1816.] Circular Appearance of the Ellipse. 205 
Articuie III. - 
A Demonstration that the Ellipse, when viewed in a certain Position, 
appears circular.. By 8. 
(To Dr. Thomson.) 
SIR, Jan. 8, 1816, 
Iris well known that the circle, if looked upon obliquely, will be 
projected into an ellipse ; but-I am not aware that the converse of 
this proposition has been demonstrated by showing that an ellipse, 
if viewed in a certain position, will appear circular. This has been 
established in the following theorems; but it was hot the primary 
object with which they were drawn up. They were occasioned b 
the wish of getting a scalene cone turned truly ina lathe. Many 
good workmen assured me that it was impossible ; and all the cones 
of this kind which I have seen (with the exception of that which I 
am about to mention) have been cut by hand. In thinking on the 
subject, it appeared that the best way would be to get a cone turned 
in the first instance with an elliptical base; but here again 1 met 
with a difficulty which was not surmounted till within these few 
months, by a very ingenious friend, who devised the means of 
executing exactly what I wished. Bya mechanical method of trial, 
he afterwards found where he might cut this elliptical cone ob- 
liquely, so that the base should become a circle; but it seemed 
more satisfactory to investigate the problem mathematically, and I 
here send you the result. 1t would be unjust, however, not to add 
that the first lemma is taken from Dr. Robertson’s Conic Sections 
(8vo. Oxford, 1802), and that several parts of the proposition were 
suggested by a recollection of the methods used in the first book of 
that valuable treatise. 8. 
Lemma \.—If from any point, I, of a straight line, NB (Plate 
XLV.* Fig. 1), a perpendicular, [ L, be drawn; and if in all cases 
the rectangle under NI, I B, shall be equal to the square of I L, 
the curve passing through N L B shall be a semicircle. 
For bisect N B in C, and join C L.° Then (5, ii.) NI, IB, + 
{ C* = C B’; therefore by hypothesis I L? + I C? = C B®; but 
(47, i.) 1L? + 1C*? = C L*; consequently CB? = CL*% The 
same would hold wherever the point I is taken; therefore the locus 
of the points L must be a curve, which would be generated by the 
extremity, L, of the given straight line, C L, when its other extre- 
mity, C, is fixed, and the straight line revolves about it. 
Lemma 2.—Let V AB (Fig. 2) be an isosceles triangle, andO G 
a straight line longer than the base; then if VT be taken, such 
that O G’ — A B* : OG? :: VB? : V 'T2, we shall have A B?: 
OG*:: AT, TB: VT. For AB: OG? : VT?—VB?: 
V‘l*; and when VC is drawn perpendicular to (and therefore 
bisecting) the base, V'T? — V B® = (47,1.) VC°+ CT? -—- VC 
— CB = CT*—CB = AT, TB (6, ii.). 
* The lower division of the Plate, 
