line, instead of being added 
 to, is diminished, then the points at which the 
 branches are joined together must all project to- 
 wards the convexity of the curves ; and each 
 branch becomes like a terminated form, not natu- 
 rally leading either into that which precedes, or that 
 which follows, so that it is situated as if it did not 
 belong to any general curve, but was only contiguous 
 to those forms beside it. This mode of composition 
 appears to have been signified in the characters of 
 the Fates, the daughters of Erebus and night. 
 Their business was to determine the limits of time, 
 fortune, or any thing else, to individuals ; and this 
 corresponds with the power of searching out par- 
 ticular modes of extension, defined and separated, 
 as it were, from those which precede and follow 
 them. To Clotho, Lachesis, and Atropos, may be 
 ascribed also the power of retardation ; because, 
 in each new branch, the same number of angles 
 will trace a form of smaller extent or circuit than 
 would have been found if the same proportion of 
 the lines had been continued. Therefore, in the 
 succession of branches, the extent or circuit of the 
 curve, has a decreasing proportion of the same 
 number of angles; and this is the same as the 
 power of retardation. This power must be sup- 
 posed to alternate in a manner peculiar to itself; 
 each branch crossing itself once, and terminating 
 
 
96 VERTUMNUS OR THE FATES. 
 
 in a form which is crossed by the succeeding branch 
 springing from it. Thus Lethe, the river of obli- 
 vion, passing through the deepest shades, soothed 
 them with the changing current of its waters, which 
 appearing to stop in successive eddies, always flow- 
 ed on in a new direction. This mode of compo- 
 sition has most relation to the elementary power of 
 Vulcan or definition, because of its relation to ter- 
 minated forms. Of the character of the ancient 
 Argives nothing is now known but that they were 
 considered as the favourite people of Juno. To 
 the Fates may be ascribed the sense of contrariety 
 in those things which are united at a common 
 boundary. 
 
 One of the greatest poets of antiquity is said to 
 have been Palamedes the Argive, who was co- 
 temporaneous with the Trojan war, and from whom 
 Homer is supposed to have borrowed many things. 
 But to the same class with the Fates may be re- 
 ferred Pindar, whose poetry, besides celebrating 
 the good fortune of individuals, relates mostly to 
 the peculiarities of different countries, divinities, or 
 races of mankind, and corresponds with the power 
 of defining, or of specifying, what is peculiar to 
 each, and what each obtains. The Fates had the 
 task of ascertaining what lots or conditions were to 
 fall to individuals, and Vulcan may be conceived 
 as intellectually discriminating the natures of ab- 
 stract powers. To the Fates may also be symbo- 
 
VERTUMNUS OR THE FATES. 97 
 
 lically ascribed the parrot as a type of articulation ; 
 because each vocal power, in the structure of words, 
 is like an abstract rule which may be prolonged to 
 any extent, before the vowel is closed or shut in ; 
 but the knitting of these different vocal powers to- 
 gether is articulation, and corresponds with Vul- 
 can, or the conjunction of different abstract rules in 
 that which is one ; for it is essential to articulation 
 that all the component vocal powers be limited. 
 Among the Asiatic nations, the mode of composi- 
 tion belonging to the Fates seems to have been 
 exemplified in the Phoenicians, who, like the Ar- 
 gives, were a mercantile people, dwelling upon the 
 sea coast, and skilled in maritime affairs. Thales 
 is said to have been born at Miletus, but his pro- 
 genitors were Phoenicians, and he was understood 
 to belong to that nation. In modern times, the 
 same character seems to have belonged to the Ve- 
 netians, also a mercantile people, dwelling on the 
 border of the sea ; and the peculiar talent shewn 
 by their painters, who excelled in colouring, also 
 corresponds with the power of Vulcan, because the 
 play of each simple colour is one abstract rule, and 
 the art of blending colours is the same as the con- 
 junction of different rules in that which is one, or 
 the conjunction of different straight lines in one 
 triangle or polygon. To the Fates may also be 
 ascribed the power of observing the change of 
 times, and of marking the boundaries of periods 
 
98 VERTUMNUS OR THE FATES. 
 
 differing in kind ; and this was perhaps signified 
 in the character of Vertumnus, the god of the sea- 
 sons. To this class may also be referred the osten- 
 tatious peacock, the favourite bird of Juno, assign- 
 ed to her by hypothesis or imputation. The Ve- 
 netians were characterized by a taste for buffoonery ; 
 and this also belongs to the Fates, whose portion 
 is to know the definitions and properties of all the 
 passions, and their respective tendencies ; and this 
 is the source of pantomime, which, in a philoso- 
 phical point of view, is the source of the most re- 
 fined intellectual pleasure. But mankind are gene- 
 rally too confined in their views, and too much un- 
 der the influence of particular affections, to perceive 
 the advantage of any such general taste, and they 
 most frequently are inclined to regard buffoonery 
 as an insult to the particular feelings which rule 
 in their minds for the time, or which habitually 
 belong to their characters as individuals. This, 
 however, shews the excellence of the powers of de- 
 finition as a source of reproof to the ignorant and 
 selfish, and as a commemoration of the magnificent 
 variety of powers which exist in the universe, and 
 which, in ancient times, was exemplified in the as- 
 semblage of the different nations at the Olympic 
 games. 
 
 Of those four powers which are produced from 
 the application of the power of Diana, or angle, to 
 the hyperbola, the Fates, being disjoined and apart 
 
VERTUMNUS OR THE FATES. 99 
 
 from the rest, should be placed first, then the mode 
 of composition which belongs to Proserpina, then 
 Pluto, and last the power of Perseus, in which the 
 particular branches all resume the evolving form 
 originally belonging to the hyperbola. 
 
100 
 
 CHAP. V. 
 
 ON THE INTERMINABLE FORMS OF TRANSVERSE 
 PROGRESSION. 
 
 SECT. I. 
 
 ON THE POWERS OF CONTINUITY AND SYNTAX. 
 
 Erichthonius or Hyperion. 
 
 In the foregoing modes of composition, a series of 
 angles having been already applied, the applica- 
 tion of the next power, Vulcan, to the curves, 
 which are guided by angles, will only have the ef- 
 fect of making the branches cross, and produce a 
 new form of the transverse mode of progression, 
 which belongs to Bacchus. The power of Vulcan 
 may be applied to such a form as that of Perseus * 
 and each branch may cross by one of the straight 
 lines in the series of guiding angles being caused 
 to turn back, and make the complement of the same 
 angle on the other side of the preceding straight 
 line. This change of course in the middle of 
 
ERICHTHONIUS OR HYPERION. 101 
 
 each branch would make the whole series quit the 
 form of the general curve of Perseus, and stretch 
 forward by alternating, as in the mode of progres- 
 sion which belongs to Bacchus. But, in this case, 
 there being no addition to the thickness of the suc- 
 cessive branches, the series would never be stopped, 
 but would extend to infinitude, and is the same 
 as interminable continuity. 
 
 This mode of composition has most relation to 
 the elementary power of Apollo ; and it was per- 
 haps signified in the character of Hyperion, the ce- 
 lestial charioteer, and also in Erichthonius, the in- 
 ventor of the chariot ; which is the same as draw- 
 ing or the continuity of a line, in which the ad- 
 vancing point retains its connexion with the ex- 
 tension thrown behind. The mode of composition 
 nearest to this is a branch returning across itself 
 in the direction contrary to its own previous ex- 
 tension, and so blending contrary tendencies into 
 one form, which appears in the intersections of the 
 parallel lines, as in the mode of composition which 
 belongs to Bacchus ; but the power of Bacchus 
 consists always of finite series of branches ; while 
 this mode of composition which belongs to Erich- 
 thonius consists of one endless series, and therefore 
 corresponds with the infinitude of continuity. In 
 a straight line, any two points are contraries, but 
 they are blended into one by a perfect intermediate 
 continuity of extension, in the same manner as 
 
102 ERICHTHONIUS OR HYPERION. 
 
 the horses are bound to a chariot by the traces ; 
 and thus Erichthonius continues for ever uniting 
 contraries in the form of each successive branch 
 which crosses. 
 
