VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. l63 



ocean, having passed over the Azores; and that its southern limit reached the 

 isle of Ushant, and the northwest coasts of Britanny, between Brest and Mor- 

 laix; and dividing our islands of Guernsey and Jersey, just touched on the 

 promontory of Normandy, called Cape la Hogue. And that after it had quitted 

 England, and traversed the German ocean, it fell on Jutland on the south side, 

 and Norway on the north ; and thence proceeded to the eastwards over Sweden, 

 Finland, &c. 



It remains now to consider the figure, position, direction, velocity, and 

 magnitude of the shadow, as it passed over us. And first, as to the figure, it 

 is obvious that the shadow of the moon being a cone, and the earth's surface 

 sufficiently spherical, the apparent shadow on the earth will be the common 

 intersection of a cone and sphere, which is a figure hitherto little considered by 

 geometers, and not being in piano, it is not to be exactly described, except in 

 the spherical or conical surface. How to find the points of this curve in all 

 cases, is taught by P. Coursier, in a very scarce Latin book, printed at Dijon ia 

 Burgundy, and published at Paris in the year l663; nor do I hear of any other 

 author that has handled the same subject since, though capable and worthy of 

 further improvement.* By what he there delivers, prop. 11, 12, lib. 1, it will 

 be easily understood, that the convexity of so small a part of the earth's sur- 

 face as the shadow commonly occupies, can produce only an inconsiderable 

 eff^ect; so that without sensible error we may take it for a plane, and the section 

 for a true Apollonian ellipsis, whose transverse axis, by reason of the smallness 

 of the angle of the cone, will be to its conjugate, nearly as radius to the sine 

 of the sun's altitude at its centre, especially if he be considerably elevated. But 

 when he is near the horizon, it will be necessary to have regard to the true 

 figure, by reason of the great length to which the transverse axe is extended, 

 and particularly when the shade is entering on or leaving the earth's disk. Of 

 these perhaps a fuller account may be given on a future occasion. 



As to the position of the axis of the shadow, it is manifest that it must always 

 lie in the plane of a great circle of the earth passing through the axis of the 

 cone of the shade: and therefore it will be only requisite to obtain the azimuth 

 and altitude of the sun, at the place where the centre of the shade at any time 

 is found, to determine the situation of the axe and species of the ellipse required. 

 Thus, the middle of the eclipse at London having been observed at 9*^ lO'" 45% 

 by the given latitude and declination we find his azimuth about 59°, and altitude 

 40° 46', that is just 40° high at the centre of the shadow. Therefore the trans- 



* In some cases, the figure or cun^e, of the intersection of a cone and sphere, is a circle. See a 

 dissertation on the nature and properties of such intersections, in Dr. Hutton's tracts, mathematical 

 and philosophical, published in 1786, viz. props. 7, 8, 9, and their corollaries, p. 88. &c. 



Y 2 



