^{page 152 note 1} Collectio, Ed. by Hultsch, , Liber VII, p. 670.Google Scholar In what follows all references will be to this edition.

^{page 152 note 2} Heath, T. L., The Works of Archimedes, Cambridge, 1897,Google Scholar p.c. Chapter V. (p. c.–cxxii.) of this work is entitled “On the Problems known as NETΣEIΣ.” Hereafter when we quote Dr. Heath it will be with reference to this chapter.

^{page 153 note 3} The case of this problem for the internal bisector with the further(unnecessary) supposition of given perimeter was proposed for solution in Leybourn's Mathematical Repository, O.S. No. 12, Dec. 20, 1801, III., 69, and solved geometrically in No. 14, May 1,1804, III., 188–9. Francoeur, , Cours Complet de Mathématiques Pures, Paris, 1809, Tome I., p. 309–10 (4th Éd. 1837, I., 356–7),Google Scholar discussed the problem algebraically under the following form :-(Problem B): From the point S at the extremity of the diameter of a circle perpendicular to a chord, E₂F₂, draw a line SBD such that the part BD between the chord and arc be of given length. Cf. F.G.M., , Exercices de Géométrie, 4th Ed., Tours and Paris, 1907, p. 170–1, also p. 163, 701Google Scholar. See also Notes 34, 41.

^{page 153 note 4} To economise space when indicating references in what follows, I give here a complete list of the portions of Huygens' writings which deal with our problems. Nine pieces are to be found in the Library of the University of Leyden in the manuscripts “Codex Hugeniorum, No. 12.” Only two of these have been published, but all the others have been placed at my disposal through the courtesy of the librarian, Dr S. G. de Vries, and Professor D. J. Korteweg of the Huygens' Commission at Amsterdam. The other references are to three letters to Van Schooten and four problems in De Circuli magnitudine, etc.

”Travaux Mathématiques Divers de 1650—Oeuvres Complètes, La Haye, Tome XI., 1908.

1. “No. IV.,” p. 226–7.

2. “No. VIII..” p. 239–42.

“Travaux Mathématiques Divers de 1652 et 1653” (MSS. unpublished at present, although later to appear in Oeuvres Complètes, Tome XII.

3. “ No. II.,” dated 30th Jan. 1652.

4. “No. IV.,” dated 1652.

5. “No. VI.,” dated 9th Feb. 1652 and 19th Oct. 1653.

6. “No. VII.,”dated Feb. 1652.

7. “No. VIII.,” dated 14th Feb. 1652.

8. “No. IX.,” dated 17th Feb. 1652.

9. “No. XIII.,” dated 16th Aug. 1652.

“Correspondanoe” Oeuvres Compulètes, La Haye, Tome I., 1888.

10. “No. 164,” dated Oot. 1653, p. 244–5.

11. “No. 166,” dated 23rd Oct. 1653, p. 247–51. Modification of H. 4,5.

12. “No. 168,” dated 10th Dec. 1653, p. 256–7. Modification of H.9.

De Circuli magnitudine inventa accedunt ejusdem problematum quorundum illustrium constructiones, Amsterdam, 1654. [Another edition, Feneburg, 1668.Google Scholar This work was included in Huygens' Opera Varia, 1724, Tome 2.]

13. “Prob. IV.,” p. 56–7. Slightly modified from H.11.

14. “Prob. V.,” p. 57–8 idem.

15. “Prob. VI.,” p. 59–61. Slightly modified from H. 5, 9, 11, Cf. Note 39.

16. “Prob. VII.,” p. 62–69. Slight modification of H. 9, 11, 12. References to this list will be by such an abbreviation as” H. 9,” which indicates the piece of 16th Aug. 1652.

^{page 154 note 5} Pappus, III., 58–63 ; IV., p. 242 et seq.

page 154 note * See folding-out plates.

^{page 155 note 6} For geometrical proof Pappus or Heath may be consulted. Conti gives an analytic proof in Enriques' Fragen der Elementargeometrie, II., 207–08, Liepzig, 1907.Google Scholar

^{page 155 note 7} P. 272, et seq.

^{page 155 note 8} Zeuthen, , Die Lehre von dem Kegelschnitten im Altertum, Kopenhagen, 1886, p. 267.Google Scholar The topics of this paper are touched on in various places, p. 267–283.

^{page 155 note 9} Mathesis, 03 1889. IX., 96–7. Analytic treatment is added by J. Neuberg.Google Scholar

^{page 156 note 10} Newton found a solution with a hyperbola, in still a third way — Arithmetica Universalis, 1707; Second Edition, London, 1722, p. 290–92.Google Scholar

^{page 157 note 11} This rare little book, so important in the history of Algebra and Spherical Triangles, was reprinted at Leyden in 1884, with an introduction by Bierens de Haan. The pages are unnumbered.

