Fn and Ft: DEPRPC1B MLBLL763
Author: Anonymous ("secundum Chilston")
Title: On the three manners of proportions
Source: London, British Library, Lansdowne 763, ff. 120v-122v.
[-f.120v-] Thus ouerpassid the rwlis of Proporcions and of their Denominacions. nou shal ye vnderstonde that as Proporcion as a Comparison betwene diuerse quantiteis oythir numbris. So is Proporcionalitas a Comparison eythir a likenes be .2o. Proporcions in .3re. diuerse quantiteis atte last. the whech quantiteis or numbris been callid the Termis of that proporcionalite. And whan the ferst terme. passithe the .2de. than [it supra lin.] is callid the ferst excesse. And whan the .2de. terme passith the thirde. than it is callid the .2de. excesse.
So ther be .3e. maner of Proporcionalitees scilicet Geometrica. Arithmetrica. and Armonica. Proporcionalitas. Geometrica is whan the same Proporcion is betwen the ferst terme. and the .2de. that is betwene the .2e. and the .3de. Whan al tho Proporcions be like. As betwene .8.4.2. is proporcionalitas. Geometrica. For proporcio Dupla is the ferst. And so is the .2de. 9. to .6. 6. to .4. sesquealtera. 16. to .12. 12. to .9. sesquetercia. 25 to .10. and 10 to .16. sesque quarta. 36 to .30. 30 to .25 sesque quinta. and so forthe vpward encresing the noumbir of difference be one. The noumbir of difference. and the excesse is all one. Whan the ferst numbir eythir terme passith the .2de. eythir the .2de. the .3de. than aftir the lasse excesse or difference shal that Proporcion be callid bothe the ferst and the .2de. As .9.6.4. the lasse difference is .2o. and Aliquota that is namyd be .2o. is callid the .2de. or Altera. Put than to the excesse or difference one vnite more and that. that is the more difference. and tho tweyn Proporcions be than bothe Icallid [-f.121r-] Sesquealtera. Than take the most numbir of. tho .3. termys. and encrese a noumbir aboue with the more difference that was before than hast thu .9. and 12. whois difference is .3. Encrese than the more numbir be .3. and one vnite scilicet be .4. than hast thu .16. so here be .126.96.36.199. in proporcionalite. Geometrica. Wherof bothe Proporcions be callid Sesquetercia. aftir the lesse difference. Werk thus forthe endlesli and thu shalt fynde the same. sesque sexta. sesque septima. sesque octaua. sesque .nona. sesque decima. sesque undecima.
Anothir general rwle to fynde this proporcionalite that is callid Geometrica is this. Take whech 2o. numbris that thu wilt. that be immediate. and that one that passith the othir be one vnite. Multiplie the one be the othir. and euerych be himselfe. and thu shalt haue .3e. termys in proporcionalite geometrica. and eyther proporcion shal be namyd in genere superparticularis. be the lasse numbir of tho .2o. that thu toke ferst. exemplum. As .3. 4. Multiplye .3. be himselfe and it makyth .9. Multiplye .3. be .4. and it makith .12. Multiplie .4. be himselfe and it makith .16. Than thus thu hast .188.8.131.52. in proporcionalite geometrica. and thus thu shalt fynde the same what .2o. numbris immediate that euer thu take.
And take this for a general rwle in this maner proporcionalite that the medil terme multiplied be himselfe is neyther more ne lesse than the .2o. extremyteis be. eche multiplied be othir Exemplum. As .12. multiplied be himselfe is 12 tymes 12. that is 144 And so is .9. tymes .16. [-f.121v-] or 15 tymys 9. that is al one. And this rwle faylith neuer of this maner proporcionalite in no maner of kende of proporcion. Asay whoso wil.
Proporcionalitas. Arithmetrica. is Whan the difference or the excesse be like id est Whan the more numbir passithe the .2de. as moche as the .2de. passith the .3de. and so forthe. yf ther be mo termis than 3e. exemplum. 6.4.2. The ferst excesse or difference is .2o. betwene .6. and .4. and thus the 2de. betwene .4. and .2.
