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own weight; the legs should never be so far extended as to form an obtuse angle with the paper or plane on which they are used. The ink and crayon points have a joint just under that part which fits into the compasses; by this they may be always so placed as to be set nearly perpendicular to the paper; the end of the shank of the best compasses is framed so as to form a strong spring, to bind firmly the moveable points, and prevent them from shaking. This is found to be a more effectual method than that by a screw. Two additional pieces are often applied to these compasses; these, by lengthening the leg, enable them to strike larger circles, or measure greater extents, than they would otherwise perform, and that without the inconveniences attending longer compasses. When compasses are furnished with this additional piece, the moveable leg has a joint, that it may be placed perpendicular to the paper. The bow compasses are a small pair, usually with a point for ink; they are used to describe small arches or circles, which they do much more conveniently than large compasses, not only on account of their size, but also from the shape of the head, which rolls with great ease between the fingers. . Of the drawing pen and protracting pin. The pen part of this instrument is used to draw straight lines: it consists of two blades with steel points fixed to a handle; the blades are so bent that the ends of the steel points meet, and yet leave a sufficient cavity for the ink; the blades may be opened more or less by a screw, and, being properly set, will draw a line of any assigned thickness. One of the blades is framed with a joint, that the points may be separated and thus cleaned more conveniently. A small quantity only of ink should be put at one time into the drawing pen, and this should be placed in the cavity between the blades by a common pen or feeder; the drawing pen acts better if the pen by which the ink is inserted be made to pass through the blades. To use the drawing pen, first feed it with ink, then regulate it to the thickness of the required line by the screw. In drawing lines, incline the pen a small degree, taking care, however, that the edges of both the blades touch the paper, keeping the pen close to the rule, and in the same direction during the whole operation. The blades should always be wiped very clean before the pen is put away. These directions are equally applicable to the ink point of the compasses, only observing, that when an arch or circle is to be described of more than an inch radius, the point should be so bent that the blades of the pen may be nearly perpendicular to the paper, and both of them touch it at the same time.
The protracting pin is only a short piece of steel wire with a very fine point fixed at one end of the upper part of the handle of the drawing pen. It is used to mark the intersection of lines, or to set off divisions from the plotting scale and protractor.
OF THE SECTOR,
Amid the variety of mathematical instruments that have been contrived to facilitate the art of drawing, there is none so extensive in its use or of such general application as the sector. It is a universal scale, uniting, as it were, angles and parallel lines, the rule and the compass, which are the only means that geometry makes use of for measuring, whether in speculation or practice. The real inventor of this valuable instrument is unknown; yet of so much merit has the invention appeared, that it was claimed by Galileo, and disputed by nations. This instrument derives its name from the tenth definition of the third book of Euclid, where he defines the sector of a circle. It is formed of two equal rules called legs; these legs are moveable about the centre of a joint, and will, consequently, by their different openings, represent every possible variety of plane angles. The distance of the extremities of these rules are the subtenses or chords, or the arches they describe. Sectors are made of different sizes, but their length is usually denominated from the length of the legs when the sector is shut. Thus a sector of six inches when the legs are close together forms a rule of twelve inches when opened; and a foot sector is two feet long when opened to its greatest extent. In describing the lines usually placed on this instrument, I refer to those commonly laid down on the best six-inch brass sectors. But as the principles are the same in all, and the differences little more than in the number of subdivisions, it is to be presumed that no difficulty will occur in the application of what is here said to sectors of a larger radius. The scales, or lines graduated upon the faces of the instrument, and which are to be used as sectoral lines, proceed from the centre, and are, 1. Two scales of equal parts, one on each leg, marked LIN. or L. Each of these scales, from the great extensiveness of its use, is called the line of lines. 2. Two lines of chords, marked Cho, or c. 3. Two lines of secants, marked SEC. or s. A line of polygons, marked Pol. Upon the other face the sectoral lines are, 1. Two lines of sines marked sIN. or s. 2. Two lines of tangents, marked TAN. 3. Between the lines of tangents and sines there is another line of tangents to a lesser radius, to supply the defect of the former, and extending from 45° to 75°. Each pair of these lines, except the line of polygons, is so adjusted as to make equal angles at the centre, and consequently at whatever distance the sector be opened, the angles will be always respectively equal. That is, the distance between 10 and 10 on the line of lines will be equal to 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. Besides the sectoral scales, there are others on each face placed parallel to the outward edges, and used as those of the common plain scale. There are on the one face, 1. A line of inches. 2. A line of latitudes. 3. A line of hours. 4. A line of inclination of meridians. 