 In this mode of composition, each successive 
 branch, as in the form of Perseus, may be trans- 
 posed into a smaller scale, to prevent the excessive 
 expansion of the form, which would otherwise take 
 place as formerly in the Gorgons ; and which was 
 probably signified in the giant Typhon, a furious 
 whirlwind, whose powers at first terrified and dis- 
 concerted all the gods. Or thus the inconsiderate 
 Phaethon, having obtained from Phoebus the boon 
 of being permitted to drive his chariot, could not 
 sufficiently restrain or control the horses, and was 
 struck from his car, to prevent the conflagration 
 of the heavens. But by the successive diminutions 
 of scale, this form contracts and p asse into infinite 
 refinement, so that the added branches, becoming 
 always less, scarcely make any visible addition to the 
 extent of the whole form. This mode of composition 
 being the same as the elementary power of Apollo, 
 partakes of the nature of a list or catalogue of par- 
 ticulars, and becomes the same as genealogy and 
 the power of determining the order of particulars 
 as to priority in time or place. To Erichthonius 
 therefore may be ascribed historical narration; 
 and this character seems to have belonged, among 
 the ancient nations, to the Egyptians, who were 
 
ERICHTHONIUS OR HYPERION. 103 
 
 remarkable for retaining among them the memory 
 of the past, and for continuing the same manners 
 and opinions which they had derived from the re- 
 motest antiquity. The Egyptians also excelled in 
 construction, which is the art of binding together 
 different particulars in the same continuity ; and, 
 therefore, construction, and the power of following 
 out a method which has been begun, are always 
 found together ; and to this class may be referred 
 those mathematicians who study the generation of 
 any simple curve as a list or genealogy of points 
 bound together in one continuity, and having their 
 places according to a certain rule. Every abstract 
 rule changes and passes through successive states 
 as it is continued, and, therefore, Erichthonius is 
 also the power of metamorphosis^ or the interme- 
 diate connexion between a state or condition which 
 has been left and another which has been passed 
 into. Thus Ovid, at the beginning of his poem, 
 asks the gods to enable him to deduce a perpetual 
 song from the beginning of the creation. In the 
 Scripture, the power of genealogy and historical 
 narration is shewn in Moses, who belonged to the 
 tribe of Levi, and who was lifted from the waters 
 of the Nile. To Erichthonius must also belong 
 the syntax or construction of words in language. 
 To the same class with the Egyptians may per- 
 haps be referred the European Thebans, deriving 
 
104 ERICHTHONIUS OR HYPERION. 
 
 their origin from Cadmus and Hermione, who were 
 changed into dragons^ that is, powers of continuity. 
 But both Bacchus and Hercules were born in 
 Thebes, from its relation to the transverse mode 
 of progression. Pindar and Hesiod, although 
 born in Bceotia, were both foreign to the Theban 
 character, which rather belonged to Herodotus. 
 The proverbial dulness of the Boeotians may have 
 arisen from too much tenacious continuity. To 
 this class may be referred the duck, the goose, and 
 the swan, which are similar in their character to 
 the protracted dragons of Bceotia ; and the swan 
 was sacred to Apollo. But the same power was 
 perhaps also exemplified in the Rhodians, who 
 seem to have worshipped Apollo as a sort of ever- 
 lasting Bacchus. Aristophanes, a native of Rhodes, 
 was an instance of that power of ridicule which de- 
 pends upon the art of forced construction, or the 
 union of contrary tendencies. Thus the horses of 
 Erichthonius, in first getting into motion, seem as 
 if they were going to leave the chariot behind 
 them, but it is compelled to follow. To this power 
 may also be ascribed sophistry and paradox, as ex- 
 emplified in Hume, w r ho was also an historian, and 
 Rousseau, one of whose favourite studies was edu- 
 cation, in which the mind is carried on through 
 successive states or conditions ; and perhaps the 
 character of the Swiss, in general, may be founded 
 
ERICHTHONIUS OR HYPERION. 105 
 
 on the same power of sustained continuity which 
 belonged to the ancient Egyptians. Regularity of 
 habits made the Swiss nation much esteemed as 
 servants throughout Europe. This is also the 
 character of Ganymede, the cupbearer of Jove ; but 
 Hebe was a form of the limited series which be- 
 longs to Bacchus ; and, in reference to its termi- 
 nation, she is said, in bearing the goblet, to have 
 stumbled before the assembled divinities, and dis- 
 pleased them ; after which Ganymede was made 
 cupbearer in her stead. In Asia, the same char- 
 acter seems to belong to the Chinese, who are 
 equally tenacious of habit, while the progress of 
 time carries it always into greater refinement. The 
 Theban Niobe, boasting that she equalled Latona 
 in beauty, had her children shot by Apollo and 
 Diana; and, in long weeping and refining upon 
 her grief, was changed into a stone. To this 
 power may also be ascribed the power of refining 
 the quality of one substance or material, as exem- 
 plified in the Chinese pottery, and the colours 
 given to it ; for these shew the art of refining 
 single hues, which are as single rules or powers of 
 continuity. And Titian, in painting, may be 
 classed rather as a Swiss than as a Venetian, since 
 it is the opinion of the skilful, that his chief ex- 
 cellence is in the clearness and strength of single 
 hues, such as that of the flesh. Among the 
 
 e 2 
 
106 ERICHTHONIUS OR HYPERION. 
 
 apostles, the power of Erich thonius seems to have 
 belonged to St Matthew, who is the chief of the 
 Evangelists as to history ; and who was of the 
 tribe of Levi, and called from sitting at the receipt 
 of the custom. But the chief characteristic of this 
 mode of composition is self-will and reliance on the 
 power of habit ; from whence its relation to edu- 
 cation. 
 
107 
 
 SECT. II. 
 
 ON THE DEDUCTION OF COMPOSITE HYPERBOLIC 
 BRANCHES. 
 
 Theseus. 
 
 The application of the power of Vulcan also pro- 
 duces another kind of infinite series, which is part- 
 ly similar to that of Erichthonius, but essentially- 
 different in power. For if the crossing, instead 
 of taking place in the middle of each branch, be 
 employed always at the point where the last 
 branch terminates and the new one begins, in this 
 case the transverse form of progression will still 
 be continued ; but each hyperbolic form will in- 
 tersect the parallel courses of the last branch. 
 Therefore, in this deduction, the parallel courses 
 of each branch, instead of being crossed, as former- 
 ly, by a farther part of the same curve, will be 
 measured against a new form, entirely separate 
 and distinct. This mode of composition is the 
 same as the power of transition into forms belong- 
 ing to a different rule or generation ; because the 
 forms which cross are not like those in the power 
 
108 THESEUS. 
 
 of Bacchus or Erichthonius, merely different parts 
 of the same curve or rule ; but, on the contrary, 
 they exemplify the power of arbitrary composition, 
 in bringing together forms which are not continua- 
 tions of the same rule. This mode of composi- 
 tion, therefore, has most relation to the elementary 
 power of Minerva, or arbitrary composition ; and 
 it was probably signified in Theseus, the hero of 
 Athens. The characteristic of this mode of com- 
 position is the same as intellectual freedom, and 
 agility in escaping from deduction, or from the 
 remainder of what belongs to the same rule. 
 Therefore it becomes as wit, which disconcerts 
 the powers of tediousness, and will not long bear 
 with sameness of method. But, in regard to the 
 blending of powers foreign to each other, one 
 characteristic of the Athenians was their love of 
 strangers, and the encouragement which foreigners 
 had to come among them. The number of famous 
 characters really born in Athens was small in 
 comparison with the number of those who only 
 resided there. Theseus himself was born in the 
 Peloponnesus. Solon was a native of Salamis ; 
 Diogenes was a native of Sinope ; Xeno came 
 from Cyprus ; Aristophanes from Rhodes ; Aris- 
 totle was a Macedonian ; Theophrastus a Lesbian; 
 Chrysippus was born in Cilicia ; and to this list 
 might be added many more. These persons were 
 cherished by the Athenians, as if for their own 
 