^{page 157 note 12} Nouv. Édition, Paris, 1887, p. 70–71;Google Scholar also Oeuvres, ed. by Tannery, Adam et, VI., 461–463, Paris, 1902.Google Scholar

^{page 157 13} Geometria à Renato des Cartes anno 1637, gallicè edita ; nunc autem cum Notis Florimondi de Beavne … in linguam Latinam versa & Commentariis illustra operâ atque studio Francisci à Schooten.

^{page 159 note 14} This is a distinguishing characteristic of the “plane”case.

^{page 160 note 15} Cf. Reference by DE LA Hire in the preface to his Traité des Epicycloïdes, 1694.

^{page 160 note 16} Letters, dated Nov. 19 and Dec. 3, 1678, from De Vaumesle to Huygens. They were published by P. J. Uylenbroek in his Christiani Hugenii aliorumque seculi XVII., virorum celebrium exercitationes mathematicae, etc., La Haye, 1833; Fasc. II., p. 42–51. See also Oeuvres Complètes VIII. 115–7; 125–7.

^{page 160 note 17} This work was first published at Paris in 1720. The problems are treated as “ Ex. 2,” Book X.

^{page 161 note 18} Arithmetica Universalis, 1707. Second edition, Lond., 1722, See. IV., Problem XXIV., p. 148–151.Google Scholar See also the edition of Noel Beaudeux, Paris, 1802, Tome II., p. 141–2, 56–59.

^{page 162 note 19} Guisnèe, , Application de l' Algebre a la Geometrie, Paris, 1733, p. 54.Google Scholar

^{page 162 note 20} Kästner, , Geschichte der Kunste und Wissenschaften, etc. Bd. III. Göttingen, 1799.Google Scholar

^{page 162 note 21} Cantor, M., Vorlesungen über der Geschichte der Mathematik, II., 809–11, 2^teAuf., Leipzig, 1899.Google Scholar As Cantor only refers to Ghetaldi in connection with the rhombus problem, it is evident that he did not know its interesting history. Ritt, G. (Problèmes de géométrie, et de Trigonometrie, Neuvième Edition, Paris, 1894, p. 311–16Google Scholar) chose AF=x, and was led to a reciprocal equation.

²² Gelcich, E., Zeitschrift für Mathematik und Physik, XXVII., Suppl., p. 213–4, 1882.Google Scholar “Eine studie ueber die Entdeckung der analytischen Geometric, mit Berüchsichtigung eines werkes des Marino Ghetaldi… aus dem Jahre 1630.”

²³ (1), (2), (3), (5), (6), (7) are given by Franck, Momenheim et, Examens et Compositions de Mathématiques, Paris, 1862, p. 6–28;Google Scholar (4) was given by Transon, , Nouvelles Annales de Mathematiques, 1847, VI., 458–61.Google Scholar

²⁴ Cf.Combette, E., cours d' Algèbre Éméntaire, Paris, 1882, p. 504–519;Google Scholar also Cours d' Algèbre, Paris, 1891, p. 396–411.Google Scholar

²⁵ Momenheim et Franck. Cf. Note 23.

²⁶ Fontés, , Nouvelles Annales den Mathématiques, 1847, VI., 180–5.Google Scholar

²⁷ Transon. Cf. Note 23.

²⁸ Gergonne, J. D., Annales de Mathématiques Pures et Appliquées (Gergonne), 01, 1820, X., 204–16.Google Scholar

²⁹ Pappus, VII., p. 778 et seq.

³⁰ Contrary to what Zeuthen states (l.c., p. 281, cf. Note 8) Pappus mentions(p. 670) the rhombus as one of the “plane” νε⋯σεις which the Greeks had solved; not only this, but he spoke of “two cases,” which evidently cover all possible solutions of the problem of Apollonius. What must have been the solution of Apollonius for the case of the lines E₂F₂, E₃F₃. does not seem to have been explicitly pointed out before. If a point G′ be taken on DB produced such that G′W=GW (where W is the centre of BD), exaotly the same construction may be employed on substituting G′ for G. For further comment on this construction see §30. The method employed by the Greeks is, then, now evident.

