the former subdivisions or a 100th of the I 
primary divisions; and if each of the pri- 
mary divisions -express 10, then each of the 
first subdivisions will express 1, and each of 
the 2d, f 6 ; and if each of the primary di- 
visions represent 100, then each of the first 
subdivisions will be 10, and each of tiie 2d 
will be 1, &c. 
Therefore to lay down a line, whose length 
is expressed by 347, 34 f- or 3 whether 
leagues, miles, chains, See. 
On the diagonal line, joined to the 4th of 
the first subdivisions, count 7 downwards, 
reckoning the distance of each parallel 1 ; 
there set one point of the compasses, and ex- 
tend the other, till it falls on the intersection 
of the third primary division with the same 
parallel in which the other foot rests, and 
the compasses will then be opened to express 
a line of 347, 34 or &c. 
Those who have frequent occasion to use 
scales, perhaps will find, that a ruler with the 
20 following scales on it, viz. 10 on each face, 
will suit more purposes than any set of simply 
divided scales hitherto made public, on one 
ruler. 
One. Side. — The divisions to an inch. 
10, 11, 12, 15, 164, 18, 20, 22, 25. 
Other Side — The divisions to an inch. 
23, 32, 36, 40, 45, 50, 60, 70, 85, 100. 
The left-hand primary division, to be di- 
vided into 10 and 12 and 8 parts; for these 
subdivisions are of great use in drawing the 
parts of a fortress, and of a piece of cannon. 
It will here be convenient to shew, how 
any plan expressed by right lines and angles, 
may be delineated by the scales of equal 
parts, and the protractor. Suppose three 
adjacent things, in any right-lined triangle 
being given, to form the plan thereof. 
Example. Let ABC (fig-. 10,) be a triangular 
field, the side AB = 327 yards ; AC 208 
yards ; and the angle at A rr 44^ degrees. 
Construction. Draw a line A B at plea- 
sure ; then from the diagonal scale take 327 
between the points of the compasses, and 
lav it from A to B; set the centre of the 
protractor to the point A, lay off 44£ degrees, 
and by that mark draw A C ; take with the 
compasses from the same scale 208, lay it 
from A to C, and join C B ; so shall the parts 
of the triangle A B C, in the plan, bear the 
same proportion to each other, as the real 
parts in the field do. 
The side C B may be measured on the 
same scale from which the sides A B, AC, 
were taken; and the angles at B and C may 
be measured by applying the protractor to 
them. 
If two angles and the side contained be- 
tween them were given. 
Draw a line to express the side (as be- 
fore) ; at the ends of that line, point off the 
angles, as observed in the field ; lines drawn 
from the ends of the given line through those 
marks, shall form a triangle similar to that 
of the field. 
Five adjacent things, sides and angles, in 
a right-lined quadrilateral, being given, to 
Jay down the plan thereof, fig. 11. 
i Example. Given Z. A = 70°; AB = 215 
links; A B = 115° ; BC = 596 links ; Z. C = 
114°. 
VOL. II. 
INSTRUMENTS. 
Construction. Draw A D at pleasure ; 
from A draw A B, so as to make with A D 
an angle of 70°: make AB=215 (taken irom 
the scales) ; Irom B, draw ft C, to make with 
A Ban angle of 115°; make BC —596; 
from C, draw C D, to make with C B an 
angle ot 114°; and by the intersection of 
C 1) with A D, a quadrilateral will be formed 
similar to the figure in which such measures 
could be taken as are expressed in the ex- 
ample. 
it three of the things were sides, the plan 
might be formed with equal ease. 
Following the same method, a figure of 
many more sides may be delineated ; and in 
tliis maimer, or some other like to it, survey- 
ors make their plans or surveys. 
The remaining lines of the plain scale are 
tlius constructed. 
Describe a circumference with any con- 
venient radius, and draw the diameters tig. 12 
AB, D E, at right angles to each other; 
continue BA at pleasure towards F ; through 
D, draw D G parallel to B F ; and draw the 
chords B D, B E, AD, AE. Circumscribe 
the circle with the square HMN, whose 
sides H M, M N, shall be parallel to A B, 
E D. 
1. To construct the line of chords. Di- 
vide the arc AD into 90 equal parts: mark 
the 10th divisions with the figures 10, 20, 
30, 40, 50, 60, 70, 80, 90 ; on D, as a cen- 
tre, with the compasses, transfer the several 
divisions of the quadrantal arc, to the chord 
AD, which 'marked with the figures corre- 
sponding, will become aline of chords. 
