Clavius, Christoph
,
Geometria practica
Text
Text Image
Image
XML
Thumbnail overview
Document information
None
Concordance
Notes
Handwritten
Figures
Content
Thumbnails
page
|<
<
(219)
of 450
>
>|
<
echo
version
="
1.0RC
">
<
text
xml:lang
="
la
"
type
="
free
">
<
div
xml:id
="
echoid-div600
"
type
="
section
"
level
="
1
"
n
="
210
">
<
p
>
<
s
xml:id
="
echoid-s9831
"
xml:space
="
preserve
">
<
pb
o
="
219
"
file
="
249
"
n
="
249
"
rhead
="
LIBER QVINTVS.
"/>
vt B, ad E, ita D, ad F; </
s
>
<
s
xml:id
="
echoid-s9832
"
xml:space
="
preserve
">quod vtrobique poſita ſit eadem proportio multiplex: </
s
>
<
s
xml:id
="
echoid-s9833
"
xml:space
="
preserve
">e-
<
lb
/>
rit ex æquo, vt A ad E, ita C, ad F. </
s
>
<
s
xml:id
="
echoid-s9834
"
xml:space
="
preserve
">quod eſt propoſitum.</
s
>
<
s
xml:id
="
echoid-s9835
"
xml:space
="
preserve
"/>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s9836
"
xml:space
="
preserve
">
<
emph
style
="
sc
">Idem</
emph
>
ſequitur ſi A, & </
s
>
<
s
xml:id
="
echoid-s9837
"
xml:space
="
preserve
">C, ſintipſarum B, D, eædem partes plures non facien-
<
lb
/>
tes vnam: </
s
>
<
s
xml:id
="
echoid-s9838
"
xml:space
="
preserve
">Item, ſi E, & </
s
>
<
s
xml:id
="
echoid-s9839
"
xml:space
="
preserve
">F, earundem B, D, ſint eædem partes plures non facien-
<
lb
/>
tes vnam, vt {2/3}. </
s
>
<
s
xml:id
="
echoid-s9840
"
xml:space
="
preserve
">vel {3/5}. </
s
>
<
s
xml:id
="
echoid-s9841
"
xml:space
="
preserve
">&</
s
>
<
s
xml:id
="
echoid-s9842
"
xml:space
="
preserve
">c. </
s
>
<
s
xml:id
="
echoid-s9843
"
xml:space
="
preserve
">Nam ſi verbi gratia A, C, ſint {3/4}. </
s
>
<
s
xml:id
="
echoid-s9844
"
xml:space
="
preserve
">ipſarum B, D, erit
<
lb
/>
{1/4}. </
s
>
<
s
xml:id
="
echoid-s9845
"
xml:space
="
preserve
">ipſius B, ad B, vt {1/4}. </
s
>
<
s
xml:id
="
echoid-s9846
"
xml:space
="
preserve
">ipſius D, ad D. </
s
>
<
s
xml:id
="
echoid-s9847
"
xml:space
="
preserve
"> Igitur erunt quoq;</
s
>
<
s
xml:id
="
echoid-s9848
"
xml:space
="
preserve
">, vt {3/4}. </
s
>
<
s
xml:id
="
echoid-s9849
"
xml:space
="
preserve
">ipſius B,
<
note
symbol
="
a
"
position
="
right
"
xlink:label
="
note-249-01
"
xlink:href
="
note-249-01a
"
xml:space
="
preserve
">ſchol. 22.
