507487LIBER VII.BZ&
, quod non eſſet ſuperpoſitum, vt ſupra oſtendimus neceſſe
eſſe, ponitur autem totam, BZ& , eſſe ſuperpoſitam ipſi, CβΛ, er-
go ita ſunt ad inuicem ſuperpoſitę, vt neutrius reſidua habeantur,
ergo ita ſunt ſuperpoſitę, vt ſibi congruant, ergo figuræ, BZ& , C
βΛ, inter ſe ſunt æquales.
eſſe, ponitur autem totam, BZ& , eſſe ſuperpoſitam ipſi, CβΛ, er-
go ita ſunt ad inuicem ſuperpoſitę, vt neutrius reſidua habeantur,
ergo ita ſunt ſuperpoſitę, vt ſibi congruant, ergo figuræ, BZ& , C
βΛ, inter ſe ſunt æquales.
Sint nunc in eodem ſchemate duæ figuræ ſolidæ quæeunque,
BZ& , CβΛ, in eiſdem planis parallelis, AD, Y4, conſſitutæ, ductis
autem quibuſcunq; planis, E6, LΣ, præfatis æquidiſtantibus, ſint
conceptæ in ſolidis figuræ, quæ iacent in eodem plano, ſemper in-
ter ſe æquales, vt, FG, æqualis, HI, & , MN, OP, ſimul ſumptæ
(ſit enim ſolida figura ex. g. BZ& , intus vtcunq; caua ſecundum
ſuperficiem, {12/ }, N, {13/ }, O,) æquales ipſi, SV. Dico eaſdem ſolidas
figuras æquales eſſe. Si enim ſolidum, BZ& , cum portionibus,
ABY, & planorũ, AD, Y4, ipſi conterminantibus, ſolido, CβΛ, ita
ſuperpoſuerimus, vt planum, AB, ſit in plano, AD, & , Y& , in pla-
no, Y4, oſtendemus (vt fecimus ſuperius circa parallelarum ipſi, A
D, conceptas in figuris planis, BZ& , CβΛ, portiones) figuras in
ſolidis, BZ& , CβΛ, conceptas, quæ erant in eodem plano, etiam
poſt ſuperpoſitionem manere in eode plano, & ideò adhuc ęqua-
les eſſe figuras in ſuperpoſitis ſolidis conceptas, & ipſis, AD, Y4,
parallelas. Niſi ergo totum ſolidum toti congruat in prima ſu-
perpoſitione, relinquentur reſidua ſolida, vel ex reſiduis compoſi-
ta in vtroq; ſolido, quæ non erunt ad inuicem ſuperpoſita, cum
enim ex. g. figuræ, QR, ST, æquentur figuræ, SV, dempta com-
muni figura, ST, reliqua, QR, æquabitur reliquæ, TV, hocq; cõ-
tinget in quouis alio plano ipſi, AD, parallelo occurrenteſolidis,
C℟Γ, CβΛ, ergo ſemper habentes reſiduum vnius ſolidi, habebimus
etiam reſiduum alterius, & patebit, iuxta methodum adhibitam in
priori parte huius Propoſitionis circa figuras planas, reſidua ſoli-
da, vel reſiduorum aggregata ſemper eſſe in eiſdem parallelis pla-
nis, vt reſidua, H℟597, ΙΓΛ, 785, eſſe in planis parallelis, E6, Y4,
ac æqualiter analoga: ſi ergo hæc reſidua adhuc ſuperponantur,
ita vt planum, EH, locetur in plano, H6, & , Υβ, in, β4, & hoc
ſempei fieri intelligatur, donec quod ſuperponitur, vt, BZ& , totũ
fuerit aſſumptum, tandem ipſum totum, BZ& , congruet toti, Cβ
Λ, niſi enim toto ſolido, BZ& , ipſi, CβΛ, ſuperpoſito, @pſa ſibi cõ-
gruerent, eſſet aliquod reſiduum vnius, vt ſolidi, CβΛ, ergo etiam
eſſet aliquod reſiduum ſolidi, C℟Γ, ſeu, BZ& , illudq; non eſſet ſu-
perpoſitum, quod eſt abſurdum, ponitur enim iam totum ſolidũ,
BZ& , eſſe ipſi, CβΛ, ſuperpoſitum, non ergo erit aliquod reſiduũ
in ipſisſolidis, ergo ſibi congruent, ergo dictæ figuræ ſolidæ, BZ& ,
CβΛ, inter ſe æquales erunt, quæ fuerunt demonſtranda.
