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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THEOREM. ARIT.
retur .20. ſcilicet et .4. certè .24. perſingulas partes diuiſo, daretur vnum proue-
niens ſex integra, & alterum vnum & quinta pars, quorum ſumma eſſet ſeptem in-
tegra cum quinta parte, tum altera parte per alteram diuiſa, daretur vnum proue-
niens quinque integrorum & alterum vnius quinti tantum, quorum ſumma eſſet
quinque integra, & vna quinta pars, minor prima reliquorum duorum prouenien-
tium per binarium.
Cuius conſiderationis cauſa, propoſitus numerus linea .q.p. ſignificetur, eius duę
partes lineis .q.x. et .x.p. tum .q.f. ſit proueniens eg diuiſione totius .q.p. per .x.p. et .
q.i. ſit proueniens eg diuiſione eiuſdem .q.p. per .q.x. adhæc .h.m. ſit proueniens,
eg diuiſione .q.x. per x.p. et .h.k. proue-
niensex diuiſione .p.x. per .q.x. patet igi-
figure: 44
[Figure 44]
tur eg .22. theoremate huiuslibri proue-
niés.h.m. minus eſſe proueniente .q.f. per
vnitaté, & proueniens .h.k. minus proue-
niente .q.i. per alteram vnitatem. Itaque .
f.q.i. minor ergo .m.h.k. per numerum binarium, quoderat propoſitum.

THEOREMA. XXXIII.

QVilibet numerus, medius eſt
proportionalis inter numerum
figure: 45
[Figure 45]
ſui quadrati & vnitatem.
Detur enim numerus propoſitus,
qui linea .a.u. ſignificetur, cuiusqua-
dratum ſit .u.e. vnitas linearis ſit .i.a.
et ſuperficialis .o. patebit eg .18. ſexti
aut 11. octaui proportionem .u.e. ab .
o. futuram duplam proportioni .u.a.
ab .i.a. ſed .i.a. et.o. eadem (ſpecie)
res sunt, tanta ſcilicet .a.i. quanta .o. vni
figure: 46
[Figure 46]
tas eſt, Itaque proportio numeri .u.e.
ab .u.a. æqualis ergo proportioni .u.a.
ab .i.a. Quare numerus .u.a. inter nu-
merum .u.e. & vnitatem, medius ergo
proportionalis.

THEOREMA XXXIIII.

HOc ipſum quem diximus & alia ratione ſpeculari licebit.
Propoſitus numerus, nunc etiam per .a.u. ſignificetur, eius quadratum per .
u.e. vnitas linearis per .a.i. productumque; .a.u. in .a.i. terminetur, ſitque; .e.i. quare
e.i. conſtabit numero íuperficiali æquali numero lineari .a.u. & eg prima fexti aut .
18. vel .19. ſeptimi, eadem ergo proportio .u.e. ab .i.e. quæ eſt .a.u. ab .a.i. ſed nu-
merus .a.u. cum numero .e.i. idem ſpecie eſt. Itaque medius eſt proportiona-
lis inter .u.e. & vnitatem.

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