that something, which a few hours before seemed to be clearly and securely grasped, has disappeared in a most bewildering fog. The only resource of the boy is then something written down or in print, and it is hoped that the necessarily short explanations given may suffice to recall the missing ideas, though want of space has precluded that fulness of illustration which is required to bring an entirely new subject home to the mind. Again, the authors of original memoirs and the larger text-books frequently give only in outline calculations which belong more properly to arithmetic than to chemistry, and boys weak in arithmetic are often puzzled to supply the intermediate steps.

The first eight sections of the introduction deal with portions of arithmetic which are subsequently required, and which experience has shewn are often omitted or not sufficiently grasped in the ordinary course of arithmetic, these may be easily read by anyone who has mastered the "first four rules," vulgar and decimal fractions, any method of working "Rule of Three" sums, and the definitions of the first book of Euclid. The following sections are devoted to the different cases in which Chemistry calls in the aid of Arithmetic without demanding a knowledge of the higher Mathematics. Common mathematical formulæ are simply given as facts to be remembered, since the proof in many cases demands greater knowledge than my readers are assumed to possess. The harder sections, which may be omitted by beginners, are marked by an asterisk as a guide to anyone who may be working without the assistance of a master.

The introduction is of necessity a compilation from all the ordinary sources, and the problems are the results of collecting for several years, at first solely with a view to private use; hence an apology may in a few cases be due to authors in whose footsteps I have too closely trodden, and without whose labours this book would never have been undertaken.

My grateful thanks are due for some problems and for much valuable advice to numerous friends, amongst whom I have special reason to mention Professor Tilden, Rev. T. N. Hutchinson, A. G. Vernon Harcourt, H. G. Madan, and W. A. Shenstone, Esqrs. I shall also be much obliged to anyone who may send corrections and suggestions to myself directly or through my publishers.

S. L.

HARROW,

June 19, 1882.

INTRODUCTION.

Sections and paragraphs marked by an asterisk may be omitted on the first reading.

ARITHMETIC.

(1) APPROXIMATION.

WHEN arithmetical processes are applied to working out the results of experiments, there are many cases in which the ordinary methods may be shortened without any loss of accuracy and with a considerable saving of labour.

All numerical results obtained by experiment are liable to more or less experimental error, so that the correct number is never obtained, but only one more or less closely approximating to it. Thus, if the diameter of a six-penny piece be measured with an ordinary foot-rule, it is found to be 75 of an inch; but owing to imperfections in the rule and in the observer the true value may be either 751 or 749 of an inch.

Experiments then being imperfect it is a misleading and useless labour to obtain results carried to a large number of

L. C. A.

1

decimal places by arithmetic, when the second or third figure may be incorrect owing to experimental error. Thus if it be required to calculate the length of the circumference of the six-penny piece from the measure of the diameter just given, it is known that the circumference of a circle is (T) 3.14159265 &c. times the length of the diameter, and hence in the given case the circumference is

•750 x 3.14159265 2.3561944875 inches.

=

But the diameter may in reality be either 749 or ·751 of an inch, and hence the circumference may be either

Since then the experiment is not sufficiently accurate to decide whether the fourth figure ought to be 3, 6, or 9, all the figures to the right of it are of no value, and in fact misleading, since they claim for the experiment an undue amount of accuracy.

A chemical experiment is a good one if the numerical results are true to 1000; in certain cases, if special precautions are taken, the results may be correct to 10000. In general then arithmetical methods may be modified in any manner to save labour so long as the working is correct to the fourth significant figure.

In approximate working it is more convenient to use decimal than vulgar fractions, because there is no trouble in reducing the former to common denominators and it is easy to estimate the error introduced by omitting any portion of the decimal. Thus if the value of π be taken as 3·14 instead of as 3.14159265, the error introduced is rather over ·001 and the result will be incorrect by about 3 of the whole.

If the portion of a decimal omitted commences with 5 or a higher figure, the last figure retained must be strengthened by the addition of 1. Thus in the case of 3.14159265, if all beyond the fifth figure is to be neglected, 3.1415 is too small by 00009, but 3.14156 is too large by only 000008, hence the error in the second case is only about of what it would have been had the last figure retained not been strengthened.

In dealing with large numbers it is frequently convenient to write them as a single figure in the unit place followed by decimals and multiplied by 10 raised to the required power. Thus a ton contains 15680000 grains, which may be written 1·568 × 107.

(2) MULTIPLICATION.

When two decimals are multiplied by the common method, a number of the decimal figures, which are so obtained, are useless owing to experimental error. The following method only makes use of sufficient figures to obtain the required degree of accuracy.

(i) Count off after the decimal point in the multiplicand (annexing ciphers if necessary) as many figures of decimals as are required in the product.

(ii) Below the last of these write the unit figure of the multiplier, and write its other figures in reversed order.

(iii) Multiply by each figure of the inverted multiplier, neglecting all figures of the multiplicand to the right of that figure, except to find what is to be carried; and let all the partial products be so arranged, that their right-hand figures stand in the column of the unit figure of the multiplier.