[1] And really, though not nominally, in the United States, where the first concepts are found in the kindergarten, and where an excellent course in mensuration is given in any of our better class of arithmetics. That we are wise in not attempting serious demonstrative geometry much earlier seems to be generally conceded.

[2] The third stage of geometry as defined in the recent circular (No. 711) of the British Board of Education, London, 1909.

[3] The closing words of a sensible review of the British Board of Education circular (No. 711), on "The Teaching of Geometry" (London, 1909), by H. S. Hall in the School World, 1909, p. 222.

[4] In an address in London, June 15, 1909, at a dinner to Sir Ernest Shackelton.

[5] Governor Hughes, now Justice Hughes, of New York, at the Peary testimonial on February 8, 1910, at New York City.

[6] The first work upon this subject, and indeed the first printed treatise on curves in general, was written by the famous artist of NÜrnberg, Albrecht DÜrer.

[7] Several of these writers are mentioned in Chapter IV.

[8] If any reader chances upon George Birkbeck's English translation of Charles Dupin's "Mathematics Practically Applied," Halifax, 1854, he will find that Dupin gave more good applications of geometry than all of our American advocates of practical geometry combined.

[9] See, for example, Henrici's "Congruent Figures," London, 1879, and the review of Borel's "Elements of Mathematics," by Professor Sisam in the Bulletin of the American Mathematical Society, July, 1910, a matter discussed later in this work.

[10] T. J. McCormack, "Why do we study Mathematics: a Philosophical and Historical Retrospect," p. 9, Cedar Rapids, Iowa, 1910.

[11] Of the fair and candid arguments against the culture value of mathematics, one of the best of the recent ones is that by G. F. Swain, in the Atti del IV Congresso Internazionale dei Matematici, Rome, 1909, Vol. III, p. 361. The literature of this school is quite extensive, but Perry's "England's Neglect of Science," London, 1900, and "Discussion on the Teaching of Mathematics," London, 1901, are typical.

[12] In his novel, "The Morals of Marcus Ordeyne."

[13] G. W. L. Carson, "The Functions of Geometry as a Subject of Education," p. 3, Tonbridge, 1910.

[14] It may well be, however, that the growing curriculum may justify some reduction in the time formerly assigned to geometry, and any reasonable proposition of this nature should be fairly met by teachers of mathematics.

[15] Professor MÜnsterberg, in the Metropolitan Magazine for July, 1910.

[16] It was published in German translation by A. Eisenlohr, "Ein mathematisches Handbuch der alten Aegypter," Leipzig, 1877, and in facsimile by the British Museum, under the title, "The Rhind Papyrus," in 1898.

[17] Generally known as Rameses II. He reigned in Egypt about 1350 B.C.

[18] Two excellent works on Thales and his successors, and indeed the best in English, are the following: G. J. Allman, "Greek Geometry from Thales to Euclid," Dublin, 1889; J. Gow, "A History of Greek Mathematics," Cambridge, 1884. On all mathematical subjects the best general history is that of M. Cantor, "Geschichte der Mathematik," 4 vols, Leipzig, 1880-1908.

[19] Another good work on Greek geometry, with considerable material on Pythagoras, is by C. A. Bretschneider, "Die Geometrie und die Geometer vor Eukleides," Leipzig, 1870.

[20] Smith and Karpinski, "The Hindu-Arabic Numerals," Boston, 1911.

[21] For a sketch of his life see Smith and Karpinski, loc. cit.

[22] Those who care for a brief description of this phase of the subject may consult J. Casey, "A Sequel to Euclid," Dublin, fifth edition, 1888; W. J. M'Clelland, "A Treatise on the Geometry of the Circle," New York, 1891; M. Simon, "Über die Entwicklung der Elementar-Geometrie im XIX. Jahrhundert," Leipzig, 1906.

[23] Riccardi, Saggio di una bibliografia Euclidea, Part I, p. 3, Bologna, 1887. Riccardi lists well towards two thousand editions.

[24] Hermotimus of Colophon and Philippus of Mende.

[25] Literally, "Who closely followed the first," i.e. the first Ptolemy.

[26] MenÆchmus is said to have replied to a similar question of Alexander the Great: "O King, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all."

[27] This is also shown in a letter from Archimedes to Eratosthenes, recently discovered by Heiberg.

[28] On this phase of the subject, and indeed upon Euclid and his propositions and works in general, consult T. L. Heath, "The Thirteen Books of Euclid's Elements," 3 vols., Cambridge, 1908, a masterly treatise of which frequent use has been made in preparing this work.