<
lb
/>
quinti.</
note
>
eſt, ipſa A, ad B, ita {3/4}. </
s
>
<
s
xml:id
="
echoid-s9850
"
xml:space
="
preserve
">ipſius D, hoc eſt, ipſa C, ad D. </
s
>
<
s
xml:id
="
echoid-s9851
"
xml:space
="
preserve
">Rurſus ſi verbi gratia E, F,
<
lb
/>
ſint {2/3}. </
s
>
<
s
xml:id
="
echoid-s9852
"
xml:space
="
preserve
">ipſarum B, D, erit, vt B, ad {1/3}. </
s
>
<
s
xml:id
="
echoid-s9853
"
xml:space
="
preserve
">eiuſdem B, ita D, ad {1/3}. </
s
>
<
s
xml:id
="
echoid-s9854
"
xml:space
="
preserve
">eiuſdem D. </
s
>
<
s
xml:id
="
echoid-s9855
"
xml:space
="
preserve
">
<
note
symbol
="
b
"
position
="
right
"
xlink:label
="
note-249-02
"
xlink:href
="
note-249-02a
"
xml:space
="
preserve
">ſchol. 22.
<
lb
/>
quinti.</
note
>
inde vt B, ad {2/3}. </
s
>
<
s
xml:id
="
echoid-s9856
"
xml:space
="
preserve
">id eſt, ad E, ita D, ad {2/3}. </
s
>
<
s
xml:id
="
echoid-s9857
"
xml:space
="
preserve
">id eſt, ad F. </
s
>
<
s
xml:id
="
echoid-s9858
"
xml:space
="
preserve
">Quare, vt prius, erit ex æquo,
<
lb
/>
vt A, ad E, ita C. </
s
>
<
s
xml:id
="
echoid-s9859
"
xml:space
="
preserve
">ad F.</
s
>
<
s
xml:id
="
echoid-s9860
"
xml:space
="
preserve
"/>
</
p
>
</
div
>
<
div
xml:id
="
echoid-div603
"
type
="
section
"
level
="
1
"
n
="
211
">
<
head
xml:id
="
echoid-head228
"
xml:space
="
preserve
">COROLLARIVM.</
head
>
<
p
>
<
s
xml:id
="
echoid-s9861
"
xml:space
="
preserve
">
<
emph
style
="
sc
">Seqvitvr</
emph
>
hinc, ita eſſe {1/4}. </
s
>
<
s
xml:id
="
echoid-s9862
"
xml:space
="
preserve
">cuiuſuis magnitudinis ad {1/3}. </
s
>
<
s
xml:id
="
echoid-s9863
"
xml:space
="
preserve
">eiuſdem, vt eſt {1/2}.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s9864
"
xml:space
="
preserve
">cuiuſuis alteri
<
emph
style
="
sub
">9</
emph
>
magnitudinis ad {2/3}. </
s
>
<
s
xml:id
="
echoid-s9865
"
xml:space
="
preserve
">eiuſdẽ. </
s
>
<
s
xml:id
="
echoid-s9866
"
xml:space
="
preserve
">Quoniã. </
s
>
<
s
xml:id
="
echoid-s9867
"
xml:space
="
preserve
">n. </
s
>
<
s
xml:id
="
echoid-s9868
"
xml:space
="
preserve
">vt oſtendim
<
emph
style
="
sub
">9</
emph
>
, ita eſt {1/4}. </
s
>
<
s
xml:id
="
echoid-s9869
"
xml:space
="
preserve
">prio-
<
lb
/>
ris magnitudinis ad {1/3}. </
s
>
<
s
xml:id
="
echoid-s9870
"
xml:space
="
preserve
">eiuſdem, vt {1/4}. </
s
>
<
s
xml:id
="
echoid-s9871
"
xml:space
="
preserve
">poſterioris ad {1/3}. </
s
>
<
s
xml:id
="
echoid-s9872
"
xml:space
="
preserve
">eiuſdem. </
s
>
<
s
xml:id
="
echoid-s9873
"
xml:space
="
preserve
"> Vtautem {1/4}.</
s
>
<
s
xml:id
="
echoid-s9874
"
xml:space
="
preserve
">
