54 MATHEMATICS, 
compasses (pl. 5, figs. 3,4). Here the legs ad and be are applied to each 
other, and have two equal slits, in which the plates f and g pass, convertible 
by the screw e into an ordinary compass joint. This screw forms, then, the 
point of rotation of the two legs, thus divided into four. Upon the leg is 
placed a graduation, determining the proportions 1,1 4, &c., between the two 
legs. Loosening the screw e, and moving the head until the index mark on 
f coincides with a certain part of the division, $ for instance, and then 
tightening the joint, we shall have a relation of 1:1 between the two sets 
of legs. The space included between the points of one set of legs, will then 
be four times that between the points of the other set, whatever be the 
opening of the compasses. As there is a definite relation between the radius 
and the side of a regular polygon, another graduation is placed upon the 
compass to indicate this relation. This graduation is so arranged that when 
the index, f, stands at 1, where all the legs are equal, and consequently 
ab = cd, ed will be the side of a hexagon, inscribed in a circle of radius, ab. 
(The side of the inscribed hexagon = radius.) If the index f stands at 15, 
as in fig. 3, then ab =radius, and cd=the side of the pentadecagon. 
Fig. 4 exhibits the compasses edgewise, showing the nut of the screw. For 
the sake of great accuracy, a micrometer screw is sometimes attached to 
this instrument, as in fig. 5. In addition to the arrangements already 
described, there is a movable nut at A, through which passes the slide &. 
To move the head, e, by a very slight amount, the slide & is attached to the 
screw b, and rendered fast. By loosening or tightening the screw J, the 
slide & will be moved backwards or forwards, and with it the head, e, to which 
it is attached. When this is done, e is screwed fast, and k brought to 2, 
where the micrometer arrangement is again brought to play in moving the 
points a, b, and consequently, c,d. We thus obtain hair proportional com- 
passes. The lengths ad and be must be perfectly equal, or else all indications 
will be erroneous. The great utility of the proportional compasses consists 
in their enabling us to enlarge or reduce all parts of a drawing to a certain 
scale, all that is necessary being to adjust (once for all) the two sets of legs 
in the same proportion as the required reduction or enlargement. 
The common compasses do not answer for long lines. In this case the 
beam compasses are to be used, pl. 5, figs. 6,7. These consist of a prismatic 
beam of wood or hollow prismatic brass rod, ab, upon which the boxes, ec, d, 
slide, capable of being fastened by clamp screws. ‘To each box is attached 
a point h, and 2, one of them being replaced, when necessary, by a lead 
tube or pen point. One of the boxes is provided with a micrometer arrange- 
ment. For this purpose a head is attached at f, through which passes the 
smooth end of the screw, e, without moving back or forwards. The nut of 
the screw is at g; so that when the screw e is turned, the spindle being 
fixed, the nut g moves forwards and backwards, and with it the box and 
point attached. 
The triangular compasses (fig. 8) serve to take off all three corners of a 
triangle at once, rendering the operation of transferring triangles one of 
great ease. This instrument has three legs, united in one head in such a 
manner, that two of the legs form ordinary compasses, in whose head, a, a 
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