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Mh ‘tsa are to eal ite: haba ip 5 oy it ad N ais ‘ 2 ie a Ns i . 4 ' ‘ vi as de By a LL bg 1 a0. 4 vy & . seep teh any ea , ks at ATLA WA ean : ON a“ ne asa me , ah ie ' » , OU S : wt) ; Ra vied Lan 4 : re SOURCE tes Ni 3h Peat ROAD a ‘ ' - ' 1a * i ‘ Hts, Saya Raa A RSME EAE +a Wey s ry" ray eae a 4 sy tataly Ons MATHEMATICS Digitized by the Internet Archive in 2022 with funding from University of Illinois Urbana-Champaign https://archive.org/details/elementsofcommer01 tate - ee aa . ape i 7 aeery, Se ne %y / ? rea a | La ; Lai ; > mi" ' Byki Wa): 7) ~) : ae ) | a ma i ah ae | | my 4 + | | ¥ ’ | ° v Th ml b NeOr ieee ae ak AT LEAST ONE OF THE EDGES OF THIS MAGAZINE HAS BEEN LEFT UNTRIMMED, BECAUSE OF AN EXTREMELY NARROW MARGIN. HERTZBERG-NEW METHOD, INC. THE ELEMENTS OF COMMERCIAL CALCULATIONS, AND AN INTRODUCTION TO THE JS¥lost Jmportant Branches OF THE COMMERCE &S PUNANCES OF THIS COUNTRY, ee By W. TATE, OF THE FINISHING ACADEMY, CATEATON*«STREET. London: PRINTED BY RICHARDS AND CO. 3, GROCERS’ HALL COURT, POULTRY. 1819, - ~ yo hid. 8 109 GK ACTS high ve PREFACE. —~) i oO onal Ss wm £ 30.3754 or £30 333533 These examples exhibit the different forms of the addition of whole, fractional, and decimal quantities. The first is proved by separating it into two parts, finding the amount of each part, and then finding the Sum of these amounts. The second is exhibited in two forms of expressing fractions, and in this we have to observe, that we can proceed only in the same manner as with the former, where we add units with units, tens with tens; so here we add the eighths with eighths. Of the different forms of finding the values of the amounts, and making unlike fractions similar, we shall take notice hereafter. In the addition of the decimal quantities, the fractional forms of expressing the values of the given quantities, and of their amount> are placed against each ; but it is to be noticed that in these forms of the equivalent fractions, they cannot be added together by the addition of their numerators. PLI LILLIE LRP EOE ERI DDE OF ARITHMETIC. 5 THE SUBTRACTION OF QUANTITIES. PLP LEO LI OLE IIT IATL The subtraction of one or more quantities from another quantity, is the determination of the remainder, or of the other part of which that quantity is considered to be com- posed. LOL DPOLADOOCL PLA ALLEL OIE Examples of the subtraction of similar and simple quantities, expressed by the same sort of numbers. £ £ £ £ 22 $3 or 837 36 3.627 or 3.62 43 iets 12 2.1582 7 8418 794 346 £14 1.4688 1.4688 346 362 —. at yer Fn 448 751 362 L 86 L£ 86 In the first of these calculations, the several quantities to be subtracted from £837 are £43, £2346, and £362, which are separately subtracted from the first sum and the successive re- mainders; but it may be otherwise performed, as in the second method, by finding the amount of the quantities to be subtracted, and subtracting this amount. In the second calculation, as the 7-8ths are greater than the 5-8ths, they are taken from an Integer or 8-8ths; the remainder, 1-8th, is then added to the 5-Sths, and the Integer is added to the 1 given; or it may. be performed by taking the 7-8ths from 1 and 5-8ths, or 13-8ths. | In the third, decimal calculation, the 2.1582 is subtracted from the (3.627 inthe manner of ordinary numbers, supposing the latter to be £3.6270 which expresses the same value. It is otherwise performed by finding what is:termed the arithmetical complement of the 2.1582, or the difference between this num- ber and 10, by subtracting, first the 2 from 10, and the 8 and the other figures from 9, which complement is thus found to be 7.8418; this is then added to the 3.627 and 10 is taken off the amount. * GOP IIL LIC IDE LIOLIIOLIN a Nes 6 GENERAL PRINCIPLES THE MULTIPLICATION OF QUANTITIES. PLL LOI LAT The multiplication of a quantity is the repetition of that quantity any given number of times; or the finding of the amount of a given quantity repeated or taken a given number of times, and it is equivalent to the Addition of so many equal quantities. PPP LOL LOL PLO LD T ODL LO! N.B. In order to obtain a clear conception of the true nature of these arithmetical operations, it should be well understood that the number which we call a multiplier has no other use or meaning, than to show how many times the given thing is to be taken, or if it were expressed as in addition, how many times the thing would be put down; and, that as quantities are the representatives of things supposed to exist, and not merely numbers, it is highly in- correct to speak of using them as multipliers; as in saying 4s are to be multiplied either by 5 yards or by 5s, when we merely intend saying, that 4s are to be multiplied by 5, or to be taken 5 times. Example of a simple, whole quantity, multiplied by 10, and by 100. Let £137 be multiplied by 10, and by 100. £ 1370 product by 10. 18700) 225 by 100. This scarcely needs any explanation, it being evident from the principles upon which numbers are composed, that the amount of 10 times any simple quantity will be expressed by the same figures, each raised one step, in the series of numbers, higher in value:— and that, to multiply by 100 raises the figures two steps, &c. PIS PPD DAD PID LDL AIG FIP Example of the multiplication of a simple, whole quantity by various Numbers. . Let £452 be multiplied by 24, by 29, and by 37. £ 3 £ p 452 by 4 452 by 29 or 452 by 30 52 by6 1808 by 6 4068 13560 2712 by6 904 Tyee Fee’. 13108 for 29 16724 for 37 £ 10848 £13108 . one ee: a OF ARITHMETIC. ri These several forms of multiplication are thus denominated.— The first is called double multiplication, and the second a long multiplication; the third is an example of the multiplication of a simple quantity by 10 times 3 in one operation, with the subtrac- tion of once the given quantity to form 29 times ; and the fourth isan example of a double multiplication, with the addition of once the given quantity to the second product, as each figure is found, to in- crease 36 times to the required number, 37 times. PLP LADD LIL POL LDA DD IO DEE Example of the multiplication of a fractional quantity. 4 PPD AD LE LOD DDE AOE ADD DDS Let £2 42 be multiplied by 3, by 7, and by 21. £ £ Ae 43 43 43 by 7 7 arr ~ TF 31 by 3 s igs £93 In the first of these calculations, 3 times 3-7ths produce 9-7ths, which are equal to 1 and 2-7ths. Inthe second, 7 times the 3-7ths produce 3 integers ; and the same is the case in the third calcula- tion. PLP LIL DAE POL PDS OOD LOLS Example of the multiplication of a Decimal Quantity. PPI FIL DL I LAD POL AAD EAE Let £37.57 be multiplied by 10, by 100, by 1000, and by 25. £37.57 by 23 £ 375.7 product by 10 Gio cae 19 f aeER oT as. 390% 100 ide LRAT B70 dei. heck. 1000 , : 864.11 It was seen in the first of the preceding Examples, that to mul- tiply by 10 has the effect of raising each figure in the given number one step higher in value, and that to multiply by 100, raises each two steps; and this is effected, when there are decimals, by re- moving the decimal point one place to the right to multiply by 10, | two places to multiply by 100, &c. ; using, as in the third multi- plication, one or more ciphers when the decimal places are not suf- | ficient. In the fourth part, the multiplication is conducted similarly ‘to when the given quantity is expressed by a whole number. Ses Ss y+ af te § GENERAL PRINCIPLES THE DIVISION OF QUANTITIES. The division of a quantity is the determination of what quantity, taken or repeated a given number of times, will produce a given quantity ; and consequently, when the Di- ' visor is a whole Number, the quantity to be found is what is termed a part, or aliquot Part, of the quantity given. - Example of the division of a simple quantity by 10, 100, Se. Let £429 be divided by 10, by 100, and by 1000. £ 42.9 or £42, quotient by 10 P'4.29 ore 14 tee le: 100 ‘D490 as eRe eres 1000 These quotients, or products of the divisions, are expressed both decimally and fractionally, with respect to those parts which cannot be expressed by whole numbers. To divide by 10 reduces each place of the given number one step lower in value ; to divide by 100, two places, &c. With respect to the figures cut off, it is evident that as the 10th and 100th parts of 1 are 1-10th and 1-100th, so the 10th part of 9 is 9-10ths, and the 100th part of 29 is 29-100ths, &c. These figures cut off, also form what are termed the remainders of the divisions, or in other words supposing the quotients should only be expressed by whole numbers, there would be these surpluses in the given quantities ;—thus the quotient of this division by 10, or the 10th of £429, would be said to be £ 42, with £9 over, or as a remainder. In many cases, as will be shown hereafter, the remainders require to be reduced into a lower denomination. x PLS FDI OPP ODD LOL DOD PDP OF ARITHMETIC. 9 Examples of the Divisions of a simple Quantity, by various whole Numbers. PPS LOD LOL OFLA VO AEA LAE Let £638 be divided by 5, by 21, and by 23. £ £ £ £ 5) 638 3) 638 ZDos8 (Wat or 23)638 (2713 A6 178 £ 1272 7) 2123 — 17 eee 178 £3045 161 17 The first of these divisions is an example of. what is termed a short division ; the second of a doudle division, or the successive divisions by two numbers, whose products make the given number ; and the third is an example of a Jong division;—it is particularly to be observed, that if the given number is neither a short divisor, nor exactly a double divisor, it can only be used as a long divisor. In the first division, the 5th part of the £3 over, produces 3-5ths of a £. Inthe second division, in the first part, the £2 over produces 2-3rds of a £, and in the second part the 7th part of £22, or of 8-3rds of a £L, produces 8-21ths of a L, for reasons which will be shown hereafter. It is frequently otherwise said, that the 2 over in the second part of the division, multiplied by 3 the first divisor, with the 2 over in the first part of the division, produces in all 8 over—the 21th part of which produces as before 8-21ths of a £. Inthe third division, as the products of 23 times the numbers found are too great to be easily managed in the mind, or to be mentally subtracted, they are expressed; the re- mainders of the subtractions are also expressed. Some contraction of this method is shown in the fourth division, where only the remainders are expressed; explaining the work in this way, we say, twice 3 are 6 from 13 leaves 7 and carry 1, twice 2 are 4 and l are 5 from 6 leaves 1. Then bringing down the 8, we. say, 7 times 3 are 21 from 28 leaves 7 and carry 2, 7 times 2 are 14 and 2 are 16 from 17 leaves 1. From the nature of the operation of a division, it is Sider that in all cases, the proof of a division is to be found from the mul- tiplication of the quotient by the divisor, when the product, if the work be correct, will be the dividend or the quantity to be divided. What was said respecting the reduction of the remainders, iu the last Example, applies also to this. Vou, I. c 10° GENERAL PRINCIPLES Example of the division of a fractional quantity. De ae POL LAL LLL Let £1232 be divided by 3, by 36, and by 57. 0 aoe 3) 1235 § 3) 1233 57 ) 1233 ( 2,3, £ 412 36 114 t 012) 412 wie) 4—— et The division of a fraction by a whole number may in all cases be done, either by the division of the numerator, or the multipli- tion of the denominator; the one method taking the required part of the number, and the other, of the value of the given parts ;— but it is to be observed, that unless the divisor is exactly con- tained in the numerator, the second method expresses the quo- tient in a more convenient manner; or instead of multiplying the denominator by the divisor, we may multiply both terms of the fraction by any number which will render the numerator exactly divisible by the given number. Thus.in the first division, and in the first part of the second division, the third part of 3-4ths is 1-4th; but in the second part, the 54 over make 21-4ths which make 84-16ths, of which the 12th part is 7-16ths. In the last division, the 93 make 39-4ths, of which the 57th part, found by multiplying the 4 by 57, is 39-228ths; or, we may say, that 39-4ths make 2223-228ths, the 57th part of which is 39-228ths. PPL LAD DAD LLP Example of the divisicn of a decimal quantity. Let £ 32.67 be divided by 5, by 12, and by 37. £ £ £ 5)32.67. 12)32.67 | 37)32.67 £ 6.534 2.7225 or 22.72! £ .883 very nearly. | In the division of a decimal quantity if there is any remainder from the, division, the process may be continued, by annexing ciphers, until either there is no remainder, or the division has been continued as far as necessary. OF ARITHMETIC. 11 THE REDUCTION OF QUANTITIES. PIPL AIL PAL LAL PAD BAS AOE The reduction of a quantity, expressed by whole num- bers, is the finding of another quantity in a different form or denomination, but of equal value. POD LIL LOL AAD ADI ADE LAL General Directions.—Consider the denomination of the given quantity, as changed into that of the quantity required; then, where the reduction is to be made into a lower denomination, multiply, or into a higher denomination, divide, by the number of the lower denomiiation which is equal to an Integer of the higher. PPI LPIPLP PID PDI ALOE LOE CaS Example of a reduction into a lower denomination. Orr PIL PDD PLD LB IS Let £376 be reduced into pence. Pence. Pence. qs 376 or 240 or 376 240 376 ata 9 Rand cae 7520 s 15040 - 15040 752 759 90240 pence- 90240 pence. 90240 pence. It may be first of all observed, that any calculation in reduction is but the working of a direct proportional calculation, as will be shown in a succeeding part; and as this process will there be further discussed, it is necessary here only to notice, that accord- ing to the principles before laid down, the product of a multipli- cation cannot be otherwise than of the same nature as the thing multiplied, and that a quantity cannot be used to express how many times a thing is to be taken. at a 12 GENERAL PRINCIPLES We may say here in explanation of these operations, that 376 pence are the value of £376 at 1d. per £, and therefore 240 times — 376 pence are the value of the same at 240 pence per pound ; or, that 240 pence being the value of £1, 376 times 240 pence are the value of £376. In the third form -we say that the £376 being considered as so many shillings, express the value at 1s. per £; and being multiplied by 20, the product expresses the value at 20s. per pound; these shillings again, being considered as so many pence express their value at 1 penny per shilling, and being multiplied by 12, the product expresses the value at 12 pence per shilling. i ; Example of a reduction into a higher denomination. PLP LIL FDP EPP LLP POD LPL Let 3265 pence be reduced into pounds. Pounds. £ Pence. 865 or : 145 £13.12.1 s 272.1 £ 13.1270" Of these two calculations, the second is the more common form ; the first may be explained in this manner. If1 penny were equal to £1, the value of 3265 pence would be 3265 pounds; but as 1 penny is the 240th part of £1, the proper value is the 240th part of the assumed value, or of £23265; the exact quotient of which is £13 and 145-240ths of a £2 :—but as the 240th part of a £ is in value id, the value of 145 of these parts is 145 pence, or 12 shillings and 1 penny. In the second form, upon the same principle, considering the 3265 pence as so many shillings, we take a 12th part to determine in shillings the value of these pence ;—this product is 272s and 1-12th of a shilling or 1 penny ;—then, assuming these 272s as so many pounds, and dividing them by 20, to find the value in pounds of these 272s, the quotient is 13-£ and 12-20ths of a £, or 12s:—thus we have as before, for the result of the reduction, £13 12s 1d. OF ARITHMETIC. 13 From what we observe in this reduction we may lay it down as a general rule, that when a division is used in this process, the value of the remainder (when divided) is so many of the original lower denomination ;—or, that upon the division of any quantity by a number which also expresses how many of a lower denomi- nation is equivalent to one of the given denomination, the value of the divided remainder is so many of that lower denomination. Thus pounds being divided by 20, the quotient of the remainder is so many shillings ;—and shillings being divided by 12, the quotient of the remainder is so many pence. Example of the reduction of cwts. and qrs. into lb. PLP LISI LILI IID AOD AIO LOE Let 32 cwt. 3 qrs. 17Ib. be reduced into Ib. Cwt.qr.lb. Cwt.qr.lb. 9 Cwt. qr. tb. Cwt. qr. lb. 32, 3.17 or 32.3.17 or 32.3.17 or 32.3.17 922 32 384 -— 333 Np y 101 131 qr. 101 32 errr i01 Ib. 3685 Lop Ib. 3685 pie ER Ib. 3685 This calculation is properly only the reduction of 32 cwt. 3 qrs. which is thus differently performed. In the first two parts the 32 is considered as so many pounds, and it is multiplied by 112 in these different forms; first, to once 32 lb. we add 111 times 21b. and then 111 times 30Ib. making in all 112 times 321b. In the second to once 32 lb. we add the same, with 10 times 32]b. and 100 times 32]b. making in all 112 times 32Ib. In the third, we take 100 times 321b. by considering the 32 is 32001b. and then adding 12 times 321b. in two places of figures to the right of the other 32, we have as before 112 times 321b. In the fourth, the cwts. are reduced into qrs. by considering them as so many qrs. and oultiplying them by 4, taking in the 3 qrs.; these are then reduced into pounds, by considering them as so many pounds and multiplying them by 28, taking in the 171b. In the first three methods, the 3 qrs. 17 Jb. are taken in as 101 Jb. 14 GENERAL PRINCIPLES FRACTIONAL REDUCTION. PPP PE PLL EL DL PLP LOL POP A fraction is said to be reduced, when it is changed in its form, without being changed in its value; or when either a fraction or whole number equivalent to it is found. PLA PIPL LLP POD The general principle of fractional reduction is, that either the’ multiplication or division of both terms of a fraction occasions no alteration in its value; for while the number of the parts is thus increased or decreased, the value of these parts is proportionately reversed : thus, j are reduced or changed into 6-8ths by multiplying both terms by 2; on the contrary, 6-8ths are reduced or changed into ¢-by dividing. both terms by 2; but it is to be observed, that-no number should be used as a divisor, to reduce a fraction, without it will exactly divide both its terms. | An improper fraction, or a fraction of which the numerator is greater than the denominator, is reduced or valued by dividing the numerator by the denominator: thus, 12-4ths are valued by dividing 12 by 4, which produces 3 units; and 29-8ths are valued by die viding 29 by 8, which produces 35. Two fractions which have different denominators, are reduced to similar fractions, or fractions having the same denominator, by multiplying both terms of each fraction, by the denominator of the other fraction ; 2-3rds and 4-5ths are therefore thus reduced : 3 multiplied in both terms, by 5, become 2° % multiplied in both terms, by 3, become 42 When the denominator of one fraction is contained in that of another fraction, they may be made similar by multiplying both terms of that fraction which has the least denominator by the number of times it is contained in the other fraction; thus, 3 are made similar to 3, by multiplying both terms by 25; the fraction then becomes £; and by the same multiplier, + is turned into 55 and thus made similar to 3.. OF ARITHMETIC. . 15 Fractional quantities are valued, either, by considering the nu- merator as so many integers, and dividing it by the denominator ; (thus, £ % are valued by dividing £5 by 8, the product or value being 12s. 6d.) or, by dividing the integer by the denominator, and multiplying the product by the numerator; thus, £3 may be valued by dividing £1 by 8, which produces 2s 6d and this multiplied by 5, produces as before, 12s 6d. SOLID LIL LOL LOE PDL LAE DECIMAL REDUCTION. SIE IE LILI LI III I IF II OID The only variation which can be made in the forms of decimal numbers, is expressing their values in the forms of fractions, or with the proper denominator, and then reducing them into lower terms, (if possible) by the division of both terms by any number which will exactly divide them. Thus .125 = 723, = 7% — 4, in which the first fractional value is divided in both terms by 5, and the produced value in the same manner by 25. On the reverse the value of a fraction is expressed as a decimal, by dividing the numerator by the denominator. Thus 3-4ths = 3~ 4 = .75 in which the division is com- menced and continued, by annexing a cipher to the 3 to reduce it into 10ths, and then to the remainder to reduce it into 100ths, and the product shows that 3-4ths are equal in value to 75-100ths. PIL DDL DDD LED PDIP OLD LLL Example of a valuation of a decimal quantity. POL LOD DDI PLD DPI DOP OOP Let the value of £ .8375 be found. £.8375 s 16.7500 Value required, s 16. Gd. d .9.0000 4 This value is found by considering the given decimal quantity as | the 10000th part of £8375, and dividing this sum of money by 10000. 16 GENERAL PRINCIPLES Example of the reduction of a simple and of a compound quane tity into decimal quantities. Let 7sand 15s 7d be expressed as decimal quantities. 20)7.£ d. 7. 94,0).18,7 £(.77916, Ke. a a ee 190 ff . 35 s. 15.5833 _220 i, alae regenera Ai . 40 £ .77916, &e. oh .16 In the first calculation 7s are considered as the 20th part of £7, the value of which isthe decimal £.35 or 35-100ths of a £. In the second part of the second, 15s 7d or 187 pence, are considered as 187-240ths of a £, which are equal to the decimal £.77916; and which, as frequently occurs with decimals, migh be continued to any extent without coming to a termination; the above value is however much more minute than is generally wanted In the first part of the second method, the pence are reduced int the decimal of a shilling, and then the shillings, with this decimal, are reduced into the decimal of a pound. te TO FIND BY INSPECTION THE VALUE OF DECIMAL PARTS OF A £2. | Directions.—Double the number of tenths for shillings, and if the: hundredths are 5 or above, take them in as another shilling. Con sider the hundredths (or the remaining hundredths, if the 5a before are taken away) as tens of farthings, and the thousandths a units of farthings, abating one when they exceed 12, and two when they exceed 37. Whenthis Rule is used, if the decimal extends to more than three places, only those 3 places are to be used, but if the remaining figures are more than 5 ten thousandths, the last cus sandth is in general to be reckoned one more. | PLO PLE LOL LOO LOL LIE AEST Let the value be found of the following decimal quantities. £ s s : ff go xen. 2 ize, .15— $ Ex. 3. .875 = 17 -6 we 4 Oe 279 = 7 + 6 5 eres 1 Al2 = fee on Ace 8 45° 9 306 — Tlie 5 Se Lk 7997 <= 11Z In explanation of these we say; that 1-10th of a £ is 2s, 2-10ths As, &c.; that 2-10ths and 5-100ths (equal to 1-20th) produce 4 and 1s,or 5s; and that as, when decimally separated, a £2 con _ OF ARITHMETIC. 17 tains 1000 parts, which are also equal to 960 farthings, 25 parts are equal to 24 farthings, or 6d; and therefore, reckoning to the nearest half above and below 25, we deduct 1 part from 12 parts to 37, and two above that number. Thus for decimal £ .279 we take .25 as 5s, and 29-1000ths as 28 farthings, or 7d. SPL LILI DDD IID LAD LAO ADD TO FIND, BY INSPECTION, THE VALUE OF SHILLINGS, PENCE, AND FARTHINGS, IN DECIMAL PARTS OF A £2. PDF LID LOD LIL LOAD DOL POT Directions.—Take half the number of shillings for the number of tenths, not noticing an odd shilling should there be one; reduce the pence and farthings into farthings, add one if they are above 12, and 2 if they are above 36; this amount will show the number of thousandths for the pence and farthings, to which add 50 for the odd shilling, if before rejected. It is evident, that if there is only one shilling there will not be any tenths; and that if there is not an odd shilling, the hun- dredths will not amount to 5. LOD ALL DDS PLP AAI OOD LAE Examples.—Let the following Sums be expressed in decimal parts of a £. el ee ee ee Ex. 1. Eg. 2: Ex. 3: ,*# ced fi ag | £ j Remote 997) a Os > ae Pirated Wh" peer | 7 6. 4ARB6 8. 5) == 93 3= 15 10 6 = .525 6 11i = .347 5 = .25 16 6 = .825 7 112 = .399 The principles of these Calculations are just the reverse of the preceding. In finding the decimal values of the last two sums of money, we say, that for 6s 112d, the half of 6 gives 3, the number of tenths ; and for 45 farthings by adding 2 to their number, we obtain 47 parts :—for 7s 113d, the half of 7 gives 3, for tenths, with 1s over ; then 113d, as before, produce 49 parts which with 50 for the odd shilling, make the 99 parts, or, in all 399-1000ths of a £. SII LOL DOL LIL APL LLL LAE Vox. I. D 18 GENERAL PRINCIPLES THE ADDITION AND SUBTRACTION OF COMPOUND QUANTITIES- * PALF LLI BLL LOD DPD LOL OPS | j ' In the addition or subtraction of any quantities, it is to be ob- f served as a general principle, that the sum or difference can only be found when the quantities are similar, and are expressed by the © same sort of numbers. : PIPL LLP DPD LOD LLP DOD FIT Examples.—Let the sum and difference of £872 14 4, and £347 17 5 be found. Nee oes oa ns ae 872 14 4 $72 14.4 347 17 5 347 17 5 Sum £1220 11 9 £524 16 11 Diff. These Calculations seem to require no explanation. SLI LIAS LOD LOD OD LDA OP DOE —— a THE ADDITION AND SUBTRACTION OF FRACTIONAL QUANTITIES. PLP DLL LDL DPI LSD ODD DOA If the given quantities are similar, but are not expressed by si- — milar Fractions, they must be reduced to a common denominator, and then the amount or difference may be found. If the given quantities are not similar, their values may be found — and used in their places. PIS AOL POLS DLO LLL OLE LOS Example.—Let the sum and difference of £42, and £ 2% be # found. £4; = LA, L275 -Sum ..... L7 or L771 or LT 6 8 Difference. 2% or Fo £2-3 4 The 3-4ths of a £ require to be changed into 9-12ths of a £ be- | fore they are used, and this is done by vary! both terms by 3. OF ARITHMETIC. 19 Example. P Let the sum and difference of £23, and s42 be found. fo er nes LA 2 8 64,4 0FR , s 43— 4 7-, + or 3% 335 Sum Difference. Here, the fractional parts of a farthing require to be made simi- lar; then the 28 and 15-35ths make 1 farthing and 8-35ths; or, in subtracting, 1 farthing, or 35 of these parts, istaken, from which the 28- 35ths are subtracted, and the 15 being added to this re- mainder, make the whole remainder 22-35ths. LIL PII IF DAD LDL PDL ALD PEE THE ADDITION AND SUBTRACTION OF DECIMAL QUANTITIES PLL LDL DOL DAG IIE LOE LOFT Example. Let the sum and difference of £3.7078 and £2.84674 he found. £ 3.7078 2.84674 rth... Scle's £6.55454 = ££6.11.1 Difference... £ .86106 = LO17 23 These decimal parts are valued by the directions in page 16; the decimal .55 producing 11s. and the 4-1000ths, 1d; also, the .85, out of the .861, produce 17s. and the remaining 11- 1000ths produce 23d POLLED PEL LOE LD ELE PLOS ; 20 . GENERAL PRINCIPLES THE MULTIPLICATION OF COMPOUND QUANTITIES. PLE PIL POL LOE FOP The multiplication of a quantity, in its strictest sense, is under- stood to imply the repetition of the whole of that quantity, a cer- tain number of times; but its meaning is more generally extended to the repetition of either the whole or a part of any quantity ; and, even when the value or amount of only one part is required to be found, the process is sometimes called a multiplication. Example of the multiplication of a compound quantity. Let £4.11.6 be multiplied by 483 and by 573. Due ard Se Ao di 6 7 32. 0. 6 tf 224. 3:.6 for 49 1. 2.10} for~ £ 223. 0. 74° for 483 fs ad Ea 4.11. 6 | ‘i oo. 1G 8 260.15. 6 Z. 2.5.90. for -ll. 57 for = £263.12. 81 for 575 The first of these calculations shows a method which is very frequently useful, viz.; that of subtracting a part from the next — higher number of times; as here, taking off 1-4th from 49 times to produce 48 times and 3-4ths. In the second, with the product of 8 times 7 times for 56 times we take in once the head line, to make 57 times ; the parts then taken for 5-8ths, are, 4-8ths as the half, and 1-8th as the 4th of 4-8ths. Cd OF ARITHMETIC. 9} Example. LIL DIL LAL AIL Let £31.16.7 be multiplied by $ and by 1423 27 ft fs. A oa SLAG, 8 or a HE RY : 4.10.112 7) 159. 2.11 5 £22.14. 83 £22.14. 83 5m a SeeLOs7 636 s. 7639 pence. 1123 9751 ) 8578597 Pence 3118 94% s. 259.10d. £12.19.10 272, _ In the first calculation, instead of taking 5 times the 7th part, ‘we take the 7th part of 5 times; a method that is generally better, in order to avoid the trouble which might arise from the multipli- cation of the fractional parts. The other method is shown in the second part of this calculation. In the second calculation, in con- sequence of having to multiply by a large number, we reduce the compound quantity into a simple quantity, and perform the calcu- Jation by a long multiplication and a long division, of which the results only are here expressed, as it would take up too much room to express them at length. PLO LISI ILI PLO EF OLE OOP >“ Gefaasie Ai 99 GENERAL PRINCIPLES Example ; Of the fractional Multiplication of a fractional Quantity. * PLD LOL PLO PDD OLE APR Let £ be multiplied by 12 or 7-4ths. (7) 35) 6 £ixl=f=: Sadat OES Git [At BOTA OF PLD LPL LAL LOD LAFLE FALE DR In the first of these methods, we multiply by 7-A4ths, or take 7. times the fourth part, upon these principles; 7 times 5-7ths. are : evidently 35-7ths; and from what was observed in page 10, the fourth part of 35-7ths is 35-28ths; hence our usual directions.are, to” multiply the numerator by the numerator, and the denominator by ’ the denominator. In the second method, we practise a method which is frequently very serviceable, viz. that of dividing one numerator and one denominator by a number which will exactly divide them, and this may be repeated as often as it can be done; the numbers thus produced are used in the place of the original numbers which are said to be cancelled, and, as in this case, when 1 is the quotient it is not necessary to express it. The calculation might otherwise be expressed, thus ; 5 ey — 35 33 9 4 pe, LEX7T=T Land ~L+4A=-L; 8¢ ¢ 35 eye . but a Tan at by multiplying both terms by 4. Bop 2 Sp i and 5 £ = 7, £=L£150 PLE PDP BBP DOL LPOPL LOL LL OF ARITHMETIC. 