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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40 - 49
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60 - 69
70 - 79
80 - 89
90 - 99
100 - 109
110 - 119
120 - 129
130 - 139
140 - 149
150 - 159
160 - 169
170 - 179
180 - 189
190 - 199
200 - 209
210 - 219
220 - 229
230 - 239
240 - 249
250 - 259
260 - 269
270 - 279
280 - 289
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300 - 309
310 - 319
320 - 329
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THEOREM. ARIT.
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47
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file
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0047
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0047
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<
p
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<
s
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xml:space
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preserve
">Sumantur enimtres numeri continui proportionales, cuiuſcunque denique pro
<
lb
/>
portionalitatis, qui in ſummam colligantur, ac poſtmodum, regula de trib. dica-
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lb
/>
mus. </
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>
<
s
xml:id
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xml:space
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">Si ſumma hæc primo numero propoſito in tres partes diuidendo reſpondet,
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cuireſpondebit vna ex tribus partibus huiuſcę
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type
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? </
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<
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xml:id
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xml:space
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">idem dereliquis duabus pa
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rti
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bus dico.</
s
>
</
p
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<
p
>
<
s
xml:id
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xml:space
="
preserve
">Exempli gratia, ſi proponatur numerus .57. diuidendus in tres continuas partes
<
lb
/>
proportionales proportione ſeſquialtera, tres numeros in eiuſmodi proportio-
<
lb
/>
nalitate diſtinctos ſumemus, vt potè .4. 6. 9. qui in ſummam collecti dabunt
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ſum- mam
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type
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mã</
reg
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.19.
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reg
norm
="
dicemusque
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type
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simple
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>
ſi .19. dant .4. quid
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dabunt
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type
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.57? </
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<
s
xml:id
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xml:space
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preserve
">vnde proueniens vnius partis erit
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num
value
="
12
">.
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12</
num
>
. </
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>
<
s
xml:id
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xml:space
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">Tum ſi dicamus, ſi .19. dat .6. quid dabit .57? </
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<
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xml:space
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<
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xml:space
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">Poſtremò, ſi
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num
value
="
19
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19.</
num
>
dat .9. quid dabit .57? </
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<
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xml:space
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preserve
">nempe .26. atque ita dabitur .18. cuius quadratum æqua-
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bitur producto reliquarum duarum partium inter ſe.</
s
>
</
p
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<
p
>
<
s
xml:id
="
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xml:space
="
preserve
">Quod vt ſciamus, numerus propoſitus in tres quaſlibet partes diuidendus ſi-
<
lb
/>
gnificetur linea
<
var
>.a.d.</
var
>
tres autem numeri dictæ proportionalitatis, lineis
<
var
>.e.f</
var
>
:
<
var
>f.g.</
var
>
<
lb
/>
et
<
var
>.g.h.</
var
>
directè inter ſe coniunctis denotentur. </
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xml:space
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">Cogitemus pariter lineam
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var
>
in
<
lb
/>
tres partes diuiſam
<
var
>.a.b</
var
>
:
<
var
>b.c.</
var
>
et
<
var
>.c.d.</
var
>
eadem cum cæteris proportionalitate, </
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>
<
s
xml:id
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"
xml:space
="
preserve
">tunc ea-
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lb
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dem erit proportio
<
var
>.a.d.</
var
>
ad quamlibet ſuarum partium, quæ eſt
<
var
>.e.h.</
var
>
ad reſponden
<
lb
/>
tem ipſius in
<
var
>.a.d</
var
>
: Verbi gratia reſpondentem
<
var
>.a.b.</
var
>
ipſi
<
var
>.e.f.</
var
>
et
<
var
>.b.c</
var
>
:
<
var
>f.g.</
var
>
et
<
var
>.c.d</
var
>
:
<
var
>g.h</
var
>
. </
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>
<
s
xml:id
="
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xml:space
="
preserve
">Di
<
lb
/>
co enim quòd ita ſe habebit
<
var
>.a.d.</
var
>
ad
<
var
>.c.d.</
var
>
ſicut
<
var
>.e.h.</
var
>
ad
<
var
>.g.h</
var
>
. </
s
>
<
s
xml:id
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xml:space
="
preserve
">Nam cum ſic ſe habeat
<
var
>.a.
