B E V 



Bhvel angle is ufcd among tlie workmen, to denote any 

 other angle befide thofe of ninety or forty.five degrees. 



'l"hij,mple Bevel (fee Plalell. GeonuHry, fy. 35.) confids 

 of two rulers moveable on a common centre, like a carpen- 

 ter's rule, with a contrivance to keep tliem fixed, at any re- 

 quired angle. The centre C mull move on a very fine axis, 

 fo as to lie in a line with the fiducial edges CB, CD of the 

 rulers, and projeft as little as poffible hefoi'p them. The fidu- 

 cial edges of the legs reprefent the fides of any given angle, 

 and their interfeftion or centre C, its angular point. A pin, 

 fi-ved in the lower ruler, and pafling through a femicircular 

 groove in the irpper, ferves, by a nut A, which fcrews upon 

 it, to fix the rulers, or legs, when they are placed at the 

 dtfired angle. 



The ufe of this inftrument may be illuflrated in the fol- 

 lowing examples : 



T. Let 3 points, A, B, C, be in the circumference of a 

 circle, which is too large to be defcribed by a pair of com- 

 paffes ; and let it be required to find any other number of 

 points in the fame circumference. Bring the centre of the 

 bevel to B (Jig, 36.), the middle point of the 3 given ones 

 A, B, and C, and holding it there, open or (hut the inftru- 

 ment till the fiducial edges of the legs lie upon the other 

 two points, and fix them there by means of the fcrew A 

 (Jig. 35.) : this operation is called fett'ing the bevel to the 

 given points. Then removing the centre of the bevel to 

 any part between B and A or C, the legs being at the fame 

 time kept upon A and C, that centre will defcribe, or be al- 

 ways found in, the arc which palfes through the given points, 

 and will thus afctrtain as many others as may be required 

 between the limits of A and C. In order to find points 

 without thoie limits, proceed in the following manner : the 

 bevel being fet, bring the centre to C, and mark the dillance 

 CB upon the left leg ; remove the centre to B, and mark 

 the dillance B.\ upon the fame leg ; then placing tlie cen- 

 tre on A, brmg the right leg upon B, and the fird mark 

 will fall upon a, a point in the circumference of the circle, 

 parting through A, B, and C, whofe dillance from A is 

 equal to the dillance BC. Removing the centre of the bevel 

 to the point a laft found, and bringing the right leg to A, 

 the fecond mark will find another poii.t a" in the fame cir- 

 eumference, whofe diftance aa" is equal to AB. By pro- 

 ceeding in this manner, any number of points may be found, 

 whofe diftanccs on the circumference are alternately BC and 

 BA. In the fame manner, by making fimilar marks on the 

 right leg, points on the other fide, as at c' and c" are found, 

 whofe diltances Cc', c'c", are equal to BA, BC refpeftively. 

 Intermediate points between any of the above are given by 

 the bevel in the fame manner with thofe between the original 

 points. 



2. Three points. A, B, and C, being given, to draw a 

 line from any one of them, tending to the centre of the 

 circle, which paflTts through tkcm all. Set the bevel to the 

 three given points A, B, and C (Jig- 3"-) ; lay the centre on 

 A, and the right leg to the point C, and the other leg will 

 give the tangent AG'. Draw AD perpendicular to AG' 

 for the line required. For BAE being =: BCA, the angle 

 E AC is the fuppkmcnt to the angle A I'C, or that to whicli 

 the bevel is let ; hence, when one leg is applied to C, and 

 the centre brought to A, the dircftion of the other leg mull 

 be in that of the tangent G'E. 



3. Three points being given as before, let it be required to 

 draw from a fourth tiven point D, a line tending to the cen- 

 tre of a circle paffing through the hi ft three points. On 

 D (fig-S^-)' with the radius DA defcribe an arc AK ; 

 fet the bevel to the three given points A, B, and C, and 

 bring its centre, always keeping the legs on A and C, to 

 fall on the arc AK, as at H ; on A and H fevenilly, with 



B E V 



any convenient radius, ftrike two arcs, croffuig each other 

 at I ; and the required line T)d will pafs through the points 

 I and D. For a line drawn from A to H w-ill be a common 

 chord to the circles AHK and ABC ; and the line ID bi- 

 fetling it at right angles, muft pafs tlixough both their 

 centres. 