23 Example. Ff Let £1.10.32 be multiplied by 1}, or 7-4ths. raid 8 £02.10; 32 or omy a 2 35. 17; or Px 7: 625 £2.12.1143 or Ss, d 1.10. 32 7 4) 10/h3.114 £ 2.12.1193 In the first method of working this proposed example, we let the given quantity stand for once, and add to it parts for the 3 quarters; saying 2-4ths are the half, and 1-4th is the half of 2-4ths. The half of 1d and 9-5ths, or of 7-5ths, or of 14-10ths, is 7-10ths; and the half of 1 penny and 7-10ths, or of 17-10ths, or of 34-20ths, is 17-20ths. In adding these fractions of a penny together, we change the 2-5ths into 8-20ths, and the 7-10ths into 14-20ths, before we can add them to the 17-20ths. The amount is 39-20ths, or 1 and 19-20ths. Inthe second method, we take the 4th part of 7 times for 13; and in taking the fourth part of 32, or of 19-5ths, or of 76-20ths, the product is 19-20ths. The trouble resulting from the fractions in the first method, may be obviated. by using their decimal values ;—thus, ie a 1110 3.4 LPe. 17 ew UG 86 £212 11 .95 4-10ths are here used for 2-5ths, and the 95-100ths, in the amount are equal to the 19-20ths in the former calculation. ? «ie OA GENERAL PRINCIPLES Example Of the Multiplication of Decimal Quantities. FLL ILL LOL ABODE FOA FADE LAF Let £80.76 be multiplied by 73, by .02 and by 8.008. 1.80.76 80.76 80.76 7 02 8.008 565.32 £ 1.6152 64608 | ee a Seo 2 seein 12,3040 64008 3,6480 L 646,72608 or L£ 595.605 or pdee Sts cire £ 646 14 61 £595 12 12 Cites S d f 2,5920 or = £ 1 12 33 nearly In the first calculation, we multiply by 7% as before, and value the product by the directions in page 16; the same mode is practised in the third calculation. In the second we have to multiply (as it. is termed) by 2-100ths, or we have to take the 100th part of twice the given quantity; the product of twice is £161.52, and the 100th part of this is £ 1.6152 ; and the same would be the product, if, instead of 2-100ths, we were to take 1-50th. In the third, we have to take 8 times and 8-1000ths, or the 1000th part of 8008 times; therefore, taking 8008 times we produce £ 646726.08, the 1000th part of which is £646.72608. From observing these effects of using decimal multipliers, we derive these general direc- tions. Find the product as in whole numbers, and from the right cut off as many places for decimals, as there are decimal places in both the multiplier and the quantity multiplied. In the second calculation, the value is found by the directions in page 15, and. not by those in page 16, merely for the purpose of showing some variation. OF ARITHMETIC. Q5, THE DIVISION OF COMPOUND QUANTITIES. PPL LIP LD DL I LL D I LOE AIS The division of a quantity is generally considered as the process of finding the amount, or value, of 4 given part of that quantity ; and, this is always the case when the divisor is a whole number: but a more general definition of a division, corresponding with the extended deiinition of a multiplication, is, that it is the reversing of a multiplication, or the method of finding what once that quan- tity must be, of which the number of times or parts expressed by the divisor, produces the quantity which is called the dividend. CLS IDOL LE LIL IID LI OE Examples of the division of a compound quantity. POL LOL LAI PII ALO LDA LAD ‘ Let £843.19.6 be divided by 42, 43, and by 100. en hate d os Ra ae | ues. gd a pesetates 6 43)843.19.6(19.12.624 £8,43.19.6 7)140.13. 3 als s 8,79 ALN, ~ 26 0. 1 1a | d 9,54 £ bi idiot 9°). G89 12s : 109 S$ d i £ 88.95% ) 28% (6d 24 The work of the second division, is expressed in the abbreviated manner described in page 9. The division by 100 is performed by cutting off the units and tens in the given number of the quan- tity to be divided, and reducing them, when necessary, into the next lower ‘denomination. Thus, the 100th part of £843 is £8 with 43-100ths of a £, the value of which with 19s is 879s; and, of this, the LOOth part is 8s and 79-100ths of @ shilling, When the valuation of a decimal, by the directions in page 16, is well understood, the division of a sum of money by 100 may be per- formed decimally, taking half the nearest even number of shillings, as the third place of decimals; as above, the 100th part of AG £ 343 19, is £ 8.440, which is equal to £ 8.8, 94. Ms y Voz. I. E :; 26 GENERAL PRINCIPLES Examples of Fractional Divisions. PLP POLI LLL OL LOVE LL ILA DL Let £ 4.16.6 be divided by 43, and by . £os-d 2 wad 43.) 4.16.6 4.16.6 4 4 4 19) 19. 6.0 3) 19. 6.0 £1. 0.34 6. 8.8 — To divide £4.16.6 by 43, is to find what quantity multiplied by 43 will produce that sum; and, here, multiplying both terms by 4, to get rid of the fraction, we divide £19.6.—by 193; that £1.0.34% isthe right product, may be proved by multiplying it — by 43. | To divide the same sum by 3-4ths, is to find that quantity, of — which 3-4ths are £4.16.6, and this is found to be £6.8.8. It may be remarked, that a divisor may be called a reciprocal or an inverted multiplier; and therefore by inverting a divisor we can make use of it as a multiplier; thus 43 or 19-4ths, as a divisor, being inverted, becomes the multiplier 4-19ths; so the divisor 3-4ths, inverted, becomes the multiplier 4-3rds. Instead of using the divisor in the above method, we may use — it in the common way, but this is generally very troublesome. Thus for Example.—Let £84 be divided by 43. 43 ) 84 10 5 ped ik : 17 hehe aa #) ee 47% 7 or . Qi 4 4 f 364 * a3 oy ae 13 - ; 19 ) 336 ( 17 43 33+ a . . 146 31 13 OF ARITHMETIC. Q7 Examples of Decimal Divisions. LAPP LDL LAD LAADL BDL LLL LBL Let £807 7s, or £807.35 be divided by 10, 100, and 10,000. First Quotient .... £ 80,735 = £8014 8st Second Quotient... 8.0735 =—=£ 8 1 54 Third Quotient.... £ .080735 = s “1 74 To divide this quantity by 10, is to remove the decimal point one place to the left; by 100, two places; and by 10000, four places, using a cipher in the last to express the proper value. LLL PID LLP LOE LAD LAE LIST Let £81 be divided by 2.1, by .36, and by .025. £ £ £ 3) 810, 6 ) $100, 5 ) 81000 7 ) 270 6) 1350 5 ) 16200 P 38 11 5 L225 £ 3240 To divide by 2 and 1-10th, we multiply both terms by 10, to get rid of the decimal fraction; to divide by 36-100ths, we multiply both terms by 100; and to divide, by 25-1000ths we multiply both terms by 1000 for similar purposes; and, generally, when there is a decimal in the Divisor, we consider the Divisor as a whole number, and correct the Dividend, by removing the decimal point in it so many places to the right, as the Divisor contains decimal places. Let £33 5 6, or £ 33.275, be divided by 1.8, by 3.607, and by 84164. £. £ 18)332.75 3607)33275 8164)332750 £. 18.486— £18 638i £946 £40 15 123 Similar corrections for the Decimals in these Divisors, are made here, as in the last Example; removing the decimal points one place for 1.8, three places for 3.607, and four places for .8164. The Divisions are performed in the usual manner, but only the re- sults are here expressed; observing, in the second and third parts, that as the Dividends become whole quantities, the remainders need not be decimally reduced. - 28 _© GENERAL PRINCIPLES OF DUODECIMAL NUMBERS AND QUANTITIES. wr LLLP LPL LLL When any whole quantity, or Integer, is separated into 12 equal parts, and those parts are each separated into 12 other parts, and so on, it is said to be duodecimally separated, and the numbers ex~- pressing the parts are called duodecimal numbers. The ordinary figures are employed for expressing these parts, the first sort of which are called primes, the next seconds, &c. and they are distinguished by the marks ', ", &c. As very particular attention will be paid to the calculations with duodecimal numbers and quantities in one of the following sections, onthe squaring and cubing the contents of surfaces and solids, we shall here only observe, that the additions and subtractions are per- formed as with shillings and pence, which is also the case, when the multiplications and divisions are made with whole numbers. Of the forms, when duodecimal numbers are employed as mul- tipliers or divisors, the following are examples. ae Let 5.10.5 be both multiplied and divided by 5. 7. ; fda, } f. in. , 5.10.5 5.7) 5.10.5 SPF —— -————— fin. , 67 )70. 5 O (1.0.723 29. 4.1 in 3 “spel Te Py Gi a i Al inches. Feet 32. 9.1 11 re api c mea ) 492 (7 23 Tn the multiplication, we have to take 5 times and 7-12ths, which latter is done by taking 7 times the given quantity, and taking the 12th part of the product, by giving to each partial product a de- nomination one step lower in value, as calling primes, seconds; inches, primes; and feet, inches, taking 1-12th in value instead of number. Instead of this form we might multiply by 5, and take parts for 7-12ths, saying, 6 are the half, and 1 is the 6th of 6. In the Division, we multiply both terms by 12, to get rid of the parts in the Divigor; and thus, divide 70 feet 5 inches by 67, in stead of 5 feet 10 in. 5-12ths by 5 and 7-12ths. . { ‘| OF ARITHMETIC. 99 OF THE COMPARISON OF QUANTITIES. PIL LILI PLO POE Two quantities are compared, either for the purpose of deter- mining how much greater or less, the one is than the other, or to determine the relation between them. As the former purpose is attained by the subtraction of one quantity from the other, it here requires no particular consider- ation, and we shall proceed to show the principles upon which the relations of quantities are expressed and determined. , aa LIL LL ILD SD ODT In the comparison of quantities it is to be observed, as general principles, that they only can be compared, which are similar and are expressed by the same sort of numbers; and that as all quantities are but expressions for some multiples or parts of an _ integer, this is made, in all cases, the medium of the comparison ; hence it results, That when the relations of two similar quantities aré to be found, the number expressing the quantity compared is to be made the numerator, and the number expressing the quantity with which it is compared, is to be made the denominator of a fraction which »will show the required relation. Thus, on comparing 5s. with 7s. we determine, that as 5s. express 5 times 1s. which is the 7th part of 7s. the former is 5 times the 7th part of the latter. LIF LOS FOF PDD DL DPD LAS PIT When the relation between two given quantities, is the same as that between two other quantities, they are said to be propor- tionals, or to form a simple proportion; thus, as the relation between 5s.and 7s. is the same as that between 15 yards and 21 yards, each second term, or consequent, being 7 times the 5th part of the first, or antecedent, these quantities are said to be in pro- portion, and are expressed by saying that as 5s. are to 7s, so are 15 yards to 21 yards. When the relation between two given quantities, is the same as that between the connected relations of more than one set of quantities, they are said to form a compound proportion, but this is merely a contraction of two or more simple proportions, as will be shown hereafter. TE =) -C- ea 30 GENERAL PRINCIPLES THE FIRST THREE TERMS OF A PROPORTION BEING GIVEN, TO FIND THE FOURTH TERM. PRE IPD AL PD PPP BPP PLE LOOT Totechnically describe the four terms of a simple proportion, we should call the required fourth term, the second consequent, the term from which it is produced the second antecedent, and the other two terms, the first antecedent, and the first con- sequent. In the arrangement of these terms, the first antecedent is made the first term, its consequent the second term; the second antecedent the third term; and the product of the calculation or the second consequent is necessarily the fourth term. By this arrangement, on comparing the second term with the first, according to the preceding principles, we find the relation between them; and hence find how many times or parts we must take of the third term to produce the fourth;—or, if the first consequent and the second antecedent change places, then the comparison is to be made between the third term and the first, and the fourth will be produced from the second term. =a : In the application of these principles, after giving some direc~ tions for the statement of the given terms, in the form of a question, the usual directions for finding the last term of a simple proportion, are thus expressed, under the denomination of ** THE RULE OF THREE.” “¢ Reduce the first and third terms (supposing the third to be compared with the first) into the lowest denomination contained in either; and reduce the second into its lowest denomination, then multiply the second and third terms together, and divide the product by the first, and the quotient will be the answer in the same name as that into which the middle term was reduced.” Independent of conveying no idea of the nature of the calcu- lation which this rule directs or why it was adopted, it certainly cannot be proved to be accurate by the test proposed in the ele- mentary principles, page 3; of performing the same operations with the things themselves, which is thus proposed to be done with their numerical representatives; it has also the particular disadvantage of being too general in its directions, to make it convenient for ordinary purposes. OF ARITHMETIC. 31 The following Rule, being founded upon the preceding inves- tigation of the principles, that determine the relations of quantities, which will consequently bear the strictest enquiry, and which has scarcely any one of the disadvantages under which the former Rule labours, is proposed to be used in its place. LOL OOD LIS AED OOP DOD DDO IMPROVED RULE OF THREE. LILLE F IID I ILI DP DDO DOS The three given quantities being arranged so that the required fourth term may be the product of the second, while the thir term is a similar product of the first ; | If the first and third terms are not similar and simple quantities, make them such, by reducing them into the lowest denomination contained in either. Then multiply the second term by the number expressing the third, and divide the product by the number expressing the first; observing, that if the second term is to be multiplied by a large number, it is in general better to reduce it, before the multiplication takes place. COIL IIS. LIL LAE LLIN N.B. The latter part of this Rule may be thus expressed. Then the number expressing the third term, isto be made the numerator, and that expressing the first term, the denominator of a fraction, which will show the relation of the third term to the first, and consequently of the fourth term to the second; to produce the fourth term, multiply the second term by this fraction or its re- duced value, and the product will be the fourth term. CPP LIP LLL LLL DDE LOO DO _ Itis also to be observed, that when the three given terms are of the same sort of quantities, the comparison may be made »ither between the second term and the first, or between the third verm and the first. aa 2 - ran Ret he ie 32 GENERAL PRINCIPLES Examples Of finding the fourth term of a Proportion. LOO FFA EOE LOO LD ODAC PLE LAE OZ. s d Ib. oz. 16 va ——- 5, 6 —~ Siar, 2 55. ozs Tndex 55 1. 7. 6 ll £15. 2. 6 fourth term. Before the third term can be compared with the first, it requires being reduced into oz. ; then it appears that the third term i: 5S times the first, and consequently that the fourth term must be "55 times the second. b The index is the fractional form of showing the relations of th third term to the first, and of the fourth to the second; and it is read by saying, it shows, that the third term is 55 times the whet of the first, and that the fourth term is consequently 55 times the whole of the second. 4 Cwt. qr. Ls a qr. di ee a3 ee sa 1B efi ab ma 1 ' 3 ut aA ; 14 qrs. Li) ee BaD , 1 pee a Index — £1. 6. 3 fourth term. 14 £8 Beh hd : } The first term here requires reduction, and as the third term is once the 14th part of the first, which the index expresses, the fourth term is the 14th part of the second. 6. 4 OF ARITHMETIC. 33 Cwt. qr- Ib. £ $ d Cwt. To’ 3 1 16 — 8110 4 —— 1 336 1630 s B12 Ib. = 19564 d 380 Ib. 19564 19564 19564 d 38,0) 219116,8 ( 5766 4 nearly. 291 s48.0 6 ask Aaa 3 936 £ 24 O 64 fourth term. 838 (4) ) 352 (4 nearly Index 1? — 2 ~ 380 — 95 or Li tts & 8110 4 by 28 326 1 4 95) 2282 9 4 Fourth term .... £ 24 O 62% nearly. In this calculation both the first and third terms require being reduced into pounds, then it appearing that the third term is 112 times the 380th part of the first; the fourth term is the same num- ber of times the same part of the second; or, which is the same, the 380th part of 112times. Instead of this, we may use the re- duced value of the index, or take the 95th part of 28 times. Having thus investigated the most important of the elementary principles of the operations of numerical quantities, and the methods of determining and expressing their relations, we shall proceed to exhibit their application to commercial calculations, occasionally repeating such of the preceding observations as may, from their importance, appear to be necessary. POD PIS LIP LED AAO AERTS Vor. I. F v0 —o THE ELEMENTS COMMERCIAL CALCULATIONS. PRI ILA LOL LDL LADD LL IAAL In the practice of Arithmetic, whether applied to Com- mercial or Scientific purposes, all calculations may be classed under the three following heads, if simple calculations, or, otherwise, under some combinations of them; viz. Addition, when the required result is to be the amount or sum of two or more similar quantities. Subtraction, when it is to be determined by their difference ; and the working of either a simple or a compound proportion, when the product is to be obtained from some single quantity, according to the relations existing between two or more given quantities; for however some of the more complicated processes may seem to require a greater extension of principles, it will always appear upon a close analysis, that they can be reduced to those above named. In the application of arithmetic to commercial purposes, the calculation must, in many cases, be guided rather by the practice of business, than by any rules or regulations which may otherwise appear to be strictly correct; and the nicety with which the results are required to be expressed, must necessarily depend upon the object, for which the calculations are made. PEL LE LAP EL LOO ELEALE LL! ON THE APPLICATION OF THE PRINCIPLES ADDITION AND SUBTRACTION. PFI LID ILL OFLA AEA AL AAS In the consideration of quantities, in the theory of arithmetic, relative to the operations of Addition and Subtraction, the only conditions required of them, are, that they should be similar ;—but in the application of these principles, we have further to consider, whether the subjects they express, are such as have the effect of one increasing or diminishing the other; thus, for Example, the charges upon goods are an addition to the cost, but a subtraction from the selling price; and, when the cost and selling price are taken into consideration, they, first, do not admit of being added together, and then as either the one or the other is the greater, the difference is either a loss or a gain. | When several quantities are to be either added or subtracted, the sums or differences of each may be either successively found, or the quantities may be united in one calculation ; or the sum of the quantities to be subtracted may be found, and the amount be taken off in one subtraction. wr LIL IID AAD AIF EXAMPLE Il. PPL IID LIL IPD I DD ADI LAIR The following sums are on hand and are tobe received and paid, during the month of January. What is the whole amount of each, -and what surplus or deficiency will exist at each period of pay- ) ment? viz. on hand Jan. 1, £584 10s; there is to be received 'Jan. 7, £47 15s 6d; Jan. 10, £143 6s 3d; Jan. 12, £817 ; lls 7d; Jan. 16, £480 10s; Jan. 24, £689 3s 5d, and Jan. (27, £372 12s; there is to be paid, on Jan. 11, £500, Jan. 17, L9G 5s, Jan. 20, £1140, and on Jan. 28, £737 12s 8d. 56 Jan. Alse ADDITION AND ‘ Receipts, &c. Payments. ge Re §) ad £ 2 @ Ls. 8451050 Jan. ld ..«: 500 4Q 40 Vode + ne ae 7 20: S00 7p 102... aks 6 as 20. ..1140, O2708 feng Ng Aye O28 ..: 737 32-98 sen, 10-90 REET sy AG mn Lf 2873.47 6 B 24,,245689 3° 5 BPI ee ve DT vice's Bie Me | O 3135 8 9 2873 17 8 £ 261 11 1 whole Surplus. 348 a 1. Jat > 1.... 58f 10) .0'..>. On nant PME ih inch t . Received. 10 cas GEO LO ae Ah LL Lii.,. «4) 600 Pe aid ihe BACs STOSLIVA OT. s Se urp ine 12s ely 1A 7 t fi ifgahes feats 16). 79e8Q° 10 0 1575 13° 4 ys 00 © Mb Gis eae. 1077 8 4 ..:. Surplus. 207. 100 0 a Lo pay: 62 11 8 .... Deficiency. mateo a Pte : .. Received. ET Rp be te Ee) 1061 15 5 ..., Less deficiency. 999 3 9 93). oF 781? Bo... Paid £261 11 1 .,... Surplus. ewe te se SUBTRACTION. 37 Example 2. LLP PPD LOS LED LILO IIIS On the following Days, the following Sums were due, to four Persons, represented by W. X. Y. and Z. What was due to each, and what was due on the whole?—June 4, to X. £8 14 3, and to -Z. £11 8 6;—June 17, toW. £117 2 6, to Y. £43 5, and toZ. £81 2 9;—July 4, toW. £ 14 6, and to X. £34 12'3; July 20, to W. £102 10 6, to X. £84 10, to Y. £33 4 3, and to Z. £81 17 7; and on July 30, toW. £44 5 6, to Y. £117 11, and to Z. £ 54 12 11. COL LOL AES wor WwW xX. Y Z ane A ee Se SS et a Se gee SR ene ne RS ER 5 2 a Oe Las .26 17 ULZ 6 2c Ge eee A REN AZ. ©5)9 O40 SL eR 20 Pe LOS Os OT S210.) OF 4 S34 2 Sh Lee res tan ee I a oe om tle og oe 47) Ler oa Ie Me to-W...275. 4 G..127 16, 6..194 0 3...229 1. 9 X EST 146.06 eres 204 “On 3 Bo 2se) FO £ 829 3 O due onthe whole. LOILOEL LEDGE REL ILICIL IIT 38 ADDITION AND SUBTRACTION. Example 3. PPP POL PPD LDL ODL PDD ODP An agent in London had consigned to him, by a planter in. Jamaica, 60 barrels of coffee, which produced £1074 13s, upon which the charges were £477 16s 4d; 10 tons of logwood which produced £142 9s 2d, upon which the charges were £46 2 2; 16 bags of pimento which produced £41 13 11, upon which the charges were 215 9s 6d; and 20 hhds. of sugar which produced £21155 13s. 6d, upon which the charges were £621 14s. It is required to find, the net proceeds of each of these sales; and, as the planter drew upon his agent, on the credit of these goods, bills to the amount of £1000, what was the balance of these transactions, and to which of these parties was it due ? et suld 60 Barrels of Coffee 1074 13 0O san Charges 477.16 4 Net Proceeds... 596 16 8 10 Tons of Logwood 142 9 2 Charges AG 2S 2s Tatelshe inate ee 96 70 16 Bags of Pimento 41 13 11 Charges G20. 90: vrei seeser eee carts 26 45 20 Hhds. of Sugar 1155 18.6 Charges OBL, 14 0% Cr aerietetole hese 533 19 6 1253 7 7 Amount of the Bills 1000 OO Due to the Planter £ 253 7 7 PPL LIL LAL LE L OED LEP 39 OF THE GENERAL APPLICATION OF THE PRINCIPLES OF PROPORTION. SIL IIL IAS WueEn any product, as the value, weight, dimensions, &c. of an object, is required to be found, a similar product must be given, from which what is required is to be obtained, by a comparison of the causes of the production, and determining the results by either the Rule of Three or Practice; using the abbreviations of com- pound Proportion for simplifying or shortening the calculation, when the required product is dependent upon the variations of more than a single set of terms. in stating the given terms of these proportions, it is usual to express them in the form of a question, the answer to which is to be the required product. In this method, the similar given product is to be the second or middle term; the cause of the production of the second, and the cause of the production of the last term, are to be made the first and third terms of the question, or the antecedents and consequents of comparison. When the statement of the question is thus made, it should be enquired, whether the third term being greater or less than the first, the answer is required to be similarly greater or less than the second ; in which case the direct rules are to be used; but when the contrary is the case, the inverse method must be substituted, or, in the comparison, the first and third terms must be inverted. LIP LAO PLILEOL LDL ELS OBSERVATIONS ON THE RULE OF THREE DIRECT. PLP PDIP PII DOP LOL POP ODI The statement of the given terms being made in the form of a_ question, the third term must always be a quantity similar to the first, or of a similar sort with it. | When the second term is also similar to the first, the compa- rison may be made with these terms, and the required Product may be obtained from the third term. In all cases where the Rule of Three is used, the terms of com parison must be expressed a& similar quantities in the same deno- mination; and therefore, if not given in this state, they must be reduced or made similar before they can be compared. When the third term is compared with the first, if the first term is expressed by unity, the number expressing the third showy how many times the second term is to be taken. If the third term is expressed by unity, the number expressing the first shows what part of the Second is to be taken to produce the fourth. If neither term is expressed by unity, the numbe expressing the third shows the number of times, and the num ber expressing the first shows the part, which is to be taken o} the second, and of the product, to obtain the answer; or, in more general terms, the second term is to be multiplied by the number of the third, and the product is to be divided by the number of the first. When the second-term is compared with the first, the preceding observations apply to the calculation, inverting the ap for the second and third terms. PLO LLL LDF The variations and contractions, which may occasionally be made, - will be noticed in the following examples. POI LOD LOL DOE LOE LOE LEAD APPLICATION OF PROPORTION. Al Example 1. LID LIPID DDD DDO OOD DODO To find how many pence may be produced from 37 shillings, at the rate of 12 pence for 1 shilling. s d s If 1 produce 12 what will 37 produce? 37 SF Answer 444 pence. 37 amount at 1d pers. Answer -444 do. at 12d. As the third term is 37 times the first, the fourth term or answer is 37 times the second. This very simple calculation is introduced ‘to show, that what is usually termed a reduction of a quantity into a lower denomination, is properly the process of working a pro- portion ; though, for purposes of convenience it is rarely expressed in the first of the above forms; it is however to be well understood, that the principles are those which are shown above, and that the usual form of describing the reduction of shillings into pence, by saying that shillings are multiplied by 12 to bring them into pence, must be interpreted by the directions in page 11, and, as in the second of the above methods ; that is, in every reduction of a quantity, if the direct or first method is not used, we must consider the denomination of the given quantity to be changed into that which is required, and as it then expresses the amount atan integral rate, we, from it, obtain the amount at the given rate. Vou. I. G 42 APPLICATION Example 2. PPP PPL IPD LOL POP LLL LEE To find how many shillings may be produced from 342 pence, at the rate of 12 pence per shilling ? © d r) _d If 12 produce 1 what will 342 produce: Ss 12 ) 342 Answer s28 9d ' It is here evident, that as 12 pence produce 1 shilling, 1 penny produces the 12th part of 1s, and consequently that 342 pence. must produce the 12th part of 342s, or of 342 times 1 shilling. In taking the 12th part of 342 shillings there are produced, 28 shillings and the 12th part of 6s, which is equal in value to 6 pence. . : The repetition of the principles of the directions given in reduction in pages 11 and 12, which is made in this and the preceding calculation, has been done for the purpose of further fixing the attention to this process, as being in no way, a devia- tion from the elements given in page 3, although for convenience sake, both the usual directions. and the forms of the calculations are very contrary to them; for it is generally said, that we divide pence by 12 to bring them into shillings, but it should be, that we take the 12th part of as many shillings as there are pence ; and @ similar explanation should be given, whenever the reduction is to be made into a higher denomination. PIL PLE LAL OOL ELI LFS OLF # OF PROPORTION. A3 Example 3. To find the value of 36 Gallons of Brandy, at £1 12 6 per Gallon. Gallon £sd Gallons If 1 produces 1126 what will 36 produce? 6 £57 10 0 Answer _ By the usual directions of the Rule of Three, the second or producing Term, is in all cases to be reduced when it is a com- pound quantity ; but this is never absolutely necessary, and it is | only convenient, sometimes, to do so, when the multiplication is to be made by a very large number, as in the following Example. LAD AIL BABE LDL POF Example 4. PLL LIL DOD DLE DOL LPO LIL To find the value of 11 cwt. 3qr. 11]b. of Tea, at 5s 3d per lb. lb. Sina cwt. qr. Ib. If 1 cost 5 3 what will 11 3°11 cost? | -63 d 1232 for 11 cwt. 1327 95 for 3qr.11 Ib. | 3981 lb. 1327 7962 ) d 83601 | s 6966 9d £ 348 6 9 Answer. | _ In this calculation, which may be more concisely performed by ‘the Rule of Practice (see Ex. 4, page 68) the third term requires ‘to be reduced into lb. before it can be compared with the first. Then, because the third term is 1327 times the first, the fourth term is so many times the second, which is here reduced into pence, to avoid the trouble of a compound multiplication. 