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*Harmonies of the World*, by Johannes Kepler, tr. Charles Glenn Wallis [1939], at sacred-texts.com

Accordingly, since we see that the universal harmonies of all six planets cannot take place by chance, especially in the case of the extreme movements, all of which we see concur in the universal harmonies—except two, which concur in harmonies closest to the universal—and since much less can it happen by chance

that all the pitches of the system of the octave (as set up in Book III) by means of harmonic divisions are designated by the extreme planetary movements, but least of all that the very subtle business of the distinction of the celestial consonances into two modes, the major and minor, should be the outcome of chance, without the special attention of the Artisan: accordingly it follows that the Creator, the source of all wisdom, the everlasting approver of order, the eternal and superexistent geyser of geometry and harmony, it follows, I say, that He, the Artisan of the celestial movements Himself, should have conjoined to the five regular solids the harmonic ratios arising from the regular plane figures, and out of both classes should have formed one most perfect archetype of the heavens: in order that in this archetype, as through the five regular solids the shapes of the spheres shine through on which the six planets are carried, so too through the consonances, which are generated from the plane figures, and deduced from them in Book III, the measures of the eccentricities in the single planets might be determined so as to proportion the movements of the planetary bodies; and in order that there should be one tempering together of the ratios and the consonances, and that the greater ratios of the spheres should yield somewhat to the lesser ratios of the eccentricities necessary for procuring the consonances, and conversely those in especial of the harmonic ratios which had a greater kinship with each solid figure should be adjusted to the planets— in so far as that could be effected by means of consonances. And in order that, finally, in that way both the ratios of the spheres and the eccentricities of the single planets might be born of the archetype simultaneously, while from the amplitude of the spheres and the bulk of the bodies the periodic times of the single planets might result.

[301] While I struggle to bring forth this process into the light of human intellect by means of the elementary form customary with geometers, may the Author of the heavens be favourable, the Father of intellects, the Bestower of mortal senses, Himself immortal and superblessed, and may He prevent the darkness of our mind from bringing forth in this work anything unworthy of His Majesty, and may He effect that we, the imitators of God by the help of the Holy Ghost, should rival the perfection of His works in sanctity of life, for which He choose His church throughout the Earth and, by the blood of His Son, cleansed it from sins, and that we should keep at a distance all the discords of enmity, all contentions, rivalries, anger, quarrels, dissensions, sects, envy, provocations, and irritations arising through mocking speech and the other works of the flesh; and that along with myself, all who possess the spirit of Christ will not only desire but will also strive by deeds to express and make sure their calling, by spurning all crooked morals of all kinds which have been veiled and painted over with the cloak of zeal or of the love of truth or of singular erudition or modesty over against contentious teachers, or with any other showy garment. Holy Father, keep us safe in the concord of our love for one another, that we may be one, just as Thou art one with They Son, Our Lord, and with the Holy Ghost, and just as through the sweetest bonds of harmonies Thou hast made all Thy works one; and that from the bringing of Thy people into concord the body of Thy Church may be built up in the Earth, as Thou didst erect the heavens themselves out of harmonies.

PRIOR REASONS

I. AXIOM. *It is reasonable that, wherever in general it could have been done, all possible harmonies were due to have been set up between the extreme movements of the planets taken singly and by twos, in order that that variety should adorn the world*.

II. AXIOM *The five intervals between the six spheres to some extent were due to correspond to the ratio of the geometrical spheres which inscribe and circumscribe the five regular solids, and in the same order which is natural to the figures*.

Concerning this, see Chapter 1 and the *Mysterium Cosmographicum* and the *Epitome of Copernican Astronomy*.

III. PROPOSITION. *The intervals between the Earth and Mars, and between the Earth and Venus, were due to be least, in proportion to their spheres, and thereby approximately equal; middling and approximately equal between Saturn and Jupiter, and between Venus and Mercury; but greatest between Jupiter and Mars*.

For by Axiom II, the planets corresponding in position to the figures which make the least ratio of geometrical spheres ought likewise to make the least ratio; but those which correspond to the figures of middling ratio ought to make the greatest; and those which correspond to the figures of greatest ratio, the greatest. But the order holding between the figures of the dodecahedron and the icosahedron is the same as that between the pairs of planets, Mars and the Earth, and the Earth and Venus, and the order of the cube and octahedron is the same as that of the pair Saturn and Jupiter and that of the pair Venus and Mercury; and, finally, the order of the tetrahedron is the same as that of the pair Jupiter and Mars (see Chapter 3). Therefore, the least ratio will hold between the planetary spheres first mentioned, while that between Saturn and Jupiter is approximately equal to that between Venus and Mercury; and, finally, the greatest between the spheres of Jupiter and Mars.

IV. Axiom. *All the planets ought to have their eccentricities diverse, no less than a movement in latitude, and in proportion to those eccentricities also their distances from the sun, the source of movement, diverse*.

As the essence of movement consists not in *being* but in *becoming*, so too the form or figure of the region which any planet traverses in its movement does not become solid immediately from the start but in the succession of time acquires at last not only length but also breadth and depth (its perfect ternary of dimensions); and, gradually, thus, by the interweaving and piling up of many circuits, the form of a concave sphere comes to be represented—just as out of the silk-worm's thread, by the interweaving and heaping together of many circles, the cocoon is built.

V. PROPOSITION. *Two diverse consonances were to have been attributed to each pair of neighbouring planets*.

For, by Axiom IV, any planet has a longest and a shortest distance from the sun, wherefore, by Chapter 3, it will have both a slowest movement and a fastest. Therefore, there are two primary comparisons of the extreme movements, one of the diverging movements in the two planets, and the other of the converging. Now it is necessary that they be diverse from one another, because the ratio of the diverging movements will be greater, that of the converging, lesser. But, moreover, diverse consonances had to exist by way of diverse pairs of planets, so that this variety should make for the adornment of the world—by Axiom I—and also because the ratios of the intervals between two planets are

diverse, by Proposition III. But to each definite ratio of the spheres there correspond harmonic ratios, in quantitative kinship, as has been demonstrated in Chapter 5 of this book.

VI. PROPOSITION. *The two least consonances, 4 : 5 and 5 : 6, do not have a place between two planets*.

For

5 : 4=1,000 : 800

and

6 : 5 =1,000 : 833.

[paragraph continues] But the spheres circumscribed around the dodecahedron and icosahedron have a greater ratio to the inscribed spheres than 1,000 : 795, etc., and these two ratios indicate the intervals between the nearest planetary spheres, or the least distances. For in the other regular solids the spheres are farther distant from one another. But now the ratio of the movements is even greater than the ratios of the intervals, unless the ratio of the eccentricities to the spheres is vast —by Article XIII of Chapter 3. Therefore the least ratio of the movements is greater than 4 : 5 and 5 : 6. Accordingly, these consonances, being hindered by the regular solids, receive no place among the planets. VII. PROPOSITION. *The consonance of the perfect fourth can have no place between the converging movements of two planets, unless the ratios of the extreme movements proper to them are, if compounded, more than a perfect fifth*.

For let 3 : 4 be the ratio between the converging movements. And first, let there be no eccentricity, no ratio of movements proper to the single planets, but both the converging and the mean movements the same; then it follows that the corresponding intervals, which by this hypothesis will be the semi-diameters of the spheres, constitute the 2/3d power of this ratio, *viz*., 4480 : 5424 (by Chapter 3). But this ratio is already less than the ratio of the spheres of any regular figure; and so the whole inner sphere would be cut by the regular planes of the figure inscribed in any outer sphere. But this is contrary to Axiom II.

Secondly, let there be some composition of the ratios between the extreme movements, and let the ratio of the converging movements be 3 : 4 or 75 : 100, but let the ratio of the corresponding intervals be 1,000 : 795, since no regular figure has a lesser ratio of spheres. And because the inverse ratio of the movements exceeds this ratio of the intervals by the excess 750 : 795, then if this excess is divided into the ratio 1,000 : 795, according to the doctrine of Chapter 3, the result will be 9434 : 7950, the square root of the ratio of the spheres. Therefore the square of this ratio, *viz*., 8901 : 6320, *i.e.*, 10,000 : 7,100 is the ratio of the spheres. Divide this by 1000 : 795, the ratio of the converging intervals, the result will be 7100 : 7950, about a major whole tone. The' compound of the two ratios which the mean movements have to the converging movements on either side must be at least so great, in order that the perfect fourth may be possible between the converging movements. Accordingly, the compound ratio of the diverging extreme intervals to the converging extreme intervals is about the square root of this ratio, *i.e.*, two tones, and again the converging intervals are the square of this, *i.e.*, more than a perfect fifth. Accordingly, if the compound of the proper movements of two neighbouring planets is less than a perfect fifth, a perfect fourth will not be possible between their converging movements.

VIII. PROPOSITION. *The consonances 1 : 2 and 1 : 3, *i.e.*, the octave and the octave plus a fifth were due to Saturn and Jupiter*.

For they are the first and highest of the planets and have obtained the first figure, the cube, by Chapter 1 of this book; and these consonances are first in the order of nature and are chief in the two families of figures, the bisectorial or tetragonal and the triangular, by what has been said in Book I. But that which is chief, the octave 1 : 2, is approximately greater than the ratio of the spheres of the cube, [303] which is 1 : √3; wherefore it is fitted to become the lesser ratio of the movements of the planets on the cube, by Chapter 3, Article XIII; and, as a consequence, 1 : 3 serves as the greater ratio.

But this is also the same as what follows: for if some consonance is to some ratio of the spheres of the figures, as the ratio of the movements apparent from the sun is to the ratio of the mean intervals, such a consonance will duly be attributed to the movements. But it is natural that the ratio of the diverging movements should be much greater than the ratio of the 3/2th powers of the spheres, according to the end of Chapter 3, *i.e.*, it approaches the square of the ratio of the spheres; and moreover 1 : 3 is the square of the ratio of the spheres of the cube, which we call the ratio of 1: √3. Therefore, the ratio of the diverging movements of Saturn and Jupiter is 1 : 3. (See above, Chapter 2, for many other kinships of these ratios with the cube.)

IX. PROPOSITION. *The private ratios of the extreme movements of Saturn and Jupiter compounded were due to be approximately 2 : 3, a perfect fifth*.

This follows from the preceding; if the perihelial movement of Jupiter is triple the aphelial movement of Saturn, and conversely the aphelial movement of Jupiter is double the perihelial of Saturn, then 1 : 2 and 1 : 3 compounded inversely give 2 : 3.

X. Axiom. *When choice is free in other respects, the private ratio of movements, which is prior in nature or of a more excellent mode or even which is greater, is due to the higher planet*.

XI. PROPOSITION. *The ratio of the aphelial movement of Saturn to the perihelial was due to be 4 : 5, a major third, but that of Jupiter's movements 5 : 6, a minor third*.

For as compounded together they are equivalent to 2 : 3; but 2 : 3 can be divided harmonically no other way than into 4 : 5 and 5 : 6. Accordingly God the composer of harmonies divided harmonically the consonance 2 : 3, (by Axiom I) and the harmonic part of it which is greater and of the more excellent major mode, as masculine, He gave to Saturn the greater and higher planet, and the lesser ratio 5 : 6 to the lower one, Jupiter (by Axiom X).

