the uranus enigma (second edition)

7/28/2019 The Uranus Enigma (Second Edition)

http://slidepdf.com/reader/full/the-uranus-enigma-second-edition 1/73

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[e` Prdgus @gihod (^`njgk @kitijg)

My Idg M`drksl`y

Njpyrihet © =14= „ =14> my Idg M`drksl`y

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It is ndll`k [e` Prdgus @gihod m`ndus` we`te`r jr gjt `xtrdt`rr`stridls l`ft teìr

teuomprigt ig jur peysins ds w`ll ds d o`ssdh` tedt s`òs tj njo` frjo te` sdo` r`hijgjf spdn` ds ^@[I’s Xjw sihgdl, jr rdgkjoly `vjlvigh systòs dgk Gdtur` njgv`rh` jg

jg` guom`r (gig`-fiftes), te` Prdgus igt`hrdl is jg` njopdnt stdtò`gt tedt òmjki`s

`v`ryteigh. It is oy m`li`f, if yju r`dk teis stjry, yju oihet dhr`` it is wedt edpp`gs if

Hypsi`s kj peysins.

Idg M`drksl`y

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Igtrjkuntijg3 D [ejrjuhe [r`dto`gt Jf Kio`gsijgdl Dgdlysis

Pgits dr` te` mdsin odhgituk`s w` us` tj o`dsur` kio`gsijgs, weine dr` qudgtiti`s w`

us` tj k`snrim` gdtur`, lib` l`ghte, odss, tio`, tòp`rdtur`, dgk `g`rhy. X` k`fig` d

biljo`t`r tj jg` t`g tejusdgkte jf te` kistdgn` frjo te` pjl` tj te` `qudtjr, d o`t`r tj m`

jg` tejusdgkte jf d biljo`t`r, d n`gtio`t`r tj m` jg` eugkr`kte jf d o`t`r. X` k`fig` ds`njgk tj m` jg` 56, ;11te jf dg @drte kdy, dgk d hrdo tj m` te` wìhet jf d numin

n`gtio`t`r jf wdt`r dt stdgkdrk tòp`rdtur` dgk pr`ssur`, weine is z`rj k`hr``s

n`gtihrdk` dgk te` pr`ssur` dt s`d l`v`l. X` k`fig` jg` k`hr`` n`gtihrdk` my sdyigh z`rj

k`hr``s is te` tòp`rdtur` dt weine wdt`r fr``z`s, dgk 411 k`hr``s n`gtihrdk` is te`tòp`rdtur` dt weine wdt`r mjils. [eis is te` o`trin systò, dgk w` nr`dt`k it ig t`ros jf

te` wjrlk w` bgjw. Dgk w` nr`dt`k it sj tedt my dssihgigh d Hr``b pr`fix, lib` Bilj

(o`dgigh tejusdgk) we`g dttdne`k tj o`t`r, o`dgs d tejusdgk o`t`rs, jr my dssihgigh

te` Hr``b pr`fix n`gti (o`dgigh eugkr`kte) is jg` eugkr`kte jf d o`t`r. [edt is d o`t`ris d eugkr`k n`gtio`t`rs, dgk d biljo`t`r is 4111 tejusdgk o`t`rs. [e` systò is us`ful

m`ndus` gjt jgly edv` w` k`riv`k te` ugits frjo peysindl prjp`rti`s, te` kiff`r`gt ugits ig

te` systò hrjw ds oultipl`s jf t`g. ^j, if w` wdgt tj wjrb wite o`dsurigh sjo`teighsodll, lib` dg dnjrg, w` k`snrim` it wite n`gtio`t`rs, dgk if w` wjrb wite sjo`teighldrh`, lib` te` kido`t`r jf te` `drte w` us` biljo`t`rs.

M`fjr` te` o`trin systò, w` o`dsur`k qudgtiti`s ig t`ros jf f``t dgk pjugks, weil`

s`njgks edv` dlwdys m``g njgstdgt tj mjte te` fjjt-pjugk systò dgk te` o`trin systò.Qrjmdmly m`ndus` s`njgks dr` k`riv`k frjo te` >60 kdy, tw`lv` ojgte, y`dr dgk te` =;

ejur kdy, weine eistjrindlly k`v`ljp`k frjo wis` ndl`gkdr odb`rs jv`r tejusdgks jf

y`drs. Mut te` fjjt odb`s littl` s`gs`, jgly njoigh p`redps frjo te` e`drsdy sej` siz` jf d bigh, dgk te` pjugk frjo Hjk bgjws we`r`. ^till furte`r, teis systò wjulk s`ò gjt

tj m` tedt prdhodtin ig tedt te`r` dr` tw`lv` igne`s ig dgk fjjt dgk ter`` f``t ig d ydrk, d

fdr nry frjo te` s`gsimility jf te` o`trin systò we`r` dll siz`s jr dojugts dr` d oultipl`jf t`g jf dgjte`r.

@urjp` us`s te` o`trin systò, lib` rjdk sihgs dr` ig biljo`t`rs, dgk sej` siz`s ig

n`gtio`t`rs. X` us` oil`s fjr jur rjdk sihgs dgk igne`s fjr sej` siz`s. Ejw`v`r,

m`ndus` jf te` prdhodtin gdtur` jf te` o`trin systò, Do`rindg dgk @urjp`dg sni`gtistsus` te` o`trin systò dlib`. Jft`g, ejw`v`r, m`ndus` te` fjjt-pjugk systò is still ig us`

ig [e` Pgit`k ^tdt`s, sni`gtist sjo`tio`s figk teòs`lv`s edvigh tj hj tj te` `ffjrt jf

njgv`rtigh kdtd te`y g``k frjo f``t dgk pjugks tj o`t`rs dgk biljhrdos.

Mut wedt if I sdik gjt jgly dr` mjte te` o`trin systò dgk te` fjjt-pjugk systò

njgg`nt`k tj te` Pgiv`rs` mut tj jg` dgjte`r ig d wdy tedt wjulk suhh`st te` `vjlutijg jf teìr k`v`ljpo`gts is kiff`r`gt tedg te` eistjry w` bgjw: [edt is wedt I s`ò tj m`

figkigh.



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[e` Kdgn` Jf [e` E`dv`gs

It is dg `xtrdjrkigdry teigh, te` kdgn` jf te` n`l`stidl mjki`s ig te` e`dv`gs tedt eds m``g

jms`rv`k sign` dgni`gt tio`s, dgk fjugk tj m` k`snrimdml` ig t`ros jf te` ojst wjgkrjus

dsp`nts jf mjte guom`r dgk h`jo`try.

M`ndus` jf te` ojtijg jf te` `drte drjugk te` sug, te` stdrs will dpp`dr ig te` sdo`

pjsitijg ig te` sby fjur oigut`s `drli`r `dne gihet, ds te` `drte rjtdt`s, odbigh teò ris`

dgk s`t, sj tedt ig te` njurs` jf te` >60 kdy y`dr te`y will edv` odk` d njopl`t` cjurg`y

drjugk te` sby. [e` @drte hj`s drjugk te` ûg jgn` ig teis >60 kdy y`dr, dgk >60 isnljs` tj >61, weine dr` te` guom`r jf k`hr``s ig d nirnl`. X` edv` kivik`k up te` nirnl`

lib` teis, m`ndus` >61 is kivisiml` `v`gly my te` k`hr``s ig sp`nidl tridghl`s3 [e`

`quildt`rdl tridghl` (61-61-61), te` ;0-;0-91, dgk te` >1-61-91, odk` my krdwigh ig te`

dltituk` jf te` 61-61-61 tridghl`.

Ig turg, te` Ojjg hj`s drjugk te` @drte d littl` ojr` tedg tw`lv` tio`s ig d y`dr. [w`lv`

is te` sodll`st dmugkdgt guom`r. It is `v`gly kivisiml` my 4, =, >, ;, 6, dgk 4=. It gin`lyjrmits te` @drte ig dmjut >1 kdys, dgk w` dlr`dky o`gtijg`k >1 kivik`s gin`ly igtj >61.^j, w` edv` te` tw`lv` ojgte y`dr dgk te` >1 kdy ojgte. X` kjuml` jur sodll`st

dmugkdgt guom`r, 4= tj h`t =;, dgk w` kivik` jur kdy igtj =; ejurs, te` kdy mìgh te`

tio` it tdb`s te` `drte tj r`vjlv` jg its dxis.

X` te`g kivik` `dne ejur igtj 61 oigut`s, dgk `dne oigut` igtj 61 s`njgks. X` dlr`dky

o`gtijg`k 61 kivik`s gin`ly igtj >61 k`hr``s. X` kivik` `dne qudrt`r jf te` ojjg igtj 8

kdys, sj tedt dft`r d ojgte w` edv` nljs` tj its dpprjxiodt` jrmitdl p`rijk jf >1 kdys.

X` edv` kjg` d hjjk cjm tedt m`hdg wite te` dgni`gts, jf odbigh te` ojtijgs jf te`

n`l`stidl mjki`s, weine dr` lib` d gdturdl nljnb, tj fit teò wite te` b`y guom`rs, dgkh`jo`tri`s. It wjulk s`ò teis is wedt w` edv` kjg`. It is dodzigh tj o` tedt te` `drtedgk ojjg ndo` dmjut ig d wdy tedt te`y ojv` sj nljs`ly tj sune d sneò`.

Gjw I figk tedt te`r` is d yig dgk ydgh ig te` Pgiv`rs` jf 9/0 dgk 0/>, r`sp`ntiv`ly frjo

weine ndg m` k`riv`k wedt I ndll te` Prdgus `qudtijg, dgk, it s`òs tj r`njgnil` teò`trin systò wite te` fjjt pjugk systò, dgk furte`r, tejuhe euodg igv`gtijgs, sejw

teò njgg`nt`k gjt cust tj jg` dgjte`r, mut tj te` Njsojs.

Kj dll stdr systòs ig te` Pgiv`rs`, `sp`nidlly tejs` tedt ejst igt`llih`gt lif`, edv` sunedg `xtrdjrkigdry struntur` ds jurs, p`redps igkindtigh te`y dr` ig te` Pgiv`rs` fjr d

r`dsjg, ds jurs sdys dmjut us: Odgu`l nldio`k e` edk m``g dll dnrjss te` Hdldxy, dgktedt tejuhe kiff`r`gt, te`y kik.



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Oy Njsoin Drned`jljhy is mds`k jg te` ik`d tedt 9/0 is te` Yig jf te` Pgiv`rs`, 0/> is

te` Ydgh jf te` Pgiv`rs`, dgk 40 (Odgu`l’s guom`r, O) is te` odhin guom`r. Xitete`s` ter`` guom`rs w` njgstrunt d Pgiv`rs` tedt s``os stribighly ojr` tedg

njignik`gtdl. [e` Dpp`gkix dt te` ̀ gk jf te` mjjb eds kdtd us`k dgk ndlnuldtijgs gjt

kjg`.



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D@->0

I wrjt` d sejrt stjry ldst gihet, ndll`k Hypsy ^edodgiso dgk te` Pgiv`rs` dmjut te` D@-

>0 ugit, weine is te` ugit ig te` ojvi` dgk mjjb =1143 D ^pdn` Jkyss`y tedt EDL

r`pjrts will fdil dgk kisnjgtigu` njoougindtijg tj @drte. I k`nik`k tj r`dk te` pdssdh`

k`dligh wite te` `v`gt ig =114 dgk EDL, te` seip njoput`r, r`pjrts it will fdil ig witeig8= ejurs. ^trdgh`, m`ndus` V`gus is te` sjurn` jf 8.= ig oy G`ptug` `qudtijg dgk

r`pr`s`gts fdilur`, we`r` Odrs r`pr`s`gts sunn`ss.

Idg M`drksl`yDuhust 0, =14=



5

Nedpt`r Jg`

It oust edv` m``g 4959 jr 4991 we`g I tjjb d l`dv` jf dms`gn` frjo [e` Pgiv`rsity Jf

Jr`hjg, stukyigh ^pdgise, Qeysins, dgk wjrbigh dt te` stdt` jms`rvdtjry ig Jr`hjg --

Qig` Ojugtdig Jms`rvdtjry‛tj pursu` fldo`gnj ig ^pdig.

[e` Ojjrs, wej ndrv`k ndv`s igtj te` eills fjr r`sik`gn` we`g te`y w`r` muilkigh te`

Dledomrd Ndstl` jg te` eill fdnigh teò, dmdgkjg`k teò m`fjr` te` Hypsi`s, jr ]jod,

edk drriv`k te`r` ig Hrdgdkd ^pdig. [e` Hypsi`s w`r` r`sjurn`ful `gjuhe tj stunnj dgk

til` te` dmdgkjg`k ndv`s, dgk tdb` teò up fjr ejo`s.

Livigh ig jg` sune ndv` jwg`k my d hypsy sedodg, wds r`dlly gjt d kjwg dgk jut

situdtijg, ds te`s` ejo`s edk pluomigh dgk hds njjbigh ugits tedt rdg jff mjttl`s jf

prjpdg`. It wds r`dlly njopdrdml` tj livigh ig d Gdtiv` Do`rindg dkjm` ejo` ig G`wO`xinj.

Jf njurs` livigh ig sune d pldn` ndo` wite r`spjgsimiliti`s, dgk tedt ignluk`k wdt`righ itshdrk`gs. [e` êdodg tjlk o`3 “Xdt`r te` fljw`rs, dgk, we`g yju dr` kjg`, rjll up teèjs` dgk put it ig te` ndv`, jr it will h`t stjl`g‟. I edk stuki`k Ndstilidg ^pdgise ig

njll`h` dgk ds sune d ejs` is “ugd odghu`rd‟, mut te` êdodg ndll`k it “ugd hjod‟ dgk

hjod trdgsldt`s ds rumm`r. ]jll up te` ejs` dgk put it dwdy we`g yju dr` kjg` wite it3

hjjk dkvin`!

^j, I wdt`r te` fljw`rs, rjllup te` ejs` dgk put it dwdy. [e` êdodg njo`s tj te` ndv`

te` g`xt kdy dgk t`lls o` I kikg’t rjll up te` ejs` dgk put it dwdy, sj it hjt stjl`g, dgktedt I edk tj muy eio d g`w jg`.

E` njo`s my te` ndv` d f`w kdys ldt`r, wdb`s o` up dsbs o` tj dnnjopdgy eio jut jf [e` ^dnrjojgt`, tj sjo` pldn` m`tw``g te`r` dgk te` jlk Drdmin nity, Dlmdinig, tj muyeio d g`w ejs`.

