geometric model, Pascal's theorem ?
Date: 1959-1960
Inventory Number: 1997-1-1620
Classification: Demonstration Model
Subject:
Maker: Walter Balcke ?
Cultural Region:
Place of Origin:
City of Use:
Dimensions:
1.5 x 35 x 27 cm (9/16 x 13 3/4 x 10 5/8 in.)
Description:
The model consists of three long thin steel pins, each with a slight hook at one end and five shorter, thicker, lighter metal pins (pins of both kinds are likely missing). There are six round brass nodes through which the pins pass. The nodes are two tiered and each tier has a hole through it. Not all holes have a pin passing through indicating the model is either missing pins, or intersections have come apart. The relative positions of the nodes and the lengths of the edges connecting them can be altered, permitting many configurations of hexagons and lines to be shown.
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 model and 1997-1-1620 are likely Balcke's representations of Pascal's Theorem and Brianchon's theorem. Neither of the models is in perfect condition any longer, with possible missing and out of place pins, missing cases, and missing accessories so it is difficult to discern which might be which. It is possible that certain of the metal pins stored in 1997-1-1637 originated from these models. The models are visual aids to understanding the geometric theorems by making visible many different cases.
Pascal's Theorem states that "if any arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration" (From the following website).
Lib.4927 in the CHSI library includes a cardboard instruction card for the model of Pascal's Theorem that states the following:
"PASCAL'S THEOREM (1639, at age sixteen) / This model shows the theorem in converse by holding in line the three intersec- / tions of extended opposite sides so that the hexagon indicates arcs of conics. / The arcs are varied by changes in the angular and lengthwise positions of the / "straight line arm" and in the length of the major side of the hexagon. In its / shorter lengths this side may be held constant by the black tab on the handle. Avoid forcing the arm. To overcome interference or binding move the arm length-wise or assist the movement of the first cylinder or other parts as required."
Brianchon's theorem, the polar reciprocal of Pascal’s' theorem, states "Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then the lines AD, BE, CF intersect at a single point" (From the following website).
Lib. 4927 in the CHSI library includes a cardboard instruction card for the model of Brianchon's Theorem that states the following:
"BRIANCHON'S CONCURRENT LINES / 'Reciprocal' of Pascal's theorem. / The hexagon, supported by the / three cross-wires, is placed around the circle or ellipse with all of / its sides tangent. / To avoid permanent bending of / elements, the wire assembly should / be adjusted to permit sufficient / folding before replacing in the case".
In a letter to Balcke, Harvard mathematics professor Joseph L. Walsh indicates that the model of Pascal's theorem is "especially simple in construction considering that it is able to show a great number of different positions; that is to say, a great number of cases of the theorem" (Lib.4927, dated January 18, 1960). Walsh also comments that the model of Brianchon's theorem "is very illuminating" (Lib.4927, dated February 29, 1960).
This model and 1997-1-1620 are likely Balcke's representations of Pascal's Theorem and Brianchon's theorem. Neither of the models is in perfect condition any longer, with possible missing and out of place pins, missing cases, and missing accessories so it is difficult to discern which might be which. It is possible that certain of the metal pins stored in 1997-1-1637 originated from these models. The models are visual aids to understanding the geometric theorems by making visible many different cases.
Pascal's Theorem states that "if any arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration" (From the following website).
Lib.4927 in the CHSI library includes a cardboard instruction card for the model of Pascal's Theorem that states the following:
"PASCAL'S THEOREM (1639, at age sixteen) / This model shows the theorem in converse by holding in line the three intersec- / tions of extended opposite sides so that the hexagon indicates arcs of conics. / The arcs are varied by changes in the angular and lengthwise positions of the / "straight line arm" and in the length of the major side of the hexagon. In its / shorter lengths this side may be held constant by the black tab on the handle. Avoid forcing the arm. To overcome interference or binding move the arm length-wise or assist the movement of the first cylinder or other parts as required."
Brianchon's theorem, the polar reciprocal of Pascal’s' theorem, states "Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then the lines AD, BE, CF intersect at a single point" (From the following website).
Lib. 4927 in the CHSI library includes a cardboard instruction card for the model of Brianchon's Theorem that states the following:
"BRIANCHON'S CONCURRENT LINES / 'Reciprocal' of Pascal's theorem. / The hexagon, supported by the / three cross-wires, is placed around the circle or ellipse with all of / its sides tangent. / To avoid permanent bending of / elements, the wire assembly should / be adjusted to permit sufficient / folding before replacing in the case".
In a letter to Balcke, Harvard mathematics professor Joseph L. Walsh indicates that the model of Pascal's theorem is "especially simple in construction considering that it is able to show a great number of different positions; that is to say, a great number of cases of the theorem" (Lib.4927, dated January 18, 1960). Walsh also comments that the model of Brianchon's theorem "is very illuminating" (Lib.4927, dated February 29, 1960).
Curatorial RemarksIt is not verified that this object was constructed by Walter Balcke for the Mathematics Department at Harvard University. However, it is constructed in the same style and same materials as other confirmed Balcke models and is very likely one of them.
ProvenanceFrom the Department of Mathematics, Harvard University.
