178 
MINUTES OF PROCEEDINGS OF 
This mark was evidently true for this particular case, and truth 
being always consistent, would be true for all other values of B — M. 
If a 9-pr. were used, a new mark, similarly obtained, would have 
to be made in the roller, this being the only change required for the 
different nature of guns. 
From 40 to 0, and again to 30, full figures are used on the top 
scale; after this come 4, 5, 6, &c.; these really mean second forties, 
fifties, &c., because, while 40 in full represents 1°, 0 represents 2|°, 
30 represents 3J, and the small 4 represents 3J°. Consequently, some¬ 
times when B —< M = 50, instead of setting to the large 50 it may be 
necessary to set to the small 5 ; but a reference to the tape mark, which 
always points out the true distance between the pieces, dispels all doubt 
as to which value should be employed. 
The rule on this head is, over 38 yards set to full figures, under 
38 yards consult the tape mark as to whether full or small figures are to 
be used. 
Theory of the Ban Test. 
The sun is at an infinite distance, consequently, when the telescopes 
are laid upon the same portion of its disc, the two angles of the sur¬ 
veyor’s triangle become = to 180°, or 100 on the range-finder 
scale. 
Theory of the Second or Four Angle Test. 
In this test we first find the two base angles of the triangle in the 
usual fashion, we then obtain the vertical angle by subtracting from the 
exterior angle of a triangle, what Euclid calls the interior and opposite 
angle. 
Now, if the instruments are in good order, these three angles will 
amount to 180° = on range-finder scale 100. If they, on the contrary, 
contain an error x, we shall have recorded for the base angles a 
quantity = true base angles + x. But the vertical angle will still 
be true, for, suppose the exterior angle contains the error y, the 
interior and opposite angle must also contain the error y , and 
then— 
as false extr. ang. = true ex. ang. + y. 
„ intr. „ = „ int. „ + y. 
:. false extr. ang. — false intr. ang. 
= in all cases true extr. ang. — true int. ang. 
= vertical angle. 
.*. Vertical angle and angles found in ordinary fashion = 1800 + error 
of instruments, or base angles found in ordinary fashion + exterior 
angle, — interior angle — 180° + error = 100 + error. 
The mathematical aspects of the question are ably given in “ Bemarks 
on Captain Nolan’s Bange-finder,” by Captain C. E. B. Leacock, B.A.:—* 
“Captain Nolan’s range-finder consists of two angle-finders, a measur¬ 
ing tape, and a calculating roller. 
See “Proceedings, R.A. Institution,” Yol. VII., p. 40, 
