510 PHILOSOPHICAL TRANSACTIONS, [aNNO ]741. 



rnxn; then take ar = am, and xr = an, and joining mr, Nr, draw at parallel 

 to them, and m, m, and n, n, being joined, draw mt, nt. Now nx : At :: nr or 

 tiiR : rN :: mR X ma : tn X am :: otr X ma : mr X an :: ma X Am : an x at; but 

 in the ultimate ratio uiA = ma, and ta is perpendicular to mn, therefore «a : At 

 :: MA*: AN X at; now if, through the centre of the circle f, there be drawn 

 the line mf, produced to meet ta, also produced in g, that is to the circum- 

 ference of the circle, then will ma^ = ta X ag; therefore tia : At :: 4g : an ; 

 therefore describe a semicircle through g and n, which will cut the line at in t, 

 from which /n being drawn, it will be a tangent to the curve, to which also no 

 is perpendicular; hence mo being joined, a parallel to it, drawn through n, will 

 be the tangent to the curve. 



And here it may be observed, that this method of drawing tangents agrees 

 with most curves. Thus let ab, fig. 2, be the conchoid of Nicomedes ; then, 

 supposing the former preparation, bp : Ft :: br or cr : e.b :: cr X cp : r^ X cp or 

 re X PH :: cp* : tp X PR ; hence the former construction is derived. 



Again, let a line cpb, fig. 3, of a given length, have its extremity c moved 

 along the line cdt, perpendicular to da, and always pass through the point p 

 in the same given line da, and so generate a curve ab. Now applying the 

 former preparation and reasoning to this, then we shall have bp : Ptiiba or 

 re : eb : cr X CP : rb X cp or bp X re :: cp* : bp X pt, as before. 



Also the method de maximis and minimis gives the greatest ordinate = \a, 

 and its absciss = ^av'3. In like manner the greatest absciss might be investi- 

 gated, but by a tedious process ; therefore find it thus : because en, fig. 1 , is 

 a tangent to the curve, the line mg, drawn from the point m through the centre 

 F, determines the point g, from which gn being drawn perpendicular to en, 

 therefore also to Aa by the hypothesis. But ng = av = ma + ap ; therefore 

 vp = MA ; but ba : am :: ma : ap ; therefore ba : pv :: vp : pa ; but pp = fv := 

 a — 2z ; therefore a : a — 2z :: a — 2z : z. Hence is easily deduced z = -^a ; 

 EN=:4a; Aa = ^\/3. Where it is to be noted, that the same point m, 

 which gives, in the line namn, the point of the greater ordinate, gives also 

 the point of the greater absciss. 



^ Machine to represent Eclipses of the Sun. By J. u4nd. Segner, Med. Physic, 

 et Mathem. Prof. Goetting, R.S.S. N" 46 1, p. 781. From the Latin. 



A projection of the arches and circles of the earth's illuminated surface, on 

 a plane, may serve well enough to show any solar eclipse ; and if the places on 

 the earth's surface, as cities, islands, &c. be inserted in the projection, and a 

 circle be added, to express the position and magnitude of the lunar penumbra. 