XII. PROPOSITION. *The great consonance of 1 : 4, the double octave, was due to Venus and Mercury*.

For as the cube is the first of the primary figures, so the octahedron is the first of the secondary figures, by Chapter 1 of this book. And as the cube considered geometrically is outer and the octahedron is inner, *i.e.*, the latter can be inscribed in the former, so also in the world Saturn and Jupiter are the beginning of the upper and outer planets, or from the outside; and Mercury and Venus are the beginning of the inner planets, or from the inside, and the octahedron has been placed between their circuits: (see Chapter 3). Therefore, from among the consonances, one which is primary and cognate to the octahedron is due to Venus and Mercury. Furthermore, from among the consonances, after 1 : 2 and

[paragraph continues] 1 : 3, there follows in natural order 1 : 4; and that is cognate to 1 : 2, the consonance of the cube, because it has arisen from the same cut of figures, *viz*., the tetragonal, and is commensurable with it, *viz*., the double of it; while the octahedron is also akin to, and commensurable with the cube. Moreover, 1 : 4 is cognate to the octahedron for a special reason, on account of the number four being in that ratio, while a quadrangular figure lies concealed in the octahedron and the ratio of its spheres is said to be 1 : √2.

Accordingly the consonance 1 : 4 is a continued power of this ratio, in the ratio of the squares, *i.e.*, the 4th power of 1: √2 (see Chapter 2). Therefore, 1 : 4 was due to Venus and Mercury. And because in the cube 1 : 2 has been made the smaller consonance of the two, since the outermost position is over against it, in the octahedron there will be 1 : 4, the greater consonance of the two, as the innermost position is over against it. But too, this is the reason why 1 : 4 has here been given as the greater consonance, not as the smaller. 1 For since the ratio of the spheres of the octahedron is the ratio of 1: √3, then if it is postulated that the inscription of the octahedron among the planets is perfect (although it is not perfect, but penetrates Mercury's sphere to some extent-which is of advantage to us): accordingly, the ratio of the converging movements must be less than the 3/2th powers of 1 : √3 but indeed 1 : 3 is plainly the square of the ratio 1: √3 and is thus greater than the exact ratio; all the more then will 1 : 4 be greater than the exact ratio, as greater than 1 : 3. Therefore, not even the square root of 1 : 4 is allowed between the converging movements. Accordingly, 1 : 4 cannot be less than the octahedric; so it will be greater.

Further: 1 : 4 is akin to the octahedric square, where the ratio of the inscribed and circumscribed circles is 1: √2, just as 1 : 3 is akin to the cube, where the ratio of the spheres is 1 : √3 . For as 1 : 3 is a power of 1 : √3, *viz*., its square, [304] so too here 1 : 4 is a power of 1: √2, *viz*., twice its square, *i.e.*, its quadruple power. Wherefore, if 1 : 3 was due to have been the greater consonance of the cube (by Proposition VII), accordingly 1 : 4 ought to become the greater consonance of its octahedron.

XIII. PROPOSITION. *The greater consonance of approximately 1 : 8, the triple octave, and the smaller consonance of 5 : 24, the minor third and double octave, were due to the extreme movements of Jupiter and Mars*.

For the cube has obtained 1 : 2 and 1 : 3, while the ratio of the spheres of the tetrahedron, which is situated between Jupiter and Mars, called the triple ratio, is the square of the ratio of the spheres of the cube, which is called the ratio of 1: √3. Therefore, it was proper that ratios of movements which are the squares of the cubic ratios should be applied to the tetrahedron. But of the ratios 1 : 2 and 1 : 3 the following ratios are the squares: 1 : 4 and 1 : 9. But 1 : 9 is not harmonic, and 1 : 4 has already been used up in the octahedron. Accordingly, consonances neighbouring upon these ratios were to have been taken, by Axiom I. But the lesser ratio 1 : 8 and the greater 1 : 10 are the nearest. Choice between these ratios is determined by kinship with the tetrahedron, which has nothing in common with the pentagon, since 1 : 10 is of a pentagonal cut, but the tetrahedron has greater kinship with 1 : 8 for many reasons (see Chapter 2).

Further, the following also makes for 1 : 8: just as 1 : 3 is the greater consonance of the cube and 1 : 4 the greater consonance of the octahedron, because

they are powers of the ratios between the spheres of the figures, so too 1 : 8 was due to be the greater consonance of the tetrahedron, because as its body is double that of the octahedron inscribed in it, as has been said in Chapter 1, so too the term 8 in the tetrahedral ratio is double the term 4 in the tetrahedral ratio.

Further, just as 1 : 2 the smaller consonance of the cube, is one octave, and 1 : 4, the greater consonance of the octahedron, is two octaves, so already 1 : 8, the greater consonance of the tetrahedron, was due to be three octaves. Moreover, more octaves were due to the tetrahedron than to the cube and octahedron, because, since the smaller tetrahedral consonance is necessarily greater than all the lesser consonances in the other figures (for the ratio of the tetrahedral spheres is greater than all the spheres of figures): too the greater tetrahedral consonance was due to exceed the greater consonances of the others in number of octaves. Finally, the triple of octave intervals has kinship with the triangular form of the tetrahedron, and has a certain perfection, as follows: every three is perfect; since even the octuple, the term [of the triple octave], is the first cubic number of perfect quantity, namely of three dimensions.

A greater consonance neighbouring upon 1 : 4 or 6 : 24 is 5 : 24, while a lesser is 6 : 20 or 3 : 10. But again 3 : 10 is of the pentagonal cut, which has nothing in common with the tetrahedron. But on account of the numbers 3 and 4 (from which the numbers 12, 24 arise) 5 : 24 has kinship with the tetrahedron. For we are here neglecting the other lesser terms, *viz*., 5 and 3, because their lightest degree of kinship is with figures, as it is possible to see in Chapter 2. Moreover, the ratio of the spheres of the tetrahedron is triple; but the ratio of the converging intervals too ought to be approximately so great, by Axiom ii. By Chapter 3, the ratio of the converging movements approaches the inverse ratio of the 3/2th powers of the intervals, but the 3/2th power of 3 : 1 is approximately 1000 : 193. Accordingly, whereof the aphelial movement of Mars is 1000, the [perihelial] of Jupiter will be slightly greater than 193 but much less than 333, which is one third of 1,000. Accordingly, not the consonance 10 : 3, *i.e.*, 1,000 : 333, but the consonance 24 : 5, *i.e.*, 1,000 : 208, takes place between the converging movements of Jupiter and Mars.

XIV. PROPOSITION. *The private ratio of the extreme movements of Mars was due to be greater than 3 : 4, the perfect fourth, and approximately 18 : 25*.

For let there be the exact consonances 5 : 24 and 1 : 8 or 3 : 24, which are commonly attributed to Jupiter and Mars (Proposition XIII). Compound inversely 5 : 24, the lesser with 3 : 24, the greater; 3 : 5 results as the compound of both ratios. But the proper ratio of Jupiter alone has been found to be 5 : 6, in Proposition xi, above. Then compound this inversely with the composition 3 : 5, *i.e.*, compound 30 : 25 and 18 : 30; there results as the proper ratio of Mars 18 : 25, which is greater than 18 : 24 or 3 : 4. But it will become still greater, if, on account of the ensuing reasons, the common greater consonance 1 : 8 is increased.

XV. PROPOSITION. *The consonances 2 : 3, the fifth; 5 : 8, the minor sixth; and 3 : 5, the major sixth were to have been distributed among the converging movements of Mars and the Earth, the Earth and Venus, Venus and Mercury, and in that order*.

For the dodecahedron and the icosahedron, the figures interspaced between Mars, the Earth, and Venus have the least ratio between their circumscribed and inscribed spheres. [305] Therefore from among possible consonances the

least are due to them, as being cognate for this reason, and in order that Axiom u may have place. But the least consonances of all, *viz*., 5 : 6 and 4 : 5, are not possible, by Proposition IV. Therefore, the nearest consonances greater than they, *viz*., 3 : 4 or 2 : 3 or 5 : 8 or 3 : 5 are due to the said figures.

Again, the figure placed between Venus and Mercury, *viz*., the octahedron, has the same ratio of its spheres as the cube. But by Proposition vii, the cube received the octave as the lesser consonance existing between the converging movements. Therefore, by proportionality, so great a consonance, *viz*., 1 : 2, would be due to the octahedron as the lesser consonance, if no diversity intervened. But the following diversity intervenes: if compounded together, the private ratios of the single movements of the cubic planets, *viz*., Saturn and Jupiter, did not amount to more than 2 : 3; while, if compounded, the ratios of the single movements of the octahedral planets, *viz*., Venus and Mercury will amount to more than 2 : 3, as is apparent easily, as follows: For, as the proportion between the cube and octahedron would require if it were alone, let the lesser octahedral ratio be greater than the ratios here given, and thereby clearly as great as was the cubic ratio, *viz*., 1 : 2; but the greater consonance was 1 : 4, by Proposition XII. Therefore if the lesser consonance 1 : 2 is divided into the one we have just laid down, 1 : 2, still remains as the compound of the proper movements of Venus and Mercury; but 1 : 2 is greater than 2 : 3 the compound of the proper movements of Saturn and Jupiter; and indeed a greater eccentricity follows upon this greater compound, by Chapter 3, but a lesser ratio of the converging movements follows upon the greater eccentricity, by the same Chapter 3. Wherefore by the addition of a greater eccentricity to the proportion between the cube and the octahedron it comes about that a lesser ratio than 1 : 2 is also required between the converging movements of Venus and Mercury. Moreover, it was in keeping with Axiom I that, with the consonance of the octave given to the planets of the cube, another consonance which is very near (and by the earlier demonstration less than 1 : 2) should be joined to the planets of the octahedron. But 3 : 5 is proximately less than 1 : 2, and as the greatest of the three it was due to the figure having the greatest ratio of its spheres, *viz*., the octahedron. Accordingly, the lesser ratios, 5 : 8 and 2 : 3 or 3 : 4, were left for the icosahedron and dodecahedron, the figures having a lesser ratio of their spheres.

But these remaining ratios have been distributed between the two remaining planets, as follows. For as, from among the figures, though of equal ratios between their spheres, the cube has received the consonance 1 : 2, while the octahedron the lesser consonance 3 : 5, in that the compound ratio of the private movements of Venus and Mercury exceeded the compound ratio of the private movements of Saturn and Jupiter; so also although the dodecahedron has the same ratio of its spheres as the icosahedron, a lesser ratio was due to it than to the icosahedron, but very close on account of a similar reason, *viz*., because this figure is between the Earth and Mars, which had a great eccentricity in the foregoing. But Venus and Mercury, as we shall hear in the following, have the least eccentricities. But since the octahedron has 3 : 5, the icosahedron, whose species are in a lesser ratio, has the next slightly lesser, *viz*., 5 : 8; accordingly, either 2 : 3, which remains, or 3 : 4 was left for the dodecahedron, but more likely 2 : 3, as being nearer to the icosahedral 5 : 8; since they are similar figures.

