The Relationship of the Harmonic

Proposition 27

The greater proportion of the motions of the Earth and Venus should have been a hard sixth between the motions at aphelion, and the smaller a soft sixth between the motions at perihelion.

For the kinds of harmonies had to be distinguished, by Axiom XX. Yet that could not have been done except by the sixths, by Part 23.

Therefore, since one of them, 5:8, has been taken by the Earth and Venus, the closest planets, which are icosahedric by XV, the other, 3-5, should also have been attributed to them.

But not between the converging and diverging extreme motions, but between the extreme motions on the same side, one between the motions at aphelion, the other between those at perihelion, by XXIV. In addition the harmony 3:5 is also akin to the icosahedron, inasmuch as both are of the pentagonic group. See Chapter II.

Here is the reason why precise harmonies are rather found between the motions of these two at aphelion and perihelion, but not between their converging motions, as in the superior planets.

Proposition 28

For the Earth the motions’ own proportion agreed with 14’.15, about; for Venus about 35:36.

For these two ought to have distinguished the kinds of harmonies, by what has already been stated. Then by XXVI the Earth in fcwt as the superior planet should have got the interval 2916:3125, that is nearly 14-15; whereas Venus, as the inferior, ought to have got the interval 243-250, that is 35-36, very nearly.

Here is the reason why these two planets have such small eccentricities, and resulting from them, small distances or proportions as their extreme motions’ own, while nevertheless Mars which is the next superior to the Earth, and Mercury which is the next inferior to Venus, have ones which are outstanding and greatest of all.

However, astronomy confirms that this is true; for in Chapter IV the Earth clearly had 14:15, and Venus 34:35, which astronomical accuracy will scarcely be able to distinguish from 35:36.

Proposition 29

The greater harmony of the motions of Mars and the Earth, that is of their diverging motions, could not have been among those greater than 5:12.

By Proposition XVII above it was not one of the lesser proportions: hut now it is not one of those greater, either. For the other common proportion of these planets, or the lesser, 2:3, multiplied by Mars’ own proportion, which by XIV exceeds 18:23, makes more than 12:23, that is, 60:123.

Therefore, multiply it by the Earth’s own proportion, 14:13, that is 36:60, by what has already been stated: the product is more than 36:123. That is very nearly 4:9, that is to say more than an octave and a tone, by a little. But the harmony next greater than an octave and a tone is 3:12, a diapason with a soft third.

Note that I do not say that this proportion is neither greater nor lesser than 5:12; but what I do say is that if it must necessarily be harmonic, no other harmony would agree with it.

Proposition 30

Mercury’s motions’ own proportion ought to have been greater than all the others’ own proportions.

For by XVI Venus’and Mercury’s own proportions combined should have made about 3:12. But Venus’own separately is only 243:230, that is 1438:1300; and that divided into 3:12, that is into 623:1300, leaves 623:1438, which is greater than a diapason with a major tone, for Mercury alone, whereas Mars’s own proportion, which among the other planets is the greatest of all, is less than the sesquialterate proportion 2:3, that is a diapente.

In fact for Venus and Mercury, the lowest planets, their own proportions combined equal thefour highest planets’own proportions combined, nearly.

For as will now immediately be apparent, Saturn andJupiter’s own proportions combined exceed 2:3. Mars’s own proportion falls considerably short of 2:3: the product is 4:9, that is 60:133. Multiply by that of the Earth, 14:13, that is 36:60: the product is 36:133, which is a little more than 3:12, and as we have just seen is the product of Venus and Mercury’s own proportions.

However, that was not sought, nor taken from some separate and special archetype of beauty, but emerges spontaneously, by the necessity of causes connected with the harmonies confirmed so far.

Proposition 31

The motion of the Earth at aphelion ought to have been in harmony with Saturn’s at aphelion over several diapasons.

For there must have been universal harmonies, by XVIII, and hence also harmony of Saturn with the Earth and Venus. But if one of the extreme motions of Saturn had been in harmony with neither of the latter, that would have been less harmonic than if both of its extreme motions were in harmony with those planets, by Axiom I.

Then Saturn should have been in harmony at both its extremes: in its motion at aphelion with one of those two planets, at perihelion with the remaining one, since there was no impediment, inasmuch as it was the motion of the first planet.

Therefore, these harmonies will be either identical in sound or different in sound, that is in either the proportion of continuous doubling or another proportion.

But both cannot be in another proportion, for between the terms of 3:3 (defining the greater harmony between the motions of the Earth and Venus at aphelion, by XXVII) there cannot exist two harmonic means, as a sixth cannot be divided into three harmonic intervals. See Book 3.

Therefore, Saturn could not have made a diapason with harmonic means between 3 and 3 with both its motions; but for its motions to be in harmony both with the 3 of the Earth and with the 3 of Venus, one of the motions must be in identical harmony with one of the terms, that is with one of the planets mentioned themselves, or in harmony over several diapasons.

Now since identical harmonies are more outstanding, they will also have to be established between the more outstanding extremes of the motions, that is between the motions at aphelion, both because they hold the position of preeminence on account of the loftiness of the planets, and because they claim the harmony 3:3, with which we are now dealing as the greater harmony of the Earth and Venus, as their own, in a sense, and as their privilege. For although that harmony also agrees with the motion of Venus at perihelion and some intermediate motion of the Earth, by XXVII, yet the beginning is from the extremes of the motions, and the intermediate motions give precedence to the chief ones. In that case since on the one hand we have the motion at aphelion of Saturn the highest planet, on the other hand it is the motion of the Earth at aphelion rather than that of Venus which must be coupled with it, because of these two, which distinguish their kind of harmony, the former is again the higher. There is also another more immediate reason: that the a posteriori arguments, in which we are now engaged, do indeed modify the a priori arguments, but only in the least im-

