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Originally appearing in Volume V02, Page 541 of the 1911 Encyclopedia Britannica.
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X153, 7s. 4d. B £I 20S. Is. 12d. 368o8d. £153, 7s. 4d. 3067s. 4d. n 0 I 2 3 6+n n 4n 0 4 8 I2 0 I 2 3 n Length of edge in inches. Volume of cube. o I 2 3 Nil. 1 cub. in. 8 cub. in. 27 cub. in. of numerical quantities, merely correspond with each other, the correspondence being the result of some relation. The volume D of a cube, for instance, bears a certain relation to the length of an edge of the cube. This relation is not one of pro-portion; but it may nevertheless be expressed by tabulation, as shown at D. 93. Interpolation.—In most cases the quantity in the second column may be regarded as increasing or decreasing continuously as the number in the first column increases, and it has inter-mediate values corresponding to inter-mediate (i.e. fractional or decimal) numbers not shown in the table. The table in such cases is not, and cannot be, complete, even up to the number to which it goes. For instance, a cube whose edge is 11 in. has a definite volume, viz.' 3'-8 cub. in. The determination of any such intermediate value is performed by Interpolation (q.v.). In treating a fractional number, or the corresponding value of the quantity in the second column, as intermediate, we are in effect regarding the numbers 1, 2, 3, . . . , and the corresponding numbers in the second column, as denoting points between which other numbers lie, i.e. we are regarding the numbers as ordinal, not cardinal. The transition is similar to that which arises in the case of geometrical measurement (§ 26), and it is an essential feature of all reasoning with regard to continuous quantity, such as we have to deal with in real life. 94. Nature of Arithmetical Reasoning.—The simplest form of arithmetical reasoning consists in the determination of the term in one series corresponding to a given term in another series, when the relation between the two series is given; and it implies, though it does not necessarily involve, the establishment of each series as a whole by determination of its unit. A method involving the determination of the unit is called a unitary method. When the unit is not determined, the reasoning is algebraical rather than arithmetical. If, for instance, three terms of a proportion are given, the fourth can be obtained by the relation given at the end of § 57, this relation being then called the Rule of Three; but this is equivalent to the use of an algebraical formula. More complicated forms of arithmetical reasoning involve the use of series, each term in which corresponds to particular terms in two or more series jointly; and cases of this kind are usually dealt with by special methods, or by means of algebraical formulae. The old-fashioned problems about the amount of work done by particular numbers of men, women and boys, are of this kind, and really involve the solution of simultaneous equations. They are not suitable for elementary purposes, as the arithmetical relations involved are complicated and difficult to grasp. XI. METHODS OF CALCULATION (i.)'Exact Calculation. 95. Working from Left.—It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right. There are several reasons for this. In the first place, an operation then corresponds more dosely, at an elementary stage, with the concrete process which it represents. If, for instance, we had one sum of £3, I5S. 9d. and another of £2, 6s. 5d., we should add them by putting the coins of each denomination together and commencing the addition with the L. In the second place, this method fixes the attention at once on the larger, and therefore more important, parts of the quantities concerned, and thus prevents arithmetical processes from becoming too abstract in character. In the third place, it is a better preparation for dealing with approximate calculations. Finally, experience shows that certain operations in which the result is written down at once—e.g. addition or subtraction of two numbers or quantities, and multiplication by some small numbers—are with a little practice performed more quickly and more accurately from left to right. 96. Addition.