213193LIBER II.
Hæc manifeſta eſt, nam habebunt baſes ipſis altitudinibus recipro-
cas, quod etiam vniuerſalius oſtenditur Vndecimo Elementorum
Propoſ. 36.
cas, quod etiam vniuerſalius oſtenditur Vndecimo Elementorum
Propoſ. 36.
THEOREMA XLII. PROPOS. XLII.
DAta recta linea terminata, vtcumque in puncto diuiſa,
poſſibile eſt ipſam ad alteram eiuſdem partium ita
producere, vt cubus compoſitæ ex propoſita linea, & adiun-
cta, ſit æqualis cubo propoſitæ lineæ, ſimul cum cubo com-
poſitæ ex adiecta, & illi conterminante portione ſectæ lineę.
poſſibile eſt ipſam ad alteram eiuſdem partium ita
producere, vt cubus compoſitæ ex propoſita linea, & adiun-
cta, ſit æqualis cubo propoſitæ lineæ, ſimul cum cubo com-
poſitæ ex adiecta, & illi conterminante portione ſectæ lineę.
Sit data recta linea, AC, terminata, diuiſaq;
vtcumque in pun-
cto, B, oftendendum eſt poſſibile eſſe ipſam ita producere ad alteram
illius partium, vt ad, C, vt cubus compoſitę ex, AC, & adiecta, ſit
æqualis cubo, AC, cum cubo compoſitæ ex eadem adiecta, & ex,
BC, portione, AC, adiectæ conterminante. Producatur ergo, C
A, ad partes, A, vt in, N, ita quod, NB, ſit tripla, BA, ſiat dein-
de, vt, NB, ad, BC, ita quadratum, BC, ad quadratum rectæ li-
neę, M, ſeorſim poſitæ: Vlterius exponatur recta, EF, æqualis com-
poſitæ ex, AC, CB, cui applicetur rectangulum æquale quadrato,
128[Figure 128]
M, excedens figura quadrata,
1129. Sex.
lem, cuius latus ſit, FH, producatur
autem, AC, verſus, C, vt in,
D, ita nempè, vt, CD, ſit
æqualis, FH. Dico cubum to-
tius, AD, æquari duobus cubis,
AC, BD. Cum. n. ſit, vt, N
B, ad, BC, ita quadratum, BC,
ad quadratum, M, ideò parallelepipedum ſub altitudine, AB, (qu
eſt, {1/3}, prædictę altitudinis, NB,) & quadrato, M, æquabitur ter-
tiæ parti cubi, BC. Quoniam verò quadratum, M, æquatur rectan-
22E. Cor. 4
Gen. 34.
huius. gulo, EHF, ideſt rectangulo ſub compoſita ex, AD, BC, & ſub,
CD, ideò parallelepipedum ſub altitudine, AB, & baſi rectangulo
ſub compoſita ex, AD, BC, & ſub, DC, æquabitur tertiæ parti
cubi, BC, addatur commune parallelepipedum ſub, BC, & baſi re-
ctangulo, BDC, erit ex vna parte hoc parallelepipedum cum, {1/3}, cu-
bi, BC, ex alia verò hæcſumma; ſcilicet parallelepipedum ſub, AB,
& ſub rectangulo ſub compoſita ex, AD, BC, & ſub, DC, vna
cum parallelepipedo ſub, BC, & rectangulo, BDC, quæ quidem
ſumma efficit parallelepipedum ſub, AC, & rectangulo, ADC,
cto, B, oftendendum eſt poſſibile eſſe ipſam ita producere ad alteram
illius partium, vt ad, C, vt cubus compoſitę ex, AC, & adiecta, ſit
æqualis cubo, AC, cum cubo compoſitæ ex eadem adiecta, & ex,
BC, portione, AC, adiectæ conterminante. Producatur ergo, C
A, ad partes, A, vt in, N, ita quod, NB, ſit tripla, BA, ſiat dein-
de, vt, NB, ad, BC, ita quadratum, BC, ad quadratum rectæ li-
neę, M, ſeorſim poſitæ: Vlterius exponatur recta, EF, æqualis com-
poſitæ ex, AC, CB, cui applicetur rectangulum æquale quadrato,
1129. Sex.
lem, cuius latus ſit, FH, producatur
autem, AC, verſus, C, vt in,
D, ita nempè, vt, CD, ſit
æqualis, FH. Dico cubum to-
tius, AD, æquari duobus cubis,
AC, BD. Cum. n. ſit, vt, N
B, ad, BC, ita quadratum, BC,
ad quadratum, M, ideò parallelepipedum ſub altitudine, AB, (qu
eſt, {1/3}, prædictę altitudinis, NB,) & quadrato, M, æquabitur ter-
tiæ parti cubi, BC. Quoniam verò quadratum, M, æquatur rectan-
22E. Cor. 4
Gen. 34.
huius. gulo, EHF, ideſt rectangulo ſub compoſita ex, AD, BC, & ſub,
CD, ideò parallelepipedum ſub altitudine, AB, & baſi rectangulo
ſub compoſita ex, AD, BC, & ſub, DC, æquabitur tertiæ parti
cubi, BC, addatur commune parallelepipedum ſub, BC, & baſi re-
ctangulo, BDC, erit ex vna parte hoc parallelepipedum cum, {1/3}, cu-
bi, BC, ex alia verò hæcſumma; ſcilicet parallelepipedum ſub, AB,
& ſub rectangulo ſub compoſita ex, AD, BC, & ſub, DC, vna
cum parallelepipedo ſub, BC, & rectangulo, BDC, quæ quidem
ſumma efficit parallelepipedum ſub, AC, & rectangulo, ADC,