But 3 : 4 indeed was not possible. For although, in the foregoing, the private

ratio of the extreme movements of Mars was great enough, yet the Earth—as has already been said and will be made clear in what follows—contributed its own ratio, which was too small for the compound ratio of both to exceed the perfect fifth. Accordingly, Proposition vii, 3 : 4 could not have place. And all the more so, because—as will follow in Proposition XVII—the ratio of the converging intervals was due to be greater than 1,000 : 795.

XVI. PROPOSITION. *The private ratios of movements of Venus and Mercury, if compounded together, were due to make approximately 5 : 12*.

For divide the lesser harmonic ratio attributed in Proposition xv to this pair jointly into the greater of them, 1 : 4 or 3 : 12, by Proposition XII; there results 5 : 12, the compound ratio of the private movements of both. And so the private ratio of the extreme movements of Mercury alone is less than 5 : 12, the magnitude of the private movement of Venus. Understand this of these first reasons. For below, by the second reasons, through the addition of some variation to the joint consonances of both, it results that only the private ratio of Mercury is perfectly 5 : 12.

XVII. PROPOSITION. *The consonance between the diverging movements of Venus and the Earth could not be less than 5 : 12*.

For in the private ratio of its movements Mars alone has received more than the perfect fourth and more than 18 : 25, by Proposition XIV. But their lesser consonance is the perfect fifth, [306] by Proposition XV. Accordingly, the ratio compounded of these two parts is 12 : 25. But its own private ratio is due to the Earth, by Axiom IV. Therefore, since the consonance of the diverging movements is made up out of the said three elements, it will be greater than 12 : 25. But the nearest consonance greater than 12 : 25, *i.e.*, 60 : 125, is 5 : 12, *viz*., 60 : 144. Wherefore, if there is need of a consonance for this greater ratio of the two planets, by Axiom I, it cannot be less than 60 : 144 or 5 : 12.

Therefore up to now all the remaining pairs of planets have received their two consonances by necessary reasons; the pair of the Earth and Venus alone has as yet been allotted only one consonance, 5 : 8, by the axioms so far employed. Therefore, we must now take a new start and inquire into its remaining consonance, *viz*., the greater, or the consonance of the diverging movements.

POSTERIOR REASONS

XVIII. Axiom. *The universal consonances of movements were to be constituted by a tempering of the six movements, especially in the case of the extreme movements*.

This is proved by Axiom I.

XIX. Axiom. *The universal consonances had to come out the same within a certain latitude of movements, namely, in order that they should occur the more frequently*.

For if they had been limited to indivisible points of the movements, it could have happened that they would never occur, or very rarely.

XX. Axiom. *As the most natural division of the kinds* [generum] *of consonances is into major and minor, as has been proved in Book 3, so the universal consonances of both kinds had to be procured between the extreme movements of the planet*s.

XXI. Axiom. *Diverse species of both kinds of consonances had to be instituted, so that the beauty of the world might well be composed out of all possible forms of*

*variety—and by means of the extreme movements, at least by means of some extreme movements*.

By Axiom I.

XXII. PROPOSITION. *The extreme movements of the planets had to designate pitches or strings* [chordas] *of the octave system, or notes* [claves] *of the musical scale*.

For the genesis and comparison of consonances beginning from one common term has generated the musical scale, or the division of the octave into its pitches or tones [*sonos*], as has been proved in Book 3. Accordingly, since varied consonances between the extremes of movements are required, by Axioms I, XX, and XXI, wherefore the real division of some celestial system or harmonic scale by the extremes of movements is required.

XXIII. PROPOSITION. *It was necessary for there to be one pair of planets, between the movements of which no consonances could exist except the major sixth 3 : 5 and the minor sixth 5 : 8*.

For since the division into kinds of consonances was necessary, by Axiom XX, and by means of the extreme movements at the apsides, by XXII, because solely the extremes, *viz*., the slowest and the fastest, need the determination of a manager and orderer, the intermediate tensions come of themselves, without any special care, with the passage of the planet from the slowest movement to the fastest: accordingly, this ordering could not take place otherwise than by having the diesis or 24 : 25 designated by the extremes of the two planetary movements, in that the kinds of consonances are distinguished by the diesis, as was unfolded in Book 3.

But the diesis is the difference either between two thirds, 4 : 5 and 5 : 6, or between two sixths, 3 : 5 and 5 : 8, or between those ratios increased by one or more octave intervals. But the two thirds, 4 : 5 and 5 : 6, did not have place between two planets, by Proposition vi, and neither the thirds nor the sixths increased by the interval of an octave have been found, except 5 : 12 in the pair of Mars and the Earth, and still not otherwise than along with the related 2 : 3, and so the intermediate ratios 5 : 8 and 3 : 5 and 1 : 2 were alike admitted. Therefore, it remains that the two sixths, 3 : 5 and 5 : 8, were to be given to one pair of planets. But too the sixths alone were to be granted to the variation of their movements, in such fashion that they would neither expand their terms to the proximately greater interval of one octave, 1 : 2, [307] nor contract them to the narrows of the proximately lesser interval of the fifth, 2 : 3. For, although it is true that the same two planets, which make a perfect fifth with their extreme converging movements, can also make sixths and thus traverse the diesis too, still this would not smell of the singular providence of the Orderer of movements. For the diesis, the least interval—which is potentially latent in all the major intervals comprehended by the extreme movements—is itself at that time traversed by the intermediate movements varied by continuous tension, but it is not determined by their extremes, since the part is always less than the whole, *viz*., the diesis than the greater interval 3 : 4 which exists between 2 : 3 and 1 : 2 and which whole would be here assumed to be determined by the extreme movements.

XXIV. PROPOSITION. *The two planets which shift the kind [genus] of harmony, which is the difference between the private ratios of the extreme movements, ought to make a diesis, and the private ratio of one ought to be greater than a diesis*,

*and they ought to make one of the sixths with their aphelial movements and the other with their perihelial*.

For, since the extremes of the movements make two consonances differing by a single diesis, that can take place in three ways. For either the movement of one planet will remain constant and the movement of the other will vary by a diesis, or both will vary by half a diesis and make 3 : 5, a major sixth, when the upper is at its aphelion and the lower in its perihelion, and when they move out of those intervals and advance towards one another, the upper into its perihelion and the lower into its aphelion, they make 5 : 8, a minor sixth; or, finally, one varies its movement from aphelion to perihelion more than the other does, and there is an excess of one diesis, and thus there is a major sixth between the two aphelia, and a minor sixth between the two perihelia. But the first way is not legitimate, for one of these planets would be without eccentricity, contrary to Axiom IV. The second way was less beautiful and less expedient; less beautiful, because less harmonic, for the private ratios of the movements of the two planets would have been out of tune [*inconcinnae*], for whatever is less than a diesis is out of tune; moreover it occasions one single planet to labour under this ill-concordant small difference—except that indeed it could not take place, because in this way the extreme movements would have wandered from the pitches of the system or the notes [*clavibus*] of the musical scale, contrary to Proposition xxii. Moreover, it would have been less expedient, because the sixths would have occurred only at those moments in which the planets would have been at the contrary apsides; there would have been no latitude within which these sixths and the universal consonances related to them could have occurred; accordingly, these universal consonances would have been very rare, with all the [*harmonic*] positions of the planets reduced to the narrow limits of definite and single points on their orbits, contrary to Axiom xix. Accordingly, the third way remains: that both of the planets should vary their own private movements, but one more than the other, by one full diesis at the least.

XXV. PROPOSITION. *The higher of the planets which shift the kind of harmony ought to have the ratio of its private movements less than a minor whole tone 9 : 10; while the lower, less than a semitone 15 : 16*.

For they will make 3 : 5 either with their aphelial movements or with their perihelial, by the foregoing proposition. Not with their perihelial, for then the ratio of their aphelial movements would be 5 : 8. Accordingly, the lower planet would have its private ratio one diesis more than the upper would, by the same foregoing proposition. But that is contrary to Axiom X. Accordingly, they make 3 : 5 with their aphelial movements, and with their perihelial 5 : 8, which is 24 : 25 less than the other. But if the aphelial movements make 3 : 5, a major sixth, therefore, the aphelial movement of the upper together with the perihelial of the lower will make more than a major sixth; for the lower planet will compound directly its full private ratio.

In the same way, if the perihelial movements make 5 : 8, a minor sixth, the perihelial movement of the upper and the aphelial movement of the lower will make less than a minor sixth; for the lower planet will compound inversely its full private ratio. But if the private ratio of the lower equalled the semitone 15 : 16, then too a perfect fifth could occur over and above the sixths, because the minor sixth, diminished by a semitone, because the perfect fifth; but this is

contrary to Proposition XXIII. Accordingly, the lower planet has less than a semitone in its own interval. And because the private ratio of the upper is one diesis greater than the private ratio of the lower, but the diesis compounded with the semitone makes 9 : 10 the minor whole tone.

XXVI. PROPOSITION. *On the planets which shift the kind of harmony, the upper was due to have either a diesis squared, 576 : 625, i.e., approximately 12 : 13, as* [308] *the interval made by its extreme movements, or the semitone 15 : 16, or something intermediate differing by the comma 80 : 81 either from the former or the latter; while the lower planet, either the simple diesis 24 : 25, or the difference between a semitone and a diesis, which is 125 : 128, i.e., approximately 42 : 43; or, finally and similarly, something intermediate differing either from the former or from the latter by the comma 80 : 81, *viz*., the upper planet ought to make the diesis squared diminished by a comma, and the lower, the simple diesis diminished by a comma*.

For, by Proposition XXV, the private ratio of the upper ought to be greater than a diesis, but by the preceding proposition less than the [minor] whole tone 9 : 10. But indeed the upper planet ought to exceed the lower by one diesis, by Proposition XXIV. And harmonic beauty persuades us that, even if the private ratios of these planets cannot be harmonic, on account of their smallness, they should at least be from among the concordant [*ex concinnis*] if that is possible, by Axiom I. But there are only two concords less than 9 : 10, the [minor] whole tone, *viz*., the semitone and the diesis; but they differ from one another not by the diesis but by some smaller interval, 125 : 128. Accordingly, the upper cannot have the semitone; nor the lower, the diesis; but either the upper will have the semitone 15 : 16, and the lower, 125 : 128, *i.e.*, 42 : 43; or else the lower will have the diesis 24 : 25, but the upper the diesis squared, approximately 12 : 13. But since the laws of both planets are equal, therefore, if the nature of the concordant had to be violated in their private ratios, it had to be violated equally in both, so that the difference between their private intervals could remain an exact diesis, which is necessary for distinguishing the kinds of consonances, by Proposition XXIV. But the nature of the concordant was then violated equally in both, if the interval whereby the private ratio of the upper planet fell short of the diesis squared and exceeded the semitone is the same interval whereby the private ratio of the lower planet fell short of a simple diesis and exceeded the interval 125 : 128.

Furthermore, this excess or defect was due to be the comma 80 : 81, because, once more, no other interval was designated by the harmonic ratios, and in order that the comma might be expressed among the celestial movements as it is expressed in harmonics, namely, by the mere excess and defect of the intervals in respect to one another. For in harmonics the comma distinguishes between major and minor whole tones and does not appear in any other way.

