On a funicular solution of Buffon's “problem of the needle” in its most general form
- J. J. Sylvester
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Assisted byJames Hammond
- SeePostscriptum, p. 205.
- The case of a straight line (the original question of theneedle) may be made to fall under this rule: for the line, asBarbier 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 figuresA 1,A 2,A 3, ... 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 withany 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 ofp 3 for this case may be expressed as the difference of the two bands,atuelgdwvbxya andatqgleuwdglsvbxya (see Fig. 19), derived from the uncrossed bandabxya roundA 1,A 3 byTwisting its rectilinear portionab right roundA 2 in opposite directions.
- Imagine a string passing fromB toC, fromC toA, fromA toD, and fromD toB. This string cannot be kept tight unless fastened by pins atA, 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 suppositionsp 2 is the algebraical length of the bandBCADB 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.
- On a funicular solution of Buffon's “problem of the needle” in its most general form
Volume 14, Issue 1 , pp 185-205
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- Kluwer Academic Publishers
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- J. J. Sylvester (1)
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- 1. Oxford