where x pt, the distance times the turn, or direction factor, and expression. Since (1) is to be a minimum, D, and D must vanish. either will suffice. This means that sum of the three roots (the symmetric function s1) vanishes in the equation when a is a cube root of 1. That is, the three angles at x are all equal to 120°. ω The argument may also be presented in its physical, instead of geometrical, aspect. The method in each case is the same. Consider a, ẞ, y as centers of attraction for constant forces of equal magnitude. The total potential of x will then be the sum of the distances from x to the three points. For the total potential to be a minimum is to have equilibrium, and the resultant force in any direction must vanish. Hence, the three angles at x are all equal to 120°. It is to be noted that the partial derivative (Dz) of potential, expresses the resultant force with direction factor included. The well known construction for (the Fermat point) x is to draw on each side aß, By, ya, an equilateral triangle, outward, calling the free vertices y', a', B'. The points y', a, x, ẞ, are concyclic, and it follows that the line y'r bisects the angle axß and coincides with the line xy. x will therefore be the intersection of the three lines aa', BB', vr'. II. SOLUTION BY R. E. MORITZ, University of Washington. = Let Pi (xi, Yi), i = Cartesian coördinates, P = 1, 2, 3, represent the angular points of the triangle expressed in (x, y) any point within the triangle, and 0; the angle which the line PP makes with the positive direction of the x-axis. Then, if D denotes the sum of the distances of P from the angular points, we have In order that D may be a minimum the partial derivatives of D with respect to x and y must be separately equal to zero; hence, - 02) = - 1, from which = 120°. Similarly, 03 - 02 = - 120°, 01 — ძვ = By squaring and then adding the resulting equations we get 2 cos (01 The problem, therefore, resolves itself into that of finding a point within the triangle at which the three sides subtend equal angles. This point is readily found as follows: Describe on two sides of the triangle segments of circles containing each an angle of 120°. The intersection point of these circles is the required point. If one of the angles of the triangle is equal to or greater than 120° the intersection point falls on or without the triangle. In that case no point within the triangle satisfies the required conditions. The following kinematic solution, while perhaps not simpler than the foregoing geometrical solution, has the advantage of being applicable to the more general problem, which I believe has never been solved, namely, Given n points in a plane, to find a point from which if lines be drawn to the n points their sum may be the least possible. Draw the triangle to a given scale on a drawing board and insert thumb tacks at the points marking the vertices. Take a flexible string and fasten one end of it to one of the thumb tacks, say to A. Pass the other end of the string through a ring R, then around the thumb tack B, then back through the ring, then around the thumb tack at C, then once more through the ring R and then back to A. Now pull the free end of the string taut until the ring assumes a fixed position. The center of the ring R will be the required point. The proof is obvious. III. NOTE BY WILLIAM HOOVER, Columbus, Ohio. The following extract from a paper by P. G. Tait1 meets the requirement fully and is of interest historically: The following problem, originally proposed by Fermat to Torricelli, To find the point the sum of whose distances from three given points is the least possible, seems to have given considerable trouble to the older mathematicians, and even in modern times (see Gregory's Examples, p. 126) to have been solved in a very tedious manner. Simpler solutions have since been given (e.g., Cambridge and Dublin Mathematical Journal, VIII, p. 92), but none, to my knowledge, so direct, as that indicated by Quaternions. The object of this note is to show the simplicity of the quaternion method. ρ If a, ẞ be the vectors of two of the given points, the origin being the third, and if p be the vector of the required point, we must have (by the conditions of the problem) for all values of Udp. Hence the versor sum in square brackets must vanish identically. The immediate interpretation is, that lines parallel to p, p − a, pẞ form an equilateral triangle. The required point is therefore in the same plane as the three given points; and their distances, two and two, subtend equal angles at it, which is the well-known solution. Equally simple is the quaternion solution of the same problem if more than three points be given. Let their vectors, to any origin, be a, ß, y, etc., and let p be the vector of the sought point. We have Hence, if unit forces act at the required point, in the lines joining it with the given points, these forces are in equilibrium. Or, in another form, a closed equilateral gauche polygon may be drawn whose sides are parallel to the lines joining the sought point with the given ones." IV. SOLUTION