portant points, because it is a question of harmony, in respect of intervals which are less than all the melodic intervals. But by the a priori arguments the motion at aphelion, not of Venus, but of the Earth, was close to the harmony of several diapasons which had to be established with the motion of Saturn at aphelion. For multiply together into a single product first the Saturnine motions’ own proportion 4:3, that is from the motion of Saturn at aphelion to its motion at perihelion, by XI; second, the proportion of the converging motions of Saturn andfupiter, 1:2, that is from the motion at perihelion of Saturn to that of Jupiter at aphelion, by VIII; third, the proportion of the diverging motions of fupiter and Mars, 1:8, that is from the motion at aphelion of Jupiter to the motion at perihelion of Mars, by XIII; and fourth, the proportion of the converging motions of Mars and the Earth, 2:3, that is from the motion at perihelion of Mars to the motion at aphelion of the Earth, by XV. You will find the total product between the motion at aphelion of Saturn and the motion at aphelion of the Earth is the proportion 1:30, which falls short by not more than 30:32, that is 13:16, or a semitone, of being 1:32, orfive diapasons. Then if a semitone, divided into parts smaller than the least melodic interval, were to be added to these four elements, there will be between the motions in the proposition of Saturn and the Earth at aphelion a perfect harmony of a fivefold diapason. But for the same motion at aphelion of Saturn to make several diapasons with the motion at aphelion of Venus, it would have been necessary by the a priori arguments to tear away almost a complete diatessaron. For if 3:3, which is be­ tween the motions at aphelion of the Earth and Venus, is multiplied by the total 1:30 produced by combining the four previous elements, the result, as if from the a priori arguments, is 1:30 between the motions at aphelion of Saturn and Venus, an interval which dijfers from 1:32, a fivefold diapason, by 32:30, that is by 16:23, which is a diapente with a diesis, and differs from a six­ fold diapason, or 1:64, by 30:64, which is 23:32, or a diatessaron minus one diesis. Therefore, the identical harmony had to be set up not between the mo­ tions at aphelion of Venus and Saturn, but between those of the Earth and Saturn, so that there would remain for Saturn a harmony different in sound with Venus. XXXII. Proposition In the universal harmonies of the planets of the soft kind, the motion of Saturn absolutely at aphelion could not have been exactly in har­ mony with the other planets. For the Earth in its motion at aphelion does not coincide with universal harmony of the soft kind, because the motions of the Earth and Venus at aph­ elion make the interval 3:3, of the hard kind, by XXVIl. However, Saturn in its motion at aphelion makes identical harmony with the motion of the Earth at aphelion, by XXXI. Then neither does Saturn coincide in its motion at aph­ elion. However, there succeeds in place of the motion at aphelion a more vigorous motion of Saturn, very close to that at aphelion, and also to the soft kind, as became apparent in Chapter VII

XXXIII. Proposition

The hard kind of harmonies and of musical scale is closely related to motions at aphelion, the soft kind to those at perihelion.*^’ For although a hard harmony is established not only between the motion of the Earth at aphelion and that of Venus at aphelion, but also between the motions of the Earth lower than aphelion and those of Venus lower than aph­ elion, right down to its perihelion; and on the other hand a soft harmony not only between the motion of Venus at perihelion and that of the Earth at peri­ helion, but also between the higher motions of Venus right up to the aphelion and the higher motions of the Earth, by XXVII; yet the proper and obvious representation of kind belongs only to the extreme motions of each planet, by XX and XXIV. Therefore, the proper representation of the hard kind belongs only to the motions at aphelion, and the proper representation of the soft kind only to the motions at perihelion.

XXXIV. Proposition

The hard kind is more closely related to the superior planet in a comparison between two, and the soft kind to the inferior.

For because the hard kind properly belongs to the motions at aphelion, and the soft kind to those at perihelion, by what has already been stated, and those at aphelion are slower and more deliberate than those at perihelion, therefore the hard kind belongs to the slower motions, the soft to the quicker.

But the superior of the two planets is more closely related to the slow motions, and the inferior to the quicker, because always in the world height is accompanied by slowness of the planet’s own motion.

Then of two which fit both kinds that which is superior is more closely related to the hard kind of scale, and that which is inferior to the soft. Further, the hard kind uses greater intervals, 4:3 and 3'3, the soft lesser intervals, 3'6 and 3:8. But in addition the superior planet also has a greater sphere and slower, that is greater, motions, and a more extended orbit; and those to which great things are appropriate in each case join in a closer relationship between themselves.

XXXV. Proposition

Saturn along with the Earth embraces the hard kind in a closer relationship, Jupiter with Venus the soft kind. For first, the Earth in comparison with Venus, and representing along with Venus both kinds, is the superior.

Therefore, the Earth chiefly embraces the hard kind, and Venus the soft kind, by what has already been stated. Now Saturn in its motion at aphelion is consonant over a diapason with the Earth’s motion at aphelion, by XXXI. Hence by XXXIII Saturn also embraces the hard kind. Second, Saturn in its motion at aphelion, by the same Proposition, cherishes the hard kind more, and rejects the soft kind, by XXXIL Therefore, it is more closely related to the hard kind than to the soft kind, because the kinds are prop­ erly represented by the extreme motions.

In that case, as far as Jupiter is concerned, in comparison with Saturn it is inferior. Then as the hard kind ought to belong to Saturn, so the soft ought to belong to Jupiter, by what has already been stated.

XXXVI. Proposition

The motion of Jupiter at perihelion should have agreed with that of Venus at perihelion in a single musical scale, but not in the same harmony as well; and much less so with that of the Earth at perihelion.

For as Jupiter ought to have belonged chiefly to the soft kind, by what has been stated previously, and the motions at perihelion are closely related to that kind, by XXXIII, therefore Jupiter by its motion at perihelion ought to have represented the scale of the soft kind, that is to say a definite position or sound in it. But the motions at perihelion of Venus and the Earth as well represent the same scale, by XXVII.

Then the motion of Jupiter at perihelion had to be associated with the motions of these latter planets at perihelion in the same tuning.

However, it could not have set up a harmony with the motion of Venus at perihelion. For as by VIII it ought to have made about 1 3 with the motion of Saturn at aphelion, that is, the note d of the system in which the motion of Saturn at aphelion made the note G, but the motion of Venus at aphelion the note e, therefore it came close to the note e within the interval of the smallest harmony.

For that is 5:6; but the interval between d and e is much less, that is to say 9:10, a tone. And although in the tuning at perihelion Venus is raised above its e in the tuning at aphelion, yet this rise is less than a diesis, by XXVIII.

However, a diesis (and something less than that) combined with a minor tone does not yet equal the interval of the smallest harmony, 5:6. Therefore, the motion at perihelion of Jupiter could not have protected the position with the motion of Saturn at aphelion except by means of 1:3 while still being in harmony with Venus. But neither could it with the Earth. For if the motion at perihelion of Jupiter has been fitted to the scale of the motion at perihelion of Venus, in the same tuning, in such a way that within less than the amount of the smallest interval it protects its interval with the motion of Saturn at aphelion, 1:3, that is to say separated from the motion of Venus at perihelion by a minor tone, that is 9:10 or 36:40 (in addition to some diapasons) on the lower side, the motion of the Earth at perihelion is of course separated from that same motion of Venus at perihelion by 5:8, that is by 25:40. Thus the motions at perihelion of the Earth and Jupiter will be separated by 25:36, in addition to several diapasons. However, that is not harmonic, as it is double 5:6, or a diapente, diminished by one diesis.

XXXVII. Proposition

The sum of Saturn’s and Jupiter’s own harmonies, 2^3, and their greater common harmony, 1:3, should have been increased by an interval equal to the interval of Venus.

