Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/248

ciprocally as the altitude IC; that is (if there be given any quantity Q and the altitude IC be called A) as $\textstyle {\frac {Q}{A}}$. This quantity $\textstyle {\frac {Q}{A}}$ call Z, and ſuppoſe the magnitude of Q to be ſuch that in ſome caſe √ABFD may be to Z as IK to KM and then in all caſes, √ABFD will be to Z as IK to KM and ABFD to ZZ as $\scriptstyle {IK^{2}}$ to $\scriptstyle {KN^{2}}$, and by diviſion ABFD−ZZ to ZZ as $\scriptstyle {IN^{2}}$ to $\scriptstyle {KN^{2}}$, and therefore, $\scriptstyle {\sqrt {ABFD-ZZ}}$ to Z or $\textstyle {\frac {Q}{A}}$ as IN to KN and therefore $\scriptstyle {A\times KN}$ will be equal to $\textstyle {\frac {Q\times IN}{\sqrt {ABFD-ZZ}}}$. Therefore ſince $\scriptstyle {YX\times XC}$ is to $\scriptstyle {A\times KN}$ as $\scriptstyle {CX^{2}}$ to AA the rectangle $\scriptstyle {XY\times XC}$ will be equal to $\textstyle {\frac {Q\times IN\times CX^{2}}{AA{\sqrt {ABFD-ZZ}}}}$. Therefore in the perpendicular DF let there be taken continually Db, Dc equal to $\textstyle {\frac {Q}{2{\sqrt {ABFD-ZZ}}}}$, $\textstyle {\frac {Q\times CX^{2}}{2AA{\sqrt {ABFD-ZZ}}}}$ reſpectively, and let the curve lines ab, ac, the toci of the points b and c, be deſcribed: and from the point V let the perpendicular Va be erected to the line AC, cutting off the curvilinear area's VDba, VDca, and let the ordinates Ez, Ex, be erected alſo. Then becauſe the rectangle Db×IN or DbzE is equal to half the rectangle A×KN or to the triangle ICK; and the rectangle Dc×IN or DcxE is equal to half the rectangle YX×XC or to the triangle XCY; that is, because the naſcent particles DbzE, ICK of the

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