Xll CONTENTS. 



584 ; Criticism of pretension of statistics, 586 ; Historical criticism, 

 588 ; Application in physics, 589 ; Clausius and Clerk - Maxwell, 590 ; 

 Mathematical representation of experimental laws, 592 ; Irreversibility 

 of natural processes, 593 ; Lord Kelvin, 594 ; " Availability " a theorem 

 in probability, 597; "Selection" as conceived by Clerk-Maxwell, 598; 

 Statistical knowledge of nature, 600 ; As opposed to historical and 

 mechanical knowledge, 603 ; Sameness and variation, 607 ; Darwin, 

 608; Galton, 609; " Pangenesis," 610; Lends itself to statistical 

 treatment, 611 ; Problem of Heredity, 613 ; Mr Bateson's historical 

 treatment, 615; " Particulate " descent, 615; Application of theory 

 of error, 618 ; Difference in application to living and lifeless units, 

 620 ; Professor Pearson : The mathematical problem, 621 ; Statistical 

 knowledge one-sided, 624 ; Critical methods, 626 ; The instrument 

 of exact research, 626. 



CHAPTER XIII. 



ON THE DEVELOPMENT OF MATHEMATICAL THOUGHT DURING 

 THE NINETEENTH CENTURY. 



History of thought, 627 ; Difference between thought and knowledge, 628 ; 

 Popular prejudices regarding mathematics, 628 ; Use of mathematics, 

 630 ; Twofold interest in mathematics, 632 ; Origin of mathematics, 

 634 ; Gauss, 636 ; Cauchy, 636 ; Process of generalisation, 638 ; Inverse 

 operations, 639 ; Modern terms indicative of modern thought, 643 ; 

 Complex quantities, 644 ; The continuous, 644 ; The infinite, 644 ; 

 Doctrine of series : Gauss, 645 ; Cauchy's Analysis, 647 ; Revision of 

 fundamentals, 649 ; Extension of conception of number, 650 ; The 

 geometrical and the logical problems, 651 ; Quaternions, 654 ; Founda- 

 tions of geometry, 656 ; Descriptive geometry, 658 ; Poncelet, 659 ; 

 Character of modern geometry, 662 ; Method of projection, 663 ; Law 

 of continuity, 664 ; Ideal elements, 664 ; Principle of duality, 665 ; 

 Reciprocity, 666 ; Steiner, 667 ; Mutual influence of metrical and pro- 

 jective geometry, 668 ; Pliicker, Chasles, Cayley, 671 ; Historical and 

 logical foundations, 671 ; Generalised co-ordinates, 673 ; Ideal elements, 

 674 ; Invariants, 676 ; Theory of forms, 678 ; Theory of numbers, 680 ; 

 Symmetry, 681 ; Determinants, 682 ; Calculus of operations, 684 ; Prin- 

 ciple of substitution, 686 ; General solution of equations, 687 ; Theory 

 of groups, 689 ; Continuous and discontinuous groups, 691 ; Theory of 

 functions, 693 ; Physical analogies, 696 ; The potential, 698 ; Riemann, 

 700 ; Weierstrass, 702 ; Riemann and Weierstrass compared, 707 ; Ex- 

 amination of foundations, 709 ; Non-Euclidean geometry, 712 ; Curva- 



