Musurgia Universalis: Unknown Combinatorial Studies in the Age of Baroque Absolutism

Bernoulli Jakob , Ars conjectandi, ed. by the brothers Thurneisen and Nikolaus i Bernoulli (Basel, 1713; reprint: Brussels, 1968); now in: Die Werke von Jakob Bernoulli, ed. by the Naturforschende Gesellschaft in Basel, iii (Basel, 1975), 107–286; German translation by Haussner Robert (Leipzig, 1899, Ostwald's Klassiker nos. 107 and 108).

Die Werke von Jakob Bernoulli, iii, 98 sq.

Bernoulli , op. cit. (ref. 1), 72f.; Haussner , op. cit. (ref. 1), no. 107, 76f.

Tacquet Andreas , Arithmeticae theoria et praxis (Louvain, 1656; 2nd ed., Antwerp, 1665), 381.

Prestet Jean , Elémens des mathématiques (Paris, 1675), 351; third ed., Nouveaux élémens des mathématiques (Paris, 1694), 134.

Mittelstrass Jürgen , Neuzeit und Aufklärung, Studien zur Entstehung der neuzeitlichen Wissenschaft und Philosophie (Berlin—New York, 1970), 425 ff.

Izquierdo Sebastian , Pharus scientiarum (Lyon, 1659), p. 4 of the “Praefatio ad Lectorem”.

Leibniz G. W. , Dissertatio de arte combinatoria (Leipzig, 1666; reprint: Frankfurt/M., 1690); now in: Leibniz G. W. , Sämtliche Schriften und Briefe, ed. by the Prussian Academy, now Academy of Sciences of the GDR (Darmstadt, Leipzig, Berlin, 1923-), vi, 1, 163–230, esp. p. 194; this fact is emphasized by Ceñal Ramón , La combinatoria de Sebastián Izquierdo (Madrid, 1974), 47.

For details see Risse Wilhelm , Die Logik der Neuzeit, vol. i, 1500–1640 (Stuttgart—Bad Canstatt, 1964), 559 ff.

de Montmort P. R. , Essay d'analyse sur les jeux de hazard (2nd ed., Paris, 1713), 387 and preface, p. xl.

Montucla J. E. , Histoire des mathématiques (2nd ed., 4 vols, Paris, 1799–1802; reprint: Paris, 1960), ii, 35.

Weingärtner Johann Christoph , Lehrbuch der combinatorischen Analysis nach der Theorie des Herrn Professor Hindenburg (2 vols, Leipzig, 1800–01), i, preface, pp. x–xi.

Knobloch Eberhard , Die mathematischen Studien von G. W. Leibniz zur Kombinatorik, Studia Leibnitiana Supplementa, xi (Wiesbaden, 1973), 6 ff.

Guldin Paul , De centro gravitatis (4 vols, Vienna, 1635–41), iv, 351–8.

ibid., 355–7.

Ernest Coumet is one of the very few scholars who dealt with the combinatorial studies of Mersenne; see for example “Mersenne: Dénombrements, répertoires, numérotations de permutations”, Mathématiques et sciences humaines, x (1972), 5–37; “Des permutations au XVIe et au XVIIe siècles”, Permutations. Actes du colloque sur les permutations, Paris, Université René-Descartes, 10–13 juillet 1972, Mathématiques et sciences de l'homme, xx (Paris—Brussels—Montreal, 1974), 277–89; Coumet's results are incorporated into the second edition of the third volume of Correspondance du P. Marin Mersenne, ed. by Tannery P. de Waard C. Pintard R. Rochot B. , iii (2nd ed., Paris, 1969), 159 and 257.

Mersenne Marin , Quaestiones celeberrimae in Genesim, cum accurata textus explicatione. In hoc volumine athei, et deistae impugnantur, et expugnantur, et vulgata editio ab haereticorum calumniis vindicatur. Graecorum, et Hebraeorum musica instauratur. Francisci Georgii Veneti cabalistica dogmata fuse refelluntur, quae passim in illius problematibus habentur. Opus theologis, philosophis, medicis, iurisconsultis, mathematicis, musicis vero et catoptricis praesertim utile (Paris, 1623).

