Fn and Ft: DEPRPA1A MNYPMB12
Title: On the nature of proportions
Source: New York, The Pierpont Morgan Library, B.12, ff. 63-65. Used by permission.
[-f.63r-] De proporc[i]onibus [m. sec. sup. lin.]
For ye schul of your proporciouns the more vndirstondynge haue aftir my symple vndirstandynge .Y. schal declare it yow. But for the names of hem be more conuenient. and more schortly sett in latyn than they wolde be in Englisch. therfore y schal leue alle the names of hem standynge stille in latyn. and schortly declare hem to yow in englisch.
Ferst ye schul vnderstande. that proporcioun is a comparisoun of .2. thyngis be enchesoun of numbre or of quantite. lyk or vnlyk eythir to othir. So that proporcioun is seyde .2. manir of wysis that is Equalitatis and inequalitatis.
Proporcioun of equalite is whanne .2. euen thyngis be likind or sett togedre in comparisoun as .2. to .2. .4. to .4. .6. to .6. and of all othir
Proporcioun of inequalite is. whanne the more thynge is set in comparisoun to the lesse. or the lesse to the more as .2. to .4. .4. to .2. .3. to .5. .5. to .3. and so of all. And this proporcioun inequalite hath .5. spycis or .5. kyndis of proporcioun. Whos names be these in general. Multiplex. Superparticularis. Superparciens. Multiplex superparticularis. [-f.63v-] Multiplex superparciens.
The firste spyce or kynde of inequalite is clepid. Multiplex. that is to sey manyfold as twyis .1. it is clepid in special. dupla. that is to sey toofold. as .2. to .1. .4. to .2. 6. to .3. and foorth with owten ende. And yf the more noumbre conteyne thryis the lesse. thanne is it clepid in special tripla. as .3. to .1. .6. to .2. .9. to .3. Yf it be foure tymis the lesse conteynid in the more. thanne is it clepid. quadrupla. as .4. to .1. .8. to .2. .12. to .3. and so foorth. quintupla. sextupla. ottupla. And so vpward euere endelesly. with owten ende.
As for othir kyndis that ther be ye schul vndirstande that ther be tweyne manir of partyis the ferste is named pars aliquota. [aliquota in marg.] the othir is pars non aliquota. Pars aliquota is whanne that partye. be any manir of multiplicacioun yeldith his hole. as whanne betwene his hole and hym. is proporcio multiplex. As an vnite is pars aliquota of euery noumbre. for be multiplicacioun of that. euery noumbre [encesit del.] encresit And as tweyne in dualite is pars aliquota of euery euene noumbre in this partye schal be namid in special aftir the noumbre be whom he is multiplyid [-f.64r-] and yeldith his hole. for if he yelde hole be multipl[i]cacioun of .2. thenne is he callid in special altera id est half. And if he yelde his hole be multiplicacioun of .3. thanne is he callid thus. tercia. id est thridde. there exemplum. .2. est tercia pars .6. et .3. of .9. .4. of .12. and if he yelde his hole be multiplicacioun of .4. thanne is he callid quarta as .2. of .8. for foure tymis .2. is .8. And if be multiplicacioun of .5. is callid quinta. of .6. sexta. and so foorth with owten ende.
Pars non aliquota.
Pars non aliquota is whanne that the partye be no manir of multiplicacioun may yelde the hole. as .2. is a partye of .5. but be not aliquota for how so euer he be multiplyid he makith not .5. euen. for if he take him twyis it makith but .4. And if he take him thryis. he passith .5. and makith .6.
Proporcio superparticularis. is whanne the more noumbre contenith the lasse. and more ouer a partye of him that is aliquota. and aftir the special name of that partye aliquota. schal that proporcioun be namid in special. as betwene .6. and .4. that is proporcio sesquealtera and that is in grw. as moche to say. as totum in latyn id est al in Englysch. so that sesquealtera is as moch say as al and half. for the more [-f.64v-] noumbre conteynith the lasse and half ouer. Moreouer. betwene .8. and .6. is proporcioun sesquetercia for the more noumbre conteynith the lasse and his thridde [and his thridde bis] part ouer. Also betwene. 10. and .8. sesquequarta. betwene .12. and .10. sesquequinta. betwene .14. and .12. sesquesexta et sic sine fine.
Proporcio superparciens. is whanne the more noumbre conteynith the lasse. and more ouer the whoch excesse or supplus is pars aliquota of the lasse noumbre. as .5. and .3. but thanne ye moste loke to that excesse whanne be the more noumbre passith the lasse. and diuide it into swych partyes that be aliquote of the lasse and loke how many ther be therof. and what ben here special names. whether they be. tercie. quarte. or quinte. and so foorth. And if ther be .2. partyis aliquote. thanne ye schail sey in special. superbiparciens. And if ther be .3. supertriparciens. And if ther be .4. superquartiparciens. et sic vlterius.
More ouer if that tho partyis be tercie. thanne schul ye put al that at the laste ende as tercias. [illeg. in marg.] if thay be quarte. quartas. and so foorth endelesly. Exemplum. betwen .5. and .3. is proporcio superbiparciens tercias. for the more noumbre contenith the lasse and .2. partyis ouer. that be tercie and they be togedre but not pars aliquota of [-f.65r- ] the lasse noumbre. Between .7. and .5. superbiparciens quintas. Also .7. and .3. superquartiparciens tercias. More ouer .9. ad .5. superquartiparciens quintas. Also .10. and .6. superb[i]parciens tercias. And loke ye take good hede. that ye diuide the excesse. into the grettest partyis aliquotas that ye mowe. as here in this laste ensaumple .4. is diuidid into .2. dualiteis. that be tercie of .6. And tak this for a general rule. that the proporcioun that is betwene .2. smale noumbres that same is betwene hem. dubblis. and treblis. quatreblis. and quinyblis. and so foorth endelesly. Exemplum. the same that is betwene .5. and .3. is betwene .10. and .6. 20. and .12. .40. and .24. .80 and .48. and so foorth.
Multiplex superparticularis. is whanne the more contenith the lasse. and a partye of him that is aliquota. as .5. and .2. est dupla sesquealtera. and so is .10. and .4. .20. and .8. .7. and .3. est dupla sesquetercia. and so is .14. and .6.
Multiplex superparciens. is whanne the noumbre contenith the lasse and partyis ouer that ben aliquote. but tho alle togedre be not on partye aliquota. as .8. and .3. is dupla superbiparciens tercias. and so is .16. and .6. .32. and .12.
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