5. A line of chords. On the other face, three logarithmic scales, namely, one of numbers, one of sines, and one of tangents: these are used when the sector is fully opened, the legs forming one line. To read and estimate the divisions on the sectoral lines. The value of the divisions on most of the lines is determined by the figures adjacent to them; these proceed by tens, which constitute the divisions of the first order, and are numbered accordingly; but the value of the divisions on the line of lines, that are distinguished by figures, is entirely arbitrary, and may represent any value that is given to them; hence the figures 1, 2, 3, 4, &c. may denote either 10, 20, 30, 40, or 100, 200, 300, 400, and so on. The line of lines is divided into ten equal parts, numbered 1, 2, 3, to 10; these may be called divisions of the first order; each of these is again subdivided into 10 other equal parts, which may be called divisions of the second order; and each of these is divided into two equal parts, forming divisions of the third order. The divisions on all the scales are contained between four parallel lines: those of the first order extend to the most distant, those of the third to the least, those of the second to the intermediate parallel. When the whole line of lines represents 100, the divisions of the first order, or those to which the figures are annexed, represent tens; those of the second order, units; those of the third order, the halves of these units. If the whole line represents ten, then the divisions of the first order are units; those of the second, tenths ; and the third, twentieths. In the line of tangents, the divisions to which the numbers are affixed are the degrees expressed by those numbers. Every fifth degree is denoted by a line somewhat longer than the rest; between every number and each fifth degree there ar four divisions longer than the intermediate adjacent ones; these are whole degrees; the shorter ones, or those of the third order, are 30 minutes. From the centre to 60 degrees the line of sines is divided like the line of tangents; from 60 to 70 it is divided only to every degree; from 70 to 80 to every two degrees; from 80 to 90 the division must be estimated by the eye. The divisions on the line of chords are to be estimated in the same manner as the tangents. The lesser line of tangents is graduated every two degrees from 45 to 50; but from 50 to 60 to every degree; from 60 to the end to half-degrees. The line of secants from 0 to 10 is to be estimated by the eye; from 20 to 50 it is divided to every two degrees; from 50 to 60 to every degree; and from 60 to the end to every half-degree. The solution of questions on the sector is said to be simple when the work is begun and ended on the same line; compound when the operation begins on one line and is finished on the other. The operation varies also by the manner in which the compasses are applied to the sector. If a measure be taken on any of the sectoral lines beginning at the centre, it is called a lateral distance. But if the measure be taken from any point in one line to its corresponding point on the line of the same denomination on the other leg, it is called a transverse or parallel distance. The divisions of each sectoral line are bounded by three parallel lines; the innermost of these is that on which the points of the compasses are to be placed, because this alone is the line which goes to the centre, and is alone, therefore, the sectoral line. We shall now proceed to give a few general instances of the manner of operating with the sector. Multiplication by the line of lines. Make the lateral distance of one of the factors the parallel distance of 10; then the parallel distance of the other factor is the product. Example. Multiply 5 by 6: extend the compasses from the
centre of the sector to 5 on the primary divisions, and open the sector till this distance become the parallel distance from 10 to 10 on the same divisions; then the parallel distance from 6 to 6, extended from the centre of the sector, shall reach to 3, which is now to be reckoned 30. At the same opening of the sector, the parallel distance of 7 shall reach from the centre to 35, that of 8 shall reach from the centre to 40, &c. Division by the line of lines. Make the lateral distance of the dividend the parallel distance of the divisor, the parallel distance of 10 is the quotient. Thus, to divide 30 by 5, make the lateral distance of 30, viz. 3 on the primary divisions, the parallel distance of 5 of the same divisions; then the parallel distance of 10, extended from the centre, shall reach to 6. Proportion by the line of lines. Make the lateral distance of the second term the parallel distance of the first term; the parallel distance of the third term is the fourth proportional. Eacample. To find a fourth proportional to 8, 4, and 6, take the lateral distance of 4, and make it the parallel distance of 8, then the parallel distance of 6, extended from the centre, shall reach tó the fourth proportional 3. In the same manner a third proportional is found to two numbers. Thus, to find a third proportional to 8 and 4, the sector remaining as in the former example, the parallel distance of 4, extended from the centre, shall reach to the third proportional 2. In all these cases, if the number to be made a parallel distance be too great for the sector, some aliquot part of it is to be taken, and the answer multiplied by the number by which the first number was divided. Thus, if it were required to find a fourth proportional to 4, 8, and 6, because the lateral distance of the second term 8 cannot be made the parallel distance of the first term 4, take the lateral distance of 4, viz. the half of 8, and make it the parallel distance of the first term 4 ; then the parallel distance of the third term 6 shall reach from the centre to 6, viz. the half of 12. Any other aliquot part of a number may be used in the same way. In like manner, if the number proposed bé too small to be made the parallel distance, it may be multiplied by some number, and the answer is to be divided by the same number. To protract angles by the line of chords. Case 1. When the given degrees are under 60. 1. With any radius on a centre, describe the arch. 2. Make the same radius a transverse distance between 60 and 60 on the same line of ehords. . 3. Take out the transverse distance of the given degrees, and lay this on the arch, which will mark out the angular distance required.