Cavalieri, Buonaventura - Geometria indivisibilibvs continvorvm : noua quadam ratione promota

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187167LIBER II. OX, vel vt aliæ figuræ ſimiles ab ipſis deſcriptæ, ſiue abipſis ſimplici-
bus, ſiue ab ipſis adiuncta quadam linea, vnde caſus iſte ad caſum Theo-
rematis præſentis, vel antecedentis deductus erit, & ideò patebit, om-
nes lineas, AE, ad omnes lineas trianguli, BCE, vel omnes figuras ſi-
miles, AE, ad omnes figuras ſimiles trianguli, BCE, ideſt vel maxi-
mas abſciſſarum, BE, ad abſciſſas omnes ipſius, BE, vel earum figuras
ſimiles eſſe, vt omnia quadrata, B F, ad omnia quadrata figuræ, BEF.
Vocabuntur autem iſtæ; Quatuor ordinum magnitudines collectæ iuxta
quatuor magnitudines proportionales vtcunque inuentas, quæ fuerunt
ex. gr. prima quadratum, OQ, ſecunda quadratum, OP, tertia, EB,
quarta, BO, magnitudines autem collecta iuxta primam. ſ. ex. gr. om
nia quadrata, BF, dicentur primi ordinis, collectæ verò iuxta ſecun-
dam. ſ. omnia quadrata figuræ, BEF, magnitudines ſecundi ordinis, col-
lectæ verò iuxta tertiam magnitudines tertĳ ordinis, & tandem collectæ
iuxta quartam magnitudines quarti ordinis, ſic igitur appellabimus hos
quatuor magnitudinum ordines. In ſupradictis autem, quod dicimus de
abſciſſis, idem intellige de reſiduis abſciſſarum, & quod de ipſis ſimpli-
cibus, idem de eiſdem adiunctis alĳs, ſiue ſint recti, ſiue eiuſdem obli-
qui tranſitus: Hoc autem Corollarium præ cæteris ſummè animaduerten-
dumeſt, ac memoriæ diligentiſſimè commendandum, ex hoc enim potiſ-
ſimas demonſtrationes tanquam ex fonte dermari ſtudioſus in ſequen-
tium Librorum lectione ſacilè comprehendet.

THEOREMA XXVII. PROPOS. XXVII.

SI duo trapezia fuerint in eadem baſi, ſumpto vnolate-
rum & ae; quidiſtantium pro baſi, & regula, & fuerint etiam
in eadem altitudine reſpectu illius baſis, & latera baſi paral-
lela fuerint & ae; qualia, trapezia erunt & ae; qualia, & omnia eo-
rundem quadrata erunt & ae; qualia.

Sint duo trapezia, AERB, IABD, in eadem baſi, AB, quæ
ſit ſumpta pro regula, cuiq; latera, ER, ID, ſint parallela, & in-
ter ſe & ae; qualia, Dico trapezia eſſe & ae; qualia, & omnia eorundem qua-
drata eſſe & ae; qualia. Producantur, AE, BR, donec ſibi occurrant,
vt in, O, & , AI, BD, donec ſimul incidant, vt in, X, & iunga-
tur, OX, quia ergo, ER, parallela eſt ipſi, AB, erunt triangula,
11Iux. diff. 1.
Sexti Ele-
ment. AOB, EOR, ſimilia, & eadem ratione ſimilia erunt triangula, A
XB, IXD, ergo vt, AB, ad, ER, velad, ID, illiæqualem, ita
erit, BO, ad, OR, vt autem, AB, ad, ID, ita eſt, BX, ad, XD,
224. Sex. El. ergo vt, BO, ad, OR, ita eſt, BX, ad, XD, ergo, OX,

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