E Memoire sur la plus grande equation des planetes. (Memoir on the Maximum value of an Equation of the Planets)

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1 Traslatio of Euler s paper E05 -- Memoire sur la plus grade equatio des plaetes (Memoir o the Maximum value of a Equatio of the Plaets) Origially published i Memoires de l'academie des scieces de Berli, 748, pp Opera Omia: Series, Volume 8, pp Reviewed i Nouvelle bibliothèque Germaique 5, 749, pp. 8-9 Traslated by Jase A. Scaramazza ad Thomas J. Osler Mathematics Departmet Rowa Uiversity Glassboro, NJ 0808 Osler@rowa.edu Itroductio to the traslatio: This traslatio is the result of a fortuate collaboratio betwee studet ad professor. Jase Scaramazza is a udergraduate mathematics ad physics major at Rowa Uiversity. Tom Osler has bee a mathematics professor for 49 years. Together we struggled to uderstad this astroomical work. Jase traslated Euler s paper from Frech to Eglish ad both he ad Tom worked to comprehed Euler s achievemet. Whe traslatig Euler s words, we tried to imagie how he would have writte had he bee fluet i moder Eglish ad familiar with today s mathematical jargo. Ofte he used very log seteces, ad we sometimes coverted these to several shorter oes. However, i almost all cases we kept his origial otatio, eve though some is very dated. We thought this added to the charm of the paper. Euler was very careful i proof readig his work, ad we foud few typos. Whe we foud a error, we called attetio to it (i parethesis usig italics) i the body of the traslatio. Other errors are probably ours. We also made a few commets of our ow withi the traslatio idetifyig our words i the same way. The separate collectio called Notes was made while traslatig this paper. This is material that we accumulated while tryig to uderstad ad appreciate Euler s ideas. I these otes we completed some steps that Euler omitted, itroduced some moder otatio ad elaborated o a few of Euler s derivatios that we foud too brief. We also added a few simple ideas ad commets of our ow ot foud i the text. A separate Syopsis is also available ad was made from a codesatio of the Notes.

2 I. I astroomy, we lear about the plaets equatios of the ceter from a heliocetric stadpoit, i which a observer is viewig them from the su. This is eeded sice a irregularity is observed as we watch plaets move across the sky from the Earth. Sometimes they move faster, the slower, ad eve stop ad appear motioless at the same poits i space. They will sometimes eve tur aroud ad begi retrograde motio. However, astroomers have realized that if plaetary movemets were observed from the su, these irregularities disappear almost etirely. A observer placed i the star would ot eve see the plaets stop or eter retrograde motio. They will see a uchagig path directly followig the order of the costellatios. Nevertheless, this movemet would ot be quite uiform. There would still iheretly be a variatio i speed. The same plaet would sometimes be observed movig faster ad sometimes slower. It is this chagig motio that astroomical tables have desigated The Equatio of the Ceter. II. Plaets as see from the Earth do ot seem to follow ay laws. We cosider our distace from them to be the cause, although it is very difficult to determie distace solely by observatio. But if the movemets of the plaets with respect to the su are recorded, ad are represeted from a supposed observer i the su, the aomalies i plaetary distaces will almost be elimiated. This is because i each revolutio, each plaet will oly oce be at its greatest distace from the su, ad oce at its closest proximity; moreover, these two poits will be diametrically opposed to each other ad uchagig i the sky. The time itervals durig which the plaet travels from the farthest distace, to the closest ad back will be costat.

3 3 The poit i the sky i which the plaet appears the farthest from the su is called its Aphelio ad the opposite poit where it is closest is called its Perihelio. Additioally, the time i which the plaet leaves from the Aphelio or from the Perihelio, ad returs back is called its period. III. The chagig distaces of each plaet from the su icely follow a costat relatio with the varyig movemet, as see from the su. Whe the plaet is farther away, it moves slower, ad whe it approaches the su, it speeds up. This is the beautiful law that Kepler discovered, ad that Newto has sice demostrated by the priciples of mechaics. More specifically, each plaet i equal times sweeps ot equal agles aroud the su, but equal areas. The law also provides irrefutable evidece that plaetary orbits are ellipses with the su at oe of the foci. The variatio i movemet is clearly regulated. Equal areas i equal times are swept i each ellipse, drawig the areas by lies straight from the plaet to the su. IV. The first thig that we ca ifer from this rule is that the greater the distace betwee the farthest ad closest poits of a plaet to the su, the more irregular its movemet would appear from the su. Whereas if a plaet would always maitai the same distace from the su, this is to say that its orbit would be a circle with the su as the ceter, the its movemet would be so regular that i equal times, it would sweep out ot oly equal areas, but also equal agles. I this case, we would therefore be able to very easily determie, by the third law (Kepler s Third Law) the distace from the plaet to the su for ay time. But as this coditio does ot exist for ay plaet, we have the custom to ideally coceive for each plaet aother