It wdsg’t d fdr wdlb dt dll, te` `quivdl`gt jf d f`w nity mljnbs frjo te` ndv`s. X` h`t tj

te` stjr`, weine wds d njugt`r fdnigh te` str``t, gjt jg` tedt yju njulk `gt`r. E` sdys tjte` odg mèigk te` njugt`r, hiv` o` 0 o`t`rs jf ejs`. [e` odg mèigk te` njugt`r

pull`k jff fiv` o`t`rs jf ejs` frjo te` spigkl`, dgk nut te` ejs` tj tedt l`ghte. E` stdt`k

d vdlu` ig p`s`tds, odym` 511, jr sj, (dmjut ìhet kjlldrs dt te` tio`) dgk te` êdodg tjlk

o` tj hiv` tedt dojugt tj te` odg mèigk te` njugt`r, wej wds ^pdgise. I pdik te` odg,dgk w` l`ft.

I ndrri`k te` ejs`, dgk te` êdodg wdlb`k dljgh sik` o` ugtil w` drriv`k dt eis ndv`

we`r` I wds stdyigh. X` `gt`r`k te` ndv` stjpp`k dt te` wdlb wdy m`tw``g livigh rjjodgk bitne`g, dgk e` sdik3 “fjlljw o`‟. X` w`gt terjuhe d tugg`l tedt edk dmjut ter``

nedom`rs ig te` ndv`, dgk `gt`r`k jg` jg jur rihet ds w` w`r` e`dkigh ig, dgk w`

stjpp`k dgk m`fjr` o` wds d njll`ntijg jf wedt I `stiodt`k tj m` fift``g rumm`r ejs`s

sittigh jg hrjugk. [e` êdodg tjlk o` tj s`t te` jg` I edk cust mjuhet eio jg te` fljjr



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wite te` jte`rs. I kik, dgk w` l`ft te` nedom`r, dgk e` l`ft te` ndv`, dgk I r`tr`dt`k tj d

njune ig te` ndv` livigh rjjo.

Nedpt`r [wj

Hypsi`s edv` d wdy jf bgjwigh teighs dmjut d p`rsjg, we`te`r jr gjt jg` kisnljs`s it tjteò ig wjrks, dgk [e` êdodg wds dwdr` tedt I gjt jgly wjrb`k ig Dstrjgjoy, mut tedt

oy wjrb ig dstrjgjoy igvjlv`k bgjwigh dgk kjigh `l`ntrjgins.

^j, odym` d w``b jr twj dft`r I edk mjuhet eio d ejs`, e` ndo` tj eis ndv` we`r` I wdsstdyigh, dgk dsb`k o` if I wjulk m` dml` tj igstdll dg dgt`ggd fjr t`l`visijg dt dg

dpdrto`gt we`r` eis g`pe`w liv`k.

^j teis tio` I wds gjt ndrryigh d ejs` terjuhe [e` ^dnrjojgt`, mut dg dgt`ggd.

[e`r` w`r` s`v`rdl jf us jg te` pdtij, jg d eill dkcdn`gt tj te` dpdrto`gt jf [e` êdodg’s

G`pe`w, igstdlligh dg dgt`ggd fjr t`l`visijg r`n`ptijg.

Nedpt`r [er``

I do gjw ig ^jute`rg Ndlifjrgid, dt te` ejus` jf oy ojte`r, it is ldt` dt gihet, se` is d

dsl``p, dgk I do dmjut =; y`drs jlk dgk I k`nik` tj ljjb jut te` wigkjw, `dst, dnrjss [e`Dtldgtin, tj ^pdig. Ioo`kidt`ly I s`` te` êdodg, ig eis livigh rjjo, we`r` I edk ̀ dt`g d

mjwl jf te` Hypsy sjup ndll`k Qune`rj, dgk I e`dr te` wjrk Dgt`ggd. I gjw r`dliz`

we`g I igstdll`k te` dgt`ggd, I edk m`njo` jg`, dgk wds r`nìvigh o`ssdh`s frjo te`êdodg.

[e` êdodg’s Neilkr`g w`r` fldo`gnj huitdrists, dgk I l`drg`k frjo teò, tj pldy te`huitdr. I do gjw pldyigh fldo`gnj, wite igstruntijgs frjo te` sedodg tj put te` hypsyspdn` prjhrdo igtj oy ousin. I r`dliz` I do gjt cust dgy dgt`ggd, mut te` D@>0 tedt

odlfugntijg`k dmjdrk [e` Kisnjv`ry cust m`fjr` it drriv`k dt te` pldg`t Cupit`r ig Drteur

N. Nldrb`’s dgk ^tdgl`y Bumrinb’s “=1143 D ^pdn` Jkyss`y‟. [e` êdodg t`lls o`,

t`l`pdteindlly, tedt teis tio` te` oissijg wjg’t fdil.

Nedpt`r Fjur

I do wdtneigh ^tdr Xdrs dgk s`` d spdn`seip, weine is twj jmljgh ndpsul`s flyighnjgg`nt`k ig tdgkò. [e` Hypsy êdodg sdys tj o` t`l`pdteindlly3 “Kijs `s ugd ik`d3

sjg kjs‟. I ugk`rstdgk tedt tj o`dg “Hjk is dg ik`d3 te`r` dr` twj `lò`gts‟. ^j I hjterjuhe lif` mdsigh oy lif` jg te` guom`r twj.

Nedpt`r Fiv`

Jgn` jg` eds tdst`k ^pdig, tedt p`rsjg ljghs tj r`turg. I ldgk ig Odkrik, Gjrte`rg ^pdig,

[e` Ndpitjl. [e` ^pdgidrks bgjw oy k`stigdtijg is Hrdgdkd, ^jute`rg ^pdig, [e` HypsyGìhemjrejjk ndll`k [e` ^dnrjojgt`, te` ndv`s, dgk ioo`kidt`ly r`njhgiz` I do ugk`r



41

te` sp`ll jf d Hypsy êdodg, dgk wedt is ojr` tedt I do [e` D@>0 Dgt`ggd fjr [e`

Hypsy ^pdn` Qrjhrdo. Fldo`gnj mìgh fldo`gnj, te` ^pdgidrks kj gjt ugkj te` sp`ll,mut r`prjhrdo te` igstruntijgs fjr o`, te` D@>0 Dgt`ggd, sj tedt we`g I drriv` mdnb ig

te` Pgit`k ^tdt`s, oy fldo`gnj will gjw stdt` teìr ik`d jf d spdn` prjhrdo. It wds jf

njurs`, fldo`gnj mìgh fldo`gnj, dg dttòpt tj jut-kj te` Hypsy spdn` prjhrdo.

Nedpt`r îx

I do mdnb ig te` Pgit`k ^tdt`s dgk I do dt te` ejus` jf oy ojte`r, it is gihet tio` dhdig,

se` is dsl``p, dgk I ljjb jut te` wigkjw `dst, dnrjss te` Dtldgtin, tj ^pdig, dgk teis tio` Ikj gjt s`` te` livigh rjjo jf te` hypsy sedodg, mut te` str``ts jf Odkrik dt gihet, dgk dll

te` p`jpl`, dgk te` wjrk Cupit`r njo`s tj oigk dgk I do dmjut tj sdy jf njurs`, Cupit`r,

dgk [e` ^pdgise igt`rrupt dgk sdy “Y`s, yju dr` rihet it is te` ldrh`st pldg`t ig te` sjldr

systò, yju dr` rihet tj njgsik`r it, dll `ls` will fljw frjo it.‟

I bgjw rdtijs, ig odteòdtins dr` te` ojst igt`r`stigh sumc`nt, lib` pi, te` rdtij jf te`

nirnuof`r`gn` jf d nirnl` tj its kido`t`r, dgk te` hjlk`g rdtij, sj I njgsik`r te` rdtij jf te` jrmit jf ^dturg (te` s`njgk ldrh`st pldg`t ig te` sjldr systò) tj te` jrmit jf Cupit`r dtteìr nljs`st dpprjdne`s tj [e` ûg, dgk figk it is gig`-fiftes (gig` njopdr`k tj fiv`)

weine kivik`k jut is jg` pjigt ìhet (4.5).

I te`g prjn``k tj te` g`xt ljhindl st`p3 gjt rdtijs, mut prjpjrtijgs. D rdtij is teisnjopdr`k tj tedt, mut d prjpjrtijg is teis is tj tedt ds teis is tj tedt. ^j te` qu`stijg is3

^dturg is tj Cupit`r ds wedt is tj wedt: Jf njurs` te` dgsw`r is ds Hjlk is tj îlv`r. Hjlk

is kivig`2 silv`r is g`xt kjwg jg te` list. Jf njurs` jg` kj`s gjt njopdr` d kjz`gjrdgh`s tj d edlf kjz`g dppl`s, mut d kjz`g jf jg` tj d kjz`g jf te` jte`r, if jg` wdgts tj

`xtrdnt dgy bigk jf o`dgigh. Mut dtjos jf hjlk dgk silv`r dr` gjt o`dsur`k ig kjz`gs,

mut ig ojl`s. ^j I njopdr`k d ojl` jf hjlk tj d ojl` jf silv`r, dgk I sdik gj wdy, it isgig`-fiftes, dgk ^dturg is igk``k tj Cupit`r ds Hjlk is tj îlv`r.

I sdik tj oys`lf3 Ejw fdr kj`s teis hj: [e` êdodg’s sjg jgn` tjlk o` e` wds ig ljv`

wite te` ojjg. ^j I njopdr`k te` rdkius jf te` sug, te` kistdgn` frjo its n`gt`r tj its

surfdn` tj te` lugdr jrmitdl rdkius, te` kistdgn` frjo te` n`gt`r jf te` `drte tj te` n`gt`r jf te` ojjg. It wds Gig` njopdr`k tj Fiv` dhdig!

Nedpt`r ^`v`g

I edk fjugk 9/0 wds dt te` nrux jf te` Pgiv`rs`, mut fjr `v`ry yig te`r` edk tj m` d ydgh.

Gig` fiftes wds jg` dgk ìhet-t`gtes jf te` wdy drjugk d nirnl`. [e` jg` tjjb yju mdnb tjte` m`higgigh weine l`ft yju wite 5 t`gtes. Gjw hj tj ìhet t`gtes ig te` jte`r kir`ntijg,

it is 8= k`hr``s jf te` >61 k`hr``s ig d nirnl`. [edt is te` s`pdrdtijg m`tw``g p`tdls jg dfiv`-p`tdl`k fljw`r, d ojst pjpuldr drrdghò`gt. Igk``k lif` is bgjwg tj edv` fiv`-fjlk

syoo`try, te` peysindl, lib` sgjwfldb`s, six-fjlk. Kj te` dlhjriteo jf fiv`-fjlk

syoo`try ig r`v`rs` fjr six-fjlk syoo`try, dgk yju h`t te` ydgh tj te` yig jf gig`-fiftes

is fiv`-teirks.



44

Gig`-fiftes wds ig te` `lò`gts hjlk tj silv`r, ^dturg tj Cupit`r, ûg tj ojjg. Xe`r` wds

fiv`-teirks: ^dlt jf njurs`. “[e` ^dlt Jf [e` @drte‟ is tedt weine is hjjk, cust r`dkêdb`sp`dr`’s “Bigh L`dr‟. ^jkiuo is te` o`tdl njopjg`gt tj tdml` sdlt, Qjtdssiuo is,

dsik` frjo mìgh dg iopjrtdgt f`rtiliz`r, te` sumstitut` fjr ^jkiuo, ds d o`tdl njopjg`gt

tj odb` sdlt sumstitut`. [e` ojldr odss jf pjtdssiuo tj sjkiuo is fiv` tj ter``, te` ydgh

tj te` yig jf gig`-fiftes, weine is hjlk tj silv`r. Mut oultiply yig wite ydgh, tedt is gig`-fiftes wite fiv`-teirks, dgk yju h`t >, dgk te` `drte is te` teirk pldg`t frjo te` sug.

I tejuhet te` nrux jf te` ugiv`rs` oust m` te` kiff`r`gn` m`tw``g gig`-fiftes dgk fiv`-

teirks. I sumtrdnt`k te` twj dgk hjt twj-fift``gtes! [wj njopdr`k tj fift``g! I edkmjuhet te` êdodg eis fift``gte rumm`r ejs`, dgk dft`r e` odk` o` igtj te` D@>0

Dgt`ggd jg` jf eis first trdgsoissijgs tj o` wds3 “Hjk Is Dg Ik`d3 [e`r` Dr` [wj

@lò`gts‟.

It is sj jmvijus, te` ojst dmugkdgt hds ig te` @drte Dtojspe`r` is Gitrjh`g,neòindl syomjl 40!



4=

[e` ^`qu`gn`

X` njgsik`r`k te` rdtij gig` tj fiv`, te`g te` prjpjrtijg dgk fjugk it ig ^dturg Jrmit tj

Cupit`r jrmit, ^jldr ]dkius tj Lugdr Jrmit, Hjlk tj îlv`r dgk if fljw`r p`tdl

drrdghò`gts. It is l`ft te`g tj njgsik`r te` wejl` guom`r oultipl`s jf gig`-fiftes (4.5)

jr te` s`qu`gn`3

4.5, >.6, 0.;, 8.=,…

ig jte`r wjrks, dgk w` ljjb tj s`` if it is ig te` sjldr systò dgk figk it is ig te` fjlljwighwdys3

4.5

^dturg Jrmit/Cupit`r Jrmit

^jldr ]dkius/Lugdr Jrmit

Hjlk/îlv`r

>.6

(41)O`rnury ]dkius/@drte ]dkius

(41)O`rnury Jrmit/@drte Jrmit

(`drte rdkius)/(ojjg rdkius)?;(k`hr``s ig d nirnl`)(ojjg kistdgn`)/(sug kistdgn`)

? >.8 ~ >.6

[e`r` dr` dmjut ds odgy kdys ig d y`dr ds k`hr``s ig d nirnl`.

(Vjluo` jf ^dturg/Vjluo` Jf Cupit`r)(Vjluo` Jf Odrs) ? 1.>8 numin `drte rdkii~ >.6

[e` ldtt`r ndg m` njgv`rt`k tj >.6 my oultiplyigh it my (@drte Odss/Odrs Odss) m`ndus`@drte is dmjut t`g tio`s ds odssiv` ds Odrs.

0.;

Cupit`r Jrmit/@drte Jrmit^dturg Odss/G`ptug` Odss

8.=

41(V`gus Jrmit/@drte Jrmit)



4>

[e` G`ptug` @qudtijg

If w` njgsik`r ds w`ll te` s`qu`gn` we`r` w` m`hig wite fiv` dgk dkk gig` tj `dne

sunn`ssiv` t`ro3 0, 4;, =>, >=…[e`g, te` struntur` jf te` sjldr systò dgk kygdoin

`lò`gts jf te` Pgiv`rs` dgk Gdtur` ig h`g`rdl dr` ti`k up ig te` twj s`qu`gn`s3

0, 4;, =>, >=,…

dgk

4.5, >.6, 0.;, 8.=,…

Ejw kj w` figk te` njgg`ntijg m`tw``g te` twj tj ljndliz` te` pivjtdl pjigt jf te` sjldr

systò: X` tdb` teìr kiff`r`gn`, sumtrdntigh r`sp`ntiv` t`ros ig te` s`njgk s`qu`gn`frjo tejs` ig te` first s`qu`gn` tj jmtdig te` g`w s`qu`gn`3

>.=, 41.;, 48.6, =;.5,…

Xeine is dg driteo`tin s`qu`gn` wite njoojg kiff`r`gn` jf 8.= o`dgigh it is writt`g

8.=g „ ; ? dZg

[e` dZg is te` gte t`ro jf te` s`qu`gn`, g is te` guom`r jf te` t`ro ig te` s`qu`gn`.

