Clavius, Christoph
,
Gnomonices libri octo, in quibus non solum horologiorum solariu[m], sed aliarum quo[quam] rerum, quae ex gnomonis umbra cognosci possunt, descriptiones geometricè demonstrantur
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LIBER SECVNDVS.
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di, neceſſe est, vt Sol exiſtens in @, hora 4. </
s
>
<
s
xml:id
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echoid-s9655
"
xml:space
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">à mer. </
s
>
<
s
xml:id
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echoid-s9656
"
xml:space
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">proijciat radium per centrum A, & </
s
>
<
s
xml:id
="
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xml:space
="
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">punctũ P, vſq; </
s
>
<
s
xml:id
="
echoid-s9658
"
xml:space
="
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">ad
<
lb
/>
planũ horologij, atq; </
s
>
<
s
xml:id
="
echoid-s9659
"
xml:space
="
preserve
">adeo ſectio, quã facit ſemicirculus horarius per P, ductus in plano horologij, indicet
<
lb
/>
horã 4. </
s
>
<
s
xml:id
="
echoid-s9660
"
xml:space
="
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">à mer. </
s
>
<
s
xml:id
="
echoid-s9661
"
xml:space
="
preserve
">vt perſpicuum eſt exijs, quę ſupra in hoc ſcholio ſcripſimus. </
s
>
<
s
xml:id
="
echoid-s9662
"
xml:space
="
preserve
">Vnde non immerito punctum
<
lb
/>
P, pro hora 4. </
s
>
<
s
xml:id
="
echoid-s9663
"
xml:space
="
preserve
">à mer. </
s
>
<
s
xml:id
="
echoid-s9664
"
xml:space
="
preserve
">ſumi poteſt, quandoquidem ſemicir culus horarius per ipſum ductus facit in horolo-
<
lb
/>
gio lineam horæ 4. </
s
>
<
s
xml:id
="
echoid-s9665
"
xml:space
="
preserve
">à mer. </
s
>
<
s
xml:id
="
echoid-s9666
"
xml:space
="
preserve
">Eodem modo ſumi poterit punctum O, pro hora 8. </
s
>
<
s
xml:id
="
echoid-s9667
"
xml:space
="
preserve
">à med. </
s
>
<
s
xml:id
="
echoid-s9668
"
xml:space
="
preserve
">noc. </
s
>
<
s
xml:id
="
echoid-s9669
"
xml:space
="
preserve
">quamuis reuera
<
lb
/>
in Aequatore indicet horam 8. </
s
>
<
s
xml:id
="
echoid-s9670
"
xml:space
="
preserve
">à mer. </
s
>
<
s
xml:id
="
echoid-s9671
"
xml:space
="
preserve
">Intelligatur iam Cylindrus, cuius baſes ſint circulus θ Y B Z, & </
s
>
<
s
xml:id
="
echoid-s9672
"
xml:space
="
preserve
">
<
lb
/>
alter a e b π, ei æqualis, & </
s
>
<
s
xml:id
="
echoid-s9673
"
xml:space
="
preserve
">oppoſitus, axis autem idem, qui axis mundi A H λ. </
s
>
<
s
xml:id
="
echoid-s9674
"
xml:space
="
preserve
">Et quoniam omnes circu
<
lb
/>
li horarij à mer. </
s
>
<
s
xml:id
="
echoid-s9675
"
xml:space
="
preserve
">vel med. </
s
>
<
s
xml:id
="
echoid-s9676
"
xml:space
="
preserve
">noc. </
s
>
<
s
xml:id
="
echoid-s9677
"
xml:space
="
preserve
">per axem mundi, ſiue Cylindri ducuntur, facient eorum plana in cylindro pa
<
lb
/>
rallelogrãma, ita vt duo latera cuiuſq; </
s
>
<
s
xml:id
="
echoid-s9678
"
xml:space
="
preserve
">ſint diametri baſium cylindri, reliqua vero duo in ſuperficie cylin
<
lb
/>
dri deſcribantur, vt à Sereno demonſtratur lib. </
s
>
<
s
xml:id
="
echoid-s9679
"
xml:space
="
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">1. </
s
>
<
s
xml:id
="
echoid-s9680
"
xml:space
="
