[Margin: Aliquot and aliquant part.]

[I.] A Part, according to Euclid (bk. 5), is a magnitude which measures a greater magnitude as [a part of] the whole; and if it measure [it] perfectly — so that, replicated some number of times, it equals its whole — it is called an aliquot part, as 2 with respect to 8. But if, when replicated, it exceeds the whole or falls short — as 2 with respect to 9 — it is called aliquant; for the binary [2], taken four times, falls short of the nine, but taken five times exceeds it.

[Margin: What are Ratio and Proportion?]

A Ratio is the mutual relation, according to quantity, of two magnitudes of the same kind. Proportion, however — which the Greeks call ἀναλογία [analogia], and some Latins “Proportionality” — [is the likeness of ratios]. [A proportion] is such, for example, between [a line] of six palms [and one of four]; for it can be expressed by numbers, so that such a proportion is between two commensurable [magnitudes], or [magnitudes] having a common aliquot part which can exactly measure each of them — of which kind [of measure] is the unit of an integer number; and therefore a rational proportion cannot [exist] except between numbers, or [magnitudes] reducible to numbers, as between 3 and 6.

[Margin: Rational and irrational proportion.]

But an irrational proportion is that which cannot be expressed by numbers, and this is found only between continuous quantities — of which kind of proportion, or incommensurability, is that between the Diameter and the side of any Square, as Euclid demonstrates (bk. 10, toward the end); for no aliquot part can be assigned between those lines, nor can their proportion be expressed by any numbers.

[Margin: Proportion of Equality and of Inequality, of greater or of lesser.]

Secondly, Proportion is divided into the Proportion of Equality (such as between 20 and 20, or between 100 and 100) and the Proportion of Inequality (which is between two unequal quantities, as between 20 and 10). And this is subdivided into the proportion of Greater and of Lesser Inequality — according as a quantity is compared, either greater with lesser as the antecedent with the consequent (e.g. 4 with 2), or lesser with greater (e.g. 2 with 4, putting the lesser in the antecedent and 4 in the consequent).

Thirdly, the Rational Proportion — both of greater and of lesser inequality — is subdivided into five genera, of which the first three contain the simple [ratios], the remaining two the compound. The simple [ratios] of greater inequality are: Multiplex, Superparticular, and Superpartient. The compound are: Multiplex-superparticular and Multiplex-superpartient. To which, if you add the prefix Sub-, you will have just as many genera of rational proportions of lesser inequality: Submultiplex, subsuperparticular, subsuperpartient, submultiplex-superparticular, and submultiplex-superpartient.

[Margin: Multiplex proportion.]

[II.] The Multiplex Proportion is the relation of a greater quantity to a lesser, [the lesser] exactly measuring the greater — of which kind is the proportion between 4 and 2, between 20 and 4, between 100 and 5. And its species are infinite in number: namely double, triple, quadruple, etc., according as the greater quantity contains the lesser twice, thrice, four times, etc.

[Margin: Superparticular proportion.]

The Superparticular Proportion is when the greater quantity contains the lesser once only, and besides one aliquot part of it — whether that part be a half, or a third, or a fourth, and so on infinitely. The species of these proportions are distinguished by the particle Sesqui as their characteristic, with the number of the aliquot part adjoined. And so the Sesquialteral proportion is so called when the greater contains the lesser once and, besides, a half of the lesser — of which kind is the proportion 3 to 2, which in Greek is called ἡμιόλιος [hēmiolios], because ἥμισυ means “half” and ὅλος “whole,” as if you should render it “whole-and-a-half.” But if the greater contains the lesser once and, further, a third, or a fourth, or a fifth, or a hundredth, or a thousandth part of the lesser, it is called a sesquitertian, sesquiquartan, sesquiquintan, sesquicentesimal, sesquimillesimal proportion, and so of the rest — some examples of which it pleases [me] here to subjoin, comparing the upper number with the lower:

[Translator’s note — the figures above are read from a small engraved “Table of Examples of Superparticular Proportions”; a few cells are only partly legible, but every pair in each row reduces to that row’s ratio (3:2, 4:3, 5:4).]