PRoporcionalitas Armonica. is Whan ther is the same proporcion betwene [the supra lin.] ferst excesse or difference and the 2de that is betwene the ferst terme and the thirde exemplum .12.8.6. Here the firste difference. betwene .12. and .8te. is .4. the 2de. betwene .8te. and .6. is .2o. than the same proporcion is betwene .4. and .2o. that is betwene .12 and .6. for eythir is proporcio dupla
These .3. proporcionalitees Boys callith medietates. id est midlis. and thei haue thes namis Geometrica Arithmetrica. Armonica. As for the maner of treting of thees .3. sciencis Gemetrye tretith of lengthe and brede of londe. Arithmetrike of morenes and acorde. and the lasnes of numbir Musike of the highnes and lounesse of voyse. Than whan thu biddist me yefe the a. middle betwene .2o. numbris .I. may aske the What maner of midle? thu wilt haue. and aftir that shal be the diuersite of myn answere. For the numbris may be referrid to lengthe and brede of Erthe or of othir mesour that longith to Gemetrie. Eythir thei may be considerid as they be numbir in hemselfe. and so they long to Arithmetrike. Eythir thei may be referrid to lengthe and shortnesse [-f.122r-] and mesure of musical Instrumentis the whech cause highnesse and lownesse of voyse and so thei longe to Armonye and to crafte of Musike. Exemplum of the ferst id est Gemetrye Of .9. and .4. yf thu aske me whech is the medle be Gemetrye .I. sey .6. for this skille. Yf here were a place of 9. fote long. and .4. fote brode. be Gemetrie that were 36. fote square. than yf thu bad me yeve the a bodi. or another place that were euyn square that is callid Quadratum equilaterum Wherein [were supra lin.] neythir more space ne lesse than is in the former place that was ferst assignyd. than must thu abate of the lengthe of the formere place. and eke as moche his brede. so that it be no lenger than it is brode. that must be. by proporcion so that the same proporcion be. betwene the lengthe of the former bodi. and a syde of the .2de. that is betwene the same side and the brede of the ferst bodi. and than hast thu the medil betwene the lengthe and the brede of the ferst bodi or place. and be that medle .a. place .4. square that is euyn therto. As in this ensaumple that was ferst assignyd .9. and .4. and .6. is the medil. And as many fote is in a bodi. or a place that is euyn .4. square .6 fote. as in that. that is .9. fote longe. and .4. fote brode id est 36 in bothe.
The .2de. Proporcionalite is opin. Why? it is callid the medil be Arithmetrike. the whech tretyth of mornesse. and lasnes of numbir inasmoche as the more numbir passith the .2de. be as moche. as the .2de. passith the .3de. Neythir more ne lesse passith .12.9. than .9. passith .6. and therfore .9. is medium [-f.122v-] Arithmetricum.
The .3de. Proporcionalite. is callid Armonica. or a medil be Armonye for this skille. Dyapason that is proporcio .2la. is the most perfite acorde aftir the Vnisoun betwene the extremyteis of the Dyapason id est the Trebil. And the Tenor wil be yeue a mydle that is callid the Mene the which is callid .a. Dyapente id est sesquealtera. to the tenor. and Dyatessaron id est sesquetercia to the treble. Therfor that maner of mydle is callid Medietas Armonica. Sequitur exemplum. A. pipe of .6. foote longe with his competent brede is a Tenor in Dyapason. to a pipe of .3. fote with his competent brede. than is a pipe of .4. fote the Mene to hem tweyne. Dyatessaron to the one. and Diapente to the othir. As thu shalt fynde more pleynli in the makyng of the Monacorde. that is callid the Instrument of plainsong the whech Monacorde. is the ferst tretyse in the begynnyng of this boke. But this sufficith for knowlech of Proporcions. Secundum. Chilston.
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