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Date: circa 1906

Inventory Number: 2006-1-0002a

Classification: Planimeter

Subject:

Maker: Jakob Amsler (1823 - 1912)

Cultural Region:

Place of Origin:

Dimensions:

12 x 47 x 63 cm (4 3/4 x 18 1/2 x 24 13/16 in.)

Accessories: instrument is in four (4) parts; the rail is in own box and is inventory # 2006-1-0002b; letter from Prof. William Brower containing the complete history of ownership and use (missing)

DescriptionThe planimeter is stored in a wooden box with a lid attached by two brass hinges along the back wall of the bottom half. The outside of the box is finished and the inside of the box is unfinished. A square card is attached inside the top half of the lid with printed handling instructions. Below, another rectangular card is attached with printed specifications for each of the different tracing pens 1, 2, and 3. The various arms and intersections of the planimeter are supported inside the box by several raised, wooden blocks, each padded with black felt. There are also felt-padded wooden blocks attached inside the top half of the box to hold the instrument in place when the box is closed. There is a metal key hole and two metal hooks on the front of the bottom half of the box.

The planimeter consists of multiple interconnected brass wheels and arms. One subsection of the instrument is an isosceles right-angle triangle. There is a large, solid brass cylinder emerging from the 90-degree vertex of the triangle. There is a brass circle fixed at each of the non-90-degree vertices of the triangle. Each circle supports one of the two main silver rollers of the device. The apparatus also includes a long, straight track with a groove on one side along a long edge. When the instrument is set up, these rollers sit in the groove causing the cylinder to slide in a straight line. There is a thin brass arm also fixed at each of the non-90-degree vertices in this isosceles triangle. Those two brass arms (of equal length) are attached together with a cross bar half way and at their other end. The first cross bar is longer than the end-most cross bar such that the arms approach one another.

The arms are attached to a complex mechanism consisting of a large brass circle skeleton and three variously-sized smaller brass circles. Each of the smaller circles sits on a silver roller and is equipped with three small, white, rotating dials with numerical increments marked in black around the circumference. Two of the dials lie horizontally and overlap one another. They are incrementally marked from 0 to 9 around the circumference. The third rotates vertically and is marked incrementally from 0 to 9. On the vertical disc, each increment is further subdivided into ten with small black lines. Next to each vertical disc is a second white plastic disc or piece with ten markings equal to those separating each increment on the partner vertical disc. The ten markings are book-ended with a 0 and a 1. Two of the three smaller circles are solid brass; the third is only a brass skeleton of a half circle. All three circles have ridges around the outside of their circumferences. The large brass circle also has ridges around the outside of its circumference. The three smaller circles rotate around the larger circle with the gear mechanism of those interlocking ridges.

The three smaller brass circles are connected to each other with a brass, isosceles right-angle triangle. The two non-90-degree vertices attach to the solid brass discs and the 90-degree vertex is attached to the smaller skeleton circle. The center of the longest edge of the triangle is attached by a short extension to the center of the big skeleton circle. The triangle is fixed; meaning that in spite of their rotation around the big circle's circumference, the relative positions of the three smaller circles is fixed; they rotate together.

Atop the large brass circle is a brass extension arm, attached with a pivot and a hinge permitting it to move up and down relative to the underlying surface, as well as left and right. The left and right (horizontal) motions of this arm create a counter-clockwise and clockwise rotation of the three small circles interlocking the main circle. There is a removable tracing pen extension attached to the extension arm. The instrument is equipped with three, interchangeable tracing arms. There is a hole into which a tracing arm is inserted and a brass screw on the top that, when tightened, holds the arm in place. On the brass arm extension, prior to the tracing arm connection screw, is a small mechanism consisting of two white dials with numerical increments marked in black and a silver roller to support the movements of the arm. One of the white discs is horizontal and marked incrementally from 0 to 9 around the circumference. The other is vertical and marked incrementally from 0 to 9 around the circumference with each increment further subdivided in ten by small black lines. Next to the vertical disc is a second white plastic disc with ten markings equal to those separating each increment on the vertical disc. The ten markings are book-ended with a 0 and a 1.

For supplementary images of a similar model and mathematical explanation of the Amsler Planimeter, click here.

The planimeter consists of multiple interconnected brass wheels and arms. One subsection of the instrument is an isosceles right-angle triangle. There is a large, solid brass cylinder emerging from the 90-degree vertex of the triangle. There is a brass circle fixed at each of the non-90-degree vertices of the triangle. Each circle supports one of the two main silver rollers of the device. The apparatus also includes a long, straight track with a groove on one side along a long edge. When the instrument is set up, these rollers sit in the groove causing the cylinder to slide in a straight line. There is a thin brass arm also fixed at each of the non-90-degree vertices in this isosceles triangle. Those two brass arms (of equal length) are attached together with a cross bar half way and at their other end. The first cross bar is longer than the end-most cross bar such that the arms approach one another.

