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History of the Nine-point Circle

Published online by Cambridge University Press:  20 January 2009

J. S. Mackay
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The earliest author to whom the discovery of the nine-pointcircle has been attributed is Euler, but no one has ever given a reference to any passage in Euler's writings where the characteristic property of this circle is either stated or implied. The attribution to Euler is simply a mistake, and the origin of the mistake may, I think, be explained. It is worth while doing this, in order that subsequent investigators may be spared the labour and chagrin of a fruitless search through Euler's numerous writings.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 11 , February 1892 , pp. 19 - 57
Copyright
Copyright © Edinburgh Mathematical Society 1892

References

page 19 note * An abstract of Euler's paper will lie found in the Proceedings of the Edinburgh Mathematical Society, IV. 51–55 (1886).

page 22 note * Republished in Steiner's Gesammelte Werke, I. 191–210 (1881).

page 22 note † Republished in Steiner's Gesammelte Werke I. 489–492.

page 24 note * I am in possession of a collection of printed mathematical papers which belonged to Wilkinson. The paper of Davies's referred to is imperfect, but is completed in Wilkinson's handwriting.

page 25 note * Quarterly Journal of Mathematics, IV. 152–4 (1861).Google Scholar

page 26 note * Might it not be that Dr Salmon had forgotten it and rediscovered it? This conjecture is made because Mr J. J. Robinson begins an article in the Diary for 1858, p. 88, by saying : “My best thanks are due to the Rev. George Salinon of Trinity College, Dublin, for having called my attention to two errors which somehow have crept into my former paper,” that is, the paper in which Feuerbach's theorem was extended.

page 30 note * One of the principal features of this simplification has been given by Mr R. D. Bohannan in the Annals of Mathematics, I, 112 (1884).

page 32 note * This simplification is given in Dr Th. Spieker's Lehrbuch der ebenen Geometrie, 15th ed., p. 216, or § 220 (1881).

page 38 note * Hence this theorem is easily inferred :

If on the circumference of the circle whose centre is O, four points A, B, C, D be taken arbitrarily, these four points will be, three and three, the vertices of four inscribed triangles to which will correspond four H points, four M points, and four G points. Now, these four points of each kind will belong to one circle whose radius will be equal to that of the given circle for the four H points, half of this radius for the four M points, and one third of it for the four G points. Besides, the centres of these three new circles will be with the point O harmonically situated on one straight line, as are the four points H, M, G, O; in such a way that the centre 0 will be the common centre of similitude of these three new circles.

page 42 note * M. Mention's notation has been slightly changed.

page 43 note * Correspondance sur l'École Pohjtechnique, I. 193 (1804).

page 46 note * I have supplied the proof that the triangles NID, LA'X are equiangular, and have omitted the proof that A'X·ND = A'D·DX. It should be added that Mr M'Dowell does not use the terms orthocentre, incentre, circumcentre or nine-point circle.

page 50 note * Because BC D2′D3′ are antiparallel, ∠ BAO is equal to the complement of ∠ ACB.

page 56 note * (10)–(16).—MrGriffiths, J. in Mathematical Questions from the Educational Times, II. 69 (1864)Google Scholar; III. 102, 76 (1865); IV. 60 (1865); V. 72 (1866); VII. 57–8, 76 (1867).

page 56 note † (17) Wilkinson, T. T. in Mathematical Questions from the Educational Times, VI. 25 (1866).Google Scholar

page 56 note ‡ (18) DrTaylor, C. in Mathematical Questions from the Educational Times, XVII. 92 (1872).Google Scholar

page 56 note § (19) MrTucker, R. in Mathematical Questions from the Educational Times III. 58(1865).Google Scholar

page 57 note * (20) Rev. Whitworth, W. A. in Mathematical Questions from the Educational Times, X. 51 (1868).Google Scholar