This same result, attributed to Apollonius, was arrived at by Horsley, Samuel in his restoration of Apollonii Pergaei Inclinationum libri duo (Oxford, 1770),Google Scholar and if doubts of the result were still held, Heath's independent research and discussion would certainly dissipate them. FLAUTI in his “Su due libri di Apollonio Pergeo detti delle inclinazioni e sulle diverse restituzioni di essi disquisizione” [1850] (Memorie di matematica e di Fisica della Società Italiana della Scienze, Modena, 1852, XXV., P. I., p. 223–36Google Scholar) gives what practically amounts to Apllonius' construction. The same is true of L'Hospital, 1704, and this is his geometrioal construction referred to in connection with the algebraic equation §11(4). Apparently independent of others D'OMERIQUE discovered this same solution, Prop. XXXII., p. 216 et seq. of his Analysis geometrica sive nova, el vera methcdus resolvendi tam problemata quam Arithmeticas Quaestiones (1698).

³¹ This method was indicated by D'Omerique (1698, cf. Note 30), and also by “Tyco Oxoniensis” in The Mathematician, No. 2, p. 105 (Lond. 1746).Google Scholar

³² This proof has been given in extenso and almost verbatim (cf. Heath) in order to illustrate the ancient mode of discussion, which we would now greatly abbreviate. Huygens gave three other geemetrical proofs with practically the same initial construction as Heraclitus. Two of these which differ little in essentials from the above are given here out of chronological order. In both it is supposed that R has been determined such that CB²+k ²=CR²; the semi-circle is described and F determined. I(H. 11, 1653). Join ER. Add CE² to the equals CB²+k ² and CR², then BC²+k ²+CE²=CE²+CR², or BE²+k ²=ER²=EF²+FR². But since FG, BE are between equally distant parallel lines at right angles to one another and equally inclined to these lines FR=BE, ∴ EF²=k ². Q.E.D. II. (H. 16, 1654). Join ER and draw FS‖DC. Since triangles BEC, FSR are similar and the sides BC, FR are equal, the side BE=FR and EC=SR. But EF²+FR²=EF²+FS²+SR²=ER²=EC²+CR². But EC²=SR². ∴ EF²=CR²–FS²=k ²(by constr.).

³³ It also appeared in a posthumous work (which has an important bearing on the history of analytical geometry–Gelcich, note 22) entitled Marini Ghetaldi Patritii Ragusini Mathematici praestanlissimi de Resolutione & Compositione Mathematica libri qainque (Rome, 1630; another edition, 1640) p. 330–2.Google Scholar The same solution in somewhat abbreviated form was given by Herigone, Pierre in Tome IV., p. 912–3 of his Cursus Mathematicus (Paris, 1634 ; another edition, 1644).Google Scholar

³⁴ Prob. II., case 5 of Ghetaldi's restitution of Apollonius : Suppose SE′Gʺ the semi-circle (Fig. 7). Produce SE′ to J such that JE′ equals half the given length BD (Fig. 5). With centre J (Fig. 7) and radius JE′ describe a circle which cuts GʺJ in Dʺ and X. In the semi-circle place a chord GʺDʺ=GʺDʺ, and produce it to meet O′E′ produced in B′, then D′B′ is equal to required length DB. For GʺDʺ·GʺX=GʺE^′2=GʺOʺ·GʺS=GʺD′·GʺB′. But GʺDʺ=GʺD′. ∴ GʺX=GʺB′, or DʺX=D′B′=DB. This is also a solution of Problem D for the external bisector D'B' of the triangle D'E'F'. Cf. Note 3. The naturalness of Ghetaldi's proof of this problem of Apollonius is the more striking if Fig. 7 be thought of as applied to Fig. 5, the singly-primed letters correspond to those unprimed, and G”with G. This comparison also suggests that the algebraic equations of the solutions of Apollonius and Ghetaldi might be the same.

³⁵ We have considered the algebraic solutions given in § 8 as the “first” and “fourth” in chronological order. Cf. Note 38.

³⁶ This construction and proof were discovered independently by Turner, John, and given in The Mathematician, No. 2 (Lond. 1746), p. 104–5.Google ScholarCf. Note 31. N. B.-GD of this solution is the same length as L'B (or LC) in hyperbola solution, §§20, 8.

³⁷ We shall return to this a little later (§ 26). Huygens gave another construction (H. 2) practically the same as the above, where instead of drawing LM parallel to BD he made CN=CL.

³⁸ In order to shorten a somewhat lengthy paper three solutions (beside the proof of the ninth) have been omitted; the seventh for the case of a square, two sides produced ; the tenth and fourteenth for the rhombus case, two sides produoed. These may be considered as other cases of the third, ninth, and thirteenth respectively. One of the most interesting features of the various published and unpublished solutions of Huygens is that we can trace the manner of their evolution to the refined forms. Only those constructions or proofs radically different have been counted as new solutions. The two algebraic equations, indicated in connection with § 9 (3) are classed as the eighth solution. The eleventh solution is § 9 (1). cf. Note 36.

