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fans　and　other　instruments　used　in　the　threshing　of　corn；　the　close

and　heavy　particles　are　borne　away　and　settle　in　one　direction；　and

the　loose　and　light　particles　in　another。　In　this　manner；　the　four

kinds　or　elements　were　then　shaken　by　the　receiving　vessel；　which；

moving　like　a　winnowing　machine；　scattered　far　away　from　one　another

the　elements　most　unlike；　and　forced　the　most　similar　elements　into

dose　contact。　Wherefore　also　the　various　elements　had　different　places

before　they　were　arranged　so　as　to　form　the　universe。　At　first；　they

were　all　without　reason　and　measure。　But　when　the　world　began　to　get

into　order；　fire　and　water　and　earth　and　air　had　only　certain　faint

traces　of　themselves；　and　were　altogether　such　as　everything　might

be　expected　to　be　in　the　absence　of　God；　this；　I　say；　was　their　nature

at　that　time；　and　God　fashioned　them　by　form　and　number。　Let　it　be

consistently　maintained　by　us　in　all　that　we　say　that　God　made　them　as

far　as　possible　the　fairest　and　best；　out　of　things　which　were　not

fair　and　good。　And　now　I　will　endeavour　to　show　you　the　disposition

and　generation　of　them　by　an　unaccustomed　argument；　which　am　compelled

to　use；　but　I　believe　that　you　will　be　able　to　follow　me；　for　your

education　has　made　you　familiar　with　the　methods　of　science。

　　In　the　first　place；　then；　as　is　evident　to　all；　fire　and　earth　and

water　and　air　are　bodies。　And　every　sort　of　body　possesses　solidity；

and　every　solid　must　necessarily　be　contained　in　planes；　and　every

plane　rectilinear　figure　is　composed　of　triangles；　and　all　triangles

are　originally　of　two　kinds；　both　of　which　are　made　up　of　one　right

and　two　acute　angles；　one　of　them　has　at　either　end　of　the　base　the

half　of　a　divided　right　angle；　having　equal　sides；　while　in　the

other　the　right　angle　is　divided　into　unequal　parts；　having　unequal

sides。　These；　then；　proceeding　by　a　combination　of　probability　with

demonstration；　we　assume　to　be　the　original　elements　of　fire　and　the

other　bodies；　but　the　principles　which　are　prior　to　these　God　only

knows；　and　he　of　men　who　is　the　friend　God。　And　next　we　have　to

determine　what　are　the　four　most　beautiful　bodies　which　are　unlike　one

another；　and　of　which　some　are　capable　of　resolution　into　one　another；

for　having　discovered　thus　much；　we　shall　know　the　true　origin　of

earth　and　fire　and　of　the　proportionate　and　intermediate　elements。　And

then　we　shall　not　be　willing　to　allow　that　there　are　any　distinct

kinds　of　visible　bodies　fairer　than　these。　Wherefore　we　must　endeavour

to　construct　the　four　forms　of　bodies　which　excel　in　beauty；　and

then　we　shall　be　able　to　say　that　we　have　sufficiently　apprehended

their　nature。　Now　of　the　two　triangles；　the　isosceles　has　one　form

only；　the　scalene　or　unequal…sided　has　an　infinite　number。　Of　the

infinite　forms　we　must　select　the　most　beautiful；　if　we　are　to　proceed

in　due　order；　and　any　one　who　can　point　out　a　more　beautiful　form　than

ours　for　the　construction　of　these　bodies；　shall　carry　off　the　palm；

not　as　an　enemy；　but　as　a　friend。　Now；　the　one　which　we　maintain　to　be

the　most　beautiful　of　all　the　many　triangles　（and　we　need　not　speak　of

the　others）　is　that　of　which　the　double　forms　a　third　triangle　which

is　equilateral；　the　reason　of　this　would　be　long　to　tell；　he　who

disproves　what　we　are　saying；　and　shows　that　we　are　mistaken；　may

claim　a　friendly　victory。　Then　let　us　choose　two　triangles；　out　of

which　fire　and　the　other　elements　have　been　constructed；　one

isosceles；　the　other　having　the　square　of　the　longer　side　equal　to

three　times　the　square　of　the　lesser　side。

　　Now　is　the　time　to　explain　what　was　before　obscurely　said：　there　was

an　error　in　imagining　that　all　the　four　elements　might　be　generated　by

and　into　one　another；　this；　I　say；　was　an　erroneous　supposition；　for

there　are　generated　from　the　triangles　which　we　have　selected　four

kinds…three　from　the　one　which　has　the　sides　unequal；　the　fourth　alone

is　framed　out　of　the　isosceles　triangle。　Hence　they　cannot　all　be

resolved　into　one　another；　a　great　number　of　small　bodies　being

combined　into　a　few　large　ones；　or　the　converse。　But　three　of　them　can

be　thus　resolved　and　compounded；　for　they　all　spring　from　one；　and

when　the　greater　bodies　are　broken　up；　many　small　bodies　will　spring

up　out　of　them　and　take　their　own　proper　figures；　or；　again；　when　many

small　bodies　are　dissolved　into　their　triangles；　if　they　become　one；

they　will　form　one　large　mass　of　another　kind。　So　much　for　their

