We may look upon geometry as a practical logic, for the truths which it studies, being the most simple and most clearly understood of all truths, are on this account the most susceptible of ready application in reasoning.--D'ALEMBERT. The advance and the perfecting of mathematics are closely joined to the prosperity of the nation.--NAPOLEON. Hold nothing as certain save what can be demonstrated.--NEWTON. To measure is to know.--KEPLER. The method of making no mistake is sought by every one. The logicians profess to show the way, but the geometers alone ever reach it, and aside from their science there is no genuine demonstration.--PASCAL. The taste for exactness, the impossibility of contenting one's self with vague notions or of leaning upon mere hypotheses, the necessity for perceiving clearly the connection between certain propositions and the object in view,--these are the most precious fruits of the study of mathematics.--LACROIX. =Bibliography.= Smith, The Teaching of Elementary Mathematics, p. 234, New York, 1900; Henrici, Presidential Address before the British Association, _Nature_, Vol. XXVIII, p. 497; Hill, Educational Value of Mathematics, _Educational Review_, Vol. IX, p. 349; Young, The Teaching of Mathematics, p. 9, New York, 1907. The closing quotations are from Rebiere, Mathematiques et Mathematiciens, Paris, 1893. FOOTNOTES: [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 Nuernberg, Albrecht Duerer. [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.