triangles in the sky

7/26/2019 Triangles in the Sky

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Triangles in the Sky: Trigonometry and

Early Theories of Planetary Motion -

IntroductionAuthor(s):

Sandra M. Caravella (New Jersey City University)

Since ancient times, human beins have observed the s!y and the movements o" the ob#ects$

sun, moon, stars, and %lanets$within it. &he reular movements o" the sun, moon, and stars

%rovided humanity with its "irst cloc!s and calendars, while the irreular but still %atternedmotions o" the %lanets ins%ired the idea that their wanderins may in"luence events here on earth.

'or these two reasons, ancient civiliations such as the abylonians, *ree!s, +ndians, Chinese,

and Mayans systematically observed the s!y and wor!ed out mathematical schemes to describewhat they "ound there, thereby establishin the science now !nown as mathematical astronomy.

Ancient mathematical astronomers in *reece and +ndia in %articular em%loyed a variety o"

eometrical models to describe the %attern o" movements within the s!y, models that were

"urther develo%ed by the +slamic civiliation. Com%utation with these models was a ma#orim%etus behind the develo%ment o" trionometry, andCo%ernicuss attem%t to sim%li"y and

re"ine them led to the -suncentered/ eometric model o" the Co%ernican 0evolution. 1or!in

with these historically sini"icant models$both ancient and Co%ernican$re2uires a ood

!nowlede o" basic trionometry, a "act which is le"t out o" most history boo!s, with the resultthat "ew %eo%le are aware o" their mathematical under%innin.

+n this %a%er + describe the %rototy%e o" the *ree! and +ndian models "or %lanetary

motion$the basic -e%icyclede"erent/ model invented by A%ollonius o" 3era. + show

how to "ind the %arameters o" this basic model and how to use the model to com%ute

%lanetary %ositions. 1e shall see that trionometry is e4actly the inredient that ma!es

such eometric models$both ancient and Co%ernican$2uantitatively use"ul.


Early Theories of Planetary Motion - The

Wandering Stars

Author(s):

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Copernicus.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Biographies/Copernicus.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Biographies/Apollonius.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Biographies/Apollonius.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Biographies/Copernicus.html


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&o understand what a %lanetary model does, we must "irst consider the s!y at niht. +n ain at

the nihttime s!y, ancient %eo%le noticed that the stars occur in distinct rou%ins, which we call

constellations. 5ne well!nown constellation is the i 6i%%er7 i" one -connects the dots/"ormed by the stars o" the i 6i%%er, the resultin sha%e loo!s li!e a lare ladle, or -di%%er./

&he i 6i%%er is actually %art o" a larer constellation !nown as Ursa Ma#or, or the *reat ear.

Another well!nown constellation is 5rion, whose belt "orms three briht stars visible in thewinter s!y o" the northern hemis%here.

As the earth moves around the sun, constellations move around in the s!y, sometimes

disa%%earin "rom view "or wee!s or months. 8ven in the course o" a sinle niht, they can be

observed to chane their %osition as the earth rotates. Ancient %eo%le observed these chanes andnoticed a very im%ortant "act$while constellations do move around in the s!y, they never

chane their sie or sha%e. &he i 6i%%er always loo!s the same, no matter when we observe it,

althouh it miht be in a di""erent %art o" the s!y or %ointin in a di""erent direction.

&he "act that constellations do not chane their sie or sha%e led ancient %eo%le to notice that

some -stars/ do not stay %ut within any constellation. &hese -stars/ became !nown as-wanderin stars,/ or %lanets. 1e now !now them to be the "ive %lanets o" our solar system that

are visible to the na!ed eye$Mercury, 9enus, Mars, Ju%iter, and Saturn.

&he ancient abylonians were the "irst to %ay care"ul attention to the wanderins o" the %lanets.Ancient %eo%le believed the %lanets to be heavenly bodies whose motions could in"luence events

on earth7 thus it was im%ortant to chart and %redict their motions throuh the stars. y care"ully

observin their wanderins over many years, the ancient abylonians observed that all the%lanets seem to move alon rouhly the same %ath throuh the stars. &hat %ath became !nown as

the ecliptic.