[…continues on p. 505 (PDF 540) with the catchword “Resi-” — the remaining superparticular species and the Superpartient proportions, still within Chapter III.]

(printed p. 505 — Chapter III continued: the complete catalogue of the genera of rational proportion, each with an engraved table of examples. The superparticular table is finished, then the Superpartient proportion is defined and tabulated, followed by the two compound genera — the Multiplex-superparticular and the Multiplex-superpartient — each with its own table of examples.)

[Header: ON THE HARMONIC SYSTEM OF THE WORLD — 505]

[Remainder of the Preceding Table — superparticular proportions]

(Residuum Tabulae Praecedentis — each proportion is shown in its base ratio and four further equivalent example-pairs, comparing the upper number with the lower.)

[Margin: Superpartient.]

[III.] The Superpartient proportion is when the greater quantity contains the lesser quantity once only, and besides several aliquot parts of the lesser which together exceed one [single] aliquot part — excepting unity, otherwise it would turn out superparticular. Such is the proportion between 8 and 5. The species of this proportion are usually distinguished by two characteristics: the one, expressed by the particles bi-, tri-, quadri-, quintu-, sextu-, septu-, octu-, etc., indicates the number of aliquot parts which the greater quantity contains [over and above the whole]; the other denominates the proportion of those very aliquot parts to the lesser quantity which they measure. For example, the proportion of the number 8 to 5 is called Supertripartiens quintas, because 8 contains 5 once, and besides three fifth-parts of that five (which are three units); and so of the rest. Examine some examples in the following table, comparing the upper with the lower number:

(In the Superquadripartiens-quintas row, the third example is printed “49 : 35,” which reduces to 7:5, not 9:5 — an apparent printer slip for 63:35.)

[Sub-heading: Compound Proportions.]

[Margin: Multiplex-superparticular.]

[IV.] The Multiplex-superparticular proportion is when the greater quantity contains the lesser several times — say twice, thrice, or four times, etc. — and besides one aliquot part of the lesser. Of which kind is the dupla-sesquiquarta: for the Novenary [9] contains the quaternary [4] twice, and besides a fourth part of the four; and therefore it is called dupla-superparticular-sesquiquarta — dupla because it contains the lesser twice, sesquiquarta because it contains besides a fourth part of the lesser. The species of this proportion, compounded from the Multiplex and the Superparticular, are therefore infinite in number; for [the one part] is expressed by dupla, tripla, quadrupla, etc., and [the other] by sesqui- with its species (sesquialtera, sesquitertia, sesquiquarta, etc.). These being combined together, there arises the proportion dupla-sesquialtera, dupla-sesquitertia, dupla-sesquiquarta, etc.; or tripla-sesquialtera, tripla-sesquitertia, tripla-sesquiquarta, etc.; and so of the rest. Look at some examples in the following little table, comparing the upper number with the lower:

[Margin: Multiplex-superpartient.]

[V.] The Multiplex-superpartient proportion is when the greater quantity contains the lesser several times, and besides several aliquot parts of the lesser which do not constitute one [single] aliquot part — of which kind is the proportion 11 to 3; for it contains the ternary [3] three times, and besides two thirds of the ternary, which do not make one [single] aliquot part (otherwise the proportion would be multiplex-superparticular). The species of this genus are infinite in number, and take their denomination partly from the species of the Multiplex proportion (namely from dupla, tripla, etc.), partly from the species of the superpartient proportion: so that, the denominations being combined, there arises dupla-superbipartiens-tertias, dupla-supertripartiens-quartas, dupla-superquadripartiens-quintas, etc.; likewise tripla-superbipartiens-tertias, tripla-supertripartiens-quartas, etc. A specimen [is] in the following table, comparing the upper with the lower:

[…continues on p. 506 (PDF 541) with the catchword “VI.” — paragraph VI completes Chapter III, after which Chapter IV (on the consonances and dissonances) begins.]

(printed p. 506 — Chapter III concludes its theory of proportion. It is proved from Euclid that there can be no more than the five genera of rational proportion; then a sub-section treats Arithmetic, Geometric, and Harmonic proportionality (“means”), defining each with examples and showing the harmonic mean embodies the chief consonances. A rule for finding three numbers in harmonic proportionality follows, with the remark that deeper music theory is omitted since the aim is only whether harmonic proportionality is found in the stars’ motions.)

