Theodosius <Bithynius>; Clavius, Christoph
,
Theodosii Tripolitae Sphaericorum libri tres
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126
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<
s
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xml:space
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">IN quadrante ABC, ſit BD, arcus grad. </
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<
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">54. </
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>
<
s
xml:id
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echoid-s4285
"
xml:space
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">ac proinde eius cõplementum
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CD, grad. </
s
>
<
s
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xml:space
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">36. </
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>
<
s
xml:id
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"
xml:space
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">quod diuidatur bifariam in H, vt vterq; </
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<
s
xml:id
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echoid-s4288
"
xml:space
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">arcuũ CH, HD, habeat
<
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grad. </
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>
<
s
xml:id
="
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xml:space
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">18. </
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<
s
xml:id
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xml:space
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">Ducatur DM, ad AB, perpendicularis pro ſinu arcus grad. </
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>
<
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xml:id
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xml:space
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">54. </
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<
s
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xml:space
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">& </
s
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<
s
xml:id
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xml:space
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">DE,
<
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<
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xlink:label
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fig-126-01
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123
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126-01
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xlink:href
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http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/YC97H42F/figures/126-01
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ad AC, perpédicularis pro ſinu arcus grad. </
s
>
<
s
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="
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">36. </
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<
s
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"
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">Iunga
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tur quoq; </
s
>
<
s
xml:id
="
echoid-s4296
"
xml:space
="
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">recta AH, quæ per lẽma in definitionibus
<
lb
/>
demonſtratũ ſecabit rectã CD, in I, bifariam, ac pro
<
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/>
inde & </
s
>
<
s
xml:id
="
echoid-s4297
"
xml:space
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">ad angulos rectos: </
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>
<
s
xml:id
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xml:space
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">eritq́ propterea CI, ſinus
<
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<
note
position
="
left
"
xlink:label
="
note-126-01
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xlink:href
="
note-126-01a
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xml:space
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">3. tertij.</
note
>
rectus arcus CH, grad. </
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<
s
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">18. </
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<
s
xml:id
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xml:space
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">Sũpta tandẽ recta EF, ipſi
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EC, æquali, diuidantur AC, AF, bifariã in G, K, & </
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>
<
s
xml:id
="
echoid-s4301
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xml:space
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<
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ex K, ad AC, perpendicularis ducatur KL. </
s
>
<
s
xml:id
="
echoid-s4302
"
xml:space
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">Dico ſi-
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num rectũ DM, arcus grad. </
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>
<
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">54. </
s
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<
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">hoc eſt, rectam AE,
<
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<
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position
="
left
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xlink:label
="
note-126-02
"
xlink:href
="
note-126-02a
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xml:space
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">34. primi.</
note
>
illi ęqualẽ, componi ex AG, dimidio ſinus totius, & </
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>
<
s
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<
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ex CI, ſinu recto arcus grad. </
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>
<
s
xml:id
="
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xml:space
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">18. </
s
>
<
s
xml:id
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xml:space
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">hoc eſt, rectam GE,
<
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(quæ cũ AG, conſtituit totam rectã AE,) ęqualẽ eſ
<
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/>
ſe ſinui recto CI. </
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>
<
s
xml:id
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xml:space
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">Item ſinũ verſum arcus grad. </
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>
<
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xml:id
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xml:space
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">72. </
s
>
<
s
xml:id
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xml:space
="
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">componi ex dimidio ſinus to
<
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tius, & </
s
>
<
s
xml:id
="
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"
xml:space
="
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">ex CE, ſinu verſo arcus CD, grad. </
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>
<
s
xml:id
="
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xml:space
="
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">36. </
s
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<
s
xml:id
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xml:space
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">hoc eſt, rectam EK, (quæ cum
<
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ſinu verſo CE, rectam CK, componit) æqualem eſſe dimidio ſinus totius, ip-
<
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ſam vero CK, eſſe ſinum verſum arcus grad. </
s
>
<
s
xml:id
="
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xml:space
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">72. </
s
>
<
s
xml:id
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xml:space
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">hoc eſt, arcum CL, (cuius ſi-
<
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nus verſus eſt CK,) eſſe grad. </
s
>
<
s
xml:id
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xml:space
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">72. </
s
>
<
s
xml:id
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"
xml:space
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">Ducta enim recta LN, ad AB, perpendicu-
<
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lari, pro ſinu arcus BL, iungantur rectæ AD, DF. </
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>
<
s
xml:id
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xml:space
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">Quoniam igitur arcus
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CH, grad. </
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<
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>
<
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">continet {1/5}. </
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>
<
s
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xml:space
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">quadrantis BC, (quòd quinquies 18. </
s
>
<
s
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">faciant 90.)
