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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