Cavalieri, Buonaventura
,
Geometria indivisibilibvs continvorvm : noua quadam ratione promota
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LIBER VI.
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xml:space
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">THEOREMA XVII. PROPOS. XVII.</
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">COmpræhenſ@m ſpatium ſub ſpirali, quæ eſt minor ea,
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quæ ſub vna reuolutione fit, nec habet terminum
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initium ſpiralis, & </
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<
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">rectis, quæ à terminis ipſius in reuolu-
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tionis initium ducuntur ad ſectorem habentem radium æ-
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qualem maiori earum, quę à termino ad initium reuolutio-
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nis ducitur, arcum verò, qui intercipitur inter duas rectas
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ſecundum eaſdem partem ſpiralis; </
s
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<
s
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echoid-s11699
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xml:space
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">habet eandem rationẽ,
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quam rectangulum compræhenſum ſub rectis à terminis
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ad initium reuolutionis ductis, vna cum tertia parte qua-
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drati exceſſus, quo maior dictarum linearum ſuperat mi-
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norem, ad quadratum maioris earundem.</
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<
s
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">In eadem antecedentis figura ſupponamus arſumptam, IS, por-
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tionem ſpiralis in vna reuolutione genitæ, quæ non habcat termi-
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num initium talis ſpiralis, a cuius extremis punctis, I, S, ſint du-
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ctæ ad, A, initium reuolutionis ipſæ, SA, IA, & </
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<
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">ſit ſector, IAR,
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cuius ſemidiameter ſit æqual s maiori ductarum, IA, AS, nempè
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ipſi, IA. </
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<
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">Dico ſectorem, IAR, ad trilineum, IAS, eſſe vt quadra-
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tum, RA, ad rectangulum, RAS, vna cum {1/3}. </
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<
s
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">quadrati, RS, (vta-
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mur conſtructis in eadem figura) Sector igitur, AIRεKA, eſt æqua-
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lis triangulo, LZ {12/ }, vt in antecedenti oſtenſum eſt, eodem modo
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probabimus triangulum, L+{13/ }, eſſe æqualem ſectori, ARεKA,
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ergo reliquus triangulus, LZ+, erit æqualis reliquo ſectori, IAR;
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</
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<
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">ſimiliter iuxta antecedentem oſtendemus ſpatium, AISMGA, eſſe
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æqualem trilineo, LΖΩ, & </
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<
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">ſpatium, ASMGA, eſſe æqualem tri-
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">14. huius.</
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lineo, LVΩ ergo reliquum ſpatium, IAS, erit æquale trilineo, LZ
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V, ergo ſector, IAR, ad trilineum, LZV, erit vt triangulus LZ+,
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ad trilineum, LZV, ideſt vt quadratum, L4, ad rectangulum ſub,
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4L3, cum {1/3}, quadrati, 34, ideſt vt quadratum, IA, vel, RA, ad re-
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ctangulum ſub, RA, A
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, vna cum {1/3}. </
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<
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">quadrati, RS, quod oſtende-
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re opus erat.</
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<
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">TRilineum, IRS, ad trilineum, ISX, erit vt, SA, cum
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{2/3}. </
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<
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">SR, ad, SA, cum {1/3}. </
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