(An et Quaenam Siderum Aspectus vim habeant ac determinationem ex Configurationibus Harmonicis)

[I.] Much about this argument I taught (bk. 7, sect. 5, ch. 8), since the aspects are affections of the planets in longitude, of which that section treated; and I seemed there to report the doctrine on this matter in such a way as not to disapprove it, because the divisions of the circle by aspects seem to have a greater kinship with the harmonic consonances than the motions of the planets do. Moreover Kepler taught (Harmonics bk. 4, ch. 5, props. 1, 2, 3) that the kinship of the Radiations (or Aspects) with the circle, its arcs, and the figures inscribable in it is greater than that of the Consonances; that the congruence of [the figures] inscribable in the circle avails more toward constituting efficacious configurations than toward consonances; and that congruence avails more toward the same than does knowability [scibilitas]. (What congruence is, and what the knowability of figures is, is indicated here in the Scholia of ch. 4.)

[Margin: The efficacious configuration.]

Further (ch. 5), by Kepler an efficacious configuration is defined: [when] the radii of two planets make such an angle as is apt to stimulate sublunary nature and the lower faculties of living things, so that each is the more excited about its own work at the time of the configuration. Then he assumes an axiom: that the arc of the Zodiac-circle which the side of a congruent and knowable figure (or star-figure) measures out is the module of an efficacious configuration, and that the angle of a knowable and congruent figure is the module or measure of an efficacious configuration. These being posited — since he had demonstrated (bk. 1) that the Diameter of the circle, the Tetragon [square], the Trigon [triangle], the Hexagon and Octagon and octagonal star, the Dodecagon and dodecagonal star, the Pentagon, the Decagon, and the pentagonal and decagonal stars are knowable, and (bk. 2) congruent — he therefore constituted 13 efficacious configurations, of which:

• the most efficacious is the Conjunction, to which corresponds the whole circuit of the circle, 360 degrees;
• next, the Opposition, because [its lines] meet in one same line (which is the most perfect congruence), to which corresponds the semicircle, 180 degrees;
• next, the Quadrate [Square] aspect, to which corresponds the quadrant, 90 degrees;
• after it, the Trine (or Trigon), to which corresponds the third of the circle, 120 degrees;
• then the Sextile (or Hexagon), to which corresponds the sixth of the circle, 60 degrees;
• then the remaining Keplerian aspects, in that order of efficacy which we present at once in a table.

[The first paragraph ends introducing the table of Kepler’s 13 efficacious aspects, which follows on p. 534 (PDF 569), within Chapter XI.]

(printed p. 534 — Chapter XI continued, Kepler’s harmonic theory of the astrological Aspects. The table of 13 efficacious configurations is given; then Kepler’s earlier axiom that the aspects answer to the consonances below the octave, which he himself refutes from meteorological experience and corrects: the aspects answer to the consonance of the aspect-arc with the whole circle. Riccioli judges the old axiom not merely insufficient but false, the comparison of consonances with aspects being geometrically unsound.)

Efficacious Configurations

(Configurationes Efficaces, from Kepler’s Harmonics bk. 4, ch. 5) — the 13 aspects, the inscribed “knowable and congruent” figure whose side generates each, and the consonance Kepler once attached to it

(Each aspect’s arc is what the side of the named regular polygon (or star-polygon) subtends in the circle: the diameter [2-gon] gives 180°, the square 90°, the triangle 120°, the hexagon 60°, and so on; the star-polygons give the “skip” arcs (the octagonal star 135°, the pentagonal star 144°, etc.). The last column shows the consonance Kepler had earlier matched to seven of these figures.)

[Margin: An old axiom of Kepler, condemned by Kepler himself.]

[II.] The same Kepler (Harmonics bk. 4, ch. 6) treats of Astrology — of which he had written in his book On the New Star (ch. 8, 9, 10), and in his responses to the physicians Helisaeus Roslin and Philip Feselius, who attacked these new aspects; and finally [what] he had said of them in his Ephemerides (pp. 33–36). For in the year 1606 he had assumed as an axiom that God the Creator drew the law of ordering the Aspects from the Harmonies of song below the octave, or that He attuned the ears of men to the celestial Aspects, [making them] judges of those concordances — an axiom coined by no one but himself (as appears from p. 34 of the Ephemerides), which he now refutes: because [if it were true] there would have to be as many aspects as there are simple consonances up to the Diapason. For [then] the Quadrate aspect should answer to the Diatessaron; the Trine to the Diapente; the Opposition to the Diapason; the Quintile to the major Third (Ditone); the Sextile to the minor Third (Semiditone); the Biquintile to the major Sixth (greater Hexachord); the Sesquiquadrate to the minor Sixth (lesser Hexachord). For if you take from the whole string as great a portion as each aspect takes from the circle, the residue of the string makes, with the whole string, the consonance assigned to that aspect.

