overview of mathematical tables in ptolemy's almagest · the mathematical tables in...

Ptolemy’sMathematical

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IntroductionPtolemy’s mathematicaltables

Tables in mathematicalargument

The “table nexus”Logical structure

Validity and applicability

Example, SolartheoryDerivation, Alm. III 5

Representation, Alm. III 6

Evaluation, Alm. III 8

Conclusion

References

Overview of Mathematical Tablesin Ptolemy’s Almagest

Nathan Sidoli

Oberwolfach, February 2011


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Conclusion

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IntroductionPtolemy’s mathematical tablesTables as an element in a mathematical argument

The “table nexus”The logical structure of the “table nexus”The validity and applicability of the approach

Example, Solar theoryDerivation, Alm. III 5Representation, Alm. III 6Evaluation, Alm. III 8

Conclusion

References


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Conclusion

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Ptolemy’s mathematical tables

The mathematical tables in Ptolemy’s Almagest are sets ofnumerical values that correspond to lengths and arcs inthe geometric models from which they are derived. (Inmost cases, this correspondence is one-to-one.)

At least in principle, the tables are produced by directderivation from geometric objects with assumed numericvalues, or from a given geometric model with specific,astronomically determined, parameters.

The tables themselves can be taken as a quantitiverepresentation of the underlying model.

The tables can be used as a tool in the evaluation specificvalues related to the underlying model.


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Tables and “functions”

Among modern concepts related to the idea of function,two are most relevant (Pedersen, 1974):

I A general relation between members of two (ormore) sets. [“Relational function.”] Usually a pair ofinverse functions.

I A computational rule that produces a single memberof the range for each member of the domain.[“Computational function.”]

I use this terminology only for our convenience. There areno ancient discussions of this sort.


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Types of Text in canonical works

Theorem Demonstration that some statement is true.(Enunciation, exposition, specification,construction, proof, conclusion.1)

Problem Demonstration that some construction canbe performed. (The same six.)

Analysis A sort of inverse problem which givesmotivation for the construction.(transformation, resolution, construction,proof.2)

Calculation Use of arithmetic and ratio theoreticoperations to produce numeric results.

Description General discussion of the diagram withsome simple inferences. (Rare but ex. Sph. &Cyl. I 39)

1These parts were identified by Proclus (5th CE).2The parts of analysis were identified by Henkel (1874).


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Types of text in Ptolemys works

Description General application of a mathematicalmodel.

Theorem Straightforward mathematical proof (whichmust be interpreted by means of the model.)

Problem Demonstration that a certain construction ispossible. (Rare, but ex. in Planisphere.)

Analysis argument by means of givens that a certaincalculation can be carried out. (Metricalresolution.)

Calculation Use of the model to produce numericalvalues.

Table List of numerical values derived from themodel and generally of use in furthercalculations.

Algorithm General description of how we use thevalues in the table.


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Types of Text related to the Tables

Calculation Use of the geometric model, numericalparameters and chord table methods toproduce corresponding sets of numbers.

Analysis Use of “metrical resolution” to show that ifcertain values are given then other valuesare also given (ie. determined on that basis).

Table An arrangement of numerical values torepresent the relationships found in thegeometric model. (“Relational functions.”)

Algorithm A general description of how we can use thevalues in the table to make certaincalculations, to solve certain problems or todetermine the apparent positions of theheavenly bodies. (“Computationalfunctions.”)


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The overall structure of the “table nexus”

The tables do not function independently, but work withthese other three types of text to form a “table nexus.”Derivation A calculation or analysis that shows that

given the geometric properties of the modeland certain numerical parameters, thenumbers in the table are determined.3

Representation A table, or series of tables, that gives anumerical representation of all of the keycomponents of the geometric model. (Eachmoving part of the model has a separateentry.)

Evaluation An algorithm that describes how the variousentries in the tables can be used to calculatephenomena that we actually see.