4 4 plaet that serves as a compaio. It completes its revolutio aroud the su i the same time period, but with uiform movemet. We furthermore suppose that this fictitious plaet would appear at the same poit i the sky as the true whe it is at the Aphelio ad the Perihelio. After these two plaets have passed by the aphelio, the false plaet will appear to go faster tha the true ad the real plaet will imperceptibly icrease its speed util it will have caught the false oe at the Perihelio. The it will pass its parter i speed, ad will leave it behid util they rejoi agai at the Aphelio. Therefore, with the exceptio of these two poits, the two plaets will be perpetually separated from each other, ad the differece betwee the positios of the two is what we call the equatio of the ceter of the plaet, or the Prostapherese. As it is easy to fid the positio of the false plaet at ay time, if the equatio that astroomical tables provide is kow, we will kow the positio of the true plaet. V. Astroomers call the mea aomaly either the distace from the false plaet to the aphelio, or the agle that the false plaet forms with the aphelio ad the su. We ca easily determie it by the time that elapsed sice the plaet s passage from the aphelio. The true aomaly is the distace from the true plaet to the aphelio, or the agle that the true plaet forms with the aphelio ad the su. Therefore, whe the plaet moves from the aphelio to the perihelio, we fid the true aomaly i subtractig the value from the equatio of the ceter from the mea aomaly; coversely, whe the plaet returs from the perihelio to the aphelio, we must add the value from equatio to the mea aomaly i order to have the true aomaly. Thus, we ca determie by the mea aomaly or by the true, the actual

5 5 distace from the plaet to the su. Cosequetly, if we determie the positio of the Earth see from the su for the same time, trigoometry shows where the plaet as see by the Earth must appear, also kow as its geocetric locatio. VI. Whe the plaet is at the perihelio or the aphelio, the (value of the) equatio of the ceter is zero. It is ecessary to realize that the (value of the) equatio grows as the plaet moves, ad later it decreases agai. There will therefore be a place where the equatio will have a maximum. Several very importat astroomical questios arise here: What is the maximum for each plaet? Ad what mea aomaly correspods to this maximum? Moreover, as the greatest value of the equatio is determied by the eccetricity of the plaet s orbit, which is the fractio whose umerator is the distace betwee the foci of the ellipse, ad whose deomiator is the major axis of the ellipse? Ad coversely, we will determie the eccetricity from the maximum value of the equatio. I am therefore goig to examie these questios, for which a rigorous mathematical solutio still does ot exist. VII. Let half the legth of the major axis of the orbit of each plaet = a, which is customarily called i astroomy the average distace from the plaet to the su. The eccetricity, or the distace betwee the foci divided by the major axis, =, ad shriks if the orbit becomes a circle. It becomes larger whe the orbit elogates from a circle. Ad if the elogatio goes to ifiity, meaig that the orbit becomes a parabola, the the eccetricity will become equal to oe. But if it becomes a hyperbola, it will be larger tha oe. The major axis beig a, the distace betwee