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">de ſectione cylindri, propoſ. </
s
>
<
s
xml:id
="
echoid-s9681
"
xml:space
="
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">2. </
s
>
<
s
xml:id
="
echoid-s9682
"
xml:space
="
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">Vt v. </
s
>
<
s
xml:id
="
echoid-s9683
"
xml:space
="
preserve
">g. </
s
>
<
s
xml:id
="
echoid-s9684
"
xml:space
="
preserve
">circulus horæ 4.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s9685
"
xml:space
="
preserve
">
<
note
position
="
left
"
xlink:label
="
note-0171-01
"
xlink:href
="
note-0171-01a
"
xml:space
="
preserve
">10</
note
>
à mer. </
s
>
<
s
xml:id
="
echoid-s9686
"
xml:space
="
preserve
">vel med. </
s
>
<
s
xml:id
="
echoid-s9687
"
xml:space
="
preserve
">noc. </
s
>
<
s
xml:id
="
echoid-s9688
"
xml:space
="
preserve
">ductus per puncta ε, P, in circulo θ γ B Z, faciet parallelogrammum, cuius vnum la-
<
lb
/>
tus eſt ε P, diameter baſis cylindri, duo vero ducuntur ex punctis ε, P, in ſuperficie cylindri, & </
s
>
<
s
xml:id
="
echoid-s9689
"
xml:space
="
preserve
">ita de
<
lb
/>
reliquis. </
s
>
<
s
xml:id
="
echoid-s9690
"
xml:space
="
preserve
">Dico latera hæc in ſuperficie cylindri deſcripta cadere in puncta ellipſis in plano horologij de-
<
lb
/>
ſcriptę nempe latera ex punctis Z, Y, circuli horæ 6. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">àmer. </
s
>
<
s
xml:id
="
echoid-s9692
"
xml:space
="
preserve
">uel med. </
s
>
<
s
xml:id
="
echoid-s9693
"
xml:space
="
preserve
">noc. </
s
>
<
s
xml:id
="
echoid-s9694
"
xml:space
="
preserve
">ducta cadere in puncta R, X,
<
lb
/>
& </
s
>
<
s
xml:id
="
echoid-s9695
"
xml:space
="
preserve
">latus ex puncto θ, Meridiani circuli ductum cadere in punctum V, & </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">latera ex punctis ε, P, ducta
<
lb
/>
cadere in puncta α L, & </
s
>
<
s
xml:id
="
echoid-s9697
"
xml:space
="
preserve
">ſic de cæteris. </
s
>
<
s
xml:id
="
echoid-s9698
"
xml:space
="
preserve
">Ducatur enim per Y Z, diametrum, quæ diametrum θ B, ad an-
<
lb
/>
gulos rectos ſecet, & </
s
>
<
s
xml:id
="
echoid-s9699
"
xml:space
="
preserve
">per axem A H, circulus horæ 6. </
s
>
<
s
xml:id
="
echoid-s9700
"
xml:space
="
preserve
">occurrens plano horologij in H. </
s
>
<
s
xml:id
="
echoid-s9701
"
xml:space
="
preserve
">Et quoniam tam
<
lb
/>
planum horologij, quàm planum huius circuli horę 6. </
s
>
<
s
xml:id
="
echoid-s9702
"
xml:space
="
preserve
">rectum eſt ad Meridianum, erit quoque eorum com-
<
lb
/>
munis ſectio, nempe linea horæ 6. </
s
>
<
s
xml:id
="
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"
xml:space
="
preserve
">ad eundem recta, ac idcirco ad meridianam lineam H B, in Meridiano
<
lb
/>
<
note
position
="
right
"
xlink:label
="
note-0171-02
"
xlink:href
="
note-0171-02a
"
xml:space
="
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">19. vndec.</
note
>
exiſtentẽ, per defin. </
s
>
<
s
xml:id
="
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xml:space
="
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">3. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">lib. </
s
>
<
s
xml:id
="
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"
xml:space
="
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">11. </
s
>
<
s
xml:id
="
