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Date: 1952-1964

Inventory Number: 1997-1-1605c

Classification: Demonstration Model

Maker: Walter Balcke

Cultural Region:

Place of Origin:

City of Use:

Dimensions:

18 x 10.5 x 1.4 cm (7 1/16 x 4 1/8 x 9/16 in.)

Material:

DescriptionThe liquid equation solver consists of a white and clear plexiglass container. In the center, vertically partitioning the container into two parts, is a deck of four laminated construction paper rulers. The rulers slide into a sleeve down the center of the apparatus. Two measurement scales are marked on each side of each ruler - all the markings are hand-drawn in black ink. The rules are white with a green strip at the top.

On either side of the rule is an empty compartment of the container. The two compartments are attached by an opening near the very top such that liquid can be transferred from one side to the other. One compartment, the larger, is triangularly shaped with one outwardly extending edge. Along that edge is a white plexiglass bar with a green strip across it that serves as a handle for the apparatus. The other compartment is rectangle-shaped, thinner than the rules, with only straight edges.

There is a small round hole on the top edge of the larger compartment, presumably once used to put liquid inside the compartments. However, it has since been filled in. Some clear liquid, slightly more viscous than water, remains in the device demonstrating its function.

The device stands on two plexiglass feet. In between them is a rotating plexiglass bar that can be rotated to various positions for extra stability.

On either side of the rule is an empty compartment of the container. The two compartments are attached by an opening near the very top such that liquid can be transferred from one side to the other. One compartment, the larger, is triangularly shaped with one outwardly extending edge. Along that edge is a white plexiglass bar with a green strip across it that serves as a handle for the apparatus. The other compartment is rectangle-shaped, thinner than the rules, with only straight edges.

There is a small round hole on the top edge of the larger compartment, presumably once used to put liquid inside the compartments. However, it has since been filled in. Some clear liquid, slightly more viscous than water, remains in the device demonstrating its function.

The device stands on two plexiglass feet. In between them is a rotating plexiglass bar that can be rotated to various positions for extra stability.

In Collection(s)

Signedunsigned

FunctionWalter Balcke built and gifted many mathematical models to the Mathematics department at Harvard University. According to substantial correspondence between mathematics professors and Balcke, the models were sometimes used in classes, circulated around the department for observation, and eventually put on display in the mathematics library.

This particular object is designed to solve various algebraic equations for specific values using liquid volume relations. Each side of each rule has a scale along each edge. Users would fill one compartment to the appropriate level and then transfer the liquid to the other side. The compartments are sized such that the liquid level after the transfer would give the solution to the equation presented on the rule currently in use. The models simultaneously solve, and visually demonstrate various algebraic relations.

The scales and equations for each side of each rule are as follows. Each scale is divided into increments of various sizes by small black lines. Only certain increments are marked with a number. The descriptions below indicate how much each scale increases or decreases in a certain number of increments. That number of increments also indicates the frequency of the numerical labels. Each scale is arbitrarily assigned a number from 1 to 4 and each side arbitrarily called a or b. The equation can, of course, only be solved for the values given on the rules, and not for any arbitrary values.

1a: Solves the equation x^2 + 4x = C. C is on the left edge, beginning with 0 at the top and increasing by 40 every two increments until 400 at the bottom. x is on the right edge, beginning with 18 at the top and decreasing by 2 every two increments until -2 at the bottom.

1b: Solves the equation x^2 - 4x = C. C is on the left edge, beginning with 0 at the top and increasing by 100 every two increments until 1600 at the bottom. x is on the right edge beginning with 42 at the top and decreasing by 4 every two increments until 2 at the bottom.

2a: Solves the equation 0.32x^2 - 6.4x = C. C is along the left edge, beginning with 0 at the top and increasing by 200 every two increments until 3000 at the bottom. x is along the right edge, beginning with 110 at the top and decreasing by 10 every two increments until 10 at the bottom.

2b: Solves the equation 1.5x^2 + 3.6x = C. C is along the left edge, beginning with 0 at the top and increasing by 10 every two increments until 150 at the bottom. x is along the right edge, beginning with 8 at the top and decreasing by 1 every two increments until -1 at the bottom.