Mersenne Marin , La vérité des sciences contre les Sceptiques ou Pyrrhoniens (Paris, 1625; reprint; Stuttgart—Bad Canstatt, 1969).

Mersenne Marin , Harmonie universelle (Paris, 1636; reprint: Paris, 1965); Harmonicorum libri, in quibus agitur de sonorum natura, causis, et effectibus: De consonantiis, dissonantiis, rationibus, generibus, modis, cantibus, compositione, orbisque totius harmonicis instrumentis (Paris, 1635, 1636).

Mersenne Marin , Traité de l'harmonie universelle (Paris, 1627 [Bayerische Staatsbibliothek Musica Theoretica 3166]); Questions harmoniques, dans lesquelles sont contenues plusieurs choses remarquables pour la physique, pour la morale, et pour les autres sciences (Paris, 1634; reprint: Stuttgart—Bad Canstatt, 1972): Questions inouyes ou recreation des scavans qui contiennent beaucoup de choses concernantes la theologie, la philosophie, et les mathématiques (Paris, 1634; reprint: Stuttgart—Bad Canstatt, 1972).

Mersenne Marin , Harmonicorum libri XII, editio aucta (Paris, 1648).

For details see Knobloch Eberhard , “Marin Mersennes Beiträge zur Kombinatorik”, Sudhoffs Archiv, lviii (1974), 356–79, esp. p. 377.

Mersenne Marin , Novarum observationum physico-mathematicarum tomus III (Paris, 1647), here Reflexiones physico-mathematicae, 168; but the number 154,600,249,126,726,147,905,433 printed there contains three typographical errors. The correct number is to be found in the Harmonicorum libri XII (dedication) and reads 1,546,007,491,267,262,147,905,433.

Mersenne , Harmonie universelle, Livre second des chants, 107.

Mersenne Marin , Cogitata physico-mathematica (Paris, 1644), 296; Klügel Simon , Mathematisches Wörterbuch oder Erklärung der Begriffe, Lehrsätze, Aufgaben und Methoden der Mathematik, fortgesetzt von Carl Brandan Mollweide, beendigt von Johann August Grunert (5 vols, Leipzig, 1803–31), i, 473; Diderot Denis and Le Rond d'Alembert Jean (eds), Encyclopédie ou dictionnaire raisonné des sciences des arts et des métiers (28 vols, Paris, 1751–72; reprint: Stuttgart—Bad Canstatt, 1966), iii (Ch-Co), sub voce ‘combinaison’ (d'Alembert), 664.

Mersenne , op. cit. (ref. 23), 203–15.

Mersenne writes: “Cum adeo multa de Combinationibus, libris Harmonicorum dixerimus, qui ad paucorum manus pervenerunt, Gallici maxime, ob exiguum exemplarium numerum, nonnullique errores in numerorum tabulis ibidem allatis contigerint, eas hic iterum absque ullis erratis accipe, cumque maxima Combinationum, seu varietatum, quibus res quotlibet sumi, vel considerari possunt, sit omnium facillima, quippe sequitur dignitates Algebraicas, seu progressionem, aut proportionem Geometricam, illam primo loco statuemus, ex qua deinde possint aliae tabulae deduci, atque separari.”

The original title is “Musurgia universalis sive ars magna consoni et dissoni in x libros digesta qua universa sonorum doctrina, et philosophia, musicaeque tam theoricae, quam practicae scientia, summa varietate traditur; admirandae consoni, et dissoni in mundo, adeoque universa natura vires effectusque, uti nova, ita peregrina variorum speciminum exhibitione ad singulares usus, tum in omni poene facultate, tum potissimum in Philologia, Mathematica, Physica, Mechanica, Medicina, Politica, Metaphysica, Theologia, aperiuntur et demonstrantur.” Birch Thomas , The history of the Royal Society of London for improving of natural knowledge (4 vols, London, 1756–57; reprint: Hildesheim, 1968), i, 76, remarks that this work was mentioned in the meeting of the Royal Society, held on 12 February 1661/62: “Mr Hill was desired to procure from his brother Kircher's secret way of music.” The pertinent footnote is somewhat inexact: “Probably that Jesuit's Musurgia universalis; sive ars magna consoni et dissoni, in decem libros digesta; qua universa doctrina et philosophia, musicaeque tam theoricae quam praticae scientia traditur, Rome 1650 two vol fol.” I have the reference for this from Mrs Brigitte Hoppe, Munich.

