Acta Mathematica

On a funicular solution of Buffon's “problem of the needle” in its most general form

J. J. Sylvester

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Assisted by James Hammond

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Acta Math., Volume 14 (1890), 185-205.

First available in Project Euclid: 31 January 2017

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1890 © F. & G. Beijer


Sylvester, J. J. On a funicular solution of Buffon's “problem of the needle” in its most general form. Acta Math. 14 (1890), 185--205. doi:10.1007/BF02413320.

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  • See Postscriptum, p. 205.
  • The case of a straight line (the original question of the needle) may be made to fall under this rule: for the line, as Barbier has observed, may be regarded as an indefinitely narrow ellipse or other oval.
  • It may be well to draw at once attention to the fact that different systems of straight lines do not necessarily cut the figures A1, A2, A3, ... in the same order; as ex. gr. if three circles touch, or so nearly touch one another that each blocks the channel between the other two, straight lines may be drawn whose intersections with any one of the three shall be intermediate to their intersections with the other two.
  • This circumstance enables us to discuss Ba. 1 and Ba. 2 simultaneously.
  • By an easy rearrangement of the bands the value of p3 for this case may be expressed as the difference of the two bands, atuelgdwvbxya and atqgleuwdglsvbxya (see Fig. 19), derived from the uncrossed band abxya round A1, A3 by Twisting its rectilinear portion ab right round A2 in opposite directions.
  • Imagine a string passing from B to C, from C to A, from A to D, and from D to B. This string cannot be kept tight unless fastened by pins at A, B, C, D. Inserting the necessary pins and tightening the string, we agree to consider the consecutive portions of the string as alternately positive and negative. On these suppositions p2 is the algebraical length of the band BCADB stretched round the pins. The method of representation by means of pinned bands may be extended to the case of two (or any number of) general figures.