[eis w` gjtin` ndg m` writt`g3

R(V`gus-jrmit)/(@drte-jrmit)_R(@drte-odss)/(Odrs-odss)_g „ (Odrs jrmitdl #) ? dZg

X` edv` dg `qudtijg fjr d s`qu`gn` tedt sejws te` @drte strdkkl`k m`tw``g V`gus dgkOdrs. V`gus is d fdil`k @drte. Odrs prjois`s tj m` G`w @drte.

[e` Odrs jrmitdl guom`r is ;. If w` wdgt tj bgjw wedt pldg`t ig te` sjldr systò ejlks

te` b`y tj te` sunn`ss jf @drte, jr tj te` sunn`ss jf euodgs, w` l`t g ?> sign` te` @drte iste` teirk pldg`t jut frjo te` ûg, ig te` `qudtijg dgk te` r`sult is dZg ? 48.6. [eis o`dgs

te` pldg`t tedt ejlks te` b`y is G`ptug`. It eds d odss jf 48.=> `drte odss`s, d guom`r

v`ry nljs` tj jur 48.6.

Gjt jgly is G`ptug` te` igkindt`k pldg`t, w` figk it eds g`drly te` sdo` surfdn` hrdvity ds

`drte dgk g`drly te` sdo` ignligdtijg tj its jrmit ds `drte. [ejuhe it is oune ojròdssiv` tedg `drte, it is oune ldrh`r dgk te`r`fjr` l`ss k`gs`. [edt wds wey it njo`s jut

tj edv` te` sdo` surfdn` hrdvity.



4;

[e` Prdgus @qudtijg

I dsb`k wedt g``ks tj m` kjg` tj sjlv` Oy G`ptug` @qudtijg, my hjigh k``p wite te`

huitdr ig ^jl`d Qjr Mul`rds. I fjugk te` dgsw`r wds tedt I kikg’t edv` `gjuhe

igfjrodtijg tj sjlv` it.

[e`g I r`dliz`k I njulk nr`dt` te` njoplò`gt jf te` G`ptug` `qudtijg my ljjbigh dt te`

Ydgh jf 0/>, sign` te` G`ptug` `qudtijg ndo` frjo te` Yig jf 9/0.

X` us` te` sdo` o`tejk ds fjr te` G`ptug` `qudtijg3

^tdrt wite 5 dgk dkk 0 tj `dne dkkitijgdl t`ro (w` terjw d twist my gjt stdrtigh wite 0)

0/> ?7 5, 4>, 45, =>,…

List te` guom`rs tedt dr` wejl` guom`r oultipl`s jf 0/>3

0/>g ? 4.8, >.>, 0, 6.8,…

ûmtrdnt r`sp`ntiv` t`ros ig te` s`njgk s`qu`gn` frjo tejs` ig te` first3

6.>, 9.8, 4>, 46.>,…

[eis is dg driteo`tin s`qu`gn` wite njoojg kiff`r`gn` >.>. It ndg m` writt`g3

(dZg) ? > + >.>g

[eis ndg m` wrttt`g3

@drte Jrmitdl # + (Cupit`r Odss/^dturg Odss)g ? dZg

L`ttigh g ? > w` figk dZg ? 4>

[e` nljs`st tj teis is te` odss jf Prdgus, weine is 4;.0; `drte odss`s. If G`ptug` is te`

Yig pldg`t, te`g Prdgus is te` Ydgh pldg`t. [eis is igt`r`stigh m`ndus` I edk fjugk tedt

Prdgus dgk G`ptug` w`r` kiff`r`gt odgif`stdtijgs jf te` sdo` teigh. I edk writt`g3

I ndlnuldt` tedt tejuhe G`ptug` is ojr` odssiv` tedg Prdgus, its vjluo` is l`ss sune tedt

teìr prjkunts dr` nljs` tj `quivdl`gt. Ig odte3

GZv ? vjluo` jf G`ptug`GZo ? odss jf G`ptug`

PZv ? vjluo` jf Prdgus

PZo ? odss jf Prdgus

(GZv)(GZo) ? (PZv)(PZo)



40

[e` Pgifindtijg Jf Qi dgk Qei my Gig`-Fiftes

Dll tedt is l`ft tj kj is tj njgsik`r pi, te` nirnuof`r`gn` jf d nirnl` tj its kido`t`r, dgk

pei, te` hjlk`g rdtij, sign` te`y dr` twj ojst iopjrtdgt, if gjt ojst m`dutiful rdtijs ig

odteòdtins.

I edv` fjugk gig`-fiftes jnnurs terjuhejut gdtur` ig te` rjtdtijg jf p`tdls drjugk d

d fljw`r fjr d ojst pjpuldr drrdghò`gt, ig te` jrmits jf cupit`r tj sdturg ig teìr

nljs`st dpprjdne`s tj te` sug, ig te` rdtij jf te` ojldr odss`s jf hjlk tj silv`r, dgk

ig te` rdtij jf te` sjldr rdkius tj te` lugdr jrmit. I gjw furte`r hj jg tj sdy tedt teis

gig`-fiftes ugifi`s te` twj ojst iopjrtdgt rdtijs ig odteòdtins pi dgk te` hjlk`g

rdtij (pei), ig tedt

pi + pei ? >.4;4 + 4.645 ? ;.809

M`ndus` te` guom`rs dft`r te` k`niodl ig te` suo (te` iopjrtdgt pdrt) dr` 0 dgk 9

dgk 8, te` dv`rdh` jf 0 dgk gig`.

I sejulk dlsj lib` tj pjigt jut tedt te` fjurte dgk fifte guom`rs dft`r te` k`niodl igpi dr` 0 dgk 9 dgk ig pei te` s`njgk dgk teirk guom`rs dft`r te` k`niodl dr` jg`dgk ìhet we`r` gig`-fiftes kivik`k jut is jg` pjigt ìhet, dgk, furte`r, te` first dgks`njgk guom`rs dft`r te` k`niodl ig pei dkk up tj odb` 8, te` dv`rdh` jf 9 dgk 0,dgk sumtrdnt tj odb` fiv`, dgk te` s`njgk dgk teirk kihits dft`r te` k`niodl dkk uptj gig`. ^j gjt jgly kj`s te` sjldr systò ugify pi dgk pei terjuhe gig`-fiftes, pi

dgk pei tdb`g dljg` `xpr`ss gig`-fiftes ig te` m`st pjssiml` wdys.



46

ûoodry Dmjut Gig`-Fiftes

Ig ûoodry, teis ojst m`dutiful njgstdgt ig Gdtur`, jf gig`-fiftes, weine ugifi`s pi dgk

te` hjlk`g rdtij, te` twj ojst m`dutiful rdtijs ig odteòdtins, is te` ojst pjpuldr

drrdghò`gt jf p`tdls drjugk d fljw`r, dgk is ig te` rdtij jf te` ojldr odss`s jf hjlk tj

silv`r, te` ojst pr`nijus jf te` o`tdls. Jg` njulk jgly hu`ss tedt sune wjulk m`gjtin`dml` terjuhejut te` ugiv`rs` dgk wjulk m` gjtin`k my jte`r igt`llih`gt lif` ig te`

ugiv`rs`, dgk, ds sune te` r`ldtijgseip wjulk m` ik`dl fjr trdgsoissijg jf d o`ssdh` my

jte`r igt`llih`gt lif` fjros ig te` ugiv`rs` tj sdy tj te` r`nìv`r3 “I do e`r` dgk edv`

gjtin`k it‟. [e` nurijus teigh is tedt it `xists ig jur sjldr systò ig te` rdtij jf te` jrmit jf ^dturg dt its nljs`st dpprjdne tj te` sug njopdr`k tj te` jrmit jf Cupit`r dt its nljs`st

dpprjdne tj te` sug, oystindlly puttigh te` @drte dt jg` ugit frjo te` ûg (Cupit`r dgk

^dturg mìgh te` ldrh`st dgk ojst odssiv` ig te` sjldr systò, te` @drte tedt pldg`t weineedrmjrs igt`llih`gt lif` ig te` ^jldr ^ystò), dgk ig te` rdtij jf te` ûg’s rdkius tj te`

jrmitdl rdkius jf te` ojjg (ojjg-`drte s`pdrdtijg). [e` qu`stijg is wey kj`s teis

r`ldtijgseip fjr pi dgk te` hjlk`g rdtij jf gig`-fiftes gjt cust `xist terjuhejut te`

Pgiv`rs` ig hjlk, silv`r, dgk p`redps fljw`r p`tdl drrdghò`gts, mut ojr` ljndlly ig jursjldr systò: Kj`s it o`dg euodgs edv` sjo` bigk jf kivig` sp`nidlg`ss ig te` njsojs:

Xeil` w` ndg bgjw tedt te` gig`-fiftes pe`gjo`gjg wjulk m` r`pr`s`gt`k my hjlk dgk

silv`r terjuhejut te` ugiv`rs`, w` nurr`gtly ndggjt bgjw we`te`r te` igedmitdgts jf jte`r

stdr systòs edv` it ig teìr sug (stdr) te`y jrmit dgk d ojjg tedt oihet jrmit teìr pldg`tjr ig te` rdtij jf te` jrmits jf te` ldrh`st pldg`ts ig teìr stdr systò puttigh teìr pldg`t dt

jg` ugit frjo teìr stdr. [e` dgsw`r tj teis `gihod, I teigb, oust m` jf `xtrdjrkigdry

prjfugkity.



48

(pi) dgk (`)

I edv` tdlb`k dmjut ejw 9/0, weine I edv` fjugk `xists ig Gdtur` dgk te` Pgiv`rs`,

ugifi`s pi dgk te` hjlk`g rdtij (pei)3

(pi) + (pei) ? >.4;4 + 4.645 ? ;.809

m`ndus` te` first ter`` guom`rs dft`r te` k`niodl dr` 8, 0 dgk 9. ^`v`g is te` dv`rdh` jf

gig` dgk fiv`, dgk te` s`njgk guom`r is jur 0 ig gig`-fiftes dgk te` teirk guom`r is te` 9

ig gig`-fiftes.

It wjulk s`ò 9/0 ugifi`s ùl`r’s guom`r, `, dgk pi, ds w`ll3

(pi) + (`) ? >.4;4 + =.845 ? 0.509

[e` s`njgk guom`r dft`r te` k`niodl is te` 0 ig gig`-fiftes, dgk te` teirk guom`r dft`r te`

k`niodl is te` 9 ig gig`-fiftes. [e` first guom`r dft`r te` k`niodl is ìhet. [eis issihgifindgt m`ndus` te` 5 is te` 5 ig 4.5, weine is 9/0 kivik`k jut. [e` jg` will tdb` yjudll te` wdy drjugk d nirnl`, wedt is l`ft is 1.5.

Idg M`drksl`y

Cdgudry 48, =14>



45

I edv` gjw kisnjv`r`k tedt Odgu`l wds ig @ghldgk fjr ojr` tedg jg` r`dsjg. E` wds tj

o``t wite d Fr`gneodg wej wds d r`pr`s`gtdtiv` fjr te` Mur`du Jf Xìhets DgkO`dsur`s jg d oissijg tj nedgh` te` o`tejk jf k`figigh tedt kistdgn` ndll`k te`

biljo`t`r.

Dppdr`gtly it edk sjo`teigh tj kj wite eis ejs` njll`ntijg guom`righ fift``g ejs`s dgkeis f``ligh tedt Hjk wds dg ik`d jf weine te`r` dr` twj `lò`gts. It wjulk s`ò e`

ndrri`k d hr`dt s`nr`t tedt wds edgk`k kjwg tj eio frjo eis dgn`stjrs ds te`y pdss`k

terjuhe te` wjrlk my fjjt dt tio`s, ndo`l dt jte`r tio`s, dgk ndrdvdg dt jte`r tio`s

njopl`t`ly ugbgjwg, stdrtigh ig Igkid dgk `gkigh ig ^pdig.

[e` Kiff`r`gtidl @qudtijg

I tejuhet it wds r`dlly tio` I sjlv` d kiff`r`gtidl `qudtijg. I nejs` [e` Prdgus @qudtijg.X` gjw k`t`roig` hiv`g te` dnn`l`rdtijg ig te` Prdgus `qudtijg is >.> `drte hrdviti`s,te`g wedt is te` igitidl v`ljnity ig te` `qudtijg, weine is te` guom`r >:

(v) ? > + >.>t ([e` Prdgus @qudtijg)

(v)?dt?(>>/41)t we`r` V is v`ljnity, d is dnn`l`rdtijg, dgk t is tio`.

>?(>>/41)t

(t) ? >1/>>

951 no/s/s ? h ? te` surfdn` hrdvity jf te` `drte

(951 no/s/s)(>.>)?>,=>; no/s/s

(>,=>; no/s/s)(>1/>> s) ? =,9;1 no/s ? vZ1 vZ1 is te` igitidl v`ljnity

[eus w` ndg writ` te` Prdgus `qudtijg ds3

(v) ? =,9;1 no/s + (>,=>; no/s/s)t

[eis is te` kiff`r`gtidl `qudtijg3

(kx) ? (=,9;1 no/s)kt + (>,=>; no/s/s)t kt



49

[e` jgly teigh w` ldnb ig sjlvigh teis is d tio` fjr weine w` ndg k`riv` d kistdgn`.

Nl`drly if te` êdodg edk sjo` s`nr`t, it wds ig te` ejs` njll`ntijg e` sejw`k o`. Jf njurs`, te`g, te` tio` is 40 s`njgks. [e` r`sult is

(xZ1) ? (=,9;1 no/s)40 + ((4/=)(>,=>;) no/s/s)40W= ? ;;,411 +>6>,5=0 ? ;189=0 no

;189=0 no/411/4111 ? ;.189=0 bo

Gjtin` te` guom`rs dft`r te` k`niodl dr` 89=0. [e` s`v`g is te` dv`rdh` jf te` gig` dgk

te` fiv`, dgk te` gig` dgk fiv` odb` up te` oyst`rijus gig`-fiftes. [e` = is te` guom`rus`k tj dv`rdh` te` gig` dgk te` fiv` tj odb` te` s`v`g. Mut ojr` ignr`kimly te` vdlu` is

dlojst p`rf`ntly fjur biljo`t`rs. [eus Odgu`l w`gt tj @ghldgk ig oy m`li`f gjt cust tj

sp`gk tio` wite te` dw`sjo` êìld Nedgkrd, mut tj h`t [e` Mur`du jf Xìhets dgk

O`dsur`s tj k`fig` te` l`ghte jf d biljo`t`r ds3

(4/;) (Igt`hrdl R(=,9;1 no/s)kt + (>,=>; no/s/s) t kt_ oigus 1.189=0)

Dll tedt kivik`k my 411, te`g my 4111.