BZ& , CβΛ, in eiſdem planis parallelis, AD, Y4, conſſitutæ, ductis
autem quibuſcunq; planis, E6, LΣ, præfatis æquidiſtantibus, ſint
conceptæ in ſolidis figuræ, quæ iacent in eodem plano, ſemper in-
ter ſe æquales, vt, FG, æqualis, HI, & , MN, OP, ſimul ſumptæ
(ſit enim ſolida figura ex. g. BZ& , intus vtcunq; caua ſecundum
ſuperficiem, {12/ }, N, {13/ }, O,) æquales ipſi, SV. Dico eaſdem ſolidas
figuras æquales eſſe. Si enim ſolidum, BZ& , cum portionibus,
ABY, & planorũ, AD, Y4, ipſi conterminantibus, ſolido, CβΛ, ita
ſuperpoſuerimus, vt planum, AB, ſit in plano, AD, & , Y& , in pla-
no, Y4, oſtendemus (vt fecimus ſuperius circa parallelarum ipſi, A
D, conceptas in figuris planis, BZ& , CβΛ, portiones) figuras in
ſolidis, BZ& , CβΛ, conceptas, quæ erant in eodem plano, etiam
poſt ſuperpoſitionem manere in eode plano, & ideò adhuc ęqua-
les eſſe figuras in ſuperpoſitis ſolidis conceptas, & ipſis, AD, Y4,
parallelas. Niſi ergo totum ſolidum toti congruat in prima ſu-
perpoſitione, relinquentur reſidua ſolida, vel ex reſiduis compoſi-
ta in vtroq; ſolido, quæ non erunt ad inuicem ſuperpoſita, cum
enim ex. g. figuræ, QR, ST, æquentur figuræ, SV, dempta com-
muni figura, ST, reliqua, QR, æquabitur reliquæ, TV, hocq; cõ-
tinget in quouis alio plano ipſi, AD, parallelo occurrenteſolidis,
C℟Γ, CβΛ, ergo ſemper habentes reſiduum vnius ſolidi, habebimus
etiam reſiduum alterius, & patebit, iuxta methodum adhibitam in
priori parte huius Propoſitionis circa figuras planas, reſidua ſoli-
da, vel reſiduorum aggregata ſemper eſſe in eiſdem parallelis pla-
nis, vt reſidua, H℟597, ΙΓΛ, 785, eſſe in planis parallelis, E6, Y4,
ac æqualiter analoga: ſi ergo hæc reſidua adhuc ſuperponantur,
ita vt planum, EH, locetur in plano, H6, & , Υβ, in, β4, & hoc
ſempei fieri intelligatur, donec quod ſuperponitur, vt, BZ& , totũ
fuerit aſſumptum, tandem ipſum totum, BZ& , congruet toti, Cβ
Λ, niſi enim toto ſolido, BZ& , ipſi, CβΛ, ſuperpoſito, @pſa ſibi cõ-
gruerent, eſſet aliquod reſiduum vnius, vt ſolidi, CβΛ, ergo etiam
eſſet aliquod reſiduum ſolidi, C℟Γ, ſeu, BZ& , illudq; non eſſet ſu-
perpoſitum, quod eſt abſurdum, ponitur enim iam totum ſolidũ,
BZ& , eſſe ipſi, CβΛ, ſuperpoſitum, non ergo erit aliquod reſiduũ
in ipſisſolidis, ergo ſibi congruent, ergo dictæ figuræ ſolidæ, BZ& ,
CβΛ, inter ſe æquales erunt, quæ fuerunt demonſtranda.