The Relationship of the Harmonic

8. Proposition: The harmony of a diatessaron can have no place among the converging motions of pairs of planets, unless the combined proportions of their own extreme motions were more than a diapente.

For let the proportion be 3:4 between the convergent motions.

Let there be, first, no eccentricity, and no proportion of their own between motions of in dividual planets, but the same motions both convergent and mean.

Then it follows that the corresponding distances, which on this hypothesis will be the radii of the spheres, constitute the square of the cube root of the proportion, that is to say 4480:5424, by Chapter III. But this proportion is already less than the proportion of the spheres of any of the regular figures.

Thus the whole interior sphere would be cut by the faces of the regular figure inscribed in any exterior sphere. This, however, is contrary to Axiom II.

Second, let the product of their oxim proportions between extreme motions be some definite amount; and let the proportion of the converging motions be 3:4 or 75-100, but the proportion of the corresponding distances be 1000:795, since no regular figure produces a smaller proportion between its spheres.

And because the former proportion, that of the motions, inverted exceeds the latter, that of the distances, by a factor of 750:795, therefore let this factor he also divided into the proportion 1000:795 in accordance with the principle of Chapter 3.

The quotient is 9434-7950, the square root of the proportion of the spheres.

Then the square of this, that is 8901:6320, or 10000:7100, is the proportion of the spheres.

Divide this by the proportion of the converging distances, 1000:795. The quotient will be 7100:7950, about a major tone. This should be as a minimum the product of the two proportions which the mean distances have to the converging distances on either side, for a diatessaron to be possible between the converging motions.

Therefore, the product of the proportions produced by the divergent extreme distances to the convergent extreme distances is about the square of that, that is 2 tones. The product for their own motions is again the square of that, that is four tones, which is more than a diapente. Therefore, if for two neighboring planets the product of their own mo­ tions is less than a diapente, a diatessaron between their converging motions will not be possible.

8. Proposition: Saturn and Jupiter should have had the harmonies 1:2 and T-3, that is a diapason and a diapente above the diapason.

For they are themselves the first and highest of the planets, and have got the first of the figures, the cube, by Chapter I of this Book; and these harmonies are the first in the order of nature and are the heads of the first families among the figures, the Bisecting or Tetragonic, and the Trigonic, by what has been said in Book I.

However, that which is the head, the diapason, 1:2 is very slightly greater than the semitriple^^’^ of the proportion of the spheres of the cube.

Thus it is appropriate for it to become the lesser proportion of the motions of the cubic planets, by Chapter III, Number 13; and in consequence, 1:3 serves as the greater proportion. However, the same conclusion is also reached as follows For if some harmony is to some proportion found between the spheres of the figures as the proportion of the apparent motions, as seen from the Sun, to the proportion of the mean distances, such a harmony will deservedly be attributed to the motions.

But it is natural that the proportion of the diverging motions should be much greater than the sesquialterate proportion of the spheres, by the end of Chapter III.

That is, it approaches the square of the proportion of the spheres; and 1:3 is also the square of the proportion of the cubic spheres, namely, as we say the semitriple. Then the threefold harmony ought to belong to the divergent motions of Saturn andfupiter. See the numerous other affinities of these proportions with the cube above in Chapter 11.

9. Proposition: Saturn’s and Jupiter’s extreme motions’ own proportions combined should have come to 2:3, about a diapente.

That follows from the foregoing proposition; for if the motion of fupiter at perihelion is triple that of Saturn at aphelion, and on the other hand the motion of fupiter at aphelion is double that of Saturn at perihelion, then on dividing 1:2 into L3, the quotient is 2:3.

X. Axiom

When there is a free choice among the others, the superior planet should have as its own proportion in its motions that which is prior by nature, or that which is of the more distinguished kind, or even that which is greater.

XL Proposition

The proportion of the motion of Saturn at aphelion to that at perihelion ought to have been 4:5, a major third, but that of Jupiter’s motions 5:6, a minor third.

For because in combination they hold 2-3, but this is not divided harmonically except into 4:5 and 5'6; therefore God the Governor-General has divided the harmony 2:3 harmonically, by Axiom I, and has given the harmonic part of it, which is the greater, and of the more distinguished hard kind, in fact masculine, to the greater and higher planet Saturn, and the lesser, 5 ‘6, to the lower, Jupiter, by X.

XII. Proposition

Venus and Mercury ought to have had the major harmony 1:4, the double diapason.

For just as the cube is the first figure of the primaries, so the octahedron is the first of the secondaries, by Chapter I of this Book.

Just as the cube, considered geometrically, is the outer, and the octahedron the inner, that is to say the latter may be inscribed in the former, so also in the world Saturn indeed and Jupiter are the beginning of the higher and outer planets, or on the outside, whereas Mercury and Venus are the beginning of the inner planets, or on the inside; and interposed between their courses is the octahedron: see Chapter 111.

Of the harmonies therefore Venus and Mercury ought also to have one which is primary and akin to the octahedron. Furthermore, among the harmonies after 1:2 and 1-3 there follows in the natural order 1:4, and it is akin to the cubic 1:2 because it has arisen from the same group offigures, that is the tetragonic, and is commensurable with it, that is the square of it; whereas the octahedron is akin to the cube and commensurable with it. Also 1:4 is akin to the octahedron, independently, on account of the quxitemary number which is within it, and in fact hidden in the octahedron the quadrangular figure, the proportion of the spheres of which is stated as semiduple.

Therefore, the harmony of its proportion is 1:4, which is by continuous multiplication in the proportion of squares, that is to say the fourth power of the semiduple: see Chapter II. Then Venus and Mercury ought to have had 1:4. And because 1:2 in the cube is the lesser harmony between the two, since the outermost location has fallen to it, there will be in the octahedron this proportion 1:4 which is now the greater harmony between the two, as it w the one to which the innermost position has fallen.

But the following is also a reason why 1:4 has here been given as the greater, not the lesser. For since the proportion of the spheres of the octahedron is semitriple, assuming that the inscription of the octahedron between the plan­ ets is perfect (though it is not perfect, but penetrates Mercury to a certain extent, which is in our favor), therefore the proportion of the converging motions must

be smaller than the sesquialterate of that semitriple proportion. But even 1:3 is plainly the square of the semitriple, and thus greater than the correct pro­ portion by the amount by which 1:4, which of course is greater than 1:3, is greater than the correct amount. Then not even the square root of 1:4 is tolerated between the converging motions.^’^’^ Therefore 1:4 cannot be the smaller octa- hedric proportion; therefore it will be the greater. Further, 1:4 is akin to the octahedric square, the proportion of the spheres of which is semiduple, in the same way as 1:3 is akin to the cube, as the proportion of its spheres is semitriple. For just as 1:3 is a power of the semitriple, that is to say its square, so also this 1:4 is a power of the semiduple, that is to say the square of its square, that is its fourth power. Hence if l’-3 should have been the greater harmony of the cube, by VIll, therefore 1:4 ought also to be the greater harmony of its octahedron.