147115TERTIVS.
li cirunferentia ad aliam partem perlongum ducta axis dicitur.
His ita poſitis addeſcriptionem uolutæ acce
dendum. Eſto linea a b. craßitudo imi ſcapi columnæ: partiatur hæc in partes decem, & octo ſuis punst is
distinctis, his addatur alia pars, quæ ſit una ex illis decem & octo partibus; ita erunt nouem, & decem p ar-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] tes, atque hæc erit longitudo, & latitudo abaci, ſcilicet a puncto a, ad punctum c. partiatur hæc longit udo
in duas æquales partes, in e. una ex his erit craſſitudo ſeu altitudo capituli cum uolutis. demittantur igitur
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] lineæ ab extremitatibus abaciad c d. diuidatur postea totum ſpatium a c. quod eſt longitudo abaci in
partes uiginti, uirgulis ſingulæ ſint notatæ, ut a prioribus illis decem & octo internoſcantur. recedamus ab
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] extremo abaco id eſt a puncto a. uerſus c. uel a puncto c. verſus a. duas partes, & dimidiam ex illis
uiginti, ut o p q. & illud ſpatium in duas partes æquas ſecetur in r. ita erit utrunque ſpatium partis unius,
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52]& quartæ. recedatur ergo ab extremo abaco ab a uel à c. in partem interiorem parte una, & quarta
1110 in i. & ab ijs ſignis demittantur catheti, hoc eſt perpendiculares lineæ, quæ ſint r g. in his frontes uolu-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] tarum deſcribendæ ſunt, & in ipſis centrum oculi collocandum, quod cyclum Albertus uocat. hæ lineæ tam
longæ ducantur, quàm priores illæ a punctis a. & c. ad d. productæ fuerint. partianturq́; in partes no-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] uem, & dimidiam, incipiendo a punctis g. verſus r. hæ partes ſingulæ prioribus illis decem & octo, in quas
abaci longitudo diuiſa erat, reſpondebunt. ex his una & dimidia dabitur abaci craſſitudini, ita abacus erit al-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] ta parte una, & dimidia; ſed abacus conſtat quadra, & cymatio. quadra eſt tertia pars altitudinis abaci. reli-
quas duas cymatium occupat: ducendum uero cymatium molliter ad figuram litteræ S. constat enim ex
duobus c. ita ut ſuperiori inferior opponatur, ſuperiori recte, ut ſcribitur formato, infra abacum octo partes
erunt. qua parte igitur finis est quartæ & dimidiæ, ibi in cathetis centra oculi figenda ſunt. eſto ibi littera o.
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] cum terminus abaci ſit a b. uel c b. poſito igitur centro in puncto o. ducatur circinatio tam magna quan-
2220 ta est una ex octo partibus his punctis concluſa l m. Cyclus ille uolutæ oculus nominatur. et ita factum erit,
quod hactenus a Vitr. præceptum habemus. quod uero dicit.
dendum. Eſto linea a b. craßitudo imi ſcapi columnæ: partiatur hæc in partes decem, & octo ſuis punst is
distinctis, his addatur alia pars, quæ ſit una ex illis decem & octo partibus; ita erunt nouem, & decem p ar-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] tes, atque hæc erit longitudo, & latitudo abaci, ſcilicet a puncto a, ad punctum c. partiatur hæc longit udo
in duas æquales partes, in e. una ex his erit craſſitudo ſeu altitudo capituli cum uolutis. demittantur igitur
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] lineæ ab extremitatibus abaciad c d. diuidatur postea totum ſpatium a c. quod eſt longitudo abaci in
partes uiginti, uirgulis ſingulæ ſint notatæ, ut a prioribus illis decem & octo internoſcantur. recedamus ab
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] extremo abaco id eſt a puncto a. uerſus c. uel a puncto c. verſus a. duas partes, & dimidiam ex illis
uiginti, ut o p q. & illud ſpatium in duas partes æquas ſecetur in r. ita erit utrunque ſpatium partis unius,
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52]& quartæ. recedatur ergo ab extremo abaco ab a uel à c. in partem interiorem parte una, & quarta
1110 in i. & ab ijs ſignis demittantur catheti, hoc eſt perpendiculares lineæ, quæ ſint r g. in his frontes uolu-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] tarum deſcribendæ ſunt, & in ipſis centrum oculi collocandum, quod cyclum Albertus uocat. hæ lineæ tam
longæ ducantur, quàm priores illæ a punctis a. & c. ad d. productæ fuerint. partianturq́; in partes no-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] uem, & dimidiam, incipiendo a punctis g. verſus r. hæ partes ſingulæ prioribus illis decem & octo, in quas
abaci longitudo diuiſa erat, reſpondebunt. ex his una & dimidia dabitur abaci craſſitudini, ita abacus erit al-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] ta parte una, & dimidia; ſed abacus conſtat quadra, & cymatio. quadra eſt tertia pars altitudinis abaci. reli-
quas duas cymatium occupat: ducendum uero cymatium molliter ad figuram litteræ S. constat enim ex
duobus c. ita ut ſuperiori inferior opponatur, ſuperiori recte, ut ſcribitur formato, infra abacum octo partes
erunt. qua parte igitur finis est quartæ & dimidiæ, ibi in cathetis centra oculi figenda ſunt. eſto ibi littera o.