<
note
symbol
="
c
"
position
="
right
"
xlink:label
="
note-249-03
"
xlink:href
="
note-249-03a
"
xml:space
="
preserve
">1. quiuti.</
note
>
poſterioris ad {1/3}. </
s
>
<
s
xml:id
="
echoid-s9875
"
xml:space
="
preserve
">ita ſunt {2/4}. </
s
>
<
s
xml:id
="
echoid-s9876
"
xml:space
="
preserve
">ad {2/3}. </
s
>
<
s
xml:id
="
echoid-s9877
"
xml:space
="
preserve
">hoc eſt, {1/2}. </
s
>
<
s
xml:id
="
echoid-s9878
"
xml:space
="
preserve
">ad {2/3}. </
s
>
<
s
xml:id
="
echoid-s9879
"
xml:space
="
preserve
">Igitur erit vt {1/4}. </
s
>
<
s
xml:id
="
echoid-s9880
"
xml:space
="
preserve
">prioris
<
lb
/>
magnitudinis ad {1/3}. </
s
>
<
s
xml:id
="
echoid-s9881
"
xml:space
="
preserve
">eiuſdem, ita {1/2}. </
s
>
<
s
xml:id
="
echoid-s9882
"
xml:space
="
preserve
">poſterioris magnitudinis ad {2/3}. </
s
>
<
s
xml:id
="
echoid-s9883
"
xml:space
="
preserve
">eiuſdem.</
s
>
<
s
xml:id
="
echoid-s9884
"
xml:space
="
preserve
"/>
</
p
>
</
div
>
<
div
xml:id
="
echoid-div605
"
type
="
section
"
level
="
1
"
n
="
212
">
<
head
xml:id
="
echoid-head229
"
xml:space
="
preserve
">PROPOSITIO II.</
head
>
<
p
>
<
s
xml:id
="
echoid-s9885
"
xml:space
="
preserve
">RECTANGVLVM ſub diametro, & </
s
>
<
s
xml:id
="
echoid-s9886
"
xml:space
="
preserve
">circumferentia maximi circuli
<
lb
/>
in ſphæra comprehenſum, quadruplum eſt circuli maximi, & </
s
>
<
s
xml:id
="
echoid-s9887
"
xml:space
="
preserve
">ſuper-
<
lb
/>
ficiei conuexæ eiuſdem ſphęræ ęquale.</
s
>
<
s
xml:id
="
echoid-s9888
"
xml:space
="
preserve
"/>
</
p
>
<
p
>
<
s
xml:id
="
echoid-s9889
"
xml:space
="
preserve
">
<
emph
style
="
sc
">Sit</
emph
>
rectangulum AB, comprehenſum ſub diametro AC, & </
s
>
<
s
xml:id
="
echoid-s9890
"
xml:space
="
preserve
">circumferentia
<
lb
/>
CB, maximi in ſphæra circuli. </
s
>
<
s
xml:id
="
echoid-s9891
"
xml:space
="
preserve
">Dico rectangulum AB,
<
lb
/>
<
figure
xlink:label
="
fig-249-01
"
xlink:href
="
fig-249-01a
"
number
="
162
">
<
image
file
="
249-01
"
xlink:href
="
http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/249-01
"/>
</
figure
>
quadruplum eſſe circuli maximi in ſphæra, & </
s
>
<
s
xml:id
="
echoid-s9892
"
xml:space
="
preserve
">ſuperficiei
<
lb
/>
conuexæ eiuſdem ſphæræ ęquale. </
s
>
<
s
xml:id
="
echoid-s9893
"
xml:space
="
preserve
">Sectis enim omnibus
<
lb
/>
lateribus bifariam in E, F, G, H, iunctiſquerectis EG, FH,
<
lb
/>
ſecantibus ſeſe in I, diuiſum erit totum rectangulum in
<
lb
/>
quatuor æqualia A I, C I, B I, D I, quodrectæ E G, F
<
note
symbol
="
d
"
position
="
right
"
xlink:label
="