[29] A contemporary copy of this translation is now in the library of George A. Plimpton, Esq., of New York. See the author's "Rara Arithmetica," p. 433, Boston, 1909.

[30] A beautiful vellum manuscript of this translation is in the library of George A. Plimpton, Esq., of New York. See the author's "Rara Arithmetica," p. 481, Boston, 1909.

[31] Heath, loc. cit., Vol. I, p. 114.

[32] The author is a member of a committee that has for more than a year been considering a syllabus in geometry. This committee will probably report sometime during the year 1911. At the present writing it seems disposed to recommend about the usual list of basal propositions.

[33] "Elementi di Geometria," Milan, 1884.

[34] See his "Elementarmathematik vom hÖheren Standpunkt aus," Part II, Leipzig, 1909.

[35] For some classes of schools and under certain circumstances courses in combined mathematics are very desirable. All that is here insisted upon is that any general fusion all along the line would result in weak, insipid, and uninteresting mathematics. A beginning, inspirational course in combined mathematics has a good reason for being in many high schools in spite of its manifest disadvantages, and such a course may be developed to cover all of the required mathematics given in certain schools.

[36] Carson, loc. cit., p. 15.

[37] Al-jabr wa'l-muqabalah: "restoration and equation" is a fairly good translation of the Arabic.

[38] Or be carried along at the same time as a distinct topic.

[39] With a single year for required geometry it would be better from every point of view to cut the plane geometry enough to admit a fair course in solid geometry.

[40] Carson, loc. cit., p. 13.

[41] Carson, loc. cit., p. 12.

[42] From the Greek ??, ge (earth), + et?e?? metrein (to measure), although the science has not had to do directly with the measure of the earth for over two thousand years.

[43] From the Arabic al (the) + jabr (restoration), referring to taking a quantity from one side of an equation and then restoring the balance by taking it from the other side (see page 37).

[44] One of the clearest discussions of the subject is in W. B. Frankland, "The First Book of Euclid's 'Elements,'" p. 26, Cambridge, 1905.

[45] "Grundlagen der Geometrie," Leipzig, 1899. See Heath's "Euclid," Vol. I, p. 229, for an English version; also D. E. Smith, "Teaching of Elementary Mathematics," p. 266, New York, 1900.

[46] We need frequently to recall the fact that Euclid's "Elements" was intended for advanced students who went to Alexandria as young men now go to college, and that the book was used only in university instruction in the Middle Ages and indeed until recent times.

[47] For example, he moves figures without deformation, but states no postulate on the subject; and he proves that one side of a triangle is less than the sum of the other two sides, when he might have postulated that a straight line is the shortest path between two points. Indeed, his followers were laughed at for proving a fact so obvious as this one concerning the triangle.

[48] T. L. Heath, "Euclid," Vol. I, p. 200.

[49] For a rÉsumÉ of the best known attempts to prove this postulate, see Heath, "Euclid," Vol. I, p. 202; W. B. Frankland, "Theories of Parallelism," Cambridge, 1910.

[50] For the early history of this movement see Engel and StÄckel, "Die Theorie der Parallellinien von Euklid bis auf Gauss," Leipzig, 1895; Bonola, Sulla teoria delle parallele e sulle geometrie non-euclidee, in his "Questioni riguardanti la geometria elementare," 1900; Karagiannides, "Die nichteuklidische Geometrie vom Alterthum bis zur Gegenwart," Berlin, 1893.

[51] This limitation upon elementary geometry was placed by Plato (died 347 B.C.), as already stated.

[52] Book I, Proposition 20.

[53] Free use has been made of W. B. Frankland, "The First Book of Euclid's 'Elements,'" Cambridge, 1905; T. L. Heath, "The Thirteen Books of Euclid's 'Elements,'" Cambridge, 1908; H. Schotten, "Inhalt und Methode des planimetrischen Unterrichts," Leipzig, 1893; M. Simon, "Euclid und die sechs planimetrischen BÜcher," Leipzig, 1901.

[54] For a facsimile of a thirteenth-century MS. containing this definition, see the author's "Rara Arithmetica," Plate IV, Boston, 1909.

[55] Our slang expression "The cart before the horse" is suggestive of this procedure.

[56] Loc. cit., Vol. II, p. 94.

[57] Address at Brussels, August, 1910.

[58] For a recent discussion of this general subject, see Professor Hobson on "The Tendencies of Modern Mathematics," in the Educational Review, New York, 1910, Vol. XL, p. 524.