For Venus by its motion at aphelion properly assists the representation of the hard kind, at perihelion of the soft kind, by XXVII and XXXIII. But Saturn by its motion at aphelion ought also to have agreed with the hard kind, and thus with the motion at aphelion of Venus, by XXXV; but Jupiter by its motion at perihelion with the motion at perihelion of Venus, by what has already been stated. Therefore, the factor of the interval which Venus makes between its aphelion and perihelion is also the factor by which it is necessary to increase the motion of Jupiter mentioned, which combined with the motion of Saturn at aphelion makes 1 3 with the actual motion of Jupiter at perihelion. But the harmony of the converging motions of Jupiter and Saturn is precisely 1:2 by VIII. Therefore, subtraction of the interval 1:2 from that, which is more than 1:3, leaves a remainder which is more than 2 3 by the sum of the intervals of each planet’s own proportions. Above, in Proposition XXVIII, Venus’ motions’ own proportion was 243:250, or very nearly 35:36. However, in Chapter IV between the motion of Saturn at aphelion and that of Jupiter at perihelion was found an excess over T3 which was a little greater, that is to say between 26:27 and 27:28. But if a single second —and I don’t know whether astronomy can detect it— is added to the motion of Saturn at aphelion, the quantity here prescribed is plainly equal to it.

XXXVIII. Proposition

The surplus factor of 243:250 in the product of Saturn’s and Jupiter’s own motions, which up to this point was established from first prin­ ciples as 2:3, had to be distributed among the planets in the following way: a comma, 80:81, from it was given to Saturn, and to Jupiter the quotient, 19683:20000, or 62:63 nearly. That this factor had to be distributed between both planets follows from XIX, so that both could coincide within a certain range with the universal har­ monies of the kind related to it. But the interval 243:250 is less than all the melodic intervals. Therefore, no harmonic laws are left by which to divide it into two melodic parts, with the sole exception of those which were needed above in Proposition XXVI for the division of a diesis, 24:25, that is to say that it should change into a comma, 80-81 (which is one, and indeed the chief, of those which are used for melodic intervals^’^^) and a quotient of 19683:2000, which is a little more than a comma, that is 62-63 nearly. However, not two commas but one comma had to be split ojf, so that the parts should not become too un­ equal, since Saturn’s and Jupiter’s own proportions are very nearly equal, ac­ cording to Axiom X extended also to melodic intervals and parts tinier than they, and also at the same time because a comma is defined by the intervals of a major tone and a minor tone, but not so two commas. Furthermore, to Saturn

as the higher and more powerful planet ought to have belonged for preference not the greater of these parts, although it did have as its own 4:5 which is the greater, but the prior and more beautiful, that is more harmonic. For in Axiom X, consideration of priority and harmonic perfection takes precedence; consideration of size comes last, because there is no beauty in size by itself Thus the motions of Saturn become 64:81, an impure major third, as we have called it in Book III, Chapter XII; but those of fupiter 6561:8000. I do not know whether it should be mentioned among the reasons for the addition of a comma to Saturn that it was to enable the ex­ treme distances of Saturn to set up the proportion 8-‘9, a major tone; or rather it came about spontaneously from the antecedent causes of the motions. You therefore have here in place of a corollary rather a reason why above in Chapter IV, page 420, the intervals of Saturn were found to embrace the proportion of a major tone, very nearly. XXXIX. Proposition In the universal harmonies of the planets, of the hard kind, Saturn could not be in harmony in its motion exactly at perihelion, nor Jupiter in its motion exactly at aphelion. For since the motion of Saturn at aphelion should have been exactly in harmony with the motions of the Earth and Venus at aphelion, by XXXI, the motion of Saturn which is more hurried than its motion at aphelion by one hard third, 4:5, will also be in harmony with those same planets; for the motions of the Earth and Venus at aphelion make a hard sixth, which by what has been shown in Book III is divisible into a diatessaron and a hard third. Then the motion of Saturn, which until this point is quicker than the motion which has now been harmonized, though by less than the amount of a melodic interval, will not be exactly in harmony. But the actual motion of Saturn at perihelion is such, because it is separated from its motion at aphelion by more than the interval 4:5, that is to say more than one comma, 80:81 (which is less than the smallest melodic interval) by XXXVIII. Therefore, the motion of Saturn exactly at perihelion is not in harmony. But neither is the motion offupiter exactly at aphelion; for it is consonant over a perfect diapason, by VIII, with the motion at perihelion of Saturn, which is not exactly consonant. Hence by what has been said in Book III it cannot itself be exactly consonant either.

XL. Proposition

To the common harmony of the diverging motions ofjupiter and Mars, 1:8, a triple diapason, confirmed by the a priori arguments, a Platonic limma had to be added

Eor between the motions at aphelion of Saturn and the Earth there had to be 1:32, that is 12:384, by XXXI; but from the motion of the Earth at aphelion to the motion of Mars at perihelion there had to be 3:2, that is 384:256, by XV; and from the motion at aphelion of Saturn to its motion at perihelion 4:5 or 12:15, together with the extra factor, by XXXVIII; and last, from the motion at perihelion of Saturn to the motion at aphelion of Jupiter 1:2, or 15:30, by VIII. Therefore, the quotient, from the motion at aphelion of Jupiter to the motion at perihelion of Mars is 30:256, after dividing it by Saturn’s extrafactor. But 30:256 exceeds 32:256, that is 1:8, by the factor 30-32, that is 15:16 or 240:256, which is a semitone. Therefore, division o f240-256 by Saturn’s extra factor, which by Proposition XXXVIII should have been 80:81, that is 2 4 0 ‘-243, leaves 243-256. But that is a Platonic limma, that is 19-20, nearly: see Book III. Therefore, a Platonic limma had to be added to the I -8. Thus the greater proportion ofjupiter and Mars, that is the pro­ portion of their diverging motions, ought to be 243:2048, which is in a way a mean between 243:2187 and 243:1944 that is between 1:9 and 1:8. O f these the former was required above by direct propor­ tion,’’’" the latter by harmonic melodicity, which is closer to hand. XLI. Proposition Mars’s motions’ own proportion was necessarily made the square of the harmonic proportion 5:6, that is to say 25:36. Eor because the proportion of the diverging motions of Jupiter and Mars had to be 243:2048, that is 729-6144, by the previous proposition, and that of their converging motions 5-24, that is 1280:6144, by XIII, therefore the prod­ uct of their own proportions was necessarily 729:1280, or 72900-128000. But Jupiter’s own proportion alone had to be 6561:8000, that is 104976:128000, by XXXVIII. Then if this proportion of Jupiter’s is divided into the product of both, the quotient is Mars’s oum proportion, 72900-104796, that is 25-36, the square root of which is 5:6. Alternatively as follows. Erom the motion of Saturn at aphelion to that of the Earth at aphelion is 1:32 or 120:3840. Erom the same motion of Saturn to the motion at perihelion of Jupiter is 1:3, or 120:360, with its excess factor surplus. Now from that to the motion at aphelion of Mars is 5’-24, or 360-1728. Therefore, from the motion of Mars at aphelion to the motion of the Earth at aphelion is the quotient, 1728:3840, divided by the extra factor in the propor­ tion of the diverging motions of Saturn and Jupiter. But from the same motion

