MATHEMATICAL INSTRUMENTS. 



eel.) to represent the end of your scale, 

 and that A C, B U, be perpendicular 

 thereto : with A B as a radius, and from 

 A as a centre, dra\y the quadrant B F C, 

 and the straight line or chord B C sub- 

 tending- that quadrant. Divide the qua- 

 drant into 90 equal parts, and from B, as 

 a centre, measure off each division suc- 

 cessively, so as to cut Ihe chord B C into 

 90 parts, all which will be unequal. Mark 

 every tenth degree, both on the quadrant 

 and on the chord, thus, 10, 20, 30, 40, 50, 

 60, 70, 80, and 90. This division will 

 make the line B C a line of chords, which 

 affords a scale of very general utility in 

 mathematics. 



The line of sines, commonly marked S, 

 shews the relation of sines to various por- 

 tions of circles. Here it is necessary to 

 state, that there are three kinds of sines, 

 viz. the sine, the co-sine, and the versed 

 sine. The sine is that perpendicular 

 which stands at right angles with the 

 chord subtending 1 an arc, and reaches 

 from it to the circumference, such as the 

 line E F ; the co-sine is a chord, such as 

 F G, which commences from the junction 

 of the sine with the circumference, and 

 is parallel with that line from which the 

 sine arises, proceeding in that direction 

 until intercepted by the perpendicular 

 A C, which terminates the quadrant ; the 

 co-sine is therefore the complement or 

 residue of the base line A B, after de- 

 ducting from its other end the amount of 

 the versed-sine B E. If from B 60 de- 

 grees be measured on the quadrant to F, 

 its sine will divide the base A B into two 

 equal parts ; so that the co-sine and 

 versed-sine will be of equal length. The 

 line of sines is therefore made on the 



aendicular A C by means of parallels, 

 e base A B, drawn from the circum- 

 ference at the parts marked 10, 20, 30, 

 Sec. degrees, which of course give a regu- 

 larly diminishing scale. 



The line of tangents is made by a con- 

 tinuation of the perpendicular B D to K, 

 and by drawing from the graduated qua- 

 drant the several lines 10, 10 ; 20, 20 ; 30, 

 30 ; &c. to that perpendicular, all point- 

 ing to the centre A. This scale regularly 

 augments, and is earned to 45 degrees 

 only. Now, by transferring all the tan- 

 gent scale, and the places of the degrees 

 thus obtained from the point A, by draw- 

 ing segments from each part respectively 

 to the perpendicular A H, we have a line 

 of secants : thus the 10 on the tangent 

 scale will be transferred to 10 beyond C 

 on the secant line, 20 to 20, and thus to 

 the end of the scale up to *90 degrees, 

 which would, however, require a great 



length of ruler. The line of tangents is 

 confined to 45 degrees ; but a line of 

 lesser tangents, from 45 to 90, is made 

 on a smaller radius. 



The line of equal parts between A and 

 B is also called the line of lines, and is di- 

 vided into 10, 100, 1000, &c. equal parts ; 

 but the indicial numerals are confined to 

 10, for we have only ten numbers on each 

 limb of the sector, made by dividing the 

 radius (or base line) A B into that num- 

 ber of equal spaces. The uses of the 

 lines above described are verv extensive ; 

 but we shall give a brief example of their 

 intentions, observing, that the line of 

 equal parts is distinguished by the letter 

 L on each limb of the sector : the line of 

 sines, by S ; the line of tangents, by T ; 

 the line of secants, by se. and the line 

 of lesser tangents, by" ta. 



N. B. In some sectors the letter C is 

 engraved close to the very centre of the 

 hinge, which centre is marked by an ob- 

 vious puncture, towards which all the 

 lines have a tendency : in using the lines, 

 the measures are to be taken from those 

 marked L. S. C. &c. on one limb to those 

 marked L. S. C. on the other limb, re- 

 spectively, they standing at an angle of 

 six degrees from their respective part- 

 ners. 



" To find a fourth proportional by the 

 line of equal parts." Say you would wish 

 to find a line proportioned to 15 as 3 is to 

 8 : on the line of equal parts take a dis- 

 tance from C with your compasses equal 

 to 15, and with that opening extend your 

 sector so as the distance between 3 and 3 

 may correspond therewith ; then measure 

 the distance thus generated between 8 

 and 8, and lay it from the point C along 

 the line of equal parts : it will fall on 40, 

 which is in the same proportion to 15 

 that 8 is to 3. And this is demonstrable 

 by common arithmetic ; for 3 being 

 f of 8, and 15 being f of 40, the solution 

 given by this scale must be correct. This 

 depends entirely on the mathematical 

 axiom ; tiz. that " parallel lines under the 

 same angle are to each other in propor- 

 tion to their respective distances from the 

 angular point." 



" To set off an angle by a line of chords 

 of 60 only," (fig. 2.) Open the sector 

 to any extent at pleasure, and with the 

 distance between 60 and 60 describe a 

 segment at least equal to the space you 

 think the angle will occupy. On the 

 same line of cords take on your com- 

 passes the number of degrees you intend 

 the angle to be, say 27, and applying one 

 leg to the commencement of your seg- 



