to 1 , and then to add 2 to the result; and 1 + ( 3 + 2 ) directs us first to add 2 to 3 , and then to add the result to 1 . Again a numerical example of equation (5) is 2 × ( 3 + 4 ) = ( 2 × 3 ) + ( 2 × 4 ) . We perform first the operations in brackets and obtain 2 × 7 = 6 + 8 which is obviously true.
BnoteB (136).–-This fundamental ratio is called the eccentricity of the curve. The shape of the curve, as
distinct from its scale or size, depends upon the value of its eccentricity. Thus it is wrong to think of ellipses in general or of hyperbolas in general as having in either case one definite shape. Ellipses with different eccentricities have different shapes, and their sizes depend upon the lengths of their major axes. An ellipse with small eccentricity is very nearly a circle, and an ellipse of eccentricity only slightly less than unity is a long flat oval. All parabolas have the same eccentricity and are therefore of the same shape, though they can be drawn to different scales.
CnoteC (204).–-If a series with all its terms positive is
convergent, the modified series found by making some terms positive and some negative according to any definite rule is also convergent. Each one of the set of series thus found, including the original series, is called "absolutely convergent." But it is possible for a series with terms partly positive and partly negative to be convergent, although the corresponding series with all its terms positive is divergent. For example, the series is convergent though we have just proved that is divergent. Such convergent series, which are not absolutely convergent, are much more difficult to deal with than absolutely convergent series.
[Note on the Study of Mathematics]Bibliography
The difficulty that beginners find in the study of this science is due to the large amount of technical detail which has been allowed to accumulate in the elementary text-books, obscuring the important ideas.
The first subjects of study, apart from a knowledge of arithmetic which is presupposed, must be elementary geometry and elementary algebra. The courses in both subjects should be short, giving only the necessary ideas; the algebra should be studied graphically, so that in practice the ideas of elementary coordinate geometry are also being assimilated. The next pair of subjects should be elementary trigonometry and the coordinate geometry of the straight line and circle. The latter subject is a short one; for it really merges into the algebra. The student is then prepared to enter upon conic sections, a very short course of geometrical conic sections and a longer one of analytical conics. But in all these courses great care should be taken not to overload the mind with more
detail than is necessary for the exemplification of the fundamental ideas.
The differential calculus and afterwards the integral calculus now remain to be attacked on the same system. A good teacher will already have illustrated them by the consideration of special cases in the course on algebra and coordinate geometry. Some short book on three-dimensional geometry must be also read.
This elementary course of mathematics is sufficient for some types of professional career. It is also the necessary preliminary for any one wishing to study the subject for its intrinsic interest. He is now prepared to commence on a more extended course. He must not, however, hope to be able to master it as a whole. The science has grown to such vast proportions that probably no living mathematician can claim to have achieved this.