MATHEMATICS. PRACTICAL OEOMETUY. [DBAWHCO INSTRUMENTS 



CHAPTER IX. 

 PRACTICAL GEOMETHY. 



fig. i. 



Tut preceding portions of this section having given, at 

 the couiiuenofment of the several Book* of Euclid, the 

 general definitions of a ;H>IM/, a line, Ac., also the 1'vttu- 

 tatrt and .ifiii't, it is unnecessary to repeat them, it 

 being sufficient for the student U> rofor to them when 

 rvquuiUi, in order to ^ive mlu a clour understanding or 

 conception of tho Problem he may at the timo have 

 under discussion. It is not here intended to give all the 

 leuis contained in a complete treatise on Practical 

 Geometry, bat 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. 



IS*TRUMKSTI}. For the purpose of performing the 

 construction of the different Geometrical figures, the 

 only instruments absolutely required are a pair of com- 

 passes, a ruler, a lead pencil, and a drawing pen. 



tpaixs. 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 movable, which at 

 any time, as occasion may require, can be replaced by 

 another containing a pencil leg (a), or a pen leg (6), tho 

 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 ; 

 tlie principal, however, 

 being to measure or 

 transfer distances, and 

 when fitted with a pen- 

 cil 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 (cid) in the 

 pencil or pen leg, so as 

 to make the joint work 

 easily. 



Rider. 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 tho 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 a 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. 



A ] * noil for drawing (Fig. 2) is generally prepared by 

 sharpening the wood and lead, so as to form a fine point, 

 similar to (Xo. 1) in the cut ; but the point (No. 2), as 

 hown in tho 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 thu 

 wther two sides, the one side showing a broad point, as 



Tb Ink commonly turd In drawing mathematical flifurm, Ac., It that 

 wvn China Ink. which, brine rubbed on a plat* or palctu-, with a 

 IK more Intlj from the pn, and drict mart quickly tlian 

 _ wriiinr Ink ; It aUu hu U adrantafe at not ruttu ur 

 Wutua- Bach on Uu paper. 



lull. 



In this way, th 

 Fig. . 



at (a), the other a fine point, as at (b). 



lead of the pencil may be 



kept close to tho ruler, and 



at the same time draw a 



fine line. A very simple 



and beautiful little iustru- : 



mi-lit, 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 (<i) into 



the hole or cone (*) ; and 



by turning it round with the 



hand against the knife edge 



(e), the point of the pencil is gradually formed, 



Fig. S. 



M 



T 



Drawing Pen. Tho drawing pen (Fig. 4), like a pencil, 

 Fig. 4. 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 dilii-n nt 

 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 to 

 it a short piece of steel with a very 

 fine point, generally called the pro- 

 tecting pin, and is used for setting 

 oil' points in a line, or marking 

 their intersections. 



Having briefly described the few instruments requisite, 

 it will be necessary to show their application, by per- 

 forming the three following simple problems, generally 

 given as Postulates (Euclid,l)efinitious, page 641), which, 

 however, by constructing, the student will the more rea- 

 dily become familiar with tho instruments and their use. 



PROBLEM L Tn drato a ttrnight line from any one point 



(A) to any other jwint (b). 



Lay one edge of the ruler upon tho point A, as shown 

 *' J - li dotted lines in the figure, 



/-A B and move the ruler round 



c l -.. ' -. ! until it coincides with tho 



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 



;race or mark left upon the surface of the paper is thu 

 straight line required. 



