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blank circles, setting off lines, &c. 2. A Port Crayon with a Black lead Pencil cut to a fine point, for drawing lines that may be easily rabbed out again, if not right. A piece of slate-pencil may also be used in this part for drawing on slate. 3. The Dotting Point, or Dotting Pen, with a small rowel, or indented wheel at the end, moving very freely; and receiving ink from the brass pen over it, communicates the same in equal and regular dots upon the paper, where dotted lines are chosen. 4. The Steel Pen, or Point, for draw. ing and describing Black Lines with Ink; for this purpose, the two parts or sides of the pen are opened or closed with an adjusting screw, ihat the line drawn may be as fine or as coarse as you please.
In the Port-Crayon, Dotter, and Steel Pen, there is a joint, by which you can set the lower part always perpendicular to the paper, which is necessary for drawing a line well, in every extent or opening of the Compasses.
In some of the better sort of instrumen s, ihese points slide into the socket, and are kept tight by a spring un the nside, that is not seen.
The Steel-Point is, sometimes, made with a joint, and furnished with a fine spring and screw; by which, when you have opened the Compasses nearly to the extent required, you can, by turning the screw, move the point to the true extent, as it were, to a hair's breadth ; which is the reason these are called Hair Compasses.
The common Compasses, at large, are not altogether so well adapted for small Drawings; and, therefore, a small sort, called Bowes, are contrived to answer all such purposes ; they consist only of a Steel Point and Drawing Pen, with a joint, and of a small length, so that 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.
OP 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 convenient 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 unscrewed, there is a fine round steel pin, or point, by which you make as nice á mark or dot upon the paper as you please, for terminating your lines in curious draughts.
The Black-lead Pencil, if good, is of frequent 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 be 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, divided into 180 degrees, aud 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 generally called a Fiducial edge, in the middle of which is a small line, or fine 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. 3. you place the centre of tho Protractor upon the angular point C, and the Fiducial edge exactly upon the line CA; then observe what number of degrees the line CB cuts
upon the graduated limb of the Protractor, and that will be the measure of the angle ACB, as required.
Secondly, Suppose it required to protract or lay off from the line AC, an angle ACB, equal 35 degrees. To do this, you place the centre of the Protractor upon the given point C, and the straight edge vpon AC, very exactly; then make a fine point, or dot, at 35 degrees on the limb at B, and, the Protractor being removed, you draw through B, the straight line CB, and it will make the angle ACB required.
Protractors, in form of a Parallelogram, or long square, are usually made in ivory or brass ; and 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 ; but the advantage scarcely con pensates the expense.
OF THE PARALLEL RULBR. The Parallel Ruler is so called, because it consists of two straight rales, connected together by two brass bars, yet so as to admit a very free motion to each ; the one ruler must always move parallelwise to the other, that is, one rule will be every where equi-distant 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 AB, and at the distance AC from it. Fig. 4, First open the rulers to a greater distance tban AC, and place the edge of one of the rules exacdy on the line AB ; then, holding the other rule (or side) firmly on the paper, you move the upper rule down from A to the point C, by which (holding it fast) you draw the line CD, which will be parallel to the given line AB, as required.
Many very useful problems in the mathematics are performed by this instrument; of which the following are examples.
Let it be required to find a fourth proportional to three right lines given, AB, BC, and AD. Fig. 3, To do this, draw the lines AC, AE,
making any angle at pleasure. Upon AC, with the compasses, set off the lines AB, and BC; and, upon AE, set off the line AD; join DB, and parallel to it draw EC, then will DE bo the fourth proportional required. For AB : BC :: AD: DE.
Again, suppose it required to divide any line AB, as another line AC is divided, fig. 6. To do this, join the extremities of each line AC; and, by these lines, the line AB will be divided exactly similar to the line AC.
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 six inches to two feet in length.
The Plain Scale, in common cases of instruments, has the following lines or scales upon it, viz. 1. A Line of 6 Inches. 2. A Line of 50 equal Parts. 3. A Diagonal Scale. 4. A Line of Chords, marked C. 5. Seven particular scales of equal parts, or Decimal Scales, of different sizes ; the numbers placed at the beginning of each, denote how many of the small divisions at the beginning are contained in one inch, viz. 10, 15, 20, 25, 30, 35, 40.
The use of the Line of Inches is the same in this as in all other Rules, viz. to take the length or dimensions of bodies in inches and tenths of an inch, in order to compute their contents.
The line of 50 equal parts, being equal to 6 inches, shows the foot to be divided into 100 of the same equal parts, and the divisions of this line are placed by those of the inches, that at may be easily seen what number in one is equal to a given number of the other ; thus 3 inches is equal to 25 parts of the 100. And 30 of these latter are equal to 3 inches and 6 tenths. This line is, therefore, often used in practical mathematics.
The Diagonal Scale is, properly, a Centissimal Scale, because by it you divide an unit into 100 equal parts; and, therefore, you can lay off, or express, any number to the 100th part of an unit, which is an exactness generally sufficient in practical business. How this is done will be easy to understand, as follows : Let AB be 1 or unit, and divide it into 10 equal parts at 1, 2, 3, 4, &c. At a proper distance BC, draw the line CD equal and parallel to AB, and divide this line CD also into 10 equal parts at a, b, c, d, &c. and these will be the 10 Diagonal Lines. Lastly, divide BC into 10 equal parts also, and number them 1, 2, 3, 4, &c. to 10 at C; then, through each of these divisions, draw lines parallel to AB, through the length of the scale, and the construction is completed. See fig. 7.
