

113 



LESSONS IN <;KU.MI.TKY._ iv. 



IN rUAiTH.'AI. QtO .tinued). 



IN addition to the mathematical instruments dcHcribod incur 

 last lesson, there in also an instrument called a I'miro 

 measuring angles upon i>;q>iT, which is NpMMBtod i" I 



-> of a omii-irdi- divided into degrees, from 0. to 

 aoh way, tho SlOtli degre boing right above tho centre, o. 

 The straight line, A H, in tho fitfiiro is tho diameter of tho semi- 

 circle, and is called tho jt<lu<-i>il (or true) edge of the protractor 

 to be applied to one of tho legs of the angle to bo measured ; 



Fig. 14. 



the arch, A M B, being tho fiducial edge to be applied to the 

 other leg. Thus, in order to measure the angle x o Y, the 

 centre of the instrument is placed on tho vertex, o, of the angle, 

 and tho edge o A on tho leg o T, so as to coincide with it 

 exactly : then the angle A o M, on tho arch A M B, determined 

 by the point M, through which tho other leg, o x, passes, is the 

 measure of tho angle x o Y. In this case, the measure appears 

 to bo nearly 45 degrees, as the figure represents divisions on the 

 arch or limb of tho protractor at every five degrees. 



This apparatus for measuring angles is sometimes engraven 

 *on the upper side of a pair of parallel rulers, and sometimes on 

 the obverse side of a plane scale. The protractor is more com- 

 monly modo so that the centre of the semicircle, and the fiducial 

 ontaining it, shall bo on tho outside of the instrument 

 rather than on tho inside, as above. 



The Plane Scale is a flat ruler with several lines of equal 

 parts, on one side divided according to certain proportional 

 parts of an inch ; and having, on the other side, the diagonal 

 scale, decimally divided so as to measure units, tens, and 

 hundreds of equal parts, with a very considerable degree of 

 exactness. Tho construction of this scale, so useful in graphical 

 (i.e., drawing) operations, such as the construction of plans, 

 maps, and charts, architectural designs, plans and sections 

 of machinery, etc., is founded on the properties of similar 

 triangles, as treated in the sixth book of Euclid. We shall 

 endeavour to give our readers a practical idea of its con- 

 struction. 



On a straight line, A (Fig. 15), divided into any convenient 



D 



E 



number of equal parts, A B, B C, C D, D E, etc., one, A B, is 

 assumed as the standard unit of measure. From the different 

 points, A, B, C, D, B, etc., perpendiculars of a convenient 

 length, as A A', B B', c c', D D', E K', etc., are drawn to the straight 

 line A E, and terminated in the straight line A' E' parallel to A E. 

 The unit A B is divided into 10 equal parts ; then the opposite 

 part, A ; B', is similarly divided ; next the perpendicular B B' is 

 divided into 10 equal parts, and through each division straight 

 lines parallel to A E or A' E* are drawn. Tho divisions of the 

 straight line A E are now marked with the numbers 1, 2, 3, etc., 



'. divisions of tho standard 

 8, 9, from B to A, to 

 parts of a unit ; and the <1 

 .lirular 11 I: . m B to B", to 



ports of a -ionn of the 



Htraight "In, those between B and A denote 



tens, and those between B and a* denote unit*. The scale a 

 r<-n<lfi-i-<l complete by drawing Htraight line* from B on B A, 

 to 1 <.n i:' A' ; from 1 on i: A, to 2 on 11' A'; from 2 on B A, to 

 3 on B' A' ; and so on, till one be drawn from 'J on B A, to A' 

 on B A'. 



By tho nature of mmilur triangles, hereafter to be explained, 

 the email part of tho parallel to the base 1 B', within the triangle 

 B 1 B', at the division marked 1, is one-tenth part of the base 

 1 B', and consequently one-hundredth part of the line A B ; the 

 small parts of tho other parallels are in succession, two- 

 hundredth, thi-f--li-nfl,-i:dthi t etc. Hence, if a straight line is to 

 be measured, take its length in the compasses, and apply it to 

 the scale from B towards E. If it measures an exact number of 

 units, say from B to E, then tho straight line may be said to 

 measure 3, 30, or 300 equal parts, according as A B is made to 

 stand for 1 unit, 1 ten, or 1 hundred. If it does not measure 

 from E to B exactly, but extends from E exactly to one of the 

 division marks between B and A, say 4, then the straight line 

 may be said to measure 3 4, 34, or 340 equal parts, according to 

 the standard unit, as before. If it does not extend from E to 

 the division marked 4 between B and A exactly, but falls some- 

 where between 4 and 5, then move the compasses downwards, 

 preserving one point always in the lino E E', and both points 

 parallel to A E, till the other point fall on the intersection of the 

 diagonal marked 4, 4, with one of the parallel straight lines 

 marked on B B', say 6 ; then the straight line may be said to 



Fig. 16. 



Fig. 17. 



measure 3*46, 34*6, or 346 equal parts, according to the standard 

 unit, as before. 



For the purposes of navigation, dialling, etc., the plane scale 

 has frequently on the side obverse to the diagonal scale just 

 described, a set of lines, besides those of equal parts, containing 

 divisions for the measurement of leagues, rhumbs, chords, 

 sines, tangents, semi-tangents, secants, lines of longitude, etc. 

 Such scales are considered the best, as they are generally 

 executed with great care. The scale called Gunter's scale haa 

 the same divisions on one side of it, as are to be found on the 

 plane scale, but of a larger size, and when well constructed, 

 admitting of greater accuracy ; but being usually made of 

 boxwood, this is seldom the case. The obverse side of Gunter's 

 scale has a set of lines representing the logarithms of the 

 numbers which denote these divisions ; by means of the loga- 

 rithmic lines, arithmetical calculations can be performed instrn- 

 mentally, that is, without the operation of the ordinary rules. 

 A modification of this instrument, called the sliding Gunter, is 

 still more ingenious in its construction, and still more useful as 

 an instrument of calculation. The explanation of these instru- 

 ments, however, belongs to a more advanced state of knowledge 

 among the generality of our readers. This we hope to reach by 

 their perseverance. 



One of tho most useful instruments in a mathematical case, 

 is the sector ; a mere sketch of its appearance is given in 1 

 It is composed of two flat rulers, movable on an axis, or jointed 

 at one end like a pair of compasses ; hence it is called by tho 

 French, compos de proportion the compasses of proportion. 

 From the centre of tho axis or joint, several scales are drawn on 

 tho faces of the rulers, so as to correspond exactly with each 

 other. The two rulers are called the legs of the sector, and 

 represent the radii of a circle ; and the middle point of tho 

 joint, its centre. It contains a scale of inches, lines of equal 