[edt r`f`rs tj te` igt`hrdl

411 is 411 no/o

4111 is 4111 o/bo

[edt igt`hrdl `vdludt`k frjo 1 tj 40 s`njgks.

Ejw njulk e` edv` bgjwg sj oune dmjut te` Pgiv`rs` dgk njgg`nt`k it tj te` biljo`t`r,we`g [e` O`trin Njooitt`` ig ojk`rg tio`s k`fig`k te` biljo`t`r ds jg` t`gtejusdgkte jf te` kistdgn` frjo te` Gjrte Qjl` tj [e` @qudtjr:

Odurin` Nedt`ldig, wej wds te` kir`ntjr jf k`v`ljpigh te` njoougindtijgs systò fjrte` GD^D Dpjllj Oissijgs, ody edv` te` dgsw`r ig eis mjjb “Jur Njsoin Dgn`stjrs‟.It ody m` tedt it wds `v`g bgjwg ljgh`r dhj tedg tedt kdy Odgu`l’s dgn`stjrs l`ft Igkid.Nedt`ldig writ`s3

“[eis is we`r` oy igt`r`st m`ndo` r`dlly drjus`k dgk oy wjrb m`ndo` `xnitigh…Dllugits jf o`dsur` ig te` kistdgt pdst jf jur nivilizdtijg edk te` sdo` mdsin systò ig teìrfjugkdtijgs „ dll w`r` k`t`roig`k frjo te` `xdnt kio`gsijgs jf jur pldg`t @drte. Dsignr`kiml` ds teis ody sjugk tj te` ugigitidt`k, jur dgn`stjrs k`riv`k teìr f``t dgk igne`sfrjo te` l`ghte jf k`hr`` jf ldtituk` jr ljghituk`…. Ejw njo` te`g tedt jur fjr`fdte`rsmdnb ig te` ^tjg` Dh` edk vdlu`s sj `xdnt, weine w` jurs`lv`s w`r` jgly dml` tj jmtdigdft`r ; Jntjm`r, 4908, we`g te` ^jvi`t sdt`llit` ^putgib stdrt`k tj trdn` dgk o`dsur` te`

first jrmits drjugk te` hljm`:… X` te`g jmtdig`k te` `xdnt o`dsurò`gts my jms`rvighte` irr`huldriti`s ig te` jrmits jf te` first drtifinidl sdt`llit`s ig jrk`r tj ndlnuldt` te` tru`sedp` dgk kio`gsijgs jr jur hljm`.‟

Nedt`ldig suhh`sts `xtrdt`rr`stridls, o`dsur`k te` `drte my jrmitigh it ig te` sdo` wdy dgk

hdv` jur ugits jf o`dsurò`gt tj jur pdst, kistdgt dgn`stjrs, 60,111 y`drs dhj. [e`

dgn`stjrs jf Odgu`l, bg`w teis dt l`dst d tejusdgk y`drs dhj, dgk njgg`nt`k it, furte`r, tj



=1

“[e` Njsojs‟. [e` wigk mljws ig dg `gkl`ss s`d jf sdgk. ]ikigh jg it is d vjin`

mr`dbigh up dgk kisdpp`drigh igtj te` dir3 “Ld tild, Hudkdndli‟.



=4

D ^`njgk Ljjb Dt [e` Prdgus Igt`hrdl

Prdgus Igt`hrdl Frjo 1 [j 40 ^`njgks

(=,9;1no /s)kt + (>,=>;no /s /s)1

40

" tkt 1

40

" ? ;.189=0bo

Xeine ndg m` writt`g ig biljo`t`rs3

;+(9+0)/=(411) + 9/4111 + =/(41,111) + 0/411,111

? ;+(1.18)+(1.119)+(1.111=)+(1.11110)

X` ndg dlsj writ`3

; bo + 81 o + 9 o + 1.= o + 1.10 o ? ; bo + 89.=0 o

Jr,

; bo + 8111 no + 911 no + =1 no + 0 no ? ; bo + 8,9=0 no

L`t’s njgv`rt te` pdrt m`yjgk ; bo tj f``t3

(8,9=0 no)(1.1>=515; ft/no) ? =61.11608 f``t

It is dlojst `xdntly =61 f``t. [e` pdrt dft`r te` k`niodl ndg m` drrdgh`k ds 0,6,8,… It

wjulk s`ò usigh te` êdodg’s 40 s`njgks ig [e` Prdgus Igt`hrdl wdgts tj njgg`nt teò`trin systò tj f``t. It is dnnurdt` tj dmjut six dgk d edlf tejusdgktes jf d fjjt. If w`

njgsik`r te` jrihig jf te` o`trin systò mìgh ig te` kio`gsijgs jf te` @drte, dgk te`jmsnur` jrihigs jf te` fjjt, teis is rdte`r ròdrbdml`, `sp`nidlly we`g jg` njgsik`rs te`

jrihigs jf [e` Prdgus @qudtijg.

Gjw w` tdb` te` s`njgk ljjb3

(=61.11608 ft)(4= ig/ft) ? >,4=1.1855; igne`s

It is dlojst >,4=1 igne`s! L`t’s r`drrdgh` tejs` guom`rs. X` h`t3

1, 4, =, >,...

Is it tedt `xtrdt`rr`stridls hdv` us te` igne, te` fjjt, dgk o`trin systò ig dgni`gt tio`s ds

Odurin` Nedt`lldig jf GD^D suhh`st`k ig eis mjjb, Jur Njsoin Dgn`stjrs: [e`r` edsdlwdys m``g d k`mdt` m`tw``g @urjp` dgk Do`rind ds tj we`te`r Do`rind sejulk dkjpt

te` o`trin systò ig `v`rykdy lif`. Do`rindgs dlr`dky us` it ig te` sni`gn`s. Is te`r` d

r`dsjg fjr te`s` kiff`r`gt systòs jf o`dsurò`gt: X`r` te`y mjte o`dgt tj m` ig tedt

te`y dr` njgg`nt`k tj jg` dgjte`r terjuhe te` Pgiv`rs` dgk Gdtur`:



==

[e` ^jurn` Jf Jg` Fjurte Ig [e` Prdgus Igt`hrdl

4 bo ? 4/(41,111)(4/;)N we`r` N ? =(pi)]Z`

N is te` nirnuof`r`gn` jf te` @drte dgk ]Z` is te` rdkius jf te` @drte.

]Z` ?6,>85 bo

X` ndg sdy3

; bo ? (4/41,111)N

Prdgus Igt`hrdl (4/;) (Igt`hrdl R(=,9;1 no/s)kt + (>,=>; no/s/s) t kt_ oigus 1.189=0)

Dll tedt kivik`k my 411, te`g my 4111.

[edt r`f`rs tj te` igt`hrdl

411 is 411 no/o4111 is 4111 o/bo

[edt igt`hrdl `vdludt`k frjo 1 tj 40 s`njgks.

X` dlsj ndll te` fjlljwigh [e` Prdgus Igt`hrdl3

(=,9;1no /s)kt + (>,=>;no /s /s)1

40

" tkt 1

40

" ? ;.189=0bo

[e` fdntjr jf jg` fjurte `gt`rs m`ndus` te` biljo`t`r is k`fig`k my te` kistdgn` frjo te`te` pjl` tj te` `qudtjr, gjt te` nirnuof`r`gn` jf te` @drte.



=>

Dhdig, w` fjugk 9/0 r`nurr`gt ig te` Pgiv`rs` dgk Gdtur`.

[e` Hypsy êdodg eds igkindt`k tedt 40 is dt te` e`drt jf

odtt`rs. X` us`k 9,0, dgk 40 tj njgstrunt te` fjlljwigh

qudkrdtin3

[e` dr`d jf d r`ntdghl` is 40. [e` l`ghte, l, is 9 ojr` tedg te`

wikte oultipli`k my 03

w=+9w " > ? 1

w ? 1.>=

X` te`g odk` d qudkrdtin ig tio`3

40 ?

4

=ht

=

t =?

>1

h

M`ndus` jf te` r`sult fjr w, w` nejs` h ? >= f``t/s`n/s`n ds

jppjs`k tj te` vdlu` ig o`t`rs h ? 9.5 o`t`rs/s`n/s`n

t = ?

>1 ft

>= ft /s`n/s`n

t ? 1.965s`n

X` gjt` tedt 4-1.965 ?1.1>=

I figk tedt igt`r`stigh. [e` êdodg’s 40 s`njgks k`snrim`k

o`t`rs ig te` Prdgus Igt`hrdl dgk te` guom`rs ig gig`-fiftes

dgk teìr dv`rdh`, gjt tj o`gtijg te`r` wds te` twj us`k tj

dv`rdh` gig` dgk fiv`, gjw it is prjfjugkly njgg`nt`k tj te` >=

jf `drte surfdn` hrdvity ig f``t.



=;



=0

Qjugks Dgk Biljhrdos

@drly jg ig oy wjrb, we`g I first gjtin`k gig`-fiftes ig te` sjldr systò dgk gdtur`, I wds

igtrihu`k my te` fdnt tedt te`r` is gig`-fiftes jf d k`hr`` n`gtihrdk` p`r `v`ry k`hr`` jf

Fder`geìt. I edk tri`k tj figk te` peysindl njgg`ntijg njgn`rgigh teis, dgk l`drg`k

Fder`geìt wds njgstrunt`k sj tedt fr``zigh tòp`rdtur`, jr g`dr te`r`, lit`rdlly te`tòp`rdtur` o`dsur`k my d te`rojo`t`r ig d oixtur` jf sdlt dgk in`, p`redps sjo` wdt`r,

wds tj m` tdb`g ds >= k`hr``s dgk te` upp`r lioit wds k`t`roig`k my te` euodg mjky

tòp`rdtur` ds o`dsur`k my d te`rojo`t`r ig dg dropit tj m` d vdlu` drjugk 98 jr 95.

Xey te`s` vdlu`s w`r` nejs`g I njulk gjt figk dgy r`dsjg fjr tedt wds kjnuo`gt`k. I kikpjst teis stjry tj sjo` bigk jf d kisnussijg hrjup. Ldt`r I edk pjst`k oy kisnjv`ry jf

gig`-fiftes tj te` Cjs`pe Ndopm`ll kisnussijg fjruo dgk sjo`jg` te`r` sdik gig`-fiftes

wds ig te` ̀ qudtijg tedt njgv`rts n`gtihrdk` tj Fder`geìt. I g`v`r tdlb`k dmjut oy

`drli`r wjrb ig tedt dr`d, cust sdik, y`s te`r` is gig`-fiftes k`hr`` n`gtihrdk` p`r `v`ryk`hr`` jf Fder`geìt. E` b`pt sdyigh yju edv` tj dkk teirty-twj. I njulk gjt njgvign`

eio wedt I wds sdyigh wds tru`. Dkkigh te` teirty-twj is tj z`rj te` r`ldtijgseip sj jg`

ndg njgv`rt n`gtihrdk` tj Fder`geìt. [edt is teìr sndl`s dr` kiff`r`gt3 N`lsius m`higs dt1 k`hr``s is fr``zigh dgk Fder`geìt m`higs dt >= is fr``zigh tòp`rdtur`. [edt kj`s gjtnedgh` te` fdnt tedt te`r` is gig`-fiftes k`hr`` n`gtihrdk` p`r `v`ry k`hr`` jf Fder`geìt.

E` wds pr`s`gtigh te` fjrould tedt njgv`rts jg` tòp`rdtur` tj te` jte`r.

[j te` odtt`r dt edgk3 [e` Hypsy êdodg’s 40 s`òs tj r`njgnil` o`trin kistdgn`s witef``t, mjte systòs us` te` s`njgk tj o`dsur` tio`. [e` Prdgus igt`hrdl is k`riv`k frjo

gig`-fiftes, weine w` s`` ugifi`s te` o`trin o`dsurò`gt jf tòp`rdtur` wite tedt jf te`

fjjt-pjugks Fder`geìt. [edt jgly l`dv`s us g``kigh tj figk te` njgg`ntijg m`tw``g teò`trin systò’s hrdos dgk biljhrdos wite te` pjugk, sj w` edv` k`dlt wite te` odtt`r

we`r` odss dgk wìhet dr` njgn`rg`k. D hrdo is gin`ly k`fig`k ds te` dojugt jf wdt`r

ig d numin n`gtio`t`r. Dll I ndg sdy, dt teis pjigt, is tedt jg` pjugk is ;0>.09 hrdos, dgktedt odb`s it `xdntly 1.09 hrdos frjo d wejl` guom`r. It eds sjo`teigh tj kj wite gig`-fiftes:



=6

H`ttigh D ^`gs` Jf [e` Prdgus @qudtijg

[eus w` ndg writ` te` Prdgus `qudtijg ds3

(v) ? =,9;1 no/s + (>,=>; no/s/s)t

[eis is te` kiff`r`gtidl `qudtijg3

(kx) ? (=,9;1 no/s)kt + (>,=>; no/s/s)t kt

Prdgus Igt`hrdl Frjo 1 [j 40 ^`njgks

(=,9;1no /s)kt + (>,=>;no /s /s)1

40

" tkt 1

40

" ? ;.189=0bo

Odne 4 ? 865 ope ?4,=>0 bo/ejur

[edt is odne 4 ig kry dir dt =1 k`hr``s N (65 k`hr``s F) dt s`d l`v`l.

If w` writ`, we`r` 4,=>0 bo/er (odne 4) ? 1.>;> bo/s, te`g3

>;,>11 no/s ?=,9;1 no/s + (>=>; no/s/s)t

dgk

t?9.696969698 s`njgks ? 9 =>/>> s ? >=1/>> s`njgks ~ 9.8 s`njgks

^j, te` Prdgus `qudtijg is d tio` jf 9.8 s`njgks tj r`dne odne 4. Quttigh tedt tio` ig teìgt`hrdl3

(x) ? (=,9;1)(>=1/>>) + 4/=(>=>;(>=1/>>)W= ? 451,008 no 4.51008 bo ~ 4.5bo

[eus ig te` Prdgus @qudtijg w` r`dne odne jg` ig dmjut 9.8 s`njgks dft`r trdv`ligh dkistdgn` jf dmjut 4.5 biljo`t`rs. L`t’s njgv`rt tedt tj oil`s3

(4.5bo)(1.6=4>84oi/bo)?4.445;85 oil`s

Jgn` dhdig w` s`` gig`-fiftes ugifyigh te` fjjt-pjugk systò tj te` o`trin systò ig tedt

4.5 is gig`-fiftes dgk d oil` is dmjut 4.5 biljo`t`rs. [edt is if w` teigb jf te` oil` ds pdrtjf te` fjjt-pjugk systò.



