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The present work on Trigonometry is intended first for students who are beginning the subject but are hoping to proceed to more advanced mathematics, and secondly for those who are wishing to revise their study of Trigonometry and to extend it beyond the limits of an ordinary elementary text-book.

It is in Trigonometry that the student is first introduced to almost all the principles which lie at the foundation of mathematics. Here negative and (so-called) imaginary symbols receive a use and meaning; symbols of quantity, of function, and of operation are distinguished from one another; ideas relating to limits, and to the indefinitely small and indefinitely great are brought into prominence; methods of approximation and errors of measurement are recognised; and the principle of continuity is made of service. With the purpose of bringing out clearly the train of reasoning required to establish and expound these principles, it has been thought desirable to make short digressions into Geometry, Algebra, and the Theory of Equations.

A description of the plan of the work may be given. It is divided into two parts, headed, respectively, Geometrical and Algebraical. In the first, Geometrical applications and methods are prominent; in the second, the purely theoretical and analytical side is developed. The first part culminates in the properties of the points and circles connected with triangles and rectilineal figures. A


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chapter has been introduced here on the Geometry of the Triangle which is intended to connect the purely geometrical with the trigonometrical treatment. Here symmetry and completeness demand the introduction of the cosine-circle (whose centre, the symmedian point, has the same relation 'to the centre of gravity that the circumcentre has to the orthocentre). The other related points and circles associated with the names Lemoine, Tucker, Brocard naturally follow. In the treatment of this branch of the subject, the writer is mainly indebted to Prof. Casey's 'Sequel to Euclid': but the theorems and proofs are arranged and formulated anew, and the terminology of ‘Modern and Analytical Geometry is as far as possible avoided. The useful term 'antiparallel is, however, introduced from the above text-book. The chapter is of course a mere introduction, which it is hoped will be useful for students interested in the modern developments of Geometry.

The algebraical part deals with Logarithms, the application of signs to Trigonometry, developments of the formula for the sums of angles, factorisation and summation. All the formulæ are first established in a simple manner independently of De Moivre's Theorem. Several of the methods of proof in Chapter xiv. on Multiple Angles are new. The treatment of infinite series is introduced by a few comprehensive theorems on convergency of series, which will prepare the student for some of the particular difficulties attaching to trigonometrical series. A modification is given of Euler's proof of the Binomial Theorem, by which it is made to depend directly on the Index Theorem. In this way the indeterminateness of fractional powers, which has to be noted in the expansions arising from De Moivre's Theorem, can be treated in close connection with the Binomial Expansion

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The introduction of imaginaries (and of De Moivre's Theorem) is postponed to Chapter xxi., where a careful determination of the values of expansions in the case of fractional indices is given. Here also an attempt is made to justify the use of imaginary indices. In the last chapter an interpretation of 1(-1) is given, by which it is hoped that the transition has been made easy from the point of view of Argand to that of Hamilton's quaternions.

To students who are reading the subject for the first time it should be explained that the chapters have been arranged with a regard rather to logical sequence than to gradation in difficulty. Hence a modification in the order of reading the book might be useful to many students. Thus Chapter 11. might be postponed until Chapter xix. is reached : for the proofs in Chapter V. might be substituted those in Chapter XIII., and any part of Part II. might be studied before Chapters IX. and x. are mastered.

The Rev. J. P. Taylor, M.A., gave in the Quarterly Journal of Mathematics, Vol. XIII., p. 197, an extremely simple proof that the nine-points circle touches the inscribed and escribed circles. I have ventured to introduce a proof on p. 139 of this theorem, which is in substance borrowed from the above contribution, but has been simplified by the omission of any reference to the method of Inversion.

For much time and trouble spent in correcting the proofs I have to express gratitude to my brother G. W. Johnson, M.A., late scholar of Trinity College, Cambridge, whose careful accuracy

has done much to free the work from errors. Corrections of such mistakes as remain and suggestions for improvement will be gratefully received.


March 28, 1889.

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