Kepler's Approach to the Oval of 1602,from the Mars Notebook

This notebook comprises the bulk of vol. xiv of the Kepler manuscripts in the Archives of the Russian Academy of Science in St Petersburg. Most of the pages are numbered in Kepler's hand, with some unnumbered insertions. Cross references to these insertions show that they were made by Kepler , and that the notebook is (with some minor perturbations) in the order in which Kepler left it.

Page numbers in the Mars notebook are identified with one or two numbers: The folio number for vol. xiv as a whole (preceded by P), and Kepler's page number (preceded by K). All pages have a folio number, but not all were numbered by Kepler.

A transcription and an annotated translation of these pages is in Donahue W. H. , “Kepler's first thoughts on oval orbits: Text, translation, and commentary”, Journal for the history of astronomy, xxiv (1993), 71–100.

Wilson Curtis A. , “Kepler's derivation of the elliptical path”, Isis, lix (1968), 5–25; Aiton E. J. , “Kepler's second law of planetary motion”, Isis, lx (1969), 75–90; Stephenson Bruce , Kepler's physical astronomy (New York, 1987), 88–89; Donahue W. H. , “Kepler's invention of the second planetary law”, The British journal for the history of science, xxvii (1994), 89–102.

K435=P307^r, in Frisch Christian , Joannis Kepler astronomi opera omnia (8 vols, Frankfurt, 1858–70), viii, 229.

This term is used in the Mars notebook on K389=P280, and in a letter to Maestlin Michael dated 10/20 December 1601 (Letter number 203 in Johannes Kepler Gesammelte Werke, xiv, 203–8). The hypothesis was evidently first applied to the Earth's orbit.

On K451=P315^r. Although this page cannot be precisely dated, it must have been written between 10/20 December 1601 and 28 March/7 April 1602. For the basis of the dating, see Gingerich Owen , “Kepler's treatment of redundant observations”, in Internationales Kepler-Symposium Weil der Stadt 1971 , ed. by Krafft F. Meyer K. and Sticker B. (Hildesheim, 1973), 309–14, and Donahue , “Kepler's first thoughts on oval orbits” (ref. 3), 73.

K471=P325^r.

K474=P376^v.

These Mars-Sun distances had been found in the course of establishing the Earth's orbit. The method is presented in Chapters 26–28 of the New astronomy. The angle at the Earth is known from observation and the well-established solar theory. The angle at the Sun was obtained from solar theory and the “hypothesis vicaria” which had been developed in an earlier part of the Mars notebook and was presented in Chapter 16 of the New astronomy. Although this hypothesis gave incorrect Mars-Sun distances, it was reliable for the longitudes. The Earth-Sun distance had been worked out in the part of the Mars notebook preceding the “Capitalis demonstratio”: cf. Gingerich , “Kepler's treatment of redundant observations” (ref. 7), 313.

K475=P327^r: Radius Martis nimis parvus.

K477–8=P328.

K478=P328^v: Videtur haec methodus non sufficere subtilitati negocii. Nam distantiae possunt variationem admittere.

K479–81=P329^r–P330^r.

K481=P330^r: Vis movendi, quae tempus pro lege celeritatis habet physica est, qualis in sole est. Vis quae quantitatem aliquam pro lege habet, ⌞non tempus⌟ mere geometrica est, et in nutu consistit, qualis est in planetis. Vis solis assimilatur vi vitali in Corde animalis. Vis planetarum assimilatur vi animali in cerebro.

Sol circulum affectat circa se, Planeta accessu et recessu a sole circul….

The words “not time” are an insertion by Kepler.

The chief sources for the theory of intelligent movers for the planets are Aristotle , Metaphysics, Λ.8; Pseudo-Dionysius, The celestial hierarchy, Chapter 15, 337D–340A, in Pseudo-Dionysius: The complete works, translated by Luibheid Colm (New York, 1987), 190; and Aquinas Thomas , Summa theologica, Part I, Q. 110, Art. 2. For a thorough discussion of the full range of mediaeval views, see Grant Edward , Planets, stars, and orbs: The medieval cosmos, 1200–1687 (Cambridge, 1994), 514–68.

New astronomy, Chapter 2 has an extensive discussion of planetary movers.

K482=P330^v: Radius ♂ nimis magnus, antea nimis parvus.

“Hic esset justus. Considera causam. Differunt non in coaequatis, non in simplicibus, sed in eccentricis anomaliis hae tres distantae.” The equated anomaly (‘anomalia coaequata’) is the angle about the Sun; the simple anomaly (also called ‘the mean anomaly’) is the angular measure of the time, and the eccentric anomaly is the angle about the centre of the eccentric circle.

