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the divisions of the hour-circle, or between two of them; and in this 
way the time can be easily estimated to half a minute, supposing the 
hours to be subdivided into spaces equal to three minutes each. 
But there may be situations of the sun, or of the dial, or of the ob- 
server, or there may be peculiarities of his vision, which would render 
the ascertainment of the hour by the foregoing method impracticable. In 
such cases a new method becomes necessary, to attain precision, the 
object of which is practically to efface the penumbra, the great enemy 
to all gnomonic observations. 
After unsuccessfully trying a number of experiments, I found that, 
by the employment of the following appendage, definiteness may be 
given to the spread and weak part of the shadow, and the penumbra re- 
duced to an imperceptible size. It consists ofa hoop of brass about two 
inches in diameter, and three-quarters of an inch in depth, soldered at 
one end to a circular plate of brass of the same diameter, a sufficiency of 
the centre of which has been cut out for the insertion of a double concave 
lens about an inch and a quarter in diameter. The lens should be of 
about four inches imaginary focus, and of equal concavity on both sides ; 
its precise centre should be found, and permanently marked, by a me- 
thod well known to opticians. On this centre a very small dot is to be 
made by applying to it the section of a common pin dipped in strong 
Indian ink. A handle completes the instrument. 
To use this, it is to be placed with its open end opposite the sun, so 
that the whole bottom shall be illuminated, and that no part of the hoop 
shall cast its shadow within; for then the axis of the lens will fairly be 
pointed towards the sun. If the lens be held in this position over the 
hair-style at such a distance (within an inch or two) as gives the thin- 
nest shadow of the hair, a very large, somewhat dark, oval willappear on 
the dial-plate, surrounded by a broad luminous border. The shadow of 
the hair-style will now appear broken into two parts near the middle of 
its length; the parts diverge from each other, one being in the luminous 
border, the other in the dark oval. The shadow of the Indian ink dot 
will appear somewhere in the dark portion as a small oval spot, very 
weak, and surrounded by penumbra. If the lens be moved to the right 
or left, as the case may require, the two dislocated parts of the shadow 
of the hair-style will approach each other, and join end to end, forming 
one straight black line, which must be made to bisect the shadow caused 
by the dot of Indian ink. This bisection can be done by estimation with 
very great accuracy, provided the sunlight be sufficiently strong. The 
straight black line, thus generated by the reunion of its parts, is now 
destitute of any perceptible penumbra; is longer and stronger than be- 
fore, and definite. It is capable of reaching and even passing the most 
distant figures on the' dial-plate, and is so narrow that by means of it 
the observer may estimate half minutes with correctness; whereas, in its 
natural state it would fill up a space equal to three minutes, 
The use of the Indian ink dot on the lens is this: although by mov- 
ing the lens one way or the other, the broken ends of the two shadows 
may be united as already described, they may not truly coincide, yet the 
eye may not be adequate to detect a minute want of coincidence; but 