—There is no difference in principle between addition (or subtraction) of numbers and addition (or subtraction) of numerical quantities. In each case the grouping system involves rearrangement, which implies the commutative law, while the counting system requires the expression of a quantity in different denominations to be regarded as a notation in a varying scale (§§ 17, 32). We need therefore consider numerical quantities only, our results being applicable to numbers by regarding the digits as representing multiples of units in different denominations. When the result of addition in one denomination can be partly expressed in another denomination, the process is technically called carrying. The name is a bad one, since it does not correspond with any ordinary meaning of the verb. It would be better described as exchanging, by analogy with the " changing " of subtraction. When, e.g., we find that the sum of 17s. and 18s. is 35s., we take out 20 of the 35 shillings, and exchange them for £1. To add from the left, we have to look ahead to see whether the next addition will require an exchange. Thus, in adding £3, 17s. od. to £2, 18s. od., we write down the sum of £3 and £2 as £6, not as £5, and the sum of 17s. and 18s. as 15s., not as 35s. When three or more numbers or quantities are added together, the result should always be checked by adding both upwards and downwards. It is also useful to look out for pairs of numbers or quantities which make 1 of the next denomination, e.g. 7 and 3, or 8d. and 4d. 97. Subtraction.—To subtract £3, 5s. 4d. from £9, 7S. 8d., on the grouping system, we split up each quantity into its denominations, perform the subtractions independently, and then regroup the results as the " remainder " £6, 2S. 4d. On the counting system we can count either forwards or backwards, and we can work either from the left or from the right. If we count forwards we find that to convert £3, 5S. 4d. into £9, 7s. 8d. we must successively add £6, 2S. and 4d. if we work from the left, or 4d., 2S. and £6 if we work from the right. The intermediate values obtained by the successive additions are different according as we work from the left or from the right, being £9, 5s. 4d. and £9, 7s. 4d. in the one case, and £3, 5s. 8d. and £3, 7s. 8d. in the other. If we count backwards, the intermediate values are £3, 7s. 8d. and £3, 5s. 8d. in the one case, and £9, 7s. 4d. and £9, 5s. 4d. in the other. The determination of each element in the remainder involves reference to an addition-table. Thus to subtract 5s. from 7s. we refer to an addition-table giving the sum of any two quantities, each of which is one of the series os., Is., 19s. Subtraction by counting forward is called complementary addition. To subtract £3, 5S. 8d. from £9, 10s. 4d., on the grouping system, we must change Is. out of the Ios. into 12d., so that we subtract £3, 5S. 8d. from £9, 9s. 16d. On the counting system it will be found that, in determining the number of shillings in the remainder, we subtract 5s. from 9S. if we count forwards, working from the left, or backwards, working from the right; while, if we count backwards, working from the. left, or forwards, working from the right, the subtraction is of 6s. from Ios. In the first two cases the successive values; (in direct or reverse order) are £3, 5S. 8d., £9, 5S. 8d., £9, 9S. 8d. and £9, Ios. 4d.; while in the last two cases they are £9, Ios. 4d., £3, 10S. 4d., £3, 6s. 4d. and £3, 5S. 8d. In subtracting from the left, we look ahead to see whether a 1 in any denomination must be reserved for changing; thus in subtracting 274 from 637 we should put down 2 from 6 as 3, not as 4, and 7 from 3 as 6. 98. Multiplication-Table.—For multiplication and division we use a multiplication-table, which is a multiple-table, arranged as explained in § 36, and giving the successive multiples, up to 9 times or further, of the numbers from r (or better, from o) to Io, 12 or 20. The column (vertical) headed 3 will give the multiples of 3, while the row (horizontal) commencing with 3 will give the values of 3 X I, 3 X 2, . . . To multiply by 3 we use the row. To divide by 3, in the sense of partition, we also use the row; but to divide by 3 as a unit we use the column. 99. Multiplication by a Small Number.—The idea of a large r) =ap. ro'+(aq+bp)IO3+(ar+bq+cp) Io2+(br+cq) 1o+cr. Hence the digits are multiplied in pairs, and grouped according to the power of to which each product contains. A method of performing the process is shown here for the case of 162.427. multiple of a small number is simpler than that of a small multiple of a large number, but the calculation of the latter is easier. It is therefore convenient, in finding the product of two numbers, to take the smaller as the multiplier. To find 3 times 427, we apply the distributive law (§ 58 (vi) ) that 3.427 =3(400+20+7)=3.400+3.20+3.7. This, if we regard 3.427 as 427+427+427, is a direct consequence of the commutative law for addition (§ 58 (iii) ), which enables us to add separately the hundreds, the tens and the ones. To find 3.400, we freat too as the unit (as in addition), so that 3.400=3.4.100= 12.100= 1200; and similarly for 3.20. These are examples of the associative law for multiplication (§ 58 (iv) ). too. Special Cases.