&o visualie the ecli%tic we can thin! o" the s!y as a hue -celestial s%here/ with the earth in thecenter and with the stars %lastered to its inside sur"ace (see 'iure below). At any one time onlyhal" the s%here is visible7 the other hal" is below the horion. Under this model the ecli%tic is a

reat circle on the s%here7 at any one time hal" o" it stretches across the s!y in a rouh east to

west direction.

'iure


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&he ecli%tic is more accurately de"ined as the a%%arent annual %ath o" the sun throuh the stars.

As everyone !nows, the sun rises in the east and sets in the west every day. &his a%%arent motion

o" the sun is caused by the earths west to east rotation. ;owever, there is a second motion o" thesun that is not as obvious, but was observed and trac!ed by the abylonians. +" you o out in the

early evenin and observe the western horion #ust a"ter sunset when the stars are "irst becomin

visible, there will be some constellation o" stars there. (+n reality, the air %ollution and arti"icialliht o" modern times obscure the stars at the horion #ust a"ter sunset, but in ancient times, "rom

whence these ideas oriinate, these stars were easier to see.) +" you did this over the course o" a

com%lete year, you would observe that the constellation in that %art o" the s!y at that time o" theevenin chanes. +n ancient times you would see (rouhly) the constellation Ca%ricorn in

January, A2uarius in 'ebruary, 3isces in March, Aries in A%ril, &aurus in May, *emini in June,

Cancer in July,


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'iure =

+n 'iure =, the blue line is the line o" siht "rom the earth to the sun, and the -a%%arent sun/ is

the a%%arent %osition o" the sun within the "ar distant bac!round o" the stars. Animatin the"iure we see that as the earth revolves around the sun, the blue line o" siht rotates around the

earth, which causes the a%%arent sun to revolve around the blue circle (the ecli%tic). ;ence, as

seen "rom earth, the sun a%%ears to circle the earth around the ecli%tic once a year.

&he %lanets also a%%ear to circle the earth around the ecli%tic, each at its own rate, movinenerally "rom west to east. &he averae time it ta!es "or one com%lete cycle ranes "rom one

year "or Mercury and 9enus to almost B years "or Saturn. ;owever, the motion o" a %lanet alon

the ecli%tic is "ar "rom uni"orm. &his is illustrated by animatin 'iure B, which simulates the =year #ourney o" Ju%iter around the ecli%tic.
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'iure B

As 'iure B illustrates, as a %lanet travels around the ecli%tic, it %eriodically slows down, sto%s,

and reverses its direction "or a short time, movin "rom east to west instead o" west to east. &hesesocalled retrograde motionswere observed and studied by the ancient abylonians.

+n order to 2uanti"y their observations o" %lanetary motions, the abylonians divided the ecli%tic

into BDE, with E ta!en to be the a%%arent location on the ecli%tic o" the sun at the s%rine2uino4. &he measure in derees o" a %lanets a%%arent %osition on the ecli%tic is calledits longitude. Usin astronomical instruments, it was %ossible "or early astronomers to measure a

%lanets lonitude and thereby trac! its motion around the ecli%tic thouh time. A %lanetary

model, as conceived o" by the *ree!s, is a eometric model that re%licates this observed motiono" the %lanet around the ecli%tic. Such a model must aree with observational data and in

%articular must ade2uately account "or the %lanets retrorade motions.


Early Theories of Planetary Motion - TheBasic Modern Model

Author(s):



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1e %ause here in our account o" history to ive the basic modern e4%lanation "or the motions o"

the %lanets described in Section B. As seen "rom earth, the motion o" the %lanet alon the ecli%tic

is actually a combination o" two motions$the real motion o" the %lanet around the sun, and ana%%arent motion caused by the motion o" the earth. &hese two motions combine to ive the west

to east circuit o" the ecli%tic with retrorade motions described in Section B. Animatin 'iure F

illustrates how this ha%%ens "or the %lanets Mercury and 9enus, the socalled -inner %lanets/which lie between the earth and the sun.

'iure F

'iure G illustrates the situation "or the -outer %lanets/ includin Mars, Ju%iter, and Saturn.