THESEUS. 109 
 
 taste in having among them forms entirely remote* 
 and incapable of having been derived from the 
 same source. Among the modern nations of 
 Europe, the French approach most to the charac- 
 teristics of this mode of composition, and shew 
 the same talent for considering the temporary 
 union of forms with others, not having a common 
 derivation. This is the power of analysis, as 
 shewn in Samuel Johnson, Jeffrey, and other 
 modern critics. To Theseus may also be ascribed 
 the love of transitions, which, by this mode of 
 composition, are found without number, in a de- 
 duction which extends to infinitude. Thus, by 
 a stroke of the hammer of Vulcan, there was 
 brought forth the immortal or interminable 
 Athana, clad in armour, and also bearing her 
 shield as a superabduction. The name Pallas, 
 signifying vibrating, may apply to the blending 
 of two different motions or rules. To this mode 
 of composition may also be referred doubt or in- 
 credulity, because of the difference of the forms 
 which are blended. These do not belong to the 
 same rule, or spring from the same source ; and, 
 therefore, their union is the same as the feeling of 
 ambiguity, as shewn in the character of St 
 Thomas, whose name signifies Geminus, double. 
 Scepticism, according to some, was the distin- 
 guishing characteristic of the atomical Democritus. 
 A sense of the presence of different elements in 
 
110 THESEUS. 
 
 that which appears as one leads to analysis, or the 
 inclination to separate and discriminate them ; 
 from whence the intellectual character of Pallas, 
 whose wisdom was the same as the power of re- 
 solution, and corresponds with the science of che- 
 mistry. Her favourite bird the owl assumed its 
 position in some ivied niche or dilapidated win- 
 dow of an uninhabited tower, and, during the 
 silence of night, seemed tranquilly to enjoy the 
 pleasures of speculation or scepticism. Its eyes 
 were directed to the ground, where, by the moon- 
 light, it distinctly perceived every blade of grass 
 that entered into the composition of the matted 
 sod, or received a foreign lustre from being gar- 
 nished with a drop of dew. Perhaps the fox cau- 
 tiously glided past, as if trusting that the sound 
 of his steps could not be distinguished or separ- 
 ated from the murmur of a distant torrent which, 
 being lost among chasms of rocks, seemed unwill- 
 ing to announce that it retained any activity dur- 
 ing the hours of repose, and rather moderated and 
 restrained its voice, as if in soothing complaisance 
 to the rest of the world. But, in the meantime, 
 the favourite of Pallas remained awake and atten- 
 tive to the sounds that were stirring, never failing 
 to solve and discriminate those which occurred to- 
 gether. The last result of analysis is the sense 
 of vacancy, negation, or silence, after all the ele- 
 ments in a compound have been exhausted or 
 
THESEUS. Ill 
 
 taken away. In reference to this, the Athenians 
 stamped upon their coin the figure of an empty 
 jar, lying upon its side, to shew that the water 
 which it had contained was all withdrawn. The 
 reason is, that a composition has no element which 
 is peculiar to itself, or which remains and consti- 
 tutes its essence ; and, therefore, when the ingre- 
 dients are separated, the result is negation, or the 
 composition ceases to exist. This sublime feeling 
 of silence and negation belongs peculiarly to Mi- 
 nerva or Theseus, and is connected with scepti- 
 cism, to which there belongs a refined intellectual 
 pleasure. The application of synthetical power to 
 the fine arts produces a firm style, in which the 
 component elements are not concealed or disguised, 
 but allowed to appear, as in antique statues, of 
 which the lineaments are not soft and natural, but 
 rather composed of parts exemplifying different 
 rules, like separate mathematical forms. Hence 
 the style which belongs to the power of Minerva 
 is founded on the beauty of such composition as is 
 visible and intelligible. This was probably the 
 character of Apelles, whose style in painting 
 pleased the ancients most. The same is the case 
 in architecture, in which the forms united are 
 foreign to each other, and evidently generated 
 from different rules. To the same class with 
 Theseus may be referred the Oreades, or moun- 
 tain-nymphs ; because every hill is a form acci- 
 
112 THESEUS. 
 
 dental or extraneous to the general level of the 
 region in which it is situated ; nor is the mountain 
 "a continuation of that form which belongs to the 
 neighbouring plains. It may be doubted whether 
 the true Athenian character belonged to Plato ; 
 for his compositions, instead of exemplifying ra- 
 pidity, fire, and freedom of transition, are for the 
 most part excessively protracted and tedious, and 
 seem rather to belong to the power of Erich- 
 thonius, or continuity, which is more allied to long 
 arguments. Socrates was accustomed to fasten on 
 some proposition acknowledged by the person with 
 whom he was conversing ; but Socrates seems to 
 have been a form of Pan or repetition. Nor 
 should this excite astonishment, for it is probable 
 that even the universal Shakspeare, distinguished 
 for his knowledge of the more vulgar constitution 
 of human nature as fixed and unprogressive, was 
 no other than a form of the universal Pan; to 
 whom belongs rotatory motion, and the repetition 
 of cycles. 
 
 From continuity like this the power of Theseus 
 must always seek to get free ; because the charac- 
 teristic of Pallas is the arbitrary union of forms 
 derived from totally different sources, and only 
 allied by accidental situation, in which they still 
 remain foreign to each other. This talent was 
 shewn in Voltaire, whose agility was like that of 
 Myrinna, whose tomb was an hill celebrated by 
 
THESEUS. 113 
 
 Homer as the place where the Trojans drew out 
 their troops. The characteristic of Voltaire was 
 freedom of transition. Thus, he describes a per- 
 son travelling from world to world through the 
 aerial medium, and struck with astonishment at 
 the dissimilarity of the modes of existence which 
 he is thus enabled to compare and measure against 
 each other. He receives from a learned man a 
 book which is to teach him the last results of all 
 inquiry, and when opened it is found to be an 
 universal blank ; which is like the negation found 
 by analysis, or by successively taking away the 
 constituent parts of any composition. The power 
 of analysis also gives birth to clearness and per- 
 spicuity of arrangement. This characteristic was 
 shewn in music by Haydn, whose mode of com- 
 bining is such as to exhibit the elements of musical 
 form in the utmost clearness, but in an order 
 which does not sooth or flatter the ear by much 
 continuity. The music of the ancient Athenians 
 must have had most resemblance to that of Haydn, 
 which is the result of analysis, and presents the 
 elements of musical form, as it were, perspicuous- 
 ly and apart. To the power of Minerva or The- 
 seus there may also be ascribed, as a type of agili- 
 ty, the monkey, which, in its natural state, lives 
 among the branches of trees, and leaps from one 
 to another so easily, that it is capable of passing 
 through a whole forest without ever touching the 
 
114 THESEUS. 
 
 ground. The reverse of this was shewn in An- 
 taeus, who gathered new strength every time he 
 touched the earth, and who, like David Hume, 
 seems to have been a form of Erichthonius or con- 
 tinuitv. 
 