—The following are some special rules: (i) To multiply by 5, multiply by lo and divide by 2. (And conversely, to divide by 5, we multiply by 2 and divide by Io.) (ii) In multiplying by 2; from the left, add t if the next figure of the multiplicand is 5, 6, 7, 8 or 9. (iii) In multiplying by 3, from the left, add 1 when the next figures are not less than 33 . . . 334 and not greater than 66 . . . 666, and 2 when they are 66 . . . 667 and upwards. (iv) To multiply by 7, 8, 9, 11 or 12, treat the multiplier as 10-3, 10-2, 10-1, 10+1 or 10+2; and similarly for 13, 17, 18, 19, &c. (v) To multiply by 4 or 6, we can either multiply from the left by 2 and then by 2 or 3, or multiply from the right by 4 or 6; or we can treat the multiplier as 5—1 or 5+1. rot. Multiplication by a Large Number.—When both the numbers are large, we split up one of them, preferably the multiplier, into separate portions. Thus 231.4273 = (200+30+ I) 4273=200.4273+30.4273+1.4273. This gives the partial products, the sum of which is the complete product. The process is shown fully in A below, A B 854600 128190 231 4273 231 987063 987063 10 -042730 and more concisely in B. To multiply 4273 by 200, we use the commutative law, which gives 200.4273 = 2 X10 0(4273= 2X4273XIoo=8J46X100=854600; and similarly for 30.4273. In B the terminal es of the partial products are omitted. It is usually convenient to make out a preliminary table of multiples up to I0 times; the table being checked at 5 times (§ Too) and at ro times. The main difficulty is in the correct placing of the curtailed partial products. The first step is to regard the product of two numbers as containing as many digits as the two numbers put together. The table of multiples will then be as in C. The next 'step is to arrange the multiplier and the multiplicand above the partial products. For elementary work the multiplicand may come immediately after the multiplier, as in D; the last figure of each partial product then comes immediately under the corresponding figure of the multiplier. A better method, which leads D E 4273!231 o8546 12819 04273 0987063 up to. the multiplication of decimals and of approximate values of numbers, is to place the first figure of the multiplier under the first figure of the multiplicand, as in E; the first figure of each partial product will then come under the corresponding figure of the multiplier. 102. Contracted Multiplication.—The partial products are sometimes omitted; the process saves time in writing, but is not easy. The principle is that, e.g., (a. 102+b. Io+c) (p. Io2+q. I0+ 4273 X 0987063 200 0854600 X - 200 132463 30 128190 x - 230 04273 1 04273 x - 231 0000 The principle is that 162.427=100.427+60.427+2.427= 1.42700+6.4270+2.427; but, instead of writing down the separate products, we (in effect) write 42700, 4270 and 427 in separate rows, with the multipliers 1, 6, 2 in the margin, and then multiply each number in each column by the corresponding multiplier in the margin, making allowance for any figures to be " carried." Thus the second figure (from the right) is given by 1+2.2+6.7=47, the t being carried. 103. Aliquot Part.—For multiplication by a proper fraction or a decimal, it is sometimes convenient, especially when we are dealing with mixed quantities, to convert the multiplier into the sum or difference of a number of fractions, each of which has 1 as its numerator. Such fractions are called aliquot parts (from Lat. aliquot, some, several). This can usually be done in a good many ways. Thus *=t—4iand also =2+3, and 15%='15=T16+26 — = e +41 . The fractions should generally be chosen so that each part of the product may be obtained from an earlier part by a comparatively simple division. Thus 2 +- --g?i is a simpler expression for T I than z ++o-. The process may sometimes be applied two or three times in succession; thus (t —*) (1—D, and -H-=4' i = (I—4) (1+-)• 104. Practice.—The above is a particular case of the method called practice, but the nomenclature of the method is confusing. There are two kinds of practice, simple practice and compound practice, but the latter is the simpler of the two. To find the cost of 2 lb 8 oz. of butter at Is. 2d. a lb, we multiply Is. 2d. by 2T~ = 22. This straightforward process is called "compound" practice. " Simple " practice involves an application of the commutative law. To find the cost of n articles at La, bs. cd. each, we express La, bs. cd. in the form L(a+f), where f is a fraction (or the sum of several fractions); we then say that the cost, being nX£(a+f), is equal to (a+f)X£n, and apply the method of compound practice, i.e. the method of aliquot parts. 105. Multiplication of a Mixed Number.—When a mixed quantity or a mixed number has to be multiplied by a large number, it is sometimes convenient to express the former in terms of one only of its denominations. Thus, to multiply £7, 13s. 6d. by 469, we may express the former in any of the ways £7.675, °o of £1, 1534s., 153•5s., 307 sixpences, or 1842 pence. Expression in £ and decimals of £t is usually recommended, but it depends on circumstances whether some other method may not be simpler. A sum of money cannot be expressed exactly as a decimal of £I unless it is a multiple of 4d. A rule for approximate conversion is that Is.