44 | APPLICATION Example 5. PLE L LIE LLL LDL LDL DDD DOD If a piece of plate weighing 35 ounces is sold for £13 12 84, what is the rate at which it is charged per ounce ? OZ. ee ae a OZ. If 35 produce 13 12 8£ what will 1 produce? 7) 2 14 64 quot. by 5. Answer s 7 94+ per ounce. This third term being the thirty-fifth part of the first, the answer is the thirty-fifth part of the second. PLE LIE LOL LOD LOI LOL LAE Example 6. PLL PDL LID DOL LDL LID BARI To find the value of 1 Piece of Calico, when 100 Pieces cost . £315 8 4. | Pe; ms Pe: If 100 produce 315 8 4 what will 1 produce? —— s 3,08 — d 1,00 Answer £3 31 SS Yor this mode of dividing £315 8 4 by 100, see page 25; _ but, as is there shown, the result of this division may be obtained — at sight, by valuing the decimal quantity, £3.154 at £3 3 12_ the 4 being obtained from the half of 8, the number of the shillings. wll ‘* » ‘ PLE LOL LL OE OLE LED LODE OL h we f OF PROPORTION. Lb Example 7. SLD LED LIS FIL II OLS LIS _ What is the value of 1 ounce of metal when 4lb. avoir- dupois cost £10 5 4? Ib. Ls 6 ; OZ. | If 4 cost 10 5 4 what will 1 cost? 64 oz. 8) 1 5 8 quot. by 8. Answer s 3 22 per oz. This first term requires being reduced into ounces, before the third can be compared with it; and as the third term is then the 64th part of the first, the fourth term is the 64th part of the second, and it is calculated by dividing by 8 times 8 for 64. POS LIL LISD LAD LAD DOD PO? Example 8. PIP LDP ODP AAI AOD DAD AIE _ To find at what rate per tb. that article was purchased, of which 37 cwt. cost £84 10 8. aperd cwt. Po wk Daal: tbs tf 37 ~ cost (84 9. 8 what will IT” cost: 2 EO Ls a aay! 9 SF =( 2 5 8. Answer. 10 S ) 209 (5 24 d ) 296 (8 The third term being here the 37th part of the first, the fourth erm is the 37th part of the second. In dividing the second term xy 37, it is repeated to avoid confusing the statement of. the, juestion. No reduction is here necessary before the division is commenced, * AG APPLICATION Example 9. PPP PDL PLE ODP PPO PED LEAL To find at what rate that article was purchased, of which cwt. 11 3 11 cost £348 6 9. cwt. qr. Ib. LK d | Ib. 111k cost 348 6 9 what will 1 cost? it 1327 ) 6966 ( 5s 3d Answer. 111 331 s 95 Be 1327 lb. ) 3981 (3 eet } In this Example we are obliged to reduce the £ 348 into shillings, before we can proceed to find thé value of the 1327th part by the division by that number, but it is quite unnecessary to reduce these shillings into pence previous to the division’ being made. Example 10. LOI DLPOPP PE LBPL LOD OLE LLL To find what will be 73 days’ wages, when those for 1 year ar 14 guineas. days 3 days If 365 produce 14 14 what will 73 produce? 1 qr ee: WL) tenths s Bie eis 1 Cwt - 28 eS Diss oe alee 11 Cwt. RE ER AMAA S, - Por SLT 7. XS BATEED H. .Gicue Sark Lee eee BO ee ae MP SAB 00 TD sistem he ists Grteadi 30 ( or ) Cwt. qr. Ib. A AS 1) 111 111 95 ....- for 3 gr11 fb. ee Ib. 1327 oe \£ z--1327 amount at 20s per Ib. roe.) 88t 13) at Ss £ 348 6 9 required value. —— Sa The second method is so much more simple than the first, thai although it requires a few more figures, it is a far preferable wa y itbelongs to the following part, or the Indirect method. PRACTICE. . 69 PARTI: PLL DIF DAI OPS PAE OF THE INDIRECT METHOD. w~_er Where the required product is obtained from an assumed product. LILI LILLIA BLD EP DL IS CASE I. PL LDL LIS PDL AAD ADI LAD Where the given quantity contains no denomination lower than _ the given valued integer. Example 1. PIP LILI LID LALO LAE DOL LE L To find the amount of 1627 1b, at 63d per Ib. d | | Zz +e-- 1627 value of 1627 Ib, at 1d per Ib. LISS 6 2 el beeen, (i 4062 eeee et ee eo eoee aw = GLUSRO ET uh lace. 62 s 915 25 £ 45 15 24 or s 4 .... 1627 value of 1627 Ib, at 1s per Ib. eee bie OF ee 6d 10%. <84.. 0% erect 2 s 915 23 ; £45 15 2% required amount. ‘ or £ 2 ao-+++ 1627 value of 1627 Ib, at it per lb. H ilo (ak dh dae . 6¢ ou ud: err 3 £45 15 24 required amount. it is to be observed, that in this indirect method we may assume a yalue at any integral rate, but it is better not to ae 1 take it too high, on account of the increased difficulty of the calculation; nor too low, on account of its requiring more figures. rn PRACTICE. than are consistent with the requisite expedition; though in general this way is the easiest. It is also commonly better not to use a lower rate than the highest integer of the given product ; but when the product is given in terms of large value, it ae . sometimes be as well to depart from this regulation. PIF LIL ALIS POO BOE LDL LAE Example 2. PLP LID ALI AIC APP APE LOL To find the value of 257 cwt, at £2.17. 3 per cwt. £ ais s+ Fees» 207. Nalue at £1 per cwt. BIA Dies Sues 20 ee 198. 1) Maden od 10s DA ae sees 5s 4 Beit: pe) eS 25 eee: Tee es ae 3d £735 13 3 required value. —$—___— or S 257 value at 1s per cwt. a s 14713 3 £ 735 13 3 required value. Though the second of these plans requires the expression of a few more figures than the first, it requires less mental calcu- lation, and thus, of the two, it is the more expeditious method. DPA LLLP LAL PPD POL In this as in the preceding part, or the direct method, the quantity whose product is sought, sometimes requires a reduction before that product can be obtained, unless, as before shown in Example 4, page 68, the rate is increased, which is seldom a proper method. PRACTICE. val Example 3. PDA PLD LIS LES III III AIS To find the value of 312 cwt, at 64d per Ib. 312 lb. 312 312 312 34944 Ib. s tte eres 34944 value at 1s per lb. Ce CRAVE Yk Sateen 6d ALLY lis RO —3d #19656, ~ 41,000.) 63 £ 982 16 required value. S 56 value of 1 cwt. at 6d per Ib aie it Me OePS —3 x eh Cee eee 63 Ay 312 value of 312 cwt, at £1 per cwt. shally «wn OsGI0n, 8, JOD - V). £3. AGD LORRY SPP les s 3. £2 982 16'sv 2: omens £3 3s In the above Example, it so happens, that the second method is the better of the two; yet it is but seldom that the given quan- tity is without lower denominations, or that the given rate can be so easily increased, or when changed, so easily used as it is above ; it is therefore better not to try the experiment, unless it j can be instantly determined, that it will be successful. | 79 PRACTICE. | : Example A. PRI LOL LDL DR D LDL DLL DLE To find the value of cwt. 47 3 17, at 7}d per lb. cwt. qr. Ib. A7 317 wi 444 101 for 3 qr. 17 Ib. Ib. 5365 S 4.... 5365 value at 1s per lb. Ze. 2682 8B oe abot d, . ere ed eed she UA 1 d. Boos wlll oz ane 2 pel as ia a Al 2. Pee ce hy Ue 73 £164 17 27 amount required. The remainders from the pence are reduced into 8ths, instead of farthings, to admit of their being accurately determined. When the fraction is in eighths of a penny, many persons find it easier to assume the value of the whole at 1d, and then the parts for the eighths will only require to be taken out of a penny, which frequently cannot be done when the above method of assuming the value ata higher rate is practised : thus, d \ Z-+++ 5365 amount at 1d per Ib. avon fe: 7d x wanted 2124 2) Ay 1 Brae fens £59 oe a eoseer 72 / s 3297 22 £ 16417 27 . 4 ea ee PRACTICE. "3 CASE II. ee When the given quantity contains denominations lower than the given valued integer. PPA LOL LEL POL LAI AAPL OLE The calculations in this branch of the indirect method, may be performed in two ways, according as the product for the lower denominations may be better found, either in a separate calcu- lation, or in the same calculation with that for the higher part. LIL III LIL LIL ADS AID AAO Example 1. PPP LDL IAI ILD PLO LAD POP | To find the value of cwt. 217 3 17, at £2 17 6 per cwt. | £ 217 value of 217 cwt, at £ 1 percwt. GGds ts. wrstenvera tak £3. SeMtTAct?, 27 Tol O ties nee ‘2s 6d 623 17 6 for 217 cwt. at 2 17 6 \ Ugh gen cota | TAS Ae gk Ors 8 21... 16 lb. GOR ee, Le £626 9 4 required amount. nS ae Sa. ss _ It is to be observed, that the indirect method is used, for the purpose of avoiding the trouble of multiplying a compound quantity i by a large number; as here, by the direct method we should have to multiply £2 17 6 by 217. 7° In this indirect mode of calculation, it is very common to express the work by saying, we multiply 217 cwt by £2 17 6, and take ‘parts for 3 qr 17 lb out of £2 17 6; and the manner in which the calculation is usually expressed, somewhat sanctions the error of the former part of this description; for instead of referring men- ‘tally to the given value, it is usual to express it, as if it were a multiplier, under the head line of the performance: thus— WV ox. I. L rE PRACTICE. Example 1. repeated. al Ss 1,.217 valueof 217 cwt, at 1s per cwt. Fs Bevevcene S70 1519 . Pitts at 575 1085 HOS pee Y at 6d 12477 6 ; Sesyeid sip Oe « 29.0, bueelors? at 14 45...... 1 qr Pita: Sieg A. 16 Ib eR 1 1b S91 25208 2 L£ 626 9 4 required amount. By keeping the directing quantity constantly in sight, this form may to some appear more easy than the preceding method ; there can be no objection to the use of it, when the real principles of the calculation are thoroughly well known; otherwise it is apt to lead a person into the very erroneous expression of saying, he multiplies cwt-217 by 57s 6d, and produces 124778 6d, a thing quite impossible to be done. wr Example 2. To find the value of cwt 334 2qr 14 lb, at £2 8 per cwt og > 334 value of 334 cwt, at £1 per cwt. a8, Lodeat edt fim t Devieity o& LES L2H > SETS FO 2 8s 1 4 for 2 qr ey 14 lb £803 2 required amount. : Sometimes ina calculation like this, it is considered easier to take the rate as a simple quantity, and find the value without having to take parts for any lower denominations ;-as, here, taking 334 8 for the value of 334 cwt at 1s, and multiplying it by 48 for 4853. but in this case the following is a better form. c PRACTICE. ve Example 2.—Repeated by the direct method. s 48 value of 1 cwt. 334 2672 1336 : » of 334 cwt. ‘ 24 for 2qr. Gea te ie Os s 16062 £803 2 required amount. _ The two preceding Examples may be considered as calculated bya compound of the indirect and direct methods; and as this complication is frequently very troublesome, different ways have been devised to facilitate these operations, by consolidating the ‘two parts—as in the following Examples. Example 3. PIL PLI LS LIL LOL AOE LOO AOR To find the amount of 1712 yards 3 qr, at 4s 6d per yard. £ t \z 4..1712 15 value of 1712yds 3qr at £1 | Beek: > Sa 8 eae koe era, OF As Eee rh. Yaeus chatels aapausiauni mans. © * 6d £385 7 42 required amount. ne +..1712 value of 1712 yards at £1 esa OS 4 Pore ae Oe. ede . fee 4s PLO, 2 ema sie ct ee + & oalgh 6d £385 7 44 required amount. —— | The former of these ways appears more concise, but it is not lways the easier of the two. — “6 PRACTICE, Example 4. wratrtyr POD LIE LFS LAI PAL To find the amount of 117 Ib. 10 oz. at 4s 7d per Ib. sia + 117 72 value at 1s per lb. rhe d OS ee, ene A As a Latend Be 04 Sh a. ee 6d 0. OED... Rs 1d 5539 11 £26 19 11 required amount. In the valuation of the oz. Avoirdupois, at 1s per Ib, each oz. is 3 farthings, and 10 oz. are 30 farthings or 73d. PRP PAIL LIL FIP DAE FASE Example 5. PAE LOL LOL ADE DAD DLE To find the amount of 47 Ib 11 oz 8 dwt, at 5s 3d per oz. lb oz dwt q ay al Tiacoes Oz 575 8 dwts. 8 1 .. 575 8 valueat £ 1 per oz CM se oe eepe 5s GOES EO LO i 3d £151 O 10 required amount. 3 es LLP LLL DDE DOOD DE LAD LIE , Example 6. PRL LE LOE FOECDEL A ELE LAT To find the value of cwt 217 3 17 at £ 2 17 6 per cwt. £Losd i 4 ,, 217 18 0.42 amount at £ 1 per cwt 3 ; * Wen... | peed i a67 et eee £ 3. Br 6419.05 “yee 2s 6d. ee £ 626 19 4.26 required amonut. PRACTICE. an | The value of the 3qrs, by the following rule, is 15s; for | 17.Ib, we say, twice 17 are 34’and 2, for 2 sevens, are 36 _ pence, or 3s; making in all 18s. For the 3lb above the 2 sevens or 14lb, 42-100ths of a penny are annexed. LIED ID LLL “a, Rute, For the valuation of Cwt, qr, and Ib, at 20s per cwt. PLS IDL LIS AIF PAF AID SIS Reckon every cwt as £2 1, every quarter as 5s, and every Ib as 2d; with 1d for every 7lb, and 14 parts for every Ib ' above the 7, 14, or 21. The principle of the rule for the valuation of the Ib is, that at £ 1 per cwt, every lb is worth 2d and 1-7th; for which | 7th we add 1d for every 7 1b, and for the remaining lb we use 14-100ths which are very nearly its decimal value. If further correctness should be necessary, we may add an extra part for 2, 3,4, and 51b, and 2 parts for 61b, but this nicety is rarely required in calculations of business. LIS FIL IID PDL ADIL ALD PAF Example 7. PID PIL LOI LED AOD AED LOAF To find the amount of 143 tons 11 cwt 3qr 141b, at £ 27 10s per ton. : i... 143 11 10} amount at £ 1 per ton. | 430 15 721 cot) 8:3 SST Fa) Oiu:7h- Soe £ 27 7 git oy Us OG Up SR ee he s 10 £ 3948 16 62 required amount. | In this example, every cwt is rekoned 1s, every qr 3d, _ and the 14]b, 12d. | 7 ng "PRACTICE: 7 When the quantities given with the tons, are such as it would be troublesome to take parts for out of aton, the calculation may possibly be rendered easier by reducing the tons into cwt, and finding the value of the whole at an equivalent price per cwt. 0 ee ee ee i Example 8. PPL LID PDE LIL LOE PAD LOAD To find the amount of 34 tons, 7cwt 1 qr 18lb at £ 22 10s | per ton. Tons cwt qr lb 34 7 118 at £22 10s perton; or, Owe! 687) TiS taet VT 258 6 perry, By: netted i....687 8-2 .56 valueat £1 per cwt. SH VS 16) 2a rad oe 28 6d. £673 6 9 nearly. For the 18 tb, reckoning 2d per lb, and an extra 2d for the two sevens or 14 lb, we have 38d, or 3s 2d3 which, with 5s for the 1 qr, produces the 8s 2d; to which we annex 56 parts, or 100ths, for the remaining 4 Ib. e Example 9. OPP POL LIE PEI ALE AP EAT 4 To find the amount of £571 10s 6d at5s 7d per £. te af z--+-571 10 6 amount at 20 s. as 142 17 4 ame sires «1670.0 5 8 4 14.5. 9 Ree ee 6d Bi !1, epee 1 d. 2 pe 5 DRG ay, ce Sea ee 58 7d. ‘ This very useful and simple style of calculation, may, when the i: parts for the given rate are either numerous or intricate, sometimes — be rendered a little easier, by using the decimal expression for the } given quantity; and it is principally in this and the following © forms of Practice Calculations, that decimals are of much utility. PRACTICE. ng Example 10. PIL LOL LIL LEP AIF DDD DDD To find the amount of £164 13 114, at 4s 73d per £. fe 3-+--164.696 amount at 20s per £. ee ty hl er ete 4 s, 4 ABEL A052, 6d $..A%-1<0293. Pe . id. URES aarti 4d Required amount £ 38.2574 = £ 38 5 2 nearly. Or, by the usual method. ee ie ----164 13 114 amount at 20s per £. ol pee POUSe AS ge ENTS SS 4s, 4) 2 peg: Benge sap naag ee a, DO tee, Sy a, 1nd. RE SO | ates OE oT Required amount £38 5 2 nearly. | + The decimal valuation of the 13s 117d, is performed: by the Rule in page 17. In the calculation, a fourth place is used for the sake of more precision in the result, in which, the decimal .257 is valued by the Rule in page 16. PLE LL ILE DIT INID DL ! | Example 11. SIS LPI LPP LOD LOD DLE OOO To find the amount of £877 at 8 Guineas per Cent. £ 8.77 amount of £877 at 1 per Cent. Bean sve VoD de 8 £ £ 73.668 = £73 13 ° 45. s0 PRACTICE. This isa form of calculation frequently occurring in Marine — Insurances, in which no sum lower than pounds is used in the © principal. If there were shillings and pence, they might be valued asin the first part of the preceding, and as in the following Example. The amount at 1 per cent is obtained, decimally, by dividing the given sum by 100; that is the amount of £887 at 1 per cent, is 8 and 87-100ths of a £. PAF PII PDF FPL DPE LOD AAP Example 12. werner PLP LLL To find the amount of £872 16 7, at 7; Guineas per cent. Bu 1... 8,72829 amount at £1 percent. 3 DO MeO Me ise em ere as £8 per cent. puptact 1001 we eae 2s 6 d per cent. £ 68.735 = £68.14. 8: amount required. Seven guineas and a half, being 2s 6d, or the eighth of a pound, less than © 8, the difference of the amounts at these two rates, produces the amount required. In the first line it is necessary to use the 5 places of deci- mals ; but in the second and third lines, three piaces only are — wanted, as the fourth and fifth would produce no effect upon the result in sterling money. POP LL PPP LOR ODO LDO DOL The preceding calculations comprise all the forms that generally occur in the practice of business; and if the different principles — upon which they are wrought are well understood, it is presumed — that no difliculty can ever be experienced in those out of the way | exercises, which are sometimes given to young persons as trials of their skill. In order, however, to assist them in such expe-— riments upon their information, and for this purpose only, the following calculation is subjoined. PRACTICE. SI Example 13. PLS LILI LIF LEE LIE LIED To find the value of 37 tons 13 cwt lqr 17Ib 11202, at £8113 74 per ton. ree al ¢-- 8113 74 value of 1 ton. 6 mou Lao 6 ee ot Vale eg a ose stem pte Wal Reais. 40. 37 tons. Be AO IG OF... cece ewe crc ec cree nc ecvies 10 cwt. Bie OS Ae ee che ee eee oe woe vn eles deligiote vials 2 cwt. § 27 1+. 625<% & or 15680-25088ths... lelqr eA Ape UT Dy ODL 4S Hisense wel F02¢. Spas waa 16 Ib. es: 9495446:|) Shy WohO3Saie. yelenawen 1 lb. Feces As.12723 Sat he PO LOD. Oe Tonka a. 8 OZ. oe LSUSRLS ae Soy es IOTO.S aux Rie on: 2 on. ered: £4 HADOCIRS Rte “AL 6OTSN. HA. ie 28 1 oz BOL7 ROT RSS Te EAS ET .octS. Slatsiel. 5 on. £ 3076 18 94.1729 or 4338-25088ths In this calculation, in order to avoid the very great trouble at- tending upon the division of the remainders from the 3rds, when _ expressed as Fractions, their amount, at first, is calculated de- cimally to five places; their actual values as fractional parts of the third of a penny, are afterwards shown in the two side columns ; in the second of which they are made similar, and the amount is found to be 54514-25088 ths of 1-3rd of 1 penny, or 2-3rds and 4338-25088 ths; these two-thirds are then added in with the other thirds, and the amount, 11-3rds, produces 3 pence and 2-3rds. The performance of such a Calculation by practice is so very troublesome, particularly when the exact fractional value is required, . that recourse had better be had to the Rule of Three, stating the the question; thus: ton fe) Boat. tons cwt qr lb oz If 1 produce 81,13 73 what will37 131 17 113 produce? 250880-7ths, 0z 58810-3rds of d -9450754-7 ths of oz. 9450754 250880 ) 555798842740 ~ d 7384653 = £ 3076 18 92 A238, perme Vou. I. M e's) 9 PER CENTAGES. PPE LOLA SI OED LED LOE LOD Under this class are usually included those calculations, denominated, Commission and Brokerage, Interest, Discount, the Stocks, &c. but as many of these branch out into other forms, and require particular regulations, they will be separately consi- dered under their different heads. POL LAO LID AED ALD LA LEP Of the general calculations of per centages, or the determination of the amount of given sums of money at given rates per cent, several examples have been given in the preceding application of the Rules of Three and Practice; but as many of the usual rates are, what are termed aliquot parts of the principals, or the amounts on which they are computed, we shall here show how such divisions may be mentally performed, or how the result can be obtained from the given amount, without any intermediate — calculation ; commencing with the rate of 10 per cent. LIL LIP ADA ALL LPL LED LOE In Per Centage Calculations it will be noticed, that 100 per Cent produces the whole of the Principal. _ Ree RAL BOUOG Aa Cie ISie ok PRG One Third. , v PIE Ghre sda > dip okats «sate hoeetnts .... One Fourth. ‘ PASE sits, see ere eeeee ..» One Fifth. S 162.056. eee S. 5k. See Oke One Sixth. ye eee eee ee iste Son 88 ere sp yr ret One Eighth. ponies LO i ae Sor mer aMel, cos Seti Cutie uric ae One Tenth. 2 5 Beery oss tented i > ae 600 O O .. produce Wy ts L007) ORO Poe Om Or cer. es Fete 75> OO St) 140: gO}. 8S. . 1 Proof ....£107 9 0,625 The explanation of the mode of producing the pounds and shil- lings, at 25 per cent, having been given to the preceding Example, to explain the rest of this operation, it is necessary only to observe, that dividing the pence by 4, &c; we estimate the 40th of a penny at 25-1000ths, which is its decimal value, and the quotient of the division of pence by 4, for 40, is necessarily so many 10ths of apenny. The reason of multiplying 3-10ths of a penny by the number of the shillings, is, that the 40th of a shilling is the 40th of 12 pence, or the 10th of 3d, or 3-10ths of a penny. Thus for 5s 7d dividing 7 by 4, we produce 1 and 3 over, for which 3 | we reckon 75-1000ths; then multiplying 5 by 3, and taking in the — 1, we produce 16-10ths, or 1 penny and 6-10ths ; to which we add 3d for the 10s. a ‘PER CENTAGES: Q7 TO CALCULATE 1 PER CENT. PLL LDL LOD LAI FEE LAE LAD For the Pounds—Separate the units and the tens of pounds in the principal, and the other figures will show the required number of pounds. For the Shillings—Double the figure of tens in the given pounds, and add 1 ifthe unit figure should be 5 or above. For the Pence—Consider the units of pounds (or the remainder if above 5) as so many tens, and half the given number of shillings, asso many units of farthings, subtracting 1 from 12 to 37, and 2 when above that number. N.B. As this rule produces the required amount only to the nearest farthing, when greater accuracy is required, the division by 100 must be performed at length. PIP BAD API ADD PDA LD E LAE Example. Principals. Per Centages. es te: a £ eyed GOO, 0 DO «% «produces oot: -- ODE 700 20% ©) ES kW Ere Pee ioe 50 7, 1073 812 0 0 at, wee oo SOs 73418 8 Su hee 7 6112 51215 7 WA oe ti. 5.2 63 Yh hak, ee ee are 817 64 ndaenem th F2.%. |, Progiqut © 42°19 ds The directions for this per centage are founded upon those for the valuation of the decimal parts of a £, given in page 16 ; con- sidering, that as shillings are so many 20Oths of a pound, half their number will.be their value in 10ths; and as the 100th part of a shilling does not produce half a farthing, we reckon to the nearest even number of shillings, as being more than sufficiently minute. Thus £512 15 7 iscalled £512 and 8-10 ths, the 100th part of which, for 1 per cent, is £5 and 128 thousandths, which pro- duces, by the above quoted rule, £5 2 633 the exact amount | found by dividing the £512 15 7 by 100, being £5 2 63 and 68-100ths of a farthing. ‘ , 88 PER CENTAGES. TO CALCULATE } PER CENT. PPL PID LLL PLL DDD AOD LAID Cut off the unit figure of the given pounds, and the other pounds will show the per centage in shillings. q Consider the units of pounds, with half the number of the given shillings, as so many tens and units of eighths of pence; deducting 1 out of every 25. GOEDEL APL LODE LOE LIE FIT N.B. When this rule is used for a number of pounds only, we may reckon the unit figure as so many pence, with the addition of 1 d for 1, 2, 3,4, 5, 6, and 7 pounds, and 2 d for8 and 9 pounds. PLD LL DDL DOLE DDO DAE AIS Example. PII OOO DDL OLE LOD DDL LIF Principals. Per Centages. Pt red oN ae | 600,720,060 |. 0..¢ spreoducesateiee so) 0770 75D Mey UF... 2 0's « AP een I a 2. 1D a) 812 ga eB j.. eee Mt. cee, 4 1 3} for,27 lesa TIRES 8 | U8. ase Sree 313 5% — 49 less 2. BAP Mie bo Bid oe Bele, 211 33 — 28 less I BS7,0o) Le ese tre npdclalx wedi Ete 4 8 92 — 77 less 3. . £42998 2 1 ..... .Proof.....-£21 9 98 — 81 less 3s | ¥ The second part of this rule is upon the principle of the Jast5 using 8ths instead of 4ths, this rate per centage being half the last. The first part is upon the principle of $, or 10s, per cent pro ducing 1s for every £ 10. | ; GLE LIE PLL IFG BALES LIL LIA : ? . PER CENTAGES. 39 TO CALCULATE } PER CENT. LIF PDD LIP DIS LEE IIO IT This per centage is sometimes thus calculated. Multiply 5s by the number of hundreds of pounds in the principal, or take 1-4th of so many pounds, and reckon 6d forevery £10, and for the fractional parts of £2 10. When greateraccuracy is required, the following rule maybe used. PLD DIL IIS LID LIL DIO IIE Multiply 5s, &c, as before, and add to the product as many pence, as there are tens of pounds in six times the remainder of the principal. DOP LOL ADI LAE LIE PEE LAL Example. Principals Usual per centages § Products by the rule fe ed Ls a GOO PRG sca sk ss LITO * ars aoe 1100 FOO” GRU fe tag © bo Te BFP GMa cess 1 17 6 812 14°6 ..-...6- SIAOT aes cies welate ors 7384 18 8 .-ceeoee PAG G6 Gaseernk ate vie 1 16 9 Blt 8G 7 6s. sues Oe ee 1 58 BaP LG Ae a's en wa By Ae Ooo s\s' oe!» Se Soke Ie LAZIB Zl ..ceees “94 DEDUL Sica: Greta a.c £10150 One quarter or 5s per cent, is 6d forevery 10 pounds, or the 10th part of the same number of pence as of 6 times the number of the pounds. Thus for £734 18 8, we say, 6 times 10s are 3 pounds, carry 3, 6 times 4 are 24 and 3 are 27, carry 3; (as for 30, the 27 being nearer 30 than 20,) 6 times 3 are 18 and 3 are 21, say 21 pence or 1s 9d, which with £1 15 for 7 times 5s, produce £1 16 9. | Vou. I. | N 90 PER CENTAGES. TO CALCULATE g PER CENT. In the usual calculations of 1-8 th per cent, a sum not exceeding, £ 25 is usually charged 1s Od 3-50 (io) eles cae said end ob 19 7.5! oe ee oe 2s 0d Eee SLOO- Sota ca Yr ea hs 2s 6d But when greater accuracy is required the following rule may be used. Multiply 2s 6d by the number of hundreds of pounds in the principal, and add to the product as many pence, as there are tens of pounds in three times the remainder of the principal. POP LIL PDL ED PAL DDD LDP Example. Principals Usual per centages Products by the rule 2 NR A! MP Rays | BPs eit DU A CEs ce totals ine 1D) “vee vcsbste ts ap UO 1 j.:1 6 WN 0 ET 9 PRS LD Otis Sa bette 18 9 tN Ae 0 be Se Li ORAS scsib bees os 1) CO; a4 Chol DS EB A Se LOO Diiiase tele oe 18 4 DRS ay ay Ea LB (Gris ites oe 12 10 BAT Gaetan 66 %s"n Lggden Bie vegies alt) Lipa abe pA ie Lar PIOOL '., . 6 eee ees One eighth per cent is calculated upon the principle of 1 half of 4 per cent ;—thus for one eighth per cent on 734 18 8, we * hy 3 times 4 are 12, carry 1, 3 times 3 are 9 and 1 are 10, say 10 pence, which with 175 6d for 7 half crowns, produce inall 18s 4d. » PER CENTAGES. 9] TO CALCULATE zo PER CENT. SLD IIL AID LAA ODA LOL ODD Cut off the hundreds, tens, and units of pounds, for the three highest places of decimals. The shillings in the principal, if above 10, are to be called another pound. PPP IID LID ODL DE PLD DBD LDR Example. Principals Per centages wou0mO. O..5. produce... £°042)0 moe OOF S Sa. Bars Sanayi -~ 150 ee AL tO. ws cretasnes etch: - 163 —for12 Ma419 8 eee CDOs See oan Tgp ead 1 let Leer ee Ge kmh -103 — 12 Ban 13064 Powe: tt ee ae = 720i) Le 38 less 2 £ 4298 Pele AOR DIOOL oe toe 5 114 — A8 less 2 This is calculated upon the same principle as one per cent, 3 figures being cut off, and. the shillings being taken to the nearest number of pounds. Thus 1 percent upon £ 887 13 4 produces 888-1000 ths of a pound, which are equal to $17. 9d. PIS LOD LOD DOP OOF LOD DDD. The principles upon which 1-10th, as well as 1 per cent, is calculated, are frequently useful in abbreviating the work of those per centages, which require a further calculation. Thus, for example, 4 per cent upon a sum of money is 1 per cent upon 4 times the amount of the principal. As, for4 per cent on £2 89 10 6 product by 4 £358 2 0 making £ 3 11 74 so, for 34 per cent on £ 67 5 6 2) 470 18 6 product by 7 £235 93 making £2 7 14 92 PARTNERSHIP, OR DISTRIBUTIVE PROPORTION. PPP IPI LIF DDL POD LEFT The Proceeds of any Concern in which several persons are jointly engaged, are usually distributed according to either the articles of Partnership or the custom of trade, by one of the four following methods. PHI LDL EPL LAD OLD DLO OD LAD First method——By Proportional Shares. Second method——By Fractional Shares. Third method——By Finding the Rate per Cent of the whole Proceeds, and then finding the amount of each Person’s Capital, at that Rate per Cent. Fourth method——By Finding the Rate in the Pound of the whole Proceeds, and then finding the amount of each Person’s Share, at that Rate per Pound. PPP PIP BILDP?S POL PPP LPP LIL Of these different methods, it is to be observed, that the first is but very seldom practised; and that some variations are occasionally made in the second, by the use of an interest account for the ad- justing of temporary variations in the capitals employed in trade. In some very few instances, also the first and second methods re- quire the products to be proportioned, both according to the - amount of the shares and the times they are employed, which forms a variety of calculations, usually called Double Partnership, or Compound Distributive Proportion ; of which, from their very rare occurrence, it is here thought sufficient to give only one specimen, in the Sixth of the following Examples. The use of rates per cent is very frequent in Marine Insurances, but the proceeds of Estates, &c. are usually distributed by the fourth method, of a rate in the pound. DISTRIBUTIVE PROPORTION. 93 Example 1. LOD LOL EOI ELIE LES Supposing the capitals employed in trade by A, B, and C to be £ 3200, £2500, and £2000, and the amount of the gain to be £1754 12 8; It is required to find the amount of each person’s share, according to the amount of his capital. e 3200 | 2500 2000 Phi vt tid £ If £7700 produce 1754 12 8 what will 3200 produce? Answer....£ 729 311 A’s gain. £ 9. it {> If 7700 produce 1754 12 8 what will 2500 produce ? Answer.... £ 569 13 9 _ Bis gain. £ A aaa te BE £ If 7700 produce 1754 12 8 what will 2000 produce : O C’s gain. 569 13 9 Bs gain. 729 311 A’s gain. £1754 12 8 Whole gain. Answer...