388 Dr» R. Hey’s Propositions concerning 
directing points, for all between i and the furthest, will be 
between L and G. 
The use of the law found, is this. When, in practice, it 
would be tedious to find * accurately, in position and length, 
the axes for each circle ; the artist may, by this law, know 
pretty nearly the directions of the axes for all the circles, of 
the rank for which he has constructed the triangle LOA or 
FOA. Those axes, howsoever seldom they may converge -f 
to a point, are always parallel to lines which do. In addition, 
he may find, by perspective, such particular points as any case 
may suggest. If ST be the diameter perpendicular to the di- 
recting line, st is readily found : which passing through v, so 
being a diameter of the ellipse, its middle point is one point in 
each axis. Or, some centroids V may be found ; v/hence the 
elliptic centres r, by easy operations of perspective. If the 
circles are equal, it may suffice, after V found in a few of the 
nearest circles, to take the remaining lines EV (in the same 
rank) equal, or to diminish them by the view. And, if the 
circles (in each rank) be also at equal intervals, their centroids 
may often be considered as at equal intervals ; or, after a few 
of the nearest. Whence the points with increased facility. 
If the mean point in any circle, as ix, be within ON, find 
the directing point R of the major axis : which may be more 
convenient than that of the minor, as nearer J to A. But this 
case cannot happen, with the mean point lying beyond a pic- 
ture perpendicular to the objective plane ; unless the perpen- 
dicular distance (by the scale assumed for any delineation) of 
* By prop. VI. f See Cor. 2 to Lemma. 
I If M be in the circumference ON, the directing points of the two axes are equi- 
distant from A ; by the construction of prop. VI. 