= •05 of £1, and that 21d.= •oi of £I. For accurate conversion we write •I£ for each 2S., and •ooi£ for each farthing beyond 2s., their number being firstincreased by one twenty-fourth. 106. Division.—Of the two kinds of division, although the idea of partition is perhaps the more elementary, the process of measuring is the easier to perform, since it is equivalent to a series of subtractions. Starting from the dividend, we in theory keep on subtracting the unit, and count the number of subtractions that have to be performed until nothing is left. In actual practice, of course, we subtract large multiples at a time. Thus, to divide 987063 by 427, we reverse the procedure of § tor, but with inter- mediate stages. We first construct the multiple-table C, and then subtract successively zoo times, 3o times and r times; these numbers being the par- tial quotients. The theory of the pro- cess is shown fully in F. Treating x as the unknown quotient corresponding to the original dividend, 427 427 427 69174 F 4273 200 30 C 4273 8546 12819 4273 1 - 04273 2 - 08546 3 - 12819 231 4273 231 08546 231 I2819 04273 0987063 r 6 2 we obtain successive dividends corresponding to quotients x—200, x—23oand x—231. The original dividend is written as 0987063, since its initial figures are greater than those of the divisor; if the dividend had commenced with (e.g.) 3 . . . it would not have been necessary to insert the initial o. At each stage of the division the number of digits in the reduced dividend is decreased by one. The final dividend being 0000, we have x—231 = o, and therefore x= 231. 107. Methods of Division.—W::at are described as different methods of division (by a single divisor) are mainly different methods of writing the successive figures occurring in the process. In long division the divisor is put on the left of the dividend, and the quotient on the right; and each partial product, with the remainder after its subtraction, is shown in full. In short division the divisor and the quotient are placed respectively on the left of and below the dividend, and the partial products and remainders are not shown at all. The Austrian method (sometimes called in Great Britain the Italian method) differs from these in two respects. The first, and most important, is that the quotient is placed above the dividend. The second, which is not essential to the method, is that the remainders are shown, but not the partial products; the remainders being obtained by working from the right, and using complementary addition. It is doubtful whether the brevity of this latter process really compensates for its greater difficulty. The advantage of the Austrian arrangement of the quotient G H 4273 4273 2 2 0987063 0987063 08546 08546 lies in the indication it gives of the true value of each partial quotient. A modification of the method, corresponding with D of § tor, is shown in G; the fact that the partial product 08546 is followed by two blank spaces shows that the figure 2 represents a partial quotient 200. An alternative arrangement, corresponding to E of § 1o1, and suited for more advanced work, is shown in H. 108. Division with Remainder.—It has so far been assumed that the division can be performed exactly, i.e. without leaving an ultimate remainder. Where this is not the case, difficulties are apt to arise, which are mainly due to failure to distinguish between the two kinds of division. If we say that the division of 41d. by 12 gives quotient 3d. with remainder 5d., we are speaking loosely; for in fact we only distribute 36d. out of the 41d., the other 5d. remaining undistributed. It can only be distributed by a subdivision of the unit; i.e. the true result of the division is 3151d. On the other hand, we can quite well express the result of dividing 41d. by 1s (=12d.) as 3 with 5d. (not " 5 ") over, for this is only stating that 41d.=3s. 5d.; though the result might be more exactly expressed as 3As. Division with a remainder has thus a certain air of unreality, which is accentuated when the division is performed by means of factors (§ 42). If we have to divide 935 by 240, taking 12 and 20 as factors, the result will depend on the fact that, in the notation (20) (12) of § 17, 935=3 r7 11. In incomplete partition the quotient is 3, and the remainders 11 and 17 are in effect disregarded; if, after finding the quotient 3, we want to know what remainder would be produced by a direct division, the simplest method is to multiply 3 by 240 and subtract the result from 935. In complete partition the successive quotients are 77'—2 and 30122=3 o• Division in the sense of measuring leads to such a result as 935d.