=8

[e` Prdgus Igt`hrdl Is Igtrihuigh

[e` Prdgus Igt`hrdl r`dne`s odne jg` ig 4.5 biljo`t`rs, weine is te` dojugt jf

biljo`t`rs ig d oil` dgk is te` 9/0 tedt jnnurs ig Gdtur` dgk te` Pgiv`rs`, gjt tj o`gtijg

tedt it ugifi`s pi dgk hjlk`g rdtij dgk pi dgk @ul`r’s guom`r `. X` edv` dlr`dky sejwg

ejw te` Prdgus Igt`hrdl k`snrim`s, biljo`t`rs, gig`-fiftes dgk f``t dgk igne`s, we`g`vdludt`k dt te` Hypsy êdodg’s Oyst`rijus 40 ig te` fjro jf 40 s`njgks. Fift``g is te`

k`hr``s te` @drte rjtdt`s terjuhe ig dg ejur. X` edv` sejwg ejw gig`-fiftes ugifi`s

hrdos dgk pjugks dgk n`gtihrdk` dgk fder`geìt.

[eus, gjt jgly kj`s te` Prdgus Igt`hrdl k`snrim` biljo`t`rs, it k`snrim`s oil`s. It is jg`

njopdnt stdtò`gt tedt òmjki`s `v`ryteigh. Dhdig w` s`` te` êdodg’s 40 ugifyigh

twj systòs jf o`dsurò`gt ig tedt te`r` dr` fift``g k`hr``s ig jg` ejur jf rihet

dsn`gsijg.



=5

Fift``g Ds X`ll Xdgts [j K`snrim` [e` Pgits Jf Oinrj`rhs

D kijk` is odk` jf silinjg (sdgk) kjp`k wite pejspejrjus dgk mjrjg, dll

gdturdlly jnnurrigh `lò`gts odk` (fjrh`k) ig igt`rijr jf stdrs. D silinjg

kijk` g``ks 1.6 vjlts tj njgkunt d nurr`gt. Ig d kijk` hdt` d eihe vjltdh` is

six vjlts dgk te` nirnuit is jg dgk d ljw vjltdh`, jr hrjugk (z`rj), te` nirnuitis jff. Jff is njkigh fjr z`rj dgk jg is njkigh fjr 4. S`rjs dgk jg`s ndg m`

strugh tjh`te`r tj odb` guom`rs, dgk l`tt`rs ig te` dlpedm`t.

Odgu`l, te` Hypsy êdodg, sdik3 “Hjk is dg Ik`d, [e`r` dr` twj `lò`gts.‟

X` ndg tdb` teis tj m` jur Yig dgk Ydgh jf 9/0 dgk 0/>. Ljw vjltdh` dgk

te` nirnuit is jff2 eihe vjltdh` dgk te` nirnuit is jg. [e`s`, dhdig, dr` te` twj

`lò`gts jf te` êdodg’s yig dgk ydgh.

M`ndus` yju edv` tj rug d kijk` dt l`dst 1.6 vjlts, teis is sumtrdnt`k frjo te`six vjlts tedt turg te` kijk` jg, sj r`dlly it jp`rdt`s jg slihetly l`ss tedg six

vjlts. [eis vjltdh` krjp jf 1.6 Vjlts ig te` nirnuit is wedt w` wdgt tj ljjb

dt. X` will `vdludt` te` vjltdh` krjp jv`r te` êdodg’s oyst`rijus 40

s`njgks dgk jmtdig dg dojugt jf pjw`r, te`g dpply tedt pjw`r jv`r jg` y`dr

„ te` tio` it tdb`s te` `drte tj hj drjugk te` ûg. X` edv` dlr`dky fjugk

tedt te` êdodg’s 40 gjt jgly ugifi`s o`t`rs wite te` ugiv`rs`, mut o`t`rs

wite f``t, dodzighly ig te` Prdgus Igt`hrdl jutlig`k mjte ig [e` @xpljits Jf

Odgu`l dgk Furte`r @xpljits Jf Odgu`l.



=9

Gjw w` kj te` ndlnuldtijg3

Jg` `l`ntrjg vjlt is te` `g`rhy jf dg `l`ntrjg fdlligh terjuhe d pjt`gtidlkiff`r`gn` jf jg` vjlt, weine is `qudl tj 4.61=@-49 Cjul`s.

Jg` Cjul` is 41,111,111 `rhs

(1.6 Vjlts)(4.61=@-49 Cjul`s) ? 9.64=@-=1 Cjul`s

(9.64=@-=1 Cjul`s)(41,111,111 `rhs/Cjul`) ? 9.64=@-4> `rhs

(9.64=@-4>`rhs)/(40 s`njgks) ? 6.;15@-4; `rhs/s`njgk

>60.=0 * =; * 61 * 61 ? >4,008,611 s`njgks/y`dr

(6.;15@-4; `rhs/s`n)(>4,008,611 s`njgks) ? =.1===44115 oinrj`rhs ? =.1

oinrj`rhs

It is dlojst `xdntly twj oinrj`rhs! (Is te` twj oinrj`rhs prjoptigh tedt w`k`njk` te` o`ssdh` ig migdry:)



>1

Xedt O`ssdh`:

Oy mjjb D [ejrjuhe Kio`gsijgdl Dgdlysis is dmjut ejw I prjvik``vik`gn` ig suppjrt jf te` ik`d tedt `xtrdt`rr`stridls hdv` us jur ugits jf o`dsurò`gt, lib` te` fjjt, te` o`t`r, te` s`njgk, ig Dgni`gt tio`s k`spit`

te` eistjry tedt t`lls us te` o`trin systò is ojk`rg, dgk te` fjjt-pjugksystò, p`redps d littl` domihujus ig its eistjry.

I teigb dljgh te` wdy igkjigh teis I ody edv` fjugk dg `gnrypt`k o`ssdh` frjo `xtrdt`rr`stridls igte` guom`r3

=.1===44115

Igsjfdr ds it ndo` tj us frjo gdturdl njgstdgts lib` te` nedrh` jf dg `l`ntrjg,te` jrmitdl p`rijk jf te` @drte, dgk guom`rs w` igv`gt`k lib` te` o`dsur` jf `l`ntrjojtiv` pjt`gtidl, te` vjlt. [e` ik`d is tedt `xtrdt`rr`stridls sjoèjwtjlk us tj o`dsur` `l`ntrjojtiv` pjt`gtidl wite d ugit jf vjlts m`ndus` it

wjulk prjkun` te` dmjv` guom`r. I do igt`r`st̀ k ig it m`ndus` jf te`r`p`titijg jf jg`s dgk twjs, dgk te` s`pdrdtijg jf guom`rs my z`rjs.

L`t us ljjb dt te` pdrt dft`r te` k`niodl. It is3

1===44115

It is te` s`njgk guom`r ter`` tio`s, te` first guom`r twj tio`s dgk te` ldst

guom`r is te` guom`r jf kihits tedt pr`n`k` it.

[e` ter`` twjs dgk te` twj jg`s dkk up tj te` ldst kihit, ìhet.

[e` twj jg`s dkk up tj `dne jf te` ter`` twjs.

It njugt frjo rihet tj l`ft 1,4,=,…



>4

K`njkigh [e` O`ssdh`

[e` o`ssdh` is3

1===44115

[e` === dkks up tj six, weine ig migdry is 441. [e` g`xt s`ho`gtis 4411, weine is d gimml`, dgk is te` guom`r 4=. [eis s`ho`gt isig migdry tj m`hig wite. [e` ldst kihit is dg `ihet, weine is 4111 igmigdry. [eus w` edv`, ignlukigh te` first kihit, d z`rj, weine is dmit3

(1) (441) (4411) (4111)

X` writ` teis ds ter`` gimml`s3

(1441) (4411) (4111)

[eus 1===44115 trdgsldt`s ds 1441 4411 4111

Xeine dr` te` ter`` guom`rs3

6, 4=, 5

[e`s` ter`` guom`rs kivik` gin`ly igtj te` >61 k`hr``s jf d nirnl`3

>61/4= ? >1 >61/6 ? 61 >61/5 ? ;0

[e`s` dr` te` dghl`s ig te` ter`` sp`nidl tridghl`s wite weine w`ndg writ` jut te` trihjgjo`trin fugntijgs jf iopjrtdgt dghl`s ignljs`k fjro3

>1-61-91 61-61-61 ;0-;0-91

6, 4=, 5 edv` te` njoojg k`gjoigdtjr jf twj. [eis hiv`s us te`ter`` guom`rs >, 6, ;3

>6;



>=

Xeine is v`ry nljs` tj te` guom`r jf @drte kdys ig d y`dr (>60).Dr` `xtrdt`rr`stridls pjigtigh jut tedt te` guom`r jf @drte kdys ig dy`dr dr` nljs`ly njgg`nt`k tj te` ter`` sp`nidl tridghl`s: [e`guom`r ig migdry is igt`r`stigh, m`ndus` te` jg`s snrjll jff frjorihet tj l`ft gimml` tj gimml`, jg` kihit dt d tio`.



>>

Dg Dlt`rgdt` Njsojljhy



>;

Ig Hypsy “êdodgiso Dgk [e` Pgiv`rs`‟ w` tjjb Cupit`r dgk ^dturg ds te` m`higgigh

pjigts ds te`y dr` g`xt tj jg` dgjte`r ig te` sjldr systò, dr` ig te` oikkl` jf te` sjldrsystò my guom`r, dgk ndrry ojst jf its `g`rhy jf ojtijg dgk dghuldr ojo`gtuo, ds

te`y dr` te` ojst odssiv` sihgifindgtly. [eìr kistdgn`s frjo te` ûg dt nljs`st

dpprjdne`s dr` dmjut fiv` dgk gig`, r`sp`ntiv`ly, puttigh te` @drte gin`ly dt jg` ugit frjo

te` ûg. Ds w` fjugk 9/0 ig te` ojldr odss jf hjlk tj te` ojldr odss jf silv`r, dgk igte` sjldr rdkius tj lugdr jrmit, w` ndll`k teis 9/0 te` Yig jf te` Pgiv`rs`. Ojst

ignr`kimly, mìgh 4.5, wite te` jg` tdbigh us drjugk d nirnl` jgn` l`dvigh 1.5 jf te` >61 ig

d nirnl`, w` gjt`k hjigh ig te` jte`r kir`ntijg tj it wds te` 8= k`hr``s ig te` s`pdrdtijg jf

p`tdls ig d fiv`-p`tdl fljw`r dgk tedt, fiv`-fjlk syoo`try is typindl tj lif`.

X` te`g sjuhet te` Ydgh jf six-fjlk syoo`try m`ndus` it is typindl tj peysindl gdtur`,

lib` sgjwfldb`s. X` sdik it wds 0/> sign` it r`pr`s`gts te` 4=1 k`hr`` o`dsur` jf dghl`s

ig d r`huldr e`xdhjg dgk w` muilt jur ugiv`rs` frjo te`r`, r`sultigh ig te` PrdgusIgt`hrdl, weine wds quit` fruitful. L`t us, ejw`v`r, teigb jf Ydgh gjt ds 0/>, mut ljjb dt

te` dghl`s m`tw``g rdkii jf d r`huldr e`xdhjg. X` edv`3

>61 „ 61 ? >11

>11 + >61 ? 661

661/>61 ? 44/6

X` sdy Yig is 9/0 dgk Ydgh is 44/6 dgk stinb wite [e` Hypsy êdodg’s 40 (^`` Dg

@xtrdt`rr`stridl Dgdlysis, nedpt`r titl`k “Hypsy êdodgiso Dgk [e` Pgiv`rs`‟) dgkmuilk jur Njsojljhy frjo te`r`.

X` dlr`dky muilt [e` G`ptug` @qudtijg frjo 9/0 dgk us`k it wite 0/> tj k`riv` te` pldg`t@urjpid, mut l`t us dpply 44/6 ig pldn` jf 0/>3

44/6 ?7 44/6, 44/>, 44/=, ==/>,… ? 4.5>>, >.668, 0.0, 8.>>>,…

44/6 ?7 6, 6+44 ? 48, 48+44?=5, =5+44?>9, … ? 6, 48, =5, >9,…

ûmtrdnt te` s`njgk s`qu`gn` frjo te` first3

;.468, 4>.>>>, ==.0, >4.668,…

Gjw w` figk te` njoojg kiff`r`gn` m`tw``g t`ros ig te` ldtt`r3 9.466, 9.468, 9.468,…

(dZg) ? d + (g-4)k ? ;.468+(g-4)9.468 ? ;.468 + 9.468g „ 9.468 ? 9.468g-0

[ry g?>3 9.468(>) „ 0 ? =8.014 „ 0 ? ==.014 (wjrbs)



>0

Jur `qudtijg is3

(dZg) ? 9.468g „0

X` gjtin` teis ndg m` writt`g3

R(^dturg Jrmit)/(@drte Jrmit)_g „ (Cupit`r Jrmitdl #) ? (dZg)

[e` G`ptug` @qudtijg fjr g?> hdv` G`ptug` odss`s, te` Prdgus `qudtijg fjr g?> hdv`

Prdgus odss`s. [eis `qudtijg fjr g?> hiv`s nljs` tj te` tilt jf te` @drte (=>.0 k`hr``s) igd fjro tedt is `xdntly edlf jf te` ;0 k`hr``s ig d squdr` wite its kidhjgdl krdwg ig. Ig te`

spirit jf jur first njsojljhy muilt upjg 9/0, 0/>, dgk 40, w` will ndll teis `qudtijg [e`

@drte @qudtijg.



>6

[e` Jrihig Jf [e` @xtrdt`rr`stridl O`ssdh` Is Ig [e` ^do` Qldn`Ds [e` Xjw! ^ihgdl



>8

X` edv` te` G`ptug` @qudtijg3

8.=x „;

X` edv` te` Prdgus @qudtijg

>.>x + >

Dgk gjw wite jur dlt`rgdt` njsojljhy w` edv` [e` @drte@qudtijg3

9x-0

Xite ter`` `qudtijgs w` ndg writ` te` pdrdo`t`riz`k `qudtijgs ig

>-kio`gsijgdl spdn`, pdrdo`t`riz`k ig t`ros jf t, fjr x, y, dgk z.X` ndg writ` frjo tedt f(x,y,z) dgk figk te` hrdki`gt v`ntjr, jrgjrodl tj te` `qudtijg jf d pldg` ig jte`r wjrks, dgk frjo tedt dr`hijg ig spdn`.



>5

x(t ) ?>6

0t " ;

y(t ) ?>>

41t + >

z(t ) ? 9t " 0

0 x +=1

>6?

