7343LIBER PRIMVS.
@ur {400/1000}.
Et in D C, {68/100}.
vnius partis rectæ AB, ſiue {63/1000}.
totius lineæ AB;
cum
quælibet centeſima particula vnius partis decimæ ſit {1/1000}. Atque in hunc mo-
dum progredilicebit ad decimas vnius decimæ ex illis {6/10}. quæin particula DC,
continentur. nimirum ad fractionem à 10000. denominatam, & c. ſed ſatis mi-
hi videtur ad partes milleſimas peruenire.
quælibet centeſima particula vnius partis decimæ ſit {1/1000}. Atque in hunc mo-
dum progredilicebit ad decimas vnius decimæ ex illis {6/10}. quæin particula DC,
continentur. nimirum ad fractionem à 10000. denominatam, & c. ſed ſatis mi-
hi videtur ad partes milleſimas peruenire.
Demonstratio hic eadem eſt, quæin gradibus, ac Minutis.
Eandem
enim proportionem habet recta 10. partium ad 1. partem, quamrecta A E, ad
particulam D C, cum vtrobique proportio ſit decupla, ideoque permutando
erit vt recta 10. partium ad rectam A E, ita 1. pars ad particulam D C. Quamo-
brem ſicut in A E, continentur {6/10}. rectæ A B, decem partium, & inſuper par-
ticula F E, reſpectu vnius partis lineæ A B, decem partium, ita quoque in par-
ticula data D C, continebuntur {6/10}. vnius partis, & inſuper talis particula re-
ſpectu vnius decimæ qualis eſt F E, reſpectu vnius partis lineæ A B, decem par-
tium, & c.
enim proportionem habet recta 10. partium ad 1. partem, quamrecta A E, ad
particulam D C, cum vtrobique proportio ſit decupla, ideoque permutando
erit vt recta 10. partium ad rectam A E, ita 1. pars ad particulam D C. Quamo-
brem ſicut in A E, continentur {6/10}. rectæ A B, decem partium, & inſuper par-
ticula F E, reſpectu vnius partis lineæ A B, decem partium, ita quoque in par-
ticula data D C, continebuntur {6/10}. vnius partis, & inſuper talis particula re-
ſpectu vnius decimæ qualis eſt F E, reſpectu vnius partis lineæ A B, decem par-
tium, & c.
Itaqve, vt vides, duabus operationibus ad milleſimas partes perueni-
11Denominatio
facilis fractio-
num milleſi.
marum. tur, ac ſi latus totum A B, in 1000. partes ſectum eſſet, quod conſideratione
dignum eſt. Et duo quidem numeratores decimarum eo crdine poſiti, quo
inuenti ſunt, dant Numeratorem centeſimarum: cui ſi præponatur ad ſiniſtram
numerus integrarum partium ante datam particulam exiſtentium, conflabitur
Numerator milleſimarum. Vt in ſuperiori exemplo, quia ante datam parti-
culam D C, reperiuntur 4. partes, inuentæque ſunt {6/10}. & {8/10}. erit tota fra ctio
{468/1000}. Sic ſi inuentæ eſſent pro aliqua particula in octaua parte, {7/10}. {0/10}. conſti-
tueretur fractio {870/1000}. Item ſi in tertia parte inuentæ eſſent {0/10}. {7/10}. fieret fractio
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42] {307/1000}. & ſic de cæteris. Quod ſirecta A B, id eſt, latus vtriuſque vmbræ diui-
deretur in 100. partes, reperientur partes milleſimæ vnica operatione; ſi nimi-
rum particula data in vna centeſima decuplaretur: quia tunc intelligerentur ſin-
gulæ centeſimę in denas particulas ſub diuiſæ. Sed quia particula data in cen-
teſima aliqua parte perparua eſt, vt vix circino capi poſsit, accipiemus eam cũ
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42] vna centeſima, vel duabus, & ex decuplo abijciemus 10. vel 20. centeſimas, vt
reliquę centeſimæ exhibeant decimas vnius centeſimę. Eadem ratione, quan-
do data particula in linea 10. partium eſt perpuſilla accipienda erit reliqua parti-
cula maior decies. Nam reliqua pars rectę AB, à B, vſque ad finem illius par-
ticulę decuplę dabit numerum decimarum in propoſita minore particula, & c.