=£3, 17s. Iid.; we may, if we please, express the 17s. 11d. as 215d., but there is no particular reason why we should do so. 109. Division by a Mixed Number.—To divide by ,a mixed number, when the quotient is seen to be large, it usually saves time to express the divisor as either a simple fraction or a decimal of a unit of one of the denominations. Exact division by a mixed number is not often required in real life; where approximatedivision is required (e.g. in determining the rate of a " dividend '), approximate expression of the divisor in terms of the largest unit is sufficient. r to. Calculation of Square Root.—The calculation of the square root of a number depends on the formula (iii) of § 6o. To find the square root of N, we first find some number a whose square is less than N, and subtract a2 from N: If the complete square root is a+b, the remainder after subtracting a2 is (2a+b)b. We there-fore guess b by dividing the remainder by 2a, and form the product (2a+b) b. If this is equal to the remainder, we have found the square root. If it exceeds the square root, we must alter the value of b, so as to get a product which does not exceed the remainder. If the product is less than the remainder, we get a new remainder, which is N—(adb)2; we then assume the full square root to be c, so that the new remainder is equal to (2a+2b+c) c, and try to find c in the same way as we tried to find b. An analogous method of finding cube root, based on the formula for (a+b)3, used to be given in text-books, but it is of no practical use. To find a root other than a square root we can use logarithms, as explained in § 113. (ii.) Approximate Calculation. 111. Multiplication.—When we have to multiply two numbers, and the product is only required, or can only be approximately correct, to a certain number of significant figures, we need only work to two or three more figures (§ 83), and then correct the final figure in the result by means of the superfluous figures. A common method is to reverse the digits in one of the numbers; but this is only appropriate to the old-fashioned method of writing down products from the right. A better method is to ignore the positions of the decimal points, and multiply the numbers as if they were decimals 2734 3 between • 1 and I .o. The method E of 314' 59 • § for being adopted, the multiplicand and the multiplier are written with a space after as many digits (of each) as will be required in the product (on the principle explained in §101); and the multiplication is performed from the left, two extra figures being kept in. Thus, 0859 to multiply 27.343 by 3'14I5927 to one decimal place, we require 2+1+1=4 figures in the product. The result is o85.9=85.9, the position of the decimal point being determined by counting the figures before the decimal points in the original numbers. 112. Division.—In the same way, in performing approximate division, we can at a certain stage begin to abbreviate the divisor, taking off one figure (but with correction of the final figure of the, partial product) at each stage. Thus, to divide 85'9 by 3'1415927 to two places of decimals, we in effect divide •o859 by '3,1415927 to four places of decimals. In 9 42 the work, as here shown, a o is inserted in front of the 859, on the principle explained in § 106. The result of the division is 27.34. 113. Logarithms.—Multiplication, division, involution and evolution, when the results cannot be exact, are usually most simply performed, at any rate to a first approximation, by means of a table of logarithms. Thus, to find the square root of 2, wP have log 112 =log (2i) =1 log 2. We take out log 2 from the table, halve it, and then find from the table the number of which this is the logarithm. (SEE LOGARITHM.) The slide-rule (see CALCULATING MACHINES) is a simple apparatus for the mechanical application of the methods of logarithms. When a first approximation has been obtained in this way, further approximations can be obtained in various ways. Thus, having found .12 = 1.414 approximately, we write ,/2 =1.414-1-0, whence 2=(1.414)2+(2.818)0+02. Since 02 is less than 4 of 0820 29 027 34 10 94 0 27 14 2 3141 5927 2734 0859 00 0628 32 230 68 219 91 10 77 135 r 26 (•001)2, we can obtain three more figures approximately by dividing 2—(I.414)2 by 2.818. 114. Binomial Theorem.—More generally, if we have obtained a as an approximate value for the pth root of N, the binomial theorem gives as an approximate formula p.(N=a+B, where N= ap+pap-1B. 115. Series.