84. A ladder 25 ft. long stands upright against a wall; find how far the bottom must be pulled out from the wall so as to lower the top 1 foot.

85. A ladder 26 ft. long stands upright against a wall; find how far the bottom of the ladder must be pulled out to lower the top 2 ft.

86. If the diagonal of a rectangular field is 100 yards, and one side is 80 yards, what is the area?

87. The area of a chess board having 6 squares along each side is 108 sq. inches. Find, to six places of decimals, the length of a side of one of these squares.

88. At 10s. a rod, what is the difference in the cost of fencing a field 20 rods square and another field containing the same area which is 40 rods long?

89. Find the square root of the third power of 6 correct to four places.

90. Raise 015 to the third power, and extract the square root of the power.

91. A room is half as wide as high, and twice as long as wide. If the floor contains 3362 sq. ft., what is the cost of papering the walls at 9d. per sq. yd. ?

92. There are two rectangular fields of equal area. mile long and 990 ft. broad, and the other a square. length of a side of the square.

One is

Find the

as

93. Extract the square root of '000003330625. 94. A rectangular piece of ground 9 acres 1 r. 163 per. is broad as long. What is the distance round it, and from one corner to the opposite ?

95. Two trees stand on opposite sides of a stream 10 ft. wide. The height of the tree is to the width of the river as 8:4, and the width of the stream is to the height of the other tree as 4:5. What is the distance between their tops?

96. Prove by the ordinary method that 2.76=√7.654.

97. Assuming that area of circle÷34 will give the radius, find the radius of a circle containing 17 sq. ft. 16 sq. in.

98. The area of a circle is said to be the square of the radius multiplied by 3.14159. Supposing a circular pond to contain 2 acres, what is its diameter ?

Now Ready. Fifth Edition. Revised. Globe 8vo.
Examples separately, 3s. Answers, 6d.

4s. 6d.

ARITHMETIC FOR SCHOOLS

BY THE

REV. J. B. LOCK, M.A.

SENIOR FELLOW AND BURSAR OF GONVILLE AND CAIUS COLLEGE, CAMBRIDGE
FORMERLY MASTER AT ETON

London

MACMILLAN AND CO., LIMITED

NEW YORK: THE MACMILLAN COMPANY

The present reprint January 1899 has been carefully revised (see pp. 164, 180, 318 (i) ).

It should be noticed that the 400 Miscellaneous Examples are, on the average, more difficult and require more work than the Examples for Exercise; they are intended to be used as home exercises for fairly advanced students. Many of them do not come out neatly,' and the last hundred are difficult questions in the working of which the Student will find an opportunity of testing his knowledge of the theory of approximation and of practising methods of abbreviation.

The Mental Exercises which are scattered throughout the book are intended to accustom the pupil as to the language of the subject and should be worked with as little help from the pen as possible.

I would again call attention to the importance of the notation, £4. 78. 6d.

to express the ratio of £4. 78. 6d. to £3. 8s. 9d.; this notation £3. 88. 9d. is now thoroughly recognised as Arithmetical and will be found admirably adapted to modern methods of ratio.

It is hoped that the multiplicity of Examples will not tempt teachers to use this book for purposes of practising their pupils in mere unreasoning calculation. Arithmetic should be studied from all of the following points of view :

(1) As the means of rapid and accurate calculation.

(2) As the instrument by which those engaged in practical science may work out their results to any required degree of approximation.

(3) As a preliminary part of Mathematics, exercising accurate thought and ingenuity, and demanding sound logical treatment.

I venture to call attention to the Key and Companion to this Edition; in it I have endeavoured to shew the working of the more difficult questions in a manner which may render it useful to Teachers and to Students without being a temptation to indolent Schoolboys.

The Examples (nearly 8000 in all) have now been published in a separate volume.

When a star* is prefixed to an article or to a question, it is intended to suggest that what follows is of some exceptional difficulty and may therefore be in some cases omitted on a first reading of the book.

As in the first Edition so in the revised edition I have to express my obligations to the Rev. H. C. Watson, M.A., of Clifton College, for very valuable advice and criticism; to him the book owes many of its best methods and the removal of many blemishes.

The following points may be noticed :

:

I. The metric system is introduced directly after English weights and measures, and examples are given which can be worked without any special reference to decimals. See pages 58 to 66.

II. For advanced pupils pp. 70, 71, 79, 80, 81, will be found very useful.

It

III. The examples on vulgar fractions are very numerous. may be pointed out that in the text a number such as 18 is never struck out or cancelled" (see p. 99) thus 18, but thus 18.

£8. 15s.

IV. The notation (p. 109) for concrete fractions 4. 10. will be found to be of very great importance. The "dot method" of expressing a ratio

or fraction has practically fallen into disuse, and in many modern Arithmetics the idea of ratio is almost entirely lost sight of. But no student can be considered to have grasped the fundamental principle of arithmetic who does not understand the idea of ratio.