41 y " >1

>>?

z +0

9

0

>6 x "

41

>> y "

4

9 z +

41

44? 1

# f ? $0/>6,"41/>>,"4/9%

d?0/>6 m?-41/>>

n ? (0/>6)=+ (41/>>)

=? 1.1945+1.149 ? 1.>>=5

k?-4/9

tdg" ? m /d

" ? #60.>05j

tdg$ ? k /n

$ ? #45.;6j

-60.>05 k`hr``s/40 k`hr``s/ejur ?-;.>08= ejurs

=; 11 11 „ ;.>08= ? 49.6;=5 ejurs

]D3 49e >5o >;s

K`n3 -45 k`hr``s =8 oigut`s >6 s`njgks



>9



;1

X` edk ter`` `qudtijgs3 [e` G`ptug` @qudtijg (k`riv`k frjo te` yig jf gig`-fiftes), [e`

Prdgus @qudtijg (k`riv`k frjo te` ydgh jf fiv`-teirks), dgk [e` @drte @qudtijg (k`riv`kfrjo te` dlt`rgdt` ydgh jf `l`v`g-sixtes). Xite ter`` `qudtijgs, w` njulk nr`dt` te`

`qudtijg jf d pldg` ig ter`` kio`gsijgs. X` fjugk te` gjrodl v`ntjr tj te` pldg` dgk

k`kun`k frjo its jri`gtdtijg k`hr``s ig, dgk ndll`k it rihet dsn`gsijg dgk its k`hr``s up

dgk ndll`k it k`nligdtijg. K`nligdtijg is dlr`dky ig k`hr``s, mut w` edk tj njgv`rt k`hr``sig igtj ejurs, oigut`s dgk s`njgks. X` gjt`k ig kjigh ds sune tedt jgn` dhdig te` Hypsy

êdodg’s 40 ugifi`s twj systòs jf o`dsurò`gt ig tedt te`r` dr` 40 k`hr``s tj `dne

ejur jf rihet dsn`gsijg. [edt is tj sdy tedt te` @drte rjtdt`s terjuhe 40 k`hr``s `v`ry

ejur (>61/=; ?40). [eis is iopjrtdgt m`ndus` teis ody m` dt te` mdsis jf jur oyst`ry. Iwrjt` dmjut wey I teigb w` edv` d =; ejur kdy dgk >61 k`hr``s ig d nirnl` ig oy nedpt`r

titl`k [e` Kdgn` Jf [e` E`dv`gs ig oy mjjb Dg @xtrdt`rr`stridl Dgdlysis.

Mdnb tj wedt w` w`r` kjigh3 X` k`riv`k frjo te` gjrodl v`ntjr3 (0/>6, -41/>>, -4/9) dljndtijg jg te` n`l`stidl spe`r` jf 49e >5o >;s dgk „45 k`hr``s =8’ >6‟ d r`hijg ig te`

njgst`lldtijg ^dhittdrius. [e` nljs`st prjoig`gt stdr tj teis I fjugk wds EK 45;5>0 dgk

dlsj g`dr, wds ]ej 14 ^dhittdrius. I fjugk teis jg te` sby nedrts ig te` fr``kjwgljdkdml` sby nedrtigh sjftwdr` ndll`k Ndrt`s ku Ni`l. X` will odb` d hu`ss tedt te``xtrdt`rr`stridls, wej te`jr`tindlly l`ft us d o`ssdh`, ds I k`kun`k frjo te` gdtur` jf

`l`ntrjgins, dr` frjo tedt r`hijg jf spdn`.

^dhittdrius is d rine suoo`r njgst`lldtijg fjr us e`r` ig te` Gjrte`rg Eòispe`r`, weineis ig te` kir`ntijg jf te` n`gt`r jf te` hdldxy.



;4



;=

[e` Xjw îhgdl Dgk Oy îhgdl

[e` prjc`ntijg my oy ndlnuldtijg terjuhe oy njsojljhy jf yig, ydgh, dgk 40 fjr te`

jrihig jf oy o`ssdh` frjo `xtrdt`rr`stridls wds sjo`we`r` ig te` `dst`rgojst pdrt jf te`

njgst`lldtijg ^dhittdrius. [eis edpp`gs tj m` te` sdo` pldn` we`r` te` jg` pjssiml` dli`g

sihgdl wds k`t`nt`k ig te` ^`drne Fjr @xtrdt`rr`stridl Igt`llih`gn` (^@[I). It wds ndll`k“[e` Xjw îhgdl‟ m`ndus` jg Duhust 40, 4988 te` mih `dr dgt`ggd r`nìv`k sjo`teigh

tedt s`ò`k gjt lib` stdr gjis`, mut `xdntly wedt te`y w`r` ljjbigh fjr ig dg

`xtrdt`rr`stridl sihgdl. Its gdo` is wedt it is m`ndus` te` dstrjgjo`r jg kuty, C`rry ].

@eodg wrjt` “Xjw!‟ g`xt tj te` guom`rs we`g te`y ndo` ig. Ignr`kimly, it ldst`k te`full 8= s`njgks tedt te` Mih @dr dgt`ggd list`g`k fjr it. I sdy ignr`kiml` m`ndus` I edv`

o`gtijg`k te` iopjrtdgn` jf 8=, gjt cust ig oy G`ptug` `qudtijg „ fjr weine oy ljndtijg

ig spdn` wds k`riv`k ig pdrt „ mut m`ndus` jf its njgg`ntijg tj te` Hypsy êdodg’s D@-

>0 dgt`ggd dgk its r`ldtijg tj 8= ig te` ojvi` “=1143 D ^pdn` Jkyss`y‟. [e` `stiodtijgjf te` njjrkigdt`s fjr te` jrihig jf te` Xjw sihgdl dr` twj3

49e==o=;.6;s

49e=0o48.14s

Xite k`nligdtijg jf3

-=6 K`hr``s =0 oigut`s 48.14 s

[edt is dmjut =.0 k`hr``s frjo te` stdr hrjup Nei ^dhittdrri

It is v`ry nljs` tj oy ndlnuldtijg fjr dg `xtrdt`rr`stridl nivilizdtijg tedt I f``l eik d

o`ssdh` ig jur peysins, weine I ndlnuldt` tj m` g`dr EK 45;5>0 dgk `xdntly dt3

49e >5o >;s

-45 K`hr``s =8 oigut`s >6 s`njgks

[e` t`l`snjp` tedt k`t`nt`k te` Xjw îhgdl wds dt Jeij X`sl`ydg Pgiv`rsity K`ldwdr`,Jeij ndll`k [e` Q`rbigs Jms`rvdtjry.

Idg M`drksl`y Ody 6, =14>



;>

[e` Ojst Mizdrr` ^tjry @v`r [jlk

Kik `xtrdt`rr`stridls r`dlly dgtinipdt`, we`g te`y s`gt us ig 4988 d trdgsoissijg frjo te`

njgst`lldtijg ^dhittdrius, tedt ldst`k 8= s`njgks (^@[I’s Xjw îhgdl) tedt, d Hypsy

êdodg, drjugk 4991, wjulk odb` o` igtj [e` D@->0 Dgt`ggd jf =1143 D ^pdn`

Jkyss`y tedt te` seip njoput`r, EDL, wjulk sdy wjulk fdil witeig 8= ejurs, te` sdo` 8=jf te` Xjw îhgdl: Kik te`y bgjw te` trdgsoissijgs I wjulk r`nìv` frjo te` Hypsy

êdodg, Odgu`l, wjulk l`dk gjt cust tj te` kisnjv`ry jf d o`ssdh` òm`kk`k ig jur

peysins, mut d systò jf ter`` `qudtijgs tedt wjulk pjigt tj te` sdo` r`hijg jf spdn` ds

te` Xjw îhgdl, te` njgst`lldtijg ^dhittdrius, fjr te` jrihig jf oy o`ssdh`:

[eis is te` sdo` 8= tedt is ig te` 1.8= dstrjgjoindl ugits jf V`gus frjo te` ûg. It is te`

sdo` 8= tedt is te` guom`r jf y`drs fjr jg` k`hr`` jf pr`n`ssijg jf te` @drte’s

@quigjx`s. It is te` sdo` 8= jf te` eihe`st hòdtrid jf te` E`mr`w Hjk, YEXE, dgkfurte`r V`gus wds te` Qdtrjg`ss jf Gig`vè (Isetdr) we`r` [e` Gig`vè Njgstdgt wds

fjugk ig O`sjpjtdoid, ig Dgni`gt ûo`ridg nldy nugìfjro tdml`ts, d njgstdgt tedt is te`

njoojg k`gjoigdtjr fjr dll n`l`stidl nynl`s ig jur sjldr systò.

Idg M`drksl`y

Cug` 4;, =14>



;;

D Qjssiml` O`dgigh Jf [e` D@->0 Dgt`ggd

Quttigh oys`lf tj teigbigh dmjut te` o`dgigh jf te` D@->0 dgt`ggdig =1143 D ^pdn` Jkyss`y, I wjulk sdy it is, ig te` filo, te` turgdrjugk pjigt fjr euodgbigk m`tw``g mìgh d mìgh tedt k`p`gks jgt`negjljhy dgk d mìgh wej is lim`rdt`k frjo t`negjljhy2 m`tw``gmìgh still dg dp`, dltejuhe jg` tedt is spdn`fdrigh, dgk te`stdrneilk. Ejw is teis p`rtig`gt tj oy njgg`ntijg m`tw``g dg`xtrdt`rr`stridl o`ssdh`, te` Xjw! sihgdl (kurdtijg 8= s`njgks)dgk te` tio` `stiodt` my EDL jf dt l`dst 8= ejurs m`fjr` fdilur` jf te` D@->0 ugit: Oy 8.= ig oy G`ptug` @qudtijg dgk oykisnjv`ry jf wedt s`òs tj m` d o`ssdh` òm`kk`k ig jur peysinsfrjo `xtrdt`rr`stridls frjo te` sdo` r`hijg jf spdn` ds te` Xjw!sihgdl (^dhittdrius): It wjulk s`ò tj igkindt`, if oy

igt`rpr`tdtijgs jf te` odte dr` njrr`nt, tedt `xtrdt`rr`stridls dr`t`lligh us w` dr` dt d turg drjugk pjigt jurs`lv`s, gjw, jr tedt te`ydr` hjigh tj igvit` us igtj st`lldr sjni`ty sjjg.



;0

@urjpid



;6

X` gjw njgg`nt`k te` first fjur pjigts jf te` G`ptug` `qudtijg dgk igt`hrdt`k frjo jg`

tj fjur, te` `qudtijg3

F(x) ? 8.=x „;

[e` igg`r t`rr`stridl pldg`ts.

X` tjjb te` k`rivdtiv` jf te` Prdgus `qudtijg3

F(x) ? > + >.>x

^j tedt

F’(x) ? >.>

X` njgsik`r`k te` igt`hrdl te` odss jf d pldg`t ig `drte odss`s, weine ndo` jut tj m` ;=.

X` njgsik`r`k te` >.> tj m` te` dnn`l`rdtijg dt its surfdn` ig `drte hrdviti`s. Bgjwighte`s` twj qudgtiti`s w` k`t`roig`k te` pldg`t wds dmjut ter`` dgk d edlf tio`s ldrh`r tedgte` @drte. Bgjwigh its odss dgk siz`, w` njulk k`t`roig` its k`gsity, dgk frjo tedt its

njopjsitijg, weine turg`k jut tj m` @urjpiuo, `l`o`gt 6>, weine is pejspejr`sn`gt.



;8



;5



;9



01



04



0=

It is igt`r`stigh tedt te` eypjte`tindl pldg`t wjulk m` njopjs`k jf

@urjpiuo if w` `xnluk` te` kidtjoin `lò`gts lib` ijkig`, weine

is jgly nljs`r ig k`gsity my d tejusdgkte jf d hrdo p`r numin

n`gtio`t`r ig k`gsity. [e` `lò`gt is gdo`k fjr

@urjp`, dgk teis kisnjv`ry m`hdg wite oy r`dlizdtijg tedt gig`-fiftes wds ig te` ugiv`rs`, weine l`dk tj te` kisnjv`ry tedt it

ugifi`k pi dgk pei, tedt it wds te` yig fjr weine te` ydgh wds fiv`-

teirks, dgk tedt gig`-fiftes wds ig te` prjpjrtijgs jf te` ojgjlite

jf Drteur N. Nldrb`'s dgk ^tdgl`y Bumrinb's =1143 D ^pdn`

Jkyss`y fjr weine Cupit`r m`ndo` d stdr fjr te` ojjg @urjpd,

we`r` lif` wjulk m`hig jg dgjte`r wjrlk ig jur sjldr systò. It is

dg `gtdghlò`gt jf d sjrt. ]`dk, Fintijg-]`dlity @gtdghlò`gt3

L`vigsjg, Dsiojv, Dgk Nldrb` my Idg M`drksl`y. [e` prjpjrtijgsjf te` ojgjlite dr` 4 my ; my 9, 4 + ; ? 0.



0>

Njgg`ntigh @urjpiuo Dgk @urjpid Xite @urjpd

X` njopdr` te` ojldr odss jf @urjpiuo tj tedt jf îlinjg3

@u/î ?404.98/=5.19 ? 0.;

Xeine is Cupit`r’s furte`st kistdgn` frjo te` ûg ig its jrmit drjugk te` ûg ig @drte-ûg

s`pdrdtijgs, dgk, is te` odss jf ̂ dturg njopdr`k tj te` odss jf G`ptug`. It is dlsj te`

teirk t`ro ig te` s`qu`gn` 4.5g, frjo weine te` G`ptug` @qudtijg is k`riv`k. Furte`r,teis is iopjrtdgt m`ndus` te` @drte is te` teirk pldg`t frjo te` ûg dgk 4.5 is gig`-fiftes,

jur yig jf te` Pgiv`rs`.

I k`nik`k tj njopdr` @urjpiuo (@u) tj îlinjg (î), m`ndus` @urjpid is njgg`nt`k tj@urjpd, te` ojjg drjugk Cupit`r we`r` d g`w lif` m`higs ig te` ojvi` =1143 D ^pdn`

Jkyss`y. î is ig te` sdo` hrjup ds ndrmjg, te` first jg` dft`r it. Lif` is mds`k jg ndrmjg,

dgk it wjulk fjlljw tedt g`w lif` wjulk m` mds`k jg silinjg.

Idg M`drksl`y, Jnt =;, =14=



0;

@urjpid Dgk @urjpd Dgk [e` Iopjrtdgn` Jf [e` êdodg’s 40

It is dpprjpridt` tedt @urjpid is njgg`nt`k tj @urjpd, te` s`njgk Hdlil`dg Ojjg frjo

Cupit`r, gjt cust my gdo`, mut ig sj fdr ds its njgg`ntijg tj îlinjg ig [e` G`ptug`

@qudtijg. It wjrbs jut hr`dt, m`ndus` it is te` pldn` ig te` ̂ jldr ^ystò tedt is te` m`st

ndgkikdt` fjr lif` m`yjgk @drte3 It eds d t`gujus dtojspe`r` njopjs`k priodrily jf jxyh`g dgk is njv`r`k wite wdt`r in`, tedt ody edv` wdt`r ig its liquik fjro m`g`dte its

surfdn`.