—A number can often be expressed by a series of terms, such that by taking successive terms we obtain successively closer approximations. A decimal is of course a series of this kind, e.g. 3'14159 . . . means 3+1/IO+4/102+1/103+5/104+ 9/IOS+ .. . A series of aliquot parts is another kind, e.g. 3'1416 is a little less than 3 -1 7 —r~. Recurring Decimals are a particular kind of series, which arise from the expression of a fraction as a decimal. If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur. The interest of these series is, however, mainly theoretical. 116. Continued Products.—Instead of being expressed as the sum of a series of terms, a number may be expressed as the product of a series of factors, which become successively more and more nearly equal to 1. For example, 3.1416=3X i MI =3XiiH=3X$}X$$$$=3(1+s1r) (1 —TAT). Hence, to multiply by 3.1416, we can multiply by 3-i, and sub-tract 25100 (_ •0004) of the result; or, to divide by 3.1416, we can divide by 3, then subtract -h of the result, and then add -si g of the new result. 117. Continued Fractions.—The theory of continued fractions (q.v.) gives a method of expressing a number, in certain cases, as a continued product. A continued fraction, of the kind we are considering, is an expression of the form a+ b+ c+d+ &c. where b, c, d, ... are integers, and a is an integer or zero. The expression is usually written, for compactness, a+b+ c+ d+ &c. The numbers a, b, c, d, . . . are called the quotients. Any exact fraction can be expressed as a continued fraction, and there are methods for expressing as continued fractions certain other numbers, e.g. square roots, whose values cannot be expressed exactly as fractions. The successive values 6, a I, ... ,obtained by taking account of the successive quotients, are called convergents, i.e. convergents to the true value. The following are the main properties of the convergents. (i) If we precede the series of convergents by i- and -h-, then the numerator (or denominator) of each term of the series a ab+i 1, b after the first two, is found by multiplying the numerator (or denominator) of the last preceding term by the corresponding quotient and adding the numerator (or denominator) of the term before that. If a is zero, we may regard b as the first convergent, and precede the series by -& and -. (ii) Each convergent is a fraction in its lowest terms. (iii) The convergents are alternately less and greater than the true value. (iv) Each convergent is nearer to the true value than any other fraction whose denominator is less than that of the convergent. (v) The difference of two successive convergents is the reciprocal of the product of their denominators; ; e. ab+ 1 _ a _ 1 , and abc+c+a ab+r—1 g —~ i 1.b be +i b b(bc+ It follows from these last three properties that if the successiveIn certain cases two or more factors can be combined so as to produce an expression of the form I=k, where k is an integer. I I I For instance, 3.141 5927=3(1+3T) (I—22.106) (I+333'113) . .; but the last two of these factors may be combined as (1 1 ). 22.113 Hence 3.5415927=1'Bi'2fsa • (i.) Systems of Measures.l 118. Metric System.—The metric system was adopted in France at the end of the 18th century. The system is decimal throughout. The principal units of length, weight and volume are the metre, gramme (or gram) and litre. Other units are derived from these by multiplication or division by powers of to, the names being denoted by prefixes. The prefixes for multiplication by so, 102, ro3 and Io4 are deca-, hecto-, kilo- and myria-, and those for division by lo, Io2 and io3 are deci-, centi- and milli-; the former being derived from Greek, and the latter from Latin. Thus kilogramme means i000 grammes, and centimetre means -r of a metre. There are also certain special units, such as the hectare, which is equal to a square hectometre, and the micron, which is u of a millimetre. The metre and the gramme are defined by standard measures preserved at Paris. The litre is equal to a cubic decimetre. The gramme was intended to be equal to the weight of a cubic centimetre of pure water at a certain temperature, but the equality is only approximate. The metric system is now in use in the greater part of the civilized world, but some of the measures retain the names of old disused measures. In Germany, for instance, the Pfund is 2 kilogramme, and is approximately equal to IT lb English. 119. British Systems.—The British systems have various origins, and are still subject to variations caused by local usage or by the usage of particular businesses. The following tables are given as illustrations of the arrangement adopted elsewhere hi this article; the entries in any column denote multiples or sub-multiples of the unit stated at the head of the column, and the entries in any row give the expression of one unit in term of the other units. LENGTH Inch. Foot. Yard. IChain. Furlong. Mile. I 1$ 31~ is Ts3s B$a~. 22 I a s8 ifs 5A 2 36 3 1 z3 5 1~ 2za Svea 792 66 22 I 330 A 7920 66o 220 20 I 6336o 528o 176o 8o 8 I
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