24 
very small circles may be nicely drawn with 
them, as they are to be conveniently moved 
and turned about in the hand, by a short 
stem or shaft. 
ILL Of the drawing pen and pencil. The 
drawing-pen is only the common steel pen 
at the end of a brass rod, or shaft, of a con- 
venient length, to be held in the hand for 
drawing all kinds of straight black lines by 
the edge of a rule. The shaft or handle has 
a screw in the middle part ; and, when un- 
screwed, there is a fine steel round pin or 
point, by which you make as nice a mark 
or dot upon the paper as you please, for 
terminating your lines in curious draughts. 
The black-lead pencil, if good, is of fre- 
quent use for drawing straight lines, and for 
supplying the place of the drawing-pen, where 
lines of ink are not necessary ; it is also often 
substituted for the common pen in writing, 
figuring, &c. Because in all cases, if what 
Te drawn with it be not right, or does not 
please, it may be very easily rubbed out with 
a piece of crumb-bread, and the whole new- 
drawn. 
IV. Of the protractor. The protractor 
is a semicircle of brass, A D B, divided into 
180 degrees, and numbered each way from 
end to end of the semicircle by 10°, 20°, 30°, 
&c. The central line is the external edge 
of the protractor’s diameter, or straight side, 
sloped down to the under side, and is gene- 
rally called a fiducial edge; in the middle of 
which is a small line or line notch in the 
very edge, for the centre of the protractor. 
The uses of the protractor are two: (1.) To 
-measure any angle proposed. (2.) To lay 
down any angle required. 
For example: suppose it required to find 
what number of degrees are contained in 
the angle ACB (fig. 4) ; you place the centre 
of the protractor upon the angular point C, 
and the fiducial edge exactly upon the line 
C A ; then observe what number of degrees 
the line C B cuts upon the graduated limb 
of the protractor, and that will be the mea- 
sure of the angle A C B as required. 
Secondly, suppose it required to protract 
or lav off from the line A C, an angle A C B, 
equal to 35 degrees. To do this, you place 
the centre of the protractor upon the given 
point C, and the straight edge upon A C 
very exactly ; then make a fine point or dot 
at 35 degrees on the limb at B, and the pro- 
tractor being removed, you draw through B 
the straight line C B, and it will make the 
angle A C B required. 
Protractors in form of a parallelogram, or 
long square, as a E F h fig. 3, are usually 
made in ivory or brass; are more exact 
than the common semicircular ones, for 
angles to 40 or 50 degrees, because at and 
about each end, the divisions (being farther 
from the centre) are larger. 
V. Of the parallel ru er. The parallel 
ruler is so called, because as it consists of 
two straight rules, connected together by 
two brass bars, yet so as to admit a very free 
motion to each: the one ruler must always 
move parallel-wise to the other, that is, one 
rule will be every where equidistant from the 
other, and by this means it becomes naturally 
fitted for drawing one or more lines parallel 
to, or equally distant from, any line proposed. 
The manner of doing which is thus: 
Let it be required to draw a straight line 
parallel to a given line A B, and at the dts- 
INSTRUMENTS, 
tance AC, from it. (fig. 5.) First open the 
rulers to a greater distance than A C, and 
place the edge of the rulers exactly on the 
line A B ; then holding the other rule (or 
side) firmly on the paper, you move the up- 
per rule down from A to the point C, by 
which (holding it fast) you draw the line 
C D, which will be parallel to the given line 
A B as required. 
Many very useful problems in the mathe- 
matics are performed by this instrument, of 
which the following are examples. 
Let it be required to find a fourth pro- 
portional to three right lines given, A B, 
B C, and A D (fig. 6). To do this, draw the 
lines AC, A E, making any angle at plea- 
sure. Upon A C with the compasses set off 
the lines A B and B C ; and upon A E set 
of the line A D ; join D B, and parallel to ; 
it draw E C, then will D E be the fourth 
proportional required. For A B : B C : : 
A D : D E. 
Again, suppose it required to divide any j 
line, A B, as another line A C "is divided 1 
(lig. 7). To do this, join the extremities of ! 
each line C B, and paral'el to C B draw 
El, Ell 1)6, through the given points' 
DEF in the line AC; and by these lines 
the line A B will be divided exactly similar | 
to the line A C. 
The parallel ruler is seldom put into a 
case of instruments, but those of the larger 
and better sort ; being generally sold by 
itself of various sizes, from 6 inches to 2 feel ; 
in length. 
Of the plain scale. The lines generally j 
drawn on the plain scale, are these following: 
Marked 
I. Lines of equal parts. 
E. P. 
II. Chords. 
Clio. 
