G. S. Carr, M.A. (“The Textbook that Unleashed Ramanujan’s Genius”)
THE work, of which the part now issued is a first instalment, has been compiled from notes made at various periods of the last fourteen years, and chiefly during the engagements of teaching. Many of the abbroviated methods and mnemonic rules are in the form in which I originally wrote them for my pupils.
The general object of the compilation is, as the title indicates, to present within a moderate compass the fundamental theorems, formulae, and processes in the chief branches of pure and applied mathematics.
The work is intended, in the first place, to follow and supplement the rise of the ordinary text-books, and it is arranged with the view of assisting the student in the task of revision of book-work. To this end I have, in many cases, merely indicated the salient points of a demonstration, or merely referred to the theorems by which the proposition is proved. I am convinced that it is more beneficial to the student to recall demonstrations with Bach aids, than to read and re-read them. Let them be read once, but recalled often. The difference in the effect upon the mind between reading a mathematical demoustration, and originating one wholly or partly, is very great. It may be compared to the difference between the pleasure experienced, and interest aroused, when in the one case a traveller is passively conducted through the roads of a novel and uncxplored country, and in the other case he discovers the roads for himself with the assistance of a map.
In the second place, I venture to hope that the work, when completed, may prove useful to advanced students as an aide-meirwire and book of reference. The boundary of mathematical science forms, year by year, an ever widening circle, and the advantage of having at hand some condensed Btatement of results becomes more and more evident.
To the original investigator occupied with abstruse researches in some one of the many branches of mathematics, a work which gathers together synopticaUy the leading propoBitions in aU, may not therefore prove unacceptable. ALler hands than inine undoubtedly, might have undertaken the task of making such a digest ; but abler hands might also, perhaps, be more usefully employed,—and vith this reflection I have the less hesitation in commenciiig the work myself. The design which I have indicated is eomevhat comprehensive, and in relation to it the present essay may be regarded as tentative. The degree of success which it may meet witli, and the suggestions or criticisms which it may call forth, will doubtI”; i have theu- effect on the subsequent portions of the work. With respect to tlie abridgment of the demonstrations, I may remark, that while some diffuseness of explanation is not only allowable but very desirable in an initiatory treatise, concisencss is one of the chief requirements in a work intended for the purposes of revision and reference only. In order, however, not to sacrifice clearness to conciseness, mucli more labour has been expended upon this part of the subject-matter of the book than wiU at first sight be at all evident. The only palpable result being a compression of the text, the resiilt is so far a negative one. The amount of compression attamed is illustrated in the last section of the present part, in which more than the number of propositions usually given in treatises on Geometrical Conics are contained, together with the figures and demonstrations, m the space of twenty-four pages.
The foregoing remarks have a general application to the work as a whole. With the view, however, of makmg the earlier sections more acceptable to beginners, it •wiU be found that, in those sections, important principles have sometimes been more fully elucidated and more illustrated by examples, than the plan of the work would admit of in subsequent divisions.
A feature to which attention may be directed is the uniform system of reference adopted throughout aU the sections. With the object of facilitating such reference, the articles have been numbered progressively from the commencement in large Clarendon figures; the breaks which will occasionally be found in these numbers having been purposely made,in order to leave room for the insertion of additional matter, if it should be required in a future edition, -without disturbing the original numbers and references. With the same object, demonstrations and examples have been made subordinate to enimciations and formulse, the former beiug printed in small, the latter in bold type. By tlieso aids, tlic interdcpendt’nco of propositions is more readily sliown, and it Lccomes ca8y to trace the connuxion between thcorcms in different branches of mutliomatics, witliout the loss of time which would be incurred in turning to separate treatises on the subjects. The advantage thus gained will, however, become more apparent as the work proceeds. The Algebra section was priuted some years ago, and does not quite correspond with the succeeding ones in some of the particulars named above. Under the pressure of other occupations, this section moreover wag not properly revised before going to press. On that account the table of errata will be found to apply almost exclusively to errors in tliat section; but I trust that the list is exhiuistivo. Great pains have been taken to secure the acciiracy of the rest of tlie volume. Any intimation of errors w-111 bo gladly received.
I have now to ackiiowledgc some of the sources from which the present part has been compiled. In the Algebra, Theory of Equations, and Trigonomctry scctioiis, I am liirgely indebted to Todhuntcr’s well-known treatises, tlie accuracy aiiil completeness of wliich it would be suporfluous iu mo to dwull upon.
In the section entitled Elementary Geometry, I liave added to simpler propositioiis a selection of tlicorcms from Town. seiid’s ifodrru Geometry and Saliuon’s Conic Sections.
In Geometrical Couics, the liue of domonstration followrd ngrecs, in the main, with that adopted iu Drew’s treatise on tlio subject. I am inclined to think tliafc the method of tliat author cannot bo much improved. It is true thiiA soine iinportant properties of the ellipse, which arr arrived at…..
This, along with many other math posts is done in memory of Richard Askey.