Syntagma Mathesios; Containing the Resolution of Equations with a New Way of Solving Cubic and Biquadratic Equations Analytically and Geometrically - Also the Universal Method of Converging Series, After an Easy and Expeditious Manner Wherein Also Are Tre (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1745 Excerpt: ...Equations so constructed this Difference only arises, that in the former Equations, by Reason of the last Term being wanting in the preceding Equation, is always pl--ql--sl--tl = o, or / = sl--ql--p7-). Therefore from the Centre E, and with the Radius EB + (ER1) ST'-VS1) having described the Circle CKxf, one of the Roots in the former Equation vanishes to nothing. And these are demonstrated after the following Manner. Having constructed the continued Points, and CP produced, if there is Occasion for it, till it cut AM in H, CH will be the ordinate of the Parabola to the Diameter AH, and therefore CH1 = AL x AH = AH, because AL = i. But CH = CP+AG, and AH = GB+BP, and therefore CP2 + 2AG x CP + AGa = GB-f-BP. Now from the Point C, let fall the Perpendicular CD to BP, which meets El in the Peint I drawn parallel to BP. Therefore by similar Triangles CDP and TVS, it will be ST; VS: . CP: VSCP = DP. O 1 And ST: VT:: CP: = CD. And therefore CP1 + 2AG x CP = BP = DP-f BD =- VRvfP vs ST + BR IE' or C+ 2AG x Cp--WF CP--BR=--IE, ButTECE1--ClCE1--CD-VT' VT1 x (Jp1--VT1--2CD xVT = VT--ST1 2ST CP = (because = ST"--CE'--Cp 4.-"CP1--ST+SV--2STx CP+ CP, ST2 bT therefore which will be equal to the Square of the Side VS CFl + 2 AG x CP CP--BR-And this Equation is reduced to the Terms p, q, r, s, t, as the veryEquation was proposed. E. D. Hence it appears, that any Biquadratic Equation may be constructed innumerable Ways by the Parabola, for an indefinite Value of that Quantity, which we laid was assumed at Pleasure. But the Case is the most simple, by making VS=/=o, and then the Construction is changed, if you regard the Thing into this common one, in which the right Lines CP, sf are Representatives of the Roots Perpendiculars to the Axis. And the Equati...