MATHEMATICAL INSTRUMENTS. 
Triangle-compasses are made for the pur- 
pose of ascertaining three points, in the 
same manner as the common biped-com- 
passes ascertain only two. This is affected 
by a third leg, which may be taken off at 
pleasure, working like a gin for raising 
weights; or like the legs of a theodolite- 
stand, and having a hinge at right angles 
with that where it joins the top of the com- 
passes. By this simple contrivance the 
added leg may be made to incline to the 
right or left of the direction given by the 
upper hinge. 
Where work is to he executed on a large 
scale, viz. projecting meridians in maps, it is 
necessary to have a veiy large pair, such as 
are known at the makers by the designation 
of beam-compasses: these may be had 
in separate cases, with covers sliding in 
grooves, and to the extent of full two feet 
in length. In some professions such are in- 
dispensable, but it is far more convenient to 
have a branch, or elbow-joint, to the shift- 
ing compasses ; the upper part fitting info 
the socket made for receiving the several 
limbs, and its lower end being socketed in 
the same manner ; so that one leg of the 
compasses may be made, to any extent, 
longer than the other. As all the additional 
joints have a hinge, the excess of length 
may be made subservient to any direction, 
by being bent downwards, so as to stand at 
right angles with the paper; as in such 
case the other leg ought also, in order to 
prevent its shifting, or cutting the surface. 
The elbow-joint is often given in a flat case 
of instruments. 
The drawing pen is ordinarily about six 
inches in length, and is made on the same 
principle as that intended for circular 
operations ; in general, this unscrews in 
the centre, and disengages the upper part 
of the handle, to which a fine steel needle 
is attached, the use of which is to mark 
down, by the slightest puncture, those 
points that require peculiar delicacy. The 
small flat steel turn-screw has one end nar- 
rowed that it may fit the screw-heads in 
the hinges of compass limbs ; while the 
other, by means of two studs that fit into 
corresponding holes on the side of the joint 
connecting the two limbs of the compass, 
serves to tighten or to relax them at 
pleasure. 
The common parallel rules made to fit 
into cases, consist of two slips, moving upon 
four pivots ; i. e. one at each end of two 
metal plates, whereby the slips are always 
kept at a perfect parallel. This chiefly de- 
pends on the perfect equality of lengths in 
the metal slips, and their being placed at 
exactly the same angle, at points equidis- 
tant from the edges of the slips. Some 
parallels have a third piece, which folds be- 
tween the two above described, and requir- 
ing an additional pair of metal hinges, 
which meet on them as a centre. This is 
certainly a very great convenience, inas- 
much as it extends the scope of the instru- 
ment, and gives a third parallel ; but the 
slightness of the middle piece subjects it to 
warp ; and, at all events, demands great 
care in using, so as not to wrench the 
pivots, or to bend them into an improper 
direction, whereby the whole work would 
be falsified. In drawing lines with this 
ruler, observe the following instructiqns : 
When a parallel is to be made above the 
line to which you apply the ruler, let the 
limbs be Closed, then press firmly on the 
bottom slip, by two fingers placed at least 
two inches apart, and Clear of the hinges ; 
slide the upper limb gently from you, by 
means of the metal stud in its centre, until 
you bring it up to the point through which 
the parallel is to be drawn. When you 
would make a parallel below any given line, 
the slips should be separated (keeping the 
upper limb well pressed by two fingers), un- 
til you bring the upper or lower edge of the 
other limb, as may prove most convenient, 
to the point through which the parallel is to 
be made. Or you may open your ruler to 
its full extent ; first, placing its upper thin 
edge along the original line, and pressing on 
the lower limb, then draw the upper one 
down to the desired point. Both the edges 
of this ruler are chamfered on one side 
only ; whence one edge lays very flat to the 
paper, so as to guide with great exactness, 
and serving. excellently for pencilled lines; 
while the other, being raised from the 
paper, gives greater security from blotting 
when ink is used, but requires a very steady 
hand, and a no less accurate eye. 
Parallel rulers are sometimes made to 
move on wheels, with graduated edges, 
shewing the parts of inches over which they 
pass. The theory is excellent, but we find 
in this many practical inconveniences ; such 
rulers being easily turned out of their pro- 
per directions, by any little inequality on 
the surface of paper, or by the smallest 
deviation from perfectly even propulsion, 
or retraction. Besides, the axis being 
necessarily made loose, so as to allow great 
freedom of motion, it is obvious the wheels 
cannot always preserve an exact level 5 