³⁹ P. 102–4. cf. also pp. 44–51, 101, 443. I am indebted to J. S. Mackay, Esq., LL.D., for this reference. Leslie reproduced H. 15. The modifications in H. 5 are considerable.

⁴⁰ Cirodde, P. L., Leçons de Géométrie Analtique. Paris, 1843 p. 104–109.Google Scholar

⁴¹ This is practically the construction of Momenheim et Francois-Franck (p. 25, cf. Note 23), who consider Heraclitus' problem as a particular case of Problem D (introductory paragraph). See also Giannattasio, F., Atti della reale Accademia della Scienze, sezione della Societa Reale Barbonica, Naples, II. Parte I., p. 49–50, 1825,Google Scholar and Note 3. Cf. Bulletin des Sciences Mathématiques et Physiques, 13^e annëe, Mai, Juillet, 1908, p. 237–239, 297–300.Google Scholar Problem D, explicitly stated for the internal bisector, was proved geometrically by Diestkrweg, W. A., Die Bücher des Apollonius von Perga De Inclinationibus wiederhergestellt von Sam. Horsley nach dem Lateinischen frey bearbeitet, Berlin, 1823, p. 41–44.Google Scholar

⁴² “ Nuovo Soluzione Geometrica di un Problema del 1° libro della Inclinazione di Apollonio Pergeo.” Atti della reale Accademia della Scienze sezione della Societa Reale Barbonica, Naples, II., Parte I., 45–19, 1825.

⁴³ L'Intermédiaire Mathématiciens, Mars, 1908, XV., 71–2 ; in answer to Question No. 3309 (XIV, 266–7).

⁴⁴ Acta Eruditorum Lipsiensis, Jan. 1692, p. 33. Opera Omnia, 1742, III., 447. Bernoulli found the equation in its expanded form. The form was first given by Hermann in a letter to Leibnitz, dated Nov. 22, 1715. Cf. Leibnitz-Gerhardt, , Mathematische Schriften, IV., 407–408, 1859.Google Scholar

⁴⁵ Spozione de metodo delle equipollenze, Modena, 1854, p. 189–191;Google Scholar translation, Nouvelles Annales de Mathématiques, 1874 (2), T. XIII., p. 229–230. This carve is parallel to a hypooycloid.Google Scholar

⁴⁶ Crelle's Journal, Bd. LV., s. 362, 1858,Google Scholar or Gesammelte Werke, II., 668–9.Google Scholar This theorem seems to have been rediscovered by Midzuhara, J., and published in Journal of the Mathematico-Physical Society of Tokyo, IV., 1899 or 1900.Google ScholarCf. Mathematical Papers from the Far East. Ed. by Mikami, Y., Leipzig, 1910, p. 135.Google Scholar

⁴⁷ cf. Note by Neuberg, J., Mathesis, 08 1889, IX., 183–4.Google Scholar Midzuhara gives as solution (Cf. reference Note 46) the intersections of this ellipse and the hyperbola which is the locus of all points O got by varying k. That the locus of O is a hyperbola was indicated by Smith, C. in his Elementary Treatise on Conic Sections, London, 1892–Cf. Ex. 20, p. 85, Ex. 2, p. 162, Ex. 12, p. 163.Google Scholar It is also implied in Newton's discussion (Cf. Note 10).

A solution by K. Tsuruta was also given in the Journal of the Mathematico-Physical Society in Tokyo. Vol. IV.Google Scholar (Cf. Note 46). It is dated May 1889, and is made to depend upon the theorems : (1) the locus of the middle point of a line of fixed length sliding between two fixed lines is an ellipse; (2) “The locus of the centre of gravity of the triangle formed by a straight line through a given point with two given straight lines coplanar with the point is a hyperbola.” The construotion is then given as follows :

“Describe an ellipse (1) taking the constant length= ⅔ of the given length [k]; then describe a hyperbola (2), the given point being taken as the fixed point.

“Next describe another ellipse similar to and concentric with the ellipse already described, the former bearing to the latter the ratio of similitude of 3 :2. The minor ellipse will in general intersect with the hyperbola in four points.

“Again draw from the intersection of the given straight lines the four radii vectores through these points, and let them intersect the ellipse in 0, Oj, 02, Os. Then the straight lines BO, BO₁, B0₂, B0₃have their segments included between the given straight lines equal to the given length.”

Finally, it may be remarked that Hayashi, T., in Journal of Physics School in Tokyo for 12 1900, X., 1–4Google Scholar (Cf. Note 46), discusses the general problem and arrives at the equation of Gergonne (§15 (5)). He then considers and, apparently, because of guess work arrives at the erroneous conclusion that “our problem is geometrically insoluble.”

⁴⁸ This term has been chosen simply as a convenient one for this paper