Sept. 20, 1883] 



NATURE 



495 



for the other area the boundary will be a circle ; if for one it is an 

 equilateral triangle, then for the other it will be an equilateral 

 triangle. 



I have been speaking of an imaginary variable (,v + iy), 

 and of a function <J> (x + iy) = X + <Y of that variable, but 

 the theory may equally well be stated in regard to a plane 

 curve : in fact the * + iy and the X + t'Y are two ima- 

 ginary variables connected by an equation ; say their values 

 are u and v, connected by an equation F («, v) = o ; then, 

 regarding 11, v as the coordinates of a point in piano, this will 

 be a point on the curve represented by the equation. The curve, 

 in the widest sense of the expression, is the whole series of 

 points, real or imaginary, the coordinates of which satisfy the 

 equation, and these are exhibited by the foregoing corresponding 

 figures in two planes ; but in the ordinary sense the curve is the 

 series of real points, with coordinates «, v, which satisfy the 

 equation. 



In geometry it is the curve, whether denned by means of its 

 equation, or in any other manner, which is the subject for con- 

 templation and stu iy. But we also use the curve as a repre- 

 sentation of its equation — that is, of the relation existing between 

 two magnitudes x, y, which are taken as the coordinates of a 

 point on the curve. Such employment of a curve for all sorts of 

 purposes — the fluctuations of the barometer, the Cambridge 

 boat races, or the Funds — is familiar to most of you. It is in 

 like manner convenient in analysis, for exhibiting the relations 

 between any three magnitudes x, y, 2, to regard them as the co- 

 ordinates of a point in space ; and, on the like ground, we 

 should at least wish to regard any four or more magnitudes as 

 the coordinates of a point in space of [a corresponding number 

 of dimensions. Starting with th'5 hypothesis of such a space, 

 and of points therein each determined by means of its coor- 

 dinates, it is found possible to establish a system of M-dimen- 

 sional geometry analogous in every respect to our two- and 

 three-dimensional geometries, and to a very considerable extent 

 serving to exhibit the relations of the variable-.. 



It is to be borne in mind that the space, whatever its dimen- 

 sionality may be, must always be regarded as an imaginary or 

 complex space such as the two- or three-dimensional space of 

 ordinary geometry ; the advantages of the representation would 

 otherwise altogether fail to be obtained. 



I omit s ime further developments in regard to geometry ; and 

 all that I have written as to the connection of mathematics with 

 the notion of time. 



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

 mathematics in any of the questions of common life or of physi- 

 cal science, but rather of the obligations of mathematics to 

 these different subjects. The consi leration which thus presents 

 itself is in a great measure that of the history of the development 

 of the different branches of mathematical science in connection 

 with the older physical sciences, astronomy and mechanics : the 

 mathematical theory is in the first 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 ques- 

 tion. Geometry and algebra must, I think, be con-idered as 

 each of them originating in connection with objects or questions 

 of common life — geometry, notwithstanding its name, hardly in 

 the mea-urement 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 geometri- 

 cal forms corresponding to the last two of these, the words 

 atyoipa and k£vos, our cone and sphere, 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 ti have arisen from the sort of easy puzzles 

 in regard to numbers which may be made, either in the pic- 

 turesque forms of the Bija-Ganita with its maiden with the 

 beautiful locks, and its swarms of bees amid the fragrant blos- 

 soms, and the one queen bee left humming around the lotus 

 flower ; or in the more prosaic forrn in which a student has pre- 

 sented to him in a modern textbook a problem leading to a 

 simple equation. 



The Greek geometry may be regarded as beginning with Plato 

 (B.C. 430-347) : the notions of geometrical analysis, loci, and 

 the conic sections are attributed to him, and there are in his 

 "Dialogues " many very interesting allusions to mathematical ques- 

 tions : in particular the passage in the "The;etetus," where he 



affirms the incommensurability of the sides of certain squares. 

 But the earliest extant writings are those of Euclid (B.C. 285): 

 there is hardly anything 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 prin- 

 ciples of the Theory of Numbers, including the theory of incom- 

 mensurables. We have next Apollonius (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 

 (including therein hydrostatics): bis dictum about the lever, his 

 " Eiip7)Ko," and the story of the defence of Syracuse, are well 

 known. Following these we have a worthy series of names, 

 including the astronomers Hipparchus (B.C. 150) and Ptolemy 

 (A.D. 125), and ending, say, with Pappus (a.d. 400), but con- 

 tinued by their Arabian commentator , and the Italian and other 

 European geometers of the sixteenth century and later, who 

 pursued the Greek geometry. 



The Greek arithmeiic was, from the want of a proper nota- 

 tion, singularly cumbrous and difficult ; nnd it was for astrono- 

 mical purposes superseded by the sexagesimal arithmetic, attri- 

 buted to Ptolemy, but probably known before his time. The 

 use of the present so-called Arabic figures became general 

 among Arabian writers on arithmetic 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 Dio- 

 phantus (a.d. 150), in fact a work on the Theory of Numbers 

 containing questions relating to square and cube numbers, and 

 other properties of numbers, w ith 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- 1 20S), 

 and successfully cultivated in the fifteenth and sixteenth centuries 

 by Lucas Paciolus, or De Burgo, Tartaglia, Cardan, and Ferrari. 

 Later on we have Vieta (1540-1603), Harriot, already referred 

 to, Wallis, and others. 



Astronomy is of course intimately connected with geometry ; 

 the most simple facts of observation of the heavenly bodies can 

 only be staled in geometrical language ; for instance, that the 

 stars describe circles about the Pole-star, or that the different posi- 

 tions of the sun among the fixed stars in the course of the year form 

 a circle. For astronomical calculations 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 con- 

 sisting of twelve books on the chords of circular arcs ; we have 

 (a.d. 125) Ptolemy's " Almagest," the first book of which con- 

 tains a table of arcs and chords with the method of construc- 

 tion ; and among other theorems on the subject he gives there 

 the theorem afterwards inserted in Euclid (Bock VI. Prop. D.) 

 relating to the rectangle contained by the diagonals of a quadri- 

 lateral inscribed in a circle. The Arabians made the improve- 

 ment 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 .he 

 form in which it is used in modern trigonometry : the before- 

 mentioned theorem of Ptolemy, or rather a particular 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 fifteenth and sixteenth centuries 

 a series of mathematicians who with wonderful enthusiasm and 

 perseverance calculated tables of the trigonometrical or circular 

 functions, Purbach, Muller or Regiomontanus, Copernicus, 

 Keinhold, Maurolycus, Vieta, and many others ; the tabulations 

 of the functions tangent and secant are due to Reinhold and 

 Maurolycus respectively. 



Logarithms were invented, not exclusively with reference to 

 the calculation of trigonometrical tables, but in order to facilitate 

 numerical calculations generally ; the invention is due to John 

 Napier of M rchiston, who died in 1618 at sixty-seven years of 

 age ; the notion was based upon refined mathematical reasoning 

 on the comparison of the spaces described by two points, the one 

 moving with a uniform velocity, the other with a velocity 

 varying according to a given law. It is to be observed that 

 Napier's logarithms were nearly but not exactly those which are 

 now called (sometimes Napierian, but more usually) hyperbolic 

 logarithms— those to the base e; and that the change to the 

 base 10 (the great step by which the invention was perfected for 

 the object in view) was indicated by Napier but actually made 

 by Henry Briggs, afterwards Savilirn Professor at Oxford (d. 

 1630). But it is the hyperbolic logarithm which is mathematically 

 important. The direct function e* or exp. x, which has for its 

 inverse the hyperbolic logarithm, presented itself, but not in a 



