502 



SCIENCE. 



[Vol. II., No. 36. 



been inserted in this edition, embracing modern 

 investigations of importance on tliis subject. 



(To be continued.) 



OBLIGATIONS OF MATHEMATICS TO 

 PHILOSOPHY, AND TO QUESTIONS OF 

 COMMON LIFE. ^ — 11. 



I SAID that I would speak to you, not of the utility 

 of the mathematics in any of the questions of com- 

 mon life or of physical science, but rather of the 

 obligations of mathematics to these diSereiit sub- 

 jects. The consideration which thus presents itself 

 is, in a great measure, that of the history of the de; 

 velopinent of the different branches of mathematical 

 science in connection with the older physical sci- 

 ences, — astronomy and mechanics. The mathemati- 

 cal theory is, in the tirst instance, suggested by some 

 question of common life or of physical science, is 

 pursued and studied quite independently thereof, 

 and perhaps, after a long interval, comes in contact 

 with it, or with quite a different question. Geometry 

 and algebra must, I think, be considered as each of 

 them originating in connection with objects or ques- 

 tions of common life, — geometry, notwithstanding 

 its .name, hardly in the measurement of land, but 

 rather from the contemplation of such forms as the 

 straight line, the circle, the ball, the top (or sugar- 

 loaf). The Greek geometers appropriated for the geo- 

 metrical forms corresponding to the last two of these 

 the words o^oipu and kuvoc;, our sphere and cone ; and 

 they extended the word ' cone' to mean the complete 

 figure obtained by producing the straight lines of the 

 surface both ways indefinitely. And so algebra would 

 seem to have arisen from the sort of easy puzzles in 

 regard to numbers which may he made, either in the 

 picturesque forms of tlie Bija-Ganita, with its maiden 

 with tlie beautiful locks, and its swarms of bees 

 amid the fragant blossoms, and the one queen-bee 

 left humming around the lotus-flower; or in the more 

 prosaic form in which a student has presented to him 

 in a modern text-book a problem leading to a simple 

 equation. 



The Greek geometry may be regarded as beginning 

 with Plato (B.C. 430-347). The notions of geometri- 

 cal analysis, loci, and the conic sections, are attributed 

 to liim ; and there are in his ' Dialogues ' many very 

 interesting allusions to mathematical questions, — in 

 particular the passage in tlie ' Theaetetus ' where he 

 aflirms the incommensurability of the sides of certain 

 squares. But the earliest extant writings are those 

 of Euclid (B.C. 285). There is hardly any thing in 

 mathematics more beautiful than his wondrous fifth 

 book; and he has also, in the seventh, eighth, ninth, 

 and tenth books, fully and ably developed the first 

 principles of the theory of numbers, including the 

 theory of incommensurables. We have next Apol- 

 lonius (about B.C. 247) and Archimedes (B.C. 287- 

 212), both geometers of the highest merit, and the 

 latter of them the founder of the science of statics 



^ Address of Professor Caylet before the British association. 

 Coucluded from No. 35. 



(including therein hydrostatics). His dictum about 

 the lever, his ' EvprjKa,' and the story of the defence 

 of Syracuse, are well known. Following these we 

 have a wortliy series of names, including tlie astrono- 

 mers Hipparchus (B.C. 150) and Ptolemy (A.D. 125), 

 and ending, say, with Pappus (A.D. 400), but con- 

 tinued by their Arabian commentators, and the Ital- 

 ian and other European geometers of the sixteenth 

 century and later, who pursued the Greek geometry.' 

 The Greek arithmetic was, from the want of a 

 proper notation, singularly cumbrous and difficult; 

 and it was, for astronomical purposes, superseded by 

 the sexagesimal arithmetic, attributed to Ptolemy, 

 but probably known before his time. The use of 

 the present so-called Arabic iigures became general 

 among Arabian writers on aritlimetic and astronomy 

 about the middle of the tenth century, but it was 

 not introduced into Europe until about two centuries 

 later. Algebra, among the Greeks, is represented 

 almost exclusively by the treatise of Diopliantus 

 (A.D. 150), — in fact, a work on the theory of num- 

 bers, containing questions relating to square and 

 cube numbers, and other properties of numbers, with 

 their solutions. This has no historical connection 

 with the later algebra introduced into Italy from the 

 east by Leonardi Bonacci of Pisa (A.D. 1202-1208), 

 and successfully cultivated in the fifteenlli and six- 

 teenth centuries by Lucas Paciolus, or de Burgo, 

 Tartaglia, Cardan, and Ferrari. Later on, we liave 

 Vieta (1540-1603), Harriot, already referred to, Wal- 

 lis, and otliers. 



Astronomy is, of course, intimately connected with 

 geometry. The most simple facts of observation of 

 tlie heavenly bodies can only be stated in geometri- 

 cal language; for instance, that the stars describe 

 circles about the Pole-star, or tliat the different posi- 

 tions of the sun among the fixed stars in the course 

 of the year form a circle. For astronomical calcula- 

 tions it was found necessary to determine the arc 

 of a circle by means of its chord. The notion is as 

 old as Hipparchus, a work of whom is referred to as 

 consisting of twelve books on the chords of circular 

 arcs. We have (A.D. 125) Ptolemy's 'Almagest,' 

 the first booli of whicli contains a table of arcs and 

 chords, with the method of construction ; and among 

 other theorems on the subject, he gives there the 

 theorem, afterwards inserted in Euclid (book vi. 

 prop. D), relating to the rectangle contained by the 

 diagonals of a quadrilateral inscribed in a circle. The 

 Arabians made tlie improvement of using, in place of 

 the chord of an arc, the sine, or half- chord of double 

 the arc, and so brought the theory into the form in 

 which it is used in modern trigonometry. The before- 

 mentioned theorem of Ptolemy, — or, rather, a par- 

 ticular case of it, — translated into the notation of 

 sines, gives the expression for the sine of the sum 

 of two arcs in terms of the sines and cosines of the 

 component arcs, and it is thus the fundamental 

 theorem on the subject. We have in the fifteentli 

 and sixteenth centuries a series of mathematicians, 

 who, with wonderful enthusiasm and perseverance, 

 calculated tables of the trigonometrical or circu- 

 lar functions, — Purbach, Miiller or Kegiomontanus, 



