362 



THE CIVIL ENGINEER AND ARCHITECrs JOURNAL. 



[Decembeb, 



number fiJ,^lre(l on tlie riyht-lmnrl, and unity to that right-hand 

 fif^ure; — and by placing tlie fiilcium at U, the ratio becomes this 

 same diffcrenre to unity, wliich is figured on the left-hand. 



TIius, supposing tlie boxes adjusted to the divisions figured 



1 3 ; then, with the fulcrum at B, the ratio 1 : 3 holds good ; — 



with the fulcrum at C, the ratio is (3 1) : 3; — and with the ful- 

 crum at I), it is (3 1) : 1. 



On the other hand, it appears that all the divisions, which have 

 the ratios expressed by niimbers, each greater than unity, are de- 

 termined by placing the fulcrum at D; and it follows, that, chang- 

 ing the fulcrum to C, we obtain a new ratio, which consists of the 

 ium of the two numbers to the riyht-hand number; and by chang- 

 ing it to B, it becomes the sum of the two numbers to the left-hand 

 number. Thus, supposing the boxes to be adjusted to the divi- 

 sions marked 5 6, this is the proportion when the fulcrum is at 



D; transferring it to C, the ratio becomes 11:6; and changed to 

 B, it is 11 : 5. 



This property increases the number of ratios as figured on the 

 instrument, and is of valne in adjusting it to the most convenient 

 position as regards the drawing, copy, size of drawing-table, &c. 



Instead of selecting a certain number of ivrbitrary ratios, as is 

 the usual course with instrument-makers, and dividing the bars to 

 conform to them, it has been proposed to make the divisions equal 

 ])arts of b for ,r, and of c for y — the value of each part being very 

 small, such as ttbtj'') according to the size of the instrument; and 

 to adjust the boxes, for any required ratio, according to values 

 which might then be readily ascertained from the equations given 

 herein. 



Thus, suppose it was required to reduce a plan on the scale of 

 100 feet to the inch, to a copy on the scale of 12 chains to the inch; 



then 12 ch. = 7fl2 feet, and — — |^, or i nearly. Referring to the 



table of equations, three of them only belong to the case of re- 

 ducing a drawing, and we can determine at once that the selection 



lies between equations 1 and .5. 



By equation 1, x=— = f|5 b, 

 ~9; and by 



which corresponds nearly to the usual division, 1- 



equation 5, ^ := — -^ = ^S b, which corresponds nearly to the 



usual division, 1 8. Now, with the instrument as it is usually 



divided, the error of adjustment made by adopting the conditions 

 of equation 1, would be rb^b; and for those of equation 5, it would 

 be j!TTjb nearly; and the accurate adjustment is a matter dejiend- 

 ing altogether upon repeated and careful trials: — whereas, if the 

 bars were divided into equal parts as suggested, the adjustment 

 would be determined at once, thus — 



100 



792 



z 

 500 



631. 

 500' 



or, 



100 

 692 



500 



72-25 

 oOO 



If the decimals were neglected, the error by adjusting to the divi- 

 sion 63, would be sTTooi'', ""•! to the division 72, it would be ttjtoo*- 

 Thus the adjustment would be more readily made, and, by the aid 

 of a vernier, it need not be necessary to neglect decimals. 



The adjustment of the three tubes in a straight line has been 

 already shown as essential to the correct working of the instru- 

 ment, and it should be made by means of a steel straightedge; the 

 drawing-board or table should be smooth and true, as a plane sur- 

 face; the Pentagraph should move in every direction with the 

 most perfect freedom; and the point of the pencil should be ex- 

 actly in the axis of the tube which contains it, — this is ascertained 

 by turning it round with the finger and thumb, and cutting the 

 point with care, until the mark it makes on paper, after so turning 

 it round, is a mathematical point, and not a diminutive circle. 

 The drawing, and the paper to receive the copy, should be pinned 

 to the table, and should be so placed that the angles of the rhom- 

 boid formed by the bars need not become very acute during any 

 part of the process. With proper attention to these points, the 

 Pentagraph will be found to be an instrument, when well made, 

 for copying drawings on the same or different scales, deserving 

 greater confidence from the draughtsman than is usually accorded 

 to it. For obvious reasons, this confidence may be more complete 

 when the copy is to be on a reduced scale. 



The drawing to be dealt with may be one of considerable size, 

 and can oidy be brought within range of the instrument in limited 

 portions at one time. To connect the various partial copies to- 

 gether, corresponding lines should be traced on each of them, as 

 well as on the drawing, of considerable length, in order that the 

 various parts may occupy their just positions, or that the drawing 

 and paper receiving the copy may be accurately shifted. 



Although it may be evident that the lines and curves traced by 

 any two of the points B, D, C, which are in motion simultaneously, 

 are precisely similar, the consideration of the curves traced or 



passed over by the other points at tlie joints or pivots of the in- 

 strument, may not be wholly speculative or without use. These 

 curves will now be discussed in a general sense. 



LetAa = Dc=ri; aD=Ac=c; cC=c'; andaB=l'. 

 Also, su](posing the fulcrum at D, take this as the origin of co- 

 ordinates, the axes being DX. DY, and B, D, C, being always in 

 the same straight line; then the co-ordinates will be expressed as 

 follows: — 



X, y, for the point C 



x', y\ for the point c 



x", y", for the point A 



tj', z' t for the point a 



V, r, for the point B 



By the properties of similar triangles — 

 y'-y y"-y' 



and. 



From equation 1, 

 From equation 2, 



c + c' 



1. 



By the properties of right-angled triangles — 



b- = x"- + y''^ 5. 



and, {y'—yy- + {x—x'y = c'- .... 0. 



Now, expanding equation 6, and making substitutions, according 

 to equations 3, 4, and 5, we obtain — 



(jr5 + y-)(e' — c) + i-(c' + c) = e'-(e' + c) + 2c'(yy" + a;x") . 7. 

 for one of the equations connecting the curves traced by the points 

 C and A. The second is derived from equation 5, by substituting 

 in it the values of x and y, given by equations 3 and 4: hence — 



(cjr + cV)-+ (cy + cy')= = 4"(e' + c)- . . 8. 



By similar reasoning we obtain for the corresponding equations 

 which connect the curves traced by B and A — 



(j)-' + 2=)(A'-4) + c°-(4' + 4) = i'-(A' + i)-24'(M'" + zy") . 9. 

 and, {lv-b'x"f-\-{b'y"-hzf^tr{b'-k-bf . . 10. 



Now these two equations are of the same form as those connect- 

 ing the curves traced by the points C and A, and they differ merely 

 in a change of sign, which obviously arises from the different posi- 

 tions of the points C and B, in reference to the axes. Hence we 

 see that any curve traced by the point B, is similar to the curve 

 traced by the point C; the ratio being obviously that of b' : 6, or 

 of c : c. 



In the same way we may arrive at equations to connect the 

 curves traced or passed over, simultaneously, by the points C, c, or 

 c. A, &c. 



To apply these equations, — let us take the 30-inch Pentagraph 

 already mentioned, and adjusting the boxes to the divisions marked 



1 2 on the bars D, B, fix the fulcrum at D, and with C trace a 



straight line which shall pass through D : it is required to deter- 

 mine the curve passed over by the point A. 



From the construction of the instrument, 



4 = 4'; (4' + 4)^(c— c') ; andc = 4'; .'.c'^b. 



Then equation 7, becomes 2b{yy"-^xx") = 0. Now, if the line 

 traced by C is assumed as the axis of x, its equation will be ^ =: 0; 

 .-.'ibxx" = 0; consequently, x"=^ ; and this being the equation of 