Here, as in most quotations from the Mars notebook, Kepler's insertions have been included, and his deletions omitted, without comment, and some of his pronouns have been replaced by their antecedents (in brackets) for the sake of clarity. The text here reads: Qui dicit, planeta describere perfectissimum in mundo circulum, is Planetae tribuit moderationem virtutis propriae cum virtute solari, et statuit, quod ex aequabili virtute solis (quoad proportionem distantiae) et ex inaequali Planetae conficiatur circulus: Sic, ut planeta ad sensum eius gradus de virtute solis in qua de praesenti movetur, suam propriam virtutem intendat vel remittat. At in planetae arbitrio (hoc est, in [col. 3] hujus ipsius propriae virtutis usurpatione[)] situm est, in quo quolibet tempore, virtute solis planeta moveatur. Ergo qui dicit, quod planeta virtutem suam variet ad suum ipsius arbitrium, is hoc dicit, quod planeta virtutem suam variet ad suum ipsius arbitrium, quod est absurdum, <…> Secundo: Qui dicit, quod planeta suam virtutem non variet, sed accessus et recessus suos a sole metiatur tempore, is rectius quidem sentit. Nam planetam liberat a consideratione alienae in sole virtutis, sed tribuit illi sensum temporis, quod necesse est eam facultatem, quae accessus et recessus conficit metirj. At qua re is metietur. Tempus est quippiam arithmeticum, circuli sunt geometricum quippiam. At planeta accessu et recessu suo intendit circulum. Tertio: Qui dicit planetam metiri accessus et recessus suos anomalia eccentrj, is rectissime sentit. Liberat enim eum a numeratio temporis, cujus mensuram non habet, facultatem non habet. Nam facultas eius est, continue moverj, at quod per numerationem temporis movet, non movet continue. Dicit ergo talis hoc: Planetam metirj accessus et recessus suos confecto in mundo spacio. Confectum autem seu conficiendum spacium tenet, geometrico intuitu fixarum et solis, quibus adminiculis et aphelium et nodos suos promovet. Hac ergo re mensura utitur.

P331^r, not numbered by Kepler : Ex inventis indubiis distantiis Eccentricitatem et Aphelium colligere, motusque medii correctionem.

Probatur hic tamen competere has distantias in circulum.

In genere facile apparet Apsidas esse circa 29 Leonis Aquarii ante 8 Virginis. Cumque in 15 Tauri sit brevior in 17 Scorpii longior Erunt igitur longitudines mediae ultra 15 Tauri, 17 Scorpii.

Quod vero attinet angulos, incertum, quos assumamus. Non illos ex locis Eccentricis, sunt enim parvi, et faciunt viam ovalem: Non illos ex temporibus et anomalia media. Sunt enim nimii. Intermedios assumendos scie; utrum praecise incertum.

Hoc tamen est proximum cogitationibus meis, ut qui sunt anguli anomaliae simplex, eas statuam ad centrum Eccentrici.

Funkenstein Amos , “The dialectical preparation for scientific revolutions”, in The Copernican achievement, ed. by Westman Robert S. (Berkeley, 1975), 165–203.

On K520=P351^v Kepler wrote in the margin, “This at last on Easter” (“Haec demum die pascali”). In 1602, Easter fell on the same day for both Lutherans and Catholics , March 28/April 7. The page containing this computation is K530=P356^v.

The “hypothesis vicaria” resulted from Kepler's attempt to find the best possible parameters for an eccentric circular orbit with an equant. The result was an unequally divided eccentricity: The equant had an eccentricity of 0.18564 while the eccentric's eccentricity was 0.11332. This hypothesis computed longitudes accurate to within two minutes of arc, but its planet-Sun distances were wrong. Kepler used it as a check on other hypotheses throughout his work on Mars.

The full text of this passage (K530=P356^v, K531=P357^r, P357^v (not numbered by Kepler ), and K532=P358^r), with an introduction, translation, and commentary, appears in Donahue , “Kepler's first thoughts on oval orbits” (ref. 3), 71–4.

K531=P357^r; cf. Donahue , “Kepler's first thoughts on oval orbits” (ref. 3), 78–4 and 84.

These paragraphs are still on K531=P357^r.

K532=P358. Again, insertions are included, and deletions omitted, without comment, and words in brackets have been substituted for pronouns for the sake of clarity. The Latin text and a detailed discussion of this passage is in Donahue , “Kepler's first thoughts on oval orbits” (ref. 3), 81 and 93–95.

P358^v, not numbered by Kepler.

New astronomy, Chapter 45, p. 215 of the 1609 edition.

Donahue , “Kepler's invention of the second planetary law” (ref. 4), 89–102.