ON PLANIMETERS. 509 



older forms, which, when well made, were necessarily expensive, out of the 

 field. 



Jacob Amsler, as I have been told by the late Prof. Hesse, was at the 

 time a student at Konigsberg, where Prof. Franz Neumann encouraged 

 his students to work at the lathe and otherwise use their hands. He thus 

 was enabled to make his first instrument with his own hands in about 1854. 

 His first publication about it is ' Ueber die mechanische Bestimmung des 

 Flacheninhalts, der statischen Momente und der Tragheitsmomente ebener 

 Figuren, insbesondere iiber einen neuen Planimeter,' ' Vierteljahrsschrift 

 der naturforschenden Gesellschaft in Ziirich,' 1856 ; also in Moigno's 

 journal, 'Cosmos,' February 29, 1856. This planimeter is so well known 

 that no description is necessary beyond what has been said already about 

 planimeters of Type III. Many thousands of them have been manufactured 

 by Amsler at his works in Schaff'hausen, and though in England many 

 are sold with the name of an English firm engraved on them, practically all 

 have come from Schaffhausen. 



The instrument has practically remained unaltered since its invention. 



It is made either with a rod QT of invariable length, giving the area, 

 say, in square inches, or with a rod of which the length may be changed so 

 that the same instrument can be set to give the area in different units. 



To the latter Amsler has added on the top of the rod a pointer, and 

 another on the ' sleeve ' in which the rod slides. These are at a distance 

 equal to that between the tracer T and the joint at Q. It is thus possible 

 to set the instrument so that the length I of the rod equals the greatest 

 extension, parallel to a given line, of the area. The reading of the instru- 

 ment is then proportional to the mean height of the area perpendicular 

 to that line. This is especially useful for finding the mean pressure directly 

 from an indicator diagram. 



Mr. Druitt Halpin has added a simple locking gear to the recording 

 wheel, so that the instrument can be taken up without moving the wheel. 

 This has the advantage that the instrument can be placed in a good light 

 for reading, but it is also particularly useful in cases where the mean 

 pressure, as determined from a great number of indicator diagrams, is 

 required. The instrument may be set to zero, then locked, placed on the 

 paper, and one diagram after the other run over, the wheel being always 

 locked whilst the diagrams are being changed. The final reading divided 

 by the number of diagrams used gives the mean at once. 



For the mere facility of reading another recent improvement is of far 

 greater importance. It consists in replacing the reflecting surface of white 

 metal on which the graduation used to be engraved by the matt white of 

 celluloid with black lines on it. 



Of the many theories of Amsler's planimeter which have been given — 

 and their name is legion — most make use of the Integral Calculus. 



J. Amsler starts with geometrical considerations similar to those which 

 I have given at the beginning. With these I became originally acquainted 

 through Culmann's 'Graphische Statik.' They give, in my opinion, th& 

 quickest access to the real nature of all planimeters. 



The other theories serve as examples, and some as very good illustra- 

 tions, of the use of curvilinear co-ordinates. 



Here it may be remarked that the integration as performed by the 

 planimeter gives really a line-integral, and many of the proofs which start 

 with the evaluation of an element of the area are examples of transforming 

 an integral over an area into a line-integral over the boundary ; hence they 

 are simple examples of Stokes' Theorem. 