- £ 455 15 nor LOO LIS Example 2. 4 PAP PDI IIS LEIP LID ILLS To find what four persons are to receive out of £3151 10, A’s share being 4, B’s4, C’s 4, and D’s 4. £3" a 3151 10 O whole Gain. 1050 10 O AV’s share. ogni 0 B's ©. yy ONY GU eR id TE 525 5 O D's £3151 10 O Proof. 94 PARTNERSHIP, OR Example 3. PLE LOL LIE LOL PLP POP In the articles of partnership entered into between three persons, it was stipulated that A’s share should be 5, B’s 4, and C’s 2, but that at the end of three years, C’s share should be increased to gz Without any mention being made of a reduction in the shares of the other partners. The profits of the first period were disall tributed according to the above regulation, and after the business | had been conducted for a length of time, without any regular settlement, a dispute arose on the closing of the concern, respecting | the distribution of a profit of £37466 16; which being submit- | ted to an arbitration, it was decreed that each share should be so | | | | | | | proportionately diminished, that the fractions expressing the three shares should amount to unity. It is required to find what each person had to receive. A’s proposed share —% DOS ee eos drs ah. ;3- ! Amount 12-11 ths. 08 AA ee, = Therefore to determine the shares of the gain, we say, if 12- 11ths produce £37466 16 what will 5-11ths produce, &c; or, as the shares thus become so many 12 ths, instead of 11 ths, we may calculate the produce of them in the following manner : wees) ca 37466 16 O Sum to be distributed. Ne I 8 Sarge ee Sr WS he £15611 93 \.40A’sushare’ 09521 9ths: £12488 18 8 B’sditto .... 4-12ths. £ 9366 14 0O C’s ditto .... 3-12 ths. PPAF ODP PEP OLOLOP OOS DISTRIBUTIVE PROPORTION. 95 Example 4. PILL PLO DDD LD OLDE LOD ODDS Upon a Policy of Insurance, on which there was a loss of £21154 8, the following sums had been subscribed $ Viz. Hymn. <. 2 200, B. 100. 6 Fae 100. Dy. 500. K. 500. Bao 20 G. . 150. It is required to find the amount of loss, which each person had to sustain. £ £ os £ ty 1750). -pay 1154 8 what will 100 pay? 35 ie ) 2308 16 2 97) 46115 2 Rate per Cent £ 6519 4 nearly. | me Sd | A £ 200 at £65 19 4 per Cent. ... 131 18 8 | BP LOO os a oe ee ic iemOae LOM 4 | Be LOUR e gg eas . 6519 4 ERE DOLS 2 GEN OLe Day! Seale uy” 329 16 8 io a EG eae Cee ake ee - 829 16 8 as ee SO 4s Sonics oA Yay ten id) 131 18 8 it eh ESO) senyct MLO dh TONE 9819 O LITse 3) 4 _In finding the rate per cent, we take the first term as 35, and the third term as 2 half hundreds, and we reckon the rate at the nearest penny. In consequence of this, the amount of the separate losses, exceeds the given loss by 4 pence, but as fractions of a penny are never used in such estimations of per centages, this trifling difference becomes often unavoidable ; and it may be fur- ther observed, that it is the general practice, to give the difference in favour of the person who is to receive the amount, or to reckon any fraction in the rate as another penny. 96 PARTNERSHIP, OR z Example 5. POL LPL ADS OOP AOD OIA The whole amount of the demands upon an estate, being L 8482 16 4, while the net amount of the different assets is only £3351 6 3; at what rate in the pound must this be distributed, and what would be the amount of A B’s dividend whose debt was — £879 6 5? oe ee eed 4 i Rees ay If 8482 16 4 pay 3351 6 3 what will 1 pay? ao Ag s d 8489817 ) 3351312,0 ( .395 = 7 103, rate per £. 80664690 43013370 Le Paste +0 4....879 6 5 amount at 20s. 4.) RU0L Oe TER eek 5s 42.6.9100 eset 286 d. Pereira the Cae 3 d. 4. SedGi lle Be 00 ae lid. Re Pew) , £ 347 14 7 Amount of the Dividend. Tn calculations of this nature, if the product cannot be exactly ob- tained, it is generally taken at the next lower farthing; and, in” estimating the dividends upon debts, ‘the farthings or half pence upon the amounts are rejected. The calculation of the rate, is usually better performed by deci ‘ shillings, pence, &c, by the directions in pages o' 4 mals, valuing the 16 and 17. CLO FIL ELE FEL OEE LIA LAT DISTRIBUTIVE PROPORTION. 97 Example 6. LOD IOS el OF COMPOUND PARTNERSHIP, OR DISTRIBUTIVE PROPORTION. —~_—wor Twelve men have been employed upon a piece of work, for which they have to receive £57 12, and which is to be divided _amongst them, according to the rates of their respective shares, and the time they have been respectively employed; in the fol- lowing manner ; required the amount of each person’s Share? The foreman has 5 Shares,and hasbeenemployed 72 hours. One overlooker 3 Shares ....... POIs oes fk 65 hours. One Ditto..... 3 Shares 9 7m. Ae for’. . ere. 57 hours. Kight men each 2 Shares ....eachfor .....'.. 72 hours. OSE ne 2 Sharesi a? Wan. . POT ee ides 64 hours. 7 Suen GO isa & AsO Ow sua eek > Los Lf PPR” eH Wscmt it eg | Te Ky Bee Se =o Loe G40 562°, Ba => 128 2006 anes If 2006 produce 57 12 what will 360 produce? &c. or, £5712 X 360 + 2006 = £10 6 9 £57.12 x 195 = 2006 = Sb2 aU can? 12% ~ 171 2006. — 418 2 £ 57 12 xX 1152-= 2006 = 33 1 7 between 8 men: eure 128 <-2000,.\— &, Loo £.57 12 O In some few calculations of such compound distributive pro- portions, it is easier to find the value of one part, and multiply it, separately, by each number of parts. Thus, the 2006th of £57 12, is the decimal £2. 028714, nearly; and this multiplied by 360, 195, &c. will produce each of the above products; but it is not often that the calculation can be rendered sufficiently correct, with any great abbreviation of labour. Vot. I. O PER CENTAGES OF PROFIT AND LOSS. PLL LDIF PPA PPL POL LDD LOO Profit and Loss are the excess and deficiency of the pro-— duce of sums of money employed in trade, the excess being called a Profit, and the deficiency a Loss; and, in deter-_ mining the Per Centages of these, the calculation is to be made upon the original amount. PLO LIL LOL LOD PLD OLD LID Example 1. POL DLE LOL POL LIE LIED DOP Suppose goods are bought at 80s the cwt, and it is required to find at what price they must be retailed per Ib, to yield a profit of 15 per cent? s qio---- 80 cost per cent. 5.-.- 8 gain at 10 per cent. <2 ae GE. ice s 92. selling price. Ib $ Ib If 112 produce 92 what will 1 produce? 4) 93 for one qr. Fae Seo ANS WEE. «ss ste 10 d nearly. ; Answers to such calculations as these can seldom be attained with any precision, as the above price, of 10d a pound, yields a greater profit than 15 percent. It may be also observed, that they are scarcely ever made for any purpose of practical utility, as the selling price of any article must rather depend upon what can be obtained for it, according to the general market price, than upon any determination of its yielding a stated per centage profit. PER CENTAGES OF PROFIT & LOSS. 99 Example 2. To find the per centage profit upon goods purchased at 75s the cwt, and retailed at 9}.d the lb. d 9+ selling price per lb. Sui are ee eetiat se Alb § SQ et teuy chads 3 « 112 1b 75 O cost. s —- ae ’ If 75 gain 11 4 what will 100 gain? | ee 5--»- 100 by 11 1100 33 6 8 for 4d & ) t133"6> 3" | 15) 22613 4 Answer £ 15 2 3 nearly; Gain per cent. The principles of this calculation are too evident, to need any explanation. The work of the latter part may be abbreviated, in this manner. Considering 11s 4d as 11s and 1-3rd, and then considering the denominations of both the first and second terms as changed into £, we compare the third term with the first, and finding it to be 4-3 rds of the first, we produce the answer by taking 4-3 rds of the second: thus, 2s. a ll 6 8 4 3) 45.6 8 £ 45—2-—3 nearly. } } ; } | | DA OLOOLODDIO DOO DOL OOD 100 PER CENTAGES OF Example 3. PPP PIPL PPI PPP PLP LDP POP If goods which cost 56s the cwt, were sold at the differen) | prices of 63s, and 67s the cwt, what would be the difference in | the rates of the gain per cent ? — The difference in the selling price, being 4s, we say, s s Te If 56 produce 4 what will 100 produce? £ } A...100 | | Answer £ 7 2 10 nearly, difference per cent. a ne PIF PDP PPP LDP POP DOG PDP PROOF. PIS PDL LPL LIED PPP LO P LOY The first gain being 7s, and the second 11s, The first gain is 1-8th, or £12 10 O per cent. and the second, 11-56ths, or .. 19 12 10 Difference, as above, £ 7 2 10 percent. The reverse of these calculations, or the determination of the cost of an article, from the selling price being given, with either a per centage gain or loss, is even of less practical utility than the preceding ; of the method in which questions of this nature may be resolved, the following calculations are Examples. FPP PLL LOL LOL LOL OIL OLDS Shalt PROFIT AND LOSS. 101 Example 4. PPS LIDDELL LDL OAD LOD DDT To find at what rate, goods which sell at s22 6 per cwt, must be purchased to yield a profit of 125 per cent. £ gf tug If 1124 require 100 what will 22 6 require San. 5-22 6 cost. 2 6. gain (124 out of 112}.) Answer, s 20 0 required rate per cwt. s Proof }..20 cost per cwt. 2 6 gain at 12} per cent. s 22 6. selling price. ———— LLL PLL LIL Example 5. PPAF PPD LIL DDD DDL DDL To find at what rate goods were purchased per cwt, when they sold at 83s 6d per cwt, and a loss was incurred of 5 Pani gine, Gi Pauanui’ oe: what will 83 6 require? Or, if 19 require per cent. oy a 83 6 20 19 ) 1670 O Answer s 87 11. required cost per cwt. cost. Proof. age eds LI 5 loss at 5 per cent. s 83 6. selling price. The loss is here 1-19th of the selling price, or remainder, and 1-20th of the cost, or the original sum. 102 BAB@ BR» OR THE EXCHANGING OF COMMODITIES. PILI LIP LOL LLE DOE DLL PDF The bartering of commodities in this country, is scarcely ever practised, but when, without entering into any statement of cal- culation, the whole of one lot of articles, is sold or exchanged for another ; when, it may not be so, the value and quantity of the goods sold and to be received, may be easily determined by the Rules of Three and Practice; or abbreviations can sometimes be made, by the use of the Rule for Inverse Proportion ; of which the following calculation is an example. PPP PLL OPP OLED OE LEP Example. PLP LPL LE DELO LOL LOD Supposing 87 lb of any article, valued at 18d per Ib, is to be exchanged for another article worth only 11d per Ib; it is required to find what quantity of the latter, will be equivalent to the former. d Ib d If 18 produce 87 what will 11 produce? 18 yas 1566 Ib 142-4. Answer. It is evident, that the less which may be the rate of the article to be received, the greater must be the quantity which will be re- quired to Ban up an equivalent value, and therefore, that the calculation belongs to Inverse Proportion. It might otherwise be worked by finding the value of the 87 Ib at 18 d per Ib, which is 1566 pence, and then saying, If 11d produce 1b what will 1566d produce ? The answer to which is 142-4, lb, as above. 103 rc ON THE ALLOWANCES OF DRAFT, TARE, AND TRET, ON Wels GROSS WEIGHT PPL IPL IIL DDL OLD IS DDL LDP Gross weight is the weight of goods with the packages, &c, before any deduction is made. Tare is either the weight of the hhd, bag, &c, in which the goods are contained, or an allowance made for that weight. The allowances for tare, are, either Real tare, or the actual weight of the package. Customary tare, or the established allowance for that weight. Average tare, or the weight of one, or the average weight of two or more hhds, &c, selected out of a lot of goods, and used as _ the allowance upon each hhd, &c, in that lot. Proportionate tare, or an allowance at a certain rate per . cwt, percent, &c. SuPER-TARE is an allowance upon a customary or proportionate _ tare, when the gross weight exceeds the usual weight upon which _ those allowances were calculated. Drarrt is an allowance upon the gross weight, intended to compensate for the loss sustained by the extra weights, which are | necessary to turn the scale, when the goods are sold in smaller quan- _ tities, than those of which they are purchased of the importer. Trert is an allowance of 4lb in every 1041b, on a few ar- _ ticles, principally drugs, as a compensation for waste, dust, &c. PID LILI I DD OL IAI LD LOL LIL REMARKS. Besides the preceding there is an extra allowance, upon a very few articles, called Cloff. By this title super-tare, or super-draft used to be distinguished. 104 ON THE ALLOWANCES Thus, in some departments of trade, the regulations for draft that when the gross weight is less than 56 lb, there is no draft to be allowed ; when less than 1 cwt, 1 1b; when less than 3 cwt, 2 lb; and when 3 cwt, and above, 4 1b; therefore if the general weight _ of the goods runs from 1 cwt to 3cwt, the general draft will be 2 lb per package, and, if any should exceed 3 cwt, the 2 1b extra which is to be allowed for each, is by some called a Cloff, but the term is very nearly obsolete. It is thus, however, that the allowance of Cloff has been erroneously stated at 2 lb per 3 cwt, upon the whole gross weight. Upon Cocoa an allowance for damage is admitted, under denomination of garble; and upon tobacco, there is an allowance called shrinkage, for loss of weight by the evaporation of moisture. Other deductions are also occasionally made for damage, &c, which are rather matters of private agreement between the buyer and seller, than any particular regulation of business. With respect to the allowance of Tret, it was formerly very ge- neral upon articles imported and sold by the pound ; but it now exists, with a very few exceptions, only upon drugs which are not imported either from the Levant or the East Indies; and of these, only upon suchas are sold by the pound Avoirdupois, with perhaps the single exception of blue or Roman vitriol. In some of the out ports, and in some departments of trade in London, there are many variations with respect to several of these allowances. In some cases, to the retail trader, there is no other — allowance than the actual tare. With the Customs and Excise no draft or tret is allowed, nor any other deduction than the tare, except, in coffee. and cocoa nuts, upon which an allowance not exceeding 2lb per cent is made for damage, or loss of weight, if it has existed, at the time when those goods are taken from the warehouses; also upon Tobacco, there is an allowance of 2 lb per Hhd, for draft ; but as this article is weighed net, the 2 Ib are deducted from the weight in the scale, or the net is called 2 Ib less, and consequently no statement. is made of it, in specifying the weights ; allowances are all made for refuse, damage, or deteriora- tion; as, those of 1-7ths in Salad Oil for sediment; 1-3rd in — Pickles for liquor; and from 6 to 8 per cent for leaves, in the weight of Succus Liquoritiz, &c. In the calculations of these allowances, actual or average allowances are generally deducted before proportionate allowances are calculated ; when the amount is 4 a lb or above, it is generally 4 OF DRAFT, TARE, &c. 105 called another pound; but in some small and valuable articles, the calculation is made in halves and quarters of pounds. In taking the weights of goods, if the beam does not turn with the weight then in the scale, the amount is in general taken at a pound less ; but in the weighing of goods at the East India ware- houses, the two ounce extra weight which is allowed by the Com- pany, is taken out of the scale when the beam is equally sus- pended, and if it then turns, the officers of the Customs and Excise, generally take the weight in the scale, but the Com- pany take it, as above, or at a pound less; which is also the practice with the Excise, in the weighing of Teas. The Custom of the East India Company with regard to Draft, is to allow 1 1b, besides the two ounce weight, on ail packages weighing gross above 28 lb; and on all articles except Tea, if the estimated tare is 281b, or above, 11b more is al- lowed, as super-tare. The super-tares allowed by the Company upon Tea, are, upon quarter chests, 1 lb, when the gross is 841b or above, and when the average tare is exactly a number of pounds or without any fraction. _ Upon half chests, 11b, whether the tare is with or without any fraction, which of itself, as before observed, would make the tare another pound. Upon whole chests, when the average tare is exactly the _ pounds at which it is taken, 2 lb super-tare is allowed ; otherwise 1 Ib for super-tare, and 1 Ib for the fraction. The Officers of the Customs allow only the fraction as another _ pound, but the Excise, on Teas, allow as the Company. In the silk trade, the allowances for draft, tare, and super-tare, _have been regulated by a general agreement of the principal _ persons in that business, and they depend upon the places whence the silk is imported, and the different weights of the bags or bales. |The allowances on silk at the East India House, are regulated ina similar manner. The general regulations which are adopted at the Custom House, / and which are also acted upon by the Officers of the Excise, for _ determining the tares of goods, are the following. Goods which can be taken out of their packages, without much _ damage or trouble, are either weighed net, or the actual weight of the package is determined; the former is always the case with Coffee. vor. I. P =e 106 ON THE ALLOWANCES Goods in large quantities, in packages which may differ from each other, but not very materially, are generally averaged in their - tares, by the Officer selecting one, and the owner, or the person acting for him, another package, and then taking half of the two | weights as the average tare, reckoning the fractions of the pound — in favour of the importer, or as another pound. For goods, in bales or linen packages, either certain established — rates of tare are allowed, according to the weight, or else per centage, or per cwt rates. ‘Liquids which pay duty or importation by their weight, as the essential oils, &c, are usually contained in tinned copper cases, and the weights are agreed between the Officer and importer, according to their usual weights, generally allowing any trifling difference to be in favour of the Merchant; and for those liquids as fish and seed oils, &c, which cannot conveniently be emptied to determine the weight of the cask, when instead of taking their gauge to determine their contents in gallons, itis chosen by the importer, to have them weighed, and to ascertain their measure by allowing 7% lb to the gallon, the tare is determined. by a per centage, according to certain established rates. In some instances, the tares are found from the aainel foreign weights, either from the description of them upon the envelope of the article, or from the merchant’s invoice; these weights are reduced into English pounds, either from certain estimated and fixed relations, or they are proportioned by a comparison of the foreign weights, both the gross and tare, and thence from the gross English weight, the relative English tare is determined. This again is sometimes converted into an average tare, by finding the relation of the original tare to the original gross, and expressing it in parts ; as in weighing Otto of Roses, the Turkish tare of one bottle may he as much as 15-16ths of the whole Turkish weight, and therefore the _ net weight is considered to be only 1-16th of the gross, for this and similar bottles in the same importation. So also with many other articles which cannot be turned out to” estimate either an actual or an average tare, the allowances are settled by long established custom at either a certain weight, or a certain proportionate part according to the gross. As with Serons of Indigo, the tare for each is 18, 22, or, 26 Jb, accord- ing as the gross under 1 cwt, 1 qr; 2 cwt, 2qr3 or above the last weight ; and with capers, when the casks weigh 5 cwt and above, the allowance for casks and pickle is 1-3rd of the gross, but if under, the allowance is 2-5ths. In taring Saccharum Saturni, = OF DRAFT, TARE, &c. 107 5-times the weight of the lid, is esteemed a fair allowance for the tare of the cask.—As there is frequently a doubt, when the allow- ance is stated at a per centage, whether it should be for 100 Ib, or lcwt or 112 lb, it is explained in Mr Smyth’s Practice of the Customs, that if the entry be made in cwts, the term means per cwt ; butif in single pounds, then it means per 1001b; but to this rule, there are some exceptions ; as in the entry of bristles, for which the duty is payable on the dozen Ib, and on which the allowance for casks, mats, ropes, strings, or drums, varies from 2 to 16 percent ; but although the gross weight in cwt requires to be brought into Ib, the tare is calculated at 2 lb, &c, per cwt, and not per 100-Ib. It is from the First Edition of this accurate and intelligent work that some of the above observations have been extracted. Itis in all cases to be observed, that if the merchant should be dissatisfied with the allowances made by the officers, he has the power of obliging the article tare to be taken, by emptying the cask, &c; but as in many instances this could be done only with _ yery considerable trouble and much liability to injury, it would _ necessarily be at his own expense and risque. At the West India and London Docks, there are printed tables of the rates of the allowances for the tare and draft, by which the Officers of those establishments are invariably guided. In general, they nearly agree with the regulations of the Customs, but they are so very numerous, that the limits of this work preclude any attempt to enumerate them, and it would also answer very little useful purpose, as Merchants rarely ever find it necessary to question the correctness of the statements; it may be supposed that, in general, some person in their employment has superin- tended the landing and weighing of the goods, and seen that the ) allowances are such as are prescribed by the established rules ; from _ this decision there scarcely lies any appeal, but should it be neces- » sary tomake any reference or enquiry, it can always be effected to the offices of those companies, where no private authority _ would have the least influence. LO EOE L AIO ELE OPE LEE COO 108 ON THE ALLOWANCES Example 1. To find the net weight and the value of the 10 following Hhds of Maryland Tobacco. cwt qr lb No 1 Gross 8 ,2 23 SS RE IS 5. ae D Mies ,0 13 ga ———_—____ —————. |b. 3233 at 4 10..78 4 9 OS eee Cwt 8 0O 21 RPE wie € up) 5 60! n ope » tie t- 316 ye OA Barb 21 24 2 2 Draft, 1 lb per Bag...... [Pood 24 1 O Tare, 4 lb per cwt........ BD ise: Cwtt. esis 15-2619 lb. TOE ie wet ome eiei's 1003 s ad 25187 at 1 10.,230 16° Q One Canister, Essence of Lemon. Cwt....- 3 18 Pares17 lb; draft, 1 Ib...... 18 $s Ib 802 at 21... 8415 9 Total amount.... £ 909 18 0 GIL OIL IDE LOL LEE LOA Vou. I Q li4 ON THE ALLOWANCES Example 4. PPP DEL DIL FDA PP LAA PL DDS To find the weight, and the number of gallons at 74 lb each, in the following 25 jars of salad oil. . cwt qr Ib cwt qr lb POR RO ee 6. LE INGLY. ss aoe oe pt Unk i At PASEO pay te Ae ee! Os stew ae. ee ee , Be ee al Lda. cot Se tae Bae FS Ra Be Pee 22 - 2 2 24 LESS aoe oe ko y Ae ee ae Ws) 2a ys 2) 12) ae 1B iat, oiaee renee 12 Pes We ae iD Pee 3 de BAe ee oa bore Dictae eect hein 16 oe 2 15 Moh se 2 YO a0 - 2 2 Ff ay ee er em Aa he 8 OM AEB Be 13 ae due JO 18 se Sk 12 jars $1 11° 9 Broa 2 2 20 3 Ga 34.0 220 13 jars 34 0 Q G5 .bs18 780....(See page 13). . 46 ¢ 7326 Deduct..2442 One-third—for jars. A884 Deduct.. 697 One-seventh—for foot or sediment. . Ib 4187 net. POR AIST Gal. 13952 Gallons. .558 This method of reducing Ib into gallons of 74 lb each, is easier than that used in the 2nd Example ; but if the remainder should be required, as it was in that, some care is necessary in making the - proper estimation. Thus, above, the remainder is 22, or 8-3rds of 10ths, or 8-30ths, or 4-15ths ; therefore, haJf the remainder, in thirds, is the number of 15ths. OF DRAFT, TARE, &c. 115 Example 5. PPS LIF AIL AID AOL AID AIS To find the net weight, and the number of Barrels of 314 Gal- lons each, at the rate of 9lb per Gallon, in the following Barrels of American last Pitch. cwt qr lb oi k fg ee) ee Lt Xe x ee) OG hey 5s Pe k eee Sikes 38 ee PPT e Bs lt, 24 Gross 34 O17 Tare 5 O O — 56 lb each. 20" OY LZ 348 1 bre Ib 3265 Gallons 362 (4 of 3265) 315 .. 10 Barrels. egies ee ee 3464... 11..... (from 362) Barrels 11 154 Gallons. These Gallons may be reduced into Barrels, by reducing them first into~ half Gallons, and then into Barrels, reckoning 63 half Gallons to a Barrel: thus, Gall 362 oe 63 ) 724 ( 11 Barrels. 94 31 halves — 154 Gallons. PLP PADDLED DDD DOP PID ALF Ex. 4 and 5 are from Smyth’s Practice of the Customs. 116 | ? COMMISSTON AND BROKERAGE. PLEO LLL DP LDL APP ODL LPL LOL Commission and Brokerage are charges made by persons acting as Agents, or Brokers, as remunerations for their skill and trouble in executing the business entrusted to their management. PLL LOL LED PLL POL LIED PPL An Agent, sometimes called Factor, is the representative of his: employer, and in most cases, particularly in the foreign trade, the business he transacts is conducted in his own name, and as upon his own account. The charge of commission is generally a matter of specific agreement between the two parties ; when no arrangement of this sort exists, it is regulated by the. custom of that particular branch of trade. In general shipments, the usual commission is 24 per cent, bu! where much trouble is required to be taken in the selec- tion of goods, &c, from 5 to 10 per cent is often charged. In the sales of consignments, the common allowance is 24 per cent ; except in the sale of bullion, &c, when the amount is consi- derable, and but little comparative trouble is required to be taken; under which circumstances the charge is only + per cent. When the Agent acts under what is called a del credere commission, or when he undertakes to guarantee the receipt of the money for the goods he sells, an extra commission is allowed for his risque. In cash transactions, the charges upon this head are very different in their amount. When a great deal of trouble attends the account, = per cent is charged on the receipts, and the same upon the pay- ments ; in some instances the charges are only one half, or one quarter of this rate, or 4 or 4 per cent; and many persons only charge a commission upon the payments,: unless the bills which may be remitted, require tobe discounted, to meet the principal’s drafts, when the expense of the brokerage is compensated by an addi- tional commission. - COMMISSION AND BROKERAGE. “17 A Broker is a person employed as an intermediate or mutual Agent, who by avowedly not acting for himself, or upon his own responsibility, serves as a witness to the contracts between the two parties. _ In the City of London, particular distinctions exist with regard to Brokers. For most purposes of the trade of the metropolis, in which Brokers are employed, it is necessary they should be such as have been sworn to the faithful execution of their office, before the Lord Mayor and Court of Aldermen, when they have a silver Medal delivered to them, certifying their employment. For some purposes such brokers are not required, and the distinc- tions seem to be these ; in all engagements or contracts to purchase or dispose of property, whether they are verbal or written, when a sworn Broker is employed, his official notice, called a‘contract note, to the party to whom the contract is to be rendered binding, with his entry of the particulars in a book, which it is a part of his _ duty to keep, has the same effect, in case of a litigation, as would attend a written agreement stamped and witnessed ; on the other hand, asin the practice of Marine Insurances, when a legal instru- ment in writing immediately binds the contractor to the perform- ance of his engagement, the intermediate Agent, though usually stiled a Broker, is not one as above, who is required to be a sworn Broker; and similarly, in the sale of goods in an open warehouse, where they can be immediately delivered or rejected, Brokers are very rarely employed to dispose of them. As the effect of a contract thus made is so very binding, in order to prevent any mistakes in the agreement, it is allowed that the party against whom it would operate, should have a reasonable time for giving it a due consideration ; this is generally considered to be before the commencement of business the next morning, and if the contract note is then returned, no legal claim exists against him ; but it is hardly necessary to notice what must be the consequences upon a person’s credit, of such a procedure, unless very justifiable reasons exist for so doing. Almost every department of business has its particular Brokers, who by confining their attention solely to that subject are supposed to possess such a superior degree of information, either of the quality of the goods to be sold, or where a market is to be found for them, as may render their services highly valuable to those by whom they are employed, as well as prevent them considerable trouble. Thus besides Stock Brokers, whose traffic is in the public . 