Benedetti, Giovanni Battista de, Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte], 1585

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IO. BAPT. BENED.
quis, qua ratione fractus numerus .c.i. minor ſit in ſuo integro .d.b. fracto .a.i. in
ſuo integro .a.b. aut fracto .a.c. in ſuo integro .a.d. conſideret is quo pacto pro-
portio .c.i. ab .d.b. minor ſit proportione .a.i. ab .a.b. et .a.c. ab .a.d. hac ratione. Ma-
nifeſtum eſt eg prima ſexti de quantitate
continua, aut .18. ſeptimi Euclidis de diſcre
figure: 2
[Figure 2]
ta, proportionem ipſius .d.i. ab .d.b. eſſe ſi-
cut .a.i. ab .a.b. & cum .c.i. minor ſit .d.i.
velut pars ſuo toto, proportio, c.i. ab .d.b.
minor ergo proportione .d.i. ab .d.b. eg .8.
quinti, quare minor ergo pariter proportio-
ne .a.i. ab .a.b. eg .12. eiuſdem vnà etiam pro-
portio .c.i. ab .d.b. minor ergo .a.c. ab .a.d.
eg eiſdem cauſis, medio .c.b. eg quas pa-
tet ratio, cur fracti diuerſarum denomina-
tionum ab vnicam reducantur. Cur etiam
numeros integros in partes fractis ſimiles
frangere liceat, quæ omnia eg ſubſequenti
figura facilè cognoſci poſſunt.

THEOREMA II.

QVae ſit ratio, cur ch, qui numeros, fractos diuerſarum denominationum col-
ligere volunt, & in ſummam redigere, multiplicent vnum eg numerantibus
per denominatorem alterius, & poſtmodum denominatores adinuicem, quorum
vltimum productum, commune eſt denominans duorum priorum productorum,
quæ collecta in ſummam efficiunt quem quærebatur.
Qua in re ſciendum eſt, denominantes conſiderari tanquam partes vnius eiuſdẽ-
q́ue magnitudinis quantitatis continuæ, linearum (verbigratia) a.b. et .a.d. æqualium
in longitudine, quarum .a.b. in quatuor partes diuidatur, et .a.d. in tres. Quare ſi colli-
gere voluerimus duo tertia cum tribus quartis, multiplicabimus .a.c. duo tertia,
cum .a.b. diuiſa in 4. partes, produceturq́ue .c.b. octo partium ſuperficialium, de-
hinc multiplicando .a.i. tres quartas cum .a.d. diuiſa in .3. partes producetur .i.d. pri
mis ſingulis æqualis, nouem partium ſuper
ficialium, multiplicata deinde a.b. diui-
figure: 3
[Figure 3]
ſa in .4. partes per .a.d. in .3. diuiſa, produ-
cetur quadratum .d.b. in continuo, in 12.
partes diuiſum, quem ergo totum commune
ſingulis productis, quorum primum erat .c.
b. Quare .c.b. ita ſe habet ab totum .d.b. ſi-
cut .a.c. ab .a.d. eg prima ſexti in continuis,
aut .18. ſeptimi in diſcretis quantitatibus,
et .d.i. ab .d.b. ſicut .a.i. ab .a.b. eg eiſdem
propoſitionibus. Collectis deinde parti-
bus producti .c.b. cum partibus producti .
d.i. manifeſtè depræhendetur eiuſmodi
ſummam componi eg partibus vnius totius
communis ſingulis earum.

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