 In this mode of composition, as in the preced- 
 ing, the angles which guide each successive branch 
 must be supposed to be produced on a smaller 
 scale to prevent the expansion of the form, which 
 would otherwise become unlimited. The fall or 
 contraction of breadth must be supposed to take 
 place in the middle of each branch, and the same 
 breadth to be continued to the middle of the next 
 branch ; so that there will be no change of breadth 
 at the intersection of the two branches. The un- 
 limited expansion of this form was perhaps signi- 
 fied in the character of Orestes, who, having put 
 to death his mother, was thenceforth deprived of 
 repose. His situation became a favourite subject 
 for tragedy, and the unhappy criminal, flying 
 across the stage, was seen pursued by the furies 
 with their scourges of snakes. When continued 
 extension passes into another rule, it in a manner 
 destroys or terminates the rule or mode of exten- 
 sion which precedes it ; and the ancients seem to 
 have thought that such a transition resembles 
 parricide, because, in theory, it makes an end of 
 the previous generation ; but Solon made no law 
 against parricide, he said because he disbelieved 
 
THESEUS. 
 
 115 
 
 in that crime. But, by the transitions of the 
 branches into smaller scales, this form passes into 
 infinite refinement, and the successive branches, 
 which increase its extension, contract and diminish 
 so fast, that the whole deduction scarcely appears 
 to advance ; while, at the same time, it continues 
 to trace out an interminable series of composite 
 forms ; and Pallas, like Perseus, wears the head 
 of the Gorgon upon her shield. Theseus is the 
 same as the Ephraim of the Jews, and Hercules is 
 Judah. 
 
 To this mode of composition must be referred 
 the philosophy of Locke and others, as to the as- 
 sociation of ideas ; which is similar to the ancient 
 Epicurean philosophy, both as to feeling and as to 
 theory ; in the first place, as to feeling, because 
 the sense of enjoyment which it cultivates relates 
 to those things which are agreeably assembled for 
 the present ; and it relates neither to the past nor 
 the future, since it is not the prolongation of any 
 one rule or continuous form, but is like the sense 
 of ingredients well blended, and producing the 
 state or condition of the mind for the time being. 
 In theory, the association of ideas, according to 
 Locke, and the concourse of atoms of different 
 kinds and figures, according to the method of Epi- 
 curus, evidently belong both to composition. The 
 discriminating and separating of these according 
 to their supposed natures is a sort of analysis. 
 
116 THESEUS. 
 
 But the power of analysis leads always to that 
 sense of vacuity or negation which Locke sup^ 
 posed to exist in the mind previous to external 
 impressions. But Theseus, after having found a 
 silence, or vacuity, again resorts to composition, 
 and reassembles the scattered elements. To The- 
 seus may also be ascribed the assemblage of mu- 
 tual auxiliaries, a characteristic which appeared in 
 all the achievements of him as the power of syn- 
 thesis. It is perhaps also the Chimaera, an union 
 of forms without real continuity, but breathing 
 fire, because the transverse mode of progression 
 has a certain relation to the power of Vulcan. 
 This power must always want the freshness of 
 nature and life ; but, at the same time, it has ad- 
 vantages which distinguish it from all the other 
 powers. Therefore Horace, after Jove, ranks 
 Pallas next, Proocimos illi tamen occupavit Pallas 
 honores ; and, in Greece, the Athenians were con- 
 spicuous beyond the other nations. 
 
117 
 
 CHAP. VL 
 
 ON THE PROGRESSIONS OF DOUBLE FORMS. 
 SECT. I. 
 
 ON THE POWERS OF FLUCTUATION. 
 
 Pollux or Eridanus. 
 
 The next elementary power to be applied is that 
 of Neptune, which has two angles, or two planes 
 extending at the same time. This power will 
 have the effect of resolving such a curve as that 
 of Perseus into two separate parts. One of the 
 curves must be the ruling one, and must be 
 supposed to be guided by a certain series of 
 angles, such as that of Perseus ; and if, on the 
 outside, there be made to project from the point 
 of each angle a straight line equal to that which 
 is the shorter side of the angle, and if the tops of 
 these projecting lines be connected by straight 
 lines, there will be produced another and outer 
 series of angles, running collateral to those of the 
 ruling curve, and capable of tracing successive 
 branches opposite to those in the ruling curve, 
 
118 POLLUX OR ERIDANUS. 
 
 Now, since, in this figure, quadrangles are built 
 on each line which guides the ruling curve, and 
 these quadrangles are all similar figures, therefore 
 the uppermost sides of the successive quadrangles 
 are in the same proportion to each other, as the 
 successive lines which constitute their bases, and 
 which also constitute the series of angles guiding 
 the ruling curve, which is the inner. And, from 
 the structure of the figure, it also follows that, in 
 the outer series, the lines are always joined to- 
 gether at the same angle as in the inner series. 
 Therefore the outer curve consists of successive 
 branches, each of which is, in reality, the same 
 form as the inner branch opposite to it, but pro- 
 duced on a larger scale, and diverging from the 
 inner branch. 
 
 But, in this mode of composition, the ruling 
 curve should not be indebted for its form to the 
 same power as Perseus, but may take a new form 
 peculiar to itself; and this is by at each branch 
 shortening the later straight line by a quantity 
 less in continued proportion, so as to leave the 
 later line always greater than the preceding. In 
 this figure the successive branches are e volutes, 
 but become always slow r er in quality, while, in 
 the form of Perseus, they become always more 
 rapid. The mode of composition now described 
 corresponds w r ith the elementary power of Juno ; 
 because, in the two collateral series, the very 
 
POLLUX OR ERIDANUS. 119 
 
 same form is always found opposite, and in a dif- 
 ferent place. Between the two forms relation 
 passes to and fro, and like the waters of a river 
 between its banks. This is the mythological 
 character of Pollux, the god of pugilism, which 
 art depends upon the opposition of forms, part for 
 part. If the two series of forms were continued 
 always on the same scale, they would continually 
 diverge ; but, to prevent the excessive expansion 
 of the form, it is necessary that each successive 
 pair of branches be transposed into a smaller scale, 
 for instance a sixth ; and this will greatly change 
 the course of the outer series which is diverging ; 
 because, the inner series being taken as a guide 
 for the other, it is evident that, after each trans- 
 position into a smaller scale, the top of the last 
 projecting line (which is on a larger scale) must 
 be connected with the top of the next (which is 
 on a smaller scale), by a line which makes a great 
 and sudden descent towards the inner curve, carry- 
 ing down the outer curve along with it. After 
 this descent the outer curve will be again pro- 
 duced, and will diverge as before But, by means 
 of the diminutions of scale, the general course 
 of the two series will become that of involutes 
 passing into infinite refinement. This mode of 
 composition is also the mythological character of 
 Eridanus or the Po, the image of which is also 
 seen flowing among the stars, because it drowned 
 
120 POLLUX OR ERIDAXUS- 
 
 the burning rays of Phaethon after he was pre- 
 cipitated from the chariot of the sun. 
 