<
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</
s
>
<
s
xml:id
="
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xml:space
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">continebit arcus CD, {2/5}. </
s
>
<
s
xml:id
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xml:space
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">eiuſdem quadrantis, ac proinde proportio arcus
<
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CD, ad arcum BC, erit vt 2. </
s
>
<
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xml:id
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">ad 5. </
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>
<
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">Eſt autem, vt arcus CD, ad arcum BC, ita
<
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<
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position
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xlink:label
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note-126-03
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xlink:href
="
note-126-03a
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xml:space
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">33. fexti.</
note
>
angulus CAD, ad rectum angulum BAC. </
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>
<
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xml:space
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">Igitur proportio anguli CAD,
<
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ad angulum rectum BAC, erit quoque, vt 2. </
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>
<
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">ad 5. </
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>
<
s
xml:id
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xml:space
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">ac proinde angulus CAD,
<
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continebit {2/5}. </
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<
s
xml:id
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">vnius anguli recti. </
s
>
<
s
xml:id
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xml:space
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">Cum ergo tres anguli trianguli CAD, con-
<
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tineant {10/5}. </
s
>
<
s
xml:id
="
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xml:space
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">vnius recti, hoc eſt, æquales ſint duobus rectis, ſintq́ue inter ſe
<
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<
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position
="
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xlink:label
="
note-126-04
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xlink:href
="
note-126-04a
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xml:space
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">32. primi.</
note
>
æquales duo anguli ACD, ADC; </
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>
<
s
xml:id
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xml:space
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">continebit vterque eorum {4/5}. </
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<
s
xml:id
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">vnius recti.
<
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</
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<
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xml:space
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<
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xlink:label
="
note-126-05
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xlink:href
="
note-126-05a
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xml:space
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">5.primi.</
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>
Et quoniam angulus DFC, angulo DCF, eſt æqualis, quòd & </
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>
<
s
xml:id
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xml:space
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">rectę DF, DC,
<
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<
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xlink:label
="
note-126-06
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xlink:href
="
note-126-06a
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xml:space
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">5.primi.</
note
>
æquales ſint; </
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>
<
s
xml:id
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xml:space
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">(cum enim DE, EF, latera trianguli DEF, æqualia ſint lateri-
<
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bus DE, EC, trianguli DEC, angulosq́ue ad E, contineant æquales, vtpo-
<
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te rectos; </
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>
<
s
xml:id
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xml:space
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">æquales erunt baſes DF, DC,) continebit quoque angulus DFC,
<
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<
note
position
="
left
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xlink:label
="
note-126-07
"
xlink:href
="
note-126-07a
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xml:space
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">4. primi.</
note
>
{4/5}. </
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<
s
xml:id
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xml:space
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">vnius recti; </
s
>
<
s
xml:id
="
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"
xml:space
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">ac proinde reliquus angulus DFA, ex duobus rectis, hoc eſt, ex
<
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{10/5}. </
s
>
<
s
xml:id
="
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xml:space
="
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">vnius recti, continebit {6/5}. </
s
>
<
s
xml:id
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xml:space
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">vnius recti. </
s
>
<
s
xml:id
="
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xml:space
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">Cum ergo angulus DAF, oſtenſus
<
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ſit continere {2/5}. </
s
>
<
s
xml:id
="
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xml:space
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">vnius recti, & </
s
>
<
s
xml:id
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xml:space
="
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">omnes tres anguli in triangulo AFD, conti-
<
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<
note
position
="
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xlink:label
="
note-126-08
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xlink:href
="
note-126-08a
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xml:space
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">32. primi.</
note
>
neant {10/5}. </
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>
<
s
xml:id
="
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xml:space
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">vnius recti, continebit angulus ADF, {2/5}. </
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>
<
s
xml:id
="
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xml:space
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">vnius recti, propte-
<
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reaq́ue angulo DAF, æqualis erit. </
s
>
<
s
xml:id
="
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xml:space
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">Quare æqualia erunt latera DF, AF.
<
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</
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>
<
s
xml:id
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<
note
position
="
left
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xlink:label
="
note-126-09
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xlink:href
="
note-126-09a
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xml:space
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">6. primi.</
note
>
Cum ergo recta DF, rectæ DC, oſtenſa ſit æqualis, erit & </
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>
<
s
xml:id
="
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xml:space
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">recta AF, rectæ DC,
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æqualis: </
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>
<
s
xml:id
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xml:space
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">ideoque & </
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>
<
s
xml:id
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xml:space
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">k F, medietas ipſius AF, ipſi CI, medietati ipſius DC,
<
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æqualis erit.</
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<
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</
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<
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<
s
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">RVRSVS quoniam AK, KF, æquales ſunt; </
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>
<
s
xml:id
="
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xml:space
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">additis æqualibus EC, FE,
<
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erit recta compoſita ex Ak, EC, æqualis rectæ KE: </
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>
<
s
xml:id
="
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xml:space
="
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">ac proinde KE, medie-
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tas erit ſemidiametri AC; </
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>
<
s
xml:id
="
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xml:space
="
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">quandoquidem AC, diuiſa eſt in duas partes æqua
<
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les, quarum vna eſt KE, altera vero, recta ex AK, EC, compoſita. </
s
>
<
s
xml:id
="
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xml:space
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">Eſt igi-
<
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tur KE, æqualis ipſi CG. </
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>
<
s
xml:id
="
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xml:space
="
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">Ablata ergo communi recta GE, remanebunt
<
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æquales GK, EC. </
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>
<
s
xml:id
="
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xml:space
="
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">Eſt autem EC, ſumpta ipſi EF, æqualis. </
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>
<
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xml:id
="
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xml:space
="
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">Igitur & </
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>
<
s
xml:id
="
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xml:space
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">
<
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GK, ipſi EF, æqualis erit; </
s
>
<
s
xml:id
="
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"
xml:space
="
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">additaque communi recta FG, erit EG, ipſi FK,
<
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æqualis, hoc eſt, ipſi CI, cui oſtendimus ſupra rectam k F, eſſe æqualem. </
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>
<
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