It remains for us to inquire which ones of the intervals set forth are preferable —whether the diesis, the simple diesis for the lower planet and the diesis squared for the upper, or the semitone for the upper and 125 : 128 for the lower. And the dieses win by the following arguments: For although the semitone has been variously expressed in the musical scale, yet its allied ratio 125 : 128 has not been expressed. On the other hand, the diesis has been expressed variously and the diesis squared somehow, *viz*., in the resolution of whole tones into dieses,

semitones, and lemmas; for then, as has been said in Book III, Chapter 8, two dieses proximately succeed one another in two pitches. The other argument is that in the distinction into kinds, the laws of the diesis are proper but not at all those of the semitone. Accordingly, there had to be greater consideration of the diesis than of the semitone. It is inferred from everything that the private ratio of the upper planet ought to be 2916 : 3125 or approximately 14 : 15, and that of the lower, 243 : 250 or approximately 35 : 36.

It is asked whether the Highest Creative Wisdom has been occupied in making these tenuous little reckonings. I answer that it is possible that many reasons are hidden from me, but if the nature of harmony has not allowed weightier reasons—since we are dealing with ratios which descend below the magnitude of all concords—it is not absurd that God has followed even those reasons, wherever they appear tenuous, since He has ordained nothing without cause. It would be far more absurd to assert that God has taken at random these magnitudes below the limits prescribed for them, the minor whole tone; and it is not sufficient to say: He took them of that magnitude because He chose to do so. For in geometrical things, which are subject to free choice, God chose nothing without a geometrical cause of some sort, as is apparent in the edges of leaves, in the scales of fishes, in the skins of beasts and their spots and the order of the spots, and similar things.

XXVII. PROPOSITION. *The ratio of movements of the Earth and Venus ought to have been greater than a major sixth between the aphelial movements; less than a minor sixth between the perihelial movements*.

By Axiom XX it was necessary to distinguish the kinds of consonances. But by Proposition XXIII that could not be done except through the sixths. Accordingly, since by Proposition XV the Earth and Venus, planets next to one another and icosahedral, had received the minor sixth, 5 : 8, it was necessary for the other sixth, 3 : 5, to be assigned to them, but not between the converging or diverging extremes, but between the extremes of the same field, one sixth [309] between the aphelial, and the other between the perihelial, by Proposition XXIV. Furthermore, the consonance 3 : 5 is cognate to the icosahedron, since both are of the pentagonal cut. See Chapter 2.

Behold the reason why exact consonances are found between the aphelial and perihelial movements of these two planets, but not between the converging, as in the case of the upper planets.

XXVIII. PROPOSITION. *The private ratio of movements fitting the Earth was approximately 14 : 15, Venus, approximately 35:36*.

For these two planets had to distinguish the kinds of consonances, by the preceding proposition; therefore, by Proposition XXVI, the Earth as the higher was due to receive the interval 2916 : 3125, *i.e.*, approximately 14 : 15, but Venus as the lower the interval 243 : 250, *i.e.*, approximately 35 : 36.

Behold the reason why these two planets have such small eccentricities and, in proportion to them, small intervals or private ratios of the extreme movements, although nevertheless the next higher planet, Mars, and the next lower, Mercury, have marked eccentricities and the greatest of all. And astronomy confirms the truth of this; for in Chapter 4 the Earth clearly had 14 : 15, but Venus 34 : 35, which astronomical certitude can barely discern from 35 : 36 in this planet.

XXIX. PROPOSITION. *The greater consonance of the movements of Mars and the*

[paragraph continues] *Earth, viz., that of the diverging movements, could not be from among the consonances greater than 5 : 12*.

Above, in Proposition XVII, it was not any one of the lesser ratios; but now it is not any one of the greater ratios either. For the other common or lesser consonance of these two planets is 2 : 3, when the private ratio of Mars, which by Proposition XIV exceeds 18 : 25, makes more than 12 : 25, *i.e.*, 60 : 125. Accordingly, compound the private ratio of the Earth 14 : 15, *i.e.*, 56 : 60, by the preceding proposition. The compound ratio is greater than 56 : 125, which is approximately 4 : 9, *viz*., slightly greater than an octave and a major whole tone. But the next greater consonance than the octave and whole tone is 5 : 12, the octave and minor third.

Note that I do not say that this ratio is neither greater nor smaller than 5 : 12; but I say that if it is necessary for it to be harmonic, no other consonance will belong to it.

XXX. PROPOSITION. *The private ratio of movements of Mercury was due to be greater than all the other private ratios*.

For by Proposition XVI the private movements of Venus and Mercury compounded together were due to make about 5 : 12. But the private ratio of Venus, taken separately, is only 243 : 250, *i.e.*, 1458 : 1500. But if it is compounded inversely with 5 : 12, *i.e.*, 625 : 1500, Mercury singly is left with 625 : 1458, which is greater than an octave and a major whole tone; although the private ratio of Mars, which is the greatest of all those among the remaining planets, is less than 2 : 3, *i.e.*, the perfect fifth.

And thereby the private ratios of Venus and Mercury, the lowest planets, if compounded together, are approximately equal to the compounded private ratios of the four higher planets, because, as will now be apparent immediately, the compounded private ratios of Saturn and Jupiter exceed 2 : 3; those of Mars fall somewhat short of 2 : 3: all compounded, 4 : 9, *i.e.*, 60 : 135. Compound the Earth's 14 : 15, *i.e.*, 56 : 60, the result will be 56 : 135, which is slightly greater than 5 : 12, which just now was the compound of the private ratios of Venus and Mercury. But this has not been sought for nor taken from any separate and singular archetype of beauty but comes of itself, by the necessity of the causes bound together by the consonances hitherto established.

XXXI. PROPOSITION. *The aphelial movement of the Earth had to harmonize with the aphelial movement of Saturn, through some certain number of octaves*.

For, by Proposition xviii, it was necessary for there to be universal consonances, wherefore also there had to be a consonance of Saturn with the Earth and Venus. But if one of the extreme movements of Saturn had harmonized with neither of the Earth's and Venus’, this would have been less harmonic than if both of its extreme movements had harmonized with these planets, by Axiom I. Therefore both of Saturn's extreme movements had to harmonize, the aphelial with one of these two planets, the perihelial with the other, since nothing would hinder, as was the case with the first planet. Accordingly these consonances will be either identisonant 1 [*identisonae*] or diversisonant [*diversisonae*], *i.e.*, either of continued double proportion or of some other. But both of them cannot be of some other proportion, for between the terms 3 : 5 (which determine the greater consonance between the aphelial movements of the Earth and Venus, by Proposition XXVII) two harmonic means cannot be set up; for the sixth cannot be

divided into three intervals (see Book III). Accordingly, Saturn could not, [310] by means of both its movements, make an octave with the harmonic means between 3 and 5; but in order that its movements should harmonize with the 3 of the earth and the 5 of Venus, it is necessary that one of those terms should harmonize identically, or through a certain number of octaves, with the others, *viz*., with one of the said planets. But since the identisonant consonances are more excellent, they had to be established between the more excellent extreme movements, *viz*., between the aphelial, because too they have the position of a principle on account of the altitude of the planets and because the Earth and Venus claim as their private ratio somehow and as a prerogative the consonance 3 : 5, with which as their greater consonance we are now dealing. For although, by Proposition XXII, this consonance belongs to the perihelial movement of Venus and some intermediate movement of the Earth, yet the start is made at the extreme movements and the intermediate movements come after the beginnings.

Now, since on one side we have the aphelial movement of Saturn at its greatest altitude, on the other side the aphelial movement of the Earth rather than Venus is to be joined with it, because of these two planets which distinguish the kinds of harmony, the Earth, again, has the greater altitude. There is also another nearer cause: the posterior reasons—with which we are now dealing—take away from the prior reasons but only with respect to minima, and in harmonics that is with respect to all intervals less than concords. But by the prior reasons the aphelial movement not of Venus but of the Earth, will approximate the consonance of some number of octaves to be established with the aphelial movement of Saturn. For compound together, first, 4 : 5 the private ratio of Saturn's movements, *i.e.*, from the aphelion to the perihelial of Saturn (Proposition XI), secondly, the 1 : 2 of the converging movements of Saturn and Jupiter, *i.e.*, from the perihelion of Saturn to the aphelion of Jupiter (by Proposition VIII), thirdly, the 1 : 8 of the diverging movements of Jupiter and Mars, *i.e.*, from the aphelion of Jupiter to the perihelion of Mars (by Proposition XIV), fourthly, the 2 : 3 of the converging movements of Mars and the Earth, *i.e.*, from the perihelion of Mars to the aphelion of the Earth (by Proposition XV): you will find between the aphelion of Saturn and the perihelion of the Earth the compound ratio 1 : 30, which falls short of 1 : 32, or five octaves, by only 30 : 32, *i.e.*, 15 : 16 or a semitone. And so, if a semitone, divided into particles smaller than the least concord, is compounded with these four elements there will be a perfect consonance of five octaves between the aphelial movements of Saturn and the Earth, which have been set forth. But in order for the same aphelial movement of Saturn to make some number of octaves with the aphelial movement of Venus, it would have been necessary to snatch approximately a whole perfect fourth from the prior reasons; for if you compound 3 : 5, which exists between the aphelial movements of the Earth and Venus, with the ratio 1 : 30 compounded of the four prior elements, then as it were from the prior reasons, 1 : 50 is found between the aphelial movements of Saturn and Venus: This interval differs from 1 : 32, or five octaves, by 32 : 50, *i.e.*, 16 : 25, which is a perfect fifth and a diesis; and from six octaves, or 1 : 64, it differs by 50 : 64, *i.e.*, 25 : 32, or a perfect fourth minus a diesis. Accordingly, an identisonant consonance was due to be established, not between the aphelial movements of Venus and Saturn but between those of Venus and the

[paragraph continues] Earth, so that Saturn might keep a diversisonant consonance with Venus. XXXII. PROPOSITION. *In the universal consonances of planets of the minor scale the exact aphelial movement of Saturn could not harmonize precisely with the other planets*.

For the Earth by its aphelial movement does not concur in the universal consonance of the minor scale, because the aphelial movements of the Earth and Venus make the interval 3 : 5, which is of the major scale (by Proposition XVII). But by its aphelial movement Saturn makes an identisonant consonance with the aphelial movement of the Earth (by Proposition XXXI). Therefore, neither does Saturn concur by its aphelial movement. Nevertheless, in place of the aphelial movement there follows some faster movement of Saturn, very near to the aphelial, and also in the minor scale—as was apparent in Chapter 7.

XXXIII. PROPOSITION. *The major kind of consonances and musical scale is akin to the aphelial movements; the minor to the perihelial*.

For although a major consonance [*dura harmonia*] is set up not only between the aphelial movement of the Earth and the aphelial movement of Venus but also between the lower aphelial movements and the lower movements of Venus as far as its perihelion; and, conversely, there is a minor consonance not merely between the perihelial movement of Venus and the perihelial of the Earth but also between the higher movements of Venus as far as the aphelion and the higher movements of the Earth (by Propositions XX and XXIV). Accordingly, the major scale is designated properly only in the aphelial movements, the minor, only in the perihelial.