3The table is, in fact, not always derived by the method Ptolemygives (Newton 1985, Van Brummelen 1993).


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The validity and applicability of the approach

Ptolemy gives little discussion of his methods and thereare no general arguments that they are valid or effective.Any argument for the validity of these methods must bebased on the elements of the “table nexus.”

For example, (1) an understanding of how thecomponents of the geometric model move can be derivedfrom studying the table, (2) the proof that the tablerepresents a certain kind of function can be seen from thefact that the different columns refer to specificcomponents of the model, (3) an understanding of whateach of the components of the table mean must be basedon an appreciation of the figure, (4) an argument that thealgorithms using the the table actually produce apparentmotion must be based on an assessment of how thecomponents relate to the diagram.


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Overall structure: Alm. III 1–8

Alm. III 1 Astronomical argument that the tropical solaryear is constant. Calculation of its length.

Alm. III 2 Table of solar mean motion.Alm. III 3 General description of the models, proof of

their equivalence.Alm. III 4 Given some observations (season lengths),

derivation of the parameters of the solarmodel(s). (Due to Hipparchus.)

Alm. III 5 Derivation, using calculation and analysis oftwo of mean longitude (κ̄), apparentlongitude (κ) and equation of anomaly (α),given the other.

Alm. III 6 Table of solar anomaly.Alm. III 7 Astronomical establishment of epoch.Alm. III 8 Algorithm describing the use of the tables.


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Overview of the table nexus, Alm. III 5, 6 & 8

The three components of the table nexus are given inAlm. III 5, 6 & 8.

Alm. III 5 shows how to calculate α, given the parametersof the eccentric model and κ̄ = 30◦. Metrical resolution isused on the eccentric model to show that for any othervalues one of κ̄, κ or α, the other two are also given. Thewhole argument is then repeated for the epicycle model.

Alm. III 6 is the table, delineated into additive andsubtractive corrections.

Alm. III 8 gives an algorithm for using the parameters ofthe model along with the values found in the solar tablesof mean motion and anomaly (Alm. III 2 & 6) to find theapparent (or true) position of the sun.


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Overview of Alm. III 5

AB

C

D

H

E

G

Z

L!

"

"

Calculation

By Alm. III 4 we know that whereZC = 60p, DC = 2; 30p [parametersof the model], and we assume that∠ECZ = κ̄ = 30◦.

We then use chord table methodsto show that ∠DZK = α = 1; 9◦

and ∠ADB = κ = 28; 51◦.


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Overview of Alm. III 5

AB

CK

D

H

E

G

Z

!"

"

Metrical resolution

We argue that given any one of thethree angles (κ̄, κ or α) the othertwo are given.

For example: Suppose arc AB(∠CDL = κ) is given. Then CD : CLis given [chord table]. And sinceCD : CZ is given [parameter of themodel], CZ : CL is given [Data 8].Hence, ∠CZL = α is given [chordtable], and ∠ECZ = κ̄ is given[external angle].

Similar arguments are given forthe others.


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On the role of tables . . .

Although using a single combined table, as anephemerides, would be simpler, Ptolemy is quite clearthat this is not an appropriate goal for the mathematicalastronomer.

Almagest III 1: “We think that the task of themathematician ought to be to show all the heavenlyphenomena being reproduced by uniform circularmotions, and that the tabular form most appropriate andsuited to this task is the one which separates theindividual uniform motions from the anomalisticmotions which only seems to take place, and is due to thecircular models. The apparent paths of the bodies arethen demonstrated (ĆpodeiknÔousan) by the combinationof these two motions into one.”4

4Adopted from Toomer (1984, 140).


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On the table of anomaly . . .

Given that he has shown that any one of the three κ̄, κand α are enough to determine the other two, it would bepossible to set up the tables in a number of differentways. (We can take this argument as equivalent toshowing that the tables will represent pairs of inversefunctions.)