118 COMMISSION AND BROKERAGE. funds; Discount Brokers, who are engaged in procuring discount — of bills; Exchange Brokers who negotiate Foreign: Bills of | Exchange ; Ship and Insurance Brokers, who dispose of ships and | effect insurances ; there are Tea Brokers, Fruit Brokers, East In- — dia and West India produce Brokers, &c, in great numbers. In the public sales of goods only sworn brokers can legally officiate, and such has become the custom of trade, that at all the | ' . principal public sales, particularly those of the East India Com-_ pany, none but brokers bid for the articles to be sold; any other person certainly possesses the right of doing so, but from the com- petition he would create, his interest would suffer a material injury. Hence in public sales, brokers act as principals, and a disclosure of the real purchasers and sellers is seldom made; the brokers becoming completely responsible for the due performance of the conditions of the sale. Thus in the East India sales, if the actual purchaser or the person to whom he may dispose of his right to the goods, does not either pay the deposit within the stipulated time, or pay the remainder of the purchase money before the expiration of the prompt, or the latest day when the money must be paid, the broker must do so, or he will be declared a defaulter, and be ever afterwards precluded from bidding at the Company’s sales. When merchants themselves dispose of their goods, as upon the — Exchange or without any witnesses to their bargains, it is neces- sary, for mutual: security, that the broker of the seller should send the purchaser a contract note, which, if not returned, as before specified, within a reasonable time, becomes binding upon the purchaser to fulfil the engagement. The following are the usual forms of contract notes. PPS IDI PPL ODE LOL EDO LOR London, 15th April 1819. Sold for Account of Messrs. Roberts, Barker, & Co. To Messrs. Boyes & Ingleden, C.B. 45. 20 Hhds. Demerara Sugar, per British Tar, at 70s 6d per cwt. Your most obedient servants, French, Rogers, & Co. COMMISSION AND BROKERAGE. 1119 London, 13th April, 1819. | Bold for Account of Messrs. Jones & Lloyd, To Messrs. Preston, Boyes & Co. Lt Kighteen puncheons, Leeward Island Rum, ex Clio, at three shillings and two pence per gallon, and 5s per puncheon Landing Charges to be taken at the Landing Strength, and gauges ; - and paid for per prompt one month. Commission 2s 6d per puncheon. John Roberts, Broker. PLF LIL AIE ILI AAD AAT LIF The regular charges for brokerage vary from 1-10th tol per | cent, which is the charge for selling goods by public sale, and, which under particular circumstances may also be increased ; as where the broker undertakes to be answerable for the party to | whom credit is to be given, as well as where extraordinary trouble is occasioned. In some cases, the brokerage is a particular charge © on each article sold, and not a per centage. It is generally understood, that a broker is not to be paid by both parties; his compensation is usually aliowed by the seller, but when the purchaser employs him to search the market for the _ particular goods he may want, the broker has a right to make him a charge for his extra trouble, and it is usual also even in these _ cases, for the broker to receive his half per cent or any other _ general allowance from the seller, as a douceur for bringing him » a customer. When a discount is allowed for early payment, it is considered proper to calculate commission or brokerage, only upon the net amount ; but this is not a universal practice. PIL LOL LAE DIL AIDE DER Some public sales are declared to be “ by candle,” an expres- sion which now means the public sale of goods in the evening, or by candle light; formerly the expression was * by inch of candle,” from the custom of allowing a limited time for bidding ; viz, during the time a wax taper was burning an inch, but this method has _ long been obsolete. PID LIL PLP DLP LDL APF As many calculations of Commission and Brokerage, will be _ found interspersed throughout this work, we shall here give only _ the one following. 120 COMMISSION AND BROKERAGE. Example. To find the amount of the commission at 24 per cent, and of the brokerage at 4 per cent, upon the amount of 100 bags of cotton wool, weighing as follows, and sold at the following prices 5 a discount of 1$ percent, being allowed for cash in 14 days from the day of delivery; or in 1 month from the day of sale. cwt qrs lb d 50 bags Pernambuco, weighing, gross 69 2 11 at 21% per Ib. Bint =~ IOMOFALA, 20. ysfopss Ps aleieaty {46 3 17 at 163 per Ib. cwt qr Ib Gross 69 2 11 Draft 1 22..1 lb per bag Ge) 17, 1 25..4 |b per cwt. ies Cwt 66 2 20..7468 lb at 213 d...... 672 17 11 cwt qr lb Gross 146 3 17 Draft 1: 22 146 1° 923 Tare 5 O 26 Cwt 141 0 25..15817 Ib at 162d ....1103 17 11 \ cca oe i 1776 15 10 Discount 14 percent 2613 1 Amount of the Goods £1750 2 9 Commission at 24 percent £43 15 1 Brokerage at 1 per cent £815 O VOOELOILILDVOO REI DED GDR AVERAGE PRICES. LFF IIL FFE FPA LPIA APA LPS When several quantities of the same sort of goods have different rates of value, the rate which the whole would sell for, to produce the same amount, is called an average Rate or Price. ; PLL ALLO FLIP EE PL EOL DL LED Of two articles, viz. Corn and Raw or Muscovado Sugar, average prices are officially computed and published, for determining the action or discontinuance of certain legislative provisions. Of Corn, the average prices of all that has been sold in the twelve maritime districts of England and Wales, are made up weekly by the inspectors of the markets in particular towns, and transmitted to the receivers of the corn returns; from which-returns of the average prices of the towns, the average prices of the district are obtained, and they are then transmitted to the chief officers, or col- lectors of the customs. Of those average prices that are made up for the six weeks preceding the 15th day of the months of February, May, August, and November, an aggregate average price for all the twelve districts is estimated, and the importation for home consumption is either prohibited or allowed, from one period to the other, according to the prices of these returns. The importation is permitted when the prices are, For Wheat, at or above .... 80s per quarter. Rye, Pease or Beans... 538 per quarter. - Barley, &c........% *,.. 40s per quarter. Maisie sae. oT. 27s per quarter. but when the average prices are below these, the prohibition is strictly enforced, with respect to all corn, &c, coming from any places between the Rivers Eyder and Bidasoa ; whence only European importations are usually made. But, for importations from the British Plantations in North America, the prices at which they Vou. I. R 122 AVERAGE PRICES. . are admitted, are, at or above the following, viz, for Wheat 675; Rye, &c, 44s; Barley, &c, 33s;'and Oats 22s per quarter;. however, both European and American corn, &c, during the con- | tinuance of the prohibitions, may be imported and warehoused in the King’s Warehouses, either for exportation, or for-home con- . sumption when the rise in the prices permits the ports and. the - warehouses to be opened. | am The average prices of Raw sugar, are estimated from the weekly returns of sales, made to the clerk of the.Grocers’ Company in London, and verified before the Lord Mayor.or one of the Alder- men; these returns are made by either the importer. or his broker or agent, but rarely otherwise than by the brokers and they are required to be made on or before Wednesday in éach week. . This average price as well asthat of corn, &c. is published every Saturday in the London Gazette, the only. paper officially ac- knowledged in this country. F rom these weekly averages, other | averages are calculated. and published, and when it’ appears, that independent of any duties of customs, ‘the average of the ayerage prices for 4 months preceding. the 5th of January, May and Sep- tember respectively, are below the following prices, the following deductions on the original War Duty, now rendered permanent,’ may be permitted to be made by order of the Treasury~ é If below 49s..:..../.1s per cwt. ASG. 3. 200 28 Hee 47s." 3s The bounties on the exportation of the refined Sugar were for- merly regulated by the average prices, - but they are now -inde- pendent of them, and are fixed at certain rates. . For ordinary mercantile purposes, average prices are estimated to determine what price, or rate, may be required, or be given. for the whole of several articles having different rates of value ; and as many of these bargains are concluded almost instantly, it is some- times of great consequence to be quickly able to determine on — which side the advantage lies; of this further notice will be taken in the explanation of the second of the following examples. PIE LODPLOP POD PLOD LOD OOD AVERAGE: PRICES. 123 Example ir ew wveeon To find the average price of the following quantities of com. 850 Quarters at 78s WORD cn 9, 5 wis at $83 Gana Gets ay. at $4 Qrs 1778 s Ss $ t 83 84 “4 4 65 390 332 420 624 7 504 a hs 2884 5460 : aa 6630 ee | 5460 i qrs lewearir 5 qr | ioe if 178 produce 14414 s what will 1 produce ? : rhe sas jobs. d _ 178 ) 14414 ( 80 112 Average price required, ort 174 s Pee aR oT eT 308 130d ) 520 ( 3 nearly The same result as the above would be produced if the full num- _ ber of quarters was used instead of using only the tens ; of which, in the third, we take 65 as the nearest number. POE OLS OOF LEE OEE Ite 124 AVERAGE PRICES. Example 2. PPA LDL IDF DIDO OP ALE LIL To find the average price per lb, of the following Hides, sold at the following prices : viz, 350 hides weighing 987341b, at 74d per Ib. BOOTS vik st ade aint erate whee IDs eate 1 OB, BPO ec iihus nc sho ge Sa OOO Re tpuat og d sper ih, 920 264988 lb. d 98734 lb, at 74d produce 715821 87410 Ib, at 8 d>...:... 699280 78844 Ib, at 94d ....... 729307 Ib If 264988 lb produce 2144408 what will 1 produce? d d 264988 ) 2144408 ( 87, nearly, Answer. 24504 ( or ) d d d 4.35 30 1.197 75 8 94 245 240 243 9 250 7 254 254 250 If 92 produce 744 what will 1 produce? d d 92 ) 744 ( 8, nearly, Answer. 3 In articles which run nearly of the same weight, it is seldom of any use to perform the calculation so minutely, as in the first of the above methods; the second method determines the average amount with all the precision that is required in business ; indeed ina cal- culation similar to the above, where the quantities at each rate are not extremely different, an average rate would most probably be -determined by adding the 74d, 8d, and 94d together, and taking 4 of the amount, 245d, which gives $d or 85d per lb, as the average price, ' 125 INTEREST. a ee ee Interest is the premium paid or charged for the use of a sum of money, in consideration of the advantages, of which the person to whom it belongs is supposed to be deprived. LPP DIF PDL ALDL OLA ODL LAI ‘Interest is either simple or compound. Simple Interest is the produce of a fixed sum, for any length of time. ? Compound Interest is the produce of a sum increasing, at stated periods, by the addition of the simple interest, first upon the origi- nal sum, and afterwards upon each subsequent amount. LEI PAL ADOT FOLD AOL FUE LAE _ Interest is also distinguished as being either legal or usurious. Legal Interest is the highest rate that is allowed to be taken by the laws of the country; which, at present, in Great Britain is the ~ allowance of £5 per cent, or for the use of £ 100 for a year, called 5 per cent per annum; at which rate it was fixed by act of par- liament, inthe year 1714. ‘The first regulation of this nature ap- pears to have been made in the year 1545, in the reign of Henry VIII; when the rate was restricted to 10 percent, and which was lowered in 1625, in the reign of James I, to £ 8 per cent. In 1645, during the period of the commonwealth, the rate was re- duced to £6 percent ; at which it continued, with some fluc- tuations, to the above stated period in the reign of Queen Anneg ‘ when the present rate was established. From these restrictions Ireland still continues so far exempt, as that the rate remains as it was fixed by an act of its own par- liament, at 6 per cent. P26 INTEREST. But although 5 per cent is the utmost interest which can be. taken for money lent in Great Britain, yet if a contract which_ carries interest should be made in a foreign country, our, courts of. law would direct the payment of interest, according to the laws-.of the country in which the contract’ was made: thus American, ; Turkish, and Indian interest have been allow ed, to the amount CF even 12 per cent. In the East Indies, and the Isle of France, the rate of interest is fixed at 12 percent; but in many parts of the East the rate is considerably higher. Generally throughout the West Indies the rate of interest is 6 per cent. In the United States, the interest on Bonds and other similar securities varies in the different States . from £ 5 to 8 per cent, but 6 per cent is the more general rate. On accounts current between merchants, interest is generally cal- culated at only 5 per cent. . For the established rates of interest in the different parts of the | Continent of Europe, see the department of Exchanges in this Work ; but it may be observed, that although the fixing of a max- imum of interest, is generally considered to be for the purpose of — preventing individuals from taking an undue advantage of the necessities of others; its particular object in many countries. where. itis adopted, appears to be rather to afford greater facilities to the . Governments in negotiating their loans, and consequently we find . that where no public debts exist, there i is seldom any limitation of interest. ; Usurious interest is, as before observed, the charging of a higher. than the legal rate upon the sum lent, or which is attended -with - the same result, the’ charging of legal interest upon a greater sum, or the charging or receiving of any additional compensation, ex- cept such as is in conformity with long established custom, and which has received the sanction of legal decisions. a Thus it is not held to: be usurious, for a person to charge y £5 5s asthe compensation for lending another person £2 100 for 1 year, upon the security of a Bill of £ 105, which would become - due at the end of that period; because, custom has sanctioned this mode of calculating the interest upon the amount: to’ be received, instead of upon the sum advanced; but it would be clearly so, if the £5 5s were otherwise charged for a twelve months use of £100. Also, it is the practice of country Bank- ers to charge 4 per cent besides the postages, stamps, and interest, INTEREST. 197 - for the trouble of discounting Bills, and this they are authorised to do; though, with London Bankers the same charge would subject them to the penal consequences of usury, or to the cancel- ment of the debt, and the forfeiture of three times the value of » the money, or other things lent. ‘The charging of interest on the Balances of running accounts, though they may ‘consist of both principal and interest, is also - Jegal ; as suppose an account in which various sums of money had . been paid and received, and interest was mutually charged and allowed; if, at the usual time of balancing, it appeared that £1000 was owing upon the géneral account, and 2 50 upon the interest account, in the calculations in the next account, interest might properly be charged upon the £ 1050, but this compounding of the interest with the principal could not be continued in a succeeding account, if it continued closed, or if there was no other item than the above balance. In the consideration of all cases of interest, it is supposed that the person lending the money ‘has the power of claiming the re- payment ; but when this power is relinquished, .and more parti- cularly when the receipt of the interest is made dependent upon ‘some contingent event, as the death of the. borrower, &c, the ‘interest is then called an annuity, and the rate ‘becomes unlimited. Of this sort are Life Annuities, and the redeemable and ter- .minable annuities of Government, which will be hereafter noticed. In Commercial Calculations, 365 days are always considered as 1 year; and consequently the 365th part of the interest for oF year, is reckoned as the interest for 1 day; therefore in the » making up-of Accounts Current, where the account is made up from the 31 st of December in the preceding year, to the end of December in a Leap year, the interest upon any. sum from the 1st of January, as well as from the 31 st of December, i is reckoned as 12 months, and not for the one 366 days, and for the other “365 days. “> . In most foreign calculations, where the interest ‘is computed in months and days, only 360 days are reckoned toa year. In Commercial Calculations in this country, the interest for months and years is usually calculated on the exact amount, but in calculating the interest for a number of days, it is usual to reckon 10s and above in the principal, as another pound, and to reject from the calculation any less sum; this also is frequently done in calculations for a number of months. - ee 128 INTEREST. TO FIND THE AMOUNT OF INTEREST, For any number of years, months, or days. PP LILI LADS PDD FOL LDL PDT The general directions are—Find the amount of £100 for the given time, and then from this amount, by either the Rule of Three or Practice, find the amount of the given sum; or, FOR ANY NUMBER OF YEARS, PIF LIF LOL BEAL AOL LDF LDL At £5 per cent,—The interest for 1 year being 1-20 th of the principal, for any number of years take so many 20 ths. At any other rate—Multiply the principal by the number of the years, and this product by the number of pounds in the rate per cent, and divide the last product by 100. Or, find the amount at £5 per cent, and then proportion the product to the given rate. | PPL LILI FEL LAL OPEL LID LD LP FOR ANY NUMBER OF MONTHS. At 5 per cent—Multiply the principal by the number of the months, and divide the product successively by 12 and by 20. At any other rate—Find the amount at £ 5 per cent, and pro= portion the product to the given rate. PLL LOL POLL ADDL LD DDL ODD FOR ANY NUMBER OF DAYS. PAL LIL EPIL DALE DL LD DLP LL FH At 5 per cent—Multiply the principal by the number of the days, and divide the product by 7300. At any other rate per cent—Multiply the principal by the num-— ber of the days, and this product by twice the rate per cent, and divide the last product by 73000. For these divisions by 7300 and 73000 we may substitute this process ; add to the product its 1-3 rd, with 1-10 th of that third, ; and 1-10th of that 10th, and divide the amount by 10000 for | 7300, and by 100000 for 73000, which amount is to be corrected, if necessary, by subtracting 1 farthing for every £ 10 of interest. INTEREST. 129 Or, for 5 percent.—Multiply as many shillings as there are pounds in the principal by the number of the days, taking parts, if requisite, for the shillings and pence in the principal, and divide the amount by 365. And in either way, when the rate is not 5 per cent, the product may be first found at 5 percent, and then this may be propor- tioned for the given rate. PPP PIPPI LL LISD LDP DPF OD GS The principles upon which these directions are founded, will be explained after each of the following calculations. lh Example 1. PPE LIL LDL LOL LAD LOD ADL To find the amount of the interest on £2 850, for 5 years, at 4 per cent per annum. The amount of £ 100 for 5 years at £ 4 percentis 20 BE Se Le Ae i ee 160 50 et nee: ee ORS 10 £170 Or, £ 850 20) S14 GS 100 ) 17000 £ 170 amount at £ 4 per cent. In the second of these calculations, asthe interest for 1 year at ' 1 per cent, isthe 100th part of the principal, the interest for 5 years, is 5-100ths, and at 4 per cent 4 times 5-100ths, or 20-100ths, for which we might take 1-5th.—Or, we may say, that at 4 per cent per annum, the interest for 1 year is 1-25th of the principal, and for 5 years it is 5-25ths, or 1-5th. WoL, 1. § 130 INTEREST. Example 2 To find the amount of the interest on £ 850, for 5 years, at 6, per cent per annum. Yee ae The amount of £ 100 for 5 years at 5 per cent,is.. 25 00 for ue BODE! . genes ace den OOD 50 ceeoe ee ower ee rene 12 10 O £212 100 | £ Or, Zz .. 850 os £212 10 amount at 5 per cent. At 5 per cent, the interest for 1 year being 1-20th of the — principal, for 5 years it is 5-20ths, or 1-4th. PIO LIL LOL BLL OL LOLS ag The amount of £100 for 5 years, at 6 per cent, is.. 30 DE AEROO. «sco ee ee 240 Up vies ety es pate inf i= 15 £ 255 23 Or, 850 SO 16, %.5 100. .25500 £ 255 amount at 6 per cent. In the second calculation, we take 30-100ths of the principal ; viz. 5-100 ths for 5 years at 1 per cent, and 6 times this for the amount at 6 per cent. INTEREST. 13] Example 3. YY ner err IIS ADO LID DDL To find the amount of the interest on £ 640 10, for 5 months, at 4, 5, and 6 per cent. = £ s d 640 10 or, 6401 5 5 12..3202 10 d 3202! mn. 200° LPG ; S.. 266 10£ Amount £13 6 104 at 5 percent. Soks 36.) TOs ee 2° 13 42 .. 1 per cent. 10 13 6 .. 4 per cent. £16--0.3 ... 6 per cent. ' The principles upon which this calculation is performed, are, that as 1 month is 1-12th of a year, and that as at 5 per cent the interest for 1 year is 1-20 th of the principal, the interest at that rate for any number of months, is so many 12ths of 1-20th of the principal ; or what is the same the 1-20th of so many 12 ths. | In the second mode of calculation, as the interest for 1 year is -1-20th of the principal, or 1s in the pound, the amount of the ) interest for one month is 1-12th of 1-20th, or 1d in the pound; and, consequently, for 5 months it is 5 times that amount. When the interest is found for 5 per cent, 1-5th of the amount gives 1 per cent, which is subtracted for 4 per cent, and added for | 6 per cent. N. B. It is evident that if the number of the months were 2, 3, 4, or 6, we might take 1-6th, 1-4th, 1-3rd, or 1 half, ‘instead of taking so many 12 ths. | PRE PIL PDIPFCPOLIO PDI AIDA 132 INTEREST. Example 4. LDL POA ODD ADA LED DDL AID To find the amount of the Interest on £ 763 15 6 for 90 days, at 4, 5 and 6 per cent, per annum. For 4 per cent For 5 per cent 2 , af 763.775 by 90 7637775 68739.750 by 8 5 -. 68739.750 i aa Lae 1 4 .. 549918.000 To +9 tree a 2291 A, .. 183306 To ee e830 patti 1833 £ 9.4172 L7. 53387 £9 8 4 for 5 percent s 10.67 PS Se ener BR 4 ene Or ess eer £7 30. 5-24 penpent £11 60 .. 6 percent Converting the shillings and pence into the decimal parts of a £, by the directions in page 17, for 4 per cent, we multiply, first by 90 for 90 days, and then by 8 for the double rate per cent ; this product is divided by 73000 by the first, tenth, and tenth rule as it is styled, reducing the remainder first into shillings, and_ then into pence. At 5 per cent, the multiplication by the double per centage is omitted, and the product is divided as before by 7300. The decimal in this amount is valued by the directions in page 16, as the amount at 4 per cent might have been. To proportion — | this amount at 5 per cent for the other per centages, we take 1-5th | for 1 percent, and subtract the product for 4 per cent, and add it for 6 percent. The 6 per cent might also be calculated, as the 4 per cent, by multiplying by 90 and by 12 ( for the double per centage ), &c, but it is not considered necessary to perform it. PIL ELE LAA PEE LOL ALE OLE INTEREST. 133 In explanation of this plan of finding the interest upon a given sum for a number of days, it is to be observed, that as one day is the 365th part of a year, and as, at 5 per cent, the interest for 1 year, is the 20th of the principal, the interest for 1 day is the 365th part of 1-20th, or the 7300th part of the principal ; and, consequently, for any number of days it is so many 7300ths ; hence the directions, for 5 per cent, which are varied for any other rate per cent in the following manner. Suppose the amount to be found at 5 per cent, and we wish to find the amount at any other rate, we may either take so many fifths of the amount at £ 5 per cent, as are expressed by the given rate, or what is the same, we may take a double number of tenths. Instead of performing this in two separate calculations, we can combine them, by multiplying by the double rate on the one part, and dividing by 10 times 7300 or 73000 on the other; by which method we still possess the means of using the approximation given in the directions, as a substitute for dividing by 7300 or 73000. The principal, upon which this taking of a third, tenth, and. tenth is founded, is that the decimal .00137 or 137-1000000 ths, is very nearly equivalent to the fraction 1-7300th, as may be deter- mined by dividing 1 reduced into a decimal, by 7300; and for 137-1000000 ths, or 1 and 37-100ths of 1-10000th, we may either multiply by the 137, or we may calculate as in practice, by letting the given line stand for once, and then taking parts for 37-100ths, saying 334 are the 3rd, 34 are the 10th of 334, and + is the 10th of 34. The amount is then divided by 10000 for the denominator of the decimal. The same form applies to the division by 73000, except that the divisor upon the amount is 100000 ; that is, in the former we cut off 4 of the right hand figures, and in the latter 5 figures. The reason for deducting 1 farthing in every £ 10, or 1 farthing from 5 to 15, and 2 from 15 to £ 25, is, that the above decimal is a little greater than the fraction 1-7300th, and therefore it requires this subtraction to render the result more exact. To shew this more fully we will calculate, by these directions, the following example, in which the time being 73 days, or the 5th of a year, the amount is previously known to be 1 per cent on the principal, or £100. LAP PPD PILI AS PILI DLL IID Ca INTEREST. Example 5. SLI PIP ALLE LOL DDD ALE LDP To find the amount of the Interest at 5 per cent per annum, on £10000 for 73 days, £ 10000 by 73 73.0000. by 137 21 a tera Lag ik PT eer f £ 100.010 aa ( or ) We 10000 by 73 . 73.0000 . 24.33331 oe 2.43331 94334 an ot ote £ 100.010 making £ 100 and 10-1000ths, or, nearly 10 farthings above the true amount, which is very nearly 1 farthing in every £10. Hence the reason of the direc- tions for making this allowance, but they should rather be, to deduct 1 farthing from £ 5 to £15, and 2 from £ 15 to £ 25, &e. N. B. In this calculation, the first form takes less figures, but in general the second is the more convenient. The thirds are expressed for the purpose of showing the exact amount. ee INTEREST. , 135 Example 6. To find the amount of the interest at 4 per cent per annum, on £875 14 6 for 35 days. 875s 35 4375)! fo, 0 '945 2625 17 6 (the $ of 3538) for 10s 7 — (the 3 ofdo.) for 45 - 10 (the $ of 4s) for .. 6d parti all es s d 365 ) 30650 4 ( 83 113 nearly. 1450 355s ) 4264 ( lid 249 d ) 996 ( 2 260 Ls. .d Product 4 3 113 for 5 percent. 16 94 for 1 per cent. £3 7 2% for 4 per cent. By those who are not very well acquainted with decimal cal. culations, this method, which is much practised in the Stock Ex. change, may be found both shorter and easier than the other ; its ‘principles are evidently the assuming of the amount of the given ‘sum for one year, as being 1s in the pound on the principal, and then taking 35-365 ths of the amount, for 35 days: excepting, that instead of determining the year’s interest on the 14s 6d, we take, asit may be said, 35 half shillings for the 10s, and 35-5 ths of a ‘Shilling for 4s, &c, which is more easily expressed in the cal- culation by considering the 35 (after it has been used as a mul- tiplier) as 35s, and taking parts out of it, for 14s and 6d as out fa £. ; 136 INTEREST. When, asin the preceding case, we can easily find component | multipliers for the number expressing the days, the calculation may be performed as in the following method. a CaS pone 875 14 6 by 5 4378 12 6 by 7 365) s 30650 7 6 s83 113 as before (or ) BOR) Bey, Bry 375 14 6 by 5 4378 12 6 by 7 1, ..30650 si... 10216 ef 08d. 102 £4,1989 £4 3 112 In this, the division by 365 having been worked before, it is not expressed at length, and it is to be observed, that as the product of the pounds is considered as so many shillings, if when the re- mainder is reduced into pence, much nicety is required, there are as many pence to be taken in, as may be considered propor- tionate for the rest of the product out of a £2; as above, for7s 6d or nearly the third of a £, to take in 4d or the third of a shilling. 7 The second form is sometimes a more convenient way of cal- culating the same, by the other part of the directions, without expressing the given sum decimally; but, in commercial calcu lations of interest for a number of days, as the given amount is seldom taken otherwise than to the nearest pound, as before ob- served, in future in similar calculations this only will be the method adhered to. When several sums are due the same time, it is usual to find the interest on the amount, instead of upon the separate sums, as in the following Example. INTEREST. 137 Example 7. LIL LDP LID DAF AOD PPA ODO To find the amount of the interest from the 17th of September, to the 23 rd of December, on the following sums ; viz, on, sa 253 84 12 i Zo lt ‘Ad 6 MeN a L£ 3,3753 or £37 6 The interest is here found upon £2 254, the 13s being reckoned as £1. In the making up of Accounts-Current, in which interest is charged and allowed for the sums paid and received, there is some diversity in the principles, as well as in the forms of the accounts. | The two principal of the former variations, are, the charging and allowing of interest on every sum paid and received, and then taking the balance of this interest account; or, the balancing of the account at each payment or receipt, and charging or allow- ing interest on the balance. Vou. I. 'C 138 _ INTEREST. Some difference also exists in the rate of the interest; as with some houses it is a practice to charge 5 per cent on their advances, and to allow only 4, or even only 3 per cent upon the balances in their hands; some also charge interest upon their advances, and do not allow it upon any surplus they may possess. In the statement of these accounts, the principal difference in the forms exists, in making out the interest accounts either with the general account, or else separately from it; in stating the amounts of the interest, or else only the products of the principals multiplied by the number of the days; and in reckoning the time in days, or else in months and days. PAS DAD LID OLDE LOL DOL DOL Example. PPAF DAF SDI LAID LIA LDAP PDD To determine the balance of the following account of payments and receipts, made up finally to December 31, 1818,-the first Bayt being balanced to June 30, of the same year. PLA LIL IAP LEAD OBE DOL LEE 1817 Se en TEL 1818 Lf iy ee Dec 31, Balance 874 10 O....Jan23, Received 841 6 3 Jan ill, pM Paid Ahad) WO Wee ee 98557... He Toad Om Maraliarccs yy ss 600 0 wr 2 eo Ae. Oe ee ee Jimells... 