But, says Kepler, in his Meteorological observations it was found that sublunary nature is stirred even by the Semisextile aspect (which intercepts a twelfth of the circle), although, a twelfth of the string being removed, the residue of eleven parts does not consonate with the whole; and, on the contrary, that nature is not sensibly stirred by the Sesquiquadrate (which intercepts three-eighths, or 135°), although, three-eighths of the string being removed, the remaining five parts do consonate with the whole. Hence, stirred [by this], he corrects his axiom, and teaches that the proportion of the aspects answers not to the major concordances [by the residue] but to the consonance of the aspect-arc with the whole circle. For example, the Trine does not answer to the Diapente, but to the Diapason-Epidiapente [octave + fifth = a twelfth, 1:3]; for the Trine is between planets distant a third of the Zodiac (120°), not the residue (240°), and between 3 and 1 is the consonance Diapason-Epidiapente. The Quadrate answers not to the Diatessaron, but to the Disdiapason [double octave, 1:4], being between [planets] distant 90° (a fourth), and between 4 and 1 is the Disdiapason. The Quintile is between [planets] distant 72° (a fifth), and between 5 and 1 is the Diapason-with-Ditone [octave + major third, 1:5], which not all admit among the true consonances. So too the Sextile answers not to the minor Third but to the Disdiapason-Epidiapente [1:6]; the Biquintile not to the major Sixth (as we supposed above) but to a [consonance] composed of the major Third and the Diapason; the Sesquiquadrate not to the minor Sixth but to one composed of the Diatessaron and the Diapason — as is clear from the proportion of the part to the whole (the Sextile distant a sixth of the Zodiac; the Biquintile two-fifths = 144°; the Sesquiquadrate three-eighths = 135°): for between 6 and 1 is the Disdiapason-with-Diapente; between 5 and 2, the Diapason-with-Ditone; between 8 and 3, the Diapason-with-Diatessaron. Let the preceding table be emended, then, as in the following, to which we have added three consonances omitted by Kepler.

The Aspects and their consonance with the whole circle

(the corrected table — Aspect · intercepted Zodiac degrees · consonance of the arc with the whole circle of 360°)

(The consonance is the ratio of the aspect’s arc to the whole 360°, all verifiable — 180:360 = 1:2 (octave); 90:360 = 1:4 (double octave); 120:360 = 1:3 (twelfth); 60:360 = 1:6; 135:360 = 3:8 (eleventh); 150:360 = 5:12; 72:360 = 1:5; 108:360 = 3:10; 144:360 = 2:5. The three crosses (Octile 1:8, Semisextile 1:12, Decile 1:10) mark arcs giving no received consonance. The Conjunction (360°, the whole circle) heads the list without a proper ratio.)

[Margin: The falsity of the aforesaid axiom.]

[III.] These and other things — having become more cautious than himself — Kepler teaches in that ch. 6; among which the notable ones are these, which it pleases [me] to state in his words, with our little explanation added… [namely, that] a finite straight line, truncated or prolonged, remains a straight line; but a circle truncated does not remain a circle: whence it follows that a proportional section of two straight lines is possible, but not of two arcs of one circle. … That a true mathematical and causal comparison of the Concordances with the Aspects may stand, [the old axiom] must plainly be overthrown — since it is not only insufficient, but [also false] …

[The catchword “Con” points to p. 535 (PDF 570), continuing Riccioli’s discussion, within Chapter XI.]

(printed p. 535 — the last page of Book IX. Chapter XI concludes: Kepler distinguishes consonances from aspects as two nations sprung from the same fatherland, Geometry — his notions of a world-soul Riccioli flags as inadmissible without correction — and his final three-order ranking of the aspects is tabulated. Riccioli’s closing judgment: since several aspects answer to no consonance, this harmony was not God’s rule; celestial “music” is metaphor and analogy only. The book ends with FINIS.)

(conclusion, and the end of Book IX)

…[Con]sult now, in our chapter 4, the first Table of Consonances, and you will see that there are not a few [consonances] to which no aspect is here attributed, and in turn that there are some aspects here to which the crosses appended indicate that no Consonance answers.

[Margin: Kepler distinguishes the Aspects from the Consonances.]

