Cardano, Geronimo - Opvs novvm de proportionibvs nvmerorvm, motvvm, pondervm, sonorvm, aliarvmqv'e rervm mensurandarum, non solùm geometrico more stabilitum, sed etiam uarijs experimentis & observationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, & in V libros digestum. Praeterea Artis Magnae, sive de regvlis algebraicis, liber vnvs abstrvsissimvs & inexhaustus planetotius Ariothmeticae thesaurus ... Item De Aliza Regvla Liber, hoc est, algebraicae logisticae suae, numeros recondita numerandi subtilitate, secundum Geometricas quantitates inquirentis ...

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1cum ſuis duplicatis ſequentibus, & in c, & in d, & in reliquis pa
riter conduplicatis ſuis ſequentibus ex altera, quod fit ex h in b ſe
mel, in c ter, in d quinquies, in e ſepties, in f nouies, in g undecies,
in h tredecies, detractis ergo rurſus quod fit ex h in b ſemel, & ex
h in c d e f g h bis relinquetur, quod fit ex h in c, & duplo ſequen
tium, & d & duplo ſequentium, & e & aliarum pariter: & ex alia
parte, quod fit ex h in c ſemel, & in d ter, & in e quinquies, in f ſe
pties, in g nouies, in h undecies. Ab his rurſus detractis, quòd fit
ex h in c ſemel, & in ſequentes bis, relinquetur h in d ſemel cum ſuis
ſequentibus bis, & in e ſemel cum ſuis ſequentibus & in f, & in g &
in h pariter, & ex alia parte, quod fit ex h in d ſemel, in e ter, f quin
quies, g ſepties, h nouies, ab his rurſus detraho, quod fit ex h in d
ſemel, & in ſequentes bis, relinquetur ex una parte, quod fit ex h
in e f g h cum duplo ſequentium ex alia, quod fit ex h in e ſe
mel, f ter, g quinquies, h ſepties, & ſimiliter ab his detractis, quod
fit ex h in e ſemel, & bis in ſequentes, relinquetur ex una par
te; quod fit ex h in f ſemel, & in g h bis, & in g ſemel, & in h bis,
& in h ſemel, & ex alia, quod fit ex h in f ſemel, in g ter, in h quin
quies. Iterum detractis, quod fit ex h in f ſemel, & in g h bis com
muniter relinquetur, quod fit ex h in g ſemel, & in h bis, & in h ſe
mel, & ex alia parte quod fit ex h in g ſemel, & ex h in h ter. Sed
iſta, quæ relicta ſunt iam, ſunt manifeſtè æqualia, ergo etiam pri
ma aggregata ab initio fuere æqualia, ergo & æqualia illis qua
drata a b c d e f g h his, quæ fiunt, ex h in eaſdem quantita
tes cum duplo producti b in i, cin k, d in l, e in m, f in n, g in o,
h in p, ſed iam his quadratis a b c d e f g h demonſtrata ſunt eſſe du
pla quadrata h p, g o, f n, e m, d l, c k, b i, cum duplo quadra
ti a, ergo quadrata omnium quantitatum ſecundi ordinis cum
quadrato a rurſus repetito, & producto h in aggregatum quanti
tatum primi ordinis ſunt tripla quadratis quantitatum primi ordi
nis pariter acceptis, quod fuit propoſitum, & fuit Archimedis in li
bro de lineis ſpiralibus, & ego adieci hic propter modum demon
ſtrandi, qui eſt elegantiſsimus, & procedit ex principijs arithmeti
cis, & diuerſis à communibus, & ideo non reuoluitur, ut ſolent re
liquæ quæſtiones.

In 5. Elem.
Prop. 12.

Lib. 6. Ele.
Prop. 17.

Propoſitio uigeſima.

Cùm fuerint quatuor quantitates, fueritque ſecunda æqualis ter
tiæ, aut primæ æqualis quartæ, erit proportio primæ ad quartam,
aut tertiæ ad ſecundam producta ex proportionibus primæ ad ſe
cundam, & tertiæ ad quartam.

Cor^{m}.

Cùm enim quantitates hæ non fuerint ęquales, conſtat per

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