0 oo b ROL Il Sb ux Se a Be 1818 oe ead 1818 J iptnal July 22,.. Paid.. 8410 O Septid8,-¢sii/ie. 2250''0-/ 0 Now 4 7seee wit. 01450 60) 10 Dees er 4.22 13-90 . July 22, Received 500 O O s»- Atag 28, (05.07 by Sh O POP PIP LOE LID LG IGP L LD A INTEREST. 139 FIRST METHOD OF STATING THE ACCOUNT. SIP PPA DID LID ADA LO ODD Dr Side. Cr Side. Cash, Interest. Cash, Interest. £s ad £s ad ws soa £sa On 874 10 0 for 6mo...21 17 3 On e4l 6 8 for 1528 Days18 4 1 140 0 0..170days.. 3 5 ro a0. OE > ISS Bp 7 pee. L054 oo ae eS ew | B42. (a) OU TON, ie SETAE O mae. Bo. TS wen arty - 16 3 RS-34 G6; - 5018450 2 1996 5 5 — 1546 8 9 27 15 6 Int on Payments 34 11 3 379 16 8 Balance of Cash, Receipts..27 15 6 1926 5 5 Balance of Interest... 6 15 9 to June 30, Balance of Cash.. 379 16 8 in favor of the Receipts. Whole Balance £ 386 12 5 Due June 30. INTEREST ACCOUNT CONTINUED. Foal fy 4d a ots £ iad On 386 12 Sfor 6mo...9 13 4 On 500 O 0 for 162 days 11 111 84 10 O.. 162days:.117 9 Syl AO UES 180 tt | Sorrel mou. O° O-.af04 7. Re 30243 ae. i 382 5 : ROO Poe da ets, eevee pee SS Pad MEMES a GEG SeekOT Ons aes « ES 8 EY. 1193-45. § Gew-"n"s"""——= —_—_—_—_—_—_————ee —_—_—_————— Int. on Payments..17 16 7 4 Receipts, ..... 16 13:0 Balance of Interest ..... wel 3 27 BO Ol Gaghiis cons 63 $62 5 5 Whole Balance..383 9 © Due Dec. 31, 1818. In the making up of this Account, the time is calculated from the dates of the payments and receipts to June 30, and Dec 31, each inclusive ; and the interest on the first sums on the Dr side of each part, is charged in months instead of days. The balance of the first Interest Account is added to the balance of its Cash Account, and the Interest in the second part is charged on this amount, producing the effect of Compound Interest. “ =~. INTEREST. SECOND METHOD OF STATING THE ACCOUNT. PPO LPL PLDI POL POL LOD LOL Dr Products. Cr Products. | ois Days. £L sd Days. | 874 10 O for 181 ..158375 841 6 3 for 158..132878 | PAO eeO..70is... 170 4. ae g800 940. 0 +:0'..«. LOS e678 G00 400/707... 105. 7863000 342 8800.5 2°79) roe Sine oei....) 19 %R 5928 122° 14-6 < "50S eee 1926 5 5 1546 8 9 Cr 202766 1996 8 oor Dr Products ..251103 : Cr Products ..202766 Balance of Cash £ 379 16 8 | Balance of Interest.. 48337 -— 7300, produces... 6 12 5 | Whole balance £386 9 1 PL LADD PIDLLLE LLL LLL POP F INTEREST ACCOUNT CONTINUED. PPP LAI BLO ORI Pir Camas Vit Days. ee BS eee Days. | 386 9 1 for 184..71024 500 O O for 162..81000 | SO Or Wan 162..13770 31110 O.... 130..4056@m) CY AAR eal (pea pe 104..26000 “811 10 O Cr. 121560. 45060) 10 A, |a 44..19800 1193 12 1 Dr. | 22 13 0 et ee | RE pa eae hie Balance. eece £382 2-3 Dr Products. . 130594 Cr Products. . 121560 ‘ Balance... +. 9034 -— 7300 produces 1 4 9 Whole Balance... £ 383 6 10. In this Account, as before, though all the sums are expressed in shillings and pence, they are reckoned at either the next higher or lower pound. A difference of 2s 2d exists between the final results of this and the former method, arising from trifling dif-— ferences in valuing the former products to the nearest penny, and | from reckoning the first sums, for months in one case, and for days in the other. 7 INTEREST. | 141. THIRD METHOD OF STATING THE ACCOUNT. Dr Cr 6 sc da eo sid £ ae d ore ee 20, NGOs, cee = 81,17; 3 84. 6 3..5mo 8 days18 8 Il , 140 0 0..5mo 20 da 3 6 0 240 O 0..5 mo 3 days 5 2 ‘0 600 0 0..3mo 14 da 813 0 342 8 0O..2mo18 days 3 13 10 Mee Ae Sine sce ce 19 da - 16 3 122 14 6..1 mo 20days - 17 0 £1926 5 5 ———— 1546 8 9 Cr. £28 1 9 a. 379 16 8 Bal of Cash. Interest on Payments...... 3412 6£1926 5 5 Receipts: cas 0490) HA) kh Balance of Interest ....... aC 1079 Balance of Cash ........ 379 16 8 Whole Balance........ £° 386 7 5 INTEREST ACCOUNT CONTINUED. £-s d fe od LS id Ss tur 386 7 5..6mo...... 913 2 500 O 0..5 mo. 9days1l 0 8 24.10 , 02.5 mo. 9 da. l-17.. 4 311 10 0..4mo 8days 510 8 250 0'0..3mo1l2da 310 9 a 1614 450 0 0..l mol3da 213 6 Stl-10> OG 22 13 (0..0 —_———. 382 O 5 Balance of Cash. £1193 10 5IntonPymts 17 14 9£1193 10 5 Receipts... 16 11 4 Balance of Interest... 1 3 5 Do of Cash .. 382 0 5°. Whole Balance... £ 383 3 10 ——————————_—___—__—__——. The time is reckoned backwards from June 30, and Dec 31, to dhe dates of the Account; and in calculating the interest, the months are considered as so many 12ths, and the days asso many 365ths of a year. With respect to the days, some persons con- sider them always asso many 30 ths of a month, and others, as so many 28 ths, 30ths, or 31ths as there are days inthe month in which they begin. The calculation of the interest for the months, is here made on the exact sums, and for the days, on the nearest pound in the principal. . The amount of this balance differs from the first by 5s 2d. anit oe 142 FOURTH METHOD OF MAKING UP THE ACCOUNT. Dec 31,1817 Dr Jan 11, 1818 Dr Dr Ys Bares @ Dr A PN lag, ©) g Cr Mar 17 .... Dr Dr pit? +... Cr Dr May 1t,".,. Cr Dr June 1l.... Dr to June 30.. Dr Interest. . Dr Balance.... £ 386 8 INTEREST. PIL PLL LOSE ODD LBL LOE ODD FIRST PART. GIS LAL LED LOOP PEP ADP ODO ee 874 10 140 O 1014 10 841 6 6 12 Dr d Products O for 11 days.. .9620 O O for 12 days. .12174 3 9. for 5 days,...866 0 Sifor 4s dayes. . ne .s% s.. 0 % 9 for 26 days. .13862 0 9 for 29 days.. «5532 6 3 for 31 days...2110 ' . 8 for 19 days...7217 SOP EAE G 51381 preses 3207 PLE LEFLLE GLE LE LOGE LOD Cr Products 2 gt 48174—-7300—6 12 — 1818. June 30.... Dr @aly 22...: Dr Dr Cr Cr mee 2S .....Cr Cr Sept 18 .... Dr Cr eel? «...< Dr Dr Mec 31 .:.. Dr Balance of Cash Interest. . INTEREST. 143 FOURTH METHOD CONTINUED, SECOND PART. Dr Cr , fae ef Products Products 386 8 8 for 22 days...8502 84 10 O 470 18 8 J00. 0. 0 mks BE LOL OS UMY MetieN's ried ee he ne 930 i leks Bs Joo S40 "11 Pitor 2Qo-dayssie sels pes S 8855 250 O O VO ede: LOGO OVS so o 156 DISCOUNTS. Example 4. OF THE FOURTH METHOD OF DISCOUNT. PAF DDLP LDIF POE FDP DE LOE ¢ The directions for these calculations are, to find the interest upon £ 100 for the given time, and add it to £ 100; then to say, if this amount yields the preceding discount, what will the given sum produce? the proportionate answer to which will be required discount. Or, tosay, if the above amount produces £100, what will the given sum produce? the proportionate answer to which will be the present worth, and which subtracted from the given sum or principal, will show the required discount. - PREPPED IDI LILI DAI DAE DDL N.B. Instead of -£ 100, any other sum, as £1, may be used. PRO LPO LOD DDE LDL LDL AEA What is the amount of the present worth of an annuity of £350, payable at the end of one year, allowing a rebate or discount at the rate of 5 per cent, per annum. & 23 fe If 105 produce 100 what will 350 produce? 20 100 _ 20 21) 7000 1052 ~——- Answer £333 6 8 fer gd ( or ) mip eaee 000, 0...0> Principal 16 13 4 Rebate £333 6 8 Present worth. DOP POP BIE LID DOLD OD OL OD 4 _ 3 a ? DISCOUNTS. 157 Example 5. POP PLO LDPE LAD To find the present worth of £275 for 87 days, allowing a rebate of 5 per cent, per annum. i re | 100 by 87 cae 700 te... 2000 a ee aU 29 £1.1919 or £1 3 10 nearly. Le a £ If 101.192 produce 100 what will 275 produce? £ 101.192 ) 27500 Answer £271 15 2 Present Worth. Le ¢. a ad Proof 271 15 2 Present Worth 87 7300 ) 23643 £ 3 4 10 Interest on P. W. for 87 days. £275 O O Principal. In the working of this example, the interest for £ 100 for the 87 days is expressed in a decimal, and this amount is added to £100. Inthe proof, we find the interest on £271 15 2 for 87 days, (as in page 136 ), and adding this to the present worth, we produce the principal, which, in all calculations of discount, is considered to be composed of the present worth and its interest. Hence the reason of the directions. POP ODODE PDD LAGI ODP DDD 158 DISCOUNTS. Example 6. LILI POP PLO LAS _ To find the rate per cent, per annum, which is made of money employed in the discounting of bills at 5 per cent per annum, at the different periods of 1, 2, 3, and 12 months. - ees 8 to Pde ge The interest on 100 for 1 month is, 8 4 leaving net 99 11 8 ee iE ee a eee 99 3 4 SO aataee tented Re PVG ORAS ge oe 98 15 O i LA aA Yas Ohne niso.s SDL) aa For 1 month. If £99 11 8 produce 8s 4d what will £100 produce? Answer 8s 4},3d nearly, for 1 month ; which renewed monthly, produces about £5 O 5 per annum; but, ; admitting the possibility of employing the interest of each month, as well as the principal, the profit per annum of £100 would be about £5 2 9. Pod hete set pe lnosne For 2 months. If £99 34 produce 16s 8d what will £100 produce? Answer 16s 95,3d, nearly, for 2 months, and £5 O 10 per annum ; or improved each 2 months, with the interest, as before, the £100 would produce nearly £5 3 per annum. PIS PPE LAP EPP LOL ALO DAL For 3 months. If £9815 produe £1 5 what will £100 produce? Answer £1 5 33 nearly, or £5 1 3 per annum, but which improved with the interest every quarter, as before, the £100 would produce per annum £5 3 23. LIL FILE LOE LD LE LPO LP OP LOD For 12 months. If £95 produce £5 what will £100 produce? Answer £25 5 3 nearly. PLP LDF DP PE PP LDP LDP OOP On comparing the results of these calculations, it appears, that the advantage which arises from the usual method of calculating the interest upon the future, instead of the present payment, is very trifling in any short period of time ; even admitting that con- stant opportunities of employing money in this way were to be afforded. DISCOUNTS. 159 Asa part of the foregoing calculation affords an opportunity of showing, how greatly a continued proportion may be abbreviated by the use of Logarithms, we shall exhibit the mode of operation. In the first statement we say, If £99 11 8 produce £2 100 what will £ 100 produce ? or, if d 23900 produce Ren orcs £ produce and, consequently, on comparing the second term with the first, we find, that the answer will be 240-239 ths of £100, which will be the amount produced by £100 laid out for 1 month; this is also to be improved ina similar manner for another month, and therefore the next statement becomes, if 239 d produce 240 d, what will the amount just found produce, the answer to which shows the amount at the end of the second month; and thus the calculation is continued for 10- other statements, the constant multiplier being 240-239 ths; and which may be expressed in one calculation ; thus, 240" Baan s= £105 2 9 nearly. £100 X Now as multiplication may be performed by the addition of the Logarithms of the numbers, and division by the subtraction, to perform this by Logarithms, we have to add to the Logarithm of 100, 12 times the Logarithm of 240, and to subtract from the amount 12 times the Logarithm of 239 ; or, what is the same thing, to add to the Logarithm of 100, 12 times the difference between the Logarithms of 240 and 239. As the tables belonging to this work are hardly sufliciently extensive for so great a multiplication, for the sake of greater precision we shall use Dr. Hutton’s Tables. Log of 100......-. 2,0000000 Log of 240... .2.3802112 30 5 30273783979 Ue aor ear l2 ——. O,Orlsoup 2.0217596 = 105.138 The product is therefore £105. 2. 9 LIL IIL IDLO PIIIL IOP _ 4 160 DISCOUNTS. Example 6. PRP PAIS LAL ALI AAD LAE LADLE To find the Rebate on £ 500, for 7, 13, and 16 months, at the rate of 8 percent per annum. 100 £ for 7 months at 8 percent is £ 42 1S months ss seine koe £L 83 16 mioutiis 31. sd ie eh By. £102 £ £ £ If 1042 produce 42% what will 500 produce? or 157 7 rf 157 ) 3500 Rebate for 7 months £22 5 10 / POP PIP LID POEL LOD LOE ALS 29 au 2g If 1082 produce 83 what will 500 produce: or 163 13 13 163 ) 6500 Rebate for 13 months £239 17 6 POO PLE LOL LADO LLE PEL LAG 25 ia £ If 1102 produce 10% what will 500 produce? or 83 8 38 83 ) 4000 Rebate for 16 months £48 3 10 The particular form of this calculation, is perhaps never to be met with in the practice of arithmetic.in this country, but it is the method upon which the allowance of rabais, or discount is calcu- jated in some places abroad, particularly at Hambro’, for which see ** Exchanges.” PEOPLE DIL ELL OIL OOF 161 AVERAGE TIMES OF PAYMENT. PII LOL ELI ODI ADD AOD BBL When several sums of money are due at different periods, and a time is found, at which the interest on the whole amount would be the same as the amount of the interest upon the different sums, for their respective times, it is called an equated or ayerage time of payment. DLL DDIL DI I LDF DOLLA DE AOE The principal use that is made of these calculations, is in the making up of Account-Sales, when the goods contained in the same account have been sold at different periods, and when interest is charged upon the amount of the advances. In which case, in- stead of making the calculations for the different periods of time, an average time of payment is usually found, up to which time the interest is charged. PLO PI OL IL AOL LAE LAE The performance of these calculations is much abbreviated, by using only the products of the principals multiplied by the number of months or days, instead of calculating the actual amount of the interest ; and therefore the usual directions are, Multiply each principal by the number of the months or the days which it has to run, and take the amount of these products ; then say, as the amount of the different given sums, is to the amount _ of the products, so is 1 month or 1 day, to the time required. PPL PDI DIL OL BIO LOE Vou. I. Xx 162° AVERAGE TIMES i Example 1. ¥ To find an average time of payment for the following sums, due at the following periods ;— £7,800 at 2 ‘months. £640 at 3 months. and £2 900 at 5 ‘months. BOO MR Gia st ree Le Sd Se ieee Loe 900. 2X). 30 ee aa oUe £ 2340 £ 8020 we month. fe, If 234,0 require 1 what will 802,0 require ? months. 234,0 ) 802,0 ( 3 mos. 13 da. Answer. 100 234 ) 3000 ( 13 days. nearly, 660 SPI LOL LOL LAD OLE ALE LOAF Average time of payment for the £2 2340, is 3 months 13 days. PIPL LID LOOP ODD DOE LOAD DAF PROOF. d The Interest on 800 for 2 months is 1600=£613 4 On “C20 OWE Soe ten Cie 1996 1S: 9 2G On, FOO? AE TET 20, 200 4500= 18 15 O of £33 8 4 The Interest on 2340 for 3 months is £29 5 O LOT GLO MUdyS. 2 Teun oot Amount asabove £33 8 4 — OF PAYMENT. 163 Example 2. wer a To find an average time for the following payments ;—viz. £876 10 O due the Ist May 840 16 O and 914 12 O 28th May, or 27 days from May 1. . 17th June, or 47 days from May 1. ES 876 L Se be hate eee ee Le oto. =, 470 = «65005 £ 2632 £ 65712 £ day a If 2632 require 1 what will 65712 require: days. 2632 ) 65712 ( 25 days nearly. 13072 May 1, with 25 days, makes May 26th, the time required. SPL IOL LID LOL DEL DEL LOD PROOF. \ ssn a The Interest on £ 841 for 27 days is..3 2 2 915 for 47 days is..5 17 10 B00 The interest on £ 2632 for 25 days, is £9 O 3, nearly the same as above. In the performance of such calculations, the time is usually reckoned from the day when the first payment becomes due. 164 ON THE STANDARDING OF GOUD AND SLOVER, DOF ODPL BBS LAI PAL OOP Gold and silver when completely refined, or separated from the other metals united with which they are usually found, are more difficult of management, and at the same time are less serviceable, than when they have certain degrees of mixture or alloy ; it has consequently always been found expedient to use them in this latter state, but, in this country, in order to.prevent too great a debase- ment in their value, the legislature has at different times inter- posed its authority, and fixed certain standards of purity below which no articles of gold and silver plate, with a few trifling exceptions, should be permitted to be exposed for sale. In former times, the fineness of gold was principally determined by what is called its touch, and which, where the value is not very considerable, is still practised; it is done in the following manner. Small bars, or needles of various known combinations of pure metal and alloy being provided, a part of the metal to be tried is rubbed ~ upon a stone, hence called a touch-stone, and portions of such of the barsasseem toapproach its quality, are rubbed off by the side, so as to make the quantity of each as nearly equal as possible ; some nitric acid, or aqua-fortis, is then poured over to pre- cipitate the baser metals, and from the residuum, or the pure gold remaining upon the stone, a judgment is formed of its previous purity. Combined with this, some opinion is founded upon its colour and general appearance, and which must principally be the guide in the touch of silver. As experiments of this nature must ne- cessarily be very uncertain in their results, a more accurate process is resorted to, for the general determinations of the quantity of alloy in the given mass; and this operation which is called an _ assay, is thus conducted. A small quantity of either metal, gene= ' STANDARDING OF GOLD, &c. 165 rally about 12 grains of silver, and 8 grains of gold, is submitted to the action of a powerful fire, by which the whole is rapidly melted, and the alloy becoming disengaged from the fine metal, is either volatilized, or absorbed in the bone cup, in which the subject of the assay has been placed ; the process then being stopped, the fine metal is weighed, and from the difference between its present and its former weight, a comparative estimate is formed of its standard purity. This is generally all that is required in the assaying of silver, but to separate the fine gold, it is necessary to submit it to what is styled the wet test, or to dissolve the silver with which it may be united, by the means of nitric acid; and both silver and gold may be more accurately assayed by the chemical tests, or precipi- tants, than they can be by means of the furnace, though, on ac- count of the expense being greater, it is not resorted to, unless the purpose of making the assay renders such precision requisite. PIE The following are a few of the most important particulars, con- nected with this subject. Gold and Silver without any alloy are termed fine or pure. Standard Gold is such as contains 22 parts of fine gold, with 2 parts of alloy. ‘These parts are termed carats, and are of no par- ticular weight. Eighteen-carat gold, or gold of the new standard, used only in the manufacture of watch cases, contains 18 parts of gold to 6 of alloy. Standard Silver is such as contains 222 parts of fine silver, with 18 parts of alloy; these parts are termed penny weights, and the standard is called of 11 0z 2 dwts. New Sterling Silver, but rarely used, contains 230 parts of fine silver, to 10 of alloy ; or it is of 11 oz 10 dwts. It was made the standard of this country in the year 1696, but the restriction to this fineness was taken off in the year 1720, when the old standard was restored. Gold and Silver partings are masses in which gold and silver are. intermixed in considerable quantities; if the gold is most, the mass is called gold parting ; otherwise, it is called silver parting ; but in these states, though in many parts of the world these metals are generally found combined, they are but seldom articles of merchandise. £e 166 STANDARDING OF The manner in which an assayer makes his report of the pure metal he finds, is to describe the metal assayed to be a certain quantity either better or worse than standard; which quantity is technically called its betterness or worseness. The portion that is weighed for assaying is called a pound, but it actually weighs only from 6 to 12 grains or upwards, differing perhaps with every assayer. The pound weight, for gold, is divided into 24 carats, and each carat into 4 grains; for silver, the pound is separated into 12 ounces each of 20 dwts; and the lowest weights used are quarter grains for gold, and a half dwt for silver. Instead of figures, assays are reported with the following letters j i ij ij Q Qi Qu Qij ie e Gf ecij eie je A ai, a: | 6 Pa ere hy, 8 Ee (8 ee Be Ro in precisely a similar manner of combination to the Roman Numerals. To these characters are to be added, the following abbreviations. — Ob, for obolus, fort a dwt, of Silver. B, for betterness; W, for worseness ; and S, for standard. PLD LPP ALD LOG BLL PDD LOLOL Thus a report of a Silver assay in this form, dwts B eQ, ob, means, that its betterness is 155 dwts, or, that in a pound of this mass, instead of the pure silver being 222 dwts, with 18 of alloy, which is the standard fineness, it is 154 dwts more, or 2374 dwts pure, with only 24 of alloy. PLE PLDI LD DD PPL AOD LOL So also a report of the quality of a Gold assay, in this form, car grs Wittig ' i732 means, that the worseness is 4 carats 23 grains, or that in one pound of this metal, instead of there being the standard quantity — of 22 carats pure, with 2 carats of alloy; there are here only 17 carats 1{ grains pure, with 6 carats 23 grains of alloy. ‘ POL FOL EVE POP LAE LPL LLL GOLD AND SILVER. 167 It should have been observed, that when either gold or silver is imported and is to be sold in a very impure state, as in grains or small pieces, it requires to be melted and refined, or the dross to be so far separated, as to bring the gold or silver nearly to standard purity. ‘In an unwrought state, these metals are called bullion ; after being refined, gold is generally cast in bars or flat pieces; and silver in thicker masses, with sloping sides, called ingots. When Gold and Silver are manufactured into articles of plate, before they are finished, and can be offered for sale, they are taken to one of the assay offices, appointed by different acts of Parlia- ment, to be proved that they are not inferior to the requisite standards, when the duties which have been imposed are also to be paid. The different public assay offices in these Kingdoms, are those under the control and management of the Goldsmiths’ Companies in the following places. London, Edinburgh, Dublin, Bir- mingham, Sheffield, York, Exeter, Bristol, Chester, Norwich, and Newcastle upon Tyne. Every manufacturer has his own die with his peculiar marks, initials, &c, with which his plate is stamped previous to its being sent to the office; where, by these marks having also been properly registered or stamped upon a sheet of brass, which is kept there, his plate can be well identified. With the article to be assayed the amount of the duty is sent, and a trifling sum for the expense of the process. If the article is found not to be of sufficient purity, it is kept for a short period, and then being broken up, it is delivered back to its owner with the duty that has been paid ; but this very rarely happens, as previous to the manufacture, the metal is assayed, and its proper purity determined ; if the contrary is the case, a part of the article is clipped or filed off, and is put into what is called the diet box. | CLI FELL ALOL IS LFA ALE DES 168 STANDARDING OF The following stamps are then impressed. DOLD LIL AIL PAD LOD AIL 1st.—A Lion, rampant, to show that the article is of standard fineness; — or, A Crown with the figures 18, to show it to be Gold of the purity, allowed for only watch cases, called new standard. The figure of a Woman, called Britannia, to show it is of that standard of silver called the new sterling ; — or POL LAL LIL AL EE BEEBE ALT 2nd,—A letter, to show the year when made, in the following order, as appears by the engraved table in the office of the Goldsmith’s Company. for 1796 A for 1801 F. 1... for sii 7B 2 G vo 2 -R 3 & Saul Ria 3 45 9 D AAA 9 O 4T 1800 E 5 K 1810 P 5 U for 1816 4a. for 1 82Toeh gor 1826 1: for 1831 q 7 »b 2 2 7 m 2 x 8 ¢ 3 h 8 on 3s 9 d 4 j 9 oO 4 1820 e 5 k 1830 p 5 u for 1836 4 for 1841 ¥. for 1846 W@. for1851 @Q An) 2 € 7 DP 2 R 3s ¢€ 3 8 MR 3 D 9 D 4 3 9 © 4 13840 € 5 1850 jp 5 & PILL OOO LO DIE LIF OAL LDS These letters are changed when the diets are proved. GOLD AND SILVER.: 169 ; ? 8rd. The Hall, mark, or the mark of the Assay Office, to show where the makers stamp is registered; and where the plate has been assayed, and the duty paid. For London—The Office stamp is what is called a Crown, but it is a Leopard’s Head Crowned ; the Leopard being the ancient supporter of the British Arms. Edinburg.... a Thistle. Dublin... a Harp. Sheffield .... a Crown. Birmingham.. an Anchor. Newcastle.... 3 Castles. And so for the other register assay offices, the arms, or pecu- liar distinctions of the respective Towns. A Beas ee 4th. The King’s Head, to show the duty has been paid. This stamp has been varied in some slight particulars, at the times when the variations have been made in the duties, by which the periods are ascertained for which the different drawbacks are allowed. PLL PLO L AE LOE LOD DAE LEP It has been mentioned, that after the assay has been made, a portion of the whole mass is put into diet boxes; this is done for the purpose of their contents being melted once a year, viz, in London on the 28th of May ; and, on the following day, assay is made of these amalgamations of the clippings, when a jury of the Goldsmith’s Company is sworn to make a true report of the ‘assays, to determine whether or not, upon the whole, the plate which has been passed by the Company, has been of sufficient purity according to the respective standards. The Chancellor of the Exchequer presides upon this occasion, and the record of the assays is deposited in that Court. The diet boxes from the country Offices, are annually sent to the Mint, and proved there. “A similar ceremony, called the trial of the Pix, takes place at the Mint, in proving the purity of the coins of the realm ; only, there, the Lord Chancellor, assisted by several members of the Privy Council, directs or superintends the proceedings. From the process of any official assay, as well from the payment of the duty, some small articles which might be damaged by the stamping, and which do not each weigh ten penny-weights, are. Vou. I. Z — 170 STANDARDING OF : ve’ < exempt ; thus for exantple, seals, being formed of several iene pieces afterwards soldered together, but of which each does not weigh more than ten penny-weights, pay no duty, are not required to be assayed and stamped, and can consequently be made of metal of any inferior quality. Small articles of plate are also usually not stamped with the Hall mark. The reports of silver which are made by the assayers of Goldsmith’s Hall, with those of Mr. Johnson, Maiden-lane, Wood-street, the Commercial Assay Master of London, differ from those of the Mint by 24 dwts; that is, the Mint reports state the exact quantity of fine silver which the mass contains, while theirs uniformly state it at 2} dwts, (more correctly it should be 2 dwts 6 grs), less than real quantity. As this circumstance fre- quently excites some surprise, and as many persons, particularly foreigners, sometimes entertain a suspicion of its being an unfair eee . practice, to correct this impression, we think it proper to enter into the following explanation. In the process of an assay, from some cause which has not yet been explained, it is found that a portion of the fine silver, to the above amount, enters into the copel or cup in which the assay is made, and, consequently, that there is this deficiency in the quan- tity which should remain. Thus, if a pound of silver, which by the chemical precipitants is ascertained to be pure, be subjected to the process of an assay, and afterwards weighed, there will be found only a very little more than 11 oz 171 dwts. Now it is the established practice of London assaying, to make the report from the exact quantity of fine silver found, without allowing for the deficiency which is known thus to exist, and therefore silver which is reported 2} dwts worse, “is understood by the trade to be standard silver, and as such when manufactured would pass the assay at the Hall. The manufacturers and dealers have been so long accustomed to this method, that, however desirable, no change could be made, without, for a length of time, giving rise to serious mistakes. It is to be further observed, that when silver is sold by any other than the Mint assay, the above deficiency is taken into consideration by all those who are aware of its existence ; and also that gold is not liable to any similar dimi- nution. ‘ ; ie GOLD AND SILVER. 171 THE FORMS OF GOLD AND SILVER PARTINGS. PPP LDIFS LILI III IIL LAI IE There are two forms in which assayers make their reports of ~ gold and silver partings, one of which is, to show how much better or worse the whole mass is than either the gold or silver stand- ards, and also how much there is of the opposite metal. Thus, a Silver Parting Ingot reported,— Worse, 4 0z; Fine Gold, 2 0z in the pound, means, that the whole mass contains only 7 0z 2 dwts of pure metal in each pound, out of which 2 oz are fine gold. So, also, a Gold Parting Ingot 4 reported,— Better, 1 carat; Fine Silver 10 dwts in the pound. means, that out of a pound of 24 carats, or 12 oz, there are 93 carats or 11 oz 10 dwts of fine metal, of which 10 dwts are fine silver. | PLAIIF LIL DI IDOE DIDI BIE The other form, which is much the better of the two, is to report the real quantity of each metal found. Thus the foregoing reports in this way would be, For the Silver Parting, Fine Silver ......... 50% 2 dwts. Fine Gold......... both eae and of course the remainder, 4 0% 18 dwts, would be alloy. For the Gold Parting. Fine Gold.v . ics a4 . 22 carats. Fine Silver........5. 