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'iure G

Note that in both models the earths orbit and the %lanets orbit are in the same %lane (the %lane

o" the ecli%tic). &he blue line sement re%resents the line o" siht "rom the earth to the %lanet7 theintersection o" this line with the ecli%tic thus re%resents the a%%arent %osition on the ecli%tic o"

the %lanet as seen "rom earth. &his %oint is called the-a%%arent %lanet./

&his account o" %lanetary motions is, o" course, a sim%li"ied version o" what actually occurs.3lanets and the earth orbit the sun in elli%ses, not circles, and chane s%eeds as they travel, unli!ethe uni"orm s%eed shown in the animation. Also their orbits are not all in the same %lane, but are

slihtly inclined to each other. ;owever, the elli%ses are almost circles, the s%eeds are almost

uni"orm, and the %lanes are almost all the same. &hus this sim%le model ives a crudea%%ro4imation to actual %lanetary motions, and is a ood start towards a more accurate one.

'urthermore, these considerations do not a""ect the basic e4%lanation iven by the model "or the

a%%arent motions o" the %lanets includin retrorade motions.


Early Theories of Planetary Motion - The

Basic ncient Model

Author(s):



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&he modern e4%lanation described in Section F is an e4am%le o" a eometric model "or the

motions o" the %lanets$the a%%arent motions alon the ecli%tic are e4%lained in terms o" actual

motions on eometric ob#ects (circles). ;owever, this e4%lanation assumes a movin earth,which contradicts both intuition and -common sense./ Usin #ust the evidence o" their ordinary

senses, it a%%eared to ancient %eo%le, as it does to us, that the earth is motionless while

everythin else in the s!y$sun, moon, %lanets, and stars$moves in various ways. &he ancient*ree!s and +ndians there"ore souht a eometric model that accounted "or the motions o" the

%lanets while leavin the earth at rest.

Such a model was %robably "irst invented in the third century .C.8. by the reat *ree!

eometer A%ollonius o" 3era>B, %. =D=. (A%ollonius is best !nown "or his study o" the conicsections$%arabolas, elli%ses, and hy%erbolas$"ound in his boo! Conics.) +n A%olloniuss model

the %lanet moves uni"ormly around a circle called an epicycle, while the center o" the e%icycle

moves uni"ormly around the earth on a larer circle called thedeferent. 'iure D illustrates thisbasic model7 as can be seen, the model can enerate retrorade motions 2uite well.

'iure D

+n circular motion, the time it ta!es an ob#ect to com%lete one "ull revolution around its circle iscalled itsperiod.&here are two %eriods in the basic ancient model: the %eriod o" the e%icyclecenter as it revolves around the de"erent, and the %eriod o" the %lanet as it revolves round the

e%icycle. &he "irst %eriod re%resents the time it ta!es the %lanet, on averae, to return to the same

%oint o" the ecli%tic7 this is called thesidereal period.&he latter %eriod could be ta!en to be thetime between successive returns o" the %lanet to the %oint Qin the diaram7 as such it re%resents

the time between the middles o" successive retrorade motions. &his %eriod is called thesynodic

period.
http://www-history.mcs.st-and.ac.uk/Biographies/Apollonius.htmlhttp://www.maa.org/sites/default/files/images/upload_library/46/Caravella_article/Figures/Figure_6.htmlhttp://www-history.mcs.st-and.ac.uk/Biographies/Apollonius.htmlhttp://www.maa.org/sites/default/files/images/upload_library/46/Caravella_article/Figures/Figure_6.html


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&he basic model has three %arameters. &hese are: rHR,where ris the radius o" the e%icycle

andRis the radius o" the de"erent7 the sidereal %eriod T7 and the synodic %eriod S. +n 'iure D,

drain the %oints labeled rHR,T, and Swill chane these %arameters. &he reader should observehow varyin these 2uantities chanes the %lanets motion, in %articular the s%acin, duration, or

sie o" the retrorade arcs.