XXXIV. PROPOSITION. *The major scale is more akin to the upper of the two planets, the minor, to the lower*.

[311] For, because the major scale is proper to the aphelial movements, the minor, to the perihelial (by the preceding proposition), while the aphelial are slower and graver than the perihelial; accordingly, the major scale is proper to the slower movements, the minor to the faster. But the upper of the two planets is more akin to the slow movements, the lower, to the fast, because slowness of the private movement always follows upon altitude in the world. Therefore, of two planets which adjust themselves to both modes, the upper is more akin to the major mode of the scale, the lower, to the minor. Further, the major scale employs the major intervals 4 : 5 and 3 : 5, and the minor, the minor ones, 5 : 6 and 5 : 8. But, moreover, the upper planet has both a greater sphere and slower, *i.e.*, greater movements and a lengthier circuit; but those things which agree greatly on both sides are rather closely united.

XXXV. PROPOSITION. *Saturn and the Earth embrace the major scale more closely Jupiter and Venus, the minor*.

For, first, the Earth, as compared with Venus and as designating both scales along with Venus, is the upper. Accordingly, by the preceding proposition, the Earth embraces the major scale chiefly; Venus, the minor. But with its aphelial movement Saturn harmonizes with the Earth's aphelial movement, through an octave (by Proposition XXXI): wherefore too (by Proposition XXXIII) Saturn embraces the major scale. Secondly, by the same proposition, Saturn by means of its aphelial movement nurtures more the major scale and (by Proposition XXXII) spits out the minor scale. Accordingly, it is more closely related to the major scale than to the minor, because the scales are properly designated by the extreme movements.

Now as regards Jupiter, in comparison with Saturn it is lower; therefore as the major scale is due to Saturn, so the minor is due to Jupiter, by the preceding proposition.

XXXVI. PROPOSITION. *The perihelial movement of Jupiter had to concord with the perihelial movement of Venus in one scale but not also in the same consonance; and all the less so, with the perihelial movement of the Earth*.

For, because the minor scale chiefly was due to Jupiter, by the preceding proposition, while the perihelial movements are more akin to the minor scale (by Proposition XXX), accordingly, by its perihelial movement Jupiter had to designate the key of the minor scale, *viz*., its definite pitch or key-note [*phthongum*]. But too the perihelial movements of Venus and the Earth designate the same scale (by Proposition XXVIII); therefore the perihelial movement of Jupiter was to be associated with their perihelial movements in the same tuning, but it could not constitute a consonance with the perihelial movements of Venus. For, because (by Proposition VIII) it had to make about 1 : 3 with the aphelial movement of Saturn, *i.e.*, the note [*clavem*] *d* of that system, wherein the aphelial movement of Saturn strikes the note *G*, but the aphelial movement of Venus the note *e*: accordingly, it approached the note *e* within an interval of least consonance. For the least consonance is 5 : 6, but the interval between *d* and *e* is much smaller, *viz*., 9 : 10, a whole tone. And although in the perihelial tension [*tensione*] Venus is raised from the *d* of the aphelial tension yet this elevation is less than a diesis, (by Proposition XXVIII). But the diesis (and hence any smaller interval) if compounded with a minor whole tone does not yet equal 5 : 6 the interval of least consonance. Accordingly, the perihelial movement of Jupiter could not observe 1 : 3 or thereabouts with the aphelial movement of Saturn and at the same time harmonize with Venus. Nor with the Earth. For if the perihelial movement of Jupiter had been adjusted to the key of the perihelial movement of Venus in the same tension in such fashion that below the quantity of least concord it should preserve with the aphelial movement of Saturn the interval 1 : 3, *viz*., by differing from the perihelial movement of Venus by a minor whole tone, 9 : 10 or 36 : 40 (besides some octaves) towards the low. Now the perihelial movement of the Earth differs from the same perihelial movement of Venus by 5 : 8, *i.e.*, by 25 : 40. And so the perihelial movements of the Earth and Jupiter differ by 25 : 36, over and above some number of octaves. But that is not harmonic, because it is the square of 5 : 6, or a perfect fifth diminished by one diesis.

XXXVII. PROPOSITION. *It was necessary for an interval equal to the interval of Venus to accede to the 2 : 3 of the compounded private consonances of Saturn and Jupiter and to 1 : 3 the great consonance common to them*.

For with its aphelial movement Venus assists in the proper designation of the major scale; with its perihelial, that of the minor scale, by Propositions XXVII and XXXIII. But by its aphelial movement Saturn had to be in concord also with the major scale and thus with the aphelial movement of Venus, by Proposition XXXV, but Jupiter's perihelial with the perihelial of Venus, by the preceding proposition. Accordingly, as great as Venus makes its interval from aphelial to perihelial to be, so great an interval must also accede to that movement of Jupiter which makes 1 : 3 with the aphelial movement of Saturn—to the very perihelial movement of Jupiter. But the consonance of the converging movements of Jupiter and Saturn is precisely 1 : 2, by Proposition VIII. Accordingly,

if the interval 1 : 2 is divided into the interval [312] greater than 1 : 3, there results, as the compound of the private ratios of both, something which is proportionately greater than 2 : 3.

Above, in Proposition XXVI, the private ratio of the movements of Venus was 243 : 250 or approximately 35 : 36; but in Chapter 4, between the aphelial movement of Saturn and the perihelial movement of Jupiter there was found a slightly greater excess beyond 1 : 3, *viz*., between 26 : 27 and 27 : 28. But the quantity here prescribed is absolutely equalled, by the addition of a single second to the aphelial movement of Saturn, and I do not know whether astronomy can discern that difference.

XXXVIII. PROPOSITION. *The increment 243 : 250 to 2 : 3, the compound of the private ratios of Saturn and Jupiter, which was up to now being established by the prior reasons, was to be distributed among the planets in such fashion that of it the comma 80 : 81 should accede to Saturn and the remainder, 19,683 : 20,000 or approximately 62 : 63, to Jupiter*.

It follows from Axiom XIX that this was to have been distributed between both planets so that each could with some latitude concur in the universal consonances of the scale akin to itself. But the interval 243 : 250 is smaller than all concords: accordingly no harmonic rules remain whereby it may be divided into two concordant parts, with the single exception of those of which there was need in the division of 24 : 25, the diesis, above in Proposition XXVI; namely, in order that it may be divided into the comma 80 : 81 (which is a primary one of those intervals which are subordinate to the concordant) and into the remainder 19,683 : 20,000, which is slightly greater than a comma, *viz*., approximately 62 : 63. But not two but one comma had to be taken away, lest the parts should become too unequal, since the private ratios of Saturn and Jupiter are approximately equal (according to Axiom X extended even to concords and parts smaller than those) and also because the comma is determined by the intervals of the major whole tone and minor whole tone, not so two commas. Furthermore, to Saturn the higher and mightier planet was due not that part which was greater, although Saturn had the greater private consonance 4 : 5, but that one which is prior and more beautiful, *i.e.*, more harmonic. For in Axiom X the consideration of priority and harmonic perfection comes first, and the consideration of quantity comes last, because there is no beauty in quantity of itself. Thus the movements of Saturn become 64 : 81, an adulterine 1 major third, as we have called them in Book III, Chapter 12, but those of Jupiter, 6,561 : 8,000.

I do not know whether it should be numbered among the causes of the addition of a comma to Saturn that the extreme intervals of Saturn can constitute the ratio 8 : 9, the major whole tone, or whether that resulted without further ado from the preceding causes of the movements. Accordingly, you here have, in place of a corollary, the reason why, above in Chapter 4, the intervals of Saturn were found to embrace approximately a major whole tone.

XXXIX. PROPOSITION. *Saturn could not harmonize with its exact perihelial movement in the universal consonances of the planets of the major scale, nor Jupiter with its exact aphelial movement*.

For since the aphelial movement of Saturn had to harmonize exactly with the aphelial movements of the Earth and Venus (by Proposition XXXI), that movement of Saturn which is 4 : 5 or one major third faster than its aphelial will also

harmonize with them. For the aphelial movements of the Earth and Venus make a major sixth, which, by the demonstrations of Book iii, is divisible into a perfect fourth and a major third, therefore the movement of Saturn, which is still faster than this movement already harmonized but none the less below the magnitude of a concordant interval, will not exactly harmonize. But such a movement is Saturn's perihelial movement itself, because it differs from its aphelial movement by more than the interval 4 : 5, *viz*., one comma or 80 : 81 more (which is less than the least concord), by Proposition XXXVIII. Accordingly the perihelial movement of Saturn does not exactly harmonize. But neither does the aphelial movement of Jupiter do so precisely. For while it does not harmonize precisely with the perihelial movement of Saturn, it harmonizes at a distance of a perfect octave (by Proposition VIII), wherefore, according to what has been said in Book III, it cannot precisely harmonize.

XL. PROPOSITION. *It was necessary to add the lemma of Plato to 1 : 8, or the triple octave, the joint consonance of the diverging movements of Jupiter and Mars established by the prior reasons*.

For because, by Proposition XXXI, there had to be 1 : 32, *i.e.*, 12 : 384, between the aphelial movements of Saturn and the Earth, but there had to be 3 : 2, *i.e.*, 384 : 256, from the aphelion of the Earth to the perihelion of Mars [313] (by Proposition XV), and from the aphelion of Saturn to its perihelion, 4 : 5 or 12 : 15 with its increment (by Proposition XXXVIII); finally, from the perihelion of Saturn to the aphelion of Jupiter 1 : 2 or 15 : 30 (by Proposition VIII); accordingly, there remains 30 : 256 from the aphelion of Jupiter to the perihelion of Mars, by the subtraction of the increment of Saturn. But 30 : 256 exceeds 32 : 256 by the interval 30 : 32, *i.e.*, 15 : 16 or 240 : 256, which is a semitone. Accordingly, if the increment of Saturn, which (by Proposition XXXVIII) had to be 80 : 81, *i.e.*, 240 : 243, is compounded inversely with 240 : 243, the result is 243 : 256; but that is the lemma of Plato, 1 *viz*., approximately 19 : 20, see Book III. Accordingly, Plato's lemma had to be compounded with the 1 : 8.

And so the great ratio of Jupiter and Mars, *viz*., of the diverging movements, ought to be 243 : 2,048, which is somehow a mean between 243 : 2,187 and 243 : 1,944, *i.e.*, between 1 : 9 and 1 : 8, whereof proportionality required the first, above; and a nearer harmonic concord, the second.

XLI. PROPOSITION. *The private ratio of the movements of Mars has necessarily been made the square of the harmonic ratio 5 : 6, viz., 25 : 36*.

For, because the ratio of the diverging movements of Jupiter and Mars had to be 243 : 2,048, *i.e.*, 729 : 6,144, by the preceding proposition, but that of the converging movements 5 : 24, *i.e.*, 1,280 : 6,144 (by Proposition XIII), therefore the compound of the private ratios of both was necessarily 729 : 1,280 or 72,900 : 128,000. But the private ratio of Jupiter alone had to be 6,561 : 8,000, *i.e.*, 104,976 : 128,000 (by Proposition XXVIII). Therefore, if the compound ratio of both is divided by this, the private ratio of Mars will be left as 72,900 : 104,976, *i.e.*, 25 : 36, the square root of which is 5 : 6.

