DOI Page / Citation link:
https://doi.org/10.11588/diglit.20658#0022
8
OF DRAWING,
The lines of sines, tangents, semitangents, and secants-, are used in the projection of the sphere*
and are of the same nature with those of the same names on the sector, hereafter described.
The sector is, as are many others treated in this chapter, property a mathematical drawing
instrument only: but as it obtains a place in our magazine case, it is necessary to give a brief de-
scription of it. It is the most-universally useful instrument of its kind, and of general application:
uniting in itself both the rule and compass, angles and parallel lines* This instrument is formed
of two equal flat rules, united together by a joint. The best and most useful are those formed of
brass, and about six inches in length when shut: by the different openings of the two legs of the
sector, it represents every possible variety of plain angles; the distance of the extremities of the
legs shewing the chords or subtenses of the arches they describe. There are three kinds of scales
or lines graduated upon the sector: namely, sectoral, plain, and logarithmic lines. The sectoral
lines are, I. two scales of equal parts, one on each leg. of the sector,-marked teiN. or l. (each
of these scales, from its extensive use, is called the Urn of lines;) 2. two lines of chords, marked
gho. or c. ;' 3. two lines of secants, marked sec. or s.; 4. a line of polygons, marked pol. : and
upon the other face of the instrument are, 5. two lines of sines, marked sin. or s-. ; 6. two
lines of tangents, marked tan. or t. : and between the line of tangents and sines there is,
7. another line of tangents, to a less radius, extending from 45 degrees, where the former line of
tangents finished, to 75 degrees. The plain lines or scales on the sector, are placed parallel to the
outward edges, and are on one of the faces (figure 5,) h. a line of inches ; 2. a line of latitudes ;
3. a line of hours ; 4. a line of inclinations-of meridians : and, 5. a line of chorda. On the other
face (figure 4,) are three logarithmic lines•; 1. of numbers; 2. of sines; and, 3. of tangents :
these three latter are used in trigonometry ; and in using them, the sector must be quite open,
as these lines run on both legs.
Questions are resolved on the sector b}r means of the compasses. In the use of this instrument
it must be observed, that when the question is resolved by taking dimensions- on one line only, it
is said to be simple; but when the operation is begun, on one line, and finished on another, the
question is compound. And if the measure be taken on any of the sectoral lines, beginning at the
center, it is called a lateral distance; but if the measure be taken from a point in one line to its
corresponding point in the fellow line on. the other leg, it is called a transverse ox parallel distance.
And it must be remembered, that the compasses must be always applied to the middle line of the
three, through which the graduations-run on each scale.
The uses of the sector are so many and various, that it is impossible to give them all a place
here : a few of its most general properties-may however just be mentioned.
Each pair of lines, except the line of polygons, make equal angles at the center, however the in-
strument be opened, and according to the nature of these lines : thus, the distance between 10 and
10 on the line of lines will be equal to the distance between 60 and 60 on the line of chords, 90
and 90 on the line of sines, and 45 and 45 on the line of tangents ; which is the same proportion
those lines respectively bear, when actually laid down upon a circle, the distance between 10 and 10
on the line of lines being considered as the radius. The line of lines is divided into 10 equal parts,
each of which is subdivided into 10 other equal parts; and each of these again subdivided into
two parts. The numbers on this line may have any value assigned them at pleasure ; as units,
tens, or hundreds, 8tc. When the first divisions on this-line signify hundreds, the second division
will represent tens, the third five units each, or half ten. The numbers affixed to the other sectoral
scales
OF DRAWING,
The lines of sines, tangents, semitangents, and secants-, are used in the projection of the sphere*
and are of the same nature with those of the same names on the sector, hereafter described.
The sector is, as are many others treated in this chapter, property a mathematical drawing
instrument only: but as it obtains a place in our magazine case, it is necessary to give a brief de-
scription of it. It is the most-universally useful instrument of its kind, and of general application:
uniting in itself both the rule and compass, angles and parallel lines* This instrument is formed
of two equal flat rules, united together by a joint. The best and most useful are those formed of
brass, and about six inches in length when shut: by the different openings of the two legs of the
sector, it represents every possible variety of plain angles; the distance of the extremities of the
legs shewing the chords or subtenses of the arches they describe. There are three kinds of scales
or lines graduated upon the sector: namely, sectoral, plain, and logarithmic lines. The sectoral
lines are, I. two scales of equal parts, one on each leg. of the sector,-marked teiN. or l. (each
of these scales, from its extensive use, is called the Urn of lines;) 2. two lines of chords, marked
gho. or c. ;' 3. two lines of secants, marked sec. or s.; 4. a line of polygons, marked pol. : and
upon the other face of the instrument are, 5. two lines of sines, marked sin. or s-. ; 6. two
lines of tangents, marked tan. or t. : and between the line of tangents and sines there is,
7. another line of tangents, to a less radius, extending from 45 degrees, where the former line of
tangents finished, to 75 degrees. The plain lines or scales on the sector, are placed parallel to the
outward edges, and are on one of the faces (figure 5,) h. a line of inches ; 2. a line of latitudes ;
3. a line of hours ; 4. a line of inclinations-of meridians : and, 5. a line of chorda. On the other
face (figure 4,) are three logarithmic lines•; 1. of numbers; 2. of sines; and, 3. of tangents :
these three latter are used in trigonometry ; and in using them, the sector must be quite open,
as these lines run on both legs.
Questions are resolved on the sector b}r means of the compasses. In the use of this instrument
it must be observed, that when the question is resolved by taking dimensions- on one line only, it
is said to be simple; but when the operation is begun, on one line, and finished on another, the
question is compound. And if the measure be taken on any of the sectoral lines, beginning at the
center, it is called a lateral distance; but if the measure be taken from a point in one line to its
corresponding point in the fellow line on. the other leg, it is called a transverse ox parallel distance.
And it must be remembered, that the compasses must be always applied to the middle line of the
three, through which the graduations-run on each scale.
The uses of the sector are so many and various, that it is impossible to give them all a place
here : a few of its most general properties-may however just be mentioned.
Each pair of lines, except the line of polygons, make equal angles at the center, however the in-
strument be opened, and according to the nature of these lines : thus, the distance between 10 and
10 on the line of lines will be equal to the distance between 60 and 60 on the line of chords, 90
and 90 on the line of sines, and 45 and 45 on the line of tangents ; which is the same proportion
those lines respectively bear, when actually laid down upon a circle, the distance between 10 and 10
on the line of lines being considered as the radius. The line of lines is divided into 10 equal parts,
each of which is subdivided into 10 other equal parts; and each of these again subdivided into
two parts. The numbers on this line may have any value assigned them at pleasure ; as units,
tens, or hundreds, 8tc. When the first divisions on this-line signify hundreds, the second division
will represent tens, the third five units each, or half ten. The numbers affixed to the other sectoral
scales