3a: Solves certain squares and square roots. N^2 is on the left edge, beginning with 0 at the top and increasing by 500 every five increments until 2500 at the bottom. N is on the right edge, beginning with 50 at the top and decreasing by 10 every five increments until 0.

3b: Solves certain squares and square roots. N^2 is on the left edge, beginning with 0 at the top and increasing by 5 every five increments until 25 at the bottom. N is on the right edge, beginning with 5 at the top and decreasing by 1 every five increments until 0 at the bottom.

4a: Solves certain squares and square roots. N^2 is on the left edge, beginning with 0 at the top and increasing by 10 every five increments until 100 at the bottom. N is on the right edge, beginning with 10 at the top and decreasing by 1 every five increments until 0 at the bottom.

4b: Solves certain squares and square roots. N^2 is on the left edge, beginning with 0 at the top and increasing by 0.1 every five increments until 1.0 at the bottom. N is on the right edge, beginning with 1.0 at the top and decreasing by 0.1 every five increments until 0 at the bottom.

This particular object is designed to solve various algebraic equations for specific values using liquid volume relations. Each side of each rule has a scale along each edge. Users would fill one compartment to the appropriate level and then transfer the liquid to the other side. The compartments are sized such that the liquid level after the transfer would give the solution to the equation presented on the rule currently in use. The models simultaneously solve, and visually demonstrate various algebraic relations.

The scales and equations for each side of each rule are as follows. Each scale is divided into increments of various sizes by small black lines. Only certain increments are marked with a number. The descriptions below indicate how much each scale increases or decreases in a certain number of increments. That number of increments also indicates the frequency of the numerical labels. Each scale is arbitrarily assigned a number from 1 to 4 and each side arbitrarily called a or b. The equation can, of course, only be solved for the values given on the rules, and not for any arbitrary values.

1a: Solves the equation x^2 + 4x = C. C is on the left edge, beginning with 0 at the top and increasing by 40 every two increments until 400 at the bottom. x is on the right edge, beginning with 18 at the top and decreasing by 2 every two increments until -2 at the bottom.

1b: Solves the equation x^2 - 4x = C. C is on the left edge, beginning with 0 at the top and increasing by 100 every two increments until 1600 at the bottom. x is on the right edge beginning with 42 at the top and decreasing by 4 every two increments until 2 at the bottom.

2a: Solves the equation 0.32x^2 - 6.4x = C. C is along the left edge, beginning with 0 at the top and increasing by 200 every two increments until 3000 at the bottom. x is along the right edge, beginning with 110 at the top and decreasing by 10 every two increments until 10 at the bottom.

2b: Solves the equation 1.5x^2 + 3.6x = C. C is along the left edge, beginning with 0 at the top and increasing by 10 every two increments until 150 at the bottom. x is along the right edge, beginning with 8 at the top and decreasing by 1 every two increments until -1 at the bottom.

3a: Solves certain squares and square roots. N^2 is on the left edge, beginning with 0 at the top and increasing by 500 every five increments until 2500 at the bottom. N is on the right edge, beginning with 50 at the top and decreasing by 10 every five increments until 0.

3b: Solves certain squares and square roots. N^2 is on the left edge, beginning with 0 at the top and increasing by 5 every five increments until 25 at the bottom. N is on the right edge, beginning with 5 at the top and decreasing by 1 every five increments until 0 at the bottom.

4a: Solves certain squares and square roots. N^2 is on the left edge, beginning with 0 at the top and increasing by 10 every five increments until 100 at the bottom. N is on the right edge, beginning with 10 at the top and decreasing by 1 every five increments until 0 at the bottom.

4b: Solves certain squares and square roots. N^2 is on the left edge, beginning with 0 at the top and increasing by 0.1 every five increments until 1.0 at the bottom. N is on the right edge, beginning with 1.0 at the top and decreasing by 0.1 every five increments until 0 at the bottom.

Curatorial RemarksThis object is clearly visible in a Polaroid photograph taken in August 1961. The photograph is labeled "Exhibit of mathematical models and devices designed and constructed by Mr. Walter H. Balcke of Winchester, Mass. Department of Mathematics, Harvard University. August 1961" (CHSI library, Lib.4927). The display was set up in the Harvard Mathematics Department Library.

ProvenanceFrom the Department of Mathematics, Harvard University.

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