6 6 the foci will be = a, ad the distace from each focus to the ceter = a. Cosequetly, the distace from the aphelio to the su will be = a a = a ( ) ad the distace from the perihelio to the su = a a = a ( ). The the semimior axis will be = a ad the parameter s half will be a ( ). ( We do ot uderstad this highlighted phrase.) VIII. These thigs beig supposed, for a give time let the passage of the plaet past the aphelio, have the mea aomaly = x, ad the correspodig true aomaly = z. The (value of) equatio of the ceter, as we have see, will be = x z. Let the distace from the plaet to the su = r. To express the relatio betwee the mea aomaly ad the true aomaly, it will be ecessary to employ a ew agle holdig a kid of middle betwee x ad z, which Kepler amed the eccetric aomaly. Let this eccetric aomaly = y. Elsewhere, I determied, with the mea aomaly x, the true aomaly z, ad with the distace deoted by r, that without questio we have x = y si y ; cos y cos z ; ad r a( cos y). Cosequetly, we will have cos y si z si y cos y ; ad ta z si y cos y ; from which, by the eccetric aomaly y, we ca fid the mea aomaly ad the true, ad the distace from the plaet to the su. With these formulas, it is easy to calculate the Astroomical Tables for plaetary movemets. IX. Before fidig the maximum value of the equatio, it will be coveiet to start with the followig problem that ca have some use i astroomy:

7 7 Fid the mea aomaly ad the true, whe the distace from the plaet to the su is equal to the average distace a. As here r must be = a, it will be ecessary that, ad it follows that the eccetric aomaly y will be = 90. We the compute from the average aomaly (Kepler s equatio ) x = For this computatio, the additio of the sie term, with the radius here equal to oe, we must look i a similar circle the arc = (same umber as agle i radias), ad the agle (coverted to degrees) that measures this arc must be added to 90 to fid the sought average aomaly x. (See Notes for more details.) Or, as is a umber less tha oe, we must treat it as a sie, ad subtract from its logarithm. The correspodig umber to the logarithm that remais will provide the agle expressed i secods. But the true aomaly z that correspods to this eccetric aomaly y = 90 will be such that =, ad z= A = 90 - A. (Euler uses Acos to mea the iverse fuctio arccos.) Let m be the agle for which the sie =, ad we will have z = 90 - m ad the ceter equatio i this case will be = + m = + A. This is why if the distace from the plaet to the su r is equal to the average distace to the su, what happes whe the plaet is i the cojugated axis (mior axis vertex) of its orbit, the mea aomaly x will be = 90 + A ad the ceter equatio = + A. X. I have begu with this problem because i this case, the expressio hardly differs sigificatly from the maximum of the equatio (of the ceter), whe the eccetricity is very small, which early happes i all plaets. I preseted i the Dissertatio o Plaetary Movemets & The Orbit of the Su, the

8 8 T.VII. of the Memoires of the Academy of Petersburg, the followig solutio of the problems o the maximum of the equatio. The true aomaly, as I showed i this paper, ca be expressed by a ifiite series, i the followig maer: If is a very small fractio, z will be approximately = ad because x is =, (should be x y si y ) the value of the equatio will be = si y. The equatio will be greatest whe y is = 90, i which case r becomes = a. However if the eccetricity of the plaet is ot so small, as happes with Mercury, this evaluatio will deviate a little from the truth. This is particularly importat whe determiig the period of some comet ad recordig the comet s movemet i the tables as we do the plaets. This calculatio will deviate very much from the truth, for the maximum of the equatio will occur cosiderably far from the place where the distace of the plaet to the su is equal to half the major axis. XI. For all these cases, we must deduce the maximum of the equatio by usig the method of maximums ad miimums, rather tha usig the formulas that are oly approximately true. Thus, to determie the average aomaly ad the true, we must first kow the eccetric aomaly. I will start with the followig problem. Beig give the eccetricity of a Plaet, fid the eccetric aomaly that correspods to the maximum of the equatio of the ceter.

9 9 Let the mea aomaly = x, ad the true aomaly = z, the the ceter equatio is = dx x z. This equatio will become the greatest whe dx dz 0, or whe dz. Callig the eccetric aomaly y, we will have as we have see earlier, ad cos y cos z. Differetiatig results i, ad cos y dysi y dysi y ( ) dysi y dz si z, or just as well dz si z. (See ( cos y) ( cos y) sectio VIII.) But is = si y, ad therefore we have cos y dz dy. cos y Sice dx must be equal to dz, we obtai the equatio: cos y, ad cos y 4 the we get cos y ad cos y 4. Let y = 90 +, ad we will have si 4, or si ( 4 )(, from which it seems that ) the eccetric aomaly is a little greater tha i the previous case, where it was y = 90. XII. As before we let m si, ad we ow have cosm. Thus give the eccetricity, the agle m will be kow. We will therefore have cos m si, si m ad cos cosm cosm si m cosm. But if the eccetricity is much less tha oe, as happes i all plaets, we will have