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+n a similar manner, addin the B=year cycle to the Gyear cycle results in a still more accurate

FIyear cycle, and the FIyear cycle added to the B=year cycle ives the even more accurate IK

year cycle: Mars com%letes a%%ro4imately F= sidereal cycles and BI synodic cycles ina%%ro4imately IK years. &his last IKyear cycle ives a synodic %eriod o" =.BG years and a

sidereal %eriod o" . years.

Cyclical data "or each %lanet is iven below. &his data was !nown to the abylonians and was

used by Claudius 3tolemyin theAlmagestto initially develo% his models >B, %%. GG=. +nhis "inal models, 3tolemy modi"ied the data somewhat to ive better lonterm results >=, %%.

F=BF=F.

Mercury: FD sidereal cycles and FG synodic cycles in FD years

9enus: sidereal cycles and G synodic cycles in years

Mars: F= sidereal cycles and BI synodic cycles in IK years

Ju%iter: D sidereal cycles and DG synodic cycles in I years

Saturn: = sidereal cycles and GI synodic cycles in GK years

y observin the times o" the e2uino4es and solstices, ancient astronomers were able to

a%%ro4imate the number o" days in one year. &he modern value "or this is well !nown: there are

BDG.=F days in one year. &his im%ortant %arameter toether with the data above determines thesidereal and synodic rates o" motion (in derees %er day) "or each %lanet. 'or e4am%le, Mars

com%letes F= sidereal cycles in IK years. &here"ore in IK years it com%letes F= 4 BDE in IK 4

BDG.=F days. 6ividin, this ives a sidereal rate o" .G=FEHday.

&able ives the sidereal and synodic %eriods and rates o" motion "or each %lanet determined by

the cyclical data listed above.

Sidereal 3eriodT(in

years)

Sidereal 0ate

(EHday)

Synodic 3eriodS(in

years)

Synodic 0ate

(EHday)

Mercury .KD .BI B.

9enus .KD .D .DD

Mars . .G=F =.F .FD=
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Ju%iter . .BB .K .K=

Saturn =K.G .BBF .F .KG=

&able



!etermining r"#

Author(s):


+n addition to the %eriods o" the %lanets, ancient astronomers observed two other ross "eatures o"

their motions$the lenths and times (durations) o" their retrorade arcs. 'or a sinle %lanet thelenth and time o" its retrorade arcs vary somewhat7 the "ollowin table ives a%%ro4imateaverae values that a%%ear to be close to those !nown to 3tolemy >.


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Ju%iter E =

Saturn IE F

&able =

0evisitin'iure D, we see that "i4in Tand Sand varyin the ratio rHRchanes the lenths o"

the retrorade arcs. , ancient astronomers may have used such data, %lus a little trionometry, to"ind an initial estimate o" the ratio rHR"or each %lanet. 'or the outer %lanets$Mars, Ju%iter, and

Saturn$this method o" determinin rHRis somewhat similar to the method actually used by

3tolemy in theAlmagest. 'or the inner %lanets$Mercury and 9enus$3tolemy in

theAlmagestcom%uted this %arameter by a 2uite di""erent method which involved usin the%lanets socalled -reatest elonation/ "rom the sun.

&o com%ute rHRby usin the data in &able =, we start at a time when the e%icycle center, the

%lanet, and the earth are all in a straiht line, with the %lanetPat the %ointP> on the e%icycle

(see 'iure I).
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'iure I

+n the "iure, Cis the location o" the e%icycle center at this time andAis the location o" the

a%%arent %lanet on the ecli%tic. At this %oint the %lanet is in the middle o" its retrorade arc.Animatin 'iure I shows the %lanets motion "rom the middle o" the retrorade arc until the

end. At that %oint Cre%resents the new location o" the e%icycle center andAre%resents the new

location o" the a%%arent %lanet. &rionometry now enters the %icture, as ris side CPandRissideECo" trianleECP7 thus to "ind the ratio rHR, we must solve the trianleECP"or the ratioo" its sides CPHEC.

Since the %lanet moves uni"ormly on the e%icycle, the arcPP has measure e2ual to the synodic

rate o" the %lanet (see &able in Section D) multi%lied by hal" the time o" its retrorade arc ivenin &able = above. Since arcPPand anlePCPhave the same measure, this determines

anle Cin trianleECP. Similarly, since the e%icycle center moves uni"ormly on the de"erent,

arc CC has measure e2ual to the sidereal rate multi%lied by hal" the time o" the retrorade arc.