4. Three points being given as before, together with a 

 fourth point, to find two other points, fuch, that a circle 

 pafTmg through them and the fourth point, (hall be concen- 

 tric to that paffing through three given points. Draw Ac 

 and Cc tending to the centre, by a former problem ; fet 

 the bevel to the three given points A, B, and C ; bring the 

 centre of the bevel to D, and move it upon that point till 

 its legs cut off equal parts AN, CQ^ of the lines Aa 

 and Cc ; and N and Q^will be the points required. For, 

 fi'ppofing lines drawn from A to C, and from N to (^ the 

 fegmenta ABC and NDQjvill be fimilar ones ; and confe- 

 quently, the angles contained in them will be equal. 



y. Two lines tending to a diftant point being given, and 

 alfo a point in one of them; to find tivo other points (one 

 of which muft be in the other given hnc), fuch, that a cir- 

 cle paffing through thefe three points may have its centre 

 at the point of interfeftion of the given hues. Draw EH 

 fA'^-39-) at right angles to AB, and mnke FH = FE ; 

 fet the bevel to the angle GDO, and keeping its legs or the 

 points H and E, bring its centre to the hne AB, which will 

 give the point I. 



An improved bevel is exhibited in_/V. 40. by which the 

 arcs of circles of any radius, without the hmits attainable 

 by a common pair of compafTes, may be defcribed. It 

 confifts of a ruler AB, compofed of two pieces rivetted 

 together near C, the centre or axis, and of a triangular part 

 CI' ED. The axis is a hollow focket fixed to the triangu- 

 lar part, about which another focket, fixed to the arm CB 

 of the ruler AB, turns. Thefe fockets are open in the 

 front for part of their length upwards, as reprefentcd in the 

 feftion at I, which fhews the point of a tracer, or pin, fitted 

 for Aiding in the focket. The triangular part is furniilied. 

 with a graduated arc DE, by which and the vernier at B, 

 the angle DcB may be determined to a minute. In tliis arc 

 is a groove, by means of which, as well as by the nut and fcrew 

 at B, or fome fimilar contrivance, the ruler AB may be 

 fixed in any required pofition. A fcale of radii is put on 

 the arm CB, by which the inftrument may be ftt to defcribe 

 arcs of given circles, not lefs than 20 inches in diameter. 

 In order to fet the inftrument to any given radius, the 

 number exprcffing it in inches on CB, is brought to cut a 

 fine hue drawn on CD, parallel and near to the fiducial edge 

 of it, and the arms are faftencd in that pofition by the fcrew 

 at B. Two heavy pieces of lead or brafs G, G, made in 

 form of the feftor of a circle, the angular parts being of 

 ileel, and wrought to a true upright edge, as Ihewn at H, 

 are ufed with tins inftrument, whofe arms are made to 

 bear againll thofe edges when the arcs are drawn. The 

 under fides of thefe ftftors are furniflied with fine lliort 

 points to prevent them from Aiding. The fiducial edges of 

 the arms CA and CD, are each divided from the centre C 

 into 200 equal parts. This inftrument might be furnilhed 

 with fmall caftors, like the pentagraph ; but little buttons, 

 fixed on its under fide, near A, E, and D, will enable it to 

 Aide with fufticient eafe. 



The ufe of this inftrument may be exempUfied in the fol- 

 lowing problems : 



I. To defcribe an arc, which fhall pafs through three 

 given points. — Place the teCtors G, G', with their angular 

 edges over the two extreme points ; apply the arms of the 

 bevel to them, and bring at the lame time its centre C, that 

 is, the point of the tracer, or pen, put into the focket, to the 



third 