10 0 4 *3 4*8 *3*7 4*8* *3*7* 4*8** etc. It follows that the agle, by which the eccetric aomaly y surpasses beig a right agle, will be expressed as si 4 *3 4*8 3 *3*7 4*8* 5 *3*7* 4*8**6 7 etc. Thus by kowig the eccetricity, we ca easily fid the agle, ad the the eccetric aomaly y = The cosie of the agle is therefore cos etc. XIII. Havig solved the problem of determiig the eccetric aomaly y correspodig to the maximum value of our equatio, we ca determie the correspodig mea ad true aomalies. But it is expediet to fid each of them separately. Beig give the eccetricity, fid the mea aomaly that correspods to the maximum value of the equatio. The eccetric aomaly for this case is y = 90 + ad si 4. Because, we will have. But if we wat to express the excess of this agle above 90 by, sice is equal to si si si etc., we 6 40 will have etc., whose value beig substituted i place of ad of foud above, the mea aomaly will be

11 x etc.. But if is ot a quatity so small that these series coverge rapidly eough, the it will suffice to use the expressio previously foud, which is easy to work with i calculus. XIV. Before the maximum of the equatio of the ceter ca be determied, we must also fid the true aomaly. Beig give the eccetricity, fid the true aomaly that correspods to the maximum of the equatio. The eccetric aomaly i this case is foud to be y 90 kowig that si 4. Cosequetly, whe we let the true aomaly equal z, we will have cos z cos y cos y cos y 4 si y. The cosie beig positive shows that z Let z 90, ad it will be that si si ( ) 4 3. Thus, by kowig the eccetricity, we will fid the agle. But if is a very small fractio, we will almost have ( ) * 4*8 4 3**5 4*8* 6 3**5*9 4*8**6 8 etc., from which we will get si etc.. Ad the same agle will

12 determie itself by the formula: si si si etc. We will the 6 40 have etc XV. Now if we subtract the true aomaly from the mea aomaly, we will have the maximum value of the equatio of the ceter. Beig give the eccetricity of the plaet s orbit, fid the maximum of x - z. For this maximum, we have foud the mea aomaly with, ad we have also foud the true aomaly with. The maximum will be =. But if i the case where is a small eough fractio, we wat oly a maximum that approaches the truest possible value. From the above results we have: But whe the distace from the plaet to the su is equal to half the major axis, the equatio is = Thus the maximum surpasses this by a quatity =

13 3 XVI. Sice we have foud, the distace from the plaet to the su whe its equatio is at a maximum, will be, a distace that is always less tha half of the major axis. From this we ca easily determie usig the eccetricity, the mea aomaly ad the distace from the plaet to the su that correspods to the maximum. But if the maximum is give as m, ad if we wat to reciprocally fid the eccetricity, the problem becomes very difficult, ad ca oly be foud by approximatio. Because we foud the equatio, by which we must fid the value of ; there is o other way to solve the equatio tha to try differet values of, ad deducig the maximum from there. We will discover i fact by this method first the boudaries betwee which the true value of is cotaied, ad followig the same route, we will make the limits still closer, util fially by the rules of iterpolatios we ca calculate the true value of the eccetricity. XVII. But if the eccetricity is ot very large, (meaig our approximate formulas ca be used without error) we will be able to directly fid the eccetricity by the give maximum. Beig give the maximum, fid the eccetricity of the plaet s orbit. Let the maximum = m, ad the eccetricity =, we will have, from which we calculate by iversio Here we must express the greatest value of the equatio m i fractios of the radius (this meas i radias rather tha degrees),

14 4 which is doe by covertig the agle m ito secods. To the logarithm of the umber (m i secods) we add , because we will thus have the logarithm of the umber m (i radias). The mea aomaly x that correspods to the maximum will be We will approach closely eough to the mea aomaly, if at 90 degrees we add five eighths of the maximum. XVIII. To clarify the applicatio of these solutios to astroomical calculatio, we will take for a example the orbit of Mercury. The astroomical tables have the eccetricity equal to. We will the have ;. If the distace from Mercury to the su is equal to its semi-major axis, or if we make the eccetric aomaly = 90, the mea aomaly x will become =. From this, we fid the agle. From, we subtract to get this logarithm: , which correspods to the umber From this we have = Thus the mea aomaly is. (The symbol 3 5 meas 90 degrees.)but the true aomaly i this case is. Now, from which we get. From this the equatio becomes = , which is early two miutes less tha the maximum. XIX. But to fid the maximum, we make the followig calculatios

