410 REPORT— 1872. 



pass along the horizontal line 1 2, it will unwind that proportion of the 

 revolution, put in on going from C to D, that is represented by the length of 

 the line 1 2. If, now, it be made to rise from 2 to 3, it wiU have a " rate " of 

 revolution equal to the length of the line C D — 1 2, and a " quantity " equal 

 to the height of the line 2 3. If it now be carried along the horizontal line 3 4, 

 another portion of the revolution given by C D will be taken out ; and then 

 if it be made to rise from 4 to 5, a further portion of a revolution will be put in, 

 having for its " rate " the length of the line C D — D 4, and for its " quantity " 

 the height of the line 4 5. This may be followed through aU the steps into 

 which the hypotenuse has been broken up, and then it wiU be found, as is 

 obvious, that the sum of all the horizontal lines 1 2, 3 4, 5 6, &c. is equal 

 to the length C D, and that the travcx'siug of them will therefore have 

 unwound all the revolution that the passage along C D had put into the 

 wheel E. ; but it will also be found that the sum of all the vertical Uncs 

 2 3, 4 5, 6 7, &c. is equal to D A ; and therefore the " quantity " of revolu- 

 tion given to the wheel H will be equal to that which it would have had, had 

 it passed up the line D A, while the means of the lengths of C D — 1 2, C D — 

 D 4, C D — D 6, &c. will exactly equal the half of C D, and thus the condition of 

 the wheel E, in relation to the index S will, when it arrives by the zigzag path 

 at B, be ijrecisely the same as it would have been if it had gone by the way of 

 the rectangle C C C" B, C C being half of C D. A large number of very small 

 steps have been taken in lieu of the straight line hypotenuse D B. Obviously 

 a greater number of much smaller steps, or an infinite number of infinitely 

 little steps, may be substituted, until the traverse ceases to be made along steps 

 at all, and becomes one along the slope line D B, in which condition of things 

 the wheel R at any part of the traverse of the tracer along the hypotenuse 

 is making a revolution compounded of the "rate" due to its horizontal distance 

 from C, and of a " quantity " equal to the rise from D. The " quantity " re- 

 mains constant during the whole journey, but the "rate" regularly diminishes, 

 and the mean of all the "rates" is that due to the proportion that half the 

 length of the line C D bears to N T, the length of Q. 



Now if it has been proved that this elementary planimeter, no matter 

 where anchored, can act efficiently in ascertaining the area of rectangles and 

 of triangles, it is self-evident that it could truly ascertain the area of any other 

 figure, because tliere is no figure from that of the regular circle to that of the 

 most irregular boundary which cannot be represented by an indefinite number 

 of straight lines lying at various angles — that is to say, a circle is only a polygon 

 of an infinite number of sides, all equal ; and any irregular figure may be 

 divided into an indefinite number of sides, most probably unequal. 



It may now be said that the elementary planimeter has been shown to have 

 its pivot N attached to the gnide-block M working up and down in the straight 

 groove 0, that that gi'oove has been sketched with its axis either in the pro- 

 longation of B C or in a position parallel to B C, whereas in the actual 

 planimeter there is no such straight groove at all ; but the pivot N is at the 

 end of a radius rod, which in its movement causes N to pass through the arc 

 of a circle, and that that arc may have its chord in almost any position in 

 relation to the line B C, and thus there are disturbing causes in the planimeter 

 as manufactured which do not exist in the elementary planimeter. The 

 answer to this objection, which at first sight appears so well-grounded a one, 

 is that these differences between the real and the elementary planimeter may 

 be left out of consideration altogether, as they really have no eflect whatever 

 upon the action of the implement. This can be made clear in a very fe^v 

 words. 



