those volumes into a continued series. To each number also will be found annexed, in the Indian figures inclosed in a parenthesis, the number of the same proposition as it was numbered in the first miscellaneous treatise of the year 1787. The additional theorems are as follow. SECTION I. OF THE ELLIPSE. THEOREM XII (5). The Difference between the Semi-transverse and a Line drawn from the Focus to any Point in the Curve, is equal to a Fourth Proportional to the Semi-transverse, the Distance from the Centre to the Focus, and the Distance from the Centre to the Ordinate belonging to that Point of the Curve. For, draw AG parallel and equal to ca the semi-conjugate; and join co meeting the ordinate de in H. Then, by theor. 2, CA2: AG2 :: CA2 - CD: DE': DH2; 2CF.CD + CD2; Also FD = CF CD, and FD2 = CF2 consequently And the root or side of this sqnare is FE = CA - CI = AL. In the same manner is found fE = CA + CI = HI. Q. E. D. Corol. 1. Hence cí or CA - FE is a 4th proportional to CA, CF, CD. Corol. 2. And fE - FE = 2C1; that is, the difference between two lines drawn from the foci, to any point in the curve, is double the 4th proportional to CA, CF, CD. THEOREM XIII (11). If a Line be drawn from either Focus, Perpendicular to a Tangent to any Point of the Curve; the Distance of their Intersection from the Centre will be equal to the Semi- transverse Axis. For, through the point of contact E draw Fe, and fe meeting FP produced in G. Then, the GEP = FEP, being each equal to the fep, and the angles at P being right, and the side PE being common, the two triangles GEP, FEP are equal in all respects, and so GE = FE, and GP = FP. Therefore, since FP = FG, and FC = rf, and the angle at F common, the side CP will be = G or AB, that is CP = And in the same manner cp = CA or CB. Q.E.D. Corol. 1. A circle described on the transverse axis, as a diameter, will pass through the points P, p; because all the lines CA, CP, Cр, св, being equal, will be radii of the circle. CA or CB. Corol. 2. CP is parallel to fe, and cp parallel to FE. Corol. 3. If at the intersections of any tangent, with the circumscribed circle, perpendiculars to the tangent be drawn, they will meet the transverse axis in the two foci. That is, the perpendiculars PF, pf give the foci F, f. THEOREM XIV (12). The equal Ordinates, or the Ordinates at equal Distances from the Centre, on the opposite Sides and Ends of an Ellipse, have their Extremities connected by one Right Line passing through the Centre, and that Line is bisected by the Centre. B 2 That That is, if CD = CG, or the ordinate DE = GH; For when CD = CG, then also is DE = GH by cor. 2, th. 1. But the ∠D = ∠G, being both right angles; therefore the third side CE = CH, and the DCE = GCH, and consequently ECH is a right line. Corol. 1. And, conversely, if ECH be a right line passing through the centre; then shall it be bisected by the centre, or have CE = CH; also DE will be = GH, and CD = CG. Corol. 2. Hence also, if two tangents be drawn to the two ends E, H of any diameter EH; they will be parallel to each other, and will cut the axis at equal angles, and at equal distances from the centre. For, the two CD, CA being equal to the two CG, CB, the third proportionals cr, cs will be equalalso; then the two sides CE, ст being equal to the two CH, cs, and the included angle ECT equal to the included angle HCS, all the other corresponding parts are equal: and so the LT=∠s, and Te parallel to us. Corol. 3. And hence the four tangents, at the four extremities of any two conjugate diameters form a parallelogram circumscribing the ellipse, and the pairs of opposite sides are each equal to the corresponding parallel conjugate diameters. For, if the diameter eh be drawn parallel to the tangent TE or us, it will be the conjugate to EH by the definition; and the tangents to e, h will be parallel to each other, and to the diameter EH for the same reason. THEOREM XV (13). If two Ordinates ED, ed be drawn from the Extremities e, e, of two Conjugate Diameters, and Tangents be drawn to the same Extremities, and meeting the Axis produced in TandR; Then 1 1 Then shall CD be a mean proportional between cd, dr, and cd a mean proportional between CD, DT. Corol. 2. Hence also CD : cd :: de : DE. And the rectangle CD. DE = cd. de, or Corol. 3. Also cd2 = CD. DT, CDE = A cde. and CD2 = cd.dr. Or cd a mean proportional between CD, DT; THEOREM XVI (14). 1 The same Figure being constructed as in the last Theorem, each Ordinate will divide the Axis, and the Semi-axis added to the external Part, in the same Ratio. Corol. 1. Hence, and from cor. 3 to the last, it is, CD2, cd2. Corol. and ca2 DE2 + de2. Corol. 3. Further, because CA2: ca2:: AD. DB or cď2 : DE2, therefore CA : ca :: cd: DE. likewise CA: ca :: CD : de. THEOREM XVII (15). If from any Point in the Curve there be drawn an Ordinate, and a Perpendicular to the Curve, or to the Tangent at that Point: Then, the Dist. on the Trans, between the Centre and Ordinate, CD: Will be to the Dist. PD:: As Sq. of the Trans. Axis : That is, CA: ca2 :: DC: DP. For, by theor. 2, CA2 : ca2 :: AD.DB: DE, CD, DT AD. DB; or Ac2 ça2: DC THEOREM XVIII (18). If there be Two Tangents drawn, the One to the Extremity of the Transverse, and the other to the Extremity of any other Diameter, each meeting the other's Diameter produced; the two Tangential Triangles so formed, will be. equal. |