note-249-04
"
xlink:href
="
note-249-04a
"
xml:space
="
preserve
">33. primi.</
note
>
rectis A D, A C, parallelæ ſint. </
s
>
<
s
xml:id
="
echoid-s9894
"
xml:space
="
preserve
">Ac proinderectangulum A B, rectanguli C I,
<
lb
/>
quadruplum erit. </
s
>
<
s
xml:id
="
echoid-s9895
"
xml:space
="
preserve
">Eſt autem rectangulum C I, contentum ſub C E, ſemidia-
<
lb
/>
metro, & </
s
>
<
s
xml:id
="
echoid-s9896
"
xml:space
="
preserve
">ſemicircumferentia C F, circulo maximo, cuius nimirum diameter
<
lb
/>
A C, ęquale, vt lib. </
s
>
<
s
xml:id
="
echoid-s9897
"
xml:space
="
preserve
">4. </
s
>
<
s
xml:id
="
echoid-s9898
"
xml:space
="
preserve
">capit. </
s
>
<
s
xml:id
="
echoid-s9899
"
xml:space
="
preserve
">7. </
s
>
<
s
xml:id
="
echoid-s9900
"
xml:space
="
preserve
">Nume. </
s
>
<
s
xml:id
="
echoid-s9901
"
xml:space
="
preserve
">1. </
s
>
<
s
xml:id
="
echoid-s9902
"
xml:space
="
preserve
">demonſtratum eſt. </
s
>
<
s
xml:id
="
echoid-s9903
"
xml:space
="
preserve
">lgitur rectangu-
<
lb
/>
lum A B, circulimaximi quadruplum eſt. </
s
>
<
s
xml:id
="
echoid-s9904
"
xml:space
="
preserve
">Et quia eiuſdem circuli maximi qua-
<
lb
/>
drupla eſt ſuperficies conuexa ſphærę, per propoſ. </
s
>
<
s
xml:id
="
echoid-s9905
"
xml:space
="
preserve
">31. </
s
>
<
s
xml:id
="
echoid-s9906
"
xml:space
="
preserve
">lib. </
s
>
<
s
xml:id
="
echoid-s9907
"
xml:space
="
preserve
">1. </
s
>
<
s
xml:id
="
echoid-s9908
"
xml:space
="
preserve
">Archimedis de ſphę-
<
lb
/>
ra, & </
s
>
<
s
xml:id
="
echoid-s9909
"
xml:space
="
preserve
">Cylindro: </
s
>
<
s
xml:id
="
echoid-s9910
"
xml:space
="
preserve
"> ęquale erit rectangulum A B, conuexæ ſuperficiei, quod
<
note
symbol
="
e
"
position
="
right
"
xlink:label
="
note-249-05
"
xlink:href
="
note-249-05a
"
xml:space
="
preserve
">9. quinti.</
note
>
rat demonſtrandum.</
s
>
<
s
xml:id
="
echoid-s9911
"
xml:space
="
preserve
"/>
</
p
>
</
div
>
<
div
xml:id
="
echoid-div607
"
type
="
section
"
level
="
1
"
n
="
213
">
<
head
xml:id
="
echoid-head230
"
xml:space
="
preserve
">COROLLARIVM.</
head
>
<
p
>
<
s
xml:id
="
echoid-s9912
"
xml:space
="
preserve
">EX demonſtratione liquet, rectangulum ſub diametro cuiuſuis circuli, (et-
<
lb
/>
iamſi non ſit maximus in ſphęra,) & </
s
>
<
s
xml:id
="
echoid-s9913
"
xml:space
="
preserve
">circumferentia eiuſdem, quadruplum eſſe
<
lb
/>
ipſius circuli. </
s
>
<
s
xml:id
="
echoid-s9914
"
xml:space
="
preserve
">Eadem enim ſemper demonſtratio adhibebitur.</
s
>
<
s
xml:id
="
echoid-s9915
"
xml:space
="
preserve
"/>
</
p
>
</
div
>
</
text
>
</
echo
>