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Á rocket Case of Mathematical Instruments contains the
following Articles : 1. A pair of Plain COMPASSES. 2. A pair of Drawing ComPAsses. with its several parts. 3. A DRAWING Pen and POINTER. 4. A PROTRACTOR, in form of a SEMICIRCLE, or, sometimes, of a PARALLELOGRAM. 5. A PARALLEL RULER. 6. A Plain SCALE. 7. SECTOR : besides the Black LEAD PENCIL for drawing Lines.
1. OF THE PLAIN COMPASSES, FIG. 1. THE use of the Common or Plain Compasses is, 1. To draw a Blanik Line AB, by the edge of a ruler, through any given point or points CD, &c. 2. Take any extent or length between the points of the Compasses, and to set it off, or apply it successively upon any line, as from C to D, fig. 2. 3. To take any proposed line CD between the points, and by applying it to the proper Scale, to find its length. 4. To set off Équal Distances upon a given line, by making a dot with the point at each, through which to draw parallel lines. 5. To draw any blank circle, intersecting Arches, &c. 6. To lay off an angle of a given quantity upon an arch of a circle froni the line of chords, &c. 7. To measure any arch, or angle, upon the chords, &c. 8. To construct any proposed figure, in plotting or making plans, &c. by setting off the quantity of the sides and angles from proper Scales. In short, the use of the Compasses occurs in every branch of pracical mathematics, as we shall see more particularly hereafter.
II. OF THE DRAWING COMPASSES. These compasses are chiefly designed for drawing circles and circular arches ; and it is often necessary they should be drawn with different materials ; and, therefore, this pair of compasses has, in one
i of its legs, a triangular Socket and Screw, to receive and fasten the following parts or points for that purpose, viz. 1. A Steel point, which, being fixed in the socket, makes the Compasses then but a plain pair, and has all the same uses as just now described in drawing blank circles, setting off lipes, &c. 2. A Port Crayon with a Black. lead Pencil cut to a fine point, for drawing lines that may be easily rubbed out again, if not right. A piece of slate-pencil may also bo 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, ibat 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, ibese 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.
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 Pencił, 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.
OF THE PROTRACTOR. The Protractor is a semicirclo of brass, divided into 180 degrees, and 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 the 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 upon 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, ara 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 conr pensates the expense.
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 than AC, and place the edge of one of the rules exacty 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 ju 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. 5, 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 be 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 lino 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.
OF THE PLAIN SCALE. The Plain Scale, in comm 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 1,3 inches. or 1,73 ; set one foot of the compasses in the third parallel under late, 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 fg is 73 parts of 100