at aphelion of the Earth to the motion at perihelion of Mars is 3:2, that is 3840:2560. Then between the motions of Mars at aphelion and perihelion the quotient will be the proportion 1728:2560, that is 27:40, or 81:120, divided by the extra factor mentioned. But 81:120 is a comma less than 80:120, or 2:3. Then if a comma were divided into 2:3, and the excess factor mentioned (which by XXXVIII is equal to Venus’ own proportion) were also divided into it, the quotient is Mars’s own proportion. But Venus’own proportion is a diesis diminished by a comma, by XXVI. Now a comma and a diesis diminished by a comma make a whole diesis, 24:25. Then if you divide 2:3, that is 24:36, by a diesis, 24:25, the quotient will be Mars’s own proportion, 25:36 as before; and the root of that, 5:6, is allotted to the intervals, by Chapter ///.*^‘ Look, here is another reason why above, in Chapter IV, page 424, the extreme distances of Mars were discovered to embrace the har­ monic proportion

XLII. Proposition

The greater common proportion of Mars and the Earth, or that of their diverging motions, was necessarily made 54:125, less than the harmony 5:12 confirmed by the a priori arguments. For Mars’s own proportion had to be a diapente, from which a diesis was removed, by the previous proposition. However, the common proportion of the converging motions of Mars and the Earth, or the lesser common proportion, had to be a diapente, 2:3, by XV. Last, the Earth’s own proportion is a doubled diesis, from which a comma has been removed, by XXVI and XXVIII. Now of these elements is composed the greater proportion, or that of the diverging motions, of Mars and the Earth; and it comes to two diapentes (or 4:9, that is 108:243) together with one diesis which is mutilated of a comma, that is together with 243:250. That is, it comes to 108:250, or 54:125, that is 608:1500. But that is less than 625:1500, that is, than 5:12, by the factor of 608:625: and that is nearly 36:37, less than the smallest melodic interval.

XLIII. Proposition

The motion of Mars at aphelion could not agree with a universal har­ mony; yet it was necessary for it to be in accord to a certain extent with the scale of the soft kind. For because the motion at perihelion of fupiter holds the position of d in he high tuning in the soft kind, and in fact between it and the motion of Mars at aphelion there had to be the harmony 5:24, then the motion of Mars at aphelion holds the position of the impure f in the same high tuning. 1 say, im­ pure; for in Book III, Chapter XII, when the impure consonances were enumer­ ated, and removed from the composition of the systems, some were omitted which do exist in the actual simple natural system. Thus the reader should write in, after the line which finishes thus, ‘‘81:120,” the following: if you divide this by 4:5 or 32:40, the quotient is 27:32, a narrow soft third, which is be­ tween d a n d / o r cq and e or a and c even in the simple octave.*’’^ And in the table below that the following should occupy the first line: For 5:6 there is 27:32, undersize. From which it is evident that in the natural system the genuine note f, as it is defined in accordance with my basic principles, constitutes with the note d an undersize or impure soft third. Therefore, since between the motion at perihelion of Jupiter, set up on the genuine note d, and the motion at aphelion of Mars, there is a perfect soft third above the double diapason, and not an undersize one, by XIII, it follows that Mars by its motion at aphelion signifies the position which is one comma higher than the genuine note f. Thus it will hold nothing but the impure f; and so it is in accord with this scale, not directly but at least to a certain extent. However, it does not enter a universal harmony, either pure or impure. For the motion of Venus at perihelion holds the position e in this tuning. But there is a dissonance between e and f, as they are neighbors. Therefore, Mars is in dissonance with the motion of one of the planets, that is with the motion of Venus at perihelion. But it is also in dissonance with the other motions of Venus, for they lag by one comma less than one diesis. Hence, since between the motion of Venus at perihelion and that of Mars at aphelion there is a semitone and a comma, therefore between the motion of Venus at aphelion and the motion of Mars at aphelion there will be a semitone and a diesis (dis­ regarding the octaves), that is a minor tone, which is still a dissonant interval. Now the motion of Mars at aphelion is in accord with the scale of the soft kind to that extent, but not with that of the hard as well. For since the motion at aphelion of Venus agrees with e of the hard kind, whereas the motion of Mars at aphelion (disregarding the octaves) has been made higher than e by a minor tone, therefore the motion of Mars at aphelion in this tuning would necessarily fall as a mean between f and fg, making with g (which in this tuning is taken by the motion of the Earth at aphelion) the interval 25:27, which is plainly unmelodic, that is a major tone from which a diesis has been subtracted. In the same way it will be proved that the motion of Mars at aphelion is also at odds with the motions of the Earth. For because with the motion of Venus at perihelion it makes a semitone and a comma, by what has been said, that is 14:15, but the motions of the Earth and Venus at perihelion make a soft sixth, 5:8 or 15:24, by XXVII, therefore the motion of Mars at aphelion with the motion of the Earth at perihelion (with octaves added to the former) will make 14:24, or 7:12, an unmelodic interval, still less harmonic, as is 7:6 also.

For anything between 5:6 and 8:9, as 6:7 is in this instance, is dissonant and unmelodic. But neither can any other motion of the Earth be in harmony with the motion at aphelion of Mars. For it has been stated above that it makes with the motion of the Earth at aphelion 25:27, which is unmelodic (disregarding the octaves): but in this case from 6'7 or 24:28 up to 25:27 all intervals are less than the smallest harmonic interval.

XLIV. Corollary It is therefore clear from this proposition XLIII, on Jupiter and Mars, and from X X X IX on Saturn and Jupiter, and from XXXV I on Jupiter and the Earth, and from X X X II on Saturn, why above in Chapter V it was discovered that neither did all the extreme motions of the planets fit perfectly a single natural system or musical scale, nor did all those which fitted a system in the same tuning divide up the positions in a natural pattern, or produce a purely natural succession of melodic intervals. For the reasons why individual planets acquired individual harmonies, why also all the planets acquired universal harmonies, and last, why the universal harmonies also acquired two kinds, hard and soft, are prior; and these being granted, now any kind of accommo­ dation to a single natural system is prevented. But if those reasons had not necessarily taken precedence, there is no doubt that a single system, and a single tuning to it, would have embraced the extreme motions of all the planets; or if two systems were needed, for the two kinds of melody, hard and soft, the actual order of the natural scale would have been expressed not only in the one scale of the hard kind but also in the other scale of the soft kind. Therefore, you have here the reasons, promised in the said Chapter V, for the dis­ agreements over very small intervals, smaller in fact than all the melodic interval

XLV. Proposition

The greater common proportion of Venus and Mercury, a double di­ apason, and also Mercury’s own proportion, by Propositions XII and XVI confirmed by a priori arguments above, had to be multiplied by an interval equal to the interval of Venus, in such a way that Mercury’s own proportion became a perfect 5T2, and thus Mercury was in har­ mony in both its motions with the motion of Venus at perihelion alone.

