Description of a Platometer, an Instrument for Measuring the Areas of Figures drawn on Paper. Invented by JOHN SANG, Kirkcaldy.* (With a Plate.)

The instrument consists of a cone C (fig. 1), on the axis of which are fixed two wheels B B, which roll over the paper. The wheels are of equal diameter, so that the cone is carried forward or backward in a straight line, as is also the frame A, in which the axis of the cone is geared. The frame A

has its edges fitted to receive the friction rollers R R R R, which carry another frame F in a right or left direction parallel to a side of the cone. The light frame F carries with it an index-wheel I, the circumference of which is kept in contact with the cone by means of springs or weights, and it also carries the tracing point P. When the tracing point is moved to the right or left, the index-wheel slides along the surface of the cone without revolving. When the tracer is moved forwards or backwards, the wheels roll over the paper, and the rotatory motion which they give to the cone is imparted to the index-wheel. When the tracer is moved in any intermediate direction, the index-wheel receives both a rotatory and lateral motion.

* Read before the Society, and Instrument exhibited, 12th January 1852. VOL. IV.

It will be seen that, from this construction, if the point P be conducted completely round the boundary of any figure, the motion which the index-wheel receives will be proportional to the area of that figure.

Fig. 2.

A

Let A B (Fig. 2) be a very narrow section of the figure, having its sides parallel to the sides of the cone, and let a b be the points at which the index-wheel touches the cone when the tracer is at A and B; also, let C be the point at which the tracer would be found if the index-wheel were brought to the apex of the cone c. The distance AB a b and BC = bc. But while the tracer moves over the short line A, the cone makes an angular motion proportional to the length of that line, and, at the same time, the circumference of the cone at a moves through a distance proportional to that angular motion multiplied by its diameter at that place, which again is proportional to A C, the distance of a from the apex of the If S were the breadth of the section A B, the motion of the circumference of the cone at a, and hence of the indexwheel, would be proportional to A Cx S. In like manner, when the tracer, in making its circuit round the figure, moves over the short line B, the motion of the index-wheel would be proportional to BC x S. That motion, however, would be in a contrary direction, so that the sum of the two would be

cone.

A CXS-BCS, which is equal to A Bx S. But A B× S is also the area of the section A B. The same thing would hold, however narrow the sections were conceived to be, for every other section of the figure; and, consequently, the motion of the index-wheel gives a quantity proportional to the whole area when the tracer is carried completely round the boundary of the figure. The divisions of the index-wheel are so arranged, that when the section A B is one square inch, the wheel indicates one.

From this, the details of the machine will be easily understood. The index-wheel must move along the line in which a plane, parallel to the paper, and passing through the axis, intersects the surface of the cone. Hence, the wheel is placed on a third light frame ƒ, provided with adjusting screws (Figs. 4, 5, 6), by means of which it can be brought into that position; and the diameter of the wheel is such, that a whole revolution indicates an aliquot part of 100. In the case of instruments Nos. 1 and 3, a revolution values 20 inches, in No. 2, 10 inches. In order to carry on the indications to 100 square inches, a second index-wheel T, of the requisite diameter, receives motion from I. The reading-glass R is placed, always ready, over the vernier of the divisions of the wheel. The handle H is attached to the instrument by means of a universal joint, in order to assimilate the effort necessary to move the tracing point, as much as possible, to that required to guide a drawing-pen. The weights of the parts are also so arranged that the tracer just slightly presses on the paper without scratching it.

As the sides of the figures which the instrument is intended to measure do not always lie in the direction A a, or A B, but in any direction taken at hazard, the index-wheel, while it is receiving its motion from the cone, may, at the same time, be continually altering its distance from the apex. It might be expected that the sliding along the cone would prevent the wheel receiving the whole rotatory motion due to it. But, on a little consideration, it will appear that the fineness of the divisions of the index-wheel does not in the least depend on the diameter of that wheel, nor on the obliquity of the cone, but solely on the distance between the apex of the