Almagest III 5: Ptolemy acknowledges that it would bepossible to arrange the table in various ways and thenstates that he will set out “the [table] having theequations of anomaly (α) laid out next to the uniform arcs(κ̄), because it is consistent with the models and bothsimple and apropos of the calculation of each of them.”5

5Heiberg 1898–1903, 251. Compare with Toomer (1984, 165).


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Almagest III 6: table of solar anomaly,6 Alm.III 6

6Vatican gr. 1594, miniscule, 9th CE (Heiberg’s B); Toomer (1984,167).


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Graph of solar anomaly, Alm. III 6


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Using the table: Alm. III 8

After finding the position of the sun at the beginning ofepoch, Alm. III 7, Ptolemy explains how to use the table tofind the sun at any given time.

Almagest III 8: “When we want to know the sun’s positionfor any required time, we take the time from epoch to thedesired moment (t), and enter with it into the table of themean motion (Alm. III 2, ∆λ̄�(t)). We add up the degreescorresponding to the various arguments, add this to theelongation from the apogee at epoch, 265; 15◦ (Alm. III 7,κ̄ at epoch), subtract complete revolutions (κ̄) and countthe result from ^ 5; 30◦ (apogee) rearwards through thesigns. The point we come to will be the mean position ofthe sun (λ̄�). . . ”7



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Using the table: Alm. III 8

Almagest III 8: “. . . Next we enter with the same number,that is the distance from apogee to the sun’s meanposition (κ̄), into the table of anomaly (Alm. III 6), andtake the corresponding amount in the third column(α(κ̄)). If the argument falls in the first column, that is if itis less than 180◦, we subtract from the mean position; butif the argument falls in the second column, that is if it isgreater than 180◦, we add it to the mean position. Thuswe obtain the true, or apparent position of the sun.”8



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The “table nexus”

As usual in the Almagest, we can take the solar theory as asort of model; however because of its simplicity, none ofthe other theories can be patterned on it exactly. In thesolar theory, each of the entries in the table can be relatedto a specific object in the model. Hence, one could arguethat the justification for the algorithm (Alm. III 8) can beread off the model itself.

In the more advanced theories, where we tabulatefunctions of more than one variable (Alm. V 8 & XI 11),individual entries in the tables will correspond tomultiple positions of the model, and interpolation will beused for the intervening positions.

Nevertheless, the justification for the final algorithm canstill be made by referring to the geometric features of themodel, although Ptolemy does not, himself, do this.Perhaps he thought the derivation was sufficient.


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Final Remarks

I The table nexus is used as a set of mathematicalmethods to model motion, or to model amathematical relationship that can be understood interms of motion.

I Irregular motion is understood in terms of functionsof regular motions. When these functions arecompounded, Ptolemy tabulates each of the series ofmotions separately, from more uniform to moreirregular.

I Each of the thee parts of the the table nexus dependson the others to provide a complete treatment ofmotion, or of a mathematical relationship that can beunderstood in terms of motion.

I Ptolemy occasionally introduces simplifyingassumptions to help in the calculation of functions ofmore than one variable, which must also beunderstood on the basis of the diagram.


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I Heiberg, J. L., 1898–1903.Claudii Ptolemaei, Syntaxis mathematica.Leipzig: Teubner.

I Newton, R. R., 1985.The Origins of Ptolemys Astronomical Tables.Baltimore: Johns Hopkins University.

I Pedersen O., 1974Logistics and the Theory of Functions: An Essay in theHistory of Greek MathematicsArchives International dHistoire des Science, 24, 29–50.

I Toomer, J. G., 1984.Ptolemy’s Almagest.London: Duckworth. (Reprinted: Princeton: PUP,1998.)


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I Van Brummelen G., 1993.Mathematical Tables in Ptolemy’s Almagest.Ph.D. Thesis, Simon Fraser University, Department ofMathematics and Statistics.

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