For because the motion of Saturn at aphelion ought to have been in har­ mony with the motion of the Earth at aphelion, that of the outermost planet, which is circumscribed about its figure and highest, with the highest motion of the Earth, which distinguishes the classes of figures, it follows by the laws of opposites that the motion of Mercury at perihelion agrees with the motion of the Earth at perihelion, that is the motion of the inmost planet, inscribed in its figure, lowest, and closest to the Sun, with the lowest motion of the Earth, the common boundary, the former indeed to mark out the hard kind of har­ monies, the latter the soft kind, by Propositions XXXIII and XXXIV. But the motion of Venus at perihelion ought to have been consonant with the motion of the Earth at perihelion in the harmony 5:8, by XXVII. Then the motion of Mercury at perihelion also ought to have been combined with the motion at perihelion of Venus in a single scale. However, by Proposition XII from the a priori arguments the harmony of the diverging motions of Venus and Mer­ cury was specified as 1:4. Then by these a posteriori arguments that had in this case to be leavened by the addition of the whole of the interval of Venus. Therefore, there is no longer a perfect disdiapason from the motion at aphelion of Venus, but from its motion at perihelion, to the motion at perihelion of Mer­ cury. But the harmony of the converging motions, 3:5, is also perfect, by Propo­ sition XV. Therefore, on dividing that into 1:4, the quotient is Mercury’s own harmony alone, 5:12, which is also perfect, but no longer (as by Proposition XVI through a priori arguments) diminished by Venus’ own proportion. Another argument. Just as Saturn and Jupiter alone are not touched at all on the outside by the dodecahedron and icosahedron, which are a married couple, so Mercury alone is not touched by the same figures inside. For they touch Mars, the Earth, and Venus, the first inside, the last outside, the middle one on both sides. Therefore, just as Saturn’s and Jupiter’s motions’ own proportions, which were supported by the cube and tetrahedron, were increased by something equal to Venus’ own proportion in corresponding shares, so in this case the solitary Mercury’s own proportion, which is contained within the octahedron, a figure allied to the cube and tetrahedron, ought to have been increased by the same factor. That is, just as the octahedron, a single figure among the secondaries, sustains the role of two among the primaries, the cube and tetrahedron — on which see Chapter I — so also among the inferior planets Mercury alone takes the place of two of the superior planets, that is to say Saturn and Jupiter. Third, just as Saturn, the highest planet, ought to have been in harmony in its motion at aphelion over several diapasons, that is, in the proportion 1:32, by continuous doubling, with the motion, also at aphelion, of the higher, and closer to itself, of the two which change the kind of harmony, by XXXI; so the other way round. Mercury the lowest planet ought to have been in harmony in its motion at perihelion, again over several diapasons, that is in the pro­ portion 1:4, also by continuous doubling, with the motion at perihelion of the lower, and similarly closer to itself, of the two which change the harmony. Fourth, only the individual extreme motions of the three superior planets, Saturn, Jupiter and Mars, agree in universal harmonies; therefore both the ex­ tremes of the lower and lone planet, that is Mercury, ought to have agreed in the same, for those in between, the Earth and Venus, ought to have changed the kind of harmonies, by XXXIII and XXXIV.

Last, in the three pairs of the superior planets perfect harmonies were found among the converging motions, but leavened harmonies among the diverging motions, also the individual planets’ own proportions; therefore in the two pairs of the inferior planets, the other way round, perfect harmonies ought to have been found not chiefly between the converging motions, nor between the diverg­ ing motions, but between motions on the same side.^^^ And because two perfect harmonies ought to have belonged to the Earth and Venus, hence Venus and Mercury also ought to have had two perfect harmonies. And the former two indeed ought to have been allotted a perfect harmony both between their motions at aphelion and between those at perihelion, because they ought to have changed the kind of harmony; whereas Venus and Mercury, as they do not change the kind of harmony, did not also require perfect harmonies between both pairs, both of the motions at aphelion and of those at perihelion. But instead of a perfect harmony of the motions at aphelion, inasmuch as it had already been leavened, there succeeded a perfect harmony of the converging motions. So, just as Venus, the superior among the inferior planets, has as its motions’ own pro­ portion the smallest of all, by XXVIII, but Mercury, the inferior of the inferior planets, has been allotted as its own proportion the greatest of all, by XXX; so also Venus’ own proportion was of all the planets’ own motions the most imperfect, or the most remote from harmonies, but Mercury’s own proportion was of all the planets’ own proportions the most perfect, that is absolute har­ mony, without leaven; and so in the end the patterns were opposite on all sides. For thus has He who is before ages and to all ages has embellished the mighty works of His wisdom: nothing is redundant, nothing is deficient, and there is no place for any criticism. How desirable are His works, and so forth, all balanced one against another, and none lacks its opposite;’^’® of every one He has established {He has confirmed with the best arguments) the goodness {their furnishing and comeliness) and who shall be sated with seeing their glory? XLVL Axiom The placing o f the solid figures among the planetary spheres, if it is unrestricted, and not prevented by the necessities of preceding causes, ought in perfection to follow the analogy of the geometrical inscrip­ tions and circumscriptions, and therefore the terms of the proportion of the inscribed spheres to the circumscribed spheres.

