Fn and Ft: DEPRPA3 TEXT
Title: On the nature of proportions
Source: Sir John Hawkins, A General History of the Science and Practice of Music, 2 vols.  (London: Novello, 1853), 1:250-51.
[-250-] Now passid al maner sightis of descant, and with hem wel replesshid, that natural appetide not saturate sufficientli, but ferventli desirith mo musical conclusions, as now in special of proporcions, and of them to have plein informacion, of the whech after myn understonding ye shall have open declaracion. But forasmoche as the namys of hem be more convenientli and compendiusli set in Latin than in English, therefore the namys of hem shal stonde stille in Latin, and as brievely as I can declare the naturis of them in English. First ye shal understond that proporcion is a comparison of two thinges be encheson of numbir or of quantitie, like or unlike eyther to other; so that proporcion is seid in two maner of wyse, scilicet, Equalitatis and Inequalitatis. Proporcion of Equalitie is whan two evyn thinges be likenyd, either sette togedir in comparison, as 2 to 2, or 4 to 4, and so of others. Proporcion of Inequalitie is whan the more thinge is sette in comparison to the lasse, or the lasse to the more, as 2 to 4, or 4 to 2, or 3 to 5, or 5 to 3; and thys proporcion of inequalitie hath five species or naturis or keendys, whois namys be these in general: 1. Multiplex; 2. Superparticularis; 3. Superpartiens; 4. Multiplex superparticularis; 5. Multiplex superpartiens. The first spece of every keende of inequalitie is callid Multiplex, that is to sey manifold, and is whan the more nombre conteynyth the lasse manyfolde, as twies 1; and that is callid in special, Dupla, id est, tweyfold, as 2 to 1, or 4 to 2, or 6 to 3, and so forthe endlesli. Yf the more numbir conteyne thries the lasse, than it is callid in special, Tripla, as 3 to 1, 6 to 2, 9 to 3; yf it be four times the lasse conteinid in the more, than it is Quadrupla, as 4 to 1, 8 to 2, 12 to 3, and so forthe. Quindupla, Sexdupla, Sepdupla, Ocdupla, and so upward endlesli. As for other keendis, ye shall understond that there be two manere of parties, one is callid Aliquota, and another is callid Non aliquota. Pars Aliquota is whan that partie be ony maner of multiplicacion yeldeth his hole, as whan betwene his hole and him is proporcion Multiplex, as a unite is Pars Aliquota of every numbir; for be multiplicacion of that, every numbir wexeth tweyne: or dualite is Pars Aliquota of every evyn numbir; and thus this partie shal be namyd in special after the numbre on whom he is multiplied and yeldeth his hole; for if he yeldeth his hole be multiplicacion of 2, it is callid Altera, one halfe; and yf he yeldeth his hole be multiplicacion of three, it is called Tertia, in the third part; Sequitur exemplum, two is the thirde part of 6, and 3 of 9, and 4 of 12; and yf he yeldeth his multiplicacion be 4, than it is called Quarta, as 2 for 8, for 4 tymys 2 is 8; and if it yeldith his hole be multiplicacion of 5, than it is callid Quinta, and of 6 Sexta, and so forth endlesli. Pars non aliquota is whan that partie be no maner of multiplicacion may yelde his hole, as 2 is a parte of 5; but he is non aliquota, for howsoever he be multiplied he makith not evyn 5, for yf ye take him twies he makith but 4; and if ye take him thries he passith and makith 6. Proportio superparticularis is whan the more numbir conteynyth the lasse; and moreover a party of him that is Aliquota, and aftir the special name of that Parties shal that proporcion be namid in special, as betwene 6 and 4 is Proporcion sesquialtera; Ses in Greek, Totum in Latin, al in Englishe, so Sesquialtera is for to sey al and a halfe, for the more numbir conteynyth al the lasse, and halfe thereof more over. Between 8 and 6 is proporcion Sesquitercia, for the more numbir conteynyth the lasse, and his thyrd part over. Betwene 10 and 8 is sesquiquarta, betwene 12 and 10 is sesquiquinta, betwene 14 and 12 is sesquisexta, et sic infinit¸. Proporcio superparciens is whan the more numbir conteynyth the lasse; and moreover the whech excesse eyther superplus is not Pars aliquota of the lasse numbir, as betwene 5 and 3. But than thou must loke to that excesse whan the more number passith the lasse, and devyde it into sweche parties that be aliquota; and loke how many there be thereof, and what is her special namys, and whether they be thyrde, fowerth, or fyfthe, and so forthe. And yf ther be two parties aliquote, than thou shalt sey in special Superbiparciens; and yf ther be three, supertriparciens; and yf ther be four, superquartiparciens, and so forthe. And ferthermore tho parties that be tercie, than thou shalt sey alwey at last ende, Tercias; and yf ther be four [-251-] Quartas, and so forth endlesli. Sequitur exemplum, betwene 5 and 3 is proporcion Superbiparciens tertias, for the more number conteynyth the lasse, and two parties over that be tercie; but they both togedir be not pars aliquota of the lasse number; betwene 7 and 5 is Superbiparciens quintas; betwene 7 and 3 is Dupla sesquitercias; betwene 9 and 5 is Superquartiparciens quintas; betwene 10 and 6 is Superbiparciens tercias: and loke ye take goode hede that ye devyde the excesse into the grettest partyes aliquotas that ye may, as here, in this last ensample, 4 is devyded into 2 dualities, that beene tercie of six. And take this for a general rewle, that the same proportion that is betwene twoe smale numberis, the same is betwene her doubles and treblis, and quatreblis, and quiniblis, and so forth endlesly. Sequitur exemplum, the same proporcion that is betwene 5 and 3, is betwene 10 and 6; betwene 20 and 12; betwene 40 and 24; betwene 80 and 48, and so forth endlesli. Multiplex superparticularis is whan the more numbir conteynythe the lasse, and a partye of him that is aliquota; as 5 and 2 is dupla sesquialtera, and so is 10 and 4; and so is 20 and 8; but 7 and 3 is dupla sesquitercia, and so is 14 and 6. Multiplex superparciens is whan the more numbir conteynyth the lasse, and the parties that be over aliquote. But thei alle togedir be not one parte aliquota, as 8 and 3 is dupla superbiparciens tercias, and so is 16 and 6, 32 and 12.
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