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Also tn Three Tarts: Graphs mid Curve Tracing. In tlie preparation of tins text-book on Analytical Geometry it has been our aim not merely to give an analytical treat- ment of curves of the second degree, but also to apply the methods of oloinontary algebra Recent homosexual jpboys ck selection c the tracing' of curves of higher degrees. Many of the curves usually classed as Higlier Plano Curves and discussed in treatises on tlio Oaleuhis are easily handled by elementary methods, and give the beginner a much bettor knowledge of the value of analysis than can bo derived from a study of tbe conic Bcctions alone.
Graphical work is now so common in the early stages of every mathematical course that it is fair to assume that every reader has some previous acquaintance with tho graphical interpretation of equations of a simple typo. Indeed, the analytical treatment of tho straight line and circle is necessary, not so much for the geometrical results as for the acquisition of facility in the use and interpretation of formulae ; only by such practice can the beginner learn to see the geometry behind the analysis, These chapters include a discussion of Harmonic Ranges and Pencils and of the usual theorems on the Circle, including Coaxal systems.
The ninth chapter contains tho equations of tho Conchoid, the Cisaoicl and the "Witch, with the usual applica- tions to the trisection of an angle and the duplication oS the cube ; experience proves that these curves arc of real interest and stimulate pupils to further study, A numhor of worked examples on loci and two sets of Miscellaneous Examples conclude this section, Chapters X,"XVII, discuss the graphical representation of equations.
The aim of these sections is to enable tho student to sketch pretty rapidly the forms of tho curvoH Tepresented by algebraic equations that are not of yovy complicated types; Recent homosexual jpboys ck selection c work on the equations of loci in tho earlier parts of the book will have suggested the necessity of this study.
Considerable stress is laid on the method ol: Many properties of tlie curves are most easily handled by the methods of Euclidean Geometry, and wo have not hesitated to adopt such Recent homosexual jpboys ck selection c when there was distinct advantage in doing so, with the result that wo have been able to incorporate the essentials of the oklor treatises on Geometrical Conics.
It is hard to justify the separation of geometrical and analytical conics ; at any rate it has seemed to us that sucli separation is totally unwaiTanted, and is even mischievous in an elementary text-book, Wo have tried to include all the important properties of conics that are of an elementary character, and to group them into a comparatively small number of theorems, so Recent homosexual jpboys ck selection c the student may not bo burdened by being confronted with propositions that aro of no special importance, The numerous Exorcises that are given in every chapter provide ample practice, both on tlie geometrical and on the analytical aspects of the treat- ment, and will, wo hope, be found useful in omphasiring the fact that, after all, the one subject, of study is geometry, even though the methods are twofold, The simplicity introduced by the use of Joachimsthars Section-Equation is, we think, sufficient warrant for the place assigned to it ; X ANAliYTIOAL aEOMETRY.
Comparatively little stress has boon laid on the General Equation of the Second Degree; its importance in an elementary coiu'se does nob seem to us to demand a fuller treatment than has been given to it. Professor Chrystal's text-books on Algebra are so funda- mental in their character that it is impo.
Mucli of our work is little more than a restatement of the ideas tliere laid down. Eliniiimtion is often a difiicult ainl tedious process, and may in many cases be dispensed so far as the repi-esentation of a curve is concerned, hy making use of Freedom Equations.
The Answers to all the Examples have been worked out by Mr, A. We would also gmtr- fully acknowledge the excellence of the work done hy Messrs. Coordinates of a Point 9. Gradient of a Straight Lino Eormuia for Gracliont SO Length of Porpcnclioular Exercises X. Anglu hutwoon Two Linos 0fi 3li. Cloi iKfid on t Points - Coaxal CiTolcs - Polo and Polar Tho Polar ns a Loons - 18 1 Tho Conolioid of Nicomcdofl M8 Oo.
Gcomotrical Problems Triseotioii of an Anglo Tho Cissoid of Dioolos J4! Tho DupUoation of Ibo Cnbo The Witoh of Agnesi The Parabola Tho Ellipse Tho Tlyporbola Conio Scotions Tho Convorso Problem Rough Form of a Graph. Grapli of Em'cma XXU. The Polyjiomial for Small Recent homosexual jpboys ck selection c of ,i; - Gradient of a Grai h t Somo Theoronia on Hoots of Equation a. Majeima and Minima Combination of two Graphs and the Chord Rnlo - General Statement of Rules.
Division by 55oi'o - iia Infinite lioot of a Simi lo Eiinatioa - - 1L7. To find Asymptotes - Tangonl at Origin J2.