For nothing is more fitting than that the physical inscription exactly rep­ resents the geometrical, as a printed work does its type. XLVIL Proposition If the inscription of the figures among the planets was unrestricted, the tetrahedron ought to have touched the sphere of the perihelion of Jupiter above exactly at its vertices, and below the sphere of the aphelion of Mars exactly at the centers of its faces. However, the cube and the octahedron, resting at their vertices each on the sphere of the perihelion of its own planet, ought to have penetrated at the centers of their faces the sphere of its interior planet, in such a way that those centers are situated between the spheres of its aphelion and perihelion. On the other hand, the dodecahedron and icosahedron, which at their vertices make contact with the spheres of perihelion of their planets on the outside, clearly ought not to have touched with the centers of their faces the spheres of aphelion of their interior planets. Last, the dodecahedric hedgehog, which stands with its vertices on the sphere of perihelion of Mars, ought to have come very close to the sphere of aphelion of Venus with the midpoints of its inverted edges,*’’” which separate the two spikes in each case. For the tetrahedron is the middle one of the primary figures, both by its origin and by its position in the world. It ought therefore, if there was no im­ pediment, to have moved apart both the regions of Jupiter and Mars equally. As the cube was upwards and further out with respect to the latter, the dodeca­ hedron downwards and further in, then it was proper for inscription within them to bring about opposite effects, between which the tetrahedron held the mean, and for one of the figures to go beyond the inscription and the other to fall short of it, that is, for one to penetrate the interior sphere to a certain extent, and the other not to reach it. And because the octahedron is akin to the cube, having an equal proportion between its spheres, but the icosahedron to the dodeca­ hedron, therefore if the cube has any element of perfection in its inscription, the octahedron ought to have had the same; and if the dodecahedron has any, so ought the icosahedron. The position of the octahedron is also very similar to the position of the cube, and that of the icosahedron to the position of the dodecahedron, because as the cube holds one boundary towards the outside, so the octahedron holds the other extreme towards the inside parts of the world, whereas the dodecahedron and the icosahedron come between. It is therefore ap­ propriate that their mode of inscription should be similar, in the former case penetrating the interior of the planetary sphere, in the latter falling short of it. However, the hedgehog, which with the tips of its spikes represents an icosahedron, with their bases a dodecahedron, ought to have filled, embraced, or arranged both regions also, that between Mars and the Earth, which are attributed to the dodecahedron, and also that between the Earth and Venus, which are attributed to the icosahedron. However, which of the opposites is ap­ propriate to which alliance, the previous axiom makes clear. For the tetrahedron, having an expressible inscribed sphere, has been allotted the place in the middle among the primaries, attended on both sides by the figures of the incommen­ surable spheres, of which the outer is the cube, the interior the dodecahedron, by Chapter I of this Book. Now this geometrical property, the expressibility of the inscribed sphere, represents in Nature the perfect inscription of the plane­ tary sphere. Therefore, the cube and its alliedfigure have inscribed spheres which are expressible only as roots, that is only in the square. Therefore, they ought to represent semiperfect inscription, in which although the actual extremity of the planetary sphere is not touched by the centers of the faces of the figure, yet at least something inside it is, namely the mean between the spheres of aphelion and perihelion, if that is possible by other arguments. On the other hand, the dodecahedron and its ally have inscribed spheres which are definitely inexpres­ sible both in the length of the semidiameter and in the square. Therefore, they ought to represent inscription which is definitely imperfect, and touches nothing whatever of the planetary sphere, that is, falling short, and definitely not making contact up to the sphere of aphelion of the planet with the centers of its faces. Although the hedgehog is akin to the dodecahedron and its ally, yet it has some similarity to the tetrahedron. For the semidiameter of the sphere inscribed in its inverted edges^^’^ is indeed incommensurable with the semidiameter of the circumscribed sphere, but instead it is commensurable in length with the dis­ tance between neighboring pairs of vertices.^^^ Thus the perfection of commen- surability of the radii is almost as great as in the tetrahedron, but in the other respect its imperfection is as great as in the dodecahedron and its ally. It is fitting, therefore, that the inscription agrees with that physically, and is neither definitely tetrahedric, nor definitely dodecahedric, but of an intermediate kind. Hence because the tetrahedron ought to have extended with its faces to the outer surface of the sphere,^^^ but the dodecahedron should have failed to reach it by a certain distance, as it is, this spiked figure stands with its inverted edges be­ tween the icosahedron’s space and the outer surface of the inscribed sphere, very nearly reaching the latter’s outer surface—if nevertheless even that figure should have been received into the fellowship of the other five, and its laws could have been tolerated by their existing laws. Yet what am I saying, ‘‘could have been tolerated’”?— which they could not do without. For if inscription which was lax and not touching agreed with the dodecahedron, what else could restrain that un­ limited laxity within the bounds of a definite amount but this subsidiary figure, akin to the dodecahedron and icosahedron, with its inscription very nearly touch­ ing, andfalling short (if however it does fall short) no more than the tetrahedron is in excess and penetrates? We shall now discuss that amount in what follows.

This reason drawn from the association of the hedgehog with the two figures which are akin to it (that is to say, for the determination of the proportion of the spheres of Mars and Venus, which they had left indefinite) is rendered very probable by the fact that the semi­ diameter of the sphere of the Earth, 1000, is found to be very close to the mean position proportionally between the sphere of the peri­ helion of Mars and that of the aphelion of Venus, as if the space which the hedgehog claims for the figures which are akin to it were divided proportionally between them, in virtue of their similarity. XLVIII. Proposition There was not pure liberty for the inscription of the regular solid figures between the planetary spheres; for it was impeded over small details by the harmonies set up between the extreme motions.’*’-^ For by Axioms I and II the proportion of the spheres of each figure ought not to have been expressed immediately by itself, but through it there had first to be sought, and fitted to the extreme motions, harmonies which were very closely related to the actual proportions of the spheres. Next, so that by Axioms XVIII and XX there could be universal harmonies of the two kinds, it was necessary to add some leaven to the greater harmonies of individual pairs, by the a posteriori arguments. Therefore, to make it possible for these to stand and to depend on their own arguments, there were required intervals a little at variance with those drawn from perfect inscription of the figures between the spheres, by the laws of the motions expounded in Chapter III. To prove that, and to show how much departure there is in individual figures through the harmonies confirmed by appropriate arguments, come, let us derive from them the distances of the planets from the Sun, by a new form of calculation not before attempted by anyone.^

Now there will be three heads of this enquiry: First, from the two extreme motions of each planet will be sought the distances, similarly extreme, of it and the Sun, and from them the semidiameter of the sphere measured by the extreme distances as appropriate to each planet. Second, from the same extreme motions, measured in the same units in all cases, will be sought the mean motions, and their proportion. Third, from the proportion of the mean motions now revealed the proportion of the spheres or of the mean distances will be investigated, and along with it that of the extremes also; and that will be compared with the pro­ portions taken from the figures. A5 far as thefirst is concerned: it must be recalledfrom Chapter III, number VI, that the proportion of the extreme motions is the square of the inverse proportion of the corresponding distances from the Sun. Therefore, since the proportion of squares is the square of the proportion of their sides, then the numbers by which the extreme motions are expressed will be considered as square, andfinding their roots will give the extreme distances. It is easy to take the arithmetic mean of those for finding the semidiameter of the sphere and the eccentricity. There­ fore, the harmonies so far confirmed have prescribed as follows:^

For the second of the proposed heads we again have need of Number XII of Chapter III, where it was shown that the number which expresses the mean motion in the proportion of the extremes is less than their arithmetic mean, and less also than their geometric mean, by half the dijference between the two means. And because we are seeking for all the mean motions measured in the same units, as all the proportions which have so far been confirmed between pairs of motions, and also all the planets’ own motions, are set out in the com­ mon measure of their lowest common fcLctor, then we should seek for the arith­ metic mean by halving the difference between the extreme motions in each case, and the geometric by multiplying one extreme by the other and extracting the root of the product. Then by subtraction of half the difference of the means from the geometrical mean the numerical value of the mean motion will be established, in each of the extreme motions’ own measure, which is easily transformed into the common measure by the rule of proportions.

