August 2, 1900] 



NATURE 



319 



study at school, and he will answer that the only men 

 he knows of who read the classics are a few famous 

 scholars and the cads who read with delight cribs of the 

 Odyssey and the Iliad ]Vi%i as if they were novels, because 

 they never had the advantage of a classical education. 

 But, of course, his mind was trained, he can always say 

 that. 



The authorities of the Science and Art Department 

 recognise that apprentices and others attending evening 

 classes may possibly benefit by a course of study very 

 different from what is necessary if students are being 

 prepared for university and other examinations. Hence, 

 in addition to their very complete orthodox courses of 

 instruction, they recognise the new method of study, 

 the most elementary part of which is beginning to get 

 crystallised in the following syllabus. There is also an 

 "Advanced" syllabus, which is too long to be published 

 here. I would advise interested persons to write to the 

 Department for copies, and also for the report on the 

 result of last year's examination, as well as for copies 

 of the examination papers and of the above-mentioned 

 summary. 



I venture to hope for criticism of this syllabus — first, 

 from men like my Cambridge friends, who are quite sym- 

 pathetic, but who think the method one fit for evening 

 classes only ; second, from men who think with me that 

 the method is one which may be adopted in every school 

 in the country, and adopted even with the one or two 

 boys in a thousand who are likely to become able mathe- 

 maticians ; third, from other men. Whatever be the 

 point of view of any critic, he must surely feel that ex- 

 haustive criticism is important, for there are many large 

 technical schools in England in which the method has 

 already been adopted, the orthodox system being quite 

 given up. I have been informed that the method is 

 spreading rapidly in Germany also. I can already see 

 from the exceedingly interesting examination results that 

 crystallisation is proceeding rapidly, and if criticism is to 

 be of value, now is the time for it. I hope also that the 

 seemingly bumptious manner in which I criticise orthodox 

 methods of teaching will not induce contemptuous in- 

 difference in men of thought. I hold a brief in the in- 

 terests of average boys and men ; my strong language 

 and possible excess of zeal are due to the fact that nearly 

 all the clever men have briefs on the other side. 



John Perry. 



PRACTICAL MATHEMATICS. 

 Elementary Stage. 



Arithvietic. — The use of decimals ; the fallacy of retaining 

 more figures than are justifiable in calculations involving 

 numbers which represent observed or measured quantities. 

 Contracted and approximate methods of multiplying and 

 dividing numbers whereby all unnecessary figures may be 

 omitted. Using rough checks in arithmetical work, especially 

 with regard to the position of the decimal point. 



The use of 5*204 x lo* for 520400 and of 5*204 x 10 -' for 

 '005204. The meaning of a common logarithm ; the use of 

 logarithms in making calculations involving multiplication, 

 division, involution and evolution. Calculation of numerical 

 values from all sorts of formulae. 



The principle underlying the construction and method of 

 using a common slide rule ; the use of a slide rule in making 

 calculations. Conversion of common logarithms into Napierian 

 logarithms. The calculation of square roots by the ordinary 

 arithmetical method. Using algebraic formula? in working 

 questions on ratio and variation. 



Algebra. — To understand any formula so as to be able to use 

 it if numerical values are given for the various quantities. Rules 

 of Indices. 



Being told in words how to deal arithmetically with a 

 quantity, to be able to state the matter algebraically. Problems 

 leading to easy equations in one or two unknowns. Easy trans- 

 formations and simplifications of formulae. The determination 

 of the numerical values of constants in equations of known 



NO. 1605, VOL. 62] 



form, when particular values of the variables are given. The 

 meaning of the expression " A varies as B." 



Factors of such expressions as x^-a^, 3^-\r\\x->rio, 

 x^-fyx- 66. 



Mensuration.— The rule for the length of the circumference 

 of a circle. The rules for the areas of a triangle, rectangle, 

 parallelogram, circle ; areas of the surfaces of a right circular 

 cyHnder, right circular cone, sphere, circular anchor ring. The 

 determination of the area of an irregular plane figure (l) by using 

 a planimeter ; (2) by using Simpson's or other well-known rules 

 for the case where a number of equidistant ordinates or widths 

 are given ; (3) by the use of squared paper whether the given 

 ordinates or widths are equidistant or not, the " mid-ordinate 

 rule " being used. Determination of volumes of a prism or 

 cylinder, cone, sphere, circular anchor ring. 



The determination of the volume of an irregular solid by 

 each of the three methods for an irregular area, the process 

 being first to obtain an irregular plane figure in which the vary- 

 ing ordinates or widths represent the varying cross sections of 

 the solid. 



Some practical methods of finding areas and volumes. 

 Determination of weights from volumes when densities are 

 given. 



Stating a mensuration rule as an algebraic formula. In such 

 a formula any one of the quantities may be the unknown one, 

 the others being known. 



Use of Squared Paper. — The use of squared paper by 

 merchants and others to show at a glance the rise and fall of 

 prices, of temperature, of the tide, &c. The use of squared 

 paper should be illustrated by the working of many kinds of 

 exercises, but it should be pointed out that there is a general 

 idea underlying them all. The following may be mentioned : — 



Plotting of statistics of any kind whatsoever, of general or 

 special interest. What such curves teach. Rates of increase. 



Interpolation, or the finding of probable intermediate values. 

 Probable errors of observation. Forming complete price lists 

 by shopkeepers. The calculation of a table of logarithms. 

 Finding an average value. Areas and volumes, as explained 

 above. The method of fixing the position of a point in a plane ; 

 the X and y and also the r and d, co-ordinates of a point. 

 Plotting of functions, such as y — ax^,y = ae'"', where a, b, n, 

 may have all sorts of values. The straight line. Determina- 

 tion of maximum aud minimum values. The solution of 

 equations. Very clear notions of what we mean by the roots 

 of equations may be obtained by the use of squared paper. 

 Rates of increase. Speed of a body. Determination of laws 

 which exist between observed quantities, especially of linear 

 laws. Corrections for errors of observation when the plotted 

 quantities are the results of experiment. 



In all the work on squared paper a student should be made 

 to understand that an exercise is not completed until the scales 

 and the names of the plotted quantities are clearly indicated on 

 the paper. Also that those scales should be avoided which are 

 obviously inconvenient. Finally, the scales should ha chosen 

 so that the plotted figure shall occupy the greater part of the 

 sheet of paper ; at any rate, the figure should not be crowded 

 in one corner of the paper. 



Geometry. — Dividing lines into parts in given proportions, 

 and other illustrations of the 6th Book of Euclid. Measure- 

 ment of angles in degrees and radians. The definitions of the 

 sine, cosine and tangent of an angle ; determination of their 

 values by drawing and measurement ; setting out of angles by 

 means of a protractor when they are given in degrees or radians, 

 also when the value of the sine, cosine or tangent is given. 

 Use of tables of sines, cosines and tangents. The solution of 

 a right angled triangle by calculation and by drawing to scale. 

 The construction of a triangle from given data ; determination 

 of the area of a triangle. The more important propositions of 

 Euclid may be illustrated by actual drawing ; if the proposition 

 is about angles, these may be measured by means of a 

 protractor ; or if it refers to the equality of lines, areas or 

 ratios, lengths may be measured by a scale and the necessary 

 calculations made arithmetically. This combination of drawing 

 and arithmetical calculation may be freely used to illustrate the 

 truth of a proposition. 



The method of representing the position of a point in space 

 by its distances from three co-ordinate planes. How the angles 

 are measured between (i) a line and plane; {2) two planes. 

 The angle between two lines has a meaning whether they do or 

 do not meet. What is meant by the projection of a Une or a 