echoid-s9707
"
xml:space
="
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">Eucl. </
s
>
<
s
xml:id
="
echoid-s9708
"
xml:space
="
preserve
">perpendicularis in puncto H. </
s
>
<
s
xml:id
="
echoid-s9709
"
xml:space
="
preserve
">Diameter igitur S T, quam ad meridia-
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0171-03
"
xlink:href
="
note-0171-03a
"
xml:space
="
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">20</
note
>
nam lineam H B, duximus perpendicularem, communis ſectio eſt plani horologij & </
s
>
<
s
xml:id
="
echoid-s9710
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xml:space
="
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">circuli horæ 6. </
s
>
<
s
xml:id
="
echoid-s9711
"
xml:space
="
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">hoc
<
lb
/>
eſt, linea horæ 6. </
s
>
<
s
xml:id
="
echoid-s9712
"
xml:space
="
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">à mer. </
s
>
<
s
xml:id
="
echoid-s9713
"
xml:space
="
preserve
">vel med. </
s
>
<
s
xml:id
="
echoid-s9714
"
xml:space
="
preserve
">noc. </
s
>
<
s
xml:id
="
echoid-s9715
"
xml:space
="
preserve
">Ac propterea latera parallelogrammi, quod circulus horæ 6. </
s
>
<
s
xml:id
="
echoid-s9716
"
xml:space
="
preserve
">in cy-
<
lb
/>
lindro facit, ducta ex punctis Z, Y, cadent in rectam S T; </
s
>
<
s
xml:id
="
echoid-s9717
"
xml:space
="
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">quandoquidem & </
s
>
<
s
xml:id
="
echoid-s9718
"
xml:space
="
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">linea horæ 6. </
s
>
<
s
xml:id
="
echoid-s9719
"
xml:space
="
preserve
">& </
s
>
<
s
xml:id
="
echoid-s9720
"
xml:space
="
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">latera hæc
<
lb
/>
in plano circuli horę 6. </
s
>
<
s
xml:id
="
echoid-s9721
"
xml:space
="
preserve
">exiſtunt. </
s
>
<
s
xml:id
="
echoid-s9722
"
xml:space
="
preserve
">Quoniam vero recta S T, ad Meridianum oſtenſa perpendicularis, per-
<
lb
/>
pendicularis quoque eſt, per defin. </
s
>
<
s
xml:id
="
echoid-s9723
"
xml:space
="
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">3. </
s
>
<
s
xml:id
="
echoid-s9724
"
xml:space
="
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">lib. </
s
>
<
s
xml:id
="
echoid-s9725
"
xml:space
="
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">11. </
s
>
<
s
xml:id
="
echoid-s9726
"
xml:space
="
preserve
">Eucl. </
s
>
<
s
xml:id
="
echoid-s9727
"
xml:space
="
preserve
">ad axem A H, in Meridiano exiſtentem; </
s
>
<
s
xml:id
="
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"
xml:space
="
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">Eſt autem & </
s
>
<
s
xml:id
="
echoid-s9729
"
xml:space
="
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">
<
lb
/>
diameter Z Y, ad eundem axem perpendicularis, (Cum enim & </
s
>
<
s
xml:id
="
echoid-s9730
"
xml:space
="
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">circulus horæ 6. </
s
>
<
s
xml:id
="
echoid-s9731
"
xml:space
="
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">& </
s
>
<
s
xml:id
="
echoid-s9732
"
xml:space
="
preserve
">Aequator rectus ſit
<
lb
/>
ad Meridianum, erit quoque corum communis ſectio Z Y, ad eundem recta, ac proinde ad axem in Meri-
<
lb
/>
<
note
position
="
right
"
xlink:label
="
note-0171-04
"
xlink:href
="