Weingärtner , op. cit. (ref. 12), x–xi.

“Question 36” reads as follows: “Peut-on apprendre à composer en Musique dans l'espace d'une heure, ou dans moins de temps?”.

Kircher Athanasius , Musurgia universalis (Rome, 1650), ii, bk 8 (vol. ii contains the last three of the ten books): “De Musurgia mirifica hoc est ars nova musarithmetica recenter inventa, qua quivis etiam quantumvis Musicae imperitus, ad perfectam componendi notitiam brevi tempore pertingere potest”, especially “Musurgiae mirificae pars prima Musurgia combinatoria”, 3–27. As to variations with repetition, Kircher calculates on p. 17 only the powers 22¹ to 22⁸, while Mersenne calculated up to 22²².

Kircher , op. cit. (ref. 31), 7: “Tabula ii combinatoria ostendens numerum mutationum rerum, in quibus non praecisa diversitas, sed quaedam sunt similes.” Kircher is doing so, of course, because he does not interpret the identical permutation as a permutation. There is an interesting hint on page 8 to his “Ars magna sciendi sive combinatoria”, which only appeared nineteen years later: “Verum hanc tabulam fusissime declaramus in Arte nostra Combinatoria, ubi eam ad quarumcunque artium et scientiarum principia applicamus, ad quam suo tempore Lector curiosus recurrere poterit.” Mersenne set up a number table in the “Liber de cantibus” (p. 135) and in the “Livre second des chants” (p. 134) which he used for calculations of combinations without repetition. He remarks explicitly that it is also useful for permutations with repetition. Though Kircher described this table already in the “Lemma 2”, he uses it once more much later in the “Problema vi” only for these permutations, not for combinations. He apparently did not understand what Mersenne really intended by this table. The first line on p. 20 is wrong, his explanation of the number 231 (p. 21) is vague and unintelligible (twenty equal and two other equal notes are arranged in different ways): “In serie vero secunda, ubi duae notae mutant filum fiunt mutationes 231 quia hae duae notae semper manentes in eadem chorda in singulis 22 notis decies mutari possunt in singulis 22 notarum seriebus et paulo amplius quae conficiunt 231 mutationes.”

Tacquet Andreas , op. cit. (ref. 4), 381; Schott Kaspar , Magia universalis naturae et artis opus quadripartitum (4 vols, Würzburg, 1657–59), iii, 686.

Ceñal Ramón , La combinatoria de Sebastián Izquierdo, “Pharus scientiarum” (1659), Disp. XXIX, De combinatione (Madrid, 1974), 112 ff.; see the review by Knobloch E. , Studia Leibnitiana, vii (1975), 156–60.

Kircher Athanasius , Ars magna sciendi sive combinatoria qua ad omnium artium scientiarumque cognitionem brevi adquirendam amplissima porta recluditur, quod uti inventum novum est, ita quoque eiusdem subsidio usuque instructus, quilibet de quavis re proposita infinitis pene rationibus disputare, omniumque summariam quandam cuiuslibet doctrinae notitiam obtinere poterit (Amsterdam, 1669), 155–62; the title of the inner side reads: Ars magna sciendi, in XII libros digesta, qua nova et universali methodo per artificiosum combinationum contextum de omni re proposita plurimis et prope infinitis rationibus disputari, omniumque summaria quaedam cognitio comparari potest.

Knittel Kaspar , Via regia ad omnes scientias et artes. Hoc est: Ars universalis scientiarum omnium artiumque arcana facilius penetrandi (Prague, 1682; 2nd ed. 1687; 3rd ed. 1691); as to Schott, see ref. 30.

Schott Kaspar , Mechanica hydraulico-pneumatica (Würzburg, 1657).

