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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 34832, 1127]*) (*NotebookOutlinePosition[ 35801, 1162]*) (* CellTagsIndexPosition[ 35695, 1155]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["PtolemyChordTable.m", "Title"], Cell[TextData[{ "Functions for the", StyleBox[" Almagest", FontSlant->"Italic"], " I 11\nby Nathan Sidoli" }], "Subtitle"], Cell[CellGroupData[{ Cell["\<\ This package calls in a number of functions which facilitate \ working with Ptolemy's trigonometry. It was written for the student of \ Ptolemy's astronomy.\ \>", "Section 1"], Cell[CellGroupData[{ Cell["Introduction", "Subsection"], Cell[TextData[{ "The main benefit of PtolemyChordTable.m is that it saves one the tedium of \ repeatedly referring to the chord table while working through Ptolemy's \ calculations. Most of the functions are trivial from a mathematical \ perspective. They are all based on, and make use of, the chord table itself. \ Thus, the most important component of this package is the chord table found \ in ", StyleBox["Almagest", FontSlant->"Italic"], " I 11. The chord table has 3 columns of rational numbers in 365 rows. The \ first column contains angles from .5 to 180 in steps of .5. The second column \ contains the chords that correspond to these angles in sexagesimal fractions, \ under the assumption that Chord(180) = 120. The third column contains the \ modifiers to the chords in the second column for each additional sixtieth of \ degree between the actual angle and the angle that we use to determine the \ chord (the greatest entry in column 1 which <= actual angle). See Pedersen \ [1974, 63 - 5] for a formal description.\n\nI have used the chord table in \ Toomer [1984, 57 - 60] which is the most accurate in print. There may, \ however, still be some errors in the table as it stands. I have included some \ functions for checking suspect entries \ (ChordTableElement(Inverse)[angle(chord)]). If an error is found please \ contact the ", ButtonBox["author.", ButtonData:>"Author", ButtonStyle->"Hyperlink"] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Ptolemy's Trigonometry", "Subsection"], Cell[TextData[{ "For the purpose of trigonometric mensuration, Ptolemy employs a single \ function: the chord. This strategy goes back at least as far as his \ predecessor Hipparchus, who is reported to have composed a work on chords \ which was probably somewhat simpler than what we find in ", StyleBox["Almagest", FontSlant->"Italic"], " I 11. I will not here go into the details of Ptolemy's trigonometry. This \ subject has been treated quite well by both Neugebauer [1975, 21 - 25] and \ Pederson [1974, 65 - 69]\n\nPtolemy derives his chord table from the \ properties of regular polygons and the application of areas, under the \ assumption that the diameter of the circle in which the chords are situated \ is 120. All of the propositions that he uses can be found in Euclid's ", StyleBox["Elements", FontSlant->"Italic"], " (II, IV, VI, & XIII), and his reasoning is very clearly layed out in ", StyleBox["Almagest", FontSlant->"Italic"], " I 10, Toomer [1984, 47 - 56].\n\t\nPtolemy is not explicit about how the \ chord table is to be used, but it is easy enough to infer the proceedure \ based on the form of the table and the examples we have in the text." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The Package", "Subsection"], Cell[TextData[{ "With no further ado, we call in PtolemyChordTable.m. (Note that, in order \ for PtolemyChordTable.m to be loaded by the following command, the file \ PtolemyChordTable.m should be placed in the AddOns/Applications subdirectory \ of the main ", StyleBox["Mathematica", FontSlant->"Italic"], " directory. Alternatively, the directory where PtolemyChordTable.m is \ located should be added to the $Path variable using ", StyleBox["AppendTo[$Path, \"DirectoryPath\"\:f3b5]", "InlineInput"], ".)" }], "Text"], Cell[BoxData[ \(Needs["\"]\)], "Input"], Cell["\<\ The basic function in PtolemyChordTable.m is \ ChordTable[angle].