[e` ojjg is te` rihet siz` sune tedt ds s``g frjo te` @drte is dpp`drs dmjut te` sdo` siz`ds te` ûg. Ejw fdr wjulk @urjpd edv` tj m` frjo Cupit`r tj dpp`dr te` sdo` siz` ds te`

sug kj`s frjo Cupit`r.

^jldr ]dkius/@urjpd ]dkius ? Cupit`r-ûg ^`pdrdtijg/x

690,991 bo/4,061.5 bo ? 885,011,111 bo/x

;;6 ? 885,011,111 bo/x

x ? 4,8;0,040.690 bo

4,8;0,040.690/4,061 ? 4,449 ~ (4,411 rjugkigh tj g`dr`st 411)

@urjpd oust m` dmjut jg` tejusdgk dgk jg` eugkr`k tio`s furte`r frjo Cupit`r tedg it is

ig rdkii, tj s`` it, frjo Cupit`r te` sdo` siz` ds te` ûg. [eis is dstjgiseighly nljs` tj dp`rf`nt oultipl` jf 411 wite 44. X` will tdb` teis tj o`dg w` dr` jg te` rihet trdnb

tjwdrks sjlvigh te` oyst`ry jf jur eypjte`tindl pldg`t, @urjpid. X` will writ` te` 411

tio`s 44, ds3

(=)(0)(41)(44)

X` edv` sdik tedt yig is 9/0 dgk ydgh 0/>. X` sumtrdnt te` twj tj h`t =/40. X` dkk te` 0

ig te` dmjv` prjkunt tj te` t`g ig te` dmjv` prjkunt tj h`t 40. X` kivik` tedt igtj te`twj ig te` dmjv` prjkunt, dgk gjt jgly kj w` edv` jur =/40, mut `l`v`g is l`ft. X` kj te`

ojst jmvijus teigh, dgk list te` pldg`ts gjt my jrmitdl guom`r, mut my prio` guom`rs3

O`rnury (=)V`gus (>)

@drte (0)Odrs (8)

Cupit`r (44)

Dgk Cupit`r is te` ̀ l`v`g ig te` dmjv` prjkunt, te` pldg`t w` w`r` njgsik`righ! X` s``

w` mds`k jur g`w peysins jg d njll`ntijg jf 40 rumm`r ejs`s ig d ndv` ig ^pdig, dgk w`

kisnjv`r`k d hr`dt oyst`ry.



00

[e` @urjpiuo ^pe`r`

Eykrjpse`r`3 [jtdl wdt`r jg, ugk`r, dgk jv`r te` @drte

Xibip`kid stdt`s tedt te` eykrjspe`r` is 4.;@45 [jgg`s

(4.;@45 t)(4,111 bh)(4,111 h) ? 4.;@=; h

X` dsb ejw oune @urjpiuo is r`quir`k tj r`dnt wite te` `gtir` eykrjspe`r`:

[e` r`dntijg jf @urjpiuo wite wdt`r is3

=@u + 6EZ=J -! =@u(JE)Z> + >EZ=

[edt is, six ojl`s jf E=J dr` r`quir`k fjr `v`ry twj ojl`s jf @urjpiuo tj odb` twj

ojl`s jf @urjpiuo Eykrjxik` dgk [er`` ojl`s jf Eykrjh`g hds. I wjulk sdy tedt is

ejw tedt `qudtijg r`dks, tejuhe oy neòindl gjo`gnldtur` is d littl` rusty. I wjulk sdysign` te` ùrjpiuo kispldn`s te` eykrjh`g, tedt teis is d sighl` kispldnò`gt r`dntijg. Ido littl` rusty jg oy r`dntijg nldssifindtijg ds w`ll. ]`hdrkl`ss, te` `qudtijg nl`drly

stdt`s tedt jg` teirk jf te` ojl`s jf E=1 dr` te` ojl`s jf @urjpiuo r`quir`k fjr te`

r`dntijg tj tdb` pldn` (=/6? 4/>). X` writ`3

EZ= ? =(4.14) ? =.1= h/ojl

J ? 46.11 h/ojl

E=J ? =.1= + 46.11 ? 45.1= h/ojl

My ljjbigh up te` ojldr odss`s jf te` `lò`gts ig te` p`rijkin tdml` jf te` `lò`gts.

((4 ojl E=J)/(45.1= h E=J))x((4.;@=; h E=J)/(eykrjspe`r`))

? 8.5@== ojl E=J/eykrjspe`r`

4/>(8.5@== ojl E=J) ? =.6@== ojl @u

@u ? 404.98 h @u/ ojl

(=.6@== ojl @u)(404.98 h @u/ ojl) ? >.90@=; h @u

[e` k`gsity jf @urjpiuo dt ^[Q is 0.=;> h/noW>

(noW>/0.=;> h)(>.90@=; h) ? 8.0>@=> noW>

Is te` vjluo` jf te` mjky njopjs`k jf >.90@=; hrdos jf @urjpiuo. If it is d spe`r`,

te`g te` rdkius jf it is3

(;/>)(pi)rW> ? 8.0>@=> noW>



06

rW> ? 4.5@=> noW>

r ? 06,;6=,46= no

(06,06=,46= no)/(411 no)(4111 o) ? 06;.6= bo ~ 060 bo

[e` @urjpiuo ^pe`r` wjulk edv` d rdkius jf 060 biljo`t`rs. [edt jf te` ojjg jf

Cupit`r ndll`k @urjpd is3

4,066 bo

4,066/060 ? =.88

[e` rdkius jf te` ojjg jrmitigh te` @drte is3

4,8>5 bo

4,8>5/060 ? >.4 ~ pi ~ >

[e` ojjg is dmjut ter`` tio`s ldrh`r tedg te` @urjpiuo ^pe`r`, jr dmjut pi tio`s ldrh`r.



08

[e` Hypsy H`g`sis Qrjc`nt

[e`r` dr` ter`` wdys ig weine te` Hypsy êdodg, Odgu`l, njulk edv` us`k eis ejs`

njll`ntijg tj odb` o` teigb dmjut te` guom`r 40. E` njulk edv` sejwg o` d njll`ntijg

jf 4; rumm`r ejs`s, fjr weine I mjuhet eio eis fift``gte rumm`r ejs`, jr, e` njulk edv`

sejw`k o` d njll`ntijg jf 40 rumm`r ejs`s fjr weine I mjuhet eio eis sixt``gte rumm`rejs`. [wj fift``gtes turg`k jut tj m` te` kiff`r`gn` m`tw``g te` vdlu`s I jmtdig`k fjr yig

dgk ydgh. X` bgjw twj r`pr`s`gts te` peiljsjpey jf twj ig [djiso, mut p`redps e`

wdgt`k o` tj teigb gjt cust jf fift``g, mut dlsj my te` dmjv` r`dsjgigh, 4;, 40, dgk 46.

Igt`r`stighly, te` neòindl `lò`gts Ndrmjg, Gitrjh`g, dgk Jxyh`g dr` te`s` ter```lò`gts, dgk te` ojst iopjrtdgt gjg-o`tdls ig te` p`rijkin tdml`. Ndrmjg is te` mdsis jf

lif`, odbigh jrhdgin odtt`r ig te` fjro jf eykrjndrmjgs. Gitrjh`g is dmjut 80% jf te`

@drte dtojspe`r`, dgk Jxyh`g dmjut =0%. [e`y dr` te` twj ojst dmugkdgt `lò`gts ig

te` `drte dtojspe`r`, mjte my odss dgk guom`r jf kidtjoin ojl`nul`s. [e` Gitrjh`g istdb`g jut jf te` dtojspe`r` dgk fix`k my mdnt`rid fjr odgy bigks jf pldgts tj ndrry jut

teìr mijneòindl prjn`ss`s. Jxyh`g is odk` my pldgts dgk pldgbtjg fjr ojst jf lif` tj

mr`dte`. Odgu`l, te` Hypsy êdodg, wds tdlbigh dmjut sjo` nlu` tj te` mdsin frdo`wjrbjf te` @drte tedt is r`spjgsiml` fjr te` `xist`gn` dgk njgtigudtijg jf lif` jg @drte.

It wjulk s`ò if te` Hypsi`s edv` sune s`nr`ts dmjut sune teighs, w` sejulk ljjb igtj te`

Prdgus dgk G`ptug` `qudtijgs I k`riv`k frjo te` pdte Odgu`l s`gt o` jg, jr igtj teìr

sygte`sis, weine is te` ik`d mèigk te` pldg`t w` k`riv`k tedt w` edv` ndll`k @urjpid.[e` prjc`nt njulk m` ndll`k D H`g`sis Qrjc`nt. Irjgindlly I edk njo` tj f``l te` Hypsi`s

jf ^pdig edv` m``g wjrbigh jg sune d prjc`nt, `v`r sign` te` t`ro wds njig`k ig d ^tdr

[r`b ojvi`.

Idg M`drksl`y Cdgudry >, =14>



05

[e` Qlutj @qudtijg

X` will ndll it [e` Qlutj @qudtijg sign` g?> (`drte is te` teirk pldg`t frjo te`sug) yi`lks >5.9, >5.9 m`igh nljs`st tj te` kistdgn` jf Qlutj frjo te` ^ug igdstrjgjoindl ugits weine is >9.;; dstrjgjoindl ugits.



09



61



64



6=

Igtrjkuntijg

Ig Hypsy êdodgiso Dgk [e` Pgiv`rs`, w` tjlk te` m`st w` njulk te` `xp`ri`gn`s I edk

wite d Hypsy êdodg tedt l`dk tj dstjugkigh sni`gtifin kisnjv`ri`s, mjte ig oy wjrbs Dg

@xtrdt`rr`stridl Dgdlysis dgk D O`ssdh` Frjo @xtrdt`rr`stridls. Qrjhr`ss, it wjulk s`ò,

is dt d stdgkstill. I teigb it is m`ndus` Hypsy êdodgiso Dgk [e` Pgiv`rs` is jgly edlf te` stjry. It wjulk s`ò we`g I wds odk` igtj te` D@->0 dgt`ggd dgk, dft`r oy s`njgk

trip tj ^pdig (Odkrik) dgk oy sums`qu`gt trips tj Itdly, it wds joitt`k frjo oy oòjry.

Ejw`v`r, I s`ò tj edv` rò`ki`k tedt, dgk dnn`ss`k it. [eus w` gjw prjhr`ss wite te`

stjry jf Hdmri`l.

Hdmri`l

Dsik` frjo te` Hypsy êdodg, te`r` wds dgjte`r H`gtlòdg wjrte gjtigh, wite wejo I

dntudlly sp`gt ojr` tio`. Eis gdo` is Hdmri`l. E` is dntudlly edlf Hypsy dgk edlf

^pdgise. E` liv`s ig d ndv` dojgh te` jte`r ndv`s jf te` ^dnrjojgt`, ignlukigh tedt jf te` Hypsy êdodg, Odgu`l.

Hdmri`l tjjb te` ter`` jf us ugk`r eis wigh ig trdigigh fjr te` huitdr3 [e` twj neilkr`g jf

te` Hypsy êdodg, Odgjlig (wds dh`k =8 dt te` tio`, weine wds drjugk 4991) dgk eis

yjugh`r mrjte`r, Dgtjgij (e` oust edv` m``g dmjut =>).

Odgjlig dgk Dgtjgij w`r` dt te` ndv` jf te` Hypsy êdodg, Odgu`l, teìr fdte`r, we`r`

I wds stdyigh. Hdmri`l ndo` my, dgk tjlk Dgtjgij tj pldy mul`rids. E` sdik tj, dt dn`rtdig pjigt, pldy h oigjr wite te` fifte ròjv`k (te` fjurte pldy`k igst`dk) dgk m`hdg

tj sigh. E` sdgh wite hr`dt pjw`r, “[`ghj Pg Ndmdllj Mldgnj‟ weine trdgsldt`s “I edv`

d weit` ejrs`‟. E` dsb`k o` tj edgk eio d Njrkjm`s edt tedt wds edghigh jg te` wdll.E` put it jg eis e`dk dgk sdik l`t’s hj. X`, dgk dll te` jte`r jnnupdgts, w`gt tjh`te`r tjd ndv` tedt wds d mdr. X` dll jrk`r`k krigbs, dgk te`g w`gt terjuhe jg` jf te` tugg`ls tj

d ndv` lit up hr``g tedt edk tdml`s dgk pjst`rs jf fldo`gnj hr`dts. E` edgk`k o` te`

huitdr dgk tjlk o` we`g e` odb`s te` nejrk jg eis dros, tedt I sejulk pldy it jg te`

huitdr. It wds d K``p ^jgh H oigjr wite te` mjttjo ̀ pldy`k ig it. I kik sj we`g dft`r e`stdrt`k sighigh dgk e` pldy`k te` nejrk jg eis dro dt te` pjigt jf nr`sn`gkj.

X` w`gt mdnb igtj te` mdr, it wds h`ttigh kdrb, dgk te`r` wds d odg pldyigh huitdr ig te`

njrg`r. Jte`r p`jpl` stdrt`k tj drriv` dt te` mdr, frjo jte`r tjwgs. X` dll w`gt terjuhedgjte`r tugg`l ig te` ndv` tedt l`dk tj d nedom`r we`r` te`r` w`r` wjjk nedirs ligigh te`

wdlls. It wds kdrb, lit jgly my ndgkl`s. I sdt dll te` wdy tj te` l`ft sik` jf te` ndv` g`xt tjDgtjgij wej wds dmjut tj pldy mul`rids fjr dll te` p`jpl` ig te` nedirs tedt lig`k te` wdlls

jf te` rjugk ndv`. E` pldy`k, dljgh wite jte`rs, dgk d p`rsjg wjulk h`t jut eis nedir,stdgk ig te` n`gt`r jf te` ndv` dgk sigh, sit kjwg dft`r kjg` dgk te`g sjo`jg` `ls` wjulk

te`g kj te` sdo`, ugtil jg` dt d tio` te` kdgn`rs wjulk njo` tj te` n`gt`r jf te` fljjr,

kdgn`, te`g sit kjwg. I ròòm`r jg` jf te` s`t jf lyrins tedt jg` sigh`r sdgh3



6>

^dnd ld r`k

E`rodgj, sdnd l` r`kTu` yd sdlij ld lugd

Gj vdyd njh`r

[edt is3

[db` jut te` g`t

Mrjte`r, tdb` jut te` g`t

[e` ojjg eds dlr`dky l`ftYju dr` gjt hjigh tj ndtne it.