In another fashion, as follows: There is 1 : 32 or 120 : 3,840 from the aphelial movement of Saturn to the aphelial movement of the Earth, but from that same movement to the perihelial of Jupiter there is 1 : 3 or 120 : 360, with its increment. But from this to the aphelial movement of Mars is 5 : 24 or 360 : 1,728. Accordingly, from the aphelial movement of Mars to the aphelial movement

of the Earth, there remains 1,728 : 3,840 minus the increment of the ratio of the diverging movements of Saturn and Jupiter. But from the same aphelial movement of the Earth to the perihelial of Mars there is 3 : 2, *i.e.*, 3,840 : 2,500. Therefore between the aphelial and perihelial movements of Mars there remains the ratio 1,728 : 2,560, *i.e.*, 27 : 40 or 81 : 120, minus the said increment. But 81 : 120 is a comma less than 80 : 120 or 2 : 3. Therefore, if a comma is taken away from 2 : 3, and the said increment (which by Proposition XXXVII is equal to the private ratio of Venus) is taken away too, the private ratio of Mars is left. But the private ratio of Venus is the diesis diminished by a comma, by Proposition XXVI. But the comma and the diesis diminished by a comma make a full diesis or 24 : 25. Therefore if you divide 2 : 3, *i.e.*, 24 : 36 by the diesis 24 : 25, Mars’ private ratio of 25 : 36 is left, as before, the square root of which, or 5 : 6, goes to the intervals, by Chapter 3.

Behold again the reason why—above, in Chapter 4—the extreme intervals of Mars have been found to embrace the harmonic ratio 5 : 6.

XLII. PROPOSITION. *The great ratio of Mars and the Earth, or the common ratio of the diverging movements, has been necessarily made to be 54 : 125, smaller than the consonance 5 : 12 established by the prior reasons*.

For the private ratio of Mars had to be a perfect fifth, from which a diesis has been taken away, by the preceding proposition. But the common or minor ratio of the converging movements of Mars and the Earth had to be a perfect fifth or 2 : 3, by Proposition XV. Finally, the private ratio of the Earth is the diesis squared, from which a comma is taken away, by Propositions XXVI and XXVIII. But out of these elements is compounded the major ratio or that of the diverging movements of Mars and the Earth—and it is two perfect fifths (or 4 : 9, *i.e.*, 108 : 243) plus a diesis diminished by a comma, *i.e.*, plus 243 : 250; namely, it is 108 : 250 or 54 : 125, *i.e.*, 608 : 1,500. But this is smaller than 625 : 1,500, *i.e.*, than 5 : 12, in the ratio 602 : 625, which is approximately 36 : 37, smaller than 625 : 1,500, *i.e.*, than 5 : 12, in the ratio 602 : 625, which is approximately 36 : 37, smaller than the least concord.

XLIII. PROPOSITION. The aphelial movement of Mars could not harmonize in some universal consonance; nevertheless it was necessary for it to be in concord to some extent in the scale of the minor mode.

For, because the perihelial movement of Jupiter has the pitch d of acute tuning in the minor mode, and the consonance 5 : 24 ought to have existed between that and the aphelial movement of Mars, therefore, the aphelial movement of Mars occupies the adulterine pitch of the same acute tuning. I say *adulterine* for, although in Book III, Chapter 12, the adulterine consonances were reviewed and deduced from the composition of systems, certain ones which exist in the simple natural system were omitted. [314] And so, after the line which ends 81 : 120, the reader may add: if you divide into it 4 : 5 or 32 : 40, there remains 27 : 32, the subminor sixth, 1 which exists between *d* and *f* or *c* and *e* 2 or *a* and *c* of even the simple octave. And in the ensuing table, the following should be in the first line; for 5 : 6 there is 27 : 32, which is deficient.

From that it is clear that in the natural system the true note [*clavem*] *f*, as regulated by my principles, constitutes a deficient or adulterine minor sixth with the note *d*. Accordingly since between the perihelial movement of Jupiter set

up in the true note *d* and the aphelial movement of Mars there is a perfect minor sixth over and above the double octave, but not the diminished (by Proposition XIII), it follows that with its aphelial movement Mars designates the pitch which is one comma higher than the true note *f*; and so it will concord not absolutely but merely to a certain extent in this scale. But it does not enter into either the pure or the adulterine universal harmony. For the perihelial movement of Venus occupies the pitch of *e* in this tuning [*tensionem*]. But there is dissonance between *e* and *f*, on account of their nearness. Therefore, Mars is in discord with the perihelial movement of one of the planets, *viz*., Venus. But too it is in discord with the other movements of Venus; they are diminished by a comma less than a diesis: wherefore, since there is a semitone and a comma between the perihelial movement of Venus and the aphelial movement of Mercury, accordingly, between the aphelion of Venus and the aphelion of Mars there will be a semitone and a diesis (neglecting the octaves), *i.e.*, a minor whole tone, which is still a dissonant interval. Now the aphelial movement of Mars concords to that extent in the scale of the minor mode, but not in that of the major. For since the aphelial movement of Venus concords with the *e* of the major mode, while the aphelial movement of Mars (neglecting the octaves) has been made a minor whole tone higher than *e*, then necessarily the aphelial movement of Mars in this tuning would fall midway between f and f sharp and would make with *g* (which in this tuning would be occupied by the aphelial movement of the Earth) the plainly discordant interval 25 : 27, *viz*., a major whole tone diminished by a diesis.

In the same way, it will be proved that the aphelial movement of Mars is also in discord with the movements of the Earth. For because it makes a semitone and comma with the perihelial movement of Venus, *i.e.*, 14 : 15 (by what has been said), but the perihelial movements of the Earth and Venus make a minor sixth 5 : 8 or 15 : 24 (by Proposition XXVII). Accordingly, the aphelial movement of Mars together with the perihelial movement of the Earth (the octaves added to it) will make 14 : 24 or 7 : 12, a discordant interval and one not harmonic, like 7 : 6. For any interval between 5 : 6 and 8 : 9 is dissonant and discordant, as 6 : 7 in this case. But no other movement of the Earth can harmonize with the aphelial movement of Mars. For it was said above that it makes the discordant interval 25 : 27 with the Earth (neglecting the octaves); but all from 6 : 7 or 24 : 28 to 25 : 27 are smaller than the least harmonic interval.

XLIV. COROLLARY. A*ccordingly it is clear from the above Proposition XLIII concerning Jupiter and Mars, and from Proposition XXXIX concerning Saturn and Jupiter, and from Proposition XXXVI concerning Jupiter and the Earth, and from Proposition XXXII concerning Saturn, why—in Chapter 5, above—it was found that all the extreme movements of the planets had not been adjusted perfectly to one natural system or musical scale, and that all those which had been adjusted to a system of the same tuning did not distinguish the pitches* [loca] *of that system in a natural way or effect a purely natural succession of concordant intervals. For the reasons are prior whereby the single planets came into possession of their single consonances; those whereby all the planets, of the universal consonances; and finally, those whereby the universal consonances of the two modes, the major and the minor: when all those have been posited, an omniform adjustment to one natural system is prevented. But if those causes had not necessarily come first, there is no doubt that either one system and one tuning of it would have embraced the extreme movements*

*of all the planets; or, if there was need of two systems for the two modes of song, the major and minor, the very order of the natural scale would have been expressed not merely in one mode, the major, but also in the remaining minor mode. Accordingly, here in Chapter 5, you have the promised causes of the discords through least intervals and intervals smaller than all concords*.

XLV. PROPOSITION.* It was necessary for an interval equal to the interval of Venus to be added to the common major consonance of Venus and Mercury, the double octave, and also the private consonance of Mercury, which were established above in Propositions XII and XIII by the prior reasons*, [315] *in order that the private ratio of Mercury should be a perfect 5 : 12 and that thus Mercury should with both its movements harmonize with the single perihelial movement of Venus*.

For, because the aphelial movement of Saturn, the highest and outmost planet, circumscribed around its regular solid, had to harmonize with the aphelial movement of the Earth, the highest movement of the Earth, which divides the classes of figures; it follows by the laws of opposites that the perihelial movement of Mercury as the innermost planet, inscribed in its figure, the lowest and nearest to the sun, should harmonize with the perihelial movement of the Earth, with the lowest movement of the Earth, the common boundary: the former in order to designate the major mode of consonances, the latter the minor mode, by Propositions XXXIII and XXXIV. But the perihelial movement of Venus had to harmonize with the perihelial movement of the Earth in the consonance 5 : 3, by Proposition XXVII; therefore too the perihelial movement of Mercury had to be tempered with the perihelial of Venus in one scale. But by Proposition XII the consonance of the diverging movements of Venus and Mercury was determined by the prior reasons to be 1 : 4; therefore, now by these posterior reasons it was to be adjusted by the accession of the total interval of Venus. Accordingly, not from further on, from the aphelion, but from the perihelion of Venus to the perihelion of Mercury there is a perfect double octave. But the consonance 3 : 5 of the converging movements is perfect, by Proposition XV. Accordingly if 1 : 4 is divided by 3 : 5, there remains to Mercury singly the private ratio 5 : 12, perfect too, but not further (by Proposition xvi, through the prior reasons) diminished by the private ratio of Venus.

Another reason. Just as only Saturn and Jupiter are touched nowhere on the outside by the dodecahedron and icosahedron wedded together, so only Mercury is untouched on the inside by these same solids, since they touch Mars on the inside, the Earth on both sides, and Venus on the outside. Accordingly, just as something equal to the private ratio of Venus has been added distributively to the private ratios of movements of Saturn and Jupiter, which are supported by the cube and tetrahedron; so now something as great was due to accede to the private ratio of solitary Mercury, which is comprehended by the associated figures of the cube and tetrahedron; because, as the octahedron, a single figure among the secondary figures, does the job of two among the primary, the cube and tetrahedron (concerning which see Chapter 1), so too among the lower planets there is one Mercury in place of two of the upper planets, *viz*., Saturn and Jupiter.

Thirdly, just as the aphelial movement of the highest planet Saturn had to harmonize, in some number of octaves, *i.e.*, in the continued double ratio, 1 : 32, with the aphelial movement of the higher and nearer of the two planets which shift the mode of consonance (by Proposition XXXI); so, *vice versa*, the perihelial

movement of the lowest planet Mercury, again through some number of octaves, *i.e.*, in the continued double ratio, 1 : 4, had to harmonize with the perihelial movement of the lower and similarly nearer of the two planets which shift the mode of consonance.

Fourthly, of the three upper planets, Saturn, Jupiter, and Mars, the single but extreme movements concord with the universal consonances; accordingly both extreme movements of the single lower planet, *viz*., Mercury, had to concord with the same; for the middle planets, the Earth and Venus, had to shift the mode of consonances, by Propositions XXXIII and XXXIV.

