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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287275EPISTOL AE. ſit quadratum ipſius .D.B. Nunc ſupponendo .A.B. ſimile .a.b. clarum erit ex diffini-
b.
ſexti, vel .18. ſeu .19. ſeptimi,
tunc cum dixerimus ſi .a.b. ita reſpondet ad .b.c. ergo .A.
B.
quare ex regula de tribus rectè fit multiplicando .
A.B.
per .b.c. productum verò diuidendo per .a.b. ex .15. ſexti vel .20. ſeptimi, cuius
Sed aliter idem poſſe fieri ſpeculatus ſum, hoc eſt multiplicando numerum .49.
rò diuidere per radicem quadratam ipſius .1000. vnde prouentus .91. erit numerus
vnius ordinis .3500. numeri propoſiti.
Cuius operationis ſpeculatio eſt iſta.
Sit .a.b. quadratum .1000. et .a.c. ſua
tum ipſius .1000. et .a.e. vnus ordo.
Sit etiam .A.B. quadratum .3500. &
A.C. eius radix et .A.D. rectangulum
ipſius numeri .3500. propoſiti, ſimile
tamen rectangulo .a.d. et .A.E. eius
vnus ordo.
Cum enim .a.b. æquale ſit
a.d. et .A.B: A.D. tunc .a.c. erit media
proportionalis inter .a.e. et .e.d. & ſic
A.C. erit etiam media proportiona
lis inter .A.E. et .E.D. per .16. ſexti,
ſeu .20. ſeptimi, & quia proporrio. A
E. ad .E.D. æqualis eſt proportioni .
a.e.
ad .e.d. cum .A.D. ſupponatur ſi-
mile .a.d. ergo proportio .A.E. ad .A
C.
ęqualis erit proportioni .a.e. ad .a.
c.
quę medietates ſunt totorum æqua-
lium, rectè igitur fiet ſi procedamus
ex regula de tribus, dicendo ſi .a.c.
correſpondet .a.e. tùc .A.C. correſpon
det .A.E. ex ſupradictis .15. ſexti. vel
20. ſeptimi.
Ratio verò quarti quæſiti per ſe
patet, quod eſt inuenire pauimentum
ſeu aream quadratam, in qua poſſint
locari quot homines volueris, ita in
ter ſe ſiti, ut vnuſquiſque occupet .
7.
pedes ipſius areę in longitudinem
et .3. per latitudinem à lateribus.
Seu ex propoſito hominum nume
ro inuenire numerum ipſorum loca-
bilem in aliqua area quadrata, ita,
vt vnuſquiſque occupet .21. pedes