Page:Newton's Principia (1846).djvu/573

LEMMA I.

To cut two right lines OR, TP, given in position, by a third right line RP, so as TRP may be a right angle; and, if another right line SP is drawn to any given point S, the solid contained under this line SP, and the square of the right line OR terminated at a given point O, may be of a given magnitude.

It is done by linear description thus. Let the given magnitude of the solid be M² $\scriptstyle \times$ N: from any point r of the right line OR erect the perpendicular

rp meeting TP in p. Then through the point Sp draw the line Sq equal to $\scriptstyle {\frac {M^{2}\times N}{Or^{2}}}$. In like manner draw three or more right lines S2q, S3q, &c., and a regular line q2q3q, drawn through all the points q2q3q, &c., will cut the right line TP in the point P, from which the perpendicular PR is to be let fall.   Q.E.F.

By trigonometry thus. Assuming the right line TP as found by the preceding method, the perpendiculars TR, SB, in the triangles TPR, TPS, will be thence given; and the side SP in the triangle SBP, as well as the error $\scriptstyle {\frac {M^{2}\times N}{OR^{2}}}-SP$. Let this error, suppose D, be to a new error, suppose E, as the error 2p2q ± 3p3q to the error 2p3p; or as the error 2p2q ± D to the error 2pP; and this new error added to or subducted from the length TP, will give the correct length TP ± E. The inspection of the figure will shew whether we are to add to or subtract; and if at any time there should be use for a farther correction, the operation may be repeated.