Hac ratione, ſi particulam CH, decuples, in cides in punctum K Ergo ſegmẽ-
tum B K, dabit {6/10}. & inſuper particulam DK, vt ſupra; cum DK, particulę FE,
ſit ęqualis. Ratio huius reieſt, quod ambę particulę DC, CH, decuplatę con-
ficere debeanttotam A B, vt conſtat.
11Denominatio
facilis fractio-
num milleſi.
marum. tur, ac ſi latus totum A B, in 1000. partes ſectum eſſet, quod conſideratione
dignum eſt. Et duo quidem numeratores decimarum eo crdine poſiti, quo
inuenti ſunt, dant Numeratorem centeſimarum: cui ſi præponatur ad ſiniſtram
numerus integrarum partium ante datam particulam exiſtentium, conflabitur
Numerator milleſimarum. Vt in ſuperiori exemplo, quia ante datam parti-
culam D C, reperiuntur 4. partes, inuentæque ſunt {6/10}. & {8/10}. erit tota fra ctio
{468/1000}. Sic ſi inuentæ eſſent pro aliqua particula in octaua parte, {7/10}. {0/10}. conſti-
tueretur fractio {870/1000}. Item ſi in tertia parte inuentæ eſſent {0/10}. {7/10}. fieret fractio
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42] {307/1000}. & ſic de cæteris. Quod ſirecta A B, id eſt, latus vtriuſque vmbræ diui-
deretur in 100. partes, reperientur partes milleſimæ vnica operatione; ſi nimi-
rum particula data in vna centeſima decuplaretur: quia tunc intelligerentur ſin-
gulæ centeſimę in denas particulas ſub diuiſæ. Sed quia particula data in cen-
teſima aliqua parte perparua eſt, vt vix circino capi poſsit, accipiemus eam cũ
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42] vna centeſima, vel duabus, & ex decuplo abijciemus 10. vel 20. centeſimas, vt
reliquę centeſimæ exhibeant decimas vnius centeſimę. Eadem ratione, quan-
do data particula in linea 10. partium eſt perpuſilla accipienda erit reliqua parti-
cula maior decies. Nam reliqua pars rectę AB, à B, vſque ad finem illius par-
ticulę decuplę dabit numerum decimarum in propoſita minore particula, & c.
Hac ratione, ſi particulam CH, decuples, in cides in punctum K Ergo ſegmẽ-
tum B K, dabit {6/10}. & inſuper particulam DK, vt ſupra; cum DK, particulę FE,
ſit ęqualis. Ratio huius reieſt, quod ambę particulę DC, CH, decuplatę con-
ficere debeanttotam A B, vt conſtat.
Ad Maiorem quo que commo ditatem pro inueſtigandis partibus decimis,
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42] hoc eſt, pro decuplanda particula propoſita, conſtrui poterit circinus dupli-
cis aperturę, in quo ſcilicet crura producta ſe mutuo interſecent, at que vna a-
pertura alterius ſit ſemper decupla, inſtar circini, quo linea duas in partes ęqua-
les diuidi ſolet. Ita enim fiet, vt accepta per minorem aperturam particula ab-
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42] ſciſſa, particula maior exhibeat eam particulam decies ſumptam, vt non opus
ſit toties circinum circumducere: qua quidem in re facile error committi
poteſt, quiillo circino, ſi recte fabricatus ſit, facilius vitatur. Sed ſine hoc
circino, idem fieri poteſt per inſtrumentum partium, quod capite pręceden-
ti conſtruximus. Nam ſi particula data circino capiatur, & ſumma diligentia ei
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42] hoc eſt, pro decuplanda particula propoſita, conſtrui poterit circinus dupli-
cis aperturę, in quo ſcilicet crura producta ſe mutuo interſecent, at que vna a-
pertura alterius ſit ſemper decupla, inſtar circini, quo linea duas in partes ęqua-
les diuidi ſolet. Ita enim fiet, vt accepta per minorem aperturam particula ab-
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42] ſciſſa, particula maior exhibeat eam particulam decies ſumptam, vt non opus
ſit toties circinum circumducere: qua quidem in re facile error committi
poteſt, quiillo circino, ſi recte fabricatus ſit, facilius vitatur. Sed ſine hoc
circino, idem fieri poteſt per inſtrumentum partium, quod capite pręceden-
ti conſtruximus. Nam ſi particula data circino capiatur, & ſumma diligentia ei
3838[Handwritten note 38]3939[Handwritten note 39]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]