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?ChordTable\)\)], "Input"], Cell[BoxData[ \("ChordTable[angle] takes an angle at the center of a circle as argument \ and returns the length of the chord under the assumption that the diameter of \ the circle is 120. NB: The angle is taken modulo 180 (Mod[angle, 180]) so a \ value will always be returned, however, if the argument > 180 this may be \ meaningless."\)], "Print"] }, Open ]], Cell["\<\ This is done by scanning the chord table for the greatest \ half-angle that is <= the argument. The chord which corresponds to this \ half-angle is used as the basis for the returned value. This basis-chord is \ augmented as the actual angle is greater than the half-angle found in the \ chord table by intervals of 1/60th. In order to find the augmentation number, \ I have elected to simply round the number of 1/60ths between the actual angle \ and the nearest table entry which is <= the actual angle. The total \ augmentation to the basis-chord is the augmentation number * column 3. Thus \ if the angle in question is 35;37,37, the line in the chord table that will \ interest us is found by \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ChordTableElement[35 + 37/60 + 37/60^2]\)], "Input"], Cell[BoxData[ \({{35.4999999999999982`}, {36, 35, 1}, {0, 0, 59, 48}}\)], "Output"] }, Open ]], Cell["\<\ The basis-chord will be 36;35,1 and the augmentation number will be \ (35;37;37 - 35;30) * 60 = 7.6166... which rounds to 8. So we add 8 * \ 0;0;59,48 = 0.132889 to our basis-chord. This gives 36.5836 + 0.132889 = \ 36.7165. All of this is simplified in the above mentioned function:\ \>", \ "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ChordTable[35 + 37/60 + 37/60^2]\)], "Input"], Cell[BoxData[ \(36.7165000000000052`\)], "Output"] }, Open ]], Cell["\<\ In order to view the result in a manner that would have pleased \ Ptolemy, we have the following function:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PtolemyForm[%]\)], "Input"], Cell[BoxData[ \({36, 42, 59, 24, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ PtolemyForm[x] returns integers followed by six sexagesimal places. \ For Ptolemy's practice of \"casting out by circles\" we have a function which \ \"neglects whole circles\" in its integer part. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PtolemyFormMod360[736 + 34/60 + 15/60^2]\)], "Input"], Cell[BoxData[ \({16, 34, 15, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ This function gives the integer part modulo 360, but leaves the \ sexagesimal fractions unaffected.\ \>", "Text"], Cell[CellGroupData[{ Cell["Right Triangles", "Subsubsection"], Cell[TextData[{ "To determine the side of a right triangle, where an angle is known and the \ hypotenuse is assumed to be 120, Ptolemy makes use of the fact that on a \ given chord the angle at the center of the circle is double an angle on the \ circumference (", StyleBox["Elements", FontSlant->"Italic"], " III 20). This is really just a modification of ChordTable[angle].\n\nThus \ if 23.465672 is an angle in a right triangle, then its chord, where the \ hypotenuse is 120, is given by" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RightTriangleSide[23.465672]\)], "Input"], Cell[BoxData[ \(47.7857685185185143`\)], "Output"] }, Open ]], Cell["We use the inverse function to find an angle given a side.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(RightTriangleAngle[%]\)], "Input"], Cell[BoxData[ \(23.4666666666666667`\)], "Output"] }, Open ]], Cell["\<\ Notice that these functions are not true inverses. This is due to \ the necessity of rounding and the use of augmentation by steps of \ 1/60ths.