MATHEMATICAL INSTRUMENTS. 
circumference into any number of equal 
parts, from four to twelve ; beeanse ♦Vom 
the figure 4 to the opposite figure 4 will 
give a chord subtending a quadrant of the 
circle; from 5 to 5 will give the side of a 
regular pentagon, or figure of five sides; 
from 6 to 6 a hexagon; and so forth. 
The line of clioi ds on the sector is known 
by the tetter C ou each limb, and measuies 
60 degrees only; though on the protractor 
it goes as far as 90, which is its full nitasiire- 
ment. This, however, is not important, as 
we can always add 30 to 60, and thus com- 
plete any figure in hand. The formation of 
the line of chords being given, its applica- 
tion will be more readily understood ; we 
shall therefore shew how they are construct- 
ed from the circle. 
Suppose the line A B (fig 1, Plate Miscel.) 
to represent the end of your scale, and that 
AC, BD, be perpendicular theieto: with 
A B as a radius, and from A as a (•entre,draw 
the quadrant B F C, and the straight line 
or chord B C subtending that quadrant. 
Divide the quadrant into 90 equal parts, 
and from B, as a centre, measure olF each 
division successively, so as to cut the chord 
B C into 90 parts, all which will be unequal. 
Mark every tenth degree, both on the qua- 
drant and on the chord, thus, 10, 20, 30, 40, 
50, 60, 70, 80, and 90. This division will 
make the line B C a line of chords, which 
affords a scale of very general utility in ma- 
thematics. 
The line of sines, commonly marked S, 
shews the relation of sines to various por- 
tions of circles. Here it is nece.>sary to 
state, that there are three kinds of sines, 
viz. the sine, the co-,sine, and the versed 
sine. The sine is that perpendicular which 
stands at right angles with the chord sub- 
tending an arc, and reaches from it to the 
circumference, such as the line E F ; the 
co-sine is a chord, .such as FG, which com- 
mences from the junction of the sine with 
the circumference, and is parallel with that 
line from which the sine arises, proceeding 
in that direction until intercepted by the 
perpendicular AC, whicli terminates the 
quadrant ; the co-sine, is therefore the com- 
plement or residue of the base line A B, 
after deducting from its other end the 
amount of the versed-sine BE. If from B 
60 degrees be measured on the quadrant to 
F, its sine will divide the base A B into two 
equal parts ; so that the co-sine and versed- 
sine will be of equal length. The line of 
sines is therefore made on the perpendicu- 
lar A C by means of parallels, to the base 
A B, drawn from the circumference at the 
par s tiiirked 10, 20, .30, &c. degrees, 
which of course give a regularly diminishing 
scale. 
Tue line of tangents is made by a conti- 
nuation of tlie perpendicular B D to K, 
and by draw ing from the graduated qua- 
di ant the several lines 10, 10 ; 20, 20 ; 30, 
30 ; iVc. to that perpendicular, all pointing 
to the centre A. This scale regularly aug- 
ments, and is carried to 45 degrees only. 
Now, by transferiing all the tangent scale, 
and the p’aces of the degrees thus obtained 
from the point A, by drawing segments 
from each part respectively to the perpen- 
dicular A H, we have aline of secants: thus 
the 10 on the tangent scale will be trans- 
ferred to 10 beyond C on the secant line, 
20 to 20, and thus to the end of the scale 
up to 90 degrees, which would, however, 
acquire a great length of ruler. The line 
of tangents is confined to 45 degrees; but 
a line of lesser tangents, from 45° to 90°, is 
made on a smaller radios. 
The line of equal parts between A and B 
is also called the line of lines, and is divided 
into 10, 100, 1000, &c. equal parts ; but the 
indlcial numerals are confined to 10, for 
we have only ten numbers on each limb of 
the sector, made by dividing the radius (or 
base line) A B into that number of equal 
spaces. The uses of the lines above de- 
scribed are very extensive ; but we shall 
give a brief example of their intentions, ob- 
serving that the line of equal parts is distin- 
guished by the letter L on each limb of the 
sector: the line of sines, by S; the line of 
tangents, by T ; the line of secants, by se.; 
and the line of les.ser tangents, by fa. 
N. B. In some sectors the letter C is en- 
graved close to the very centre of the hinge, 
which centre is marked by an obvious punc- 
ture, towards which all the lines have a 
tendency : in using the lines, the measures 
are to be taken from those marked L. S. C. 
&c. on one limb to those marked L. S. C. 
on the other limb, respectively, they stand- 
ing at an angle of six degrees from their re- 
spective partners. 
“ To find a fourth proportional by the 
line of equal parts.” Say you would wish 
to find a line proportioned to 15 as 3 is to 8 : 
on the line of equal parts take a distance 
from C with your compasses equal to 15, 
and with that opening extend your sector 
so as the distance between 3 and 3 may 
correspond therewith; then measure the 
distance thus generated between 8 and 8, 
and lay it from the point C along the line of 
