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In this we suppose the radius of the sphere to be unity; and so if, for instance,
(c.) By the method of Correcting the Angles, and Treating the Triangle as Plane.
be the sides to radius r, then the circular measures of the sides are a b c
But if we suppose that we retain all terms up to the fourth order, i, e.
a2 + b2
+ c −— a) (a + b — c) (a − b + c)
16 A, if A is the area of a plane triangle whose sides are a, b, c. (Plane Trigon. Art. 41.)
Now if E is the spherical excess, i.e. the excess of the sum of the three observed angles over two right angles, we have (Spherical Geometry, Prop. ix.)
The same clearly holds good of either of the other angles; hence the rule determines the spherical excess into three equal parts, subtract one part from each of the angles, and the triangle can then be considered plane. If 8C is equal to n" then, as before,
It is to be observed that n is very small, e.g. rarely more than 5 or 6, hence a small error in the area will produce no appreciable error. Hence A can be found on the supposition that the original triangle is plane.
JOHN F. TWISDEN.
The preceding portions of this treatise on “ The Mathematical Sciences” having given, at the commencement of the several Books of Euclid, the general definitions of a point, a line, &c., also the Postulates and Arioms, it is unnecessary again to repeat them, being sufficient for the student to refer to them when requisite, in order to give him a clear understanding or conception of the Problem he may at the time have under discussion. It is not here intended to give all the Problems contained in a complete treatise on Practical Geometry, but merely a selection of those which may be considered most useful in assisting the mechanical draughtsman, workman, or others who may be engaged in like pursuits.
Instruments.-For the purpose of performing the construction of the different Geometrical figures, the only instruments absolutely required are a pair of compasses, a ruler, a lead pencil, and a drawing pen.
COMPASSES.—The best form of compasses, or dividers (Fig. 1), are made of metal, such as brass or silver, from five to six inches in length, having steel points, and formed with one of the points or legs moveable, which at any time, as occasion may require, can be replaced by another containing a pencil leg (a), or a pen leg (b), the pen being constructed in the same manner as the drawing pen, afterwards described. The various uses to which the compasses may be applied are well known; the principal, however, being to measure or transfer distances, and when fitted with a pencil or pen to describe circles, the one with black-lead pencil, the other with ink. * The small figure (c) is an instrument used for tightening the joint of the compasses when the legs work too easily, or the reverse; the two points (e e) at the one end fitting into two small holes (e e) at the head of the compasses—the other end being used for screwing up the nails (d d) in the pencil or pen leg, so as to make the joint work easily.
RULER.-The ruler in general use is merely a bar of wood or metal, the edges being formed straight, and, for convenience, should be from six to twelve inches in length, about an inch in breadth, and is for the purpose of guiding the motion of the pencil or pen in a straight line or direction.
PENCIL.-It may perhaps be considered unnecessary to give a description of a common drawing pencil; but as there are few who, in commencing to draw mathematical figures, can form a proper point to their pencils, a few words may suffice to show the best method of doing it.
+ The ink commonly used in drawing mathematical figures, &c., is that known as China Ink, which, being rubbed on a plate or palette, with a little water, runs more freely from the pen, and dries more quickly than common black writing ink; it also has the advantage of not running or blotting so much on the paper.
A pencil for drawing (Fig. 2) is generally prepared by sharpening the wood and lead, so as to form a fine point, similar to (No. 1) in the figure; but the point (No. 2), as shown in the figure, is the best form used for mechanical drawing, and is made by cutting two sides of the wood and lead flat, and leaving but a small flat edge on the other two sides, the one side showing a broad point, as at (a), the other a fine point, as at (6). In this way, the lead of the pencil may be kept close to the ruler, and at the
same time draw a fine line. A very simple and beautiful little instrument, not much known, called the “ Pencil Cutter and Sharpener" (Fig. 3), has been invented, which forms the pencil point (No. 1), and is used by placing the pencil through the guides (a) into the hole or cone (6); and by turning it round with the hand against the knife edge (C), the point of the pencil is gradually formed.
DRAWING PEN.—The drawing pen (Fig. 4), like a pencil, is used for drawing straight lines, guided along the edge of the ruler; it is usually made in two parts—viz., the pen and the handle. The pen part consists of two blades, with steel points, so bent that the ends or points meet, but leaving a space or cavity for the ink; and in order to draw lines of different thickness, those blades can be opened more or less by a small screw. The best pens are also made with a joint to one of the blades, to admit of the pen being more easily cleaned by separating them. The other, or upper part, forming the handle of the pen, screws into the lower portion, having attached a short piece of steel with a very fine point, generally called the protracting pin, and is used for setting off points in a line, or marking their intersections.
Having briefly described the few instruments requisite, it will be necessary to show their application, by performing the three following simple problems, generally given as Postulates (Euclid, Definitions, page 47), which however, by constructing, the student will the more readily become familiar with the instruments and their use.
Fig. 4. PROBLEM I.–To draw a straight line from any one point (A) to any other point (B). Lay one edge of the ruler upon the point A, as shown in dotted lines in the figure,
and move the ruler round until it coincides with the point B; draw the point of the pencil or pen from A to B, keeping it always close to the edge of the ruler : the trace or mark left upon the surface of the paper is the straight line required.
PROBLEM II.-To prolong or produce the straight
line (AB) to any length towards (C), in a straight
Fig. . Place one edge of the ruler upon the point B at the extremity of the line AB; take ! any point in AB, such as a, so far from B that
the distance may be less than the length of the ruler; make the ruler coincide with that point a, and draw as before with the pencil or pen from B towards C, which will leave the line produced as required. This operation may be repeated as often as wished, so as to pro long a line to any extent.
PROBLEM III.—To describe a circle from a centre (O), and at any
distance (OA) from the centre (O). Place the fixed leg or point of the compasses in the centre 0, opening them until they take in the given distance OA or radius of the circle to be drawn; then move them round, describing the circumference of the required circle.
After performing the three preceding problems, it will be useful, in order to acquire a free use of the instruments in describing circles, drawing lines through points of intersection, and joining points by straight lines, to draw the accompanying figure several times, and thereafter to proceed to the construction of the following problems :
To bisect or divide into two equal parts a given straight line
Fig. 8. 1st METHOD.— When there is not sufficient space in the drawing on either side of the given line. By means of the compasses, place a leg at one of the extremities of the line, open them until the other leg is at any other point near the middle of the line ; mark this point; transfer the leg of the compasses to the other extremity of the line, and mark the distance of the other leg from this point ; if the two points so marked do not coincide, open the compasses to as near the middle of these two points as the eye will direct: proceed as before; the distance between the new points will be smaller than before, and a trial or two will enable us to bisect the line accurately.
2ND METHOD.7-JVhen the giren line (AB) occurs in or near the centre of the drawing,
* References are made to the propositions in Euclid, where several of the Problems are demonstrated, in order that the student, should he wish to investigate the Problems further, may do so by examining the methods of construction there adopted.
+ This is the method most frequently used to bisect a line ; but in practice it is not necessary to draw the whole arcs, but merely the intersections, and by means of the ruler placed on those points, to mark the centre point C. The line DC or DE, in this and the following Problem, when drawn, will also be found to be perpendicular to AB.