296 



NATURE 



[January 26, 1905 



Polar Plotting Paper. 



May I be allowed to direct the attention of all interested 

 in mathematical teaching in our schools and colleges to 

 the polar plotting paper recently prepared by Mr. Ellice 

 Horsburgh, lecturer on technical mathematics in Edin- 

 burgh University? 



The special feature of this paper is that it is ruled 

 radially with lines which subdivide the region about a 

 point into aliquot parts of a radian. There are two forms 

 of sheets now in the market. In one the origin is at the 

 centre, and the radial subdivision is carried right round 

 through four right angles. In the other, a reduced copy 

 of which is here reproduced, the origin is taken near one 

 corner, and the graduation is carried through a little more 

 than a quadrant. Dotted radial lines show the backward 

 continuation of the axis from which the radians are 

 measured, and also the axis perpendicular to it. These 

 dotted lines do not, of course, belong to the system of 

 lines dividing the region into aliquot parts of a radian. 



The radius of the fiftieth orthogonal circle is taken as 

 the unit, and on the margins just outside the proper 

 radian subdivisions small radial lines are drawn giving 

 the usual division into degrees. The two circles drawn, 

 the one on the axis as diameter and the other on the 

 dotted perpendicular of unit length, serve to give by in- 



FlG 



spection the sines and cosines of the angles given in 

 radians. 



Thus the paper contains on its own surface the means 

 of plotting with great ease the polar equations of curves 

 involving radians, sines and cosines, and a little calculation 

 will enable the student to take account of other functions. 



The first important use in the hands of the student is 

 obviously to get a clear idea of the radian as the true 

 scientific measure of angle; but a great many other im- 

 portant uses will at once occur to the teacher of practical 

 mathematics, such, for example, as finding reciprocals, 

 geometric means, mean proportionals, fourth proportionals, 

 squares, square-roots, &c. 



Another use is the evaluation of the integrals , r-dB and 

 JrdB. The former is got by simply counting the elements 



included in the area, and the latter by multiplying the 

 total angle between the initial and final radius by the 

 mean radius, the value of which may be obtained by a 

 method similar to Simpson's rule. 



From these few statements and indications the purpose 

 of Mr. Horsburgh 's patent will be readily appreciated. 

 It is doubtful if the average student, taught along the 

 usual lines, ever gets an accurate working knowledge of 

 the radian or circular measure of an angle, indispensable 

 though that is for all higher trigonometrical and analytical 



work. A few hours' systematised exercise with the polar 

 paper will do more than days of arithmetical transform- 

 ations in the usual academic stvle. C. G. Knott. 



Lissajous's Figures by Tank Oscillation. 



The oscillation of a rectangular water basin may be 

 utilised for the illustration of the composition of two 

 simple harmonic motions in two directions, perpendicular 

 to each other. 



\ light pendulum was constructed of a thin aluminium 

 rod, R (Fig. i), 10 cm. long. The bob B was made of a 

 disc of wood. On the upper end of the rod a light mirror 

 M was attached. The rod could be supported at any 

 desired point by a small gimbal G, so that the rod could 

 oscillate as a spherical pendu- 

 lum. A small brass weight 

 w was attached to adjust the 

 period of oscillation by raising 

 or lowering it to a proper 

 position. 



The bob is sunk into the 

 middle part of a suitable 

 rectangular basin, filled with 

 water to a proper depth. If 

 the basin be tilted suddenly, 

 and then let stand, the water 

 is set in an oscillation which 

 consists of two simple har- 

 monic motions in perpen- 

 dicular directions, the ratio of 

 the periods varying as the 

 ratio of the corresponding 

 sides. The amplitudes of two 

 component oscillations may be 

 varied at pleasure. If the 

 natural period of the pendu- 

 lum is considerably shorter 

 than that of the basin, the 

 bob follows very nearly the 

 motion of water, as judged by 

 the motion of fine dust par- 

 ticles suspended in water. 

 Now, if a beam of strong 

 sun-light be sent as shown in 

 the figure, the motion is projected on the ceiling of the 

 room. 



I have also obtained a photographic record of the motion 

 of a small bead attached to the upper end of a small 

 needle erected on the rod. By making the illumination 



r.j. 1 



intermittent by means of a perforated rotating disc, the 

 difference of velocities at different phases may be shown. 



The motion of a kaleidophone may be projected in a 

 similar manner. T. Terada. 



Physical Laboratory, Tokyo, December 19, 1904. 



NO. 1839, VOL. 71] 