Then from the prescribed harmonies has been found the proportion of the mean daily motions, that is to say the proportion between the values of each, in degrees and minutes, and so on, compared with each other; and it is easy to examine how closely it comes to astronomy.^^’^ The third head of what has been proposed needs Number VIII of Chapter III. For having found the proportion of the mean daily motions in individual planets, we can find the proportion of their spheres as well. For the proportion of the mean motions is the sesquialterate of the inverse proportion of the spheres. But the proportion of cubic numbers is also the sesquialterate of the proportion of the squares belonging to the same roots in the table of Clavius, which he appended to his Practical Geometry.

Hence, if our values for the mean motions are sought (curtailed, where necessary, to an equal number of figures) among the cubes in that table, they will show to their left under the heading of the squares the values for the proportion of the spheres.

Then the eccentricities ascribed above to the individual planets, in the appropriate measure of the semidiameters of each, will easily be transposed by the rule of proportions into the measure which is common to them all; and hence those values added to the semidiameters of the spheres, and subtracted from them, will establish the extreme distances of the individual planets from the Sun. However, we shall give for the semi­ diameter of the sphere of the Earth the round measure 100,000, as is customary in astronomy, with the intention that this number, either squared or cubed, always consists of pure zeros. Thus we shall also bring the mean motion of the Earth out to the number 1,000,000,000, working by the rule of proportions to make the value for the mean motion of each planet to the value for the mean motion of the Earth as 1,000,000,000 is to the new measure. So the matter can be carried through with only five cube roots, by comparing them individually with the single value for the Earth.\

In the last column, therefore, may be seen what numbers emerge to express the converging distances of the pairs of planets; and all of them approach very closely the distances which 1 have found from the observations of Brahe.^^’^ In the case of Mercury alone there is a tiny dijference. For astronomy seems to give it the distances 470, 388, and 306, which are all shorter. It seems reasonable to guess that the reason for the discrepancy is either in the paucity of obser­ vations or in the size of the eccentricity. See Chapter III. But I am hurrying on to the end of the calculation. For it is now easy to compare the proportion of the spheres of the figures with the proportion of the converging intervals.

That is, the cubic faces descend a little below the mean distance ofJupiter; the octahedric faces do not descend absolutely to the actual mean distance of Mercury; the tetrahedricfaces descend a little below the greatest distance ofMars; the edges of the hedgehog do not descend absolutely to the actual greatest distance of Venus; Imt thefaces of the dodecahedronfallfar short of the distance at aphelion of the Earth; thefaces of the icosahedron also fall far, and almost proportionately short of the distance at aphelion of Venus; and last, the square of the octahedron definitely does not fit at all.™ But there is no harm in that; for what has a plane figure to do with solids? You see therefore that if the distances of the plan­ ets are Educed from the harmonic proportions of the motions, which have been demonstrated so far, the former must necessarily turn out to be of the size which the latter permit, but not of the size which the laws of free inscribing would require, as prescribed in Proposition XLV. For this “geometrical cosmos” of per­ fect inscription was no longer close to that other “possible harmonic cosmos,” to use Galen’s words taken from the frontispiece of this Book V. That much had to be demonstrated from actual numerical calculation for the elucidation of the proposition in question. I do not disguise the fact that if I were to increase the harmony of the converging motions of Venus and Mercury by Venus’ motions’ own proportion, and in consequence to diminish Mercury’s own har­ mony by the same amount, then I should obtain by that procedure as the distances of Mercury and the Sun 469, 388, and 307, which are very precisely those indicated by astronomy. But, first, I cannot defend that diminution by harmonic arguments, for the motion of Mercury at aphelion will not agree with any musical scale; and in planets which are opposed in the world a complete pattern of opposition is not main­ tained in all properties. Second, the mean daily motion of Mercury is made too great, and so the periodic time of Mercury, which is the most certain in the whole of astronomy, is shortened too much.

Thus I stand by the harmonic commonwealth of the motions which has been assumed here and confirmed by the whole of Chapter IX. Neverthe- less, by this example I challenge as many of you as will chance to read this book and are imbued with the disciplines of mathematics and knowledge of the highest philosophy: come, be vigorous and either tear up one of the harmonies which have been everywhere related to each other, change it for another one, and test whether you will come as close to the astronomy laid down in Chapter IV; or else argue rationally whether you can build something better and more appro­ priate on to the heavenly motions, and overthrow either partly or wholly the arrangement which I have applied. Whatever contributes to the glory of Our Founder and Lord is equally to be permitted to you throughout this my book; and I have assumed that I myself am per­ mitted up to this hour freely to change anything which I could dis­ cover which was incorrectly conceived in the preceding days if my at­ tention nodded or my enthusiasm was hasty.

IL. Envoi

See Book IV, page 367. See Book IV. It was right that in the genesis of the distances the solid figures should give way to the harmonic arguments, and the greater harmonies of the pairs to the universal harmonies of all, to the extent to which the latter was necessary. By a splendid coincidence we have come to the square of the sevenfold, 49, so that like a kind of Sabbath it may succeed the foregoing six solid octaves of utterances on the structure of the heavens. Also I justly made into an envoi what could have been put earlier among the Axioms; for God also when He had now completed the task of Creation “saw all that He had done, and behold! it was very good.”*’ * There are two limbs of the envoi. The first, on the harmonies in general, is demonstrated as follows. For where there is a choice between different things which do not allow each other to have sole possession, in that case the higher are to be preferred, and the lower must give way, as far as is necessary, as the actual word “cosmos,” which means “decoration,” seems to bear witness. But har­ monic decoration is as far above the simple geometrical as life is above the body, or form above matter. For just as life completes the bodies of animate beings, because they were born to lead it, which follows from the archetype of the world, which is the actual divine essence, so motion measures out the regions allotted to the planets, to each its own, as a region has been assigned to a star so that it could move. But the five solid figures, in virtue of the word itself, relate to the spaces of the regions, and to the number of them and of the bodies; but the harmonies to the motions. Again, as matter is diffuse and unlimited in itself, but form is limited, unified, and itself the boundary of matter; so also the number of the