Finally, in the three pairs of the upper planets perfect consonances have been found between the converging movements, but adjusted [*fermentatae*] consonances between the diverging movements and private ratios of the single planets; accordingly, in the two pairs of the lower planets, conversely, perfect consonances had to be found not between the converging movements chiefly, nor between the diverging, but between the movements of the same field. And because two perfect consonances were due to the Earth and Venus, therefore two perfect consonances were due to Venus and Mercury also. And the Earth and Venus had to receive as perfect a consonance between their aphelial movements as between their perihelial, because they had to shift the mode of their consonance; but Venus and Mercury, as not shifting the mode of their consonance, did not also require perfect consonances between both pairs, the aphelial movements and the perihelial; but there came in place of the perfect consonance of the aphelial movements, as being already adjusted the perfect consonance of the converging movements, so that just as Venus, the higher of the lower planets, has the least private ratio of all the private ratios of movements (by Proposition XXVI), and Mercury, the lower of the lower, has received the greatest ratio of all the private ratios of movements (by Proposition XXX), so too the private ratio of Venus should be the most imperfect of all the private ratios or the farthest removed from consonances, while the private ratio of Mercury should be most perfect of all the private ratios, *i.e.*, an absolute consonance without adjustment, and that finally the relations should be everywhere opposite.

*For He Who is before the ages and on into the ages thus adorned the great things of His wisdom: nothing excessive, nothing defective, no room for any censure. How lovely are his works! All things, in twos, one* [316] *against one, none lacking its opposite. He has strengthened the goods—adornment and propriety—of each and every one and established them in the best reasons, and who will be satiated seeing their glory?*

XLVI. Axiom. *If the interspacing of the solid figures between the planetary spheres is free and unhindered by the necessities of antecedent causes, then it ought to follow to perfection the proportionality of geometrical inscriptions and circumscriptions, and thereby the conditions of the ratio of the inscribed to the circumscribed spheres*.

For nothing is more reasonable than that physical inscription should exactly represent the geometrical, as the work, its pattern.

XLVII. PROPOSITION. *If the inscription of the regular solids among the planets was free, the tetrahedron was due to touch with its angles precisely the perihelial sphere of Jupiter above it, and with centres of its planes precisely the aphelial sphere of Mars below it. But the cube and the octahedron, each placing its angles in the perihelial sphere of the planet above, were due to penetrate the sphere of the inside planet*

*with the centres of their planes, in such fashion that those centres should turn within the aphelial and perihelial spheres: on the other hand, the dodecahedron and icosahedron, grazing with their angles the perihelial spheres of their planets on the outside, were due not quite to touch with the centres of their planes the aphelial spheres of their inner planets. Finally, the dodecahedral echinus, placing its angles in the perihelial sphere of Mars, was due to come very close to the aphelial sphere of Venus with the midpoints of its converted sides which interdistinguish two solid rays*.

For the tetrahedron is the middle one of the primary figures, both in genesis and in situation in the world; accordingly, it was due to remove equally both regions, that of Jupiter and that of Mars. And because the cube was above it and outside it, and the dodecahedron was below it and within it, therefore it was natural that their inscription should strive for the contrariety wherein the tetrahedron held a mean, and that the one of them should make an excessive inscription, and the other a defective, *viz*., the one should somewhat penetrate the inner sphere, the other not touch it. And because the octahedron is cognate to the cube and has an equal ratio of spheres, but the icosahedron to the dodecahedron, accordingly, whatever the cube has of perfection of inscription, the same was due to the octahedron also, and whatever the dodecahedron, the same to the icosahedron too. And the situation of the octahedron's similar to the situation of the cube, but that of the icosahedron to the situation of the dodecahedron, because as the cube occupies the one limit to the outside, so the octahedron occupies the remaining limit to the inside of the world, but the dodecahedron and icosahedron are midway: accordingly even a similar inscription was proper, in the case of the dodecahedron, one penetrating the sphere of the inner planet, in that of the icosahedron, one falling short of it.

But the echinus, which represents the icosahedron with the apexes of its angles and the dodecahedron with the bases, was due to fill, embrace, or dispose both regions, that between Mars and the Earth with the dodecahedron as well as that between the Earth and Venus with the icosahedron. But the preceding axiom makes clear which of the opposites was due to which association. For the tetrahedron, which has a rational inscribed sphere, has been allotted the middle position among the primary figures and is surrounded on both sides by figures of incommensurable spheres, whereof the outer is the cube, the inner the dodecahedron, by Chapter 1 of this book. But this geometrical quality, *viz*., the rationality of the inscribed sphere, represents in nature the perfect inscription of the planetary sphere. Accordingly, the cube and its allied figure have their inscribed spheres rational only in square, *i.e.*, in power alone; accordingly, they ought to represent a semiperfect inscription, where, even if not the extremity of the planetary sphere, yet at least something on the inside and rightfully a mean between the aphelial and perihelial spheres—if that is possible through other reasons is touched by the centres of the planes of the figures. On the other hand, the dodecahedron and its allied figure have their inscribed spheres clearly irrational both in the length of the radius and in the square; accordingly, they ought to represent a clearly imperfect inscription and one touching absolutely nothing of the planetary sphere, *i.e.*, falling short and not reaching as far as the aphelial sphere of the planet with the centres of its planes.

Although the echinus is cognate to the dodecahedron and its allied figure, nevertheless it has a property similar to the tetrahedron. For the radius of the sphere inscribed in its inverted sides is indeed incommensurable with the radius

of the circumscribed sphere, but it is, however, commensurable with the length of the distance between two neighbouring angles. And so the perfection of the commensurability of rays is approximately as great as in the tetrahedron; but elsewhere the imperfection is as great as in the [317] dodecahedron and its allied figure. Accordingly it is reasonable too that the physical inscription belonging to it should be neither absolutely tetrahedral nor absolutely dodecahedral but of an intermediate kind; in order that (because the tetrahedron was due to touch the extremity of the sphere with its planes, and the dodecahedron, to fall short of it by a definite interval) this wedge-shaped figure with the inverted sides should stand between the icosahedral space and the extremity of the inscribed sphere and should nearly touch this extremity—if nevertheless this figure was to be admitted into association with the remaining five, and if its laws could be allowed, with the laws of the others remaining. Nay, why do I say "could be allowed"? For they could not do without them. For if an inscription, which was loose and did not come into contact fitted the dodecahedron, what else could confine that indefinite looseness within the limits of a fixed magnitude, except this subsidiary figure cognate to the dodecahedron and icosahedron, and which comes almost into contact with its inscribed sphere and does not fall short (if indeed it does fall short) any more than the tetrahedron exceeds and penetrates —with which magnitude we shall deal in the following.

This reason for the association of the echinus with the two cognate figures (*viz*., in order that the ratio of the spheres of Mars and Venus, which they had left indefinite, should be made determinate) is rendered very probable by the fact that 1,000, the semidiameter of the sphere of the Earth, is found to be practically a mean proportional between the perihelial sphere of Mars and the aphelial sphere of Venus; as if the interval, which the echinus assigns to the cognate figures, has been divided between them as proportionally as possible. XLVIII. PROPOSITION. *The inscription of the regular solid figures between the planetary spheres was not the work of pure freedom; for with respect to very small magnitudes it was hindered by the consonances established between the extreme movements*.

For, by Axioms I and II, the ratio of the spheres of each figure was not due to be expressed immediately by itself, but by means of it the consonances most akin to the ratios of the spheres were first to be sought and adjusted to the extreme movements.

Then, in order that, by Axioms XVIII and XX, the universal consonances of the two modes could exist, it was necessary for the greater consonances of the single pairs to be readjusted somewhat, by means of the posterior reasons. Accordingly, in order that those things might stand, and be maintained by their own reasons, intervals were required which are somewhat discordant with those which arise from the perfect inscription of figures between the spheres, by the laws of movements unfolded in Chapter 3. In order that it be proved and made manifest how much is taken away from the single planets by the consonances established by their proper reasons; come, let us build up, out of them, the intervals of the planets from the sun, by a new form of calculation not previously tried by anyone.

Now there will be three heads to this inquiry: First, from the two extreme movements of each planet the similar extreme intervals between it and the sun will be investigated, and by means of them the radius of the sphere in those dimensions,

of the extreme intervals, which are proper to each planet. Secondly, by means of the same extreme movements, in the same dimensions for all, the mean movements and their ratio will be investigated. Thirdly, by means of the ratio of the mean movements already disclosed, the ratio of the spheres or mean intervals and also one ratio of the extreme intervals, will be investigated; and the ratio of the mean intervals will be compared with the ratios of the figures.

As regards the first: we must repeat, from Chapter 3, Article VI, that the ratio of the extreme movements is the inverse square of the ratio of the corresponding intervals from the sun. Accordingly, since the ratio of the squares is the square of the ratio of its sides, therefore, the numbers, whereby the extreme movements of the single planets are expressed, will be considered as squares and the extraction of their roots will give the extreme intervals, whereof it is easy to take the arithmetic mean as the semidiameter of the sphere and the eccentricity. Accordingly the consonances so far established have prescribed:

For the second of the things proposed, we again have need of Chapter 3, Article XII, where it was shown that the number which expresses the movement which is as a mean in the ratio of the extremes is less than their arithmetic mean, also less than the geometric mean by half the difference between the geometric and arithmetic means. And because we are investigating all the mean movements in the same dimensions, therefore let all the ratios hitherto established between different twos and also all the private ratios of the single planets be set out in the measure of the least common divisible. Then let the means be sought: the arithmetic, by taking half the difference between the extreme movements of each planet, the geometric, by the multiplication of one extreme into the other and extracting the square root of the product; then by subtracting half the difference of the means from the geometric mean, let the number of the mean movement be constituted in the private dimensions of each planet, which can easily, by the rule of ratios, be converted into the common dimensions.

[319] Therefore, from the prescribed consonances, the ratio of the mean diurnal movements has been found, *viz*., the ratio between the numbers of the degrees and minutes of each planet. It is easy to explore how closely that approaches to astronomy.

The third head of things proposed requires Chapter 3, Article VIII. For when the ratio of the mean diurnal movements of the single planets has been found, it is possible to find the ratio of the spheres too. For the ratio of the mean movements is the 3/2th power of the inverse ratio of the spheres. But, too, the ratio of the cube numbers is the 3/2th power of the ratio of the squares of those same square roots, given in the table of Clavius, which he subjoined to his *Practical Geometry*. Wherefore, if the numbers of our mean movements (curtailed, if need be, of an equal number of ciphers) are sought among the cube numbers of that table, they will indicate on the left, under the heading of the squares, the numbers of the ratio of the spheres; then the eccentricities ascribed above to the single planets in the private ratio of the semidiameters of each may easily be converted by the rule of ratios into dimensions common to all, so that, by their addition to the semidiameters of the spheres and subtraction from them, the extreme intervals of the single planets from the sun may be established. Now we shall give to the semidiameter of the terrestrial sphere the round number 100,000, as is the practice in astronomy, and with the following design: because this number or its square or its cube is always made up of mere ciphers; and so too we shall raise the mean movement of the Earth to the number 10,000,000,000 and by the rule of ratios make the number of the mean movement of any planet be to the number of the mean movement of the Earth, as 10,000,000,000 is to the new measurement. And so the business can be carried on with only five

cube roots, by comparing those single cube roots with the one number of the Earth.

Accordingly, it is apparent in the last column what the numbers turn out to be whereby the converging intervals of two planets are expressed. All of them approach very near to those intervals, which I found from Brahe's observations. In Mercury alone is there some small difference. For astronomy is seen to give the following intervals to it: 470, 388, 306, all shorter. It seems that the reason for the dissonance may be referred either to the fewness of the observations or to the magnitude of the eccentricity. (See Chapter 3). But I hurry on to the end of the calculation.

