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]

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6452IO. BAPT. BENED.
Sint 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. ad .c.d. eadem ſit ipſius .e.f. ad .g.h.
Tunc permutando dico eandem pro
portionem fore ipſius .a.b. ad .e.f. quæ ipſius .c.d. ad .g.h.
Nam, ex 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 ex .a.b. æquale erit
c.d. & reſiduum .e.i. æquale .g.h.
Sit igitur exempli gratia .c.d. maior .g.h. per .c.n.
vnde .n.d. æqualis erit .g.h.
quare .a.m. maior erit .e.i. per .a.K. æqualem .c.n. ex com-
muni ſcientia.
Vnde .K.m. æqualis erit .n.d. hoc eſt ipſi .g.h. hoc eſt ipſi e.i. Quare ex
communi conceptu .b.K. æqualis erit ipſi .f.e. ſed .n.d. æqualis eſt .g.h. vt dictum eſt.
Cum ergo .b.K. æqualis ſit .e.f. et .d.n. ipſi .g.h. et .a.b. maior ſit ipſa .K.b. per .a.K. æqua-
lem ipſi .c.n. per quam c.n: d.c. maior eſt ipſa .d.n. ſequitur verum eſſe propoſitum hoc
eſt, quod eadem proportio ſit ipſius .a.b. ad .e.f. quæ .c.d. ad .g.h. arithmetice ſcilicet.
87[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 erunt .4. et .3. eadem proportione, qua diuidentes.
Cuius eſt ratio numerus diuiſibilis ſignificetur rectangulis .u.x. et .n.e. diuidentes
autem ſint .u.o. et .e.o.
quare ex ijs, quæ .10.
88[Figure 88] theoremate dicta fuerunt .u.x. per .u.o. diui-
ſo dabit .x.o. & diuiſo .n.e. per .e.o. dabit .o.
n
.
Dicimus itaque eandem eſſe proportionem
o.x. ad .o.n. quæ .e.o. ad .o.u. quod patet ſub
ſcriptam figuram conſiderantibus, in qua,
ex .15. ſexti aut .20. ſeptimi, eadem propor-
tio cernitur .o.x. ad .o.n. quæ .o.e. ad .o.u.
THEOREMA LXXX.
CVR quauis quantitate, tribus
89[Figure 89] aut quatuor aut etiam pro libi-
to pluribus diuidentibus numeris di-
uifa, prouenientia eandem prorſus
inter ſe proportionem ſeruabunt,
quam ipſi diuidentes habere compe
riuntur.
Exempli gratia, proponitur nu-
merus .60. quinque numeris diuiden
dus, vtpotè .30. 20. 15. 12. 10. pro-
uenientia erunt .2. 3. 4. 5. 6. eadem

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