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amount of salary to be provided. Occasionally a teacher is engaged at a fixed guaranteed salary, which is made to include the sum payable from the Committee of Council to possessors of certificates of merit. Such arrangements tend to obscure the design of the Government in making these grants, and ought, therefore, to be in all cases avoided.

The minutes of 1846, which instituted augmentation grants, and every subsequent document of the Committee of Council bearing any reference to schoolmasters' salaries, distinctly enunciate, as the conviction of those charged with the distribution of the parliamentary fund, that the scale of remuneration heretofore provided for teachers is quite inadequate to the requirements of that position, which it is the interest of society that the instructor of even its humblest poor should occupy. It was therefore wisely determined that the certificate of merit should be more than a mere honorary testimonial to the requirements and aptitudes for their profession of which teachers might give evidence on examination. It was made the index which should regulate, as the primary condition, the amount of the annual allowance to which it gave its possessor a conditional claim. But the function of the Committee of Council is administrative only. It may protest, as it has done in the most emphatic manner, against the misapplication of the aid which it distributes; but it has not the power to enforce compliance with its equitable requirements. It is entirely optional with managers to second, or to thwart, or to ignore the design of the Education Committee in augmenting teachers' salaries. In the great majority of cases I freely acknowledge that school promoters do not sacrifice the interests of the teachers to those of the subscribers, in reference to these grants. But it nevertheless happens that, in too many cases, the difficulty of raising the requisite funds for conducting an elementary school, is made the excuse for reducing the salary of the master, and that such reduction would not liave been attempted, if it were not supposed to be compensated by the government augmentation grant to certificated teachers. As one sincerely desirous to see the work of the clergyman and the schoolmaster go hand in hand, I trust always to find the former the ever-constant champion and advocate of the interests of the latter, I trust that in their influential position as the usual chairmen of school committees, the clergy will ever raise their voice against any relaxation in local effort and personal sacrifice to supply a fit salary for the schoolmaster, even though the master should have acquired the conditional right to an annual payment from the parliamentary grant.

I am, Sir, your obedient servant,

A SCHOOLMASTER. March 26, 1851.

133

Notices of Books.

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PLANE TRIGONOMETRY, WITH The Use OF LOGARITHMS (Part 1.).

BY THE REV. J. W. COLENSO, M.A., RECTOR OF FORNCETT ST. MARY, NORFOLK, LATE FELLOW OF ST. JOHN'S COLLEGE, CAMBRIDGE. (Lon

don, Longman and Co.) FIRST BOOK ON PLANE TRIGONOMETRY, COMPRISING GEOMETRICAL

TRIGONOMETRY, AND ITS APPLICATION TO SURVEYING, WITH NUMEROUS EXAMPLES. BY G. W. JIEMMING, M.A., FELLOW OF St. John's

COLLEGE, CAMBRIDGE. (London: Taylor, Wulton, und Maberley.) The two neat little works whose titles are connected with the present notice, not only treat on the same subject, but in both it is carried to nearly the same extent. The author of the former of these treatises announces that Part II., on the higher parts of trigonometry, will be published shortly, and the author of the latter is engaged in the preparation of a short treatise on analytical trigonometry, with the methods of calculating logarithmic and trigonometrical tables.

In Mr. Colenso's work the discussion of the measurement of lines and angles, and the trigonometrical ratios of an angle occupy the first and second chapters. The trigonometrical functions considered as belonging to the subtending arc rather than to the angle are also defined, and the changes in sign and magnitude of the trigonometrical functions are traced through all the four quadrants, with reference to ratio-definitions as well as to line-definitions. In the third chapter the trigonometrical ratios of the sum and difference, multiples and submultiples of angles are elegantly investigated, and numerous interesting and useful exercises are interspersed for illustration. The probable origin of the inverse notation is alluded to thus:

“If tan A=m, then A=tan- im." “The origin of this notation is easily seen: for, if we could for a moment conceive the symbol tan separated from the angle A, we might write the above equation A=., or by the theory of Indices, A=tan-im."

The calculation of the numerical values of trigonometrical ratios, and the properties and solution of triangles are considered in the three following chapters. The useful device of throwing certain expressions into factors by means of the introduction of a subsidiary angle, and thus adapting them forlogarithmic computation, is also explained and enforced.