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] cum terminus abaci ſit a b. uel c b. poſito igitur centro in puncto o. ducatur circinatio tam magna quan-
2220 ta est una ex octo partibus his punctis concluſa l m. Cyclus ille uolutæ oculus nominatur. et ita factum erit,
quod hactenus a Vitr. præceptum habemus. quod uero dicit.
Tvnc ab linea, quæ ſecundum abaci extremam partem demiſſa erit, in interiorem partem alia re-
cedat unius, & dimidiatæ partis latitudine.
cedat unius, & dimidiatæ partis latitudine.
Ita intelligendum eſt, ut uelit nobis terminos centrorum in oculo constituere, ut uolutam restè ducamus,
quod a paucis animaduerſum eſt, putantes Vitr. nullum omnino eorum centrorum lumen dediſſe, ſed in fine li-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] bri deſcriptam formam ſimpliciter poſuiſſe. Recedatur igitur ab extremitatibus abaci in partem interiorem
unius, & dimidiatæ partis ex uiginti latitudine, uerbi gratia ad ſignum s. & ab eo ſigno catheti, & perpen-
diculares lineæ deducantur, & ita diuidantur, ut dictum eſt. dicit enim.
quod a paucis animaduerſum eſt, putantes Vitr. nullum omnino eorum centrorum lumen dediſſe, ſed in fine li-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] bri deſcriptam formam ſimpliciter poſuiſſe. Recedatur igitur ab extremitatibus abaci in partem interiorem
unius, & dimidiatæ partis ex uiginti latitudine, uerbi gratia ad ſignum s. & ab eo ſigno catheti, & perpen-
diculares lineæ deducantur, & ita diuidantur, ut dictum eſt. dicit enim.
Deinde eæ lineæ diuidantur ita, ut quatuor partes, &
dimidia ſub abaco relinquantur.
Tunc in
3330 eo loco, quilocus diuidit quatuor & dimidiam, & tres & dimidiam partem, centrum oculi ſignetur,
ducaturq́; ex eo centro rotunda circinatio, tam magna in dia metro, quàm una pars ex octo partibus
eſt, ea erit oculi magnitudine. Hæc declarata ſunt. deinde dicit.
3330 eo loco, quilocus diuidit quatuor & dimidiam, & tres & dimidiam partem, centrum oculi ſignetur,
ducaturq́; ex eo centro rotunda circinatio, tam magna in dia metro, quàm una pars ex octo partibus
eſt, ea erit oculi magnitudine. Hæc declarata ſunt. deinde dicit.
Et in ea catheto diametros reſpondens agatur.
Ducenda eſt diametro per medium' oculum, quæ re-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] ſpondeat catheto r g. ita ut r g. perpendicularis, & ad rectos angulos ſit diametro n u.
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] ſpondeat catheto r g. ita ut r g. perpendicularis, & ad rectos angulos ſit diametro n u.
Tvnc ab ſummo ſub abaco inceptum in ſingulis tetrantorum pactionibus dimidiatum.
oculi ſpa-
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] tium minuatur, donec in eundem tetrantem, qui eſt ſub abaco ueniat.
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52] tium minuatur, donec in eundem tetrantem, qui eſt ſub abaco ueniat.
Hic difficultas.
Prima igitur conditio uolutæ est, ut diuiſo oculo in partes quatuor, quas tetrantes uocat
( quod fit catheti, & diametri dictæ interſectione ) ducatur ita, ut dimidiatum oculi ſpatium minuatur in ſin-
gulis flexibus ſingulorum tetrantum. Sed boc fierinon poteſt niſi contra ſingularum actionum inueniantur.
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52]4440 Vitr. certe centrorum terminos innuit, quando dixit, {Tunc ab ea linea, quæ ſecundum abaci extremam par-
tem demiſſa erit in interiorem partem, aliarecedat unius, & dimidiatæpartis latitudinem. } Sumaturigitur
ſpatium o ſ. & transferatur ab a. uerſus c. & ab c. uerſus a. & demittantur catheti ſ x. hæ diame-
trum n u. ſecabunt in y. Spatium igitur quod est ab o. cẽtro oculi ad y. transferatur in diametro ad aliam par
tem à y. ad r. tantum erit latus quadrati quod circa o. centrum faciendum est. fiat ergo quadratum 1. 2. 3. 4.
