SECTOR. 



refponding divifions of the fcales of the fame name, the legs 

 of the feflor being in an angular pofition : but in taking 

 thefe tranfverfe dillances, it is to be obferved, that each of 

 the feveral fcales hath three parallel lines, acrofs which the 

 divifions of the fcale are marked, and that the points of the 

 compafles mult be always fet on the infide hne, or that hne 

 next the inner edge of the leg, which is the only line, in 

 each fcale, which runs to the centre. 



For the ufe of the logarithmic fcale of numbers, fee 

 Gunter's Line. 



Sector, Ufe of the Line of Lines on the. I. To divide a 

 given line into any number of equal parts ; e. g. 9. Make 

 the length of the given line, or feme known part of it, a 

 tranfverfe diftance to 9 and 9 : then will the tranfverfe 

 diftance of i and i be the ^th part of it ; or fwch a fub- 

 multiple of the ^th part, as was taken of the given line : 

 or the 4th part will be the difference between the given line 

 and the tranfverfe diftance of 8 and 8. 



Hence, 2. To make a fcale of a given length, to contain 

 a given number of equal parts ; e. g. let the fcale to the map 

 of a furvey be 6 inches long, and contain 140 poles, and let 

 it be required to open the fedfor, fo that a correfponding 

 fcale may be taken from the line of lines. Make the tranf- 

 verfe diftance 7 and 7 (or 70 and 70, viz.. ^i") equal to 3 

 inches (=i -j) ; and this pofition of the line of lines will 

 produce the given fcale. 



3. To divide a given line {e.g. of 5 inches) into any 

 affigned proportion, as of 4 to 5. Make j inches, the 

 length of the given line, a traiilverle diftance to 9 and 9, the 

 fum of the propofcd parts ; and the tranfverfe diftances of 

 the afligned numbers, 4 and 5, will be the parts required. 



4. To two given lines, viz. 2 and 6, to find a third pro- 

 portional. Take between the compafles the lateral diftance 

 of the fccond term, -viz. 6 ; let one point on the divifion 

 expreffing the firft term, w'z. 2 on one leg, and open the 

 legs of the feftor till the other point will fall on the cor- 

 refponding divifion on the other leg : keeping the legs of 

 the feftor in this pofition, take the tranfverfe diftance of 

 the fecond term, viz. 6, and this diftance is the third term 

 required, which diftance, meafured laterally from the centre, 

 will give 18, the number required : for 2 : 6 :: 6:18. 

 Otherwife, take the diftance 2 laterally, and apply it tranf- 

 verfely to 6 and 6, the fedlor being properly opened : then 

 the tranfverfe diftance at 2 and 2, being taken with the 

 compafles, and applied laterally from the centre of the 

 feftor on the fcale of lines, will give tlie third term, when 

 the proportion is decreafing ; for 6 : 2 :; 2 : -?• If the legs 

 of the fctlor will not open (o far as to let the lateral diftance 

 of the fccond term fall between the divifions expreffing the 

 firft term; then take ~, ', 4, or any aliquot part of the 

 fecond term, th.it will conveniently fall within the opening 

 of the feftor, and make fuch part the tranfverfe diftance of 

 the firft term : then, if the tranfverfe diftance of the fecond 

 term be multiplied by the denominator of the part taken 

 of the fecond term, the product will give the third term. 



5. To tliree given lines, w'z. 3, 7, and 10, to find a fourth 

 proportional. Open the legs of the feftor, till the tranf- 

 verfe diftance of the firft term, 3, be equal to the lateral 

 diftance of the fecond term, 7, or to fome part of it ; then 

 will the tranfverfe diftance of the third term, 10, give the 

 fourth term, 234, required ; or fuch a fubmultiple of it, as 

 was taken of the fecond term ; for 3:7:: 10 : 23;. 

 Otherwife, fct the lateral diftance, 7, tranfverfely from 

 10 to 10, opening the feftor accordingly ; and the tranfverfe 

 diftance, at 3 and 3, applied laterally, will give 2-r'o ; fer 

 10 : 7 :: 3 : 2^,. 