 To this mode of composition may also be re- 
 ferred pilgrimage, as the fluctuation or change of 
 that which passes into another place, like the waves 
 of a river. Among the Homeric heroes, this 
 power belonged to Ulysses, who visited every 
 different shore, and even passed to Cimmeria to 
 have conferences with the shades of the dead. 
 The crafty wisdom attributed to Ulysses may 
 refer to his shewing only one side of his thoughts 
 or intentions. But, in relation to novelty of cir- 
 cumstances, or the fluctuation which arises from 
 the change of distances, the characteristics of this 
 power are likewise exemplified in Bunyan^s alle- 
 gory of the pilgrim's progress. Pollux was always 
 represented having over his head a star, which 
 must have been a planet or wanderer. To this 
 class must also be referred the mythological char- 
 acter of Atlas, king of Mauritania, who was feign- 
 ed to support the heavens on his shoulders, be- 
 cause Uranus or Coelus is the same as relation, 
 or the power of Juno, which belongs to this mode 
 of composition. Atlas signifies unshaken, but he 
 was probably named by antiphrasis ; and his 
 character must have referred to the powers of 
 revolution and change in the heavens, by which 
 new conjunctions are brought about, because all 
 novelty of external circumstances comes originally 
 
FOLLUX OR ERIDANUS. 121 
 
 from motion. To Pollux and Atlas may there- 
 fore be ascribed agitation and change, and the 
 celestial sign of the balance, which shews the 
 effect of the same weight or power in a different 
 place. This was probably the character of the 
 ancient Carthaginians ; from a colony of whom 
 the Irish nation is supposed to be partly descend- 
 ed ; and to the same class may be referred Hymen, 
 the god of marriage, because of his relation to the 
 power of Juno ; for his torch shines with an agi- 
 tated lustre on the affairs of mankind, and has an 
 influence like that of the r>lanets over their for- 
 
 JL 
 
 tunes. In music, this power seems to have be- 
 longed to Handel, whose compositions have a 
 kind of motion and tumult which cannot be imi- 
 tated by method ; and, in the fine arts, this power 
 is characterized by the expression of fluctuation, 
 or by that kind of felicity which appears to result 
 from chance. In modern times, it has been shewn 
 in the poetry of Sir Walter Scott, which often re- 
 lates to peregrination and encounter, or to the 
 change of circumstances which is produced by the 
 powers of motion going on in the world ; or some- 
 times even to the influences of the planets. The 
 art of expressing fluctuation in the structure of 
 verse belonged, among the ancients, to Catullus, 
 the author of the Epithalamium of Peleus and 
 Thetis ; for his compositions seem to undulate, 
 and he was characterized by the love of peregrina- 
 
122 POLLUX OR ERIDANUS. 
 
 tion, and speaks of his expected pleasure in visit- 
 ing the illustrious cities of Asia : Jam mens prce- 
 trepidans avet vagari. His native city gave birth 
 also to Paul Veronese, who, in his paintings, as- 
 sembled the figures and dresses of strangers of 
 all nations who came to Venice, and in this man- 
 ner allied his productions to the agitation of exter- 
 nal occurrences. It was said of Hannibal, the 
 Carthaginian leader, that he generally did not fol- 
 low out the advantages of the victory he gained ; 
 and this trait corresponds with the want of in- 
 clination to go on according to rule. But novelty 
 must always result from there being two different 
 positions of the same form to be contrasted, as ap- 
 pears in the progression of this double form. 
 Thus, the Lombardy poplar, growing beside the 
 waters of Eridanus, when agitated by a breath of 
 wind, shews the other sides of its leaves, having a 
 different and lighter sort of green. 
 
123 
 
 SECT. II. 
 
 ON THE CONTINUED RELATION OF TWO SIMPLE 
 CURVES. 
 
 Castor or Better ophon. 
 
 The application of the power of Neptune is capa- 
 ble of producing also another double form, in 
 which the two curves proceeding together are 
 both simple, but in quality different from each 
 other. Suppose that in the inner or ruling curve 
 the same angle and the same proportion of lines 
 is retained throughout, so as to continue always 
 tracing the same form of a simple evolving hyper- 
 bola, without successive branches ; and suppose 
 that on the outside of this series of angles there 
 be made to project from the points of the suc- 
 cessive angles a series of lines diminishing accord- 
 ing to the same proportion as that in which the 
 straight lines guiding the inner curve increase ; 
 then if the tops of these projecting lines be con- 
 nected by other straight lines, an outer series of 
 angles will be produced as before, and it will be 
 capable of guiding an outer curve, which, instead 
 
124 CASTOR OR BELLEROPHON. 
 
 of diverging from the other, will continually sink 
 and approach towards it, because of the decreasing 
 proportion of the lines extending between them. 
 This outer curve will also be simple and without 
 successive branches, because it is guided by a series 
 of lines and angles generated always according to 
 the same principle ; but the outer curve will not 
 be the same form as the inner on a larger scale, 
 but will be an hyperbolic form differing in quali- 
 ty, because the successive quadrangles on which 
 it is built are continually changing their forms 
 and becoming more elongated ; because the up- 
 right lines are shortening and decreasing in con- 
 tinued proportion, while the bases and upper se- 
 ries of lines are lengthening and increasing in con- 
 tinued proportion. But all this proceeds accord- 
 ing to an unchanging principle ; and the outer 
 curve which is traced must be an hyperbola con- 
 tinually sinking and approaching to the other. 
 It is only as evolutes that two curves differing in 
 their qualities as hyperbolic forms can continue 
 to accompany each other as collateral forms ; be- 
 cause their curvature is always becoming less, and 
 bringing both of them nearer to the same standard, 
 which is a straight line. If they were traced as 
 involutes, in which the degree of curvature is al- 
 ways increasing, their incongruity of direction 
 would become more and more apparent ; and they 
 would sooner or ]ater intersect, and cease to ex- 
 
CASTOR OR BELLEROPHON. 125 
 
 tend as collateral forms accompanying each other. 
 The reverse is the case in the conjoined progress 
 of two evolutes approaching each other, and al- 
 ways coming nearer to the same standard. But 
 the breadth of the inner or lower curve would re- 
 quire to be contracted in continued proportion, as 
 in the power of Proserpina ; for otherwise the 
 thickness of the curves would cause the two to 
 meet, although the two series of angles guiding 
 them remained always different and apart. In 
 regard to length, their forms being evolutes, would 
 continually increase ; and to prevent their exces- 
 sive expansion, it becomes necessary that the forms 
 always, after a certain number of angles, be trans- 
 posed into a smaller scale, for instance a sixth. 
 Thus, the inner curve, though simple, would be 
 forced to break its continuity of extension, and 
 assume, at each transposition, the form of a curve 
 angle, having both sides convex ; and it is evi- 
 dent also that at each diminution of scale the top 
 of the last upright line (on the larger scale) would 
 require to be connected with the top of the next 
 (on the smaller scale) by a straight line making a 
 great and sudden descent towards the inner curve^ 
 the outer course being carried down along with 
 it ; after which the form of the outer curve would 
 again be produced as before, and would continue 
 gradually sinking towards the inner curve. By 
 means of the changes of course produced by these 
 
126 CASTOR OR BELLEROPHON. 
 
 successive diminutions of scale, the general courses 
 of the two curves together would be that of invo- 
 lutes, passing into infinite refinement; although 
 all the successive portions were parts of the same 
 two evolutes. 
 
 The peculiar characteristic of this power is, that 
 the relation which has once begun to flow between 
 the two different curves is always continued, and is 
 drawn out like the waters of a smooth and placid 
 river. It has relation to the pleasures of idleness, 
 or of any simple and continuous employment ; and 
 it was probably signified in the characters of the 
 Sirens, who, on the coast of Sicily, endeavoured 
 to detain Ulysses by their warbling. But it was 
 probably also the true character of Achilles, the 
 most powerful of the Grecian warriors ; because, 
 by the diminution of scales, it contracts and 
 compresses the future course of the same simple 
 curves, and becomes as strength and swiftness. 
 When Achilles withdrew in anger, because of the 
 loss of Briseis, he was contented to remain idle 
 in his tent. In his native region was the cele- 
 brated vale of Tempe, watered by the Peneus, 
 and affording such a prolongation of delight, that 
 even the inhabitants of Olympus descended to 
 waste their time in it. This mode of composition 
 has most relation to the elementary power of Ceres ; 
 because it is the continued approach of two dis- 
 similar forms, between which the difference is al- 
 