&his determines anle CECwhich "orms %art o" anleEin trianleECP.'inally, arcAAis byde"inition hal" o" the %lanets retrorade arc7 thus its measure is !nown "rom &able = above. &his

determines anleAEP7addin this to anle CECives anleEin trianleECP. Altoether wehave "ound anles CandEin trianleECP7 subtractin their sum "rom E determinesanleP.Usin the


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'rom &able =, we et the retrorade arc data: (H=)lenth L E and (H=)time L BD days.

'rom &able , we "ind that the synodic rate is .FD=EHday and the sidereal rate is .G=FEHday.

AnleECP =arcPPL (synodic rate) 4 (H=)time L D.DE.

Anle CECL arc CC =(sidereal rate) 4 (H=)time L .KE.

AnleAEP =arcAAL (H=)lenth L E.

AnlePEC =the sum o" anles CECandAEP ==D.KE.

Anle CPE =E minus the sum o" anlesPECandECP =BD.GE.

y the


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9enus .D .I=

Mars .K =. .DD

Ju%iter . . .K

Saturn =K.G . .

&able B

1ith these %arameters in hand, it is "un to revisit'iure Dand "eed in the a%%ro%riate %arameters

"or each %lanet. 0unnin the animation at the same s%eed "or each %lanet allows us to com%are

the various #ourneys o" the %lanets as they ma!e their way around the ecli%tic. 'rom the swi"tmovements o" Mercury to the lumberin #ourney o" Saturn, we see the same eneral %attern with

im%ortant individual di""erences.


Early Theories of Planetary Motion -$alculating Planetary Positions

Author(s):


Now that we have determined the %arameters o" the basic ancient model "or each %lanet, we can

calculate %lanetary %ositions. &his involves solvin the same trianle CEPthat we solved to "indthe %arameter rHR.

'irst we must initialie the model. 'iure shows two hy%othetical %ositions o" the %lanet: oneat the initial time Tand another at a "uture time T.
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'iure

At time Tthe %lanet is in the middle o" its retrorade arc and occu%ies the %ositionPon the

e%icycle. &he e%icycle center is at the %oint Con the de"erent, and the a%%arent %lanet is alinedwith the center at the %ointAon the ecli%tic. At this %oint one uses an astronomical instrument to

observe the lonitude o" the %lanet. &his ives the %osition in derees o" the a%%arent %lanet on

the ecli%tic, as measured "rom the %oint on the ecli%tic, and initialies the model. (0ecall "rom

Section B that is the a%%arent %osition o" the sun on the ecli%tic at the s%rin e2uino4.)

&he %lanets lonitude at time Tis calculated by "indin the measure o" arcAAand addin it to

the %lanets lonitude at T. &o do this we must solve trianleECP"or anlePEC. A sam%le

calculation "or Mars is shown below. 'or sim%licity in the calculation we ta!e as our unit o"measure the radiusRo" the de"erent7 with this unit o" measure, ris e2ual to the %arameter rHR.

&he solution %resented ma!es use o" modern trionometry and notation.

Example: &he motion o" Mars over the course o" a retrorade arc is observed with an

astronomical instrument, and it is determined that the middle o" the arc occurred at lonitude F.'ind the lonitude o" Mars =G days later.

Solution:

y &able in Section D. we !now that the Sidereal rate is .G=FHday and the Synodic rate is

.FD=Hday.RL unit (assumed)7 r = r/RL .DD by &able B in Section .

&he lonitude at T is iven as F and the ela%sed time "rom Tto T is iven as =G days.


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At T, arc CCL (sidereal rate) 4 (ela%sed time) L (.G=F) 4 (=G) L B and

arcPPL (synodic rate) 4 (ela%sed time) L (.FD=) 4 (=G) L G.G.

AnleECPL arcPPL G.G.

EC = RL unit7 CP = rL .DD.

y the


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. &he motion o" Ju%iter over the course o" a retrorade arc is observed with an astronomical

instrument, and it is determined that the middle o" the arc occurred at lonitude BFG. 'ind the

lonitude o" Ju%iter FB days later.

