Part 1, sect. 1.

=1640.= Many of the greatest masters of the mathematical sciences were first attracted to mathematical inquiry by problems relating to numbers, and no one can glance at the periodicals of the present day which contain questions for solution without noticing how singular a charm such problems still continue to exert. The interest in numbers seems implanted in the human mind, and it is a pity that it should not have freer scope in this country. The methods of the theory of numbers are peculiar to itself, and are not readily acquired by a student whose mind has for years been familiarized with the very different treatment which is appropriate to the theory of continuous magnitude; it is therefore extremely desirable that some portion of the theory should be included in the ordinary course of mathematical instruction at our University. From the moment that Gauss, in his wonderful treatise of 1801, laid down the true lines of the theory, it entered upon a new day, and no one is likely to be able to do useful work in any part of the subject who is unacquainted with the principles and conceptions with which he endowed it.--GLAISHER, J. W. L.

_Presidential Address British Association for the Advancement of Science (1890); Nature, Vol. 42, p. 467._

=1641.= Let us look for a moment at the general significance of the fact that calculating machines actually exist, which relieve mathematicians of the purely mechanical part of numerical computations, and which accomplish the work more quickly and with a greater degree of accuracy; for the machine is not subject to the slips of the human calculator. The existence of such a machine proves that computation is not concerned with the significance of numbers, but that it is concerned essentially only with the formal laws of operation; for it is only these that the machine can obey--having been thus constructed--an intuitive perception of the significance of numbers being out of the question.--KLEIN, F.

_Elementarmathematik vom höheren Standpunkte aus. (Leipzig, 1908), Bd. 1, p. 53._

=1642.= Mathematics is the queen of the sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.--GAUSS.

_Sartorius von Waltershausen: Gauss zum Gedächtniss. (Leipzig, 1866), p. 79._

=1643.=

Zu Archimedes kam ein wissbegieriger Jüngling, Weihe mich, sprach er zu ihm, ein in die göttliche Kunst, Die so herrliche Dienste der Sternenkunde geleistet, Hinter dem Uranos noch einen Planeten entdeckt. Göttlich nennst Du die Kunst, sie ist’s, versetzte der Weise, Aber sie war es, bevor noch sie den Kosmos erforscht, Ehe sie herrliche Dienste der Sternenkunde geleistet, Hinter dem Uranos noch einen Planeten entdeckt. Was Du im Kosmos erblickst, ist nur der Göttlichen Abglanz, In der Olympier Schaar thronet die ewige Zahl. --JACOBI, C. G. J.

_Journal für Mathematik, Bd. 101 (1887), p. 338._

To Archimedes came a youth intent upon knowledge, Quoth he, “Initiate me into the science divine Which to astronomy, lo! such excellent service has rendered, And beyond Uranus’ orb a hidden planet revealed.” “Call’st thou the science divine? So it is,” the wise man responded, “But so it was long before its light on the Cosmos it shed, Ere in astronomy’s realm such excellent service it rendered, And beyond Uranus’ orb a hidden planet revealed. Only reflection divine is that which Cosmos discloses, Number herself sits enthroned among Olympia’s hosts.”

=1644.= The higher arithmetic presents us with an inexhaustible store of interesting truths,--of truths too, which are not isolated, but stand in a close internal connexion, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed.--GAUSS, C. F.

_Preface to Eisenstein’s Mathematische Abhandlungen (Berlin, 1847), [H. J. S. Smith]._

=1645.= The Theory of Numbers has acquired a great and increasing claim to the attention of mathematicians. It is equally remarkable for the number and importance of its results, for the precision and rigorousness of its demonstrations, for the variety of its methods, for the intimate relations between truths apparently isolated which it sometimes discloses, and for the numerous applications of which it is susceptible in other parts of analysis.--SMITH, H. J. S.

_Report on the Theory of Numbers, British Association, 1859; Collected Mathematical Papers, Vol. 1, p. 38._

=1646.= The invention of the symbol ≡ by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic.--MATHEWS, G. B.

_Theory of Numbers (Cambridge, 1892),