It may be further explained that the sliding of the scale cylinder until the new number is opposite the index point really involves two operations: one sliding it till the end of the scale is opposite the index point—which subtracts the logarithm of the divisor; and the other sliding it till the next multiplier is opposite the index point—which adds its logarithm to the previous result. Hence, when the operations end with division the scale cylinder must be moved till the end of the scale is opposite the index point. The second scientific instrument to be noticed is the Polar Planimeter, invented by M. J. Amsler-Laffon, Schaffhausen, Switzerland, the object of which is to find the area of any figure by simply tracing the outline with a pointer, the instrument—of which the pointer is a part—doing all the rest; the results read off from it having to undergo only a very simple and elementary calculation to attain the desired result. [Illustration: FIG. 25. AMSLER’S POLAR PLANIMETER—(FIXED SCALE).] Planimeters are made of several forms, the two kinds illustrated by Figs. 25 and 26 being the most usual.[34] The planimeter shown by Fig. 25 represents the instrument as made to one scale only, for square inches of actual measurement. By its means the areas of, say, cross sections of ship’s hull can be ascertained in an extremely short time and with almost perfect accuracy, the readings taken from the instrument having simply to be multiplied by a multiplier consisting of the square of the number of units to the inch, corresponding to the scale on which the sections are drawn, as 4 for ½-inch scale, 16 for ¼-inch, 64 for ⅛-inch, etc. [Illustration: FIG. 26. AMSLER’S PLANIMETER—(VARIOUS SCALES).] The Planimeter shown by Fig. 26 is the instrument in a form adaptable to various scales, but does not possess any very marked advantages over the simpler form for the purposes of the naval architect or marine engineer, so that notice of it must be brief. In this form of the instrument the unit can be changed by altering the length of the arm which carries the tracer to any of the scales for which the instrument may be made available, and which are found divided upon the variable arm. The scales which are usually provided for are as follows:— 10 sq. in. = 10 square inches } 0·1 sq. f. = 0·1 square foot } 1 sq. dcm. = one square decimetre } Every total 0·5 sq. dcm. = 0·5 square decimetre } rotation of 2000 sq. m. } = 2000 square metres on a } the roller. 1 : 500 } scale 1 : 500 } 1000 sq. m. } = 1000 square metres } 1 : 500 } scale 1 : 500 }