THE SCIENTIFIC SPIRIT IN ENGLAND. 



275 



mon ; ^ nor should we forget the suggestive writings of 

 George Boole.^ The influence of these men originated 

 outside of Cambridge, and a history of mathematics at 

 that university does not contain their names,^ though the 

 ideas of which they have been the bearers have largely 

 entered into the text-books and the teaching of the Cam- 

 bridge school. 



So far I have mainly dealt with one side only on which 

 the progress of science depends, namely, the methodical 

 use of experiment, measurement, and calculation : this 



quaternions comiolex quantities 

 Avhich are compounded of a purely 

 algebraical or quantitative element 

 and three distinct elements corre- 

 sponding to the three directions or 

 dimensions of sjaace. He was the 

 first to work out this calculus, and 

 the labour occupied twenty years 

 of his life. In Hamilton's calculus 

 of quaternions, distance (or length) 

 and direction are introduced as they 

 naturally present themselves when 

 we deal with geometrical or physical 

 problems, instead of all quantities 

 being reduced to lengths, as was 

 the case in the Cartesian geometry. 

 Hamilton thus broke through the 

 conventionalism of the latter and 

 showed how the consideration of di- 

 rections in space forces us to extend 

 the original operations of arithmetic. 

 It is interesting to note how simul- 

 taneously Grassmann (see p. 243, 

 note 1) iu his ' Ausdehnungslehre ' 

 (1844) and Von Staudt in his ' Geo- 

 metrie der Lage ' (1847), quite inde- 

 jiendently worked at similar exten- 

 sions of our arithmetical and geo- 

 metrical conceptions, and how sub- 

 sequently quaternions, in which 

 Hamilton had seen a powerful me- 

 thod for solving geometrical and 

 physical problems, present them- 

 selves as a special form of the ex- 

 tended algebra and geometry elabor- 



ated from these different beginnings. 

 Whilst the practical usefulness of 

 the calculus has been demonstrated 

 by some extensive applications, as, 

 for example, to spherical trigono- 

 metry, the ideas contained in it 

 frequently without Hamilton's no- 

 tation are gradually finding their 

 way into text-books, and the strange- 

 ness which for half a century pre- 

 vented the labours of Hamilton, 

 Grassmann, and Von Staudt from 

 being generally ai^preciated, is dis- 

 appearing. A popular exposition 

 of the relation of quaternions to 

 general arithmetic is given in 0. 

 Stolz, ' Grossen und Zahlen,' Leip- 

 zig, Teubner, 1891. 



^ The excellent treatises of Sal- 

 mon on ' Higher Algebra,' ' Higher 

 Plane Curves,' ' Geometry of Three 

 Dimensions,' and 'Conic Sections' 

 have in their German translations 

 by Fiedler done a great work in 

 systematising and popularising mo- 

 dern conceptions in algebra and 

 geometry. See Giuo Loria's treatise 

 on the " Principal Theories of Geo- 

 metrj'" in the German translation 

 by Schiitte, Leipzig, 1888, p. 25, 

 &c. 



- See p. 247, note 2. 



^ See Rouse Ball, ' A History of 

 the Study of Mathematics at Cam- 

 bridge,' 1889. 