=. &he motion o" Mars over the course o" a retrorade arc is observed with an astronomical

instrument, and it is determined that the middle o" the arc occurred at lonitude B.

a. 'ind the lonitude o" Mars G days later.

b. 'ind the lonitude o" Mars G days !eforethe observation.

B. &he motion o" Saturn over the course o" a retrorade arc is observed with an astronomicalinstrument, and it is determined that the middle o" the arc occurred at lonitude =. 'ind the

lonitude o" Saturn days later.

Ans"ers

. G

=. a. G= b. =

B. =F



Trigonometry in the Basic Modern ModelAuthor(s):


&rionometry enters into the basic modern model in essentially the same way as the ancient one.

&his is because the two models are -e2uivalent/: iven a %articular %lanet, the ancient e%icycle

de"erent model "or the %lanets motion ives the same lonitudes as the modern model based on a

movin earth.

'iure demonstrates the e2uivalence "or an outer %lanet. (&o see thins more clearly, startyour e4%loration o" this "iure by hidin the trianles.)


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'iure

+n each model the lonitude o" the %lanet is iven by the rotatin anle centered at the earth7 thesides o" the anle are brown in the modern model on the le"t and blue in the ancient model on theriht. +n both cases one side o" the anle %oints in a "i4ed direction "rom the earth, while the

other side %asses throuh the %lanet. +n the modern model the earth moves in a circle, brinin

the verte4 o" the anle with it7 in the ancient model the earth and the verte4 remain stationary.

1hen set in motion, the corres%ondin sides o" the anles remain %arallel to each other7 thus thelonitude is the same under both models. Su%erim%osin the two models, we see that the motion

o" the %lanet around its e%icycle re"lects the earths actual motion, while the motion o" the center

around the de"erent re"lects the %lanets actual motion.

'iure demonstrates the e2uivalence "or an inner %lanet. Note that here the role o" the

e%icycle and the de"erent are reversed: the e%icycle re"lects the %lanets actual orbit o" the sun,while the de"erent re"lects the earths.

'iure
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Showin the trianles in the two "iures, we can ain some insiht into how this e2uivalence

a""ects com%utations within the basic modern model. +n the ancient model, !nown in"ormation

about the motion o" the %lanet around its e%icycle and the %oint Caround the de"erent translatesinto in"ormation about the trianle "ormed by the earth, the %lanet, and the e%icycle center C7

trionometry then allows one to calculate %arameters and %lanetary %ositions. +n the modern

model, the mathematical situation is virtually unchaned. Pnown in"ormation about the motionso" the earth and %lanet around their circles translates into in"ormation about the trianle "ormed

by the earth, the %lanet, and the center o" the earths orbit 7 %arameters and %lanetary %ositions

are then calculated by solvin this trianle. +n each case the crucial ste% is the ability to solve atrianle, a ste% that is im%ossible without trionometry.

Triangles in the Sky: Trigonometry and Early Theories of Planetary Motion -

Concluding Remarks

Author(s):

Sandra M Cara!ella ("e# $ersey City %ni!ersity)

&n this article #e ha!e e'lored the asic eicycle-deferent model for lanetary

motion and its modern e*ui!alent+ and ha!e seen ho# trigonometry enales these

models to gi!e a *uantitati!e descrition of the #anderings of the lanets The

lanetary ositions comuted y such simle models are not !ery accurate+ and

e!en early astronomers #ere #ell a#are of their limitations, as a result+ they

modied their models #ith !arious geometric de!ices These modications did make

the models more accurate, for e'amle+ errors in Ptolemy.s models for the outer

lanets amounted to no more than one degree in redicted longitude /01+ 2324

At the same time+ ho#e!er+ such modications led to the need for e!en more

trigonometric comutations &n the nal lanetary models of oth Ptolemy

and Coernicus+ for e'amle+ not one ut at least three triangles had to e sol!ed

to comute a single lanetary osition

The models descried in this article reresent 5ust one alication of trigonometry

to early astronomy, there are many others 6o#e!er+ considering 5ust this one

alication+ #e ha!e seen that the trigonometry re*uired in early astronomy

included kno#ledge of oth the 7a# of Sines and the 7a# of Cosines (or their

e*ui!alents)+ as #ell as the aility to comute at least the sine (or its e*ui!alent) of

