MATHEMATICAL INSTRUMENTS. 
whence the instrument must move as though 
of a conical form, and give concentrating, 
instead of parallel lines. Hence such rulers 
are deservedly discarded in most instances; 
though, for work requiring more celerity 
than accuracy, they may be found to an- 
swer. 
Protractors are chiefly made of ivory, in 
the form of a thin flat scale, or ruler, of 
which one. side is plain, excepting a very 
small nick, or mark, that points out its 
exact centre, and corre.sponds w'ith a line, 
perpendicular to it, on the opposite edge, 
marked 90, dividing the instrument 'into 
two equal and similar portions. The edges 
on three sides of the protractor are gradu- 
ated with 180 degrees, backwards and for- 
wards, the centre point 90 being a right 
angle. The protractor is used for laying 
down angles to any extent, as also for tak- 
ing their measurements : hence it is of ex- 
treme service in every branch of mathema- 
tics, and indeed of mechanism. On the 
same side, with the gradations, we generally 
find a line of chords on an extensive scale. 
We shall explain its construction when we 
treat of the sector, observing in this place, 
that by its aid we are enabled to set off any 
angle without the assistance of a protractor : 
thus, take the measurement of 60", from the 
line of chords, as a radius wherewith to 
describe any segment at pleasure, putting 
one foot of your compasses at the point 
wlience the angle is to proceed, and com- 
mencing the segment from that line whence 
the angle is to be made. Take then from 
the line of chords the number of degrees 
you intend the angle should contain ; set 
them off upon the segment from the place 
where it joins the line ; the angle will be 
thus made, leaving the centre whence the 
radius was drawn for its point, and the two 
ends of the chord that cut off the segment 
for its measurement. See Geometry and 
Dialling. 
Some protractors are made of brass, in 
the form of, a semi-circle ; they are pre- 
cisely on the same principle, but are more 
calculated for the measurement than for 
the construction of angles ; because they 
expose the directions of lines, however 
short, and enable us, by means of any right 
line instrument, laid from the centre to the 
circumference, to ascertain the angle with- 
out extending the line, as must be done 
when an ivory protractor is used to a short 
line. 
On the back of the protractor there are 
nsually six scales, marked 60, 50, 4.5, 40, 
3.5, and 30 ; meaning that the measures, or 
equal points, 1, 2, 3, Sic. respectively in- 
clude 60, 50, &c. such within the length of 
an inch; the number 1, 2, 3, &c. being con- 
sidered at 10, 20, 30, &c. of such small divi- 
sions as are placed at the commencement 
of each scale respectively. The scale mark- 
ed C, standing on the same line with that 
of 60 to an inch, is a line of chords on a 
reduced scale, for the convenience of per- 
sons working on such; and the broader 
scale, of 10 lines in depth, is of half 
and quarter inch divisions, with oblique 
scales at the two ends. These shew all the 
tenths of a half, or of a quarter of an inch 
respectively, according as the oblique line 
gives more space between it and the first 
perpendicular, as may be seen by referring 
to the figures 2, 4, 6, 8, which shew 
wt fo> of the division, and enable us to em- 
brace any number of whole divisions, and 
of tenth parts, within our compasses, with 
readiness and precision. This is intended 
chiefly for work on a larger scale, such as 
ground-plans, &c. ; though for such pur- 
poses, a scale divided into twelfth parts is 
more convenient ; since it takes feet and 
inches, instead of decimals of feet. 
It is proper to remark in this place, that 
the protractor should be prevented from 
warping, else its measurements of angles 
will not be true. When this defect has 
taken place, it will be necessary to press 
the instrument ; thereby to bring it as flat 
as possible, that the measurements may be 
accurate, by the bearings being restored to 
their proper places. 
The sector is made to fold in the middle, 
not only that it may lay in a smaller com- 
pass, but to solve many problems by means 
of the references given to various tables 
and scales that are engraved on both sides 
of each limb. AV^hen opened to its full 
length, the sector commonly measures one 
foot; each inch being numbered, and divid- 
ed into tenth parts, called lines. At the 
edge is another scale, which divides the foot 
into ten equal parts (numbered 10, 20, 30, 
&c.) because each tentli part of the foot is 
again subdivided into ten ; thus giving a 
division of the twelve inches into 100 equal 
parts. 
The fiVst scale we shall notice is that next 
to the inner edges, marked Pol. meaning 
polygon. By opening the sector to such a 
width, as may admit the radius of any circle 
to measure exactly from the figure 6, on 
one, to the figure 6 on the other limb, we 
at olice ascertain the division of that circle’s 
