Fn and Ft: DEPRPC1C MDTC516
Title: On the three manners of proportions
Source: Dublin, Trinity College, 516, ff. 129v-131v
[-f.129v-] betwene the II and the IIIde that is whenn Al proporciouns byn like as bytowen 8 4 2 is proporciounalites Geometria for proporcio dupla ys the first and so is the IIde 9 6 4 Sesquealtera 12 9 sesquetercia 25 26 16 sesquequarta 36 38 25 sesquequinta and so furth vpward encresynge thy nowmbre of difference by .1. the nowmbre of difference the excesse is al onn .1. and when the first nowmbre or terme passith the secunde or the secunde the IIIde and after the lesse excesse or difference that proporcioun schall be calde both the first and the IIde as 9 6 the laste difference is II to and pars aliquota that his name by II ys called the secunde or altera put thenn to the excesse euer difference onn [-f.130r-] vnite more and that is the more difference and tho II proporciouns be thenn called both sesquealtera thenn take the moste nowmbre of the III termes and encrese A nowmbre Abouenn with the more difference that was before and haste then 9 and 12 whose difference ys 3 encrese thenn the more nowmbre by II and on vnite more that is by 8 thenn hast thou 16 so that there be .3.9. and .12.16. In proporcionalite Geometrica wherof bothe proporcions be called Sesquetercia After the lesse difference werke thus for in lesse and thouu schalt fynde the same sesquesexta sesqueseptima Sesqueoctaua Sesquenona Sesquedecima Sesquevndecima 36 42 49 56 64 72 81 98 200 [288 corr.] 710 144 6 7 8 9 10 11 12 And other rewle generall to fynde this proporcionalite that is called Geometria ys thys take thou whych II Nowmbres that thouu wylt that benn inmediants that is that on passeth that other but by on vnite multiplice that onn by that other and euer ylke onn be hymself and thou schall haue III termes in proporcionalite Geometria and eyther proporcioun schall be named in general superparticularijs be the lesse nowmbre of the II that thou take firste Exsample III and IIII multiplite [multiplicite corr.] 3 by hymselfe and that makyth 9 multipli 3 by 4 and that makyth 12 multiplie 4 by himselfe and that makyth 16 thenn thou haste this III 9 12 16 in proporciounalite Geometria and thus thou schallt fynde the same but 4 nowmbrys that thou take inmediate and take this for a generall rewle in this maner proporcionalite that the mydil terme multiplied by hymselfe ys neyther more ne lesse then the 2 extremites bynn multiplied ywonn by other as 12 be multiplied by hymselfe that is 12 tymes 12 144 and so is 9 tymes 16 or 26 tymes 9 that is al onn and this rewle fayleth not of thys maner proporcionalite in no kynde of proporcioun assay who so wyle proporcionalitas Arsmetrica is whenn tee difference or the [-f.130v-] excesse bynn like that is when the more nowmbre passyth the secunde by as mykil as the IIde passeth the IIIde and so furst he yf ther moo termes thenn III Exemplum 6 4 and 2 the first excesse or difference ys 2 bytwene 6 and 4 and so ys the IIde bytwenn 4 and 2 proporciounalytas Armonica ys when they ys the same proporcioun bytwenn the first excesse or difference and the IIde that ys bytwenn the first terme and the IIIde Exemplum 12 8 6 3 the frist difference bytwene [i2 del.] 12 and 8 4 the secunde bytwenn 8 and 6 16 and 2 thenn the same proporcioun ys bytwene 4 and 2 that ys bytwene 12 and 6 for eyther ys proporcioun dupla The 3 proporcionalytes be calde medietates that is for to sey m[i]ddeles and they haue this names Geometrica Arsmetrica Armonica for the maner of tretinge of scientes Geometrie treti[t]h length and brede of londe Arsmetryk of mornesse and lessenes of nowmbre Musyke of highnes and of lownes and Acorde of voyce thenn whenn thou biddest me yf the a myddil bytwenn II nowmbres than may I aske the what maner of mydil thou wilt haue and efter that schall be dyuersyte of mynn ansure for the nowmbres mowe be refferre to hyse length and brede of erth or of other mesure that longeth to Geometrye or thay may be considrede as they be nomebrede in thaynnself and so yt longeth to Arsmetryk or they may refferred to length and schortnesse and mesure of musicall Instrumentys that causen highnes and lownes of voyce and yt belongeth to Armony and [to. del.] crafte of musyke Exemplum of the first that is of Geometrie 9 2 4 yf thou aske me whiche ys the medill be Geometrie Y sey 6 and that for thys skill ys yf ther were a place that were 6 fote longe and 4 fote brode thenn [thenn bis] by Geometrye yt were 36 square foot thenn [-f.131r-] if thou bid me yif the a body or Another place that were euene square that ys called quadratum Aquilaterum wherynn were neyther more spece ne lesse thenn ys in that former place that was first assigned thann I muste abate of the length of that former place And eke a mykyll hys brede so that yt be no lenger thenn yt is brede that thus be proporcioun so that the same proporcioun betwenn the leng[t]h of this forme body and a syde of the secunde yt is bytwenn the same side and brede of the first body and thenn hast thou the mydil betwene the length of the brede of the first body [and then hast thou the medille bytwenn the length of the brede of the first body bis] euery place and by that mydill a place 4 square ys euere therto as yn thys ensample that was fyrste assyned that is 9. and 4. 6. ys the medil and as many fete benn in a body or a place that ys euynn 4 square 6 fete as in that at is 9 fete longe and 4 fete brode that is 36 in both the secunde proporcionalyte ys opynn why it is called the medil by arsmetryke the whiche tretith [tretihth corr.] of morenesse and lessenesse of nowmbre in as mykil that the nowmbre passeth the secunde by as mekil as the IIde the IIIde as by nowmbre no lesse passyth .2.1.9. thenn 9 dose 6 and therfor 9 is medium Arsmetricum gilte The IIIde proporcioun is callde Armonica or a medill by armonie for this ilke diapasoun yt is proporcioun dupla yt is most perfyte Acorde after the vnisoun bytwene the extremytes of that diapasoun that is treble and tenor will be zevynn a medil that is called a mene the whych mene ys diapente that is sesquealtera to this tenor and diapasoun to diatasseroun that is sesquetercia to the trebyll therfor that Arme of medill ys called Armonica medietas Exemplum of thys A pype of vj fete longeth whiche ys competent brede ys a tenor to diapasoun to A pype of III with hys [-f.131v-] Competent brede thann is A pype of IIII fete A mene to thaymm diatasseroun to that onn and diapente to that othyr As thou schalt fynde more plenerly of the makynge of the monacorde that is calde the Instrument of pleynn songe but this suffyseth for to haue knowynge of the proporcions perfeccio largarum constet in longis longarum in breuibus breuiumque in semebrevibus 4o set minimis probacio semper constat in semibreuibus 6.
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