any gi!en angle &n resonse to such demands+ the ancient 8reeks de!eloed a

rudimentary form of trigonometry called the 9theory of chords Along #ith Euclid.sgeometry+ this theory enaled astronomers to sol!e triangles (using rocedures

e*ui!alent to the 7a#s of Sines and Cosines)+ and to comute the 9chord of any

gi!en angle (e*ui!alent to t#ice the sine of half the angle) in one-half degree

increments 6o#e!er+ the theory of chords #as clumsy and di;cult to #ork #ith+

and as techni*ues of astronomy imro!ed+ so did the need for more and more

accurate and e;cient trigonometric comutations As a result+ the theory of chords
http://www-history.mcs.st-and.ac.uk/Biographies/Copernicus.htmlhttp://www-history.mcs.st-and.ac.uk/Biographies/Copernicus.html


21/23

#as de!eloed y &ndian+ &slamic+ and later Euroean mathematicians and

astronomers into much of the asic trigonometry that #e kno# today

the Coernican Re!olution &t is fair to say that

#ithout the study of triangles in the sky+ that re!olution may ne!er ha!e occurred

Convergence



Bi%liogra&hy and 'urther #eading

Author(s):


*ood accounts o" early astronomy can be "ound in >G, >, and >K7 > and >K also containsections on early trionometry. 3resentations o" the history o" astronomy and trionometry

within the more eneral conte4t o" the history o" mathematics can be "ound in >= and >I. 'or

discussions o" early astronomy "ocusin mainly on %lanetary models and the Co%ernican

0evolution, see >B and >. &he various websites listed below o""er Java animations o" many o"the eometric models o" early astronomy, as well as eneral in"ormation %ertainin to the

sub#ect. &he %rimary source "or *ree! astronomy and trionometry is 3tolemysAlmagest7 >= isa ood 8nlish translation. 'inally, it should be mentioned that > contains classroom resource

materials utiliin various ideas "rom the history o" trionometry and astronomy.

. Scott 0. Anderson, -+ntroduction to Astronomy,


22/23

G. James 8vans, T%e #istory and Practice of Ancient Astronomy, 54"ord University 3ress, New

@or!, KK.

D. Andres 9aras +drobo, -3tolemys &heory o" Su%erior 3lanets,/ htt%:HH"acultysta"".ou.eduHH3eter.ar!erH;SC+BBH%lanet.html, accessed Se%tember G, =.

I. 9ictor J. Pat,A #istory of $at%ematics:An )ntroduction, Brded., Addison1esley, oston,

=.

. 9ictor J. Pat and Paren 6ee Michalowic, eds.,#istorical $odules for t%e Teac%ing and

*earning of $at%ematics(on C605M), Mathematical Association o" America, 1ashinton6C, =F.

K. 0osemary Pennett, -8%icycles and

6e"erents,/ htt%:HHwww.%hy.syr.eduHcoursesH#avaHdemosH!ennettH8%icycleH8%icycle.html,

accessed Se%tember G, =.

. &homas S. Puhn, T%e Copernican Re(olution:Planetary Astronomy in t%e +e(elopment of

&estern T%oug%t, ;arvard University 3ress, Cambride, KGI.

. C. M.


23/23

=. *. J. &oomer, transl.,Ptolemy2s Almagest, 3rinceton University 3ress, 3rinceton, KK.

=. *re 9an rummelen, -Animations o" 3tolemys 3lanetary

Models,/ htt%:HH"aculty.benninton.eduHvanbrumH, accessed Se%tember G, =.

htt%:HHwww.maa.orH%ressH%eriodicalsHconverenceHtrianlesinthes!ytrionometryandearlytheorieso"%lanetarymotionbibliora%hyand"urther
http://faculty.bennington.edu/~gvanbrum/http://faculty.bennington.edu/~gvanbrum/

triangles in the sky

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