III. Rhumbs. 
Ru. 
IV. Sines. 
Sin. 
V. Tangents. 
Tan. 
VI. Secants. 
Sec. 
VII. Half-Tangents. 
S. T. 
VIII. Longitude. 
Long. 
IX. Latitude. 
Lat. 
X. Hours. 
Ho. 
XI. — Inclinations. 
In. Mer. 
Ofljie lines of equal parts. Lines of equal 
parts are of two sorts, viz. simply divided, 
and diagonally divided, Plate 5. 
1. Simply divided. Draw three lines pa- 
rallel to one another, at unequal distances 
(iig. 8), and of any convenient length ; di- 
vide tli is length into what number of equal ! 
parts is thought necessary, allowing some 
certain number of these parts to an inch, such 
as 2, 2 j, 3, 4, 4-|, &c. which divisions 
distinguish by lines drawn across the three 
parallels. Divide the left-hand division into 
10 equal parts, which distinguish by lines 
drawn across the lower parallels only; but 
for distinction’s sake, let the fifth division be 
somewhat longer than the others : and it may 
not be inconvenient to divide the same left 
hand division into 12 equal parts, which are 
laid down on the upper parallel line, having 
the third, sixth, and ninth divisions distin- 
guished by longer strokes than the rest, 
whereof that at the sixth division make the 
longest. 
There are, for the most part, several of 
these simply divided scales put on rulers, 
one above the other, with numbers ou the 
left hand, shewing in each scale, how many 
equal parts an inch is divided into; such as 
20, 25, 30, 35, 40, 45, tsc. and are severally 
used, as the plan to be expressed should be 
larger or smaller. 
The use of these lines of equal parts, is to 
lay down any line expressed by a number of 
two places or denominations, whether deci- 
mally or duodecimally divided ; as leagues, 
miles, chains, poles, yards, feet, inches, &c. 
and their tenth parts, or twelfth parts ; thus, 
if each of the divisions be reckoned 1, as 1 
league, mile, chain, &rc. then each of the 
subdivisions will express part thereof; 
and if each of the large divisions be called 
10, then each small one will be 1 ; and if the 
large divisions be 100, then each small one 
will be 10, &c. 
Therefore to lay off a line 8^., 87, or 
870 parts, let them be leagues, miles, chains, 
&c. set one point of the compasses on the 
8th of the large divisions, counting from the 
left hand towards the light, and open the 
compasses, till the other point falls on the 
7th of the small divisions, counting from the 
right hand towards the left, then are the 
compasses opened to express a line, of S-L. 
87, or 870 leagues, miles, chains, &c. and 
bears such proportion in the plan, as the line 
measured does to the thing represented. 
But if a length of feet and inches was to be 
expressed, the same large divisions may re- 
present the feet, but the inches must be taken 
from the upper part of the first division, 
which (as b'etore noted) is divided into twelve 
equal parts. 
Thus if a line of 7 feet 5 inches was to be 
laid down, set one point of the compasses on 
the fifth division among the twelve, counting 
from the right hand towards the left, and 
extend the other to 7, among the large di- 
visions: and that distance laid down in the 
plan, shall express a line of 7 leet 5 inches; 
and the like is to be understood of any other 
dimensions. 
2. Diagonally divided. Draw eleven 
lines parallel to each other, and at equal 
distances'; divide the upper of these lines 
into such a number of equal parts, as the scale 
to be expressed is intended to contain ; and 
from each of these divisions draw perpen- 
diculars through the eleven parallels (tig. 9) : 
subdivide the first of these divisions into 10 
equal parts, both in the upper and lower 
lines ; then each of these subdivisions may 
be also subdivided into ten equal parts, by 
drawing diagonal lines; viz. from the 10th 
below, to the ninth above; from the ninth 
below to the eighth above; from the eighth 
below to the seventh above, &c. till from the 
first below to the 0th above, so that by these 
means one of the primary divisions on the 
scale will be divided into 100 equal parts. 
There are generally two diagonal scales 
laid on the same plane or face of the ruler, 
one being commonly half the other (fig. 9). 
The use of the diagonal scale is much the 
same with the simple scale; all the difference 
is, that a plan may be laid down more accu- 
rately by it ; because in this, a fine may be 
taken of three denominations, whereas from 
the former, only two could be taken. 
Now from this construction it is plain, if 
each of the primary divisions represent 1, each 
! of the first subdivisions will express -J-g- of 1 ; 
j and each of the second subdivisions (which 
! are taken on the diagonal fines, counting 
! from the top downwards) will express G f 