BY OTTO DUNKEL, Washington University. Suppose that at P, a point inside the triangle ABC, we have a minimum with AP = r1, BP = 12, CP = r3. Then r1 must be the minimum distance from A to the ellipse with foci B and C and passing through P, since for any point Q on the ellipse BQ + CQ = =ra+r3. But the shortest distance from an external point to a closed convex curve is orthogonal to the curve and we know from the properties of the ellipse that r1 in this case must make equal angles with the focal rays r and rs. Applying the same argument to r2, we see that 71, 72, 73 must make equal angles with one another. V. REMARKS AND HISTORICAL NOTES BY R. C. ARCHIBALD, Brown University. This problem has been already discussed in the MONTHLY by Professors Jackson and Johnson.3 Professor Jackson's discussion is about the same as Gregory's. The analytic treatment by Professor Moritz is practically identical with that given in 1853 by A. Cohen1 and in 1902 by E. 1 Proceedings of the Royal Society of Edinburgh, Vol. 6 (1866–69), 1869, pp. 165–166; also in P. G. Tait, Scientific Papers, Vol. 1, 1898, pp. 76, 77. * Volume 24 (1917), pp. 42-44. Volume 24 (1917), pp. 243–244. ▲ Cambridge and Dublin Mathematical Journal, Vol. 9, p. 92. Goursat,' and his kinematic solution is given in the latter part of Tait's paper of 1867 from which Professor Hoover quoted. Tait concludes: "This kinematical process, equally with the quaternion one whose form directly suggests it, gives easily the solution of the more general problem,To find a point such that m times its distance from A, together with n times its distance from B, etc., may be a minimum." (For three points this is solved in Stegmann-Kiepert, Lehrbuch der Differentialrechnung, 7. Aufl. 1895, p. 260.) The problem of finding a point in a plane the sum of whose distances from any number of given points is a minimum was solved by Tédenat in Annales de mathématiques pures et appliquées (1810–11), pp. 285–291. The celebrated problem of our question was formulated by Fermat' in the seventeenth century. According to Viviani3 he suggested it to Torricelli (before 1648, for Torricelli died in 1647). Torricelli discovered three solutions, one by "plane loci" (that is, with ruler and compasses), two others by "solid loci" (that is, by means of conic sections). He afterwards proposed it to Viviani in the following terms: "A triangle, each of whose angles is less than one-third of four right angles being given; to find a point from which, if straight lines be drawn to its three angles, their sum shall be a minimum." Viviani solved the problem after repeated efforts (he says, non nisi iteratis oppugnationibus tunc nobis vincere datum fuit) and his solution is given in the appendix to his Geometria divinatio.5 The problem was treated by Thomas Simpson in 1750. He gave the following construction for determining the Fermat point: Describe on BC a segment of a circle to contain an angle of 120°, and let the whole circle BCQ be completed. From A to Q, the middle point of the arc BA'C draw AQ intersecting the circumference of the circle in P, which will be the point required. Because of this construction Simson has been credited with the theorem: If on the sides of a triangle ABC, equilateral triangles A'BC, B'CA and C'AB be described externally AA', BB' and CC' are concurrent-a construction which Mr. Morley cites above. Simpson treats also the more general problem: Three points A, B, and C being given, to find the position of a fourth point V, so that if lines be drawn from thence to the three former, the sum a AP+b.BP + c.CP, where a, b, c denote given numbers, shall be a minimum. The further history of generalizations is very extensive. Also solved by P. J. DANIEL, W. W. GORSLINE, R. A. JOHNSON, H. M. ROESER, J. ROSENBAUM, L. WEISNER and the PROPOSER. 1 Cours d'analyse mathématique tome 1; English ed. by Hedrick, 1904, pp. 130-131. See also G. Humbert, Cours d'Analyse, tome 1, 1903, pp. 193-196. 2Oeuvres, Tome 1, Paris, 1891, p. 153: “Datis tribus punctis, quartum reperire, a quo si ducantur tres recta addata puncta, summa trium harum rectarum sit minima quantitas." See also Tome 3, 1896, p. 136. 3 V. Viviani, De maximis et minimis geometria divinatio. Florentiæ, MDCLIX, p. 144. 4 This careful statement is necessary for constructions indicated in solutions above. If one of the angles, A say, is equal to, or greater than, 120°, in a certain sense its vertex is to be considered as the minimum point; compare the discussion of this by Goursat, l.c., by Bertrand in Journal de mathématiques pure et appliquées, tome 7 (1843), pp. 155–160, and by Sturm in Journal für die reine und angewandte Mathematik, Vol. 97 (1884), p. 51. 