For now it is easy to compare the ratio of the spheres of the figures with the ratio of the converging intervals.

[320] For if the semidiameter of the sphere circumscribed around the figure

That is to say, the planes of the cube extend down slightly below the middle circle of Jupiter; the octahedral planes, not quite to the middle circle of Mercury; the tetrahedral, slightly below the highest circle of Mars; the sides of the echinus, not quite to the highest circle of Venus; but the planes of the dodecahedron

fall far short of the aphelial circle of the Earth; the planes of the icosahedron also fall short of the aphelial circle of Venus, and approximately proportionally; finally, the square in the octahedron is quite inept, and not unjustly, for what are plane figures doing among solids? Accordingly, you see that if the planetary intervals are deduced from the harmonic ratios of movements hitherto demonstrated, it is necessary that they turn out as great as these allow, but not as great as the laws of free inscription prescribed in Proposition XLV would require: because this κόσμος γεωμέτρικος [geometrical adornment] of perfect inscription was not fully in accordance with that other κόϐμον ἁρμόνικον ἐνδεχόμενον [possible harmonic adornment]—to use the words of Galen, taken from the epigraph to this Book v. So much was to be demonstrated by the calculation of numbers, for the elucidation of the prescribed proposition.

I do not hide that if I increase the consonance of the diverging movements of Venus and Mercury by the private ratio of the movements of Venus, and, as a consequence, diminish the private ratio of Mercury by the same, then by this process I produce the following intervals between Mercury and the sun: 469, 388, 307, which are very precisely represented by astronomy. But, in the first place, I cannot defend that diminishing by harmonic reasons. For the aphelial movement of Mercury will not square with that musical scale, nor in the planets which are opposite in the world is the planetary principle [*ratio*] of opposition of all conditions kept. Finally, the mean diurnal movement of Mercury becomes too great, and thereby the periodic time, which is the most certain fact in all astronomy, is shortened too much. And so I stay within the harmonic polity here employed and confirmed throughout the whole of Chapter 9. But none the less with this example I call you all forth, as many of you as have happened to read this book and are steeped in the mathematical disciplines and the knowledge of highest philosophy: work hard and either pluck up one of the consonances applied everywhere, interchange it with some other, and test whether or not you will come so near to the astronomy posited in Chapter 4, or else try by reasons whether or not you can build with the celestial movements something better and more expedient and destroy in part or in whole the layout applied by me. But let whatever pertains to the glory of Our Lord and Founder be equally permissible to you by way of this book, and up to this very hour I myself have taken the liberty of everywhere changing those things which I was able to discover on earlier days and which were the conceptions of a sluggish care or hurrying ardour.

[321] XLIX. ENVOI. *It was good that in the genesis of the intervals the solid figures should yield to the harmonic ratios, and the major consonances of two planets to the universal consonances of all, in so far as this was necessary*.

With good fortune we have arrived at 49, the square of 7; so that this may come as a kind of Sabbath, since the six solid eights of discourse concerning the construction of the heavens has gone before. Moreover, I have rightly made an envoi which could be placed first among the axioms: because God also, enjoying the works of His creation, "saw all things which He had made, and behold! they were very good."

There are two branches to the *envoi*: First, there is a demonstration concerning consonances in general, as follows: For where there is choice among different things which are not of equal weight, there the more excellent are to be put first and the more vile are to be detracted from, in so far as that is necessary, as the

very word ὁ κόϐμος, which signifies *adornment*, seems to argue. But inasmuch as life is more excellent than the body, the form than the material, by so much does harmonic adornment excel the geometrical.

For as life perfects the bodies of animate things, because they have been born for the exercise of life—as follows from the archetype of the world, which is the divine essence—so movement measures the regions assigned to the planets, each that of its own planet: because that region was assigned to the planet in order that it should move. But the five regular solids, by their very name, pertain to the intervals of the regions and to the number of them and the bodies; but the consonances to the movements. Again, as matter is diffuse and indefinite of itself, the form definite, unified, and determinant of the material, so too there are an infinite number of geometric ratios, but few consonances. For although among the geometrical ratios there are definite degrees of determinations, formation, and restriction, and no more than three can exist from the ascription of spheres to the regular solids; but nevertheless an accident common to all the rest follows upon even these geometrical ratios: an infinite possible section of magnitudes is presupposed, which those ratios whose terms are mutually incommensurable somehow involve in actuality too. But the harmonic ratios are all rational, the terms of all are commensurable and are taken from a definite and finite species of plane figures. But infinity of section represents the material, while commensurability or rationality of terms represents the form. Accordingly, as material desires the form, as the rough-hewn stone, of a just magnitude indeed, the form of a human body, so the geometric ratios of figures desire the consonances—not in order to fashion and form those consonances, but because this material squares better with this form, this quantity of stone with this statue, even this ratio of regular solids with this consonance—therefore in order so that they are fashioned and formed more fully, the material by its form, the stone by the chisel into the form of an animate being; but the ratio of the spheres of the figure by its own, *i.e.*, the near and fitting, consonance.

The things which have been said up to now will become clearer from the history of my discoveries. Since I had fallen into this speculation twenty-four years ago, I first inquired whether the single planetary spheres are equal distances apart from one another (for the spheres are apart in Copernicus, and do not touch one another), that is to say, I recognized nothing more beautiful than the ratio of equality. But this ratio is without head or tail: for this material equality furnished no definite number of mobile bodies, no definite magnitude for the intervals. Accordingly, I meditated upon the similarity of the intervals to the spheres, *i.e.*, upon the proportionality. But the same complaint followed. For although to be sure, intervals which were altogether unequal were produced between the spheres, yet they were not unequally equal, as Copernicus wishes, and neither the magnitude of the ratio nor the number of the spheres was given. I passed on to the regular plane figures: [322] intervals were formed from them by the ascription of circles. I came to the five regular solids: here both the number of the bodies and approximately the true magnitude of the intervals was disclosed, in such fashion that I summoned to the perfection of astronomy the discrepancies remaining over and above. Astronomy was perfect these twenty years; and behold! there was still a discrepancy between the intervals and the regular solids, and the reasons for the distribution of unequal eccentricities among the planets were not disclosed. That is to say, in this house the world, I

was asking not only why stones of a more elegant form but also what form would fit the stones, in my ignorance that the Sculptor had fashioned them in the very articulate image of an animated body. So, gradually, especially during these last three years, I came to the consonances and abandoned the regular solids in respect to minima, both because the consonances stood on the side of the form which the finishing touch would give, and the regular solids, on that of the material—which in the world is the number of bodies and the rough-hewn amplitude of the intervals—and also because the consonances gave the eccentricities, which the regular solids did not even promise—that is to say, the consonances made the nose, eyes, and remaining limbs a part of the statue, for which the regular solids had prescribed merely the outward magnitude of the rough-hewn mass.

Wherefore, just as neither the bodies of animate beings are made nor blocks of stone are usually made after the pure rule of some geometrical figure, but something is taken away from the outward spherical figure, however elegant it maybe (although the just magnitude of the bulk remains), so that the body may be able to get the organs necessary for life, and the stone the image of the animate being; so too as the ratio which the regular solids had been going to prescribe for the planetary spheres is inferior and looks only towards the body and material, it has to yield to the consonances, in so far as that was necessary in order for the consonances to be able to stand closely by and adorn the movement of the globes.

The other branch of the *envoi*, which concerns universal consonances, has a proof closely related to the first. (As a matter of fact, it was in part assumed above, in XVIII, among the Axioms.) For the finishing touch of perfection, as it were, is due rather to that which perfects the world more; and conversely that thing which occupies a second position is to be detracted from, if either is to be detracted from. But the universal harmony of all perfects the world more than the single twin consonances of different neighbouring twos. For harmony is a certain ratio of unity; accordingly the planets are more united, if they all are in concord together in one harmony, than if each two concord separately in two consonances. Wherefore, in the conflict of both, either one of the two single consonances of two planets was due to yield, so that the universal harmonies of all could stand. But the greater consonances, those of the diverging movements, were due to yield rather than the lesser, those of the converging movements. For if the divergent movements diverge, then they look not towards the planets of the given pair but towards other neighbouring planets, and if the converging movements converge, then the movements of one planet are converging toward the movement of the other, conversely: for example, in the pair Jupiter and Mars the aphelial movement of Jupiter verges toward Saturn, the perihelial of Mars towards the Earth: but the perihelial movement of Jupiter verges toward Mars, the aphelial of Mars toward Jupiter. Accordingly the consonance of the converging movements is more proper to Jupiter and Mars; the consonance of the diverging movements is somehow more foreign to Jupiter and Mars. But the ratio of union which brings together neighbouring planets by twos and twos is less disturbed if the consonance which is more foreign and more removed from them should be adjusted than if the private ratio should be, *viz*., the one which exists between the more neighbouring movements of neighbouring planets. None the less this adjustment was not very great. For the proportionality has

been found in which may stand the universal consonances of all the planets may exist (and these in two distinct modes), and in which (with a certain latitude of tuning merely equal to a comma) may also be embraced the single consonances of two neighbouring planets; the consonances of the converging movements in four pairs, perfect, of the aphelial movements in one pair, of the perihelial movements in two pairs, likewise perfect; the consonances of the diverging movements in four pairs, these, however, within the difference of one diesis (the very small interval by which the human voice [323] in figured song nearly always errs; the single consonance of Jupiter and Mars, this between the diesis and the semitone. Accordingly it is apparent that this mutual yielding is everywhere very good.

Accordingly let this do for our *envoi* concerning the work of God the Creator. It now remains that at last, with my eyes and hands removed from the tablet of demonstrations and lifted up towards the heavens, I should pray, devout and supplicating, to the Father of lights: *O Thou Who dost by the light of nature promote in us the desire for the light of grace, that by its means Thou mayest transport us into the light of glory, I give thanks to Thee, O Lord Creator, Who hast delighted me with Thy makings and in the works of Thy hands have I exulted. Behold! now, I have completed the work of my profession, having employed as much power of mind as Thou didst give to me; to the men who are going to read those demonstrations I have made manifest the glory of Thy works, as much of its infinity as the narrows of my intellect could apprehend. My mind has been given over to philosophizing most correctly: if there is anything unworthy of Thy designs brought forth by me—a worm born and nourished in a wallowing place of sins—breathe into me also that which Thou dost wish men to know, that I may make the correction: If I have been allured into rashness by the wonderful beauty of Thy works, or if I have loved my own glory among men, while I am advancing in the work destined for Thy glory, be gentle and merciful and pardon me; and finally deign graciously to effect that these demonstrations give way to Thy glory and the salvation of souls and nowhere be an obstacle to that*.

1054:1 *Smaller* (lesser) and *greater* consonances are equivalent to our modern "more closely spaced" and "more widely spaced" consonances. E. C., Jr.

1062:1 "Identisonant consonances" are such as 3 : 5, 3 : 10, 3 : 20, etc.

1066:1 See footnote to *Intervals Compared with Harmonic Ratios*, 1026.

1067:1 *Timaeus*, 36.

1068:1 Here "sixth" (*sexta*) should probably be "third" (*tertia*). E. C., Jr.

1068:2 *C* and *e* do not produce a subminor third in the "natural system." E. C., Jr.