The arms are attached to a complex mechanism consisting of a large brass circle skeleton and three variously-sized smaller brass circles. Each of the smaller circles sits on a silver roller and is equipped with three small, white, rotating dials with numerical increments marked in black around the circumference. Two of the dials lie horizontally and overlap one another. They are incrementally marked from 0 to 9 around the circumference. The third rotates vertically and is marked incrementally from 0 to 9. On the vertical disc, each increment is further subdivided into ten with small black lines. Next to each vertical disc is a second white plastic disc or piece with ten markings equal to those separating each increment on the partner vertical disc. The ten markings are book-ended with a 0 and a 1. Two of the three smaller circles are solid brass; the third is only a brass skeleton of a half circle. All three circles have ridges around the outside of their circumferences. The large brass circle also has ridges around the outside of its circumference. The three smaller circles rotate around the larger circle with the gear mechanism of those interlocking ridges.

The three smaller brass circles are connected to each other with a brass, isosceles right-angle triangle. The two non-90-degree vertices attach to the solid brass discs and the 90-degree vertex is attached to the smaller skeleton circle. The center of the longest edge of the triangle is attached by a short extension to the center of the big skeleton circle. The triangle is fixed; meaning that in spite of their rotation around the big circle's circumference, the relative positions of the three smaller circles is fixed; they rotate together.

Atop the large brass circle is a brass extension arm, attached with a pivot and a hinge permitting it to move up and down relative to the underlying surface, as well as left and right. The left and right (horizontal) motions of this arm create a counter-clockwise and clockwise rotation of the three small circles interlocking the main circle. There is a removable tracing pen extension attached to the extension arm. The instrument is equipped with three, interchangeable tracing arms. There is a hole into which a tracing arm is inserted and a brass screw on the top that, when tightened, holds the arm in place. On the brass arm extension, prior to the tracing arm connection screw, is a small mechanism consisting of two white dials with numerical increments marked in black and a silver roller to support the movements of the arm. One of the white discs is horizontal and marked incrementally from 0 to 9 around the circumference. The other is vertical and marked incrementally from 0 to 9 around the circumference with each increment further subdivided in ten by small black lines. Next to the vertical disc is a second white plastic disc with ten markings equal to those separating each increment on the vertical disc. The ten markings are book-ended with a 0 and a 1.

For supplementary images of a similar model and mathematical explanation of the Amsler Planimeter, click here.

Signedin black script on the apparatus: J. Amsler

Inscribedin black lettering on the apparatus: No 1124

FunctionThe planimeter is used to determine the area of an arbitrary, two-dimensional figure: it gives the area between the graphically defined curve and a horizontal x-axis. Users manually trace the figure in question with the attached tracing arm. The tracing motion is communicated through the device such that the large, solid brass cylinder will slide along the track. The distance that the cylinder has moved when the tracing is complete (i.e. when the tracer has returned to the point from which she started), is proportional to the area inside the figure. This motion transfer is guaranteed by the trigonometric relations of the main mechanism consisting of the large circle, and the three connected smaller circle. The white plastic discs attached to the tracing arm are used to record the area. This instrument, unlike some of its close relatives, does not have a drawing mechanism.

Instruments of this kind were later referred to as 'moment planimeters' because the integrals of f(x)^2 and f(x)^3 give respectively the static moment and the moment of inertia for the figure's area.

For supplementary images of a similar model and mathematical explanation of the Amsler Planimeter, click here.

For a detailed description of how the rolling sphere planimeter measures area, see the article in the Primary Sources.

Instruments of this kind were later referred to as 'moment planimeters' because the integrals of f(x)^2 and f(x)^3 give respectively the static moment and the moment of inertia for the figure's area.

For supplementary images of a similar model and mathematical explanation of the Amsler Planimeter, click here.

For a detailed description of how the rolling sphere planimeter measures area, see the article in the Primary Sources.

Curatorial RemarksA letter from Prof. William Brower containing the complete history of ownership and use is listed in accessories. The letter is currently missing.

Primary SourcesJ.C. Maxwell, "Descriptions of a New Form of Planometer, an Instrument for Measuring the Areas of Plane Figures Drawn on Paper," in *The Scientific Papers of James Clerk Maxwell, Vol. 1*, ed. W. D. Niven (Mineola, NY: Dover Publications, 2003), 230-237. [Reprint of original 1890 publication by Cambridge University Press.]

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