402 REPORT— 1872. 



The circumference of the wheel R is " divided," and it works against a vernier 

 at 2/; the horizontal wheel s gives ♦' tens" in square inches, the larger divisions 

 on the travelling wheel E "units," the smaller divisions on that wheel 

 " tenths," and the vernier " hundredths " of square inches. All that has to 

 be done for ascertainiug an area is to read the indices after the machine is 

 anchored and the tracer is put to the starting-point; but before it is started, to 

 book the reading, to re-read after the circuit of the figure has been made, and 

 then to deduct the first reading from the second ; the remainder gives the 

 area (in square inches and decimals) of the particular figure. 



The foregoing being, briefly stated, the construction, the manner of using, 

 and the result of that using of the planimeter, it now remains to endeavour 

 to show, as intelligibly as possible, why it is that such an implement, by 

 merely foUowtng the boundary of a figure, shoiild give with absolute 

 accuracy the area of that figure. 



Such a proposition at first sight appears to involve an impossibility. One 

 is in the habit of saying, and of most truly saying, that there is no fixed 

 relation between perimeter and area; and of saying, moreover (and also truly), 

 that not only is this the fact when areas of great irregularity are dealt with, 

 but, as regards direct proportion, it is also the fact when the most regular 

 figures (figures in all respects the same, except in their actual size) are under 

 consideration; for it is as true that the circumferences of perfectly regular 

 figures like circles bear no more fixed direct proportion to the areas of those 

 circles, unless the exact size he known, as it is true that the coast-line of 

 Norway, indented with its deep fjords, bears no more relation to the area 

 of that romantic country than the perimeter of a prosaic rectangular portion 

 of the United States bears to the square miles of prairie contained within it. 

 These things being so, it does, as has already been said, seem at first sight 

 absurd to endeavour to obtain from the traverse of a perimeter, be that 

 perimeter the most regular imaginable (and if possible stiU more absurd when 

 that perimeter may be the most irregular imaginable), the correct area con- 

 tained within it, not merely in terms of the perimeter, but in a definite stan- 

 dard measurement, such as square inches. 



As a preliminary to the investigation of the action of an elementary plani- 

 meter, let the results of the moving of a plain eyhuder in contact with a flat 

 surface, and under certain varying conditions, be considered. 



Assume a cylinder, as A in fig. 2, and that it is intended to move that 

 cylinder parallel with itself in the direction shown by the arrow, over the 

 length .V y. The cylinder may be (1st) at right angles to the direction in 

 which it is to be traversed, as in A A'. If under these circumstances the 

 cylinder be moved from x io y and brought into the position as doited at A\ 

 the motion will be entirely one of rolling, without any sliding whatever ; and 

 if there were upon the surface a trace {x y) of ink capable of making a mark 

 upon the cylinder, there woiild be found circumferentially upon it, when it 

 had reached the new position, a line, the length of which would be equal to 

 X y. (2nd) The cylinder may be placed with its axis parallel to the direction 

 of motion, as at A A" ; then no roUing action would take place, but the cylinder 

 would simply slide endways upon the surface. The cylinder would, however, 

 still bear upon it the trace x y, equal in length to the distance it had moved 

 through, but that trace would be obviously a mere straight line in the direction 

 of the axis of the cylinder. (3rd) The cylinder may be in a position inter- 

 mediate between that of A A' and A A'"' ; that is to say, may be neither at 

 right angles to the line of motion, as in A A', nor parallel with the line of 

 motion, as in A A^ but at an angle therewith, as in AA^ In this instance, 