<
lb
/>
b.</
var
>
ad
<
var
>.b.c.</
var
>
ſicut
<
var
>.e.f.</
var
>
ad
<
var
>.f.g.</
var
>
ex præſuppoſito, permutando ſic ſe habebit
<
var
>.a.b.</
var
>
ad
<
var
>.e.f.</
var
>
ſi-
<
lb
/>
cut
<
var
>.b.c.</
var
>
ad
<
var
>.f.g.</
var
>
& eadem ratione ſic ſe habe-
<
lb
/>
bit
<
var
>.c.d.</
var
>
ad
<
var
>.g.h.</
var
>
ſicut
<
var
>.b.c.</
var
>
ad
<
var
>.f.g.</
var
>
&
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0047-01
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xlink:href
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ter</
reg
>
ſicut
<
var
>.a.b.</
var
>
ad
<
var
>.e.f.</
var
>
ex quo ex .13. quinti ſic
<
lb
/>
ſe habebit tota
<
var
>.a.d.</
var
>
ad totam
<
var
>.e.h.</
var
>
ſicut
<
var
>.c.d.</
var
>
<
lb
/>
ad
<
var
>.g.h.</
var
>
aut
<
var
>.b.c.</
var
>
ad
<
var
>.f.g.</
var
>
aut
<
var
>.a.b.</
var
>
ad
<
var
>.e.f.</
var
>
per-
<
lb
/>
mutando itaque propoſitum manifeſtum erit, ipſum autem productum
<
var
>.a.b.</
var
>
in
<
var
>.c.b.</
var
>
<
lb
/>
æquale erit quadrato
<
var
>.b.c.</
var
>
ex .15. fexti aut .20. ſeptimi.</
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>
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<
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xml:space
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">THEOREMA
<
num
value
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56
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num
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.</
head
>
<
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<
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<
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style
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emph
>
aliud quoque problema indeterminatum propoſuerunt, quod ex
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more ratione à me definietur, eſt autem eiuſmodi.</
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<
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<
s
xml:id
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xml:space
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">Quomodo propoſitus numerus in tres eiuſmodi partes diuidatur, vt
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vnius æquale fit fummæ quadratorum reliquarum duarum partium.</
s
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<
p
>
<
s
xml:id
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xml:space
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preserve
">Hoc vt efficiamus tria quadrata ſeparata ſumamus,
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<
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type
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æquale ſit reliquis
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duobus; </
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xml:space
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<
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type
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<
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autem
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type
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">autẽ</
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>
radices in ſummam ſimul colligantur, tum regulam de tribus ſe
<
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quemur, ratione præcedenti theoremate demonſtrata, & rectè vt infra docebimus,
<
lb
/>
quod autem dico de quadratis, etiam de cubis, & quibuſuis dignitatibus aſſero.</
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>
</
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<
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>
<
s
xml:id
="
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"
xml:space
="
preserve
">Exempli gratia, ſi numerus diuiſibilis proponatur .30. in tres eiuſmodi partes di
<
lb
/>
uidendus, vt quadratum vnius æquale ſit ſummæ quadratorum reliquarum duarum
<
lb
/>
partium, in primis radices trium quadratorum ſumemus, ſic quomodocunque ſe
<
lb
/>
habentes, vt maius ipſorum æquale ſit ſummæ reliquorum duorum, verbi gratia .25.
<
lb
/>
16. et .9. nempe .5. 4. et .3. quæ ſi colligantur in ſummam efficiunt .12. </
s
>
<
s
xml:id
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xml:space
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preserve
">Tum ex regu-
<
lb
/>
la de tribus dicemus, ſi .12. reſpondet .30: </
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>
<
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xml:space
="
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">cui, 5. radix maior reſpondebit? </
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>
<
s
xml:id
="
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"
xml:space
="
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">nem-
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pe .12. cum dimidio.</
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>
</
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<
p
>
<
s
xml:id
="
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"
xml:space
="
preserve
">Deinde ſi dixerimus ſi .12. valet .30. quid valebit .4. radix media? </
s
>
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xml:id
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"
xml:space
="
preserve
">nempe vale-
<
lb
/>
bit .10. tertia autem minor .7. cum dimidio. </
s
>
<
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xml:id
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xml:space
="
preserve
">Itaquetota ſumma erit .30. & quadra- </
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>
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