10 dwts. And the alloy would be 10 dw?s. PPO LDOLD DE DDOIP IDE LIOILS isi > 3 172 STANDARDING OF Although the following observations, are in some respects a recapitulation of the preceding, they may serve to convey a clearer — idea of the nature of assay reports. The standard weight, which an assay master uses for the quan- — tity of metal to be tried, is called a pound ; and the smaller weights used in determining the quantity remaining from the tests, are proportioned to this pound, according to the divisions of carats, oz, &c, as before mentioned. The quantity of metal to be assayed being flattened for the sake of being easily cut, and being very accurately weighed to agree with the pound, the process of purifying it is completed, and the fine metal is weighed ; if the subject be fine silver, upon which as before stated, there would be a loss of 24 dwis, its actual _ weight would be 11 oz 174 dwts, but as the assay is required to be given accoring to a differential comparison with standard silver, or silver of 11 oz 2dwts fine, this would be reported as_ 154 dwts better; hence, in weighing the result of the assay, a proportionate standard pound weight of {1 oz 2 dwts is used, and if the beam, which in some scales will turn with the 1200 th part of a grain, should be suspended, the report is standard ; otherwise, according to whether the additional weight is required to be that info the pound scale, or into the scale with the silver, the report is So much better or worse. In asimilar manner, the weighing of an assay of gold is con- ducted. We are informed, that at the Mint and at the Hall when assay 4 is made of manufactured silver, the standard pound weight is di- minished by 24 dwts, to allow for the loss in absorption. PP PL LL PDL PLL PPI POP PDP The present Mint Standard Value of Crold i823) ouue £ 317 101 per oz, and of DUVEN. ese acm ee 66s per |b. PLE POL LL SI LPL POLI DIO LED GOLD AND SILVER. 173 The duties, per 0z, which have been paid upon the manufacture of plate, from the year 1784 to the present time, are Silver. Gold. From 1st Dec, 1784, to 5th July, 1797, - 6d «ates, “ thence ... to 10th Oct, 1804, 1s - andy Bs Me thence ... to 3lst Aug, 1815, 1s 3d te gat LGs. since that period ........... ls 6d Prana 4; Me When the duty is paid, as the article is in an unfinished state, and will be reduced in weight in completing, an allowance for this is made of one sixth of the whole weight, and the duty is paid upon the remainder. On the exportation of the article, if it has not been at all used, the whole of the duty is allowed as a drawback. Previous to the entry of plate for exportation, it formerly required to be stamped, which is now not done ; it is entered at the Custom House as other articles for drawback, (see Customs and Excise,) and on the debenture being procured from that place, instead of its being paid in the Long Room, it is taken to Goldsmith’s Hall, and the money is received the next day. The balance of the duties collected at the Hall, is paid to the Commissioners of Stamps, four times in the course of the year; and under their controul the stamps on plate are used. x The duty upon the importation of either gold or silver plate of foreign manufacture, is so heavy as almost to amount to a pro- hibition; but bullion and foreign coins may be imported duty free, and without the making of any entry. Upon exportation, no bounty is allowed, nor is any duty paid; and, up to the pre- sent time, no bullion could be entered at the Custom House for shipment, until oath or solemn affirmation was made by the owner, before the Lord Mayor and the Court of Aldermen, that the whole was foreign bullion, that no part of it had been molten from coin of the realm, or clippings thereof, or from plate which had been _manufactured in this kingdom; but from the recommendations of the Bank Committee, these restrictions are to be entirely removed. PPI BPD OPE LES FLERE E FEE 174 STANDARDING OF TO STANDARD GOLD. PAL ILL LIL LPP BALA ALA Directions—Say, as 22 carats are to the given report, so is the given quantity to the whole betterness or worseness, which is to be added or subtracted. Or, add or subtract the given report to or from 22 carats, and then say, as the sum or remainder is to 22 carats, so is the given quantity to the required quantity of Stand- ard Gold. , Or, take parts out of the given quantity, for the given report out of 22 carats, and the amount will be the betterness or worseness. PRP PLD ELL D POEL LOL PLP LOLOL Example. PPP PILD POL OL PLD LD LDPL AOL To find the quantity of Standard Gold, in an Ingot of Gold weighing 530 oz 8 dwts, at the different reports of 3. carats 23 grains, worse,—and 1 -carat 1} grain, better. oz dwt gz---- 530 8 -— at3car 2% gr, worse. 3. 48 4 8 o7 at 2 car. q. 24°°2 ‘A Wie. I 'car. aes iz 4 ae 2 gr. 3, eee 3 O 6 . zt =. 110 3 nn 88 18 O Worseness. oz 441 10 O Standard. oz dwts 2) 530 8 at 1 car 13 gr, better. 11) 265 4 -. 24 2 at 1 car, better. ea 0 Sn Nis 9 Pv10 6-2. gr 31 13 Betterness. oz 562 1 Standard. Pla Pl GOLD AND SILVER. 175 TO DETERMINE THE QUANTITY OF FINE GOLD. PEL LID LIL LAO BD E DOO OOD Directions—To or from the standard weight of 22 carats,add or subtract the given report, then say, as 24 carats is to the sum or remainder so is the given quantity to the quantity of Gold. Or, find the quantity of Standard Gold by the preceding direc- tions, and subtract 1-12 th to find the quantity of fine Gold. SLD IDL LOL AAD LOD LOE DOD Example. CLL LIL PDD LAL DOO DOO ODP To find how much fine gold and alloy in 1060 oz 16 dwts of gold reported, worse, 1 carat 14 grains. 4 gr $gr oz dwt If 192* produce 165 what will 1060 16 produce? Lor car 21216 dwts 176 standard — 22 165 11 thereport—= 1 14 gr. 165 fine gold. 192 ) 3500640 a dwts 18232 12 gr. * For 24 carats. Answer oz 911 12 12 fine gold. oz, 149" 3. 12 Alloy. (or ) oz dwt gr 2) 1060 16 O given weight. Li; }i 630 9. 8ano 3 48 4 9 nearly, for 1 carat. ytd Lael Se ee Leer. Cae Oe 1S” ert sc + gr 66 6 O Worseness. ages PONE LG O Standard Gold. 82 17 12 Standard Alloy. oz 911° 12 12 # £4Fine Gold. awe 176 STANDARDING OF TO FIND THE VALUE OF GOLD. ee A ee Directions.—The weight of the gold and the report being given, - find the quantity of standard gold, and find the amount at the given rate. GLI LOD LOL LICL AOL OOLAIS These directions apply only when the gold is sold by the standard ounce, which is frequently not the case when the quantity is small ; the price is then varied according to the report, reckon- ing each carat in the report at 4s. Example. POI LOL DEL ALI AFI PAF AL? To find the value of a bar of gold weighing 157 oz 12 dwts, reported at 2 carats 3 grains worse, and sold at £4 2 6 per oz standard. oz dwt zee--157 12 given weight. 19 14 worseness—out of 22 car. Oz 137 18 © standard. yt ieee Jones 4 i %....137:18 QO amountat £ 1 per oz. 4 RRM GON! dure bee £4 LS Tee ¥en ee he 25 *6.0 £ 568 16 9 value required. ~ By the usual method of valuing smaller quantities of gold, 11 s (for 2 car 3gr, or 11 qr grains) would be deducted from £4 2 6, making the rate £3 11 6, at which the 157 oz 12 dwt would come to £563 8 4. In the valuation of foreign gold coin, particularly Portuguese and French, whose purity is known, the price is fixed without its un- dergoing any assay or standarding. - GOLD AND SILVER. 177 TO DETERMINE THE VALUE OF GOLD PARTING. LIL ALD IDL DPD DDD - Pirections.—F ind the actual quantity of each of the fine metals, _and find their values at the given rates. PLE L LOAD A LAL LAA LAL DAT Example 4. To find the value of a gold parting ingot weighing 146 oz 18 dwts, reported 1 carat better, fine silver 3 oz 10 dwts ina pound ; at £4 8s per oz for fine gold, and 6s 4d per oz. for fine silver. carats. ) 22 standard. | 1 better. oe Carats 23 or 110z 10 dwts—fine metal in a pound. 30z 10 dwts—fine silver. Oe ee tite gold. Gold. Silver. » oz dwts oz dwts we---146 18 at 8 07. 1....146 18 at 3 oz 10dwts. Meee. (3S for 6 oz. 2.... 36 144 for 3 oz. 294 9 16 for 202. ; 6 24 for 10 dwts. Oz 97 18 16 fine gold Oz 42 17 fine silver. Lasse fc. a"! at £1 peroz..97 18 8 qeee 42 17 O 1 eee 391 14 8 4,,..10 14 3 at5s. — 49° ER 1... Meg 10 1k ——_—_—_—__——_ = 14, Site 4 Oe Value £430 18 1 Gold eee 13 11 4 Silver 5 13.~ Lhe £ 444 9 5 Whole value. Vou. I. Qa 178 STANDARDING OF TO STANDARD SILVER. Directions.—Say, as 222 dwts (for 11 oz 2 dwts) are to the re- port, so is the given quantity to the quantity of betterness or worseness ; which is accordingly to be added or subtracted. Or, Add or subtract the given report to or from 222 dwts, and say, as 222 dwtsare to the sum or remainder, so isthe given quantity to the quantity of standard silver. Example 5. PPP PIL PDIP PAE LODE DLE LOL To find the quantity of standard silver, in an ingot weighing 211 0z 18dwts, reported 124 dwts, worse. dwts dwts oz dwts If 222 produce 12: whatwill 211 18 produce? 19% 922* ) 2648 15 1I 19 worseness. Answer....0z 199 19 standard silver, Or, dwts 299 123 dwts oz dwts If 222 produce 2091 what will 211 18 produce ? idwt 444 419 4238 dwts. 419* 444* ) 1775722 dwts 3999 Answer..oz 199 19 standard Silver. * These multiplications and divisions require working at length. N. B.—It very seldom occurs that parts for the betterness, &c, can be taken out of the standard weight for silver, as it fre- quently can out of that for gold. | & , GOLD AND SILVER. 179 TO DETERMINE THE QUANTITY OF FINE SILVER. LIP PPD PIII L IID LOD Directions—To or from the standard weight of 222 dwts, add or subtract the given report ; then say, as 240 dwts are to the sum or remainder, so is the given quantity to the quantity of fine silver. Or, if the quantity of standard silver has been found, subtract | from that weight its 3-40ths. wor LOO LIL LIS IOS. Example 6. LPL LDL LOL LOL LAE FOLD DAE To find the quantity of fine silver contained in an ingot weighing 112 oz 11 dwts, reported 34 dwts better. dwts | dwts oz dwts If 240 produce 2251 what will 112 11 produce? 2251 dwts. 9254 20 ) 507600 240 ———_— 12) 25380 dwts 2115 Answer..oz 105 15 fine silver. | Oz 6 16 weight ofalloy. + Or, the quantity of standard silver in the above ingot, is found, by the directions in the last page, to be 114 oz 6 dwts. oz dwts 114 6. standard weight. 3 0 ) | 40 ) 342 18 811 subtracted. Oz 105 15 fine silver. 180 — STANDARDING OF TO FIND THE AMOUNT OF SILVER, AT A GIVEN RATE PER STANDARD OUNCE. PPO LIL AIL DAI LAE AIF LAE 4 Directions—Find the weight of standard silver, and then find its amount at the given rate. PRLS LDS LADD LOL LOL LIL DAL Example 7. PLP PALO LIL ALE ADO? LOI ALT To find the value of an ingot of silver, weighing 95 oz 10 dwts, reported 11 dwts worse, at 5s 103d per standard oz. dwts dwts oz dwts If 222 produce 11 what will 95 10 produce? iT 229 ) 1050 10 Answer oz 4 14 Worseness. OZ 90 16. Standard Silver. + 2 8 fl x - 90 16 O Value at £1 per oz. i. De eae eS OL Wy Wecato ee ap} a Cecgaens A151 82a ce Pa 10d 3 9 A £26 13 .5 Value required. N.B. Silver is sometimes bought or sold, in small quantities, by.the given weight; varying the rate of its value, from standard, by reckoning loz.10dwts 5dwts 4dwts 3dwts 2 dwts 1 or £ dwt at 6d 3d lid lid 1d $d id 4 and so accordingly increasing or decreasing the value per standard oz. Thus, the above would be rated at 31d less: and the value would be found of 95 oz 10 dwt, at 5s 73d per oz, which wom be £26 15 .2. GOLD AND SILVER. 18] TO DETERMINE THE VALUE OF SILVER PARTING. PEALE IID IAL ODS IIS Directions—Find the actual quantities of each of the fine metals, and find their value at the given prices. - Example 8. What is the value of a silver parting ingot, weighing 343 oz 10 dwts, reported 2 oz 4 dwts worse, fine gold 3 0z in alb; at 6s 6d per oz for fine silver, and £4 10 per oz for fine gold. oz dwts Standard 11 2 less ... 2 4 Worse 8 18 Fine Metal. 3 O Fine Gold. oz 5 ig Fine Silver. | oz dwts a pune EE aia (3 3 ozperlb, make, 85 174 for the fine Gold. 6ozperlb 171 15 2 dws 4... 2 17 (1-60thof 02171 15) oz 168 18. for the fine Silver. oz 254 151 of fine metal. TO FIND THE RESPECTIVE VALUES. LODO LOO LOL IF E LDS AIL Gold Silver A ae tS) Sad At £1 peroz 85 17 6 168 18 0 oak... 343 10 O ati4e:siess 15,7 e107. : 42 AS «9 Ss\G6dnn 2 2 3 2386.52 "9 Silver . 54 17 10 —— Gold 386 8 9 Whole value £ 441 6 7 i 182 : eee ABMIGATION, OIL PLL DIL LED ALPE OP LD DOO Alligation is the reversing of the methods of finding average prices, &c, or it is the determination of the quantities at given values, which are required to form one quantity at an average rate. PPL PPL LOL LOL PLDI LOD ODD Directions—The average and given rates being reduced, if neces- sary, to similar and simple quantities, link together the rates above and below the average: rate, find the difference between the numbers of each and that of the average rate, place them against the opposite rate, and they will show the numbers of the quan= tities required. Example 1. PLL ALE LOL AAO DDD LOD ODL \ The quantities being unlimited. PLL LILO LOL DDE DLE DOP LOL To find how many quarters of wheat, at the rates of 72,76, 82, and 84s are required to form an average rate of 80s. s s FD) sree ects ok 4 quarters at 72 value 288 76 wie an dets 7 EE ere, Siar ae {Ow ee 152 89 5 (gal) «salad in Spi piece: abr ahi 328 ) odie} Sat ere cua cetrr ae ESS 672 1S: “Quarters isan eae 1440 s or 80s per Quarter, ; mor.) s s IMT CMMI. fog oc 4 quarters at 72 value 288 80 76a eee is p Aas Bs aa a rs Ya ee aeati 456 82 oTete ease Bg LO £ Aon ee aa B Zia dua ee B4i fete. ae ey ALGAE 3 Sree cree!) ot Foes ye 1008 eee ee 14&12, or 26 Quarters 2080s oe or 80s per Quarter. i * ALLIGATION. 183 Example 2. PIS LIDIA LOD DA PID ALD OLD OO The amount of one of the quantities being given. SID IIL III IIS FADD IIE BBP To find the proportions of the three following Ingots of Silver, that will compose a mass of 4 dwts better, or 226 dwts fine ; viz No 1—Better, 18 dwts; No 2—Better, 12 dwts; No 3— Worse, 4 dwts, the quantity of No 2 being limited to 20o0z. No dwts mean dwts fine OZ Letter 18 2... or’... 240 xe SO te NOE 2 —Retter 12... 226°... 2s). gant S Noa See WV OT8G A oan a doe dnt tle oY, >. - 84514 Now 3 38 02. No 1 No 2 No 3 8 0Z : 8 0% 22 23 + or Quantities required 20 oz 20 o2 55 07 The fine dwts are found by adding or subtracting the reports, _ to or from the standard fineness of 222 carats ; (see page 179;) and the quantities are first found by linking No1 and 2 with No 3, proceeding according to the foregoing directions; then on comparing the required quantity of No 2, with that found at first, viz, 20 oz with 8 oz, and finding that it is 24 times, we use that - number asa multiplier to each of the quantities found by the alligation, and thus we obtain the required amounts. PLI LOE LIP LIOLEL CEL III SO Oe ee - 184 - ALLIGATION. Example 3. PPP LPI AID APIA LD LAT THE WHOLE QUANTITY BEING GIVEN. PLD DAI LILI IAS PDP PDD To find the quantities of each ingredient of the following values; viz, of 8s, 10s, 13s, and 19s per lb, which will be required to make 1 cwt, or 1121b, at the average rate of — 12s per lb. $ $ 8 72ND at 8 .. value 56 es >) 1 ge en ee 10 Fe ee 10 13 SD a Ste A eles oi AD Petey cee 26 19 ae ar By ep e ty 76 I: Toy. od ear 3% tata v0 to otetetd o Bie eG 168 s lb ib lb Ib 7 1 2 4 8 8 8 Products 561b $Ib 161b 32 1b, inall112Ib. ea ie Py li ( or ) i lb Ss s ‘ 8 L KIT) ws 008-65 B00) Btole 8 velne-64 10) 1 ESD HEL Sie, TO Le Dor ae ey ops LO) ini 10 en 9 iP PP nh ge B ie aap s i ono whee 19 4 Re RC TOE Fae 19 76 HG GHD oa D Tht 298 5 Ib Ib Ib Ib 8 1 6 4 112 112 112 112 % 19 19 19 To Products 47,3; 1b 5iglb 63575 1b 23445 Ib, inall 112 Ib. ages om ee ~ 185 OF SQUARING AND CUBING THE, CONTENTS OF SURFACES AND SOMOS. LFF LD I LILI LIL DID IDE AIS The general principles upon which the measures of surfaces and. solids are calculated, are, that The similarly situated lines, or boundaries of all similar surfaces, are to one another in the ratios of the numbers expressing their lengths ; that, The contents of all similar surfaces, are to one another in the relation of the squares of the numbers, expressing the lengths of similarly situated lines ; and that The contents of all similar solids, are to one another in the relation of the cubes of the numbers, expressing the measures of the similar lengths. Thus, in circles, the boundaries or circumferences are to one another as the diameters, and, therefore, if the diameter of one circle be double that of another, the length of the circumference of the former, will be double that of the latter. Again, with the same circles, the surfaces enclosed by the cir- cumferences being to another as the squares of the numbers ex- pressing either the diameters, or the circumferences, the surface of the former will be to that of the latter, as the square of 2 to the square of 1; oras 4 to 1; that is, the contents of a circle whose diameter is twice as large, will be 4 times as great. _ Also supposing two spheres or globes to have the diameter of one double that of the other, then the solid contents of the one will be to those of the other, as the cube of 2 to the cube of 1, or as 8 to 1; that is, the contents of the one globe, will be 8 times the contents of the other. Asa general explanation of the reason of these relations, it may be observed, that with lines there is but one dimension to be taken into consideration, viz, that of length ; with surfaces, there are two dimensions, viz, length and breadth ; and with solids, there are three, viz, length, breadth, and depth or thickness. Vou. I 2B awe 186 OF SQUARING ' Hence, also, many surfaces are to each other, in the compound relations of their lengths and breadths; and many solids, in the compound relations of their lengths, breadths, and depths. SID LIIS LOL LOL DG LDP IOS The contents of surfaces and solids are usually estimated in square and cubic feet; occasionally, a square yard containing 9 square feet, the builder’s square of 100 square feet, and the solid yard of 27 solid feet, are also employed. In determining the contents, either of surfaces or solids, the- foot is divided either into quarters and eighths, or else decimally or duodecimally ; in the latter of which cases, the first division of the foot is called inches, the second, parts; but this does not coincide with the definitions of square and solid inches, as the former are the 144th parts of a square foot, and the latter only 1728 th parts of a solid foot. PIAL IAAP PBLID DLE LLL DAP | eas od Nea GOL LLP ALA AED AOD DOF OOF OF SURFACES. The contents of rectangular surfaces are correctly determined upon the following principles. ISL AAA APA LOL DL LOL OL A surface uniformly 1 foot broad, contains as many square feet. and paits (or as they are termed inches) as there are feet and inches in the length; therefore, if the given breadth be any other than 1 foot, the contents will be as many times and 12 ths, of the above contents, as there are feet and inches in the breadth. PLD LLL ELE OBL LB 4 LOE OD AS N.B. Rectangular surfaces and solids are those whose boun- dagies are perpendicular to one another. ‘ “ PLED IL PLL OPE LOO LPO LLL For the general mode of expressing the directions, see the explanation to the following example. THE CONTENTS. 187 Example 1. PEF IS LILO FILE LIL LEP FE To find the square contents of a surface 17 feet 7 in long, and uniformly 4 feet 8 in wide; and to find the amount at 1s 7d per square foot. sqf in 11 7 contents, 1 foot broad. 4 8 AG ANT oe test ne teete Paes Re Sq feet...-54 0 8 contents required. $ oa 1 7 amount of 1 foot. ee hci ss a an 6 feet rN te see ere 54 feet. 1 for 1 in, or 8-12 ths. £4 #5 7 amount required. or, Cae foe 5 54 O 8 amountat 1s per foot DR hai 2 Po) a tan he itolgete «is 6 d. BU Gere Oia t ae 1d. Seo eae whole amount. Calculations similar to the above are commonly expressed by saying, we multiply the length by the breadth, or the measure of one line by the measure of another, a.thing truly impossible. A line multiplied can produce nothing but a line ; and it is only from one surface, that by any multiplication, another surface can be pro- duced ; besides, the multiplier, in all cases, shows only how many times or parts the given thing or measure is to be taken, and we eamnot with any propriety say, we take 4 feet 8 inches of times. | if. “OFSSQVARING Example 2. To find the square Contents, and the amount at 3id per yard, of a rectangular Surface 283 yards long, and .742 yard wide. sq- yd. a °.... .742 contents 1 yard long. Yards 21.332 whole Contents. d i .... 21.3 amount at Id per yard 3 SV 63008 doe ens a 10.6 ri Cn eA ae re * d 744 = s 6 24, amount required. This calculation is one of a similar nature to those which are made, in estimating the amount of the duties of excise upon printed goods. The width is taken in decimal parts of a yard, and in the above it is used in estimating the contents, instead of beginning with a surface deduced from the length. The general widths of printed calicoes, are, for what are termed yard wide, 67-100 ths, or nearly 2-3rds of a yard ; and for what are termed ell wide, 95-100 ths, or 19-20ths of a yard. For the — former, the lengths are generally 28 yards, and for the latter, 24 yards in each piece. THE CONTENTS. 189 To find the contents of a Trapezoid, or a surface two of whose opposite sides are parallel. PILI LID IIL OPP DFO DOO Add the lengths of the opposite parallel sides together, and take half the amount, for a medium base of an equal rectangle. LIP IID PDIP PLEO DLP PDO Example 3. PIL III DAL LOD LD DDD DOD To find the contents of a surface, having two parallel sides measuring 64 feet 10 inches, and 54 feet 4 inches; the perpen- dicular distance between them being 19 feet 10 inches. {." in. 64 10 54 4 2)119 2 feet 59 Squi.o ints t.... 59 7 contents 1 foot broad. 20 . 1191 > eh ELE 20 feet. ea Neneh Ne i iT Soa feet, (LIST ate? cia. x 19 feet 10 inches. In this calculation, the assumed contents for 1 foot broad, are expressed separately from the same numbers denoting the length of the medium base; for though the numbers are the same, they are used in expressing very different quantities; one being the measurement of length, and the other of surface; but in general for practical purposes, it is no ways necessary to do so. PDAPLDPDPDLIP PIP OLE ODD 190 OF SQUARING TO FIND THE CONTENTS OF A TRIANGULAR SURFACE. PLE BOL LAD DBD LDL LOE PPL Upon one of the sides, produced if necessary, let fall a perpen= dicular from the opposite angle, then estimate the contents of the’ triangle, as being half those of a rectangle of the same base and altitude. 4 PRPIPD LI LIAL PDL LOD LIE Example 4. t PROP OL LAL DL I LILIA | To find the contents of aright angled triangle, whose base is is. 10 feet 4 inches, and perpendicular 12 feet 6 inches. ) sq. f. in. 10 4 12 6 124 0O Dow Square feet 129 2 contents of the Rectangle. 640 FO servants 4s Triangle. N.B. When one side of a triangle is perpendicular, or at right angles, to the other side, the length of the third side, which is called the Hypotheneuse, may be thus found; reduce the meas sures of the two sides, if necessary, into similar simple quantities, which may be done, either by common reduction, or by expressing” any lower denomination, as decimals. Then square the numbers thus expressing the lengths of these sides, add the products toge= ther, and extract the square root of the amount, which root will be the length, in the above denomination, of the side required. Thus above; the baseis 124 inches. Perpendicular 150 inches. 1947 == bado Pot = CO BLA \ The square root of 37876 is : ‘ The length of the 3rd side. Inches 194.61 For the method of extracting the square root, see the introm duction to the Tables of Logarithms. THE CONTENTS. 191 TO FIND THE CONTENTS OF AN IRREGULAR, FOUR-SIDED FIGURE. ' LIP LII LOPLI ODD LD BDL OO Measure the distance in a straight line across from one angle to the opposite angle, and measure the lengths of perpendiculars let fall from the remaining angles upon this base line: the figure will be thus divided into two triangles, whose contents may be found together, as being half those of a rectangle of an equal base, and of the sum of the two altitudes. wer LPL LAD PAID Example 5. PID LID DID DLL DPD ADD ADT To find the contents of an irregular field, the length of a dia- gonal from one corner to the opposite corner, being 15 chains 60 links, and the length of the perpendiculars being 10 chains 65 links, and 6 chains 15 links. Chains 10 65 links is get 16 80 Sun Chains 8 40 half Sum. Sq chains 15.60 Contents, 1 Chain broad. 8.40 131,0400 Do 8 Chains 40 Links broad. Acres 13,104 Was wT! als Roods 0,416 13. 0. 16 Contents reqd. eee ae a ee ee Poles 16,640 — Land is generally measured with a Chain, decimally separated into 100 links; hence we multiply the square chains and parts by 8.40, as in decimal multiplication. In estimating the contents . in acres, &c, we have to observe, that, 10 Square Chains make 1 Acre.* 1 Acre contains .... 4 Roods. 1 Rood .......... 40 Square Poles. * An Acre of Land is 10 Chains long and 1 broad, or any other dimensions, the product of whose numbers makes 10, as 5 by 2, 4 by 21, &c. A chain is 22 yards long ; 80 chains make 1 mile. 192 OF SQUARING OF CIRCULAR SURFACES. IIS IIL LID PBS it having been ascertained that the dimensions of a circle whose. diameter is 1 foot, 1 yard, &c, are nearly the decimal .7854 or & 7854-10000 ths of a square foot, or square yard, &c, or, that if the circumference is 1 foot, &c, the area or surface is .07958 or 7958-10000 ths of a square foot ; and as all similar surfaces are to one another as the squares of the numbers expressing the similar lines; we have, hence, these two methods of finding the con- tents of a circle. Ist, To. multiply .7854 square feet, &c, by the square of the number of the feet, &c, in the diameter. 2nd, To multiply .07958 square feet, &c, by the square of the number of the feet, &c, inthe circumference. PID LPL LID DLL DDL ODD The area of a circle may also be found by assuming the diameter as somany square feet, &c, and multiplying them by half the number of feet, &c, in the circumference. The contents of an oval or an ellipse may be found, by multi. plying .7854 square feet, &c, successively by the numbers of the feet, &c, in each of the two diameters. POD LIL PAL LDL ODO LED It is to be remarked, that a circle is the space or surface en- closed by a line which is properly called its circumference, but — which itself is usually called a circle. PLL LIL LEE ELE POLLED Example 6.—To find the area or superficial contents of a circle — whose circumference measures 12 feet. .07958 12 . 95496 12 Square Contents 11,45952° or 114 Square feet, nearly. We here multiply the contents of a circle 1 foot in circum- ference, by 12 times 12 times, as the square of the number of the feet in the given circumference. f be 4 5 ‘s ‘sy > , 1 AND CUBING. 193 PSH T,IT. OF sOLIDs. PPP LPL LDL LD I LAI LAE IEE THE CONTENTS OF RECTANGULAR SOLIDS ARE THUS CORRECTLY CALCULATED. SPP DDD FIL DDL DAD DDD OAS If asolid be uniformly 1 foot broad, and 1 foot deep, there will be as many cubic feet and parts, in the contents, as there are feet and inches in the length; and therefore, if the given breadth and depth be any other than 1 foot, the contents will be so many times? and 12 ths, of the above, as there are feet and inches, first in the width, and thenin the depth. PLE DOF LLL LODO DIO DEL Example 1. PLI LOO LDF LOVOBDL DO LI AIF To find the cubic contents of a rectangular solid, 8 feet 4 inches long, 2 feet 6 inches wide, and 3 feet 4 inches deep; and to find the capacity of an equal hollow body, in ale gallons of 282 cubic inches each. cu. ft. in. ... 8 4 Contents 1 ft wide and 1 ft deep. Zee G RIK 16 8 for 2 feet. 4 2... 6 inches. 1,,.,20 .10 ...2 feet 6 inches wide. 62 6... for 3 feet. 6 fie'4> 2.2 4 mches. ° Solid feet 69 5 4... 3 feet 4 in depth. 12 $33 12 10000 12 120000 Solid Inches. Vou. I. ac 194 OF CUBING Example 1.—Continued. in Gall : in If 282 produce 1 what will 120000 _ produce? Gall Gall 282 ) 120000 ( 4255 Answer. 720 1570 170 Instead of calculating the cubic inches, as in the last page, they might have been found in the following manner. ft.) An: Cis 8 4 = 100 Contents for 1 in wide and 1 in deep. . 30 3000 .... 30 inches wide. 40 Solid-Inches 120000 .... 40 inches deep. And as 1 cubic foot contains 6.1276 ale gallons, and 1 cubic inch .003546 of a gallon, instead of finding the contents by the above method, we may produce them from either of these cubic contents, in the following ways. A. Gall. A. Gall. 4.. 6.1276 in 1 foot -003546 in 1 inch. 69 120000 55.1484 Gall 425.520000 367 .656 rare ¥.- 2.0425 for 4 in. ee 5106 > 9d an. 1702 .. 4 parts. Gall 425.5277 or 425! Gallons. PLD POP LOD LOOP PIE LLL THE CONTENTS. 195 TO FIND THE SOLID CONTENTS OF A CYLINDER. LILI LDL LID LOD AOE LOD The solid contents of a cylinder whose height is 1 foot, being as many cubic feet, &c, as there are superficial feet, &c, in the circular base, Find the superficial contents of the circular base, assume these feet, &c, as so many solid feet, &c, and take as many times and parts, as there are feet, &c, in the height. Or, consider the decimal .7854 as parts of a cubic foot, &c, multiply it by the square of the number of the feet, &c, in the diameter of the base, and multiply this product by that of the feet, &c, in the height or depth. OLLI L OL LE LD AOD ADD POD ODP Example 2. wer To find the solid contents of a cylinder, the diameter of whose base is 3 feet, and whose height is 7 feet 4 inches. 7854 Square Feet. 9 = 3 Squared. Square Feet 7.0686 Area of the Base. Cubic Feet. Solid Contents 7.0686 for 1 foot high. iC 49.4802 ... 7 feet. 3.3562 ... 