15 5 subtract Therefore Thus the eccetric aomaly that correspods to the maximum is. Furthermore, to fid the mea aomaly, we take subtract from ad we have Therefore, or. From the mea aomaly that correspods to the maximum, beig, we will fid, which we fid agrees with the maximum i the tables. Furthermore the true aomaly is, where, from which we deduce the followig calculatio:

16 6 = = ad = = = subtract = = Therefore = Addig = The maximum is = , which does ot differ eve a secod from the maximum represeted i the (astroomical) tables, which verifies the previous theory. So as the semi-major axis of Mercury s orbit is = 3870 = a, we will have = = = ad r will be the distace from Mercury to the Su where its equatio is the greatest.

17 7 XX. Iversely, to fid the eccetricity from the give maximum, I believed it ecessary to place here the followig table sice this calculatio ca oly be achieved by iterpolatios. I this table we fid for each hudredth of eccetricity, the correspodig maximums, as well as eccetric ad mea aomalies. The last colum also provides the logarithm of the distace from the plaet to the su whe the value of the equatio is a maximum. I fact, callig this distace = r, ad the semi-major axis = a, the we have. The last colum cotais the logarithms of the formula, that beig addig to the logarithms of the mea distace, will produce the logarithm of the desired distace r. XXI. With the help of this table, if give ay eccetricity, we will fid by iterpolatio the correspodig maximum. Thus, the eccetricity of the Earth beig = 0.069, we fid the eccetricities i the table. Eccetricity Maximum Differece We subtract from the kow value to get the differece , ad the result will be the followig proportio: 00 :.8.46 = 69 : I addig this agle 47.6 to the smaller equatio.8.45, we will have the maximum value of the equatio of the ceter of the orbit of the Earth =.56..

18 8 Eccetricity Maximum Differece The eccetricity of Mars give i the tables is = We look for the two closest eccetricities with the maximums from our table. Now the eccetricity give exceeds the smaller value by , ad from this we calculate the proportio 0000 :.8.58 = 998 : Add this agle 0.40 to the precedig value of the equatio (i the table), which is 0.9., ad we will get the maximum of the equatio of the ceter of Mars orbit = This agrees perfectly with the (astroomical) tables. XXII. The mai use of the table below will be to determie the eccetricity whe the maximum is kow. Without this table, the problem is absolutely usolvable. To demostrate this by a example, we take the maximum of Mercury, which the (astroomical) tables show as We have already remarked that this is perfectly i agreemet with the eccetricity provided by the same tables (see sectio XIX.) Let us take the two maximums that are the closest to this value: Maximum Eccetricity Differece

19 9 The let the smaller value of the equatio be subtracted from the give value of Mercury s maximum: From this we fid the proportio.9.47 : 0.0 = : This umber added to the smaller eccetricity 0.0, will give the eccetricity of Mercury s orbit = This hardly differs from what has bee observed, although we have assumed here that the value of the equatio is 4 greater (tha that calculated i sectio XIX ). I fact, this additio of 4 oly icreases the logarithm of the eccetricity by , which was previously = XXIII. It is ecessary to remark here that the two variables ad from cotiually grow as the eccetricity icreases. The third term,, is of more importace; sice its value disappears i the case = 0 ad also whe =. To fid its maximum, we must examie the differetial equatio dcos d si. Now sice si 4, or 4 si, we will have dsi d cos d 4. (This is oe of Euler s few misprits, this

20 0 term should be d 4 3.) We substitute here the precedig value, ad that will be, or. Let us suppose that, ad will become. This equatio beig solved by approximatio, we will fid that = , or p = , from there we fid that = , ad the greatest value of becomes = Also, it is useful to observe that if the eccetricity is = , the maximum value of the equatio of the ceter will be exactly = 90.

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