note-0171-04a
"
xml:space
="
preserve
">19. vndec.</
note
>
diano exiſtentem, per defin. </
s
>
<
s
xml:id
="
echoid-s9733
"
xml:space
="
preserve
">3. </
s
>
<
s
xml:id
="
echoid-s9734
"
xml:space
="
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">lib. </
s
>
<
s
xml:id
="
echoid-s9735
"
xml:space
="
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">11. </
s
>
<
s
xml:id
="
echoid-s9736
"
xml:space
="
preserve
">Eucl. </
s
>
<
s
xml:id
="
echoid-s9737
"
xml:space
="
preserve
">perpendicularis) erunt rectæ S T, Z Y, in eodem plano circu
<
lb
/>
li horę 6. </
s
>
<
s
xml:id
="
echoid-s9738
"
xml:space
="
preserve
">exiſtentes, cum ad axem ſint perpendiculares, inter ſe parallelæ. </
s
>
<
s
xml:id
="
echoid-s9739
"
xml:space
="
preserve
">Quocirca parallelogrammum
<
lb
/>
<
note
position
="
right
"
xlink:label
="
note-0171-05
"
xlink:href
="
note-0171-05a
"
xml:space
="
preserve
">28. primi.</
note
>
erit quadrilaterum, cuius latera ſunt axis A H, ſemidiameter A Z, latus cylindri ex Z, ductum, & </
s
>
<
s
xml:id
="
echoid-s9740
"
xml:space
="
preserve
">por
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0171-06
"
xlink:href
="
note-0171-06a
"
xml:space
="
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">30</
note
>
tio rectæ H T, inter H, & </
s
>
<
s
xml:id
="
echoid-s9741
"
xml:space
="
preserve
">dictum latus ex Z, ductum. </
s
>
<
s
xml:id
="
echoid-s9742
"
xml:space
="
preserve
">Eſt enim & </
s
>
<
s
xml:id
="
echoid-s9743
"
xml:space
="
preserve
">latus cylindri ex Z, ductum, axi A H,
<
lb
/>
<
note
position
="
right
"
xlink:label
="
note-0171-07
"
xlink:href
="
note-0171-07a
"
xml:space
="
preserve
">33. primi.</
note
>
parallelum, quòd illud latus & </
s
>
<
s
xml:id
="
echoid-s9744
"
xml:space
="
preserve
">axis, ſi producãtur, coniungant ſemidiametros æquales A Z, λ ξ, baſium
<
lb
/>
cylindri æqualium, quæ quidem ſemidiametri æquidiſtantes ſunt, vtpote ſectiones baſium æquidiſtantium
<
lb
/>
<
note
position
="
right
"
xlink:label
="
note-0171-08
"
xlink:href
="
note-0171-08a
"
xml:space
="
preserve
">16. vndec.</
note
>
factæ à plano circuli horæ 6. </
s
>
<
s
xml:id
="
echoid-s9745
"
xml:space
="
preserve
">à mer. </
s
>
<
s
xml:id
="
echoid-s9746
"
xml:space
="
preserve
">vel med. </
s
>
<
s
xml:id
="
echoid-s9747
"
xml:space
="
preserve
">noc. </
s
>
<
s
xml:id
="
echoid-s9748
"
xml:space
="
preserve
">Quare recta A Z, æqualis erit lateri oppoſito in dicto
<
lb
/>
parallelogrammo, hoc eſt, portioni rectæ H T, inter H, & </
s
>
<
s
xml:id
="
echoid-s9749
"
xml:space
="
preserve
">latus cylindri ex Z, ductum. </
s
>
<
s
xml:id
="
echoid-s9750
"
xml:space
="
preserve
">Eſt autem H R,
<
lb
/>
ſemidiameter circuli C R X, ſemidiametro A Z, circuli θ Y B Z, æqualis: </
s
>
<
s
xml:id
="
echoid-s9751
"
xml:space
="
preserve
">poſita enim eſt H C, æqualis
<
lb
/>
ſemidiametro A B, vel A Z. </
s
>
<
s
xml:id
="
echoid-s9752
"
xml:space
="
preserve
">Igitur latus cylindri ex Z, ductum cadit in punctum R. </
s
>
<
s
xml:id
="
echoid-s9753
"
xml:space
="
preserve
">Eodem{q́ue} pacto
<
lb
/>
latus ex Y, ductum in punctum X, cadet.</
s
>
<
s
xml:id
="
echoid-s9754
"
xml:space
="
preserve
"/>
</
p
>
<
p
style
="
it
">
<
s
xml:id
="
echoid-s9755
"
xml:space
="
preserve
">RVRSVS, quia latus cylindri ex θ, ductum cadit in lineam meridianam, cum exiſtat in Meridia-
