22 CARNEGIE INSTITUTION OF WASHINGTON. 



volumes issued, only two classes of them, selected mainly for the 

 purpose of showing trends of progress, may be referred to here. 

 The most elementary, the most essential, and hence the most 

 widely used, if not esteemed, of the sciences is arithmetic. It 

 is a fundamental requisite, in fact, of all exact knowledge. 

 Ability to add, subtract, multiply, and divide affords probably 

 the simplest test of capacity for correct thinking. Conversely, 

 inability or indisposition to make use of these simple operations 

 affords one of the surest tests of mental deficiency, as witnessed, 

 for example, by numerous correspondents who are unable to 

 or who refuse to apply these operations to the finances of the 

 Institution. But the familiar science of arithmetic lies at the 

 foundation also of a much larger and a far more complex 

 structure called the theory of numbers. This theory has been 

 cultivated by many of the most acute thinkers of ancient and 

 modern times. It has more points of contact with quantitative 

 knowledge in general than any other theory except the theory of 

 the differential and integral calculus. These two theories are 

 complementary, the first dealing with discrete or discontinuous 

 numbers and the second with fluent or continuous numbers. 

 Naturally, a subject which has attracted the attention of nearly 

 all of the great mathematicians of the past twenty centuries 

 has accumulated a considerable history. The more elementary 

 contributions of Euclid, Diophantus, and others of the Greek 

 school; the extensions of Fermat, Pascal, Euler, Newton, Ber- 

 noulli and many others in the seventeenth and the eighteenth 

 centuries; and the work of Lagrange, Laplace, Gauss, and their 

 numerous contemporaries and successors of the nineteenth cen- 

 tury, make up an aggregate which has stood hitherto in need 

 of clear chronological tabulation and exposition. This laborious 

 task was undertaken about ten years ago by a Research Asso- 

 ciate of the Institution, Professor Leonard E. Dickson, of the 

 University of Chicago. A publication under the title ''History 

 of the Theory of Numbers" has resulted, and volume I (8vo, 

 XII + 486 pp.), devoted to divisibihty and to primality of 

 numbers, has appeared during the past year; and a second 

 volume devoted to diophantine analysis is now in press. This 

 work is remarkable for its condensation of statement. It con- 

 tains more information per unit area than any other work issued 



