MATHEMATICAL INSTRUMENTS. The term "mathe matical instruments" in its widest significance includes various instruments used in drawing, surveying, astronomy, etc. We will here consider certain instruments designed to perform operations involving computation and measurement. Instruments and machines concerned with the mechanical performance of addition, subtraction, multiplication, division, etc., are described under CALCULATING MACHINES.
Instruments for Solving Equations.—Many instruments have been designed for the mechanical solution of algebraical equations. These are in general more remarkable for the in genuity displayed in their design than for the practical value of the results obtainable. No instrument of this type has been brought into extensive use, but a few of them have found a limited application in cases where a considerable number of roughly approximate results are required. (See the works of Jacob, Horsburgh and Baxandall given in the bibliography.) Planimeters.—The invention in 1814 of the first instrument for directly measuring an area bounded by an irregular curve is attributed to the Bavarian engineer, J. H. Hermann. The instru ment was improved by Lammle in 1816, and actually constructed in 1817. Tito Gonella of Florence in 1824 invented independently a similar instrument. It embodied a recording wheel which rolled on the surface of a cone, the angular motion of the wheel relative to that of the cone varying with the distance of the wheel from the apex of the cone. The position of the wheel on the cone was made to vary according to the length of the ordinate of the curve, thus the total angular rotation of the recording wheel gave the measure of area. Gonella soon afterwards replaced the cone by a disc.
The Swiss engineer Oppikofer invented in 1826 a planimeter which was similar to Gonella's first type (wheel and cone). This was first made successfully by Ernst, who improved it and made it for general sale. In 1849, Wetli of Zurich independently invented the disc type of planimeter adapted for both positive and negative co-ordinates, and the instrument was made by Starke of Vienna. The example shown in P1. I. fig. I, which was constructed about 1860, is engraved :—"Patent von Wetli & Starke, No. 103." It consists of a rotatable horizontal circular disc
with a specially prepared fine upper surface on which the register ing roller rests. The disc is mounted on a frame supported by three grooved wheels, which can roll on three parallel rails.
Beneath the disc and mounted on the frame is a horizontal rod held between two pairs of guide rollers so that it can move in a direction at right angles to the rails. By means of a thin wire wound round the axle of the disc and attached to the ends of the rod the disc is given an angular movement proportional to the longitudinal displacement of the rod.
An upright frame screwed to the other end of the base-plate carries one end of the axle and the two divided circles which record the rotation of the registering wheel. Pivoted to this upright is a light frame carrying the registering wheel at its other end, which can be raised or lowered by means of a milled headed screw; when the frame is lowered the wheel rests with a constant pressure on the disc. As the tracing point attached to one end of the rod is guided along the curve whose area is to be measured, the distance between the centre of the disc and the plane of the registering wheel is always proportional to the ordinate of the curve. The number of revolutions of the regis tering wheel gives therefore a measure of the area.
Hansen of Seeberg suggested improvements, which were em bodied in the instrument made by Ausfeld, known as the Wetli Hansen planimeter. John Sang of Kirkcaldy invented and made in 1851 a "planometer" of the wheel and cone type, which re sembled that made by Ernst. An example of Sang's instrument is shown in Pl. I., fig. 2.
It will be seen that the revolving motion of the index-wheel is in proportion to the motion of the tracer up or down the paper, multiplied by the right and left distance of the wheel from the apex of the cone; and therefore, when the tracer is made to describe any complete perimeter, the whole rotatory motion of the index wheel represents the algebraic sum of the products of ordinates to every point in that perimeter, multiplied by the increment of their co-ordinates; thus it is a measure of the included space.