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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106
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71
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rhead
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THEOREM. ARITH.
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n
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81
"
file
="
0081
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0081
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primi viatoris) tot milliaria abſoluat vnus
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<
figure
xlink:label
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fig-0081-01
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fig-0081-01a
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number
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109
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0081-01
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0081-01
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quot alter abſque vlla differentia, quæ ſigni-
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ficetur per
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>.o.s.</
var
>
neceſſe eſtitaque ex communi
<
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conceptione tot dies eſſe poſt
<
var
>.o.s.</
var
>
quotante-
<
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ceſſerant, vt exceſſus æqualis ſit defectui, qui
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ſimul collecti, iuncta etiam
<
var
>.o.s.</
var
>
duplum erunt
<
lb
/>
<
var
>d.s.</
var
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dempta vnitate, prout facilè in ſubſcripta
<
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/>
figura qui ſque per ſe ſcientificè poterit ſpecu
<
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/>
lari. </
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>
<
s
xml:id
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echoid-s936
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xml:space
="
preserve
">Quamobrem conſultum erit duplicare
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lb
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numerum
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>.o.s.</
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>
& exduplo vnitatem detrahe-
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lb
/>
re, quandoquidem dies ſupra
<
reg
norm
="
infraque
"
type
="
simple
">infraq́;</
reg
>
<
var
>.o.s.</
var
>
cum
<
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/>
die
<
var
>.o.s.</
var
>
minores ſunt duplo numeri
<
var
>.d.s.</
var
>
aut
<
var
>.o.
<
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/>
s.</
var
>
(quodidem eſt) vnitate.</
s
>
</
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</
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<
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xml:id
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type
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n
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<
head
xml:id
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xml:space
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">THEOREMA
<
num
value
="
107
">CVII</
num
>
.</
head
>
<
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<
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xml:space
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<
emph
style
="
sc
">QVod</
emph
>
<
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norm
="
ſiſecundus
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type
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context
">ſiſecũdus</
reg
>
viator
<
reg
norm
="
ordinem
"
type
="
context
">ordinẽ</
reg
>
<
reg
norm
="
ſecundæ
"
type
="
context
">ſecũdæ</
reg
>
progreſſionis arithmeticæ
<
reg
norm
="
ſeruansiter
"
type
="
context
">ſeruãsiter</
reg
>
<
lb
/>
agat, nempe ea quæ ab vno per binarium aſcendit, ſemper numerus dierum
<
lb
/>
æqualis erit numero milliarium diurnorum primi viatoris.</
s
>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s938
"
xml:space
="
preserve
">In cuius gratiam animaduertendum eſt numerus ne milliarium diurnorum pri-
<
lb
/>
mi viatoris par an impar ſit. </
s
>
<
s
xml:id
="
echoid-s939
"
xml:space
="
preserve
">Etenim ſi par eſt, primus viator in fine ſingulorum die-
<
lb
/>
rum primæ medietatis numeri omnium dierum ſecundum antecedet numero diſpa
<
lb
/>
ri milliarium; </
s
>
<
s
xml:id
="
echoid-s940
"
xml:space
="
preserve
">altero verò dimidio numero dierum, à ſecundo numero etiam diſpa
<
lb
/>
ri præteribitur, vt in ſequenti figura patet. </
s
>
<
s
xml:id
="
echoid-s941
"
xml:space
="
preserve
">Nam prima die, ſecundus ex primo
<
lb
/>
milliare vnum ex numero pari, qui à primo conficitur detrahit; </
s
>
<
s
xml:id
="
echoid-s942
"
xml:space
="
preserve
">ſecunda verò die
<
lb
/>
idem ſecundus, duo ſubtrahit milliaria exdiſpari, qui primo reliquus fuerat,
<
reg
norm
="
ſicque
"
type
="
simple
">ſicq́;</
reg
>
<
lb
/>
perpetuò diſpar remanet vſque ad vnitatem, ad quam cum peruenerint, nempe ad
<
lb
/>
illius diei exitum, quo primus ſecundum vnitate tantummodò ſuperat, manifeſtè
<
lb
/>
depræhendetur ſubſequente dieſecundum vnitate primum ſuperaturum, altera ve
<
lb
/>
rò tribus vnitatibus, prout penultima die ſecundus à primo tribus vnitatibus ſupera
<
lb
/>
batur. </
s
>
<
s
xml:id
="
echoid-s943
"
xml:space
="
preserve
">Quare neceſſe erit, tot diebus ſecundum cum primo iter agere, inchoan-
<
lb
/>
do ab ea die, qua fecundus primam ſuperabit, quot egerat dum à primo ſuperare-
<
lb
/>
tur, vt ex communi conceptione, media figura
<
var
>.A.</
var
>
depræhendi poteſt. </
s
>
<
s
xml:id
="
echoid-s944
"
xml:space
="
preserve
">Quod au-
<
lb
/>
tem ſingula dimidia dierum, dimi-
<
lb
/>
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xlink:label
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fig-0081-02
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xlink:href
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fig-0081-02a
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number
="
110
">
<
image
file
="
0081-02
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0081-02
"/>
</
figure
>
dia ſint numeri milliarium diurno-
<
lb
/>
rum primi; </
s
>
<
s
xml:id
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"
xml:space
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preserve
">patebit exſequenti fi-
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lb
/>
gura, cogitato termino
<
var
>.u.n.</
var
>
vlti-
<
lb
/>
mo progreſſionis ſuperatę à primo
<
lb
/>
vſque ad vnitatem
<
var
>.e.</
var
>
quiterminus
<
lb
/>
<
var
>u.n.</
var
>
coniunctus primo
<
var
>.o.</
var
>
nempe
<
var
>.e.</
var
>
<
lb
/>
ſemper
<
reg
norm
="
duplum
"
type
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">duplũ</
reg
>
<
unsure
/>
eſt numeritermino-
<
lb
/>
rum
<
var
>.o.n.</
var
>
vt .94. theoremate circa
<
lb
/>
finem dictum fuit. </
s
>
<
s
xml:id
="
echoid-s946
"
xml:space
="
preserve
">Sed
<
var
>.u.n.</
var
>
cum
<
var
>.e.</
var
>
<
lb
/>
numero æquali conſtat numero
<
lb
/>
milliarium diurnorum primi viatoris, ex quo ſequitur totum numerum dierum, quo
<
lb
/>
rum
<
var
>.o.n.</
var
>
dimidium eſt, æqualem eſſe numero milliarium diuruorum primi via-
<
lb
/>
toris.</
s
>
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