Dec. 12, 1878] 



NATURE 



129 



The ir.ain bar is supported and turns on the pirot 

 earned bj the triangular weight ; its position with respect 

 to the pivot is adjusted by sliding longitudinally in the 

 box fitted with clamping screw and vernier. On vertical 

 pins at the ends of the main bar turn the two equal 

 pulleys shewn ; attached to these on their under sides are 

 small boxes also fitted with verniers for the longitudinal 

 adjustment of the two transverse bars. On the similarly 

 situated ends of the transverse bars the tracer and pencil 

 point are carried. When the permanent adjustment of 

 the instrument has been made the transverse arms are 

 parallel, and the pulleys being of equal sizes any rotation 

 given to one communicates an equal rotation to the other 

 by means the flat steel band passing tightly round both ; 

 thus the parallelism of the arms is maintained in any 

 " position. If now the temporary adjustments are so made 

 that the ratio of the two parts into which the axis of the 

 pivot divides the main beam is equal to ratio of the 

 lengths of the corresponding transverse arms measured 

 from the axes of the pulleys to the pencil and tracer, it is 

 evident that each of those latter is at the apex of a similar 

 triangle, and that the line joining them passes through 

 the axis of the main pivot. Thus the path described by 

 the pencil point is similar to that described by the tracer. 

 The graduations on the bars provides the means of 

 setting the instrument in the required ratio. In the old 

 form the distance between the axes of the pulleys was 

 divided into 200 equal parts, the graduations reading each 

 way from the centre. The transverse arms were made 

 of equal length divided into 200 parts, also reading 

 each way from the centre. For enlargement the setting 

 would be on one side of the centre in each of the three 

 bars and for reducing on the other side. In the improved 

 form shown the tracer and pencil are made interchange- 

 able, and thus the graduation on one half only of each 

 bar is required, while at the same time part of the half 

 arm, B, is dispensed with, making the instrument more 

 handy. In the figure the instrument is set for reducing. 



The setting is obtained as follows :— Let - be the ratio 



A 

 of the scales of the original and reduced plan, and x 

 the reading on the graduations, then for the similar 



triangles we have ^^ - ^ = -,oxx=\oo. ^ ~ ^. The 



100 + ;ir A A-j-a 



chief improvements introduced by the author in the con- 

 struction of the eidograph consist in making the pencil 

 and tracer interchangeable, which is a considerable sim- 

 pHfication, and the introduction of the small roller under 

 the larger arm of the main beam. The improved instru- 

 ment is stated to be capable of making a reduction down 

 to one-eighth, while the old form certainly became un- 

 manageable at anything beyond one-third. 



There is perhaps no instrument whose true value is so 

 little known in the drawing-office as Amsler's Polar 

 Planimeter. The accurate measurement of an area 

 bounded by cur\-ed or irregular lines is daily required ; 

 and although this can be effected readily and correctly by 

 the aid of the polar planimeter, it is usually laboriously 

 performed by cutting up the area into triangles whose 

 areas are separately determined, or by the measurement 

 of ordmates. 



The instrument may be described with the assistance 

 of the figure (Fig. 3). The weight retaining the pin below 

 It at a fixed pomt, forms the centre about which the more 

 distant arm revolves ; to the other extremity of this arm 

 IS pivoted a rod carrying a tracing-point at its free extre- 

 mi_ty._ A small roller is mounted on this rod so that its 

 ^s IS in a line passing through the tracer and pivot at 

 2 ends. The roller is provided with a thin projecting 

 edge and is retained in contact with the paper and free 

 torotate on its axis during any motion given to the in- 

 strament. The rotation of the roller is read off from the 

 ^ouations on its rim by means of a rernier, the number 

 « wnoie revolutions being shown on the smaU dial driven 



by a worm wheel and screw-pinion on the roller-axle. 

 Any motion on the surface of the paper that is given to 

 the point of contact of the roller is resolved into two 

 components, one at right angles to the axis of the roller 

 which is recorded by^ the dial and vernier readings, and 

 the other parallel to the axis which is a sliding of the 

 roller longitudinally, and is not recorded. To measure 

 the area inclosed by a boundary- line as shown in the 

 illustration, the tracing-point is adjusted to any point of 

 the boundary', the dial and wheel are then read off; the 

 tracing-point is then carried round the boundary-line, 

 carefully following it throughout until the starting-point is 

 again reached. The dial and roller are then read off, and 

 the difference of the readings gives the actual area in 

 square inches, or any other units for which the instrument 

 has been graduated. We may now attempt an explana- 



FlG. 3. 



tion of the principle of the instrument. Consider first the 

 motion of a straight line parallel to itself. The motion 

 of the centre of the line is compounded of a motion at 

 right angles to it, and one in the direction of its length. 

 The area passed over by the line is equal to its length 

 multiplied by the distance travelled by the centre at 

 right angles to its length. 



If, however, the line be moved, not parallel to itself, 

 but into any other position, it could have been made to 

 reach this position by first moving parallel to itself until 

 its centre reached its new position, and thus, by a rota- 

 tion of the line about its centre as a fixed point, it could 

 be made to assume the position sought. If a figure re- 

 presenting this be drawn it will be seen that, when the 

 movement is small, the area passed over by the line is 

 approximately equal to its length, multiplied by the per- 



FlG. 4. 



pendicular distance traversed by its centre, as before, 

 and that, when the movement is diminished indefinitely, 

 the area described is in the limit equal to the length mul- 

 tiplied by the perpendicular distance traversed. Now let 

 a roller be mounted at the centre of the line, so as to 

 rotate about it as axis, and let it be so graduated on the 

 rim that the length of one division, multiplied by the 

 length of the line, is equal to a unit of square measure. 

 Then, as the line passes from one position to another, 

 parallel to the plane of the paper, the roller will record 

 the sum of the separate infinitesimal movements of the 

 line at right angles to its length, and will thus, by the 

 graduations on its rim, read off, say, at the point of con- 

 tact with the paper, give the area passed over by the 

 line. Let / = the length of the line. Ax = the perpen- 

 dicular distance travelled by the roller. Thei', in the 



