MATHEMATICAL INSTRUMENTS. 



graduated with 180 degrees, backwards 

 and forwards, the centre point 90 being- a 

 right angle. The protractor is used for 

 laying down angles to any extent, as also 

 for taking their measurements : hence it 

 is of extreme service in every branch of 

 mathematics, and indeed of mechanism. 

 On the same side with the graduations, we 

 generally find a line of chords on an ex- 

 tensive scale. We shall explain its con- 

 struction when we treat of the sector, ob- 

 serving, in this place, that by its aid we 

 are enabled to set off any angle without 

 the assistance of a protractor : thus, take 

 the measurement of 60, from the line of 

 chords, as a radius wherewith to describe 

 any segment at pleasure, putting one foot 

 of your compasses at the point whence 

 the angle is to proceed, and commencing 

 the segment from that line whence the 

 angle is to be made. Take then from the 

 line of chords the number of degrees 

 you intend the angle should contain ; set 

 them off upon the segment from the place 

 where it joins the line ; the angle will be 

 thus made, leaving the centre whence the 

 radius was drawn for its point, and the 

 two ends of the chord that cut off the 

 segment for its measurement. See GEO- 

 SIETKY and DIALLING. 



Some protractors are made of brass, in 

 the form of a semi-circle ; they are pre- 

 cisely on the same principle, but are 

 more calculated for the measurement than 

 for the construction of angles ; because 

 they expose the directions of lines, how- 

 ever short, and enable us, by means of 

 any right line instrument, laid from 

 the centre to the circumference, to ascer- 

 tain the angle without extending the line, 

 as must be done when an ivory protractor 

 is used to a short line. 



On the back of the protractor there 

 are usually six scales, marked 60, 50, 45, 

 40, 35, and 30 ; meaning, that the mea- 

 sures, or equal points, 1, 2, 3, &c. respec- 

 tively include 60, 50, &c. such within the 

 length of an inch ; the number 1, 2, 3, 

 &c. being considered at 10, 20, 30, &c. of 

 such small divisions as are placed at the 

 commencement of each scale respective- 

 ly. The scale marked C, standing on the 

 same line with that of 60 to an inch, is a 

 line of chords on a reduced scale, for the 

 convenience of persons working on such; 

 and the broader scale, of 10 lines in 

 depth, is of half and quarter inch divi- 

 sions, with oblique scales at the two ends. 

 These shew all the tenths of a half, or of 

 a quarter of an inch, respectively, accord- 

 ing as the oblique line gives more space 

 between it and the firsi perpendicular, as 

 may be seen by referring to the figures 



246 ,* which 



ot the division, and enable us to embrace 

 any number of whole divisions, and oi" 

 tenth parts, within our compasses, with 

 readiness and precision. This is intended 

 chiefly for work on a larger scale, such as 

 ground-plans, &c. ; though for such pur- 

 poses, a scale divided into twelfth parts is 

 more convenient ; since it takes feet and 

 inches, instead of decimals of feet. 



It is proper to remark in this place, 

 that the protractor should be prevented 

 from warping, else its measurements of 

 angles will not be true. When this de- 

 fect has taken place, it will be necessary 

 to press the instrument ; thereby to bring 

 it as flat as possible, that the measure- 

 ments may be accurate, by the bearings 

 being restored to their proper places. 



The sector is made to fold in the mid- 

 dle, not only that it may lay in a smaller 

 compass, but to solve many problems, by 

 means of the references given to various 

 tabks and scales that are engraved on 

 both sides of each limb. When opened 

 to its full length, the sector commonly 

 measures one foot ; each inch being num- 

 bered, and divided into tenth parts, called 

 lines. At the edge is another scale, 

 which divides the foot into ten equal 

 parts (numbered 10, 20, 30, &c.) because 

 each tenth part of the foot is again subdi- 

 vided into ten ; thus giving a division of 

 the twelve inches into 100 equal parts. 



The first scale we shall notice is that 

 next to the inner edges, marked Pol. 

 meaning polygon. By opening the sec- 

 tor to such a width, as may admit the ra- 

 dius of any circle to measure exactly 

 from the figure 6 on one, to the figure 6 

 on the other limb, we at once ascertain 

 the division of that circle's circumference 

 into any number of equal parts, from four 

 to twelve ; because, from the figure 4 to 

 the opposite figure 4 will give a chord 

 subtending a quadrant of the circle ; 

 from 5 to 5 will give the side of a regular 

 pentagon, or figure of five sides ; from 6 

 to 6 a hexag'on ; and so forth. 



The line of chords on the sector is 

 known by the letter C on each limb, and 

 measures 60 degrees only ; though on the 

 protractor it goes as far as 90, which is its 

 full measurement. This, however, is not. 

 important, as we can always add 30 to 60, 

 and thus complete any figure in hand. 

 The formation of the line of chords be- 

 ing given, its application will be more 

 readily understood; we shall therefore 

 shew how they are constructed from the 

 circle. 



Suppose the line A B (fig. 1, Plate Mis- 



