- These solutions were reprinted in The Mathematical Questions -proposed in the Ladies' Diary… by Thomas Leybourn, London, 1817, volume 1, pp. 280–282; The Diarian Miscellany… by Cha. Hutton, London, 1775, volume 2, pp. 104–106. One of the solutions was reprinted in The Diarian Repository, or Mathematical Register… by a Society of Mathematicians, London, 1774, pp. 359–360.
- Or AA' bisected at B; with this given Lambert showed, in 1774, that a parallel to AB can with ruler alone, readily be drawn through any point.
- Histoire de l'Académie Royale des Sciences, 1732. Paris, 1735, Memoires, pp. 1–14.
- Idem, pp. 15–16.
- The following sentences taken from different parts of the pamphlet represent fairly well the position of the author:
- “This curve of pursuit problem has estranged old friends and vexed eminent mathematicians. Many wagers have been made which professors of mathematics have been called on to settle, and their decisions have been in the negative [that is, the dog will not catch the duck], without one single line of proof to sustain their findings. A bare assertion is not satisfactory proof.”…
- “The ‘nevers’ have claimed that the pursuer's position [path] is always tangent to a straight line drawn to the corresponding position of the pursued. This is the modern theory and also a false one. Tracing the dog's path by setting off the circumference into short spaces, and setting off equal distances on the line from the dog to the duck [in its successive positions] will show that the pursuer is always on a straight line drawn from his original position to the pursued, and that the duck is caught when it has moved over a portion of the circumference equal to the diameter.”
- This idea is virtually the same as the idea, which Professor Hathaway uses (in his solution of problem 2801), of a system of polar coordinates moving with the duck.
- See the “turn-table” problem, Math. Visitor, 1878, vol. 1, p. 37; also Math. Quests. Educ. T., 1889, vol. 51, p. 157.
- We follow Newton's definition. “Differentials are corresponding limits of equimultiples of vanishing differences.” Also, “instantaneous state” is mathematically defined as that variation to which differentials are increments, just as in actual variation differences are increments. It is demonstrated, not assumed, that if a point be generating a curve, its instantaneous state generates the tangent, etc. Taking differences from a succeeding position, Q'P' = r + Δr(Fg. 1) unwrap an inextensible string QQ'P', from the arc QQ', describing the arc P'R' always normal to the string.
- are PP' = Δs, PR' = Δr + ΔS, arc R'P' = qΔΨ, r + Δr < q < r + Δr + ΔS.
- Ext end P R' and PP' in the ratio N: 1 so large that N·PP' is as near as we please to PT, when Q'P' is as near as we please to QP. The differences and figure PR'P' are vanishing, but the finite figure similar to PR'P', approaches a limit, namely, the right triangle PRT Hence the above differential values, by definition of differentials. See Science, July 11, 1919, and Feb 13, Mar. 28, May 7, June 11 July 9, 1920, for correspondence on the early history and concepts of the calculus
- This curve, so named by M. Simon (Analytische Geometrie, Leipzig, 1900, p. 316), and called also the four-leaved rose curve, and the quadrifolium, is one of the class of rose curves or rhodonees, r = a sin mθ, r = a cos mθ, discussed by Guido Grandi in letters to Leibnitz, 1713 in Philosophical Transactions 1723, and in his book, Flores geometrici ex rhodonearum et clœiarum curvarum descriptions resultantes, 1728. These curves have been also called corollae by W. J. C. Miller (Math. Questions with their Solutions, London, vol. 19, 1873 p. 60) on account of their fanciful resemblance to the petals of an open flower. It was remarked empirically m 1752 by G. B. Suardi, that the corollae can all be generated as elongated or contracted epi- or hypocycloids but the proof of this fact was first given in 1844 by L. Ridolfi. Bellavitis showed in 1852 that the pedal curve o any ordinary epi- or hypocycloid with respect to the center of its fixed circle is a rose curve; in particular the four-leaved clover is the pedal curve of the so-called four-cusped hypocycloid or asteroid. It is also the orthoptic curve of the astroid (Lambiotte, 1877) and the inverse of the circular cross curve (Kreuzcurve)—concerning which considerable has been written since it was first conceived by Terquem in 1847.—Editor.
- In the leaf of the third quadrant, for all values of k, entrances are below, exits above, the ray of angle θ given by sin θ = - 2/(k + √k2 + 8) [180° to 225°], showing a drift round behind the duck.
The American Mathematical Monthly
Volume 28, 1921 - Issue 2