1, 2, 3, 5, 6, 7, 8, 9, 10, 11

Multiple, that is double, triple, quadruple.

Superparticular, that is sesquialtera, sesquitertia, and sesquiquarta.

Superpartient, that is superbipartiens tertias and supertripartiens quartas, superquadripartiens quintas.

Multiple-superparticular, that is dupla sesquialtera, tripla sesquialtera.

Multiple-superpartient, that is dupla superbipartiens tertias, that is, 8 receives 3 two times and 2/3.

Where there is proportionality, there of necessity is proportion; but not where there is proportion is proportionality found.

Proportionality has 3 principal members, that is, harmonic, arithmetic, and geometric.

[Editorial note: Leonardo here is completing Pacioli's definitions, according to which the difference between the first extreme (³ and the mean (⁴ is half the difference between the mean and the other extreme (³ and half of the second (⁶. The modern formula for this is that the harmonic mean between the two extremes (6 and 3) is obtained by dividing their product (¹⁸ by their half-sum (4 1/²: the result is 4.]

2, 3 4 6, 1 2, 2

Harmonic: in which, the extremes being double, the difference too is double, that is, the difference which is from the extremes to the [mean at] 1/2.

2, 1 3, 2, 4 6

Harmonic proportionality is a likeness of the proportions of the extremes among themselves and the proportions that are among the differences of the said extremes -- not of the extremes among themselves, but the proportions that are among the differences of the said extremes. Thus let the extremes be 6 and 3, and let the middle term be 4. Now, from 6 to 3 is a double proportion. Now it is needful that the proportion which is from the [mean at] 1/2 to the two extremes be like the aforesaid: that is, this first being double, that the difference from 4 to 6 be double to the difference from 3 to 4; which is here found to be so, because from 4 to 6 is 2 of difference, and from 4 to 3 is one of difference, so that 2 is double to one. And on this I extend no further, but rather leave it.

Arithmetic proportionality is the equality of the differences of the extremes compared together, just as are the differences that run from 6 to 4 and from 3 to one, where the excess of the extremes in each is 2. And thus we have made, of 2 equal proportions, the proportionality of arithmetic equiparance. Geometric proportionality is a likeness of proportions of quantities integrally compared among themselves, as 6 to 3, so is found 8 to 4, which, taking the quantities entirely, make 2 doubles, that is, the proportionality of equiparance. And whoever should compare the differences, they would be in arithmetic proportion, and it would be a double.