We next allude to one of the excellent little books in "Stewart's Local Examination Series," as one in which the student may find a brief outline of the subject, but an outline only; for the work is singularly concise and compact in the statement of its propositions and proofs, and does not attempt any lengthy discussion of difficulties or illustration by means of examples. Its title is, Conic Sections Treated Geometrically; together with an Elementary Treatise on Analytical Geometry, by F. Harrison, M.A. (18., Stewart and Co.) We should have liked the work better if, as an "elementary" treatise, it had been less comprehensive and had attempted more in the shape of explanation, practical illustration, and examples. As a class-book in the hands of a competent teacher it is admirable. Private students must supplement it with other help. In one respect it is unique-it contains the subject of Polar Co-ordinates in an entirely separate part of the work, instead of mixing it up with the other class of co-ordinates. Such problems as the following are solved in this book by means of polar co-ordinates : To find the length of the perpendicular from a given point on a given straight line. Through a given point to draw a straight line parallel to a given straight line. Through a given point to draw a straight line perpendicular to x given straight line. To find the equation of the chord of a circle. To deduce the equation of the tangent from that of the chord. In the section on Cartesian Co-ordinates the axes are rectangular only, and all the propositions which are given on The Straight Line and The Circle are presented on about ten pages. The student may fittingly give his attention to such points as the following: To find the locus of the intersections of pairs of tangents that include a right angle. To find the length of the line x cosa +y sina-p=0 cut off by the circle x + y2=c2. The geometrical interpretation of the general equation of the circle. The next work is one of the well-known and able " Cambridge School and College Text-Books," published by Bell and Co., and though it is styled "elementary" it is really comprehensive and yet explanatory enough to be used alone and to fully prepare the candidate for this part of his examination. We allude to Vyvyan's Elementary Analytical Geometry (4s. 6d., Bell and Co.). We like the book exceedingly; it is clear, systematic, and elucidatory in its method. There are worked-out examples to illustrate the operations of Analytical Geometry; there are exercises in the middle of the chapters as easy applications of the book-work; more difficult but well-graduated examples at the end of each chapter; and a list of useful Formulæ at the end of the book. In the first 94 pages the candidate will find everything he will really need in both systems of co-ordinates-rectangular and polar. Without further description of the book we thoroughly commend it. Puckle's Treatise on Conic Sections and Algebraic Geometry (7s. 6d., Macmillan), although "especially designed for the use of beginners," is really a more thorough and advanced text-book than any we have yet described. Its treatment of the subject, whilst completely logical and systematic, is simple and graduated enough for the tyro who is willing at once to plunge in medias res without first having recourse to a more elementary work. The author says: My chief object was to write with special reference to those difficulties and misapprehensions which I had found most common to beginners." He has carried out this aim by attempting to remove those difficulties and explain the most common misapprehensions. His 66 'numerous examples, with hints for their solution," contribute to this end; for they are continuously and yet judiciously introduced, and illustrate well the use of the formula, which are fully investigated and discussed. Those parts of the work which can be passed over in a first reading are marked with an asterisk; and the Pass candidate may omit Chap. VI. altogether, closing his studies with Chap. VII. There are simple examples to show the method of representing loci by means of equations; the general equations are fully explained, the forms for particular cases deduced from them, and the results afterwards verified by independent geometrical considerations, especially in reference to the straight line; examples of the application of formula to the solution of geometrical problems are also given, and every part of the book required in the examination will repay attentive reading. The brief Appendix on the Properties of Quadratic Equations will remind the student how necessary is a careful application of them in Algebraic Geometry. We scarcely need describe at length Todhunter's Treatise on Plane Co-ordinate Geometry (7s. 6d., Macmillan). It is well known as admirably adapted to the requirements of the candidate for the London degree. The use of formulæ and the method of dealing with problems are not so fully illustrated in worked examples as could be desired for beginners; and yet the plan of the work is well adapted for them, inasmuch as the simpler aspects of the subject are always presented first. With this view the various propositions are generally considered in reference to rectangular co-ordinates before the other systems of co-ordinates are formally introduced. In this respect it differs from Puckle's work, which usually begins with oblique co-ordinates, and thence deduces the equations for rectangular co-ordinates. In Mr. Todhunter's book the polar equations and those which relate to oblique axes are investigated in separate sections at the end of a chapter. The former should by no means be overlooked; and the student would do well, in dealing with the different propositions in the latter work, to use the more symmetrical forms of equations to the straight line, as well as y=mx+b, which is the form most generally used by Mr. Todhunter, whose book we heartily recommend for its thoroughness, simplicity, and consecutive interest to the mind of the student. The candidate will have no need to go beyond page 105, and Chap. IV. may be omitted. PLANE TRIGONOMETRY. "Plane Trigonometry is not exactly what might be inferred from the etymology of its name."