% : ADDRESS. 21 
of the word a prime number, may very well be the product of two 
integers of the form a + b./2, and consequently not a prime number in 
-the new sense of the word. Among the incommensarables which can be 
thus introduced into the Theory of Numbers (and which was in fact first 
‘so introduced) we have the imaginary ¢ of ordinary analysis: viz. we may 
consider numbers a+bi (a and 6 ordinary positive or negative integers, 
not excluding zero), and, calling these integer numbers, establish in 
regard to them a theory analogous to that which exists for ordinary real 
integers. The point which I wish to bring out is that the imaginary 7 
‘does not in the Theory of Numbers occupy a unique position, such as it 
does in analysis and geometry ; it is in the Theory of Numbers one out of 
an indefinite multitude of incommensurables. 
I said that I would speak to you, not of the utility of mathematics 
‘in any of the questions of common life or of physical science, but rather 
of the obligations of mathematics to these different subjects. The con- 
‘sideration which thus presents itself is in a great measure that of the 
history of the development of the different branches of mathematical 
‘Science in connection with the older physical sciences, Astronomy and 
‘Mechanics: the mathematical theory is in the first instance suggested by 
‘some question of common life or of physical science, is pursued and 
-studied quite independently thereof, and perhaps after a long interval 
-comes in contact with it, or with quite a different question. Geometry 
‘and algebra must, I think, be considered as each of them originating in 
connection with objects or questions of common life—geometry, notwith- 
standing its name, hardly inthe measurement of land, but rather from the 
contemplation of such forms as the straight line, the circle, the ball, the 
top (or sugar-loaf): the Greek geometers appropriated for the geometrical 
forms corresponding to the last two of these, the words opaipa and xavoe, 
our cone and sphere, and they extended the word cone to mean the 
complete figure obtained by producing the straight lines of the surface 
-both ways indefinitely. And so algebra would seem to have arisen from 
_ the sort of easy puzzles in regard to numbers which may be made, either 
‘in the picturesque forms of the Bija-Ganita with its maiden with the 
beautiful locks, and its swarms of bees amid the fragrant blossoms, and 
the one queen-bee left humming around the lotus flower ; or in the more 
prosaic form in which a student has presented to him in a modern text- 
book a problem leading to a simple equation. 
The Greek geometry may be regarded as beginning with Plato 
(B.C. 430-347): the notions of geometrical analysis, loci, and the conic 
sections are attributed to him, and there are in his Dialogues many very 
interesting allusions to mathematical questions: in particular the passage 
in the ‘ Theeetetus,’ where he affirms the incommensurability of the sides 
of certain squares. But the earliest extant writings are those of Euclid 
‘(B.c. 285): there is hardly anything in mathematics more beautiful 
‘than his wondrous fifth book; and he has also in the seventh eighth, 