<
lb
/>
no, parallelum{q́ue} eſt axi A H; </
s
>
<
s
xml:id
="
echoid-s9756
"
xml:space
="
preserve
">quod demonſtr abitur ea ratione, qua paulo ante oſtendimus, latus ex Z,
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0171-09
"
xlink:href
="
note-0171-09a
"
xml:space
="
preserve
">40</
note
>
<
note
position
="
right
"
xlink:label
="
note-0171-10
"
xlink:href
="
note-0171-10a
"
xml:space
="
preserve
">33. primi.</
note
>
ductum eidem axi eſſe parallelum, quia nimirum latus illud ex θ, ductum, & </
s
>
<
s
xml:id
="
echoid-s9757
"
xml:space
="
preserve
">axis coniungunt ſemidia-
<
lb
/>
metros. </
s
>
<
s
xml:id
="
echoid-s9758
"
xml:space
="
preserve
">A θ, λ π, æquales & </
s
>
<
s
xml:id
="
echoid-s9759
"
xml:space
="
preserve
">æquidiſtantes; </
s
>
<
s
xml:id
="
echoid-s9760
"
xml:space
="
preserve
">erit vt B A, ad A θ, ita B H, ad portionem meridianæ lineæ
<
lb
/>
<
note
position
="
right
"
xlink:label
="
note-0171-11
"
xlink:href
="
note-0171-11a
"
xml:space
="
preserve
">2. ſexti.</
note
>
inter H, & </
s
>
<
s
xml:id
="
echoid-s9761
"
xml:space
="
preserve
">latus ex θ, ductum: </
s
>
<
s
xml:id
="
echoid-s9762
"
xml:space
="
preserve
">Eſt autem recta B A, rectæ A θ, æqualis. </
s
>
<
s
xml:id
="
echoid-s9763
"
xml:space
="
preserve
">Igitur & </
s
>
<
s
xml:id
="
echoid-s9764
"
xml:space
="
preserve
">B H, portioni di-
<
lb
/>
ctæ meridianæ lineæ æqualis erit; </
s
>
<
s
xml:id
="
echoid-s9765
"
xml:space
="
preserve
">ac propterea, cum H V, ipſi B H, ſit æqualis, cadet latus cylindri ex
<
lb
/>
θ, ductum in punctum V.</
s
>
<
s
xml:id
="
echoid-s9766
"
xml:space
="
preserve
"/>
</
p
>
<
p
style
="
it
">
<
s
xml:id
="
echoid-s9767
"
xml:space
="
preserve
">POSTREMO, quoniam iuncta recta O P, parallela eſt rectæ Y Z, ex ſcholio propoſ. </
s
>
<
s
xml:id
="
echoid-s9768
"
xml:space
="
preserve
">27. </
s
>
<
s
xml:id
="
echoid-s9769
"
xml:space
="
preserve
">lib. </
s
>
<
s
xml:id
="
echoid-s9770
"
xml:space
="
preserve
">3.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s9771
"
xml:space
="
preserve
">Eucl. </
s
>
<
s
xml:id
="
echoid-s9772
"
xml:space
="
preserve
">Nam arcus Y O, Z P, æquales ſunt, quòd vtrumque punctum O, P, quatuor horis diſtare pona-
<
lb
/>
tur à puncto B, at que adeo duabus horis à punctis Y, Z; </
s
>
<
s
xml:id
="
echoid-s9773
"
xml:space
="
preserve
">ſi per P O, concipiatur duci planum æquidiſtans
<
lb
/>
parallelogrammo per Z Y, & </
s
>
<
s
xml:id
="
echoid-s9774
"
xml:space
="
preserve
">axem ducto, faciet hoc planum in cylindro parallelogrammum, per pro-
<
lb
/>
poſ. </
s
>
<
s
xml:id
="
echoid-s9775
"
xml:space
="
preserve
">3. </
s
>
<
s
xml:id
="
echoid-s9776
"
xml:space
="
preserve
">lib. </
s
>
<
s
xml:id
="
echoid-s9777
"
xml:space
="
preserve
">1. </
s
>
<
s
xml:id
="
echoid-s9778
"
xml:space
="
preserve
">Sereni de ſectione cylindri, cuius duo latera ſunt recta O P, & </
s
>
<
s
xml:id
="
echoid-s9779
"
xml:space
="
preserve
">alia recta a b, in oppoſita
<
lb
/>
<
note
position
="
left
"
xlink:label
="
note-0171-12
"
xlink:href
="
note-0171-12a
"
xml:space
="
preserve
">50</
note
>
baſi ei reſpondens, reliqua autẽ duo in ſuperficie cylindri ex punctis O, P, ducta, quorum illud, quod ex P,
<
lb
/>
demittitur, est etiam latus parallelogrammi per ε P, ducti, adeo vt per ipſum ducatur circulus horæ 4.
