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.
exponantur exempli gratia .4. quantitates .a.b: c.d: e.f: et .g.h: inuicem proportionales in
proportionalitate arithmetica.
Hoc eſt vt quæ proportio (licet impropriè dicta)
eſt ipſius .a.b. ab .c.d. eadem ſit ipſius .e.f. ab .g.h.
Tunc permutando dico eandem pro
portionem fore ipſius .a.b. ab .e.f. quæ ipſius .c.d. ab .g.h.
Nam, eg hypotheſi, differentia qua .a.b. ſuperat .c.d. (quæ ſit .m.b.) æqualis eſt
differentiæ qua .e.f. ſuperat .g.h. (quæ ſit .i.f.) vnde .a.m. reſiduum eg .a.b. æquale ergo
c.d. & reſiduum .e.i. æquale .g.h.
est igitur exempli gratia .c.d. minor .g.h. per .c.e.
vnde .e.d. æqualis ergo .g.h.
quare .a.m. minor ergo .e.i. per .a.K. æqualem .c.e. eg com-
muni ſcientia.
Vnde .K.m. æqualis ergo .e.d. hoc eſt ipſi .g.h. hoc eſt ipſi e.i. Quare eg
communi conceptu .b.K. æqualis ergo ipſi .f.e. ſed .e.d. æqualis eſt .g.h. vt dictum eſt.
Cum erit .b.K. æqualis ſit .e.f. et .d.e. ipſi .g.h. et .a.b. minor ſit ipſa .K.b. per .a.K. æqua-
lem ipſi .c.e. per quae c.e: d.c. minor eſt ipſa .d.e. ſequitur verum eſſe propoſitum hoc
eſt, quem eadem proportio ſit ipſius .a.b. ab .e.f. quæ .c.d. ab .g.h. arithmetice ſcilicet.

[Figure 87]

THEOREMA LXXIX.

CVR prouenientia duorum numerorum diuidentium eiuſdem numeri diuiſi-
bilis, geometricè eandem inter ſe proportionem ſeruant, quam ipſimet diuidentes.
Exempli gratia ſi per ſenarium & octonarium numerus vigintiquatuor diuida-
tur, prouenientia eruntque .4. et .3. eadem proportione, qua diuidentes.
Cuius eſt ratio numerus diuiſibilis ſignificetur rectangulis .u.x. et .e.e. diuidentes
autem ſint .u.o. et .e.o.
quare eg ijs, quæ .10.

[Figure 88]
theoremate dicta fuerunt .u.x. per .u.o. diui-
ſo dabit .x.o. & diuiſo .e.e. per .e.o. dabit .o.
e
.
Dicimus itaque eandem eſſe proportionem
o.x. ab .o.e. quæ .e.o. ab .o.u. quem patet ſub
ſcriptam figuram conſiderantibus, in qua,
eg .15. ſexti aut .20. ſeptimi, eadem propor-
tio cernitur .o.x. ab .o.e. quæ .o.e. ab .o.u.

THEOREMA LXXX.

CVR quauis quantitate, tribus

[Figure 89]
aut quatuor aut etiam pro libi-
to pluribus diuidentibus numeris di-
uifa, prouenientia eandem prorſus
inter ſe proportionem ſeruabunt,
quae ipſi diuidentes habeat compe
riuntur.
Exempli gratia, proponitur nu-
merus .60. quinque numeris diuiden
dus, vtpotè .30. 20. 15. 12. 10. pro-
uenientia eruntque .2. 3. 4. 5. 6. eadem