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["General Triangles", "Subsubsection"], Cell[TextData[{ "Ptolemy has two procedures for working with general triangles that roughly \ correspond to applications of the cosine rule and sine rules; he uses these \ infrequently and rarely respectively. I have followed Ptolemy very exactly in \ writing these functions and the curious reader can consult the ", StyleBox["Almagest", FontSlant->"Italic"], " and PtolemyChordTable.m for the details. In what follows, I have in each \ case included the relevant trigonometric rule. Ptolemy, of course, makes no \ use of the trigonometric functions to achieve the same end." }], "Text"], Cell["\<\ Cosine Rule (a): (side3)^2 = (side1)^2 + (side2)^2 - \ 2(side1)(side2) (Cos[angle3]) \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?PtolemyCosRuleSide\)\)], "Input"], Cell[BoxData[ \("PtolemyCosRuleSide[angle3, side1, side2] takes angle3 and the two \ adjacent sides of a triangle as arguments and returns the other side, side3."\ \)], "Print"] }, Open ]], Cell["\<\ .Cosine Rule (b): (Cos[angle3]) = [(side1)^2 + (side2)^2 - \ (side3)^2] / 2(side1)(side2) \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?PtolemyCosRuleAngle\)\)], "Input"], Cell[BoxData[ \("PtolemyCosRuleSide[side1, side2, side3] takes three sides of a \ triangle as arguments and returns the angle between the first two sides, \ angle3."\)], "Print"] }, Open ]], Cell["Sine Rule: (side1) / Sin[angle1] = (side2) / Sin[angle2]", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(?PtolemySinRule\)\)], "Input"], Cell[BoxData[ \("PtolemySinRule[side1, angle1, angle2] takes two angles and one of the \ opposite sides as arguments and returns the other opposite side, side2."\)], \ "Print"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["An Example", "Subsection"], Cell[TextData[{ "In order to show the usefulness of PtolemyChordTable.m for the purpose of \ reading the ", StyleBox["Almagest", FontSlant->"Italic"], " we will work through the first calculation used to determine the ratio of \ the epicycle to the deferent for the first, simple anomaly of the moon, ", StyleBox["Almagest", FontSlant->"Italic"], " IV 6. A modern reading of this procedure can be found in Neugebauer \ [1975, 74 - 77], but these calculations will best be followed by referring to \ Ptolemy's own methods, Toomer [1984, 191 - 198].\n\nThe determination of the \ lunar epicycle is based on three \"ancient\" eclipse observations taken in \ Babylon. These are as follows:\n \n I \t-720 March 19/20, \"an hour after \ moonrise,\" in Virgo 24.5 degrees\n II\t-719 March 8/9, \"exactly at \ midnight,\" in Virgo 13.75 degrees\n III\t-719 September, 1/2. \"after \ moonrise,\" in Pisces 3.24 degrees\n \n From these we can determine the time \ intervals between two different sets of eclipses, namely I to II and II to \ III. The interval between the first and the second is" }], "Text"], Cell[BoxData[ \(\(t1\ \ = \ 354 + \((2 + 17/30)\)/24;\)\)], "Input"], Cell["\<\ \"reckoned in mean solar days.\" And the interval between the \ second and the third, in the same units, is\ \>", "Text"], Cell[BoxData[ \(\(t2\ = \ 176 + \((20 + 1/5)\)/24;\)\)], "Input"], Cell[TextData[{ "Ptolemy has, at this point in his argument, already derived a mean motion \ table for the moon's motion in both longitude and anomaly, ", StyleBox["Almagest", FontSlant->"Italic"], " IV 4, and he, in essence, takes figures from this table to find the \ change in longitude and anomaly over the two given time intervals: " }], "Text"], Cell[BoxData[{ \(\(deltameanlongitude1 = \ 345 + 51/60;\)\), "\n", \(\(deltaanomaly1\ = \ 306 + 25/60;\)\), "\n", \(\(deltameanlongitude2\ = \ 170 + 7/60;\)\), "\n", \(\(deltaanomaly2 = \ 150 + 26/60;\)\)}], "Input"], Cell["\<\ These figures allow us to imagine all of these eclipses as \ occurring on a single position of the moon's epicycle (or eccentric). This is \ done by allowing the arc which corresponds to change in anomaly over the \ first interval to specify an arbitrary starting point for angular \ measurements and setting the other intervals relative to this. The positions \ of the eclipses on the imaginary epicycle (or eccentric) are then A, B, and \ C.