Weidler, Johann Friedrich, Jo. Friderici Weidleri Tractatus de machinis hydraulicis toto terrarum orbe maximis Marlyensi et Londinensi et aliis rarioribus similibus in quo mensurae prope ipsas machinas notatae describuntur, et de viribus earum luculenter disseritur

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18968
2ac + 2bd = 2xy
et ac + bd = xy
iterum ſubſtitutis aequipollentibus x et y
ac+bd=Va2+b2. Vc2+d2.
facta irrationalium multiplicatione, (§ 30. Anal. finit.)
ac+bd=Vc2a2+c2b2+a2d2+b2d2
ut ſignum radicale tollatur, fiat quadratum ex ac + bd, et fiet
a2c2 + 2acbd + b2d2 = c2a2 + c2b2 + a2d2 + b2d2
deletis acqualibus
2acbd = c2b2 + a2d2,
ſequeretur ergo ex bac bypotbeſi, duo haec facta ultima aequalia
eſſe, id quod promiſcue ſumtis perpendiculis et baſibus, fieri nequit.
Interim tamen ex theor. pythagorae conſtat, omnino promiſcue ſumta
perpendicula et baſes, ſiquidem angulum rectum capiant, triangu-
lum rectangulum formare. illuſtratio exemplo ſpeciali facta rem to-
tam clariorem reddet. ſint perpendicula 8 et 14 baſes 10 et 18
per theor. pyth. erit 82 + 102 = □ bypotenuſae = 164
et 142 + 182 = □ hypot. = s20
ſed ſecundum ſuperiorem hypotheſin foret
(8+14)2 + (10+18)2 = 484 + 784 = 1268 = 684
quod manifeſto ueritati repugnat.
Attamen datur caſus, quo regula Voigtelii bene ſeſe haber, nem-
pe, quando triangula rectangula aequales babent angulos, et ſunilia
ſunt, ut ualeat analogia, (§. 68. Geom.)
a: b = c: d ubi {bc/a} = d
ſiquidem ſubſtituto in aequatione
2abcd = c2b2 + a2d2
ualore d et d2, prodit
2abc. {bc/a} = c2b2 + a2. {b2c2/a2}
boc eſt
2b2c2 = 2b2c2

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