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

 172 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 uppoe the magnitude of Q to be uch that in ome cae √ABFD may be to Z as IK to KM and then in all caes, √ABFD will be to Z as IK to KM and ABFD to ZZ as $$\scriptstyle{IK^2}$$ to $$\scriptstyle{KN^2}$$, and by diviion 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 ince $$\scriptstyle{YX \times XC}$$ 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 $$\textstyle{\frac{Q}{2\sqrt{ABFD-ZZ}}}$$, $$\textstyle{\frac{Q \times CX^2}{2AA\sqrt{ABFD-ZZ}}}$$ repectively, and let the curve lines ab, ac, the toci of the points b and c, be decribed: 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 alo. Then becaue 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 nacent particles DbzE, ICK of the area's