CASTOR OR BELLEROPHON. 127 
 
 ways becoming less, but cannot be exhausted. 
 From the continued sinking of the outer curve, 
 this power becomes the same as the love of hu- 
 mility ; and, from never passing into a succession 
 of different branches, it becomes the same as 
 homeliness and contentment with things which are 
 easily procured. This was probably the character 
 of Antisthenes, the Cynic, and his follower Dio- 
 genes, who studied to live on the cheapest possi- 
 ble terms. In modern times, Swift was distin- 
 guished for an affected humility in the choice of 
 subjects for poetry ; Hogarth shewed the same 
 taste in painting ; and Cobbett, the political writer, 
 has endeavoured to illustrate the pleasures of 
 thrift and of humble and contented labour, and 
 to shew the value of the resources which are 
 hidden in every plain material and vulgar pos- 
 session. Similar characteristics are shewn in 
 Cowper's mode of treating religious subjects in 
 poetry. But all these persons were distinguished 
 for those powers of hatred and recoil which belong 
 to Ceres ; and they were characterized also by the 
 inclination to study nature in its humblest forms, 
 or to shew the resources contained in the simplest 
 materials that can be assumed. This is also the 
 character of Southey, among the modern English 
 poets. To this mode of composition may be re- 
 ferred the Elysian fields, where the shades of 
 heroes existed in a state of pleasing idleness, 
 
128 CASTOR OR BELLEROPHON. 
 
 amidst blooming meadows, receiving light from a 
 sky of their own. Finding their happiness pro- 
 longed from day to day, they wished for nothing 
 more than to experience the progressive continua- 
 tion of the same feelings. Among the nations of 
 Europe, it would appear that some classes of the 
 Spaniards approach the nearest to this character ; 
 because they are distinguished for idleness and 
 contentment with little, when more might easily 
 be procured. Some travellers describe them as 
 spending whole days wrapt in their cloaks, leaning 
 in rows against a wall, or dozing under a tree. 
 Sicily was considered as sacred to Proserpina, be- 
 cause from thence she had been carried off by 
 Pluto ; but Proserpina is the retrogression of one 
 simple curve ; while the power of the Sirens is the 
 evolution of two different simple curves, which to- 
 gether are forced to assume the form of an invo- 
 lute. Such is the freshness and continuity of 
 vegetation, and, at the same time, its internal 
 powers of repulsion or elasticity. 
 
 But, in regard to the doubleness of the form, 
 Thessaly was said to be also the country of the 
 Centaurs. The education of Achilles was in- 
 trusted to Chiron, in whose structure appeared 
 the form both of horse and of man ; and this was 
 probably meant to signify the conjoined progress 
 of two curves altogether different in quality. To 
 the same class may be referred Castor, the tamer 
 
CASTOR OR BELLEROPHON. 129 
 
 of horses ; since the art of riding is skill in recon- 
 ciling the powers and motions of two dissimilar 
 forms. The same power belonged also to Belle- 
 rophon, who mounted the winged Pegasus, and 
 controlled him amidst the regions of the air. The 
 name of Pegasus is derived from aw, a spring, 
 because of the relation which extends like a river 
 between the two different curves ; and something 
 like this is the effort of the mind in continuous 
 poetical invention, when it finds the materials al- 
 together within itself. But to this mode of com- 
 position must be referred the art of producing the 
 bathos, or of finding something always more hum- 
 ble, easy, and soothing to human nature, by shew- 
 ing how much is contained in the most ordinary 
 materials, and in those things which are not diffi- 
 cult to be found. The Thessalians were cele- 
 brated for their skill in enchantments, perhaps 
 from the power of anticipating the future parts of 
 the same continuity, and drawing them into a 
 narrower and closer form, which is like anticipat- 
 ing the future course of any simple affection of the 
 mind, and increasing its strength by bringing al- 
 ways more of it into a smaller compass. Castor, 
 like Pollux, was represented with a star over his 
 head ; and to him, more particularly, should be 
 ascribed the power of tranquillizing the sea ; nor 
 would Bellerophon have been thrown from his 
 horse had he not attempted to soar to heaven, in- 
 
 f 2 
 
130 CASTOR OR BELLEROPHON. 
 
 stead of carrying Pegasus up into the clouds only 
 to let him gradually and smoothly sink towards 
 the terrestrial plains, where, the density of the air 
 being increased, its elasticity would prevent him 
 from ever reaching the ground. Such is the art 
 of lowering a theme. Perhaps this was also the 
 character of the Cumean or Campanian Sibyl, to 
 whom Apollo granted an endless continuation of 
 years ; but she, contracting and wearing away in 
 her bodily form, seemed at last to retain only a 
 voice. 
 
131 
 
 CHAP. VII. 
 
 ON THE POWERS OF COLLOCATION, OR THE 
 DISTRIBUTION OF FINITE PARTS OF THE 
 HYPERBOLA. 
 
 Geryon or Silvanus. 
 
 If the power of Vesta, which is equivalent to three 
 angles or planes produced at the same time, be 
 applied to the hyperbola, it must be supposed to 
 separate it into three different parts or courses, 
 each conducted by its own series of angles. But 
 the power of Vesta is also that of terminating and 
 breaking off, and it is also the power of a plane 
 coming down to fasten upon the side of another, 
 which gives the power of collocation or the idea of 
 building ; and, therefore, in this mode of composi- 
 tion, the series of angles which trace the curves 
 may be supposed to break off and leave finite parts 
 of the curves, but to retain the power of tracing 
 out the remainders of the same curves elsewhere. 
 In conformity to the power of Vesta, therefore, 
 they may be supposed to build and fasten subse- 
 quent portions of the same curves to the sides of 
 
 • f 
 
132 GERYON OR SILVANUS. 
 
 those portions which have already been placed; 
 and to the sides of this second order of branches or 
 terminated curves to add a third, and so on in con- 
 tinued progression ; and the added parts being ap- 
 plied to the points of angles, in the series which 
 leads the former portion of the curve, the portions 
 which are built upon that portion must be sup- 
 posed to cross it. To enable this distribution of 
 forms to proceed, it is evident that every succeed- 
 ing order of parts must be transposed in a smaller 
 scale, so as still to find room, and not be entangled 
 with the others. On each finite portion of the 
 curve two others must be built, that there may be 
 a continued multiplication of parts. Thus, in a 
 tree, each larger branch has smaller ones built 
 upon it, and upon them are arranged smaller 
 branches still. Without these transpositions, this 
 mode of laying out the hyperbola would produce 
 the grossest exuberance and confusion, which was 
 probably signified in the character of the Hydra 
 and the marsh of Lerna. But, by the continued 
 diminution of scales, it assumes order and refine- 
 ment ; and each portion of the curve, by also cross- 
 ing within itself twice, may acquire variety of di- 
 rection, and will not change the distribution of the 
 general form. This mode of composition was pro- 
 bably signified in Silvan us, the divinity of the 
 woods, which have an obvious relation to the same 
 form. The wife of Silvanus was named Fatua, 
 
GERY0N OR SILVANUS. 
 