passage　into　one　another。　I　have　now　to　speak　of　their　several

kinds；　and　show　out　of　what　combinations　of　numbers　each　of　them　was

formed。　The　first　will　be　the　simplest　and　smallest　construction；

and　its　element　is　that　triangle　which　has　its　hypotenuse　twice　the

lesser　side。　When　two　such　triangles　are　joined　at　the　diagonal；　and

this　is　repeated　three　times；　and　the　triangles　rest　their　diagonals

and　shorter　sides　on　the　same　point　as　a　centre；　a　single

equilateral　triangle　is　formed　out　of　six　triangles；　and　four

equilateral　triangles；　if　put　together；　make　out　of　every　three

plane　angles　one　solid　angle；　being　that　which　is　nearest　to　the

most　obtuse　of　plane　angles；　and　out　of　the　combination　of　these

four　angles　arises　the　first　solid　form　which　distributes　into　equal

and　similar　parts　the　whole　circle　in　which　it　is　inscribed。　The

second　species　of　solid　is　formed　out　of　the　same　triangles；　which

unite　as　eight　equilateral　triangles　and　form　one　solid　angle　out　of

four　plane　angles；　and　out　of　six　such　angles　the　second　body　is

completed。　And　the　third　body　is　made　up　of　120　triangular　elements；

forming　twelve　solid　angles；　each　of　them　included　in　five　plane

equilateral　triangles；　having　altogether　twenty　bases；　each　of　which

is　an　equilateral　triangle。　The　one　element　＇that　is；　the　triangle

which　has　its　hypotenuse　twice　the　lesser　side＇　having　generated　these

figures；　generated　no　more；　but　the　isosceles　triangle　produced　the

fourth　elementary　figure；　which　is　compounded　of　four　such

triangles；　joining　their　right　angles　in　a　centre；　and　forming　one

equilateral　quadrangle。　Six　of　these　united　form　eight　solid　angles；

each　of　which　is　made　by　the　combination　of　three　plane　right

angles；　the　figure　of　the　body　thus　composed　is　a　cube；　having　six

plane　quadrangular　equilateral　bases。　There　was　yet　a　fifth

combination　which　God　used　in　the　delineation　of　the　universe。

　　Now；　he　who；　duly　reflecting　on　all　this；　enquires　whether　the

worlds　are　to　be　regarded　as　indefinite　or　definite　in　number；　will　be

of　opinion　that　the　notion　of　their　indefiniteness　is　characteristic

of　a　sadly　indefinite　and　ignorant　mind。　He；　however；　who　raises　the

question　whether　they　are　to　be　truly　regarded　as　one　or　five；　takes

up　a　more　reasonable　position。　Arguing　from　probabilities；　I　am　of

opinion　that　they　are　one；　another；　regarding　the　question　from

another　point　of　view；　will　be　of　another　mind。　But；　leaving　this

enquiry；　let　us　proceed　to　distribute　the　elementary　forms；　which　have

now　been　created　in　idea；　among　the　four　elements。

　　To　earth；　then；　let　us　assign　the　cubical　form；　for　earth　is　the

most　immoveable　of　the　four　and　the　most　plastic　of　all　bodies；　and

that　which　has　the　most　stable　bases　must　of　necessity　be　of　such　a

nature。　Now；　of　the　triangles　which　we　assumed　at　first；　that　which

has　two　equal　sides　is　by　nature　more　firmly　based　than　that　which　has

unequal　sides；　and　of　the　compound　figures　which　are　formed　out　of

either；　the　plane　equilateral　quadrangle　has　necessarily；　a　more

stable　basis　than　the　equilateral　triangle；　both　in　the　whole　and　in

the　parts。　Wherefore；　in　assigning　this　figure　to　earth；　we　adhere

to　probability；　and　to　water　we　assign　that　one　of　the　remaining　forms

which　is　the　least　moveable；　and　the　most　moveable　of　them　to　fire；

and　to　air　that　which　is　intermediate。　Also　we　assign　the　smallest

body　to　fire；　and　the　greatest　to　water；　and　the　intermediate　in

size　to　air；　and；　again；　the　acutest　body　to　fire；　and　the　next　in

acuteness　to；　air；　and　the　third　to　water。　Of　all　these　elements；　that

which　has　the　fewest　bases　must　necessarily　be　the　most　moveable；

for　it　must　be　the　acutest　and　most　penetrating　in　every　way；　and　also

the　lightest　as　being　composed　of　the　smallest　number　of　similar

particles：　and　the　second　body　has　similar　properties　in　a　second

degree；　and　the　third　body　in　the　third　degree。　Let　it　be　agreed；

then；　both　according　to　strict　reason　and　according　to　probability；

that　the　pyramid　is　the　solid　which　is　the　original　element　and　seed

of　fire；　and　let　us　assign　the　element　which　was　next　in　the　order

of　generation　to　air；　and　the　third　to　water。　We　must　imagine　all

these　to　be　so　small　that　no　single　particle　of　any　of　the　four

kinds　is　seen　by　us　on　account　of　their　smallness：　but　when　many　of

them　are　collected　together　their　aggregates　are　seen。　And　the

ratios　of　their　numbers；　motions；　and　other　properties；　everywhere

God；　as　far　as　necessity　allowed　or　gave　consent；　has　exactly

perfected；　and　harmonised　in　due　proportion。

　　From　all　that