[IV.] These and the like compelled Kepler to confess (in the same bk. 4, ch. 6) that the harmonic consonances and the Aspects have indeed something in common as to origin — from the divisions of the circle — but that Music and Meteorology [astrology] are born thence in a different manner; for diverse causes concur to constitute the Aspects, and Nature has a choice of those [aspects] that are furnished with more prerogatives. But let his words be noted, I pray: “What, then, is that which sets a limit to the number of the aspects? And why is no Semiquadrate, or Octile, no Decile or Tridecile, introduced except only after the principal ones? Why is the Sesquiquadrate, ennobled by musical kinship, either omitted or held ignoble — while the Semisextile, a stranger in Music, is not only inserted, but even displayed among the first?” He at once answers himself: “Because it is not Music that forms the Aspects, but Geometry [that forms] both kinds — yet the one by some laws, the other by others. For whatever is both Harmonic in Music and Efficacious in Meteors comes from a noble figure that has some singular privileges in Geometry. But Meteorology and Music are diverse, like two nations sprung from the same fatherland, Geometry.”

And he adds that the harmonic proportions, arisen from the circle, nevertheless led out colonies that departed from the circle; but that the Aspects remained within the fatherland of the circle, and use no other laws than the circle’s — taken from the plane figures inscribed in the circle, Regular and Congruent. The other things which (ch. 7) [he treats] — concerning a faculty of sublunary nature, and a certain Soul of the whole universe, and a Soul of the Earth that would perceive the force of the aspects and be stirred to act — are neither of this place, nor to be admitted by Catholics without correction. Yet here we must select, from bk. 6 of the Epitome of Copernican Astronomy, the aspects which Kepler finally accepted, and in that degree of dignity which he afterward recognized in them by the very observation of meteorological effects.

[Margin: the falsity of the aforesaid axiom.]

[V.] Therefore (Epitome bk. 6, p. 843) he teaches that in the first degree of aspects are placed: the Conjunction, as the principle of all; the Opposition, as occurring in all three divisions of the circle; the Quadrate, as occurring in two (the area of its figure being effable [expressible]); the Sextile, because its side is effable; the Semisextile, because its side is among the ineffable of a more perfect order, and because, twelve times repeated, it encompasses a stable plane; and the Trine, since its side is effable in power — for which, consult what was said in Scholium 1 of ch. 4. In the second order he says are the Quintile and Biquintile (because, though their sides are ineffable of a worse order, they share among themselves a divine proportion, and their figures excel in congruence into solid figures), and the Quincunx (because its figure is fruitful in the congruence of planes). But the Decile and Tridecile already fall short of congruence; and the most ignoble are the Octile and Sesquiquadrate, because they are formed from sides neither effable nor of divine proportion — and though these degrees are not altogether indubitable, let this stand as the final table of the Aspects, according to Kepler’s mind:

The Order of the Aspects in Force and Dignity

(Ordo Aspectuum in Vi et Dignitate — per Kepler; aspect · intercepted Zodiac degrees · consonance of the arc with the whole circle)

It deserves consideration, indeed, that the other consonances to which some aspects answer are either a principle or a species of the Diapason — which Kepler did not notice; and that, among the consonances pertaining to the Diapason, enumerated in the table of ch. 4, there is not one which has its own Aspect in the Zodiac, except the one composed of the Diapason and the minor Hexachord (as is between 16 and 5) — and that only approximately, since 360 is not divided into 16 whole parts (a 16th of it is 22½, and five such make 112½, a number near the 108 of the Tridecile, which pertains to the Diapason-with-major-Hexachord); no wonder, then, if their efficacies cannot be told apart.

Nor, however, do I think that this [agreement] with the Diapason was proposed to God as the mode of determining those aspects — since, even without it, and without any consonance whatever, three aspects are given: namely the Semisextile, the Decile, and the Octile. But just as, in the mixture of colors, tastes, odors, and temperaments — in innumerable plants and animals, in the climacterics, in the rhythms of the pulse, in the kinds of fevers, and in very many other things — there are certain degrees expressible by number (were their nature perfectly known); and yet, because those degrees can bear the proportions due to harmonic numbers, we are not therefore bound to be anxious about hunting out a Music in them beyond the looser bounds of Metaphor and Analogy — so neither in the Periods of the planets, nor in their Motions, nor in their Bulks, nor in their Intervals: as though God had wisely made nothing except what He had subjected to the laws of Harmony.

FINIS — The End of Book IX

(LIBRI NONI FINIS)

(So ends Liber IX of the Almagestum Novum — Riccioli’s “De Mundi Systemate.” This Section V, “On the Harmonic System of the World,” closes the book. The catchword “LIBER” at the foot of the page points to Liber X (Book X), which begins a new book beyond this Section.)

[Translator’s note on the final sentence: the Latin reads “Quasi nihil à Deo factum sit sapienter, nisi quod Harmoniae legibus non subiecerit” — literally awkward (with a “non”); the sense, fitting the whole argument, is the ironic reductio: it is absurd to suppose that whatever God did NOT make conform to musical harmony was therefore made unwisely. God’s wisdom shines in the world’s order and proportion — but that order need not be a literal, audible, or strictly musical harmony.]