1SJ7] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



313 



MEASUREMENT OF ANGLES. 

 A New Method of Measuring the Degrees, Minutes, Sfc, in any Recti- 

 linear Angle, hy Compasses only, without using Scale or Protractor. 

 By Oliver Byrne. 

 Let it he requirerl to find the number of degrees, minutes, &c., in the 

 angle A B C =■ B (Fig. 1). With any radius, AC, describe a circle: then 



take A B in the compasses, and apply it from B to 1 ; from 1 to 2; from 



2 to 3 ; &c. (the numbers outside the circle are referred to). If, in applying 



the arc A B, we find that on our return to B, after n applications, we have a 



coincidence, then it is well known that the number of degrees, &c. will 



360 

 be = — . But, in the present example, after eight applications the point 

 n 



falls at 8, putting A, = from 8 to B, continue to apply the same arc or 

 opening of the compasses from 8 to^ from 9 to 10; frojn 10 to 11 ; dtc. on 

 to 16. This process is to he continued till we have the half or more than 

 half the arc A B between the last point found and B. In this case 2^ is the 

 point. Any error that may be involved in the process will be much neu- 

 tralised by thus determining the points 8, 16, 24, &c. independently. Theo- 

 retically, the arcs B, 8 ; 8, 16 ; 16,24; &c. are all equal, but practically 

 they may imperceptibly differ. We might have taken the arc B, S, and 

 applied it from 8 to 16 ; from 16 to 24 ; &c., but this process would multi- 

 ply any error that might be involved in B, 8 ; while the process just described 

 has a correcting tendency. To lessen error further, we are again to begin at 

 A, and apply the arc A B in a contrary direction, from A to 1 ; from 1 to 2 ; 

 from 2 to 3 ; &c. (the numbers inside the circle are in this case referred 'i). 

 Should the points 24 and 16 coincide, as in fig. 1, then we have 

 8 8 + A, = 360°; and 5 A, = 6; 



.-. A, = 360°- 89 = 5; .-. 1800 - 40e = fl; 



1800 



.-. 9 = 



41 



If the points 24, 24, overwrap or fall, as in fig. 2. — Then put Aj = from 24 



to 24 : this arc will be very small in most cases — in this case it is the 20th 



part of A B ; 



.-.9 = 20 A,; 89 + A, = 360°; 6A, -2A2 = a. 



From these equations, which involve the unknown quantities 8, A,, ^^ B is 



readily eliminated. 



A, = 360° — 8 9, from the second ; 



9 + 2Aj 

 and A, = , from the third. 



2160 - 48 9 = 



+ 2A, = 

 21600 



— ; since A, = — . 



10 '20 



491 



= 43° 59' ll\. 



If the points 24, 24, do not overwrap, as in fig. 3, and A, be in excess 



instead of defect, that is, that some multiple of 9 made less by A , make up 



the circumference. In this case the three equations will stand thus ; — 



59 -A, = 360°; 10 A, + A, = 9; and 29 Aj = 9. 



In this example, tke distance between 24 and 24, or A2, is found to be the 



29th part of the arc AB. 



52200 



.-. 9 = = 73° 25' nearly. 



711 



It is evident that the numbers on these figures maybe omitted in practice, 

 as none of them except the first is required ; indeed, where the points of the 

 compasses rest need not be noted, except those points that fall inside the 

 points A, B. 



Fig. 3. 



This method of measuring an angle is more accurate and expeditious than 

 may at first appear from the above lengthened details, and will often be 

 found convenient when compasses only can be obtained. A general rule 

 may be arrived at as follows : Let 



ra 9 1 A, = TT = 360° ; n A] 1 p Aj = 9 ; and j Aj = 9 ; 

 be the three equations generally expressed; p being always equal + 1 or 



- 2. 



9 = 



n g tr 



mu q z. q ^ p 



In example, fig. 2, this expression becomes 



6x20x360 



9 = ^ 



8x 6x20 + 20-2 



10 X 29 X, 360 



In example, fig. 3, 



(Q) 



43°59'i. 



73° 25'. 



5x 10 X 29-29 Vl 

 The only thing to be observed in (Q) is the sign of q. In examples like the 

 latter it is to be minus, but in those like the former plus. 



This method of measuring angles will be found more correct than the in- 

 genious one proposed by M. De Lagny, which consists in measuring angles 

 with a pair of compasses, and that too without any scale whatever, except 

 an undivided semicircle. Having any angle drawn upon paper, to measure 

 it : produce one of the sides of the angle backwards behind the angular 

 point; then with a pair of fine compasses describe a pretty large semicircle 

 from the angular point as centre, cutting the sides of the proposed angle, 

 which will intercept a part of the semicircle. Then take this intercepted 

 part very exactly between the points of the compasses, and turn them suc- 

 cessively over upon the arc of the semicircle, to find how often it is contained 

 in it, after which there is commonly some remainder; then take this re- 

 mainder in the compasses, and in like manner find how often it is contained 

 in the last of the integral parts of the first arc, which will again most likely 

 give some remainder ; find in like manner how often this last remainder is 

 contained in the former ; and so on continually, till the remainder becomes 

 too small to be taken and applied as a measure. By this means M. De 

 Lagny obtained a series of quotients, or fractional parts one of another, 

 which being properly reduced into one fraction, give the ratio of the first 

 arc to that of a semicircle ; or the ratio of the proposed angle to two right 

 angles or 180 degrees, and consequently the degrees and minutes of the 

 angle itself becomes known. 



Fig. 4. 



Suppose the angle A C B (fig. 4) be proposed to be measured. Produce 

 A C towards D ; and from the centre C, describe the semicircle A B D, on 

 which A B is the measure of the proposed angle. Take A B in the com- 



42 