[59] A more extended list of applications is given later in this work.

[60] Abu'l-'Abbas al-Fadl ibn Hatim al-Nairizi, so called from his birthplace, Nairiz, was a well-known Arab writer. He died about 922 A.D. He wrote a commentary on Euclid.

[61] This illustration, taken from a book in the author's library, appeared in a valuable monograph by W. E. Stark, "Measuring Instruments of Long Ago," published in School Science and Mathematics, Vol. X, pp. 48, 126. With others of the same nature it is here reproduced by the courtesy of Principal Stark and of the editors of the journal in which it appeared.

[62] In speaking of two congruent triangles it is somewhat easier to follow the congruence if the two are read in the same order, even though the relatively unimportant counterclockwise reading is neglected. No one should be a slave to such a formalism, but should follow the plan when convenient.

[63] Stark, loc. cit.

[64] Of which so much was made by Professor Olaus Henrici in his "Congruent Figures," London, 1879,—a book that every teacher of geometry should own.

[65] Much is made of this in the excellent work by Henrici and Treutlein, "Lehrbuch der Geometrie," Leipzig, 1881.

[66] MÉray did much for this movement in France, and the recent works of Bourlet and Borel have brought it to the front in that country.

[67] W. N. Bruce, "Teaching of Geometry and Graphic Algebra in Secondary Schools," Board of Education circular (No. 711), p. 8, London, 1909.

[68] This is the latest opinion. He is usually assigned to the first century B.C.

[69] See page 54.

[70] A Greek philosopher and mathematician of the fifth century B.C.

[71] This illustration and the following two are from C. Dupin, "Mathematics Practically Applied," translated from the French by G. Birkbeck, Halifax, 1854. This is probably the most scholarly attempt ever made at constructing a "practical geometry."

[72] This illustration and others of the same type used in this work are from the excellent drawings by R. W. Billings, in "The Infinity of Geometric Design Exemplified," London, 1849.

[73] From H. Kolb, "Der Ornamentenschatz ... aus allen Kunst-Epochen," Stuttgart, 1883. The original is in the Church of Saint Anastasia in Verona.

[74] From J. Bennett, "The Arcanum ... A Concise Theory of Practicable Geometry," London, 1838, one of the many books that have assumed to revolutionize geometry by making it practical.

[75] The figures are from Dupin, loc. cit.

[76] For a very full discussion of these four definitions see Heath's "Euclid," Vol. II, p. 116, and authorities there cited.

[77] These two and several which follow are from Stark, loc. cit.

[78] The author has a beautiful ivory specimen of the Sixteenth century.

[79] See, for example, G. B. Kaye, "The Source of Hindu Mathematics," in the Journal of the Royal Asiatic Society, July, 1910.

[80] An interesting Japanese proof of this general character may be seen in Y. Mikami, "Mathematical Papers from the Far East," p. 127, Leipzig, 1910.

[81] Special recognition of indebtedness to H. A. Naber's "Das Theorem des Pythagoras" (Haarlem, 1908), Heath's "Euclid," Gow's "History of Greek Mathematics," and Cantor's "Geschichte" is due in connection with the Pythagorean Theorem.

[82] The rule was so ill understood that Bhaskara (twelfth century) said that Brahmagupta was a "blundering devil" for giving it ("Lilavati," § 172).

[83] Bosanquet and Sayre, "The Babylonian Astronomy," Monthly Notices of the Royal Asiatic Society, Vol. XL, p. 108.

[84] This and the three illustrations following are from Kolb, loc. cit.

[85] This was in five colors of marble.

[86] The proof is too involved to be given here. The writer has set it forth in a chapter on the transcendency of p in a work soon to be published by Professor Young of The University of Chicago.

[87] These may be purchased through the Leipziger Lehrmittelanstalt, Leipzig, Germany, which will send catalogues to intending buyers.

[88] An excellent set of stereoscopic views of the figures of solid geometry, prepared by E. M. Langley of Bedford, England, is published by Underwood & Underwood, New York. Such a set may properly have place in a school library or in a classroom in geometry, to be used when it seems advantageous.

[89] The actual construction of these solids is given by Pappus. See his "Mathematicae Collectiones," p. 48, Bologna, 1660.

[90] The illustration is from Dupin, loc. cit.

[91] For the historical bibliography consult G. HolzmÜller, Elemente der Stereometrie, Vol. I, p. 181, Leipzig, 1900.

[92] The illustration is from Dupin, loc. cit.