4818GEOMETR. PRACT.
duumin arcu abſciſſo contentorum.
Et ſi quid in diuiſione fueritreſidui, illud
per 60. multiplicatum, atq; in eundem diuiſorem, hoc eſt, in 128. diuiſum, dabit
minuta graduum. Et ſi adhuc quippiam remanſerit in hac diuiſione, illud eodem
modo per 60. multiplicatum, & in eundem diuiſorem 128, diuiſum, exhibebit
ſecunda. Atq; hoc modo progrediendo, reperientur Tertia, Quarta, & c. do-
nec nihil in diuiſione ſuperſit. Tunc enim vlterius progrediendum non eſt; Sed
fatis eſt, ad ſecunda vſq; progredi. Exempligratia. Ponatur ex Quadrante P Q,
vſq; ad gradum 100. producto, qualis eſt duo decimus, filum perpendiculi ab-
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] ſcidiſſe partes 20. ex illis 128. in quas totus arcus productus diſtributus eſt. Fiat
ergo, vt 128. ad 20. ita 100. ad aliud; inuenienturq; gradus 15. ſupereruntq; in di-
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] uiſione 80. quę ducta in 60. faciunt 4800. quę diuiſa per 128. dant minuta 37, &
ſuperſunt adhuc 64. quæ ſi ducãtur in 60. & productus numerus 3840. diuida-
tur per 128. prodibunt Sec. 30. nihilq; in diuiſione ſupereſt. Arcus ergo Q 20,
vel arcus Quadrantis BC, inter C, & filum perpendiculi includit gr. 15. Min. 37.
Sec. 30. Rurſus ponamus ex octauo quadrante RS, vſq; ad gradum 96. produ-
cto filum perpẽdiculi ab ſcidiſſe partes 96. ex illis 128. quæ in toto arcu producto
continentur. Fiat ergo, vt 128. ad 96. ita 96. ad aliud: reperientur que gradus 72.
pręcisè arcui abſciſſo conuenire. Item ceciderit filum in partem 64. Quadrantis
ſextidecimi MN, vſque ad gradum 104. producti. Si ergo fiat, vt 128. ad 64. ita
104. ad aliud, producentur quo que grad. 52. præcisè. atque ita de cæteris; dum-
modo ſis memor, vt ſi quid in diuiſi onibus ſuperfuerit, reſidua diuiſionum mul-
tiplicentur per 60. & producti numeri per 128. diuidantur, vt dictum eſt.
per 60. multiplicatum, atq; in eundem diuiſorem, hoc eſt, in 128. diuiſum, dabit
minuta graduum. Et ſi adhuc quippiam remanſerit in hac diuiſione, illud eodem
modo per 60. multiplicatum, & in eundem diuiſorem 128, diuiſum, exhibebit
ſecunda. Atq; hoc modo progrediendo, reperientur Tertia, Quarta, & c. do-
nec nihil in diuiſione ſuperſit. Tunc enim vlterius progrediendum non eſt; Sed
fatis eſt, ad ſecunda vſq; progredi. Exempligratia. Ponatur ex Quadrante P Q,
vſq; ad gradum 100. producto, qualis eſt duo decimus, filum perpendiculi ab-
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] ſcidiſſe partes 20. ex illis 128. in quas totus arcus productus diſtributus eſt. Fiat
ergo, vt 128. ad 20. ita 100. ad aliud; inuenienturq; gradus 15. ſupereruntq; in di-
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] uiſione 80. quę ducta in 60. faciunt 4800. quę diuiſa per 128. dant minuta 37, &
ſuperſunt adhuc 64. quæ ſi ducãtur in 60. & productus numerus 3840. diuida-
tur per 128. prodibunt Sec. 30. nihilq; in diuiſione ſupereſt. Arcus ergo Q 20,
vel arcus Quadrantis BC, inter C, & filum perpendiculi includit gr. 15. Min. 37.
Sec. 30. Rurſus ponamus ex octauo quadrante RS, vſq; ad gradum 96. produ-
cto filum perpẽdiculi ab ſcidiſſe partes 96. ex illis 128. quæ in toto arcu producto
continentur. Fiat ergo, vt 128. ad 96. ita 96. ad aliud: reperientur que gradus 72.
pręcisè arcui abſciſſo conuenire. Item ceciderit filum in partem 64. Quadrantis
ſextidecimi MN, vſque ad gradum 104. producti. Si ergo fiat, vt 128. ad 64. ita
104. ad aliud, producentur quo que grad. 52. præcisè. atque ita de cæteris; dum-
modo ſis memor, vt ſi quid in diuiſi onibus ſuperfuerit, reſidua diuiſionum mul-
tiplicentur per 60. & producti numeri per 128. diuidantur, vt dictum eſt.