6. To dimmifti a line of four inches, in the proportion of 



10 



8 to 7. Open the feftor till the tranlverfe diltancc of 

 8 and 8 be equal to the lateral diftance of 7 : mark tl»e 

 point, where four inches, as a lateral diftance, taken from the 

 centre, reaches ; and the tranfverfe diftance taken at that 

 point will be the hne required. If the line fhould be too 

 long for the legs of the feftor, take ^, ,, or J, &c. part of 

 the given line for the lateral diftance, and the correfponding 

 tranfverfe diftance, taken twice, thrice, or four times, &c. 

 will be the hne required. 



7. To open the feftor, fo that the two fcales of lines (hall 

 make a right angle. Take the lateral diftance from the 

 centre to the divifion marked 5, between the points of the 

 compaffes, and fet one foot in the divifion marked 4, on 

 one of the fcales of fines ; and open the legs of the feftor 

 till the other foot falls on the divifion marked 3, on the 

 other fcale of fines, and then will thofe fcales ftand at right 

 angles to one another ; for the fines 3, 4, 5, or any of their 

 multiples, conftitute a right-angled triangle. 



8. To two right lines given, e.g. 40 and 90, to find a mean 

 proportional. Set the two fcales of lines at right angles; find 

 the half fum of the given lines, <u(z. 65, and the half dif- 

 ference, wz. 25, and take with the compaft^es the lateral 

 diftance of the half fum, 65, and apply one foot to the halt 

 difference, 25, the other foot tranfverfely will reach to 60, 

 the mean proportional required ; for 40 : 60 :: 60 : 90. 



Sector, Lffe of the Scale of ChorJs on the. I. To open 

 the feftor fo that the two fcales of chords may make aa 

 angle of any number of degrees, e. g. 40. Take the diftance 

 from the joint to 40, the number of degrees propoled on 

 the fcale of chords ; open the feftor till the tranfverfe 

 diftance from 60 to 60, on each leg, be equal to the afore- 

 faid lateral diftance of 40 : then do the fcales of chords 

 make the angle required. 



2. The feftor being opened, to find the degrees of its 

 aperture. Take the extent from 60 to 60, and lay it off 

 on the fcale of chords from the centre : the number, where 

 it terminates, ftiews the degrees of its opening. By apply- 

 ing lights on the fcales of chords, the feftor may be ufcd to 

 take angles, as a (urveying inftrument. 



3. To protraft or lay down an angle of any given number 

 of degrees, i. Let the number of degrees be lefs than 60, 

 vi%. 46. At any opening of the feftor, take tlie tranfvcrlc 

 diftance of 60 and 60 on tlie chords ; and with this open- 

 ing defcribe an arc : take the tranfvirle diftance of the 

 given number of degrees, 46, and !.iy this diftance on 

 the arc defcribed, marking its extremitii-s : from the 

 centre of the arc, through thefe extremities, .draw two 

 lines, and they will contain the angle required. 2. Whoi 

 the degrees given are more than 60, I'iz. 14S ; defcribe 

 the arc as before ; take the tranlverfe diftance of .; or ; of 

 the given degrees, 148, e.g. -! = 49; degrees: lay this 

 diftance on the arc thrice : and from the centre draw two 

 lines to the extremities of the arc thus diterniined, and they 

 will contain the required angle. N. B. If tlie radius of 

 the arc or circle is to be of a given length, then make the 

 tranfverfe diftance of 60 and 60, equal to that aftigiied 

 length. 



4. To find the degrees which a given angle contains. 

 About the vertex defcribe an arc, and open tiie feclor till 

 the diftance from 60 to fio, on each leg, be equal to the 

 radius of the circle ; then taking the chord of the arc be- 

 tween the compalles, and carrying it on the legs of the 

 feftor, fee what equal number, on each leg, the points of 

 the compaftes fall on : this is the quantity of degrees the 

 given angle contains. 



5. To take an arc, of any quantity, from off the cir- 

 cumference of a circle. Open the feCtor till the diftance 



from 



