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 The following are the ratios of some of the units; each unit is expressed approximately as a decimal of the other, and their ratio is shown as a continued product (§ 116), a few of the corresponding convergents to the continued fraction (§ 117) being added in brackets. It must be remembered that the number expressing any quantity in terms of a unit is inversely proportional to the magnitude of the unit, i.e. the number of new units is to be found by multiplying the number of old units by the ratio of the old unit to the new unit.

(ii.) Special Applications.

121. Commercial Arithmetic.—This term covers practically all dealings with money which involve the application of the principle of proportion. A simple class of cases is that which deals with equivalence of sums of money in different currencies; these cases really come under § 120. In other cases we are concerned with a proportion stated as a numerical percentage, or as a money percentage (i.e. a sum of money per £100), or as a rate in the £ or the shilling. The following are some examples. Percentage: Brokerage, commission, discount, dividend, interest, investment, profit and loss. Rate in the £: Discount, dividend, rates, taxes. Rate in the shilling: Discount.

Text-books on arithmetic usually contain explanations of the chief commercial transactions in which arithmetical calculations arise; it will be sufficient in the present article to deal with interest and discount, and to give some notes on percentages and rates in the £. Insurance and Annuities are matters of general importance, which are dealt with elsewhere under their own headings.

122. Percentages and Rates in the £.—In dealing with percentages and rates it is important to notice whether the sum which is expressed as a percentage of a rate on another sum is a part of or an addition to that sum, or whether they are independent of one another. Income tax, for instance, is calculated on income, and is in the nature of a deduction from the income; but local rates are calculated in proportion to certain other payments, actual or potential, and could without absurdity exceed 20s. in the £.

It is also important to note that if the increase or decrease of an amount A by a certain percentage produces B, it will require a different percentage to decrease or increase B to A. Thus, if B is 20% less than A, A is 25% greater than B.

123. Interest is usually calculated yearly or half-yearly, at a certain rate per cent. on the principal. In legal documents the rate is sometimes expressed as a certain sum of money “per centum per annum”; here “centum” must be taken to mean “£100.”

Simple interest arises where unpaid interest accumulates as a debt not itself bearing interest; but, if this debt bears interest, the total, i.e. interest and interest on interest, is called compound interest. If 100r is the rate per cent. per annum, the simple interest on £A for n years is £nrA, and the compound interest (supposing interest payable yearly) is £[(1 + r)n − 1]A. If n is large, the compound interest is most easily calculated by means of logarithms.

124. Discount is of various kinds. Tradesmen allow discount for ready money, this being usually at so much in the shilling or £. Discount may be allowed twice in succession off quoted prices; in such cases the second discount is off the reduced price, and therefore it is not correct to add the two rates of discount together. Thus a discount of 20%, followed by a further discount of 25%, gives a total discount of 40%, not 45%, off the original amount. When an amount will fall due at some future date, the present value of the debt is found by deducting discount at some rate per cent. for the intervening period, in the same way as interest to be added is calculated. This discount, of course, is not equal to the interest which the present value would produce at that rate of interest, but is rather greater, so that the present value as calculated in this way is less than the theoretical present value.

125. Applications to Physics are numerous, but are usually only of special interest. A case of general interest is the measurement of temperature. The graduation of a thermometer is determined by the freezing-point and the boiling-point of water, the interval between these being divided into a certain number of degrees, representing equal increases of temperature. On the Fahrenheit scale the points are respectively 32° and 212°; on the Centigrade scale they are 0° and 100°; and on the Réaumur they are 0° and 80°. From these data a temperature as measured on one scale can be expressed on either of the other two scales.

126. Averages occur in statistics, economics, &c. An average is found by adding together several measurements of the same kind and dividing by the number of measurements. In calculating an average it should be observed that the addition of any numerical quantity (positive or negative) to each of the measurements produces the addition of the same quantity to the average, so that the calculation may often be simplified by taking some particular measurement as a new zero from which to measure.

.—For the history of the subject, see W. W. R. Ball, Short History of Mathematics (1901), and F. Cajori, History of Elementary Mathematics (1896); or more detailed information in M. Cantor, Vorlesungen über Geschichte der Mathematik (1894–1901). L. C. Conant, The Number-Concept (1896), gives a very full account of systems of numeration. For the latter, and for systems of notation, reference may also be made to Peacock’s article “Arithmetic” in the Encyclopaedia Metropolitana, which contains a detailed account of the Greek system. F. Galton, Inquiries into Human Faculty (1883), contains the first account of number-forms; for further examples and references see D. E. Phillips, “Genesis of Number-Forms,” American Journal of Psychology, vol. viii. (1897). There are very few works dealing adequately but simply with the principles of arithmetic. Homersham Cox, Principles of Arithmetic (1885), is brief and lucid, but is out of print. The Psychology of Number, by J. A. McLellan and J. Dewey (1895), contains valuable suggestions (some of which have been utilized in the present article), but it deals only with number as the measure of quantity, and requires to be read critically. This work contains references to Grube’s system, which has been much discussed in America: for a brief explanation, see L. Seeley, The Grube Method of Teaching Arithmetic (1890). On the teaching of arithmetic, and of elementary mathematics generally, see J. W. A. Young, The Teaching of Mathematics in the Elementary and the Secondary School (1907); D. E. Smith, The Teaching of Elementary Mathematics (1900), also contains an interesting general sketch; W. P. Turnbull, The Teaching of Arithmetic (1903), is more elaborate. E. M. Langley, A Treatise on Computation (1895), has notes on approximate and abbreviated calculation. Text-books on arithmetic in general and on particular applications are numerous, and any list would soon be out of date. Recent English works have been influenced by the brief Report on the Teaching of Elementary Mathematics, issued by the Mathematical Association (1905); but this is critical rather than constructive. The Association has also issued a Report on the Teaching of Mathematics in Preparatory Schools (1907). In the United States of America the Report of the Committee of Ten on secondary school studies (1893) and the Report of the Committee of Fifteen on elementary education (1893–1894), both issued by the United States Bureau of Education, have attracted a good deal of attention. Sir O. Lodge, Easy Mathematics, chiefly Arithmetic (1905), treats the subject broadly in its practical aspects. The student who is interested in elementary teaching should consult the annual bibliographies in the Pedagogical Seminary; an article by D. E. Phillips in vol. v. (October 1897) contains references to works dealing with the psychological aspect of number. For an account of German methods, see W. King, Report on Teaching of Arithmetic and Mathematics in the Higher Schools of Germany (1903).

ARIUS ( ), a name celebrated in ecclesiastical history, not so much on account of the personality of its bearer as of the “Arian” controversy which he provoked. Our knowledge of Arius is scanty, and nothing certain is known of his birth or of his early training. Epiphanius of Salamis, in his well-known treatise against eighty heresies (Haer. lxix. 3), calls him a Libyan by birth, and if the statement of Sozomen, a church historian of the 5th century, is to be trusted, he was, as a member of the Alexandrian church, connected with the Meletian schism (see