4 inches. Solid Contents 51.8364 (or) Cubic Feet. ' .7854 6 4.7124 11 51.8364 In the latter calculation, we multiply by 66, as the product of the square of 3,or 9 by 7 and 1«3rd, 196 OF CUBING Example 3. PRP LOL LLP LDOL LL LDL LAP To find the contents in Ale and also in Wine Gallons, of an _ehiptical copper, whose base measures 27 inches by 22 inches, and the depth of whose perpendicular sides is 18 inches. Cubic inches. .7854 oa 5,4978 15,708 21,2058 22 42,4116 414,116 Cubic inches 466,5276 Contents 1 inch deep. 18 2 3732,2208 4665,276 Cubicinches $397,4968 Contents 18 inches deep. Gall - 231 ) 8397 ( 363 Wine Gall 1467 ae) Gall 982 ) $397 ( 293 Ale Gallons 2757 ~219 SEPLOD LOD PEO D ED DIL AIF . THE CONTENTS. 197 THE PRECEDING EXAMPLE REPEATED. PPP PPP LALLA LIS LILI LIS The contents of a cylinder 1 inch in diameter, having beeu found to be .002785 of an Ale Gallon. .003399 of a Wine Gallon. we may from these determine the contents of this solid from its dimensions of length, breadth, and depth,—thus, Ale Gall Wine Gall -002785 by 2. .003399 by 297 3 S -008355 .010197 9 9 .075195 by 22 .091773 by 22 3° be . 150390 . 183546 11 11 1.654290 by 18 2.019006 by 18 a v. 3.308580 4.038012 9 9 Ale Gall 29.777220 Wine Gall 36.342108 These contents agree very nearly with those produced by the preceding method. | PEODIF IEE DOD DIG GES 198 OF CUBING TO FIND THE CUBIC CONTENTS OF EITHER A PYRAMID OR A CONE, | | |) Find the contents of either a rectangular Solid, or a Cylinder | having the same dimensions, one-third of which will be contents of — the Pyramid or of the Cone. Ya Example 4. A ee | | To find the Solid Contents of a Cone, the Diameter of whose Base is 4 feet, and whose Altitude is 6 feet. cub ft Contents .7854 1 foot diameter, and 1 foot high. fs ip at Bias 16 12.5664 do A feet do 6 ; Cubic ft 75.3984 Contents of the Cylinder. Do 25.1328 Do of the Cone. LOD PLD Example 5. To find: the Solid Contents of a Pyramid whose base and alti- tude are each 100 yards. cub. yds 100 X 100 = 10000 Contents 1 yard high. 100 3 ) 1000000 Contents of the rectangular Solid. Cubic yards, 3333335 Do of the Pyramid. THE CONTENTS. 199 TO FIND THE CUBIC CONTENTS OF A SPHERE OR GLOBE. SAP LPL LAI LPP PPD DOD Take two thirds of the contents of a cylinder, of equal dia- meter and altitude. — or, Multiply .5236—10000 ths of a cubic foot or inch, by the cube of the number expressing the axis, in the same dimensions. PAS LAD AA DL DL ODE LLG LIL Example. 6. What are the contents in cubic inches, of a sphere whose axis or diameter is 21 inches, and what is the difference be-« tween these and the contents of a Globe whose diameter is 18 inches. 21% 21x 21 = 9261 is? |= 5832 . 5236 . 5236 5236 34992 31416 | 17496 10472 11664 47124 29160 Cubic inches 4849,0596 Cubic in. 3053,6352 Do in 18 in. Globe 30534 Difference 17954 If only the difference had been required, the decimal . 5236 might have been multiplied by the difference of the Cubes. The surface of a Globe or its superficial contents may be found, by multiplying 3.1416 square inches, &c, by the square of the number of the inches, &c, in the diameter.—Thus to find the contents of the surface of a Globe, 18 inches in diameter, Sq. inches 3.1416 X 18 X 18 == 1017.8784 Square inches. OF CASK GAUGING. PPP LIS PAS OLO LAE LOL LEDS To determine the contents of a Cask, the following measures are | taken ; — the length, the head diameter, and the bung diameter. From the two latter, a mean diameter is estimated, by which the contents of the Cask are reduced to those of a cylinder of this diameter. aM. f | As this is a subject of much too considerable a degree of intri- cacy, to be fully developed in this work, we must content our- selves with laying down the principal of the regulations, by which — those who are concerned in Cask Guaging, are generally guided. 1 st—With respect to the length—This is taken outside with an instrument called the long callipers, from which there is an allow- — ance of 2 inches, or more, for the thickness of the two heads, with various deductions to reduce the dimensions of the cask, to a form * which is called spheroidica]. The proper deductions are generally estimated by inspection, but in some measure they are common to the particular use of the Cask; thus on Hhds there is seldom any deduction, while on some Pipes it amounts to as much as two. inches. 2 nd—With respect tothe head diameter, it is taken externally, in general without any deductions, from the interior or where the head joins the staves, to the middle of the thickness of the opposite projecting stave, or what is termed the chimb. Very commonly, the dimensions of each head are taken, and the medium is reckoned. - 3rd—The bung diameter is taken, first, externally, with the. cross Callipers, and then internally with the bung or diagonal Rod, from which two dimensions a mean is estimated: a deduction or — additition is sometimes required, according as there may be any accidental protuberance or depression in the staves.—This mean — bung diameter being determined, the sliding brass stop in the Rod — is fixed against it, the Rod is again dipped into the Cask, and the — wet inches are thus determined. OF CASK GAUGING. 20} _ Hence, there are these dimensions in inches and tenths, chalked on the head of the Cask, viz. 1 st’ — The Length as ......... . inches 45.3 @nd — The Head diameter ........... 22. 26.— $rd— The Bung diameter .............. 30.8 ath <—The Wet Inches’... ..'....: Weck ree See With these are chalked, first the full contents, and then the ullage or actual contents, which are thus found by the sliding rule used in estimating the head diameter. The brass upon the slide, being set against the head diameter on the lower line or the line of inches on the stock, against the bung diameter on the stock, a number is to be sought on the second line, reckoning upwards, of the slide. Thus, with the preceding dimensions, the brass on the slide being set against 26 inches on the lower line of the stock, against 30.8 on the same line is 3.3 onthe second line of the slide ;—then against the same number 3.3, but on the lower line of the slide, is 29.3 inches on the stock for the mean diameter. These upper and lower lines on the slide are marked Spheroid Inches. With this mean diameter and the length, the contents are thus found. Set the brass end, or the gauge point on the slide, against the length on the top line of the stock; as here it is set against 45.3, i.e. half way between 2 and 4, and then against the mean diameter 29.3 on the top of the slide, is 132 on the top of the stock, for the contents in gallons which the cask would hold if full. CLD LPL LIF ALD LED ALT EDD The actual contents are then determined from the full contents and the wet inches, by the sliding rule, thus ; On the back. of the rule, the bung diameter 30.8 on the lower part of the slide, is set against 100 on the line marked Seg : Ly: for Segments Lying ; and against the wet inches 29.4, on the same part of the slide, is a proportionate number 99.—Then setting 99 on the upper part of the slide, to 100 on the upper part of the stock, against 132 on the same upper part of the stock, are the actual or ullage contents, on the upper part of the slide ; here they are rather above 130, at which they are reckoned, as the fractional parts of a gallon are never taken into this account. Vou. I. Qn ‘? 202 CASK GAUGING. When the difference between the wet inches and the bung | diameter, or what are termed the dry inches, does not exceed those in the following table, the actual or ullage contents are found without working themin the preceding manner. Otherwise, when Pipes exceed 6 gallons, and Hhds 4 gallons, on ullage, it is better» to cast the ullage quantity by the head rule, as a trifling difference ‘ in size, will then materially alter the calculation. SLL IAL LAD LDPE LDL POL LOE TABLES TO CALCULATE THE ULLAGE QUANTITY OF PIPES OF WINE, Ke, BY THE DRY INCHES, PLL IPL AIPA OPE BOP EAL AAD Find the difference between the wet inches, and the bung diameter and allow, / On Pipes On Butts In 10ths’ Gall In 10ths. Gall under . 1 ete A, wate 1 ¥ UU GY snk ee ee 1 Do as tOue das eres 2 EE: NOR HRP RS = Beaten os tA orld e Sagan a yea” AS BSD is 3 MS cts, eis oe. pee 3 NPA FEO. 2 ie cet Oe gM Lobia di ety “ic lo bells, 4 DO Pe LOs 6 2) OG inte bag 5 re hes Py TEU W igkad ed 5 eT LO ea ee ce ee ee 6 3, kL eehGurg ee See 6 Hhds under 68 gallons Claret Hhds In 10ths = Gall In 10ths Gall UNC Gre res eee 1 BNUGr. 6. phn Deen, 1 i hin ay Paley Dini Babb cod vm 2 Li” (6.5. kn a eee ay ee ee z. PAT Ry Ca go tos Bp 3 2 o> hoot votes ee a Agel ls va Wate SD 4 aati Big 4 Din. (0s, say. oe 4 To calculate the contents of a Cask from the length, with the head and bung diameters. Find the difference between the head diameter and the diameter, and multiply this difference, TO FI LRI DAL LDL LOG LIF ND THE MEAN DIAMETER. bung if not above 4 inches, by .68 6 inches, .. .69 if above ... Fihisis stant JF @dd the product to the head diameter, and the amount will be the mean diameter. Se CASK GAUGING. 203 TO FIND THE WHOLE CONTENTS. VIS IIS IDOLOS DDD DDO ODO Square the number -expressing the mean diameter, multiply it by the number of inches in the length, and divide the product by 294.12 for the contentsin Wine Gallons, or by 309.05 for Ale Gallons. SIL LIS IIL DIL OOD DDD ODO Example. LIS IIS III DDI ODS DDD COD From the preceding dimensions of a pipe, viz, length 45.3, head diameter 26, bung 30.8, and wet inches 29.4, to find the whole and the ullage contents in Wine Gallons. 30.8 Bung. 26. Head. A.8 Difference. ~69 432 288 orale od a Head. 29:3 Mean Diameter 29.3 29.3 879 2637 586 858,49 45.3 257547 429245 343396 29412 ) 3888959,7 ( 132 Gallons, full Contents. 94775 - 65399 \ 204 CASK GAUGING. The preceding «ample continued. PLS PPD ABE LAL DAD DAD LAE TO FIND THE ULLAGE CONTENTS. PPP ILD IID LID DDL POD ODF From the foregoing table—the deficiency corresponding 1D. sin UD less... 29.4 Or Sto... Li oe > eee Os Full contents ..132 Ullage ‘do.......130 It is to be remarked, that when the contents do not exceed 30 gallons, they are usually taken with the diagonal Rod; but when above this quantity, no dependence can be placed upon the © contents thus found; fora cask measuring 100 gallons by the rod, ~ has been found not capable of containing 70 gallons. wee POD AFL ODD LDA LDA DLE The following dimensions taken from Mr. Smyth’s Practice, will serve to determine the accuracy of the foregoing statements. Length. Head Bung. Wetin. Con. Ull con. — Brandy ..46.2 26.3 30.6 27.9 134 129 46.1 26 .6 31.6 26.7 141 129 3 Oils eae 45.4 30 35.3 32.3 176 169 45.- 30.- ie 32.4 172 167 Rum, sie 35.8 a hag 32.9 yh Py f 119 111 31.7 28.3 31.9 28.— 112 105 GPLLELLEOLEL ELE LOE ASG 205 OF SPECIFIC GRAVITY. PEI AIF FLAI LDA ADA OE LA AFT The specific gravities of bodies whether Liquid or Solid, are the yelations which the weights of equal magnitudes bear to each other, or to some common standard of comparison. In this country, from the circumstance of a cubic foot of _yain water weighing exactly 1000 oz, avoirdupois, at a ‘temperature of 60 degrees of Fahrenheit’s Thermometer, the other substances and liquids are generally compared with it, and their specific and absolute gravities become thus identified. In the determination of the specific gravities, if the objects are of such dimensions that their cubic contents can be exactly found, then, the relation which their contents bear to a cubic foot, is as their absolute weight to their specific gravity ; but as these dimen- sions can seldom be so perfectly ascertained, the more general method of determining the specific gravity, is to weigh the body if solid, in water, or in some liquid whose specific gravity is known, and also to weigh it out of the liquid, and so to attain the required purpose, by a comparison of these weights ; on the other hand if the subject is liquid, to weigh some solid in it whose specific gravity is known, and thus determine it by a method similar to the preceding. The apparatus generally used for making experiments of this nature, is an Hydrostatic balance with a weighing bottle ; but for finding the specific gravity of fluids, and small pieces of many solids, the instrument called a Hydrometer is more commonly employed. Of this we shall have occasion to speak more particu- larly, in treating of the method in which the duties of Excise are payable upon spirits, according to their strengths, or their specific gravities ; and similar to much of the matter immediately preceding, we shall here give examples of only some of the most useful calculations relative to this subject* yf . 206 SPECIFIC GRAVITY. ° Of the following table, it is to be observed, that the specific gravities are given of ouly a few of the most common objects, and that in all subjects which are capable of been expanded or con- tracted at the ordinary degrees of heat and cold, considerable variations take place according to their temperature at the time of making the experiment; in this table, the different woods are supposed to be taken dry, and the liquids at the temperature of 60 degrees of Fahrenheit’s Thermometer. Of all these it is to be- understood, thus the tabular gravity is only the mean weight of many | experiments; for, in some slight degree, every substance and : liquid, though similar in its nature, differs in its gravity. A TABLE OF SPECIFIC GRAVITLES. LILI DIILPILV LOE ODD OOF Platina fi ge id a 23.000 Logwood ......... .913 Gold—Pure ....... 19.640 Ash 3 7: MLR . 800 — Standard... 18.888 Bolin 3) 0 aout aes .600 Quicksilver........ 14.000 Birh, Sb A cm ge 0 . 550 Lead sey saliva. 11.340 Cork? iiuh SEAM A 238 Silver—Pure, ..... 11.092 Sea Water......... 1.030 — Standard... 10.536 Ale ty sae F) Aybar oy, 1.028 Copper £2415. 22).06 9.000 Vinegar ore? 2.44, 1.026 Brake da oo A8 ue 8.400 Lar .4t ASE Ae 1.015 Bardron?.. 520 ¢es 2 7.644 Rain Water ....... 1.000 CastT ror eee £40 17 OF Primage, 6d percask.... —- 5 O@ PiGIage.... css eee - 3 OF | £41 6 Contribution for general average loss.— 10 Hhds of Sugar valued per policy £300 ll 2 oe Average on £300 at £3 14 2 per cent. : £52 8 om Respecting this charge as a contribution towards a general average loss, see the department ‘* Marine Insurances.”—The trouble of collecting the proportionate amounts, belonging to ship, freight, and cargo, and reimbursing the parties owning the articles iro dae usually devolves upon the ship’s broker. VIS LIL DLO ODA DOD ODE 230 ON THE DUTIES OF CUSTOMS AND EXCISE. PILI IL LOL AAD O FE LAE ROD The Duties of Customs and Excise are contributions laid upon goods, towards defraying the expenses of the state. The Duties of Customs are payable upon the importation and exportation, and those of Excise generally upon the manufacture of particular articles in this country, though in a few instances they have hitherto been additional to the duties of customs at importation. The business of the Custom House, and of the Excise Office, with respect to importations and exportations, is generally ma- naged by brokers and agents; and in consequence of the number of the checks, which are adopted to prevent any fraud upon the revenue, the business in itself is frequently so intricate, and re- quires so much personal acquaintance with the principal officers, that scarcely any detail, however minute, would serve as a sufli- cient guide to an inexperienced person. To give some general idea of the most usual routines of this business, we shall here sub- join a description of the usual process of importations and ex- portations. PLP POF LLII LPO LAO IES OF IMPORTATIONS. PPO LLI LIL AIF FB IGO IE When a vessel arrives in any port in these kingdoms, say the port of London, possession is taken of her by an inferior officer of the Customs, called a Tide Waiter, in whose charge she remains until she has landed the whole of her cargo. The principal duty of this officer is to seize any prohibited goods, and goods not contained in the manifest, &c, and to prevent any article being taken from the vessel without sufficient authority from the proper officers. Within 24 hours after the Vou. I. 2u ship’s arrival, the captain of the ship must make his report, at the | Custom House, of the cargo, &c, in a statement called a manifest ; and certain duties connected with the vessel, must be paid before any entry can be made for the purpose of landing the goods. On the re- 234 OF THE DUTIES OF port being made, it is usual for the broker acting in behalf of the | owners of the vessel to inform the consignees of the arrival of | their goods; and the first step which is requisite for the importer to | take, is to deliver to the proper oflicer in the Long Room a bill of | entry, or an account of the goods to be landed, specifying whether | they are duty free, or upon what amount, weight, or measurement, : &c, the requisite duty is to be paid; this is called, making a prime | entry, and it formerly was the case, that this duty was to be paid | at the time when the entry was made ; but of late years permission | has been granted for most of the usual articles of importation to be | landed and lodged in warehouses appointed for that purpose, with= | out the payment of any duty, until the goods are either exported or ; are taken out for home consumption, -or until the period of two | years has elapsed, which period is generally allowed to be ex- | tended, upon the Board of Commissioners being petitioned to that effect.— When the duties, or the remainder if any were paid at the prime entry, are to be paid, a post entry is to be made, and the | goods are then out of the charge of the Customs. The acts of parlia= — ment by which these goods are thus allowed to be landed without _ paying duty at prime entry, have admitted certain places, suchas the © warehouses of the West India and London Dock Companies to be | sufficiently safe, for preventing the goods being removed without. either their being exported or the duty being paid, and various goods are allowed to be deposited in other warehouses, upon bond ~ being given for the fulfilment of the same conditions. Some very - few articles are allowed to be imported without any entry, or pay- ment of duty. Bullion and some articles of provision, are of this nature.—In the port of London, different restrictions are imposed upon goods as to the places where they may be landed. East and West India ships must go into the docks for those places, and the goods must be landed there. Wines, tobacco, &c, if not from the East and West Indies, and if above certain quantities, must be — landed in the London Docks. On those articles which are liable to duties of Excise at'importation, these duties must either be paid, or bond must be given for either the exportation or the payment of the duties at the time of taking the goods from the warehouses within ‘the period of two years; which period, as before stated, may be successively prolonged for three more years, upon the bonds } CUSTOMS AND EXCISE. 935 being renewed, and the duties upon any deficiency of either weight or measurement, being paid as at a post entry. At the expiration of the time allowed, if the duties are not paid the goods are to be sold, but if such sale would not produce suf- ficient to defray the expenses of freight and primage, if not before | paid, with the warehousing charges, and the duties of the Customs and Excise, they are directed to be destroyed in the presence of _ the officers. For the management of the importations by the East India | Company, particular acts of parliament have been passed, but they are upon the same general principles as importations by ivate persons. The goods are regularly entered previous to Tanding ; bonds are given forthe payment of the duties, and the goods in the Company’s warehouses are under the joint custody _ of the officers of the Company, and those of the Revenue. For- merly the Duties on all goods were paid by the Company, and charged by them to the purchasers at their sales; latterly, this regulation has been abolished for every article except Tea, and therefore before any goods can be taken from these warehouses, a post entry must be made, (the prime entry being made by the Company) and the duties must be paid. Three years reckoned from the day of importation, are allowed for goods belonging to the Company, to be in their warehouses before they are required to be cleared of Duties, but a further time may be granted by the Lords of the Treasury. The foregoing comprise the most important of the regulations which regard the duties payable on the importation of mer- chandize, or the suspension of the payments when secured by bonds, or by the goods being lodged in the warehouses of the public companies ; we shall now proceed to briefly exhibit the process and regulations of exportations. PL PAF LOD DOLD DIL LOL OLDS OF EXPORTATIONS. The Revenue business connected with an exportation, is very nearly the reverse in its different steps to that of an importation. At the commencement of this business, a warrant or account of the goods to be exported, is delivered to one of the cocket writers, who writes out a cocket on parchment in words at length, specifying every requisite particular of the goods to be shipped ; 236 OF THE DUTIES OF copies of this, but made out with figures, are left with the other clerks as checks upon the statement, and their signatures are obtained to the one retained ; admitting the goods to be of British growth or manufacture, and to be exported duty free, no other | process is required in the Long Room. The cocket is endorsed with a description of such things as are ready to be shipped, | and the cocket is delivered to the lighterman who exhibits it to the searcher; upon his having examined the articles and having | seen that they correspond with the cocket and endorsement, and that there is not any fraud or mal-practice, or any mis-statement which would subject the goods to seizure, he keeps the cocket — and delivers in exchange two warrants or certificates, one o | which is delivered to the Custom-house officer on board the vessel, and the other to the Captain to be kept by him until the vessel finishes her clearance outwards. Where duties are payable on exportation, they are paid in the Long Room, and a certificate or receipt is given to accompany the cocket, which must be separate for all such goods. In the exportation of goods for bounty, ‘bond is required to | be given that the goods shall not be relanded; the penalties | of which remain in force, until a certificate is obtained of the » landing of the goods at the port abroad, or until satisfactory proof — is made in case of a loss of the vessel at sea, | | | In exportations of all goods which have been imported, a cer- tificate is required to be produced of the time and other par- ticulars of the prime entry. If the goods have been warehoused and no duty has been paid, a post importation entry is to be made, when a bond is also required, for the same as the preceding purposes ; to which bond, for treble the amount of the duties, the Proprietor, Captain, and one other surety are required to be parties. Ifthe goods imported have paid the full duty, and are ex- ported within one twelvemonth of the day of such payment, the whole or part of the duty is generally claimable under the deno- mination of a drawback, and for these, separate entries, and bonds of a similar nature to the preceding, are also required. The entries of goods claiming either Excise drawback or bounty, are conducted in a similar manner to the preceding, only that 24 hours notice both of packing and of shipping the goods must be given, in order for the officer first to attend and seal them, and afterwards, if necessary, to ascertain their identity. CUSTOMS AND EXCISE. 237 The Custom House drawbacks and bounties are received’ in about six weeks after the goods have been shipped, from the cocket clerk who passed the entry ; the papers in the mean time having passed through the different offices, and every possible check having been obtained. In the case of the drawback being upon manufactured plate, the debenture or certificate only is obtained at the Custom House. (See the Standarding of Gold and Silver.) The Excise Drawbacks and Bounties are obtained at the Excise offices. a ee ee a _ Besides the duties payable to the Government of this country, - there are several smaller contributions payable either to certain public companies, or, in the port of London, to the Lord Mayor and Corporation.— These duties are, The Levant Dues, payable to the Turkey Company, Russia Dues, payable to the Russia Company, ‘ and South Sea Dues, imposed in lieu of the dues formerly claimed by the South Sea Company; these dues are accumulating to form a guarantee fund, which is to be paid the Company in exchange, or asa compensation for the surtender of their privileges to an exclusive trade in the South Seas, and with part of the Eastern side of South America. The duties payable to the City of London dre levied upon the goods of Aliens, or Denizens, or the Sons of Denizens being Alien born; they are termed,—Scavace, or a toll for per- mitting the goods to be landed, and exposed for sale.— Package, ora charge for packing, though not performed by the city officers, upon the exportation of merchandize.—BariLuage, or a charge for the surveying or delivering of goods.—Porrtacr, or Porters’ dues for landing and shipping stranger’s goods. To secure the payment of these duties to the Levant and Russia Companies, no entry can be made for the clearance of the goods with the Customs, before a certificate of such payment is ob- tained from the Company’s officer ; and, with respect to the claims of the City of London, at the time of importation or exportation, the merchant must declare, upon oath, whether the goods are upon British or Alien account; and in some instances the duties are to be paid, although British property, if not imported in British ships legally navigated. LOF LEA SERIA LEL LEA ALS 938 OF THE DUTIES OF In the preceding consideration of the duties of Customs, no_ notice has been taken of those which particularly belong to ships ; they are chiefly the Tonnage Dues, calculated upon the register Tonnage; the Trinity Dues, payable to the Corporation of the Trinity House, for defraying the expenses of Light Houses, &c, The Greenwich Hospital Dues, levied upon the wages of the seamen, with various forfeitures. ‘The Quarantine, Ramsgate, and Dover Dues, charged upon ships, but recoverable by the owners from the merchants. Most of them are but of small amount, and peculiarly belong to the business of the ship’s Broker. 4 BPEL LIDLOL LPL AEE LDL LED The rates at which the following Examples have been calculated are those at present payable; here only the amount of them has been taken; but having been imposed at different periods, they now require being made up for the separate rates when officially calculated. The present consolidation of the Duties of the Customs and Excise, very considerably simplifies the calculations of finding the amounts, when the quantity is ascertained upon which the duty isto be levied. The determination of this quantity is regulated in many instances by long established custom, sanctioned by the Commissioners of the two Boards, and in others by the directions _ of particular acts of parliament. Of the manner in which the customary allowances on the weights of goods, is made, notice has been taken in the introduc- — tion to the department of ‘¢ Draft, Tare, and Tret ;” of the me- | thods in which other peculiar calculations, are performed, the _ following examples will afford some elucidation; they are partly | taken from a valuable work before referred to, Mr. Smyth’s Prac- | tice of the Customs, and partly from the communications of different — officers of the revenue. | | POL IDD LIM OL LO EO LEI IAF ry CUSTOMS AND EXCISE. 939 Example 1. PES LIP IIL LAS LAS AIF LAD To find the amount of the duties of customs, upon the 1% following cases of Opium. Cwee dr. lb. Not ).?. 9 Sigedg 2... 0 47 q $ ssheee 270) p10 5 Aen? seen, 20 5... 3) "ae Gc. 2 a 127 Meh of Sede, f Oe. 3.0 15 eee Se5, Fe 29 fee 8. 0 SS re, a eg CTs in Poets mans an $59 3" 14 555 333 98 4018 80 ‘Tare for leaves, 2 lb per 100 Ib. Ib. 3938 Net, at 10s per 100 ]b. 1. 39,38 | - rea tem 2 | £ 19,69 reckoned £19 14 O | ~s ees te aS ee eS Tables of these rates, with much other information, have been | published by various officers of the Customs ; of which those of Mr. Mascall, Mr. Pope, and Mr. Dew, are particularly emi- | nent. 240 OF THE DUTIES OF Example 2. To find the amount-of the Customs, on the following 10 bags of West India Cotton Wool, imported in a British vessel. cwt. qr. Ib. No? ic oe 2 "4 17 Se 4] O ll eee | 1 O oD Pus winks 5 2 ane oe ee ‘ Dives oa tek 2 3 19 Ais a chatecs 2 3 a f Pee HAL es | 12 OF ot natetets elt a 4 DF Ss ale ae O FO at i ene hes ou, 210 27 333 27 3387 Ib gross. 135 Tare 4 lb per 100 Jb. Ib 3252 Net at 8s 7d per 100 Ib. | 3252 W/o d 3349 8-279 J £ 13.19 1 reckoned £ 13 19 6 ——_——w ——_—— N. B. The rate of 8s 7d per 100 Ib, is 1d perlb with 3 per cent, and for this 3 per cent, we multiply by 3 cutting off the two right hand figures. The amount of the above is £13 19 13 but the officers are allowed to take the odd pence to make up 6 d or 1s, without their being considered as a fee ; this is however at the option of the merchant, but it is perhaps, never refused. The officers account to the crown, in all cases, for the exact amount. bake CUSTOMS AND EXCISE. 941 Example 3. PIP LID LIS ODL IID DIOS To find the amount of the Duties of Customs, upon a log of Mahogany of the following dimensions. The Duty is rated at £3 16s per ton of 20 ewt, to which 40 cubic feet are considered as equal. feet. in , Length . 126s 1G Wiles. se 4 Sea Depth ..... 3, 10 cub. ft. in ater TS) EO 4 51 4 3 24 for 3 in. 4...554 64 A 218 2 for 4 feet. ° ( Subtract ) 9 1, for; 2cin, Cub. feet 209 1 —— Tons 5 9-40ths fae