In this diagonal scale (upon the plate) AB is one inch ; then, if it be required to take of 10 inches. or 1,73 ; set one foot of the compasses in the third parallel under 1 at e, and extend the other foot, or point, to the 7th diagonal in that parallel at g, and the distance eg is that required ; for ef is 1 inch, and tg is 73 parts of 100
Again, suppose it required to set off upon any line 2,37 inches ; then place one point of the compasses on the 7th parallel under 2 at h, and extend the other to the 3rd diagonal in the same parallel at i ; and the distance hi is that required. Or, if AB be 10, the distance eg is 17,3 ; and hi is 23,7. Also, if AB be 100, then eg is 173, hi is 237 ; and so on.
This diagonal scale has this centissimal division at each end, and the unit in one is just double of that at the other ; thus, if AB be I inch at one end, it is an inch at the other ; or if it be an inch in the larger, it is 4 inch in the lesser divisions, as is the case upon most of the common Plain Scales.
This unit AB may also be 1 foot, 1 yard, 1 rod, 1 mile, &c. So that every unit, in every kind of measures, is hereby estimated in 100th parts of the whole, which shews the Diagonal Scale to be a most useful invention.
On the other side of the plain scale are the seven decimal lines, which are usually Plotting Scales, because their divisions of an unit into 10 parts, being different in proportion of 4 to 1, the Surveyor has it in his power to vary the scale of his plot or plan of an estate, &c. In that ratio, in seven different drawings ; and the superfices, or sizes of the greatest and least plans, will be as 16 to 1. Or, that drawn by scale No. 10, will be sixteen times larger than the plan laid down from scale No. 40.
Also the same variety is to be had in the construction of all other geometrical figures, whether superfices or solids; and, with respect to the latter, the greatest will be to the least as 64 to 1; that is, the Architect can vary the size of his house, temple, &c. in the ratio of 64 to 1 in seven different elevations.
The last Line on the common plain scale is that of Chords; and much more used than the protractor for laying off or measuring any proposed angle. Thus, let it be required to draw the line BC to make an angle of 35 degrees with the line AC, fig. 3. To do this, set one point of the compasses in the beginning of the line of chords, and extend the other to 60 ; with that extent (as radius) place oue foot in C, and, with the other, describe the arch AD; then take from the chords 35° in the compasses, and set them on the arch from A to B ; and through B draw CB, and it is done.
Again, suppose it required to measure any angle ABC proposed, proceed thus, produce CA indefinitely, take 60° from the chords in the compasses, and, with one foot in C, strike the arch AD, cutting the leg BC in B; then take the arch AB in the compasses, and applying it upon the beginning of the line of chords, it will reach to 35°, the quantity of the angle required. But the Line of Chords is more useful on the Sector, to which we now proceed.
Note. This line of chords is so constructed on my Navigation Scales, with sexagessimal divisions, that any angle may be measured, or laid off, to a minute of a degree.
The Sector is the most useful of all Mathematical Instruments; for it not only contains all the most useful lines or scales, but, by its nature, renders them universal, as we shall show in the following description :
The lines commonly laid down upon the Sector are these. 1. A Line of Equal Parts, marked L at the end. 2. A Line of Chords to 60 degrees, marked S. 3. A Line of Sines, 90 degrees, marked C. 4. A Line of Tangents, to 45 degrees, marked T. 5. Another Line of Tangents, from 45 to 75 or upwards, marked ta, 6. A Line of Secants, marked se. 7. A Line of Polygons, marked POL.
Besides these, when the Sector is quite opened, there is, on one side, 1. Gunter Line of Artificial Numbers, n. 2. Line of Artificial Sines, s. 3. Line of Artificial Tangents, t. There is, also, a line of 12 inches, and another of the foot divided into 1000 equal parts, placed by it for the purposes already mentioned.
Before a proper idea can be formed of these sectoral lines, and their uses, we must show their construction from the circle. To this end, let AGB, fig. 8, be a quarter of a circle, divided into 90 degrees, described with the radius AC, on the centre C; let AF and CF be perpendicular to AC, in A and C; then, if the radius AC be divided into 10, 100, 1000, &c. Equal Parts, it will be the line so called upon the Sector.
If from A, a line be drawn to any part or division of the quadrant, as G at 60°, then that line AG is the chord of that arch, or of 60°. And if the line AB be drawn, it will be the chord of 90°; and, by setting one foot of the compasses in A, and extending the other to the several divisions 10, 20, 30, 40, &c. they may be transferred from the circle to the line AB, which will then be properly divided into a line of chords in 10, 20, 30, 40, &c. to 90, as on the plain scale.
If from any point G in the quadrant you let fall a perpendicular GI to the radius AC, or GH to the radius CB, then the line GI is called the Sine of the arch AG; and the line GH is the Sine of the arch GB, the complement of AG to 90°. And if all the divisions of the quadrant were transferred to CB by lines GH parallel to AB; then the line CB will be divided as a Line of Sines in the points 10, 20, 30, 40, &c. to yo, as on the Sector you see it.
By laying a rule from the centre C to the several divisions of the quadrant, 10, 20, 30, &c. it will cut the line AE in the points 10, 20,30, &c. which will be thereby divided into a Lire of Tangents; and here it must be observed, that the Line of Tangents T on the Sector, extends but to 45°, equal to AD, or BC, radius. And that the Line of Lesser Tangents t, are projected from a lesser radius, and begin from 45° at the distance of its radius from the centre of the Sector.
By drawing the line CL through the division 60°, to the line AE, it makes AL, the Tangent of 60°, and is itself the Secant of 60°, or of the angle ACL. And if, with one foot of the compasses in C, you extend the other to the several divisions in the line AE, end