[Header: BOOK IX. SECTION V. — 506]

[VI.] But that there cannot be given more Rational Proportions than the aforesaid five — both of greater and of lesser inequality — is established from this: that since any commensurable quantities whatsoever (between which there is a rational proportion) have between themselves that proportion which number has to number, as Euclid demonstrated (bk. 10, prop. 5), a greater number cannot be compared to a lesser in other ways than the five aforesaid. For the greater either contains the lesser some number of times exactly, that is, with no appendage [multiplex]; or contains it once, and besides one aliquot part of the lesser [superparticular]; or contains it once, and besides several aliquot parts of the lesser not making one [single] aliquot part [superpartient]; or contains the lesser several times, and over that one aliquot part of the lesser [multiplex-superparticular]; or, finally, contains the lesser several times, and besides several aliquot parts not making one aliquot part [multiplex-superpartient]. As regards the Denominators of [each] proportion, see our [Father] Clavius, on bk. 5 of Euclid, from p. 540 (as I have it) of the Roman edition of the year 1589.

On Arithmetic, Geometric, and Harmonic Proportionality

[Margin: Arithmetic proportionality.]

[VII.] Boethius, Jordanus Nemorarius, and many other Arithmeticians call the aforesaid proportionalities Means [Medietates].

ARITHMETIC Proportionality, or Mean, is when three or more numbers progress by the same difference — as are these numbers 4, 7, 10, 13, 16; for each of them exceeds its antecedent by three. And if this be done continuously, it is called a continuous proportionality; but if they be set with a leap and an interruption — for example, 4, 7 and 8, 11 and 30, 33 — it is called discrete.

[Margin: Geometric proportionality.]

GEOMETRIC Proportionality, or Mean, is when three or more numbers keep the same proportion among themselves — and this is, properly, Analogy or proportionality — of which kind is the subtriple in these numbers 2, 6, 18, 54, 162; which, if it thus progress continuously, is called continuous; but if it be interrupted, discrete, as in 2, 6 and 54, 162.

[Margin: Musical [Harmonic] proportionality.]

HARMONIC, or Musical, Proportionality or Mean, is when three numbers are so ordered that the proportion of the greatest to the least is the same as [the proportion] of the difference between the two greater to the difference between the two lesser — of which kind is [the case] in these numbers 3, 4, 6; for the proportion of 6 to 3 is double, and the difference of the greatest [two] is 2 [6 − 4], and of the least [two] is 1 [4 − 3], between which is likewise a double proportion. In these [harmonic] numbers, the terms neither progress by the same difference (as in the Arithmetic) nor by the same proportion (as in the Geometric). Another example of Harmonic proportionality is in these numbers 42, 12, 7. It is called Musical or Harmonic because it usually has those proportions in which the Musical Consonances consist. For in the former example [3, 4, 6]: between 6 and 4 is the sesquialteral proportion, constituting the consonance called Diapente or the Fifth; between 4 and 3 is the sesquitertian proportion, constituting the consonance called Diatessaron or the Fourth; and finally, between the extreme numbers 6 and 3 is the Double proportion, constituting the consonance called Diapason or the Octave — and so of most others, as Kepler too sets forth (bk. 3 of the Harmonics, ch. 3).

The five genera of ratio, each with example species, are displayed in a table (after Kepler, Harmonics III.3) — its first three rows printing at the foot of this page, the last two at the top of p. 507:

Hence it is clear that the aforesaid definition of harmonic proportion is not empty, since the cause of harmony is to be taken from the division of the circle into aliquot parts, according to what is to be said in chapter 4 [below].

[Margin: The properties of the three proportionalities. — The first property.]