 138 
 
 and was supposed to occasion the nightmare by 
 perching on the breasts of persons asleep. But 
 the character of Silvanus being founded on the 
 power of spreading and distributing forms in every 
 direction, and finding out new sites for them upon 
 the sides of each other, may easily be distinguish- 
 ed from the more vulgar attributes of Pan. The 
 power of Silvanus is probably the true character 
 of the people of Indostan, among whom is found 
 the streaked tiger, a form of breaking off or tear- 
 ing to pieces, as in the distribution of the parts of 
 a single curve. Among the Indians are found the 
 remains of the enormous systems of law, morality, 
 and mythology. This mode of composition has 
 most relation to the elementary power of Neptune, 
 which bears two planes having relation to a com- 
 mon basis ; and, in this mode of composition, each 
 branch has two others built upon it. One of the 
 ephithets of Neptune was Phutalmios, or produc- 
 tive. To Silvanus may also be ascribed the power 
 of melancholy and obscuration from intervening 
 parts ; like the obscuration or gloom which exists 
 in the midst of a forest. Nevertheless, this is the 
 true character of the ship Argo, which signified 
 the power of being loaded, or of carrying other 
 forms upon it. This is the first type of naviga- 
 tion ; and, among animals, the same character is 
 expressed in the bear, which, in the northern seas, 
 is sometimes found floating on large pieces of ice, 
 
134 GERYON OR SILVANUS. 
 
 as if to signify its relation to loading. But a har- 
 mony extends throughout the parts which are laid 
 out according to this mode of composition ; because 
 all the parts in each of the three curves has rela- 
 tion to corresponding parts in the other two, at 
 each transposition into a smaller scale, and through- 
 out each new order of branches which is produced. 
 The branches which correspond, however, are al- 
 ways farther separated, and are forced to seek for 
 each other through a crowd of intervening parts. 
 This mode of composition was perhaps also signi- 
 fied in three-bodied Geryon. Among the nations 
 of modern Europe, this character belongs to the 
 Germans in regard to obscurity and depth of feel- 
 ing, and to the Dutch, in regard to methodical 
 distribution, and the mere collocation of things not 
 blended in the same continuity. Thus Silvanus 
 differs from the power of Theseus. In another 
 point of view, the power of Silvanus may be as- 
 sumed as the orignial type of epic poetry, which 
 is not a mere narrative of the order of events, but 
 is rather an ample and profuse array of parts 
 spreading in every direction, although all belong- 
 ing to one subject or mode of deduction. To 
 the same class with Silvanus therefore may be re- 
 ferred the mode of construction which belongs to 
 Homer, and which exemplifies the power of rami- 
 fication and minuteness of detail, but still preserves 
 the clearest order. This power is also shewn in 
 
GERYON OR SILVANUS. 135 
 
 the Dutch school of painting, which expresses the 
 nature of woods and marshes, and, in general, the 
 various compilations and distributions of sub- 
 stance. To this class may also be referred the 
 science of geology, which had its origin among the 
 Germans. Among the Homeric heroes this seems 
 to have been the power belonging to the Sala- 
 minian Ajax, who was considered as the next 
 warrior after Achilles, but somewhat tardy and 
 stupid. To this class may also be referred the 
 philosophical system of Spinosa, who endeavoured 
 to prove that the continuation of one substance, 
 whether worldly or divine, constitutes the nature 
 of all things, and extends throughout those forms 
 which have the appearance of being separate and 
 particular. This mode of composition was, per- 
 haps, also signified in the golden fleece, which was 
 the object of the expedition undertaken by Jason. 
 The artificial use of substance was expressed in 
 the poisons and enchantments of Medea, and also 
 in her cutting GEson to pieces to restore him to 
 youth, which applies to breaking off the hyperbola 
 and laying it out in a different order. The golden 
 ram of Phryxus was capable of transporting its 
 owner through the air, because it carries parts of 
 the same curve into a new order. To this class 
 may perhaps be referred the artist Daedalus, whose 
 wings, being constructed only with wax, could not 
 bear the heat of the sun. The power of Silvanus 
 
136 GERYON OR SILVANUS. 
 
 is departure from continued derivation in joining 
 parts of the same form together ; and to Silvanus 
 may be ascribed sententiousness, which gives forth 
 wisdom in minute portions. This character be- 
 longed, among the English poets, to Pope; but 
 the same power appeared in Virgil as art in shift- 
 ing the arrangement of his words, and producing 
 beauties of collocation. But to Silvanus may also 
 be ascribed the study of conic sections, as the 
 means of comparing different parts of the same 
 curve. Mathematical reasoning depends much on 
 the power of arbitary collocation and comparison 
 of figures. Sir Isaac Newton, whose genius was 
 gross and tardy, like that of Ajax, was a native of 
 Lincolnshire, a region abounding with fens. From 
 each two later branches being fastened upon the 
 side of a former portion of the same simple curve, 
 this mode of composition is like a reference to per- 
 sonal experience or past states of the mind. This 
 constitutes the sedateness of age. The essential 
 characteristic of a man of experience and sententi- 
 ous wisdom is the power of referring to situations 
 in which he has formerly been. Thus the genius 
 of Newton fitted him for comparing the different 
 portions of the trajectories or courses through which 
 moving bodies pass. But this mode of composi- 
 tion, from each new form being speedily broken 
 off and left terminated, becomes like a feeling of 
 the vanity of all human undertakings, and gives 
 
GERYON OR SILVANUS. 137 
 
 birth to a peculiar strain of morality. Thus Solo- 
 mon, having tried every different pursuit and enjoy- 
 ment, and having accumulated vast experience as 
 an individual, declared all things to be vanity, a 
 declaration which applied only to his own kind of 
 progression as to feeling. This mode of writing 
 is studied by Lord Byron among the modern Eng- 
 lish poets. To Silvanus, however, belongs a great 
 depth and tenderness of sentiment, founded always 
 on the reference of present affections to those 
 which have preceded them. Thus the wallflower, 
 egregious in its tints, nods over the remains of 
 some dilapidated tower. At each new distribu- 
 tion of forms, by the power of Silvanus, the forms 
 brought together are more remote as to their real 
 places in the curve from whence they are taken. 
 Thus the stork and crane, and other aquatic birds, 
 unite in themselves the substances collected from 
 remote marshes. To the same class may be re- 
 ferred the literary commentators who build their 
 observations upon those of previous commentators 
 till the structure becomes overloaded. But it is 
 evident that the successive orders of forms, pro- 
 ceeding according to the power of Silvanus, are 
 capable of infinite progression. This power has a 
 certain relation to the finite ranges successively ex- 
 plored by the power of Orion or of Bacchus, who 
 was celebrated for having conquered India ; and, 
 perhaps, his attendant Silenus was in reality a form 
 
138 GEEYON OE SILVANUS. 
 
 of Silvanus. But as to the accumulation of parti- 
 cular forms, this power was evidently signified in 
 the character of Plutus, the god of wealth, who de- 
 rived his origin from Jason and Ceres ; for every 
 little portion which is broken off is gained from an 
 infinite curve, which can never be exhausted. 
 
 After this power of Silvanus, which is produced 
 from the application of the power of Vesta, this 
 series is done ; because the next power, which is 
 Mars, is the same as that by which the hyperbola 
 was originally accelerated. Therefore, after this, 
 there is only room for a composition of composi- 
 tions,. 
 
AN 
 
 INDEX 
 
 PROPER NAMES 
 
 MENTIONED IN THE FOREGOING TREATISE, 
 
 A 
 
 PAGE 
 
 
 PAGE 
 
 Acheron 
 
 ■ 83 
 
 Ashur, tribe of 
 
 ■ 34 
 
 Achilles 
 
 126 
 
 Athenians 
 
 108 
 
 Adonis 
 
 81 
 
 Atlas 
 
 120 
 
 iEsculapius 
 
 67 
 
 
 
 Ajax 
 
 134 
 
 B 
 
 
 Alcaeus 
 
 54 
 
 Bacchus 
 
 49 
 
 Anacreon 
 
 63 
 
 Bacon 
 
 64 
 
 Angelo, Michael 
 
 - 78 
 
 Bellerophon 
 
 123 
 
 Antisthenes 
 
 127 
 
 Benjamin, tribe of 
 
 45 
 
 Apollo 
 
 15 
 
 Bunyan 
 
 120 
 
 Arcadians 
 
 84 
 
 Buonaparte 
 
 71 
 
 Argives 
 
 96 
 
 Byron 
 
 137 
 
 ArgOj the ship 
 
 131 
 
 
 
 Ariadne 
 
 65,77 
 
 C 
 
 
 Aristophanes 
 
 104 
 
 Cadmus 
 
 104 
 
 Aristotle 
 
 66 
 
 Caelus 
 
 6 
 
140 
 
 INDEX. 
 