The next chapter contains a few exercises in surveying, and the usual application of trigonometry to the determination of heights and distances. The investigation of logarithmic series, and the properties of logarithms are very neatly deduced in the eighth and concluding chapter. The use of complementary logarithms is to convert the operation of subtraction into that of addition, and we think that the contraction Mr. Colenso has employed to denote this change is the best that has hitherto been devised, and we most cordially recommend its universal adoption. The contraction colog. is a most appropriate one, inasmuch

, as it is self-interpreting, and consequently not liable to be misunderstood. This little volume, consisting of 128 pages, is'enriched with numerous excellent examples, and contains much valuable and interesting matter.

Mr. Hemming's treatise is strictly confined to geometrical trigonometry, and thus the subject is placed more within the reach of those

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who have but little time to devote to the acquisition of a knowledge of this useful and interesting branch of science. After defining the trigonometrical ratios, and tracing the changes of the different functions of an angle as it varies from zero to two right angles, Mr. Hemming deduces the trigonometrical functions of more than one angle, giving several elegant geometrical Investigations, and leaving others as exercises for the student. The properties and use of logarithms are given in the third chapter, as well as the modification of formulas which require addition or subtraction, before logarithmic computation can be applied to them. This modification is effected by the introduction of new angles, termed subsidiary angles. The next chapter is entirely appropriated to the solution of triangles, and a few examples are given for exercise. The fifth chapter treats on surveying, and here we have a short description of the instruments employed, viz. the chain, the level, the theodolite, and the sextant. The principle of the vernier is also explained, and the numerous examples of the determination of leights and distances occupy nearly the whole of this division. A short chiapter on the areas of triangles and polygons, and the radii of their inscribed and circumscribed circles, concludes the volume. Mr. Hemming's little treatise will be found very useful to those students who have not leisure time to pursue the subject to any great extent. PROBLEMS IN ILLUSTRATION OF THE PRINCIPLES OF PLANE COORDIGEOMETRY.

WALTON, M.A., LEGE, CAMBRIDGE. (Cambridge: John Deighton. London : Simp

kin & Co.; G. Bell.) Coordinate geometry is an interesting and extensive branch of modern mathematical science. Invented more than two hundred years ago by the celebrated French geometer, Descartes, it has ever since been cultivated with great success by the continental mathematicians. Not many years ago the "new science" of coordinate geometry was almost unknown in this country, but several valuable treatises, exclusively devoted to the subject, and accessible to most readers, have recently appeared, diffusing a knowledge of its principles among all classes of students, and investing it with an importance in some measure commensurate with its unquestioned merits, as one of the most powerful instruments of research with which the geometer can be furnished. But while many excellent works on coordinate geometry are now in the hands of students, and while in most of these publications the theory of the science has been developed with much ability and success, the want of an ample collection of examples in illustration of principles has been felt and acknowledged for some time past. To the student who has acquired an accurate knowledge of the theory of any portion of mathematical science, a collection of examples for exercise in the application of the principles of that particular branch, judiciously selected, and arranged in a connected and classified form, cannot fail, if carefully perused, to be productive of lasting advantage. The want of such a set of examples in coordinate geometry is now happily supplied by the publication of the excellent collection of problems which forms the subject of the present notice.

The author of these problems in plane co-ordinate geometry is already well known to mathematicians by his various writings and pub

lications, especially by his two valuable collections of problems,-the one in illustration of the principles of theoretical mechanics, and the other in illustration of the principles of theoretical hydrostatics and hydrodynamics. These are both admirable productions, and the work now before us fully sustains the high reputation of its talented author. We shall briefly notice some of the most prominent features of Mr. Walton's valuable collection of problems, taking the subjects in the order of their arrangement, and premising that the author adopts the judicious plan of giving in the first part of each section of the work the solutions in extenso of several well-chosen examples, and in the remaining part of the section a selection of unsolved exercises, illustrative of that particular principle to which the section is appropriated.