Demonstratio huius operationis perſpicua eſt.
Quoniam enim eſt, (in
vltimo exemplo) vt arcus N M, vſque ad grad. 104. productus, quatenus in
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] 128. partes ſectus eſt, ad arcum N O, earundem partium 64. vt idem arcus N M,
totus productus, quatenus grad. 104. complectitur, ad eundem arcum N O, re-
ſpectu eorundem graduum; efficitur, vt ſi fiat quemadmodum partes 128, totius
arcus N M, vſque ad grad. 104. producti ad partes 64. in arcu N O, contentas ita
idem arcus N M, productus graduum 104. ad aliud, reperiantur gradus in eo dem
arcu N O, contenti, & c.
vltimo exemplo) vt arcus N M, vſque ad grad. 104. productus, quatenus in
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] 128. partes ſectus eſt, ad arcum N O, earundem partium 64. vt idem arcus N M,
totus productus, quatenus grad. 104. complectitur, ad eundem arcum N O, re-
ſpectu eorundem graduum; efficitur, vt ſi fiat quemadmodum partes 128, totius
arcus N M, vſque ad grad. 104. producti ad partes 64. in arcu N O, contentas ita
idem arcus N M, productus graduum 104. ad aliud, reperiantur gradus in eo dem
arcu N O, contenti, & c.
8.
In gratiam autem ſtudio ſorum placet hic tabellam inſerere, in qua ex re-
11Conſtructio
& vſ{us} tabel-
la pro minutis
& ſecundis. ſiduo primæ operationis regulæ aureæ, qua gradus eliciuntur, mox apparet,
quot minuta, & ſecunda illi reſi duo reſpondeant: Ita vt opus ſit ſemel tantum
regulam auream adhibere. Conſtruitur autem tabella, ſi ſingula reſidua, quæ
plura, quam 127, eſſe nequeunt, per 60. multiplicentur, productique numeri per
128. diuidantur. Atque vt ſtructura, & vſus huiuſce tabellæ facilius intelliga-
tur, apponemus vnum exemplum. Cadat verbi gratia filum perpendiculi in
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] partem 29. Quadrantis 32. ad gradum vſque 120. producti. Fiat igitur vt 128.
ad 29. ita 120. ad aliud; producenturque grad. 27. Quia verò in diuiſione ſu-
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] perſunt 24. ſub quo numero in tabella ponuntur duo hi numeri 11. 15. Prior
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] ergo dat minuta, & poſterior ſecunda; Ita vt arcus à filo abſciſſus complecta-
tur grad. 27: Min. 11. ſec. 15. Atque hæc minuta, & Secunda producuntur, ſi reſi-
duum diuiſionis, nimirum 24. ducatur in 60. & productus numerus per 128. di-
uidatur, & c. Eadem ratio eſt dereliquis tabellę numeris. Nam ſemper ſuperior
numerus eſt ille, qui in diuiſione remanſit: Inferiorum autem numerorum prior
ad minuta, & poſterior ad ſecunda ſpectat.
11Conſtructio
& vſ{us} tabel-
la pro minutis
& ſecundis. ſiduo primæ operationis regulæ aureæ, qua gradus eliciuntur, mox apparet,
quot minuta, & ſecunda illi reſi duo reſpondeant: Ita vt opus ſit ſemel tantum
regulam auream adhibere. Conſtruitur autem tabella, ſi ſingula reſidua, quæ
plura, quam 127, eſſe nequeunt, per 60. multiplicentur, productique numeri per
128. diuidantur. Atque vt ſtructura, & vſus huiuſce tabellæ facilius intelliga-
tur, apponemus vnum exemplum. Cadat verbi gratia filum perpendiculi in
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] partem 29. Quadrantis 32. ad gradum vſque 120. producti. Fiat igitur vt 128.
ad 29. ita 120. ad aliud; producenturque grad. 27. Quia verò in diuiſione ſu-
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] perſunt 24. ſub quo numero in tabella ponuntur duo hi numeri 11. 15. Prior
1515[Handwritten note 15]1616[Handwritten note 16]1717[Handwritten note 17]1818[Handwritten note 18]1919[Handwritten note 19]2020[Handwritten note 20] ergo dat minuta, & poſterior ſecunda; Ita vt arcus à filo abſciſſus complecta-
tur grad. 27: Min. 11. ſec. 15. Atque hæc minuta, & Secunda producuntur, ſi reſi-
duum diuiſionis, nimirum 24. ducatur in 60. & productus numerus per 128. di-
uidatur, & c. Eadem ratio eſt dereliquis tabellę numeris. Nam ſemper ſuperior
numerus eſt ille, qui in diuiſione remanſit: Inferiorum autem numerorum prior
ad minuta, & poſterior ad ſecunda ſpectat.