 
 
 PAGE 
 
 
 PAGE 
 
 Carthaginians 
 
 121 
 
 Elysian fields 
 
 127 
 
 Castor 
 
 123 
 
 English 
 
 64 
 
 Catullus 
 
 121 
 
 Ephraim 
 
 115 
 
 Centaurs 
 
 128 
 
 Epicurus 
 
 115 
 
 Ceres 
 
 - 38 
 
 Erichthonius 
 
 100 
 
 Chaldeans 
 
 - 93 
 
 Eridanus 
 
 119 
 
 Chimaera 
 
 116 
 
 Eros 
 
 80 
 
 Chinese 
 
 105 
 
 Etrurians 
 
 78 
 
 Chiron 
 
 128 
 
 
 
 Circe 
 
 87 
 
 F 
 
 
 Cobbett 
 
 129 
 
 Fates 
 
 94 
 
 Cocytus 
 
 89 
 
 Fatua 
 
 132 
 
 Coleridge 
 
 92 
 
 Fauns 
 
 58 
 
 Cowper 
 
 127 
 
 Flora 
 
 78 
 
 Crete 
 
 65 
 
 French 
 
 109 
 
 
 
 Furies 
 
 89 
 
 D 
 
 
 
 
 Daedalus . - 
 
 135 
 
 G 
 
 
 Dan, tribe of 
 
 25 
 
 Gad, tribe of 
 
 29 
 
 Dante 
 
 78 
 
 Galileo 
 
 78 
 
 Democritus 
 
 109 
 
 Ganymede 
 
 105 
 
 Demogorgon 
 
 88 
 
 Germans 
 
 134 
 
 Diana 
 
 18 
 
 Geryon 
 
 131 
 
 Diogenes 
 
 127 
 
 Goethe 
 
 80 
 
 Diomed 
 
 57 
 
 Graces, the 
 
 53 
 
 Dutch 
 
 134 
 
 H 
 
 
 E 
 
 
 Handel 
 
 121 
 
 Egyptians 
 
 102 
 
 Haydn 
 
 113 
 
 Elis 
 
 ■ 76 
 
 Hebe 
 
 56 
 

 INDEX. 
 
 141 
 
 
 PAGE 
 
 
 PAGE 
 
 Hercules 
 
 60 
 
 Locke 
 
 105 
 
 Hesiod 
 
 80 
 
 Lucifer 
 
 19 
 
 Hogarth 
 
 127 
 
 
 
 Homer 
 
 134 
 
 M 
 
 
 Horace - , 
 
 93 
 
 Mars 
 
 29 
 
 Hume, David 
 
 104 
 
 Matthew, St 
 
 106 
 
 Hydra 
 
 132 
 
 Medea 
 
 135 
 
 Hymen 
 
 121 
 
 Medusa 
 
 89 
 
 
 
 Meleager 
 
 54 
 
 I 
 
 
 Mercury 
 
 45 
 
 Jacob - » 
 
 . 6 
 
 Milton 
 
 86 
 
 ■Jason 
 
 135 
 
 Minerva 
 
 41 
 
 Jeffrey 
 
 109 
 
 Minos 
 
 65 
 
 Jerusalem 
 
 93 
 
 Morpheus 
 
 63 
 
 Indostan 
 
 131 
 
 Moses 
 
 103 
 
 Johnson, Samuel - 
 
 109 
 
 Muses, the three earlier 11 
 
 Joseph 
 
 42 
 
 Muses, the nine 
 
 60 
 
 Irish 
 
 121 
 
 Mytelene 
 
 54 
 
 Issachar, tribe of - 
 
 25 
 
 
 
 Italy 
 
 55 
 
 N 
 
 
 Judah, tribe of 
 
 18 
 
 Naphthali 
 
 38 
 
 Judas Iscariot 
 
 66 
 
 Nemesis 
 
 93 
 
 Juno 
 
 5 
 
 Neptune 
 
 25 
 
 Jupiter 
 
 9 
 
 Nestor 
 
 80 
 
 
 
 Newton 
 
 136 
 
 L 
 
 
 Niobe 
 
 105 
 
 Lacedaemonians 
 
 90 
 
 
 
 Latona 
 
 15 
 
 O 
 
 
 Lethe 
 
 96 
 
 Olympic games 
 
 76 
 
 Levi, tribe of 
 
 16 
 
 Oreades 
 
 111 
 
142 
 
 INDEX. 
 
 
 PAGE 
 
 R 
 
 PAGE 
 
 Orion 
 
 49 
 
 Raphael 
 
 84 
 
 Ovid 
 
 103 
 
 Reuben, tribe of 
 
 . 6 
 
 
 
 Rhea 
 
 26 
 
 P 
 
 
 Rhodians 
 
 104 
 
 Palamedes 
 
 96 
 
 Romans 
 
 55 
 
 Pan 
 
 67 
 
 Rousseau 
 
 104 
 
 Paul, St - 
 
 66 
 
 Rubens 
 
 84 
 
 Pegasus 
 
 129 
 
 
 
 Pelops 
 
 72 
 
 S 
 
 
 Perseus 
 
 88 
 
 Sappho 
 
 54 
 
 Persians 
 
 86 
 
 Saturn 
 
 46 
 
 Phlegethon 
 
 75 
 
 Scotch 
 
 69 
 
 Phoenicians 
 
 97 
 
 Scott, Sir Walter - 
 
 121 
 
 Pindar 
 
 96 
 
 Shakspeare 
 
 112 
 
 Pirithous 
 
 87 
 
 Sibyl 
 
 130 
 
 Plato 
 
 102 
 
 Sicily 
 
 128 
 
 Pluto 
 
 81 
 
 Silenus 
 
 137 
 
 Plutus 
 
 138 
 
 Silvanus 
 
 131 
 
 Po, the river 
 
 119 
 
 Simeon, tribe of 
 
 10 
 
 Pollux 
 
 117 
 
 Sirens 
 
 126 
 
 Pope, the 
 
 57 
 
 Socrates 
 
 112 
 
 Pope, the poet 
 
 136 
 
 Solomon 
 
 137 
 
 Pomona 
 
 57 
 
 Somnus 
 
 91 
 
 Priapus 
 
 56 
 
 Southey 
 
 127 
 
 Prometheus 
 
 69 
 
 Spaniards 
 
 128 
 
 Proserpina 
 
 72 
 
 Spartans 
 
 90 
 
 Proteus 
 
 26 
 
 Spenser 
 
 63 
 
 Pythagoras 
 
 86 
 
 Spinosa 
 
 135 
 
 Pythian Apollo 
 
 69 
 
 Styx 
 
 86 
 

 1IN1 
 
 PAGE 
 
 uiijL* 
 
 PAGE 
 
 Swift 
 
 127 
 
 Uranus 
 
 6 
 
 Swiss 
 
 104 
 
 V 
 
 
 T 
 
 
 Venetians 
 
 97 
 
 Tempe 
 
 126 
 
 Venus 
 
 34 
 
 Thales 
 
 97 
 
 Veronese 
 
 122 
 
 Thamyris 
 
 79 
 
 Vertumnus 
 
 94 
 
 Thebans 
 
 103 
 
 Vesta 
 
 24 
 
 Themis 
 
 12 
 
 Virgil 
 
 136 
 
 Theophrastus 
 
 54 
 
 Voltaire 
 
 113 
 
 Theseus 
 
 107 
 
 Vulcan 
 
 20 
 
 Thessaly 
 
 128 
 
 
 
 Typhon 
 
 102 
 
 W 
 
 
 
 
 Wordsworth 
 
 78 
 
 U 
 
 
 
 
 Ulysses 
 
 120 
 
 Z 
 
 
 Urania 
 
 8 
 
 Zabulon, tribe of 
 
 - 21 
 
 THE END, 
 
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