& ducantur diagoniæ 1. 3. & 2. 4. quæ in ſex partes ſingulæ diuidantur, in quibus erunt centra uolutarum col
locanda, ta ut incipiamus a ſigno 1. & dilatemus circinum uſque ad abacum, & ducamus tetrantem circuli
in exteriorem partem ab abaco ad diametrum n u. postea mutato centro in 2. ducamus alium tetrantem a dia
metro n u. ad cathetum r g. in inferiorem partem. inde mutato centro in ſigno 3. ducamus alium tetrantem
a catheto r g. ad diametrum n u. ac demum a diametro n u. facto centro in ſigno 4. ducamus alium tetrantem
5550 ad cathetum r g. ſub abaco. atque ita erit prima uolutæ circinatio abſoluta. & in ſingulis tetrantum ductio-
nibus diminutum erit dimidiatum oculi ſpatium. ita fiet ſecunda circinatio poſito circino in primo puncto diago
niæ in ſigno proximo ſigni. 1. & postea in alijs eodem ordine quo facta est primum circinatio. non abſimili mo
do fiet tertia circinatio. Sed boc erit diſcriminis omnium circinationum, quod quemadmodum prima in ſin-
gulis tetrantum ductionibus diminuebat dimidiatum oculi ſpatium, ita in tertia diminuitur ſexta, & in ſecun-
da tertia pars oculi. Latitudo uolutæ eadem ratione abſoluitur contractione circini, dimidiato oculo a prima
circinatione eiſdem centris, eodem ordine conſtitutis. ita ut par latitudo uſque in finem ſit ab utraque circinæ
tione, in fine autem contrahitur paulum, ita enim res poſtulat.
( quod fit catheti, & diametri dictæ interſectione ) ducatur ita, ut dimidiatum oculi ſpatium minuatur in ſin-
gulis flexibus ſingulorum tetrantum. Sed boc fierinon poteſt niſi contra ſingularum actionum inueniantur.
4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52]4040[Handwritten note 40]4141[Handwritten note 41]4242[Handwritten note 42]4343[Handwritten note 43]4444[Handwritten note 44]4545[Handwritten note 45]4646[Handwritten note 46]4747[Handwritten note 47]4848[Handwritten note 48]4949[Handwritten note 49]5050[Handwritten note 50]5151[Handwritten note 51]5252[Handwritten note 52]4440 Vitr. certe centrorum terminos innuit, quando dixit, {Tunc ab ea linea, quæ ſecundum abaci extremam par-
tem demiſſa erit in interiorem partem, aliarecedat unius, & dimidiatæpartis latitudinem. } Sumaturigitur
ſpatium o ſ. & transferatur ab a. uerſus c. & ab c. uerſus a. & demittantur catheti ſ x. hæ diame-
trum n u. ſecabunt in y. Spatium igitur quod est ab o. cẽtro oculi ad y. transferatur in diametro ad aliam par
tem à y. ad r. tantum erit latus quadrati quod circa o. centrum faciendum est. fiat ergo quadratum 1. 2. 3. 4.
& ducantur diagoniæ 1. 3. & 2. 4. quæ in ſex partes ſingulæ diuidantur, in quibus erunt centra uolutarum col
locanda, ta ut incipiamus a ſigno 1. & dilatemus circinum uſque ad abacum, & ducamus tetrantem circuli
in exteriorem partem ab abaco ad diametrum n u. postea mutato centro in 2. ducamus alium tetrantem a dia
metro n u. ad cathetum r g. in inferiorem partem. inde mutato centro in ſigno 3. ducamus alium tetrantem
a catheto r g. ad diametrum n u. ac demum a diametro n u. facto centro in ſigno 4. ducamus alium tetrantem
5550 ad cathetum r g. ſub abaco. atque ita erit prima uolutæ circinatio abſoluta. & in ſingulis tetrantum ductio-
nibus diminutum erit dimidiatum oculi ſpatium. ita fiet ſecunda circinatio poſito circino in primo puncto diago
niæ in ſigno proximo ſigni. 1. & postea in alijs eodem ordine quo facta est primum circinatio. non abſimili mo
do fiet tertia circinatio. Sed boc erit diſcriminis omnium circinationum, quod quemadmodum prima in ſin-
gulis tetrantum ductionibus diminuebat dimidiatum oculi ſpatium, ita in tertia diminuitur ſexta, & in ſecun-
da tertia pars oculi. Latitudo uolutæ eadem ratione abſoluitur contractione circini, dimidiato oculo a prima
circinatione eiſdem centris, eodem ordine conſtitutis. ita ut par latitudo uſque in finem ſit ab utraque circinæ
tione, in fine autem contrahitur paulum, ita enim res poſtulat.
Capitvli autem craſsitudo ſic eſt facienda, ut ex nouem partibus, &
dimidia tres partes præ-
pendeant infra aſtragalum ſummi ſcapi. Cymatio adempto abaco, & canali reliqua ſit pars.
6660pendeant infra aſtragalum ſummi ſcapi. Cymatio adempto abaco, & canali reliqua ſit pars.
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