The measures are expressed as multiples of the minor element.

The measures are expressed as parts of the major whole.

The sum of the 2 smallest of 4 consecutive measures equals the largest.

Minor element + major element = major piece

Minor element : major element

3:4

The difference between the 2 largest of 6 consecutive measures equals the smallest. Thus the difference between the major and minor whole lies two places before the minor part.

Major whole – minor whole = minor piece

The difference between the 2 largest of six consecutive measures equals the smallest. Thus the difference between the major and minor part lies 2 places before the minor piece.

Major part – minor part = minor element

We start with choosing 100 for the minor element of system I. It is the unit.

From it, we shall first derive the values for the minor elements of system II and III.

As the major whole of system II the minor element of system I is 7 times the minor element of system II plus the minor elements of system III.

7x the minor element of II + minor element of III = minor element of I

(7 x …) + minor element III = 100

(100 – minor element III) : 7 = 14

As the major whole of system III the minor element of system II is 7 times the minor element of system III plus the minor elements of system IV, the latter of which is negligible.

7x the minor element of III = minor element of II

7 x … = 14

14 : 7 = 2

Thus the minor elements of system I, II and III therefore have the values 100, 14 and 2.

100 = 7 x 14 +2

The numerical values of the measures of system III are now easy to determine.

That of the major whole is 14, that of the minor element 2.

The major whole is 4 times the minor piece.

4 x … = 14

14 : 4 = 3,5

The values of the remaining 5 measures can easily be determined with the help of two rules, namely that the sum of the two smallest of 4 consecutive measures equals the largest, and that the difference between the two largest of 6 consecutive measures equals the smallest. 

The difference of the major and minor wholes is 3,5.

14 – … = 3,5

14 – 3,5 = 10,5

The values of the remaining 5 measures can easily be determined with the help of two rules, namely that the sum of the two smallest of 4 consecutive measures equals the largest, and that the difference between the two largest of 6 consecutive measures equals the smallest. 

The difference between the major and minor part is 2 and their sum is 14.

14 – 2 = 12

12 : 2 = 6 (minor part)

6 + …  = 14

14 – 6 = 8 (major part)

The major piece, added to the minor part, makes up the minor whole.

6 + … = 10,5

10,5 – 6 = 4,5 (major piece)

Finally the major element, added to the minor piece, makes up the minor part.

3,5 + … = 6

6 – 3,5 = 2,5

The values for measures of system II can be established in the same way, but this time taking the small quanta from system III into account.

The major whole has the value 100 and the minor element 14.

The major whole is 4 times the minor piece, plus the minor element from system III.

(4 x …) + 2 = 100

(100 – 2) : 4 = 24,5

The sum of the two smallest of 4 consecutive measures equals the largest, and the difference between the 2 largest of 6 consecutive measures equals the smallest. 

The difference between major and minor whole is 24,5.

100 – … = 24,5

100 – 24,5 = 75,5

The sum of the two smallest of 4 consecutive measures equals the largest, and the difference between the 2 largest of 6 consecutive measures equals the smallest. 

The difference between the major and minor part is 14 and their sum is 100.

100 – 14 = 86

86 : 2 = 43 (minor part)

43 + …  = 100

100 – 43 = 57 (major part)

The major piece, added to the minor part, makes up the minor whole.

43 + … = 75,5

75,5 – 43 = 32,5

Finally, the major element, added to the minor piece, makes up the minor part.

24,5 + … = 43

43 – 24,5 = 18,5

The numerical values for the measures of system I can now be determined, taking into account the small quanta from systems II and III.

The major whole is 7 times the minor element, whose value is 100, plus the minor elements of systems II and III.

(7 x 100) + 14 + 2 = 716

The major whole is 4 times the minor piece, plus the minor element of system II.

(4 x …) + 14 = 716

(716 – 14) : 4 = 175,5

The sum of the two smallest of 4 consecutive measures equals the largest, and the difference between the 2 largest of 6 consecutive measures equals the smallest. 

The difference between the major and minor whole is 175,5.

716 – … = 175,5

716 – 175,5 = 540,5

The sum of the two smallest of 4 consecutive measures equals the largest, and the difference between the 2 largest of 6 consecutive measures equals the smallest. 

The difference between the major and minor part is 100 and their sum is 716.

716 – 100 = 616

616 : 2 = 308 (minor part)

308 + … = 716

716 – 308 = 408 (major part)

The major piece, added to the minor part, makes up the minor whole.

308 + … = 540,5

540,5 – 308 = 232,5

Finally the major element, added to the minor piece, makes up the minor part.

175,5 + … = 308

308 – 175,5 = 132,5

Or the major element is a minor element smaller than the major piece.

232,5 – 100 = 132,5

The ratios between the measures of a derived system are no different from those for an authentic system. The derived measures also form a geometric progression, with a ratio that, as with the authentic measures, is equal to the ground ratio.

Every derived measures has fixed ratio to the authentic measure from which it is derived, and this ratio is easily expressed arithmetically.

The difference between the major whole and its derivative equals the minor element, which is also the major whole of the smaller system.

Major whole – derivative equals = minor element

716 – … = 100

716 – 100 = 616

The major whole therefore exceeds its derivative by its corresponding measure from the smaller system, and this holds true for all measures.

Since  all measures of one system are seven times their corresponding measures from the system below, every derived measure is 6/7 of its authentic measure.

We also know that every derived measure is exactly the double of a lower authentic measure, for the double of the smallest of four consecutive measures always equals the derivative of the largest.

100 x 2 = 200

200 is the derivative of 232,5

Since  all measures of one system are seven times their corresponding measures from the system below, every derived measure is 6/7 of its authentic measure.

Numerical values for the measures of the two consecutive derived systems, Ia and IIa, can now be established, taking into account the small quanta from the lower system. For this we only need to subtract from each authentic measure its corresponding measure from the system below.

major element I – major element II = the derivative of major element Ia

132,5 – 18,5 = 114

Since  all measures of one system are seven times their corresponding measures from the system below, every derived measure is 6/7 of its authentic measure.

Building up the derived system Ia.

Derived system Ia

Derived system IIa

Derived system IIIa

The value of the minor whole can be seen as five times the minor element plus the the major piece from system II and the major part of system III

(5 x minor element I) + major piece II + major part III = minor whole I

(5 x 100) + 32,5 + 8 = 540,5

The value of the major part can be recognized as respectively four times the minor element, plus the major part from system III

(4 x minor element) + major part III = major part I

(4 x 100) + 8 = 408

The value of the minor part can be recognized as respectively three times the minor element, plus the major part from system III

(3 x minor element) + major part III = major part I

(3 x 100) + 8 = 408