geometrical proportions is infinite, the harmonies are few. For although even among the geometrical proportions there are definite degrees of limitation and shape and restriction, and not more than three can be formed by the ascription of spheres to regular figures, yet even to these is attached an accidental property in common with all the rest, that is the presupposition of an infinite possible division of quantities. Those of which the terms are incommensurable with each other also involve that in practice in a way. But the harmonic proportions are all expressible, and the terms of all of them are commensurable. Also, they have been taken from a definite and limited class of plane figures. Now infinite di­ visibility signifies matter, but commensurability or expressibility of term sig­ nifies form. Therefore, as matter strives for form,^’^^ as a rough stone, of the cor­ rect size indeed, strives for the Idea of the human form, so the geometrical proportions in the figures strive for harmonies; not so as to build and shape them, but because this matter fits more neatly to this form, this size of rock to this effigy, and also this proportion in a figure to this harmony, and therefore so that they may be built and shaped further, the matter in fact by its own form, the rock by the chisel into the appearance of an animate being, but the pro­ portion of the spheres of the figures by its own, that is, by close andfitting harmony. What has been said up to this point will be made clearer by the story of my discoveries. When, twenty four years ago, I had engaged in this study, I first enquired whether the individual circles of the planets were separated by equal distances from each other (for in Copernicus the spheres are separated, and do not mutually touch each other). Of course I acknowledged nothing as more splendid than the relationship of equality. However, it lacks a head and a tail, for this material equality provided no definite number for the moving bodies, no definite size for the distances. Therefore, I thought about the similarity of the distances to the spheres, that is about their proportion. But the same com­ plaint followed, for although in fact distances between the spheres emerged which were certainly unequal, yet they were not unequally unequal, as Copernicus would have it, nor was the size of the proportion nor the number of the spheres obtained. I moved on to the regular plane figures: they produced the distances in accordance with the ascription of their circles, but still in no definite number. I came to the five solids:^’^’^ in this case they revealed both the number of the bodies and nearly the right size for the intervals—so much so that I appealed over the remaining discrepancy to the state of accuracy of astronomy. The ac­ curacy of astronomy has been perfected in the course of twenty years; and see! there was still a discrepancy between the distances and the solid figures, and the reasons for the very unequal distribution of the eccentricities among the planets were not yet apparent. Of course in this house of the cosmos I was look- ingfor nothing but the stones — of more elegant form, but of a form appropriate to stones — not knowing that the Architect had shaped them into a fully detailed effigy of a living body. So little by little, especially in these last three years, I came to the harmonies, deserting the solid figures over fine details, both because the former were based on the parts of the form which the ultimate hand had impressed, but the figures from matter, which in the cosmos is the number of the bodies and the bare breadth of their spaces, and also because the former yielded the eccentricities, which the latter did not even promise. That is to say, the former provided the nose and eyes and other limbs of the statue, for which these latter had only prescribed the external quantity of bare mass. Hence just as the bodies of animate beings have not been made, and a mass of stone is not usually made, according to the pure norm of some geometrical figure, but something is removedfrom the external round shape, however elegant (though the correct amount of bulk remains) so that the body can take on the organs necessary to life, and the stone the likeness of an animate being, similarly also the proportion which the solid figures were to prescribe for the planetary spheres, as lower, and having regard only to a body of a particular size and to matter, must have given way to the harmonies, as much as was necessary for the former to be able to stand close and to adorn the motions of the globes. The other limb of the envoi, on universal harmonies, has a proof which is akin to the previous one. For that which chiefly makes the world perfect ought preferably to have the supreme hand in perfection; but in return it is the one from which something must be removed (if something must be removed from one or the other), because in this case it is in the secondary position. But it is the universal harmony of all which chiefly makes the world perfect, rather than the individual twinnings of neighboring pairs. For harmony is a certain relationship of unity: therefore they are united if they are all at one at the same time rather than if each pair separately agree in pairs of harmonies. So that in a conflict between the two, one or other of pairs of harmonies of the pairs of planets must have yielded so that the universal harmonies of all could stand. However, the greater harmonies of the diverging motions must have yielded, rather than the lesser harmonies of the converging motions. For if the former are diverging, therefore they are having regard not to the planets of the pair specified, but to other neighboring ones; and if the latter are converging, there­ fore the motion of one planet is tending towards the motion of the other, in ex­ change. So in the pair of Jupiter and Mars, the motion of the former at aphelion tends towards Saturn, and of the latter at perihelion towards the Earth; but the motion of the former at perihelion tends towards Mars, and that of the latter at aphelion towardsJupiter. Therefore, the harmony of the latter motions belongs more particularly to Jupiter and Mars; and the harmony of the former, diverg­ ing, motions is in a way more alien from Jupiter and Mars. Now the relationship of union which binds together pairs and neighboring pairs was less damaged if the harmony which is more alien and more remote from them were leavened, than if the harmony which belongs particularly to them were, that is the har­ mony which exists between the more closely neighboring motions of neighboring planets. However, this leavening was not very great. For the relationship was ound, by which both the universal harmonies of all the planets would stand, and those of two distinct kinds and with a certain latitude in tuning, which would equal at least a comma, and also the two individual harmonies of neigh­ boring pairs of planets would be protected: in fact perfect harmonies of the con­ verging motions in four pairs; similarly perfect harmonies of the motions at aphelion in one, in perihelion in two; but of the diverging intervals in four pairs, within a discrepancy of one diesis, the smallest interval, by which the human voice in figured melody is almost perpetually out of tune. However, in the single case of Jupiter and Mars the discrepancy is between a diesis and a semitone.

Therefore, this mutual concession on all sides holds exceedingly good.

So far, therefore, we have “delivered our envoi” on the work of God the Creator. It now remains for me, at the very last, to take my eyes and hands away from the table of proofs, lift them up to the heaven, and pray devoutly and humbly to the Father of light: O Thou who by the light of Nature movest in us the desire for the light of grace, so that by it thou mayest bring us over into the light of glory; I thank Thee, Creator Lord, because Thou hast made me delight in Thy handiwork, and I have exulted in the works of Thy hands. Lo, I have now brought to completion the work of my covenant, using all the power of the talents which Thou hast given me. I have made manifest the glory of Thy works to men who will read these demon­ strations, as much as the deficiency of my mind has been able to grasp of its infinity. My intellect has been ready for the most accurate details of philosophy. If anything unworthy of Thy intentions has been put forward by me, miserable worm that I am, born and nourished in a slough of sins, which thou wouldst wish men to know, inspire me also to set it right; if I have been enticed into temerity by the wonderful splendor of Thy works, or if I have loved my own glory among men, while advancing in work destined for Thy glory, mildly and mercifully pardon it; and last, be gracious and deign to bring about that these my demonstrations may be conducive to Thy glory and to the salva­ tion of souls, and may in no way obstruct it