Note. In the construction of this, and the 
following scales, only the primary divisions 
are drawn ; the intermediate ones are omit- 
ted, that the figure may not appear too much 
crowded. 
2. The line of rhumbs. Divide the arc 
BE into 8 equal parts, which mark with the 
figures I, 2, 3, 4, 5, 6, 7, 8, and divide each 
of those parts into quarters; on B, as a cen- 
tre, transfer the divisions of the arc to the 
chord BE, which marked with the corre- 
sponding figures, will be a line of rhumbs. 
3. The line of sines. Through each of 
the divisions of the arc A D, draw right lines 
parallel to the radius AC; and C D will be 
divided into a line of sines which are to be 
numbered from C to D for the right sines, 
and from D to C for the versed sines. The 
versed sines maybe continued to 180 degrees 
by laying the divisions of the radius C D, 
from C to E. 
4. The line of tangents. A ruler on G, 
and the several divisions of the arc A D, 
will intersect the line DG, which will be- 
come a lint* of tangents, and is to Ire figured 
from D to G, with 10, 20, 30, 40, &c. 
5. The line of secants. The distances 
from the centre C to the divisions on the 
line of tangents being transferred to the line 
CF from the centre C, will give the di- 
visions of the line of secants ; which must be 
numbered from A towards F, with 10, 20, 
30, See. 
6. The line of h alf -tangents ( or the tangent s 
of half the arcs). A ruler on E, and the 
several divisions of the arc A D, will inter- 
sect the radius CA, in the divisions of the 
’ D 
25 
semi or half tangents ; mark these with tire 
corresponding figures of the arc A 13. 
4 he semitangents on the plane scales are 
generally continued as far as the length of 
the ruler they are laid on will admit; the di- 
visions beyond 90° are found by dividing the 
arc A E like the arc A D, then laying a ruler 
by E .and these divisions of the arc A E, 
the divisions of the semitangents above 90 
degrees will be obtained on the line C A con- 
tinued. 
7. The line of longitude. Divide AH 
into 60 equal parts ; through each of these 
divisions, parallels to the radius AC, will 
intersect the arc AE, in as many points; 
from E as a centre, the divisions of the arc 
E A, being transferred to the chord EA, will 
g.ve the divisions of the line oi longitude. 
The points thus found on the quadrantal 
arc, taken from A to E, belong to the sines 
of the. equally increasing sexagenary parts of 
the radius ; and those arcs reckoned from E, 
belong to the cosines of those sexagenary 
parts. 
8. The line of latitude. A ruler on A, 
and the several divisions of the sines on C D, 
will intersect the arc BD, in as many points ; 
on B as a centre, transfer the intersections of 
the arc BD, to the right line BD ; number 
the divisions from B to D, with 10, 20, 30, 
&c. to 90 ; and B D will be a line of latitude. 
9- The line of hours Bisect the quadrant- 
al arcs B D, BE, in a, b ; divide the qua- 
drantal arc ab into 6 equal parts (which gives 
15 degrees for each hour) ; and each of these 
into 4 others (which will give the quarters). 
A ruler on C, and the several divisions of the 
arc ab, will intersect the line M N in the hour, 
Ac. points, which are to be marked as in 
the figure. 
10. The line of inclinations of meridians. 
Bisect the arc EA in c; divide the quadran- 
tal arc be into 90 equal parts ; lay a ruler on 
C and the several divisions of the arc be, and 
the intersections of the line HM will be the 
divisions of a line of inclinations of meridians. 
Of the sector. A sector is a figure formed 
by two radii of a circle, and that part of 
the circumference comprehended between 
the two radii. 
The instrument called a sector, consists of 
two fiat rulers moveable round an axis or 
joint; and from the centre of this joint se- 
veral scales are drawn on the faces of the 
rulers. 
The two rulers are called legs, and repre- 
sent the radii of a circle ; and the middle of 
the joint expresses the centre. 
The scales generally put on sectors, may- 
be distinguished into single, and double. 
The single scales are such as are common' 
ly put on plain scales, and from whence di- 
mensions or distances are taken, as have been 
already directed. 
Tire double scales are those which proceed 
from the centre; each scale is laid twice on 
the same face of the instrument, viz. once 
on each leg: from these scales, dimensions 
or distances are to be taken, when the legs of 
the instrument are in an angular position, as 
will be shewn hereafter. 