5 L.c., pp. 145–150. Viviani's method of solution is reproduced in D. Cresswell's Maxima and Minima, Cambridge, 1812, pp. 120–121; second edition, 1817, pp. 121–122. Doctrine and Applications of Fluxions, London, 1750, § 36. = BB' = 7 The first formulation of the result as here stated seems to have been by T. S. Davies in the Gentleman's Diary for 1830, p. 36. It is here shown also that AA' CC'. Problems closely related to this are frequently published, e.g., in School Science and Mathematics, Feb., 1918, pp. 170-171; Jan., 1919, pp. 86-87; April, 1919, pp. 374-375. NOTES AND NEWS. EDITED BY E. J. MOULTON, Northwestern University, Evanston, Ill. Dr. J. S. MILLER has resigned the chair of mathematics at Emory and Henry College to accept a similar position at Hampden-Sidney College. Dr. W. W. KUSTERMAN has been appointed associate in mathematics at the University of Illinois. Dr. E. P. LANE has resigned his position as instructor in Rice Institute to accept an appointment as assistant professor of mathematics at the University of Wisconsin. Mr. J. R. OSBORN and Mr. W. E. ARMENTROUT have been appointed assistants at the University of Kentucky. Dr. T. F. HOLGATE, of Northwestern University, has resumed his duties as professor of mathematics, after serving as Acting President of the University for the last three years. Dr. HOLGATE was honored with the degree of Doctor of Laws by Queen's University at the time of the installation of a new principal and chancellor on October 16. Dr. C. H. YEATON has resigned his position as instructor in mathematics at the University of Minnesota to accept an appointment as professor of mathematics at the School of Engineering of Milwaukee. Professor P. P. BOYD, of the University of Kentucky, has been elected president of the Kentucky academy of science. Since his discharge from the national service, Mr. J. E. DOTTERER has been appointed professor of mathematics and physics in Manchester (Indiana) College. At the University of Manitoba, Major N. B. MACLEAN, after active service in command of the Royal Garrison Artillery in France during the late war, for which he was awarded the Distinguished Service Order, was recalled last spring to take the directorship of the returned soldiers' tutorial course during the summer. He has now resumed his duties as head of the department of mathematics and astronomy. He has been elected a fellow of the Royal Astronomical Society of Canada of which Professor H. R. KINGSTON, in the same university, was elected president in 1919. Major N. R. WILSON, who also was in service in France, has now returned to his duties as professor of mathematics. Dr. C. P. SOUSLEY, of the Pennsylvania State College, has been promoted from assistant professor to associate professor of mathematics. Dr. FRANK SCHLESINGER, director of the Allegheny Observatory of the University of Pittsburgh, has been elected director of the Yale Observatory. Owing to the death of Mr. E. M. Reed, there is released for the general purposes of the Yale Observatory one-third of his estate, which will add to Observatory funds an estimated $60,000 during the year.. Professor F. W. BEAL, of the University of Tennessee, has been appointed assistant professor of mathematics at the University of Pennsylvania. At Cornell University, Dr. F. W. REED of the University of Illinois has been appointed instructor in mathematics. Mr. H. M. LUFKIN, Mr. H. PORITSKY, and Mr. H. A. STURGES have been appointed assistants. Dr. R. J. T. BELL, of the University of Glasgow, has been appointed to the chair of pure and applied mathematics in the University of Otago, New Zealand. It will be recalled that Dr. Bell is the author of An Elementary Treatise on Coördinate Geometry of Three Dimensions, editor of the fourth edition of Frost's Curve Tracing [1919, 201-202], and an editor of recent volumes of the Proceedings of the Edinburgh Mathematical Society. Professor A. B. NELSON, of Central University, Danville, Ky., died in November, 1918. Rev. C. J. BORGMEYER, professor of mathematics and astronomy in St. Louis University since 1896, died December 6, 1919, aged 58. He was one of the charter members of the Association. Dr. L. G. WELD, director of the Pullman (Ill.) Free School of Manual Training since 1911, died at Chicago, November 29, 1919, as the result of an infected tooth. He was in his fifty-seventh year. Dr. Weld taught mathematics for about twentyfive years at the State University of Iowa where he was professor of mathematics and astronomy and head of the department 1889-1911, dean of the graduate college 1900-1907, director of the school of applied science 1903-1905, and dean of the college of liberal arts 1907-1910. He was the author of A Short Course in the Theory of Determinants (1893), of the section on Determinants in Higher Mathematics edited by Merriman and Woodward (1896), afterwards reprinted as a separate book, and of several pamphlets dealing with the history of Iowa. Dr. G. L. DEMARTRES, professor of mathematics at the University of Lille, died in July, 1919, aged seventy-one years. His Cours de Géométrie Infinitésimale (Paris, 1913) is familiar to many Americans. A Paris announcement of his death concludes: "Il avait subi avec courage l'occupation allemande et contribué par son exemple à maintenir le fonctionnement du haut enseignement." Nature reports the death on October 5 of Mr. G. W. PALMER who was appointed senior mathematical master and master of the Royal Mathematical School at Christ's Hospital in September, 1911. |