1849.] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



361 



NOTES ON THE PENTAGRAPH. 



The Pentagraph is an instrument so well known, and described 

 in so many familiar works, that it may be thought nothing remains, 

 deserving of special remark, belonging to it. These notes have 

 not for their object anything new as regards its principle or con- 

 struction; but rather to present somewhat systematically, and in a 

 form that is useful in a practical sense, an explanation of its geo- 

 metrical proportions, and some more precise directions for its use, 

 than are found in modern treatises on mathematical instruments. 



If two similar triangles having their homologous sides parallel, 

 are subject to a change of position, and to the conditions that two 

 of the adjacent sides in each constantly retain the same length or 

 value, and are in all positions parallel, as they were at first — but 

 that the magnitudes of the angles, as well as of the third sides, 

 are variable; then these third sides will also be parallel, and will 

 retain the same ratio to each other in all positions of the two tri- 

 ang:les, because it is evident their similarity is preserved, although 

 their magnitude is changed. 



This, then, is the elementary principle of the Pentagraph; and 

 the instrument is so devised, as to connect two such triangles by a 

 simple arrangement of bars of brass, or of ebony, graduated in 

 such a manner, and provided with such joints and moveable parts, 

 as to adapt it to a great variety of proportions. 



Thus, confining our attention to geometrical figures, suppose 



ABC, uBD, to be two tri- 

 angles, so connected that 

 oB may always form a part 

 of AB; that aD may always 

 be parallel to AC; that the 

 angle at A, and therefore at 

 «, may have any variable 

 value; and that the three 

 points B, D, C, lie in the 

 same straight line in any 

 given position of the two 

 triangles. 



It is evident, that wh.itever value the angle at A may have, 

 B, D, C, once in a straight line will always be so, because, when 

 they are first adjusted, 



aB:AB::aD:AC; 



and, when from any change in the value of the angle A, we still 

 have the angle a equal to it, by reason of the parallelism of AC, 

 and oD, and of AB being a straight line; and nothing having 

 altered the magnitudes of aB, aD, AB, AC, the ratio already 

 established remains unchanged; therefore the two triangles are 

 still similar, and the angles aDB, ACB are equal; but the line DC, 

 connecting the parallels aD, AC, makes the alternate angles at C 

 and D equal; hence the vertical and opposite angles at D are equal, 

 and BDC is a straight line. 



The instrument consists of a bar of metal, or of wood, to em- 

 brace the three points A, «, B, of which the two former are the 

 centres of accurately-formed steel pivots, connecting it with two 

 other bars shaped so as to include a, A, C, and to attach them by 

 similar pivots to a fourth bar, of which the length should be ex- 

 actly equal to Aa, or as usually made equal to half of AB. The 

 tln-ee points B, D, and C, are made the centres of metal cylinders 

 or. tubes, which are accurately turned and bored, so as to be true 

 inside and out, and which each fit, with perfect precision, solid 

 cylinders of metal forming a tracing-point, a pencil-holder, and a 

 fulcrum about which the whole instrument may revolve with free- 

 dom and ease. 



This fulcrum may therefore be fixed at either B, D, or C; or the 

 pencil may occupy either of these three points; or the tracer may 

 fill either of the three tubes alternately with the pencil; — conse- 

 quently, there may be 1x2x3 = 6, different arrangements. 



The tube at C is permanently fixed to the bar AC, but the tubes 

 at B, and D, are attached to boxes, which slide along the bars aB, 

 BD, and may be secured in any desired position on the bars, by 

 means of clamp screws. The letters B, D, C, are usually engraved 

 on the ends of the several bars, in the same arrangement as they 

 stand in the figure, and they will be used in these notes to designate 

 the bars. 



The part aB of the bar B, as well as the bar D, are graduated 

 and figured, so as to indicate certain ratios which may be preserved 

 between the original of any system of lines or curves, as a draw- 

 ing, and its proportional copy, — the two being supposed to lie in 

 one geometrical plane : the principles upon which this is done may 

 now be developed. 



Premising that the sliding-boxes may be adjusted to any points 



of the bars aB, and D, we must consider the length of either and 

 both of these as variable. Let us represent 



The constant length of Aa = 6 

 „ „ of C = c 



The variable length of aB =: x 

 „ „ of D = 3/ 



Then, if the fulcrum is fixed at D, the tracer at C, and the pencil 

 at B, the ratio of the copy to the original, considered in point of 

 scale, will be that of the lines BD, DC. 



Let BD : DC :: n : m; then, x-\-b : x :: m-\-n : n:: c : y; 



, nb 



and .r = — ; 

 m 



y = 



m-\-n 



Suppose the drawing is to be copied to the same scale: then, 

 n^=rn; x^=b; y:^^c. 



In a 30-inch Pentagraph, by Troughton and Simms, from its 

 construction 6 =^ |c ; therefore, with this instrument, x = y =: gc; 

 and the divisions for this are figured 1 3. 



If the copy is to be ^rd the original, then m = 3ra; x = g6; and 

 y ■=. ^b. On the instrument, this division is figured 1 i. 



Now, if m : ra :: 12 : 11, then we have 



and y = 



226 



_ 11* 



^ - 12 ; -- . - 23 



The divisions on the instrument corresponding with these values 



are figured 11 12; and we should find, if^ n = ^m, that tlie 



corresponding divisions were figured 5 6; and so on. 



By changing the fulcrum to C, the tracer to B, and the pencil to 

 D; BC : DC :: m : «, becomes the ratio of the oeioinal to the 

 copy, and 



b : x-]-b :: 71 : m :: c — y : c ; 



hence, x = 



b{m—n) 



y 



c {m — n) 



n ' ~ m 



For examples : Suppose m = n, .\x = y ^0, which the construc- 

 tion of the instrument does not admit of. If n =: -^', we should 

 have .V = yV^j V = i^i corresponding to the divisions figured 



1 12. If n = -r-, then the divisions coiTe5])onding to the 



values of x and y, would be found figured 1 6. 



By extending this process to each of the six different cases, we 

 are enabled to arrange the results so as to offer facilities in apply- 

 ing the instrument, as follows. 



From the construction of the instrument, x cannot be greater 

 than b; nor can .r or y =: 0, or have negative values. 



Surfaces or areas may be enlarged or reduced in any ratio m : n, 

 by finding values of x and y from the formulse given above, but 

 substituting V???, Vn,, for m, and n, therein. 



Considering the equations here set forth, in connection with the 

 values of x and y, as measured on the instrument, it appears that 

 the divisions on the bars B, D, which have unity for one of the 

 ratios, are figured in reference to tlie position of the fulcrum at B, 

 their values being given by equations 5 and 6. 



And that, by changing the fulcrum to C, we obtain a set of ratios, 

 which consists, in any particular case, of the diffiixnce between the 



47 



