An Introduction to Mathematics · Alfred North Whitehead
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.
Passing to the serious treatises on the subject to be read after this preliminary course, the following may be mentioned: Cremona's Pure Geometry (English Translation, Clarendon Press, Oxford), Hobson's Treatise on Trigonometry, Chrystal's Treatise on Algebra (2 volumes), Salmon's Conic Sections, Lamb's Differential Calculus, and some book on Differential Equations. The student will probably not desire to direct equal attention to all these subjects, but will study one or more of them, according as his interest dictates. He will then be prepared to select more advanced works for himself, and to plunge into the higher parts of the subject. If his interest lies in analysis, he should now master an elementary treatise on the theory of Functions of the Complex Variable; if he prefers to specialize in Geometry, he must now proceed to the standard treatises on the Analytical Geometry of three dimensions. But at this stage of his career in learning he will not require the advice of this note.
I have deliberately refrained from mentioning any elementary works. They are very numerous, and of various merits, but none of such outstanding superiority as to require special mention by name to the exclusion of all the others.
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