[VIII.] Furthermore, the aforesaid three Proportionalities are distinguished among themselves by several notable properties, which Clavius reviews (on bk. 5 of Euclid, from p. 555):

• First, that in the Geometric the proportion of the greatest to the mean is the same as that of the mean to the least of the three numbers; but in the Arithmetic the proportion of the greatest to the mean is less than that of the mean to the least, whereas in the Musical it is greater.
• Second, that the Arithmetic has the differences of its terms equal but their proportions unequal; the Geometric, on the contrary, has the differences of the terms unequal but the proportions equal; and the Harmonic, finally, has neither the differences nor the proportions equal.
• Third, that in the Arithmetic the sum of the extremes is double the mean; but in the Geometric and the Harmonic the sum of the extremes exceeds double the mean by that number by which the difference of the greater [terms] exceeds the difference of the lesser.
• Fourth, that in the Geometric the number produced by multiplying the first into the third is equal to the square of the mean; but in the Arithmetic the first and third multiplied together produce a number which the square of the mean exceeds by that number which arises from the difference of the lesser multiplied into the difference of the greater.
• Fifth, that in the Harmonic, finally, the number generated from the multiplication of the extremes among themselves exceeds the square of the mean by that number which arises from the difference of the lesser multiplied into the difference of the greater.

The rest see in Clavius (p. 597), and [observe] how Kircher too (bk. 3 of the Musurgia, ch. 2) teaches to find three numbers of harmonic proportionality, thus:

To find three Numbers in Harmonic Proportionality

From three numbers in Arithmetic proportionality, multiply the mean by [each of] the extremes, and you will have the extremes of a Harmonic proportionality; and conversely, the extremes of a Harmonic [proportionality], multiplied into one another, generate the [harmonic] mean — as in the four following examples:

But how such proportions are to be continued, and other problems pertaining thereto are to be solved, is handed down — in the same place — by Clavius and Athanasius Kircher; by Zarlino (part 1 of the Harmonic Institutions, from ch. 31 to 44); and by Lodovico Fogliano (Musica Theorica, section 1, especially from ch. 9) — matters of which we do not now treat. [We inquire] only whether the Harmonic proportionality, or Musical Mean, is found in the motions and intervals of the stars, or is to be sought there. For, as Pietro d’Abano said (on Aristotle’s Problems): “It is the mean that generates Harmony”; since the mean of three strings, according to the ratios of the extremes, generates the sweetest concord to the ears.

It pleases [me], however, in this [part] of the chapter to subjoin, in each of the five Genera of Proportion, three examples of Harmonic proportionality among three [numbers] — the generating pair being the “Roots,” the resulting three-term harmonic mean the “Proportionality.” The table begins here and continues at the top of p. 507:

[Translator’s note — the figures are read from the engraved “Roots / Harmonic Proportionalities” table, whose final six rows print at the top of p. 507; in those rows only the second root and the harmonic triple are clearly legible, and one cell is illegible (shown as ”…”).]

[…continues on p. 507 (PDF 542) with the catchword “Radi-” (Radices, “Roots”) — the remainder of the table of harmonic proportionalities, still within Chapter III; Chapter IV (on the consonances and dissonances) follows.]

(printed p. 507 — the summary tables of Chapter III conclude, then Chapter IV begins, on the discoverers, number, and nomenclature of consonant and dissonant intervals and the division of the monochord. Riccioli traces music’s origin to Lamech’s house in Genesis 4 and recounts the legend of Pythagoras and the hammers, whence the axiom that sound answers to magnitude. He then treats which intervals Pythagoras admitted as consonances (only the simplest ratios of the Tetractys), and the two sects — the Canonici, who trust ratios, and the Harmonici, who trust the ear — with Ptolemy taking a middle way.)

[Header: ON THE HARMONIC SYSTEM OF THE WORLD — 507]

[At the top of this page the two tables of Chapter III finish (both are shown complete on p. 506): the genera-of-ratio table is completed by its last two rows — In the Multiplex-superparticular genus: Dupla sesquialtera, Tripla sesquitertia, Dupla sesquiquarta; In the Multiplex-superpartient genus: Dupla superbipartiens tertias, Dupla supertripartiens quartas, Dupla superquadrupartiens quintas — and the “Roots / Harmonic Proportionalities” table is completed by its last six rows (the harmonic triples 14·20·35; 39·60·130; 52·72·117; 33·48·88; 60·88·165; 98·…·166).]