Schott Kaspar , Cursus mathematicus (Würzburg, 1661).

The word thaumatourgos = thaumatopoios (someone who performs wonders, juggler) is formed in the same way as Kircher's mousourgia. I have tried to translate this word by ‘musical art’. Schott's original title reads: Magia universalis naturae et artis, sive recondita naturalium et artificialium rerum scientia, cuius ope per variam applicationem activorum cum passivis, admirandorum effectuum spectacula, abditarumque inventionum miracula ad varios humanae vitae usus eruuntur. pt iii is entitled: “Pars iii in ix libros digesta, quibus pleraque, quae in Centrobaryca, Mechanica, Statyca, Hydrostatica, Hydrotechnica, Aerotechnica, Arithmetica et Geometria, sunt rara, curiosa, ac prodigiosa, hoc est, vere magica, seu theoriam spectes, seu praxin, non minus varie, quam methodice pertractantur, infinitarumque inventionum mathematicarum penuarium aperitur; ut merito appellari queat hoc opus Thaumaturgus mathematicus.”

Reimmann Jacob Friedrich , Versuch einer Einleitung in die Historiam literariam insgemein und derer Teutschen insonderheit (6 vols, Halle im Magdeburg, 1708–13), iv, 149.

Schott , op. cit. (ref. 33), iii, 674.

Knittel , op. cit. (ref. 37), 7 has the headline: “Pars septima Ars universalis sciendi ac disserendi Lulliano-Kircheriana.”

Izquierdo Sebastián , Pharus scientiarum ubi quidquid ad cognitionem humanam humanitus acquisibilem pertinet, ubertim iuxta, atque succincte pertractatur. Scientia de scientia, ob summam universalitatem utilissima, scientificisque iucundissima, scientifica methodo exhibetur. Aristotelis organum iam pene labens restituitur, illustratur, augetur, atque a defectibus absolvitur. Ars demum legitima, ac prorsus mirabilis sciendi, omnesque scientias in infinitum propagandi, et methodice digerendi; a nonnullis ex antiquioribus religiose celata; a multis studiose quaesita; a paucis inventa; a nemine ex propriis principiis hactenus demonstrata, demonstrative aperte, et absque involucris mysteriorum in lucem proditur. Quo verae Encyclopediae orbis facile a cunctis circumvolvendus, eximio scientiarum omnium emolumento, manet expositus (Lyon, 1659), p. 4 of the “Praefatio ad lectorem”.

For details see Knobloch , op. cit. (ref. 13), 17.

de Lobkowitz Johann Caramuel , Mathesis biceps vetus et nova (Campagna, 1670), the second of the unpaginated pages which enumerate all of Caramuel's works.

de Lobkowitz Johann Caramuel , Primus calamus (3 vols, Rome—St. Angelo dei Lombardi, 1663–65); vol. ii: Primus calamus tomus II. ob oculos exhibens rhythmicam, quae Hispanicos, Italicos, Galilcos, Germanicos, etc. versus metitur, eosdemque concentu exornans, viam aperit. ut orientales possint populi (Hebraei, Arabes, Turcici, Persici, Indici, Sinenses, Iaponici, etc.) conformare, aut etiam reformare proprios Numeros (St. Angelo dei Lombardi, 1665; second ed., Campagna, 1668); vol. iii: Primus calamus ob oculos ponens metametricam, quae variis currentium, recurrentium, adscendentium, descendentium, necnon circumvolantium versuum ductibus, aut aeri incisos, aut buxo insculptos, aut plumbo infusos, multiformes labyrinthos exornat (Rome, 1663).

Caramuel , Metametrica, 12, “Caput x. De multiform Anagrammatismi varietate, pro numero et natura litterarum. Harum divisio et observatio.”

Caramuel , Metametrica, 13–14.

Caramuel , Rhythmica, 153, “Hypothesis vi”; Caramuel reckons n!:40 instead of nl:36.

Caramuel , op. cit. (ref. 46), i, 181, 523; ii, 934, 960–1.

Knobloch , op. cit. (ref. 13), 18.