ADDRESS. 29 



which still survive as figures of speech among ourselves, it supplied a voca- 

 bulary for the moral notions of mankind, and quickened them by <nvino- 

 them the power of expression. In this lies the great and enduring interest 

 in the fragments which remain to us of the Pythagorean philosophy. 



The consecutive processes of Mathematics led to the consecutive pro- 

 cesses of Logic ; but it was not until long after mankind had attained to 

 abstract ideas that they attained to any clear notion of their connexion 

 with one another. In process of time the leading ideas of Mathematics 

 became the leading ideas of Logic. The " one " and the " many " passed 

 into the " whole " and its " parts ; " and thence into the " universal " and 

 the "particular." The fallacies of Logic, such as the well-known puzzle 

 of Achilles and the tortoise, partake of the nature of both sciences. 

 And perhaps the conception of the infinite and the infinitesimal, as well 

 as of negation, may have been in early times transferred from Logic to 

 Mathematics. But the connexion of our ideas of number is probably 

 anterior to the connexion of any of our other ideas. And as a matter of 

 fact, geometry and arithmetic had already made considerable progress 

 when Aristotle invented the syllogism. 



Genei'al ideas there were, beside those of mathematics — true flashes of 

 genius which saw that there must be general laws "to which the universe 

 conforms, but which saw them only by occasional glimpses, and through 

 the distortion of imperfect knowledge ; and although the only records of 

 them now remaining are the inadequate representations of later writers 

 yet we must still remember that to the existence of such ideas is due not 

 only the conception but even the possibility of Physical Science. But 

 these general ideas were too wide in their grasp, and in early days at 

 least were connected to their subjects of application by links too shadowy, 

 to be thoroughly apprehended by most minds ; and so it came to pass 

 that one form of such an idea was taken as its only form, one application 

 of it as the idea itself ; and philosophy, unable to maintain itself at the- 

 level of ideas, fell back upon the abstractions of sense, and, by preference 

 upon those which were most ready to hand, namely, those of mathematics. 

 Plato's ideas relapsed into a doctrine of numbers ; mathematics into mys- 

 ticism, into neo-Platonism, and the like. And so, through many lono- 

 ages, through good report and evil report, mathematics have always held 

 an unsought-for sway. It has happened to this science as to many other 

 subjects, that its warmest adherents have not always been its best friends. 

 Mathematics have often been brought into matters where their presence 

 has been of doubtful utility. If they have given precision to literary 

 style, that precision has sometimes been carried to excess, as in Spinoza 

 and perhaps Descartes ; if they have tended to clearness of expression in 

 philosophy, that very clearness has sometimes given an appearance of 

 finality not always true ; if they have contributed to definition in theology 

 that definiteness has often been fictitious, and has been attained at the 

 cost of spiritual meaning. And, coming to recent times, although we 