-Prof. Elliott. "Trigonometry was originally, as the name imports, the science which furnished methods for determining the magnitude of the sides and angles of triangles, but it has been extended to the treatment of all theorems involving the consideration of angular magnitudes."-J. Hamblin Smith, M.A. "But the word is now used in a much more extended sense, so as to include all manner of Algebraical reasoning about lines and angles, whether parts of a triangle or not; and, particularly, when that reasoning is carried on by means of certain quantities which are called the trigonometrical Ratios or Functions of an angle."-Dr. Colenso. "In Trigonometry we apply Algebraical symbols to establish certain relations between the magnitudes of the sides and angles of plane rectilineal figures. These relations are useful for all the higher branches of Mathematics, and are specially applicable to surveying, and to determining the heights and distances of inaccessible objects."-R. D. Beasley, M.A. Although the Regulations now particularly specify three parts of this subject which were not mentioned before, yet they do not now really require more than was implied and usually understood by candidates in previous years. The "Measurements of Angles," the "Relations of their Ratios," and the "Elementary Formula" now enumerated, were virtually included in the requirement—“ Plane Trigonometry as far as to enable the candidate to solve all the cases of Plane Triangles." "The Determination of Heights and Distances" is but the practical application of the formulæ necessary for the "solution of the several cases of Plane Triangles." THE DIFFERENT MEASUREMENTS OF ANGLES. By these measurements are meant the Sexagesimal or English Method; the Centesimal or French Method; and the Circular Method, or the system in which the unit of angular measurement is the angle subtended at the centre of a circle by an arc equal to the radius of the circle. Of these three methods the last is the most important and the most difficult to remember. On this account it is essential that the candidate should begin with a clear conception of what is meant by it, the principle that underlies it, the relative size of the angle, and the connection of the circular measure of any angle with the measure of the same angle in degrees. arc radius To remember what is meant by it think of the words arc divided by radius, or of the form for this fraction expresses the ratio of the arc subtending any angle to the radius of that circle of which the arc is a portion, and this fraction denotes what is called the Circular Measure of an Angle. When the arc= radius, that is, when the fraction = 1, we have, in the number of degrees contained in such an angle, the unit of measurement above referred to. The number of degrees in this unit is 57.29577951. Suppose A° in any angle, where A may denote any number; let a be the arc of the angle, and r the radius. Then In this case m is 57°-29578; and if we wish to find the number of degrees in the angle A, we have, if the circular measure be, A° 57°29578 x 28° 64789. "There are two methods of forming an idea of the magnitude of an angle which is estimated by the fraction arc divided by radius. Suppose, for example, we speak of the angle ; we may refer to the unit of angular measurement, which is an angle containing about 57 degrees, and imagine two-thirds of this unit to be taken; or without thinking about the unit at all, we may suppose an angle is taken such that the arc subtending it is two-thirds of the corresponding radius." In the following propositions the student will see which are the main points to be investigated. The first three relate to Circular Measure. 1. The ratio of the circumference of a circle to its diameter is the same for all circles. Or, we may say: The circumferences of circles vary as their radii. C с To remember this, think of the form = or CR, or Co R, where C and c are the circumferences of two circles, and R, r are their radii. Also that 355 Observe that 113355 is the order of the figures in if we 113 begin with the denominator. 2. The angle subtended at the centre of any circle by an arc equal to the radius is an invariable angle; that is, is the same for all circles. Remember the value of this angle in degrees. It can easily be 2 right angles 180° -57°2958 nearly, shown that this angle = π 3.14159 the number of degrees in the unit of circular measure. 3. Show that the circular measure (0) of an angle is equal to the fraction which has for its numerator the arc subtended by that angle at the centre of any circle, and for its denominator the radius of that 4. If we wish to find the number of degrees in an angle A, when the unit of angular measurement is particularly assigned, then we must multiply the number of degrees contained in the unit by the circular measure of the angle. Suppose we take the half of a right angle as our unit, and the circular measure of the angle A be f Then An =m= 45° x=22°.5. a r 5. To convert the measure of any angle expressed in the Sexagesimal Method to the corresponding measure of the same angle in the Centesimal Method, and vice versa. Think of the form A° Ao = 90 100 where A° and Ag represent the number of degrees and grades respectively in the same angle. 6. To find the circular measure of an angle when the number of degrees in it are given, and vice versa. A° 0 Think of the form 180 where A° is the number of degrees in an angle, and the circular measure of the same angle. 7. To connect the circular measure of any angle with the measure of the same angle in grades. Remember the form in the angle. А 200 π 8. To find the number of grades in the angle which is the unit of circular measure. By putting 0=1, we get A = 200 200 =63.661977. NOTE. The candidate should practice himself frequently in the use of all these formulæ, and especially that of Art. 6. Whenever |