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s9780
"
xml:space
="
preserve
">Nam ex vno puncto P, vnum tantum latus cylindri duci potest, vt facile probari poteſt ex propoſ. </
s
>
<
s
xml:id
="
echoid-s9781
"
xml:space
="
preserve
">8. </
s
>
<
s
xml:id
="
echoid-s9782
"
xml:space
="
preserve
">lib. </
s
>
<
s
xml:id
="
echoid-s9783
"
xml:space
="
preserve
">
<
lb
/>
1. </
s
>
<
s
xml:id
="
echoid-s9784
"
xml:space
="
preserve
">Sereni de ſectione cylindri. </
s
>
<
s
xml:id
="
echoid-s9785
"
xml:space
="
preserve
">Nam recta quæcunque ducta ex puncto P, ad aliud punctum, quod in late
<
lb
/>
re cylindri non ſit, cadit, per dictam propoſ. </
s
>
<
s
xml:id
="
echoid-s9786
"
xml:space
="
preserve
">intra cylindrum. </
s
>
<
s
xml:id
="
echoid-s9787
"
xml:space
="
preserve
">Huius ergo parallelogrammi per O P, du
<
lb
/>
cti, & </
s
>
<
s
xml:id
="
echoid-s9788
"
xml:space
="
preserve
">Meridiani cõmunis ſectio ſit recta E Q, quæ axi A H, parallela erit, propterea quòd E Q, A H,
<
lb
/>
<
note
position
="
right
"
xlink:label
="
note-0171-13
"
xlink:href
="
note-0171-13a
"
xml:space
="
preserve
">16. vndec.</
note
>
ſectiones ſunt planorum parallelorum, nempe illorum parallelogrammorum, quæper E Q, A H, ducun-
<
lb
/>
tur, factę à Meridiano. </
s
>
<
s
xml:id
="
echoid-s9789
"
xml:space
="
preserve
">Quocirca erit vt ſemidiameter B H, ad B A, ſemidiametrum, ita B Q, ad B E;
<
lb
/>
</
s
>
<
s
xml:id
="
echoid-s9790
"
xml:space
="
preserve
">
<
note
position
="
right
"
xlink:label
="
note-0171-14
"
xlink:href
="
note-0171-14a
"
xml:space
="
preserve
">4. ſexti.</
note
>
ac proinde arcus circulorum θ Y B Z, B T V S, reſpondentes ſinubus verſis B Q, B E, ſimiles erunt, ex
<
lb
/>
lemmate propoſ. </
s
>
<
s
xml:id
="
echoid-s9791
"
xml:space
="
preserve
">1. </
s
>
<
s
xml:id
="
echoid-s9792
"
xml:space
="
preserve
">ſuperioris lib. </
s
>
<
s
xml:id
="
echoid-s9793
"
xml:space
="
preserve
">Eſt autem arcus B F, ſimilis arcui B P, quòd vterq; </
s
>
<
s
xml:id
="
echoid-s9794
"
xml:space
="
preserve
">quatuor horas com-
<
lb
/>
plectatur; </
s
>
<
s
xml:id
="
echoid-s9795
"
xml:space
="
preserve
">eſt{q́ue} B E, ſinus verſus arcus B P. </
s
>
<
s
xml:id
="
echoid-s9796
"
xml:space
="
preserve
">Igitur B Q, ſinus quoque verſus erit arcus B F; </
s
>
<
s
xml:id
="
echoid-s9797
"
xml:space
="
preserve
">ac </
s
>
</
p
>
</
div
>
</
text
>
</
echo
>