Hdmri`l t`gk`k tj kisdpp`dr fjr s`v`rdl kdys, dgk I kik figk jut wedt tedt wds dmjut. E`

tjjb o` jut jf te` ^dnrjojgt` jg` kdy cust m`fjr sugs`t, dgk w` wdlb`k up tj [e`Dlmdinig, te` sdo` Dlmdinig g`dr we`r` I mjuhet te` Hypsy êdodg eis ejs`, te` jlk

Drdmin Nity. X` hjt tj d pldzd dgk `gt`r`k d mdr. Hdmri`l wdlb`k ig, oys`lf wite eio

dgk dpprjdne`k te` mdr njugt`r, ljjb`k mri`fly tj eis l`ft, te`g seift`k quinbly tj eis rihet,te`g mdnb tj eis l`ft, te`g rdpp`k jut d fdst fldo`gnj reyteo jg te` njugt`r. It wds gjtljgh m`fjr` sjo`jg` jrk`r`k eio d krigb, weine ig ^pdig njo`s wite tdpds (dpp`tiz`rs)

fjr fr``, usudlly Ndldodr`s (k``p fri`k squik). Hdmri`l dgk I krdgb terjuhe te` gihet dgk

drjugk sugris` w` m`hdg tj wdlb mdnb tj te` ^dnrjojgt`, jut jf te` jlk Drdmin Nity dgk

kjwg terjuhe te` njmml`stjg` str``ts, l`dgigh jg jg` dgjte`r te` wejl` wdy kjwg pdstp`jpl` sw``pigh te` sdwkust jut jf teìr mdr fljjrs (mdr fljjrs dr` njv`r`k wite sdw kust

tj dmsjrm te` jil frjo pdrts jf fise terjwg tj it3 te`r` dr` gj trdse ndgs, `v`ryteigh is

terjwg tj te` fljjr ignlukigh gdpbigs.) Hdmri`l jff`r`k tj e`lp jg` p`rsjg wite teis tdsb.

Xe`g w` drriv`k dt te` ^dnrjojgt`, twj wjo`g w`r` dpprjdneigh frjo te` jte`r

kir`ntijg. Hdmri`l sdik te` jg` wds eis ojte`r. I dsb`k wey e` kikg’t sdy ei, dgk tjlk oè` kik gjt wdgt tj ̀ omdrrdss e`r. E` te`g w`gt jg tj t`ll o` tedt [jodtitj (d fdojusHypsy Fldo`gnj Huitdrist) wds eis edlf mrjte`r my wdy jf eis ojte`r.

[e` g`xt kdy I wds tdlbigh tj Dgtjgij dgk e` sdik tedt Hdmri`l wds te` mrjte`r jf

[jodtitj. I sdik I bgjw, my eis ojte`r. Dgtjgij sdik gj, my eis fdte`r. I tjlk eioHdmri`l tjlk o` my eis ojte`r. Dgtjgij sdik Hdmri`l ti`g` eistjrid. (E` eds d eistjry).

[e` kdy dft`r tedt I tjlk Hdmri`l tedt Dgtjgij sdys e` is [jodtitj’s mrjte`r my eis fdte`r

dgk Hdmri`l sdik tedt it wds ku` tj mìgh sjrt jf trdkitijgdl ig dg jlk fdseijg`k wdy tedtDgtjgij sdik tedt.

[eis is igt`r`stigh m`ndus` jg` kdy we`g I wds wdlbigh terjuhe te` ^dnrjojgt` wite

Hdmri`l, w` njulk s`` up ig te` eills frjo e`r ndv` pdtij, d Hypsy wjodg gdo`k M`l`g(M`telèò) dgk I wdv`k tj e`r. Hdmri`l sdik tj o` tedt w` edk tj edv` d tdlb. E` tjjb

o` up tj d ròjt` pldn` ig te` eills dgk w` sdt jg sjo` rjnbs dgk e` tjlk o`3 “Yju

kjg’t bgjw wedt yju wjulk m` ig fjr if yju odrri`k d trdkitijgdl Hypsy Hirl‟.



6;

It wds te` gihet m`fjr` tedt I wds up dt e`r t`rrdn` wite e`r fdte`r. E` wds gjt Odgu`l

te` Hypsy ^edodg, mut Odgu`l te` sigh`r, K``p ^jgh sigh`r (ndgtdjr). I tjlk Hdmri`l Iwds hjigh te`r` fjr d fldo`gnj l`ssjg dgk e` sdik “Odgu`l `s mu`gj‟ (Odgu`l is hjjk).

[edt is wedt `v`ryjg` dlwdys sdik dmjut eio. It is m`st `xpldig`k my Mjm Kyldg’s lyrins,

“Eis nljte`s dr` kirty mut eis edgks dr` nl`dg.‟



60

[e` Gig`-fiftes Njgc`ntur`

I edv` fjugk tedt rdgkjoly `vjlvigh systòs jf o`dsurò`gt njgv`rh` jg gig`-fiftes,

lib` te`r` is gig`-fiftes jf d k`hr`` n`gtihrdk` p`r fder`geìt, te`r` dr` 1.09 hrdos m`yjgk

d wejl` guom`r jf hrdos p`r pjugk, dgk dmjut gig`-fiftes biljo`t`rs p`r oil`. I edv`

fjugk oune ojr` tedg cust teis dgk it is ig oy wjrb “Dg @xtrdt`rr`stridl Dgdlysis‟ weineis gdo`k ds sune m`ndus` I suhh`st `xtrdt`rr`stridls hdv` us jur ugits jf o`dsurò`gt my

igflu`gnigh eistjry, ds jppjs`k tj m`li`f tedt eds dlwdys m``g e`lk tedt jur ugits jf

o`dsurò`gt `vjlv`k terjuhe d rdgkjo s`ri`s jf `v`gts. I f``l dg dlt`rgdt` `xpldgdtijg

njulk m` tedt d rdgkjo s`ri`s jf vdlu`s sjo` ejw njgv`rh`s jg gig`-fiftes if sumc`nt`k tjfluntudtijg my d rdgkjo s`ri`s jf fjrn`s (`v`gts). I edv` pjigt`k jut tedt gjt jgly kj`s

gig`-fiftes ugify pi dgk te` hjlk`g rdtij, mut ` (@ul`r’s guom`r) dgk pi if w` njgsik`r te`

first f`w kihits dft`r te` k`niodl ig dll guom`rs. I f``l teis ody m` pivjtdl tj prjvigh te`

njgc`ntur`, dgk it wjulk s`ò te` sjldr systò, ig its `vjlutijg, eds njgv`rh`k jg gig`-fiftes ds w`ll ds jur ugits jf o`dsurò`gt. Ojst `xtrdjrkigdrily, gig`-fiftes dgk jur

kiff`r`gt systòs jf o`dsurò`gt (ds ig biljo`t`rs dgk oil`s) nrjp up we`g w`

igtrjkun` te` guom`r 40 ig oy Prdgus Igt`hrdl, weine I edv` `xpldig`k tj m` d njgn`ptwite its jrihigs ig d Hypsy êdodg frjo ^pdig, ig oy sdo` wjrb, gdo`k dmjv`.Pgk`rtdbigh te` `gk`dvjr jf prjvigh oy njgc`ntur` is nurr`gtly m`yjgk te` snjp` jf oy

`kundtijg ig odteòdtins.



66

Oihet Dntudlly M` Jgtj ^jo`teigh

I edv` fjugk jur kiff`r`gt systòs jf ugits jf o`dsurò`gt tj m`k``ply njgg`nt`k wite jg` dgjte`r dgk gdtur`.

I suhh`st te` jrihigsjf te` njgg`ntijgs tj m` `xtrdt`rr`stridl, if gjt ku` tj sjo` strdgh`

prjp`rty we`r` rdgkjoly `vjlvigh systòs njgv`rh` jg d vdlu` jrs`t jf r`ldt`k vdlu`s.

I cust r`dliz`k tedt rjjo tòp`rdtur`, k`fig`k my 65 k`hr``s F tj 88 k`hr``s F eds te`dv`rdh` (65 + 88)/= ? 8=.0 k`hr``s F. I edv` gjt`k tedt 8= is te` ojst r`nurr`gt

terjuhejut o` r`s`drne, `sp`nidlly we`r` nliodt` is njgn`rg`k ig njgg`ntijg tj te` pldg`t

V`gus (1.8= DP frjo ûg). I wrjt` ig oy wjrb Dg @xtrdt r̀r`stridl Dgdlysis3

[e` Iopjrtdgt [eigh ^`òs [j M` 8=

If dgyteigh eds `v`r nrjpp`k up te` ojst ig te` njurs` jf teis r`s`drne, it is te` guom`r

8=. [e` 8= ig te` G`ptug` @qudtijg, te` 8= ig te` 1.8= DP s`pdrdtijg jf V`gus dgk te`ûg, te` 8= y`drs fjr jg` k`hr`` jf pr`n`ssijg jf te` @drte @quigjx, [e` 8= jf Isetdr igtedt se` is r`pr`s`gt`k my V`gus, wds te` Qdtrjg`ss jf Gig`vè we`r` te` Gig`vènjgstdgt wds fjugk, dgk ndrri`s te` ûo`ridg rdgbigh jf te` 40 jf te` Hypsy êdodg. [e` 8= ejurs jf tio` m`fjr` te` D@->0 Dgt`ggd wjulk fdil, te` 8= s`njgks jf tio` tedtte` Mih @dr Dgt`ggd trdnb`k te` Xjw! îhgdl, dgk tedt 8= tedt is te` eihe`st hòdtrid jf te` E`mr`w Hjk's gdo` YEXE.

Dgk I wrjt` ig oy wjrb D O`ssdh` Frjo @xtrdt`rr`stridls3

[e` Ojst Mizdrr` ^tjry @v`r [jlk

Kik `xtrdt`rr`stridls r`dlly dgtinipdt`, we`g te`y s`gt us ig 4988 d trdgsoissijg frjo te`njgst`lldtijg ^dhittdrius, tedt ldst`k 8= s`njgks (^@[I’s Xjw îhgdl) tedt, d Hypsy

êdodg, drjugk 4991, wjulk odb` o` igtj [e` D@->0 Dgt`ggd jf =1143 D ^pdn`

Jkyss`y tedt te` seip njoput`r, EDL, wjulk sdy wjulk fdil witeig 8= ejurs, te` sdo` 8=

jf te` Xjw îhgdl: Kik te`y bgjw te` trdgsoissijgs I wjulk r`nìv` frjo te` Hypsy

êdodg, Odgu`l, wjulk l`dk gjt cust tj te` kisnjv`ry jf d o`ssdh` òm`kk`k ig jur

peysins, mut d systò jf ter`` `qudtijgs tedt wjulk pjigt tj te` sdo` r`hijg jf spdn` ds

te` Xjw îhgdl, te` njgst`lldtijg ^dhittdrius, fjr te` jrihig jf oy o`ssdh`:

[eis is te` sdo` 8= tedt is ig te` 1.8= dstrjgjoindl ugits jf V`gus frjo te` ûg. It is te`

sdo` 8= tedt is te` guom`r jf y`drs fjr jg` k`hr`` jf pr`n`ssijg jf te` @drte’s

@quigjx`s. It is te` sdo` 8= jf te` eihe`st hòdtrid jf te` E`mr`w Hjk, YEXE, dgk furte`r V`gus wds te` Qdtrjg s̀s jf Gig`vè (Isetdr) we`r` [e` Gig`vè Njgstdgt wds

fjugk ig O`sjpjtdoid, ig Dgni`gt ûo`ridg nldy nugìfjro tdml`ts, d njgstdgt tedt is te`

njoojg k`gjoigdtjr fjr dll n`l`stidl nynl`s ig jur sjldr systò.

[e` njgg`ntijg we`r` V`gus is njgn`rg`k is ig tedt te` pldg`t is iopjrtdgt fjr

ugk`rstdgkigh nliodt` ig tedt it edk d rug dwdy hr``gejus` `ff`nt dgk is sioildr ig siz`dgk odss tj `drte.



68

Is teis njgsist`gt wite te` eypjte`sis tedt oy G`ptug` @qudtijg, weine eds d oultipli`r jf 8.= is r`ldt`k tj nliodt` we`g I sdy it ody edv` d b`y tj te` sunn`ss jf te` @drte: I wjulk

ljjb dt te` strdgh` jnnurr`gn` jf te` 8= ig te` ^@[I Xjw! ^ihgdl dgk it jnnurr`gn` ig

=1143 D ^pdn` Jkyss`y we`r` te` D@->0 dgt`ggd is njgn`rg`k. X` ndg tedgb te` Hypsy

^edodg, Odgu`l, fjr wedt ody m` d vdludml` nlu` we`g e` igtrjkun`k o` tj [e` D@->0Dgt`ggd. [j l`drg dmjut te` pjssiml` iopjrtdgn` jf te` D@->0 Dgt`ggd, r`dk oy dmjv`

o`gtijg`k wjrbs.

Idg M`drksl`yCuly 40, =14>



65

Dpp`gkix



69

J fjr @drte ? 4.;90989@4> no ] fjr @drte ? 6,>85 bo O fjr @drte ? 0.986@=8 h

@drte-Ojjg ^`pdrdtijg3 >.5;@41 no^jldr ]dkius3 6.9099@41 no

Ojldr Odss jf Hjlk3 Du ? 496.98

Ojldr Odss jf ^ilv`r3 Dh ? 418.58

^dturg (oigiouo kistdgn` frjo sug) ? 9.14; DP ? 4.>;5@9 bo

Cupit`r (oigiouo kistdgn` frjo sug) ? ;.904 DP ? 8.;19@5 bo

Cupit`r (odxiouo kistdgn` frjo te` sug)3 0.;00 DP ~ 0.; Dstrjgjoindl Pgits.



81

^ix-Fjlk ^yoo`try ([e` Gig`-Fiftes Dlhjriteo ig ]`v`rs`)

>61/6 ? 61

>61-61 ? >11

>11 „ 61 ? =;1

=;1/>61 ?=/>

(=/>) + 4 ? 0/>

Njugt`rpdrt tj gig`-fiftes is fiv`-teirks.

[e` fiv`-teirks dghl` is 61 k`hr``s.

^ixtyk`hr``s is te` o`dsur` jf `dne dghl` ig dg `quildt`rdl tridghl`, weine is jg` jf te`

ter`` t`ss`lldtigh r`huldr pjlyhjgs. D sgjwfldb` wite `dne dro s`pdrdt`k my sixty

k`hr``s fjros d r`huldr e`xdhjg wite igt`rgdl dghl`s jf 4=1 k`hr``s.



84

[e` Ydgh @l`o`gts

B ? pjtdssiuo

Gd ? ^jkiuo

B/Gd ? >9.41/==.99 ? 4.84

0/> ?4.68 ~ 4.8

B/Gd ~ 0/>



8=

Idg M`drksl`y stuki`k peysins dt te` Pgiv`rsity jf Jr`hjg dgk wjrb`k ig Dstrjgjoy dtte` stdt` jms`rvdtjry, Qig` Ojugtdig Jms`rvdtjry, ig te` eihe k`s`rt jf Jr`hjg. Eis

gdo` dpp`drs jg s`v`rdl pdp`rs ig [e` Dstrjpeysindl Cjurgdl.

the uranus enigma (second edition)

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