The Unitary Method is an admirably simple way of solving problems, and is carefully explained (see pages 193 to 198). But it is also most useful for the purpose of leading up to the fundamental principle of Arithmetic, namely that of ratio. It will be found that the old method taught in Colenso and known as the "dot method," which is the method of ratio, can be easily translated into the unitary method and vice versa. The first is only an abbreviation of the second. See Arithmetic, Chapter X., and particularly pages 198, 200, 204, 205.

V. Percentage is taught in a very full and graduated way, p. 217, etc. VI. Separate examples are given for interest for a given number of days; see p. 227, and generally, the Chapter on Interest and Discount is very full and complete.

VII. Compare Articles 125A, 181в, 183A.

VIII. Note Example iii., p. 248. This example illustrates the fundamental idea of Arithmetic. Almost every purely arithmetical question is answered by finding the ratio which the quantity required bears to some given quantity.

IX. Note Example v. p. 256. A similar remark (viii.) applies here.

X. Examples are given of the different Scales of Temperature; p. 260. XI. And of the Percentage Composition of Chemical Compounds; p. 262. XII. The method of teaching Profit and Loss is new. See Exercise cxxiiв, and Examples on pp. 267, 268.

XIII. On p. 304 will be found a very concise explanation of Horner's Method applied to each Root, etc.

XIV. The four hundred Examples, p. 319, called Miscellaneous Examples, require considerable calculation and are intended as a help in the preparation for the modern style of examination in which some of the questions do not come out neatly. While in the Examples for Exercise will be found a thousand examples carefully graduated, most of which are short and neat in the working. The last hundred are problems requiring some little thought and skill.

XV. Perhaps the most important point in the book is the attempt to explain the Theory of Approximation in Chapter viii. This, for students of practical science, is of very great importance. See particularly pp. 161 to 165, and 171, 172.

XVI. In connection with Approximation particular attention should be directed to Article 137A, and students should be encouraged to work all practical questions concerning money, in decimals of £1. See

Examples ii. to iv. on pp. 172, 173; Example ii. p. 237; Example ii. p. 245; Example iii. p. 248.

XVII. As regards the whole of the higher part of the book, while the methods are all based on the unitary method, the Author has endeavoured to lead the student carefully on to the idea of ratio. Thus the working of every question may, by the interpolation of the step called reduction to unity, be translated into the Unitary Method; as is seen in the Examples on Discount, Profit and Loss, and Stocks.

XVIII. As regards the early part of the book, the method of Complementary Addition will, it is hoped, some day be generally adopted in the elementary teaching in England, as it is on the Continent; so that the Italian method of division will replace the comparatively clumsy English method. See Articles 20, 23, 31, 36.

XIX. The method of multiplication by which the digits of the highest denomination in the multiplier are treated first, is well worth the consideration of Teachers. It is clearly the most scientific method, and it leads up to the idea of Approximation. It is, however, perhaps doubtful how far students who are familiar with the old method should be made to change either their method of division or of multiplication, and it is not insisted on in the book.

The "Arithmetic for Schools" is in use in the following Schools and Colleges:Eton; Westminster; Clifton; Tonbridge School; Shrewsbury; Brighton College; Highgate School; Leys School, Cambridge; Fettes College, Edinburgh.

At the following Training Colleges:-Brighton; Glasgow, Church of Scotland; Bangor; Battersea; Wandsworth; Chelsea; Stockwell.

At the following Colleges and High Schools for Girls :—Clapham Common, Clarence House; Carlisle; West Dulwich; Manchester; Truro; Durham; Richmond, Surrey; Aberdeen; Carshalton; Cheltenham: Ladies' College, and Public School.

The Arithmetic for Schools" is also in use at the following Schools:-Aberdeen, Gordon's College; Alton, Grammar School; Atherstone, Grammar School; Aspatria, Agricultural College; Arbroath, High School; Basingstoke, Queen's School; Bath: Competitive College, and Kingswood School; Bedale, Grammar School; Bedford, Modern School; Birkenhead, The School; Birmingham: Five Ways, Aston, Blue Coat School, Oratory School, and Girls' School, Camp Hill; Bishop Auckland, Grammar School; Bishop Stortford, Grammar School; Bridgnorth, Grammar School, Burnley: Grammar School, and Mechanics' Institute; Burton, Grammar School; Campbeltown, The Academy; Carmarthen, Presbyterian College; Chelmsford; Grammar School; Cheltenham: Grammar School, and Dean Close School; Chester: Modern School, Higher Grade Board School, and Arnold House; The College, Cheshunt; Chigwell, Grammar School; Clifton: Redland Hill House, and Shottery House School; Clitheroe, Grammar School; Colwyn Bay, Pen-y