1. The Straight Line.- On the straight line are given various elementary problems of an interesting character, their solutions being effected by referring the straight line both to rectangular and to oblique axes of coordinates. The use of the polar equation of a straight line is shown in the solution of one or two problems, and these are followed by numerous examples producing rectilinear loci. The problems on transversals are treated in a very perspicuous and elegant manner, both by means of the symmetrical equation of a straight line,+=1, which is termed the method of explicit parameters, and also by the method of implicit parameters, in which the equations of straight lines are expressed by u = = 0, v = 0, W = o, etc. ; where u, v, w, etc., are of the form x cos. E + y sin. ε – d. In these two extensive sections, which are entirely appropriated to the discussion of problems on transversals, the solutions are models of excellence, and worthy of the best attention of the student. This division of the work concludes with several examples in reference to the areas of rectilinear figures. Mr. Walton' makes frequent use of the equation of a straight line in the form x cos. a ty sin. a = d, since the expression for the distance of a point (x y) from this line is of the form ä сos. a + y sin. a -- -o, which is easily recollected.

2. The Circle.—The author has selected elegant and appropriate examples in this division, and their solutions are effected by taking for axes of coordinates perpendicular diameters, -any two straight lines at right angles, or oblique to each other,-and also two tangenis to tlie circle. The polar equation to the circle, and the polar equations to tangents and chords, find their application in the solution of some useful examples, and several of the succeeding sections contain exercises on poles and polars, radical axes, centres of similitude, etc., and inscribed and circumscribed polygons. Numerous examples producing circular loci complete this division.

3-5. The Parabola, Ellipse, and Hyperbola. The examples in these three parts are, as usual, selected from various sources, and admirably fitted for illustrating the principles of the conic sections. The equation of the tangent to a conic section, involving a function of the angle which the tangent makes with the principal axis of the curve, has been found very effective in facilitating the demonstration and investigation of several useful properties of conic sections, especially in thos problems of tangency which do not involve the consideration of th point of contact. Owing to its utility in examples relating to tangen

e

it has been termed the magical equation to the tangent. There is little magic in mathematics; magic squares and magic circles are unknown in Euclid's vocabulary; and we are of opinion that a more appropriate term might be devised to distinguish this useful form of the equation to the tangent. We are not, however, prepared with a more suitable phrase, and we would merely suggest the adoption of the term angular equation instead of magical equation. The equation involves the trigonometrical tangent of the inclination of the tangent to the principal axis of the curve, and the term angular equation is distinct and explicit enough to prevent its being mistaken for, or confounded with, the well-known and appropriate term polar equation.

If tan 'a be the angle which the tangent makes with the principal axis of the curve then y= ax + " is the magical (query angular) equation to the tangent to the parabola, y = 4 m x; and y = ax + (aa' + 6?) * are the magical (angular) equations to the tangents to the ellipse and hyperbola = 1.

Several of the problems have reference to poles and polars, parabolic, elliptic, and hyperbolic loci, together with others involving parabolic and elliptic envelopes, and each division is closed with a miscellaneous collection of examples, affording abundant exercise for the student.

6. Lines of the second order. This is the last division of the volume, and several of the examples are extracted from sources inaccessible to many students. The problems are here necessarily more general than in the preceding parts of the work, applying to all the conic sections, and it is enriched with the most elegant methods of solution which the united efforts of the best geometers have been able to devise,

In concluding this brief notice of a publication which has hitherto been a desideratum, we must not omit to remark that, to select and arrange in a connected and classified form, no fewer than 896 examples in illustration of the principles of plane coordinate geometry, is indeed no light task. Yet this has been accomplished by Mr. Walton in the volume before us; and whether we regard the elegance of the methods of investigation employed, or the ability with which these methods are conducted, it alike demands our warmest commendation. Indeed we have no hesitation in stating that we could not put into the hands of a student any work containing so many elegant analytical solutions as he will find in this volume, and we cordially recommend it to all who are desirous of possessing the best and neatest solutions to an extensive and valuable collection of useful and interesting problems.

SMALL BOOKS ON GREAT SUBJECTS.

NO. XVIII.

THE PHILOSOPHY OF

RAGGED SCHOOLS. Pp. 128. (London: W. Pickering, 1851.) Some of the philosophy in this little book is, to say the least, of a very questionable sort, and such as we cannot recommend. But the testimony which it bears in favour of ragged schools, when relating their origin and progress, well deserves attention. John Pounds, a poor shoemaker of Portsmouth, is mentioned as probably the first who opened a school, strictly of this description. Of him it is recorded, that when he died in 1839, the poor children, who then formed his class,

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