\ \>", "Text"], Cell[BoxData[{ \(\(arcBA = 360 - deltaanomaly1;\)\), "\n", \(\(arcBG = deltaanomaly2;\)\), "\n", \(\(arcAG = deltaanomaly2 - arcBA;\)\)}], "Input"], Cell[TextData[{ "In order to determine the apparent positions of the eclipses on the \ imaginary epicycle (or eccentric) it will be sufficient to find the \ difference between the angular span between the true zodiacal appearances of \ two eclipses and the changes in mean longitude that have occurred between the \ two observations.\n\nThe angular span is found by considering the Zodiac \ itself.\n\nI to II is Virgo 24.5 to Virgo 13.75, or 349.25 degrees.\nII to \ III is Virgo 13.75 to Pisces 3.25. or 169.5 degrees.\n\nPtolemy provides two \ different figures for the epicyclic and eccentric hypotheses (fig. 4.4 & fig. \ 4.5, pp. 193 - 194) and he will have us understand that his reasoning can \ readily be transferred between the two, however, he only carries out \ calculations for the epicyclic hypothesis. The apparent positions of the \ eclipses on the imaginary epicycle (or eccentric) marks the first difference \ between the two hypotheses, because in the case of the epicyclic hypothesis \ the earth is outside the circle upon which the moon travels while in the case \ of the eccentric hypothesis it is inside of it.\n\nAt this point it becomes \ necessary for the persevering readers to avail themselves of the figures for \ ", StyleBox["Almagest", FontSlant->"Italic"], " IV 6. " }], "Text"], Cell[CellGroupData[{ Cell["The Epicycle", "Subsubsection"], Cell["\<\ The epicyclic hypothesis distinguishes itself from the eccentric in \ that the angles between the apparent eclipses are quite small. They are found \ by taking the difference between the angular span between the zodiacal \ positions of the eclipses and the change in mean longitude over the time \ interval between them, Toomer [1984, 193].\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(angleADB = 349 + 15/60 - deltameanlongitude1;\)\), "\n", \(PtolemyForm[angleADB]\)}], "Input"], Cell[BoxData[ \({3, 24, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(angleBDG\ = \ Abs[169 + 30/60\ - \ deltameanlongitude2];\)\), "\n", \(PtolemyForm[angleBDG]\)}], "Input"], Cell[BoxData[ \({0, 37, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ Now, E is a point on the circumference of the epicycle so the \ angles at E will be half of their arcs.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(angleBEA = arcBA/2;\)\), "\n", \(PtolemyForm[angleBEA]\)}], "Input"], Cell[BoxData[ \({26, 47, 30, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(angleBEG = arcBG/2; \nPtolemyForm[angleBEG]\)], "Input"], Cell[BoxData[ \({75, 13, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(angleAEG = arcAG/2; \nPtolemyForm[angleAEG]\)], "Input"], Cell[BoxData[ \({48, 25, 30, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ AngleBEA is an external angle for triangleDEA, angleBEG is an \ external angle for triangleDEG, and angleTGE one of the acute angles in right \ triangleTGE, thus:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(angleEAZ = angleBEA - angleADB; \nPtolemyForm[angleEAZ]\)], "Input"], Cell[BoxData[ \({23, 23, 30, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(angleEGD = angleBEG + angleBDG; \nPtolemyForm[angleEGD]\)], "Input"], Cell[BoxData[ \({75, 50, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(angleTGE = 90 - angleAEG; \nPtolemyForm[angleTGE]\)], "Input"], Cell[BoxData[ \({41, 34, 30, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["The chordAG, where the diameter of the epicycle is 120, is", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(AG\ = \ ChordTable[arcAG]; \nPtolemyForm[AG]\)], "Input"], Cell[BoxData[ \({89, 46, 13, 45, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ In the right triangleEZD, where DE is 120, EZ will be found as \ follows:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EZonDE\ = \ RightTriangleSide[angleADB]; \nPtolemyForm[EZonDE]\)], "Input"], Cell[BoxData[ \({7, 6, 59, 53, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ In the right triangleEAZ we can find EZ, where AE is 120, from \ whence we can express AE in the same units as DE. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EZonAE\ = \ RightTriangleSide[angleEAZ]; \n AEonDE\ = \ 120*EZonDE/EZonAE; \nPtolemyForm[AEonDE]\)], "Input"], Cell[BoxData[ \({17, 55, 31, 38, 23, 10, 32}\)], "Output"] }, Open ]], Cell["\<\ EH is found in right triangleDEH and right triangleGEH and EH and \ GE are expressed in the same units as DE.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EHonDE\ = \ RightTriangleSide[angleBDG]; \nPtolemyForm[EHonDE]\)], "Input"], Cell[BoxData[ \({1, 17, 29, 39, 0, 59, 59}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EHonGE\ = \ RightTriangleSide[angleEGD]; \n GEonDE\ = \ 120*EHonDE/EHonGE; \nPtolemyForm[GEonDE]\)], "Input"], Cell[BoxData[ \({1, 19, 55, 30, 55, 39, 34}\)], "Output"] }, Open ]], Cell["\<\ GT and ET are found in the right triangleGTE and both of these are \ expressed in the same units as DE.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(GTonGE\ = \ RightTriangleSide[angleAEG]; \n GTonDE\ = \ GEonDE*GTonGE/120; \nPtolemyForm[GTonDE]\)], "Input"], Cell[BoxData[ \({0, 59, 47, 27, 52, 37, 21}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(ETonGE\ = \ RightTriangleSide[90 - angleAEG]; \n ETonDE\ = \ GEonDE*ETonGE/120; \nPtolemyForm[ETonDE]\)], "Input"], Cell[BoxData[ \({0, 53, 2, 17, 54, 38, 3}\)], "Output"] }, Open ]], Cell["TA and AG are now known in the same units as DE.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(TAonDE\ = \ \ AEonDE\ - \ ETonDE; \nPtolemyForm[TAonDE]\)], "Input"], Cell[BoxData[ \({17, 2, 29, 20, 28, 32, 28}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(AGonDE\ = \ Sqrt[TAonDE^2\ + \ GTonDE^2]; \nPtolemyForm[AGonDE]\)], "Input"], Cell[BoxData[ \({17, 4, 14, 8, 31, 23, 40}\)], "Output"] }, Open ]], Cell["\<\ With this we can now find DE and GE where the diameter of the \ epicycle is 120, because AG is known in these units.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DE\ = \ 120*AG/AGonDE; \nPtolemyForm[DE]\)], "Input"], Cell[BoxData[ \({631, 3, 12, 30, 57, 51, 26}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(GE\ = \ DE*GEonDE/120; \nPtolemyForm[GE]\)], "Input"], Cell[BoxData[ \({7, 0, 18, 33, 21, 7, 27}\)], "Output"] }, Open ]], Cell["\<\ BE, in the same units is found by calculating arcBE and finding its \ chord.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(arcGE\ = \ ChordTableInverse[GE]; \narcBE\ = \ arcGE\ + \ arcBG; \n BE\ = \ ChordTable[arcBE]; \nPtolemyForm[BE]\)], "Input"], Cell[BoxData[ \({117, 37, 7, 12, 0, 0, 0}\)], "Output"] }, Open ]], Cell[TextData[{ "DK, in the same units, is found through the Pythagorean theorem (", StyleBox["Elements", FontSlant->"Italic"], " I 47) and the length of the radius of the epicycle can now be expressed \ where the radius of the deferent is 60." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DK\ = \ Sqrt[3600\ + \ \((\((BE + DE)\)*DE)\)]; \n epicycleradius\ = \ 60^2/DK; \nPtolemyForm[epicycleradius]\)], "Input"], Cell[BoxData[ \({5, 13, 3, 33, 8, 50, 11}\)], "Output"] }, Open ]], Cell["\<\ To follow through with Ptolemy's considerations we find the angle \ between the second eclipse, B, and the moon's apogee, L, by dropping a \ perpendicular to from the center of the moon's epicycle to the line which \ joins the observer to the second eclipse. We use this length to work \ backwards to arcLB. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DN\ = \ \((BE/2)\) + DE; \nDNonDK\ = \ DN*120/DK; \n arcDN\ = \ ChordTableInverse[DNonDK]; \n arcLB\ = \ 180\ - \ \((arcDN/2)\) - \ \((arcBE/2)\); \n PtolemyForm[arcLB]\)], "Input"], Cell[BoxData[ \({12, 25, 29, 59, 59, 59, 59}\)], "Output"] }, Open ]], Cell["\<\ The careful reader will have noticed that the preceeding numbers \ have been, all along, in good agreement with Ptolemy, but it is still \ gratifying to note that our final figures are remarkably close to Ptolemy's. \ For the ratio, Ptolemy gives 5;13 where we have 5;13,3; and, for the angle to \ the apogee, Ptolemy has 12;24 where we find 12;25,29. All this goes to show \ what no one ever actually doubted; whatever the real means of calculation \ employed, or commanded, by Ptolemy, they were sufficiently precise. \ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The Eccentric", "Subsubsection"], Cell[TextData[{ "Although Ptolemy does not provide the calculations necessary to see the \ numeric agreement between the two hypotheses in ", StyleBox["Almagest ", FontSlant->"Italic"], "IV 6, he is aware of this possibility and provides two figures. Now, Theon \ when he chanced upon this gap, in the course of his ", StyleBox["Commentary on the Almagest,", FontSlant->"Italic"], " felt compelled to fill it. Theon's work has not yet been graced with a \ modern translation, however, Neugebauer discusses these calculations in the \ course of a dubious argument concerning the possibility that the equivalence \ of the hypotheses for the moon goes back to Apollonius of Perga, Neugebauer \ [1975, 265 - 267]. The reasoning that follows corresponds very closely to \ what Theon actually puts forward, as found in the commentary, Rome [1931, \ 1053 - 1060]. \n\nI will be very brief in laying out Theon's calculations, \ since the geometric argument follows almost exactly the same reasoning as \ that found in Ptolemy. The points of departure, however minor, will be noted \ by blue input.\n\nThe observations dictate that the initial givens will be \ the same, hence the time intervals, the changes in longitude and anomaly, and \ the arcs on the circle of the moon's orbit will be identical. The first \ difference is in the angles at which the observer would have seen the \ eclipses had they all occurred on a single position of the eccentric circle. \ Now the observer is inside the circle of the moon's orbit and these apparent \ angles must be much greater. (See fig 4.5 in Toomer [1984, 194].) These \ angles are found by considering that, if the eclipses had been observed on a \ single position of the eccentric circle, then they would have appeared to be \ distant from each other by their actual zodiacal distances less the change in \ the position of the center of the moon's eccentric circle. The change in the \ position of the moon's eccentric circle is really just the mean longitude \ corrected by the anomaly, or the difference between these two." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(deltacentertheon1 = deltameanlongitude1 - deltaanomaly1; \n deltacentertheon2 = deltameanlongitude2 - deltaanomaly2; \n deltatheon1 = 349.25 - deltacentertheon1; \nPtolemyForm[deltatheon1]\)], "Input"], Cell[BoxData[ \({309, 48, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(deltatheon2 = 169.5 - deltacentertheon2; \nPtolemyForm[deltatheon2]\)], "Input"], Cell[BoxData[ \({149, 48, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ The apparent angles on a single position of the eccentric are found \ by allowing the first interval to specify an arbitrary starting point for \ angular measurements and setting the other intervals relative to this.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(angleBDAtheon = 360 - deltatheon1; \nPtolemyForm[angleBDAtheon]\)], "Input"], Cell[BoxData[ \({50, 11, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(angleADGtheon = deltatheon2 - angleBDAtheon; \n PtolemyForm[angleADGtheon]\)], "Input"], Cell[BoxData[ \({99, 37, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(angleBDGtheon = angleBDAtheon + angleADGtheon; \n PtolemyForm[angleBDGtheon]\)], "Input"], Cell[BoxData[ \({149, 48, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ The geometry follows Ptolemy's, and these figures are all the same \ as those given for the epicycle.\ \>", "Text"], Cell[BoxData[ RowBox[{ \(angleBEAtheon = arcBA/2\), ";", "\n", \(angleBEGtheon = arcBG/2\), ";", "\n", \(angleAEGtheon = arcAG/2\), ";", "\n", "\n", \(angleEAZtheon = angleBDAtheon - angleBEAtheon\), ";", "\n", \(angleEGDtheon = angleBDGtheon - angleBEGtheon\), ";", "\n", StyleBox[\(angleTGEtheon = 90 - angleAEGtheon\), FontColor->GrayLevel[0]], StyleBox[";", FontColor->GrayLevel[0]], "\n", "\n", StyleBox[\(AGtheon\ = \ ChordTable[arcAG]\), FontColor->GrayLevel[0]], StyleBox[";", FontColor->GrayLevel[0]]}]], "Input"], Cell["\<\ In right triangles DEZ and AEZ the geometry is the same although \ the lengths are different. \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EZonDEtheon\ = \ RightTriangleSide[angleBDAtheon]; \n PtolemyForm[EZonDEtheon]\)], "Input"], Cell[BoxData[ \({92, 10, 18, 14, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EZonAEtheon\ = \ RightTriangleSide[angleEAZtheon]; \n AEonDEtheon\ = \ 120*EZonDEtheon/EZonAEtheon; \n PtolemyForm[AEonDEtheon]\)], "Input"], Cell[BoxData[ \({232, 9, 46, 49, 5, 25, 29}\)], "Output"] }, Open ]], Cell["\<\ The displacement of D, the observer, dictates a minor change in the \ geometry for right triangle DEH.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(EHonDEtheon\ = \ RightTriangleSide[180 - angleBDGtheon]; \n PtolemyForm[EHonDEtheon]\)], "Input"], Cell[BoxData[ \({60, 19, 55, 42, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(EHonGEtheon\ = \ RightTriangleSide[angleEGDtheon]; \n GEonDEtheon\ = \ 120*EHonDEtheon/EHonGEtheon; \n PtolemyForm[GEonDEtheon]\)], "Input"], Cell[BoxData[ \({62, 34, 44, 33, 47, 9, 6}\)], "Output"] }, Open ]], Cell["\<\ Finding AG in the same units as DE requires no changes in the \ geometric reasoning.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(GTonGEtheon\ = \ RightTriangleSide[angleAEGtheon]; 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\nPtolemyForm[DEtheon]\)], "Input"], Cell[BoxData[ \({54, 52, 39, 27, 35, 15, 38}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(GEtheon\ = \ DEtheon*GEonDEtheon/120; \nPtolemyForm[GEtheon]\)], "Input"], Cell[BoxData[ \({28, 37, 5, 41, 11, 32, 49}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(\tarcGEtheon\ = \ ChordTableInverse[GEtheon]; \n\t arcBEtheon\ = \ arcGEtheon\ + \ arcBG; \n BEtheon\ = \ ChordTable[arcBEtheon]; \nPtolemyForm[BEtheon]\)\)], "Input"], Cell[BoxData[ \({119, 58, 56, 54, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ A line is dropped perpendicular from the center of the eccentric \ circle to the line which joins the observer and the second eclipse, and the \ Pythagorean theorem is used to find the length between the observer and the \ center of the eccentric circle where the radius of the eccentric circle is \ 60.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DNtheon\ = \ Abs[BEtheon/2 - DEtheon]; \n KNtheon\ = \ Sqrt[60^2 - \((BEtheon/2)\)^2]; \n epicycleradiustheon\ = \ Sqrt[DNtheon^2\ + \ KNtheon^2]; \n PtolemyForm[epicycleradiustheon]\)], "Input"], Cell[BoxData[ \({5, 12, 55, 30, 11, 57, 27}\)], "Output"] }, Open ]], Cell["\<\ The angle between the second eclipse and the apogee of the moon's \ eccentric circle is found following the same sort of reasoning we used above.\ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(DNonDKtheon\ = \ 120*DNtheon/DKtheon; \n angleDKNtheon\ = \ RightTriangleAngle[DNonDKtheon]; \n angleBKNtheon\ = \ ChordTableInverse[BEtheon]/2; \n arcLBtheon\ = \ 180\ - \((\ angleDKNtheon + angleBKNtheon)\); \n PtolemyForm[arcLBtheon]\)], "Input"], Cell[BoxData[ \({12, 19, 29, 59, 59, 59, 59}\)], "Output"] }, Open ]], Cell["\<\ By these means we find figures somewhat in agreement with Theon's. \ In all fairness, however, it should be remembered that Theon's intent was to \ show that the two models are numerically equivalent, so he may have allowed \ his numbers to slide somewhat in the direction of Ptolemy's. For this ratio \ he has 5;12,43 which he rounds to 5;13, while we find 5;12,55 which would \ even more happily round; for the arc he has 12;24 where we find 12;19,29. The \ hypotheses, then, are very close and they differ only in this:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(differenceinratio\ = \ Abs[epicycleradius\ - \ epicycleradiustheon];\)\), "\n", \(PtolemyForm[differenceinratio]\ \ \)}], "Input"], Cell[BoxData[ \({0, 0, 8, 2, 56, 52, 43}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\(differenceinarc\ = Abs[arcLB\ - arcLBtheon]; \n PtolemyForm[differenceinarc]\ \)\)], "Input"], Cell[BoxData[ \({0, 6, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Bibliography", "Subsection", CellTags->"Bibliography"], Cell["\<\ Neugebauer and Paderson are have complete discussions of Ptolemy's \ methods, however, Toomer's introduction is both concise and valuable. \ Toomer's translation is the best in any language.\ \>", "Text"], Cell[TextData[{ StyleBox["\[Bullet] O. Neugebauer, ", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox["A History of Ancient Mathematical Astronomy", FontSlant->"Italic", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox[ ", 3 vols. (Berlin etc.: Springer-Verlag, 1975).\n\n\[Bullet] O. Pedersen, \ ", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox["A Survey of the Almagest", FontSlant->"Italic", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox[" (Odense University Press, 1974).", FontColor->RGBColor[0.0901961, 0.00172427, 0]], "\n\n", StyleBox["\[Bullet] Ptolemy, ", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox["Ptolemy's Almagest", FontSlant->"Italic", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox[ ", trans. G. J. Toomer (London: Duckworth, 1984). (Reprinted by Princton, \ 1998, with some corrections)\n\t\n\[Bullet] Ptolemy, ", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox[ "Claudii Ptolemaei Opera quae exstant omnia.Vol. I, Syntaxis Mathematica", FontSlant->"Italic", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox[ ", ed. J. L Heiberg, 2 vols. (Liepzig: Teubner, 1898, 1903).\n\t\t\n\ \[Bullet] M. Abb\[EAcute] A. Rome, ed., ", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox[ "Commentaries de Pappus et Th\[EAcute]on d'Alexandrie sur l'Almagest", FontSlant->"Italic", FontColor->RGBColor[0.0901961, 0.00172427, 0]], StyleBox[" (Roma: Biblioteca Apostolica Vaticana, 1931).", FontColor->RGBColor[0.0901961, 0.00172427, 0]] }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Contacting the Author", "Subsection", CellTags->"Author"], Cell["\<\ If you have any questions about, or encounter any errors in, \ PtolemyChordTable.m please let me know. Nathan Sidoli nathan.sidoli@utoronto.ca\ \>", "Text"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for X", ScreenRectangle->{{0, 1024}, {0, 720}}, WindowToolbars->"EditBar", WindowSize->{799, 743}, WindowMargins->{{24, Automatic}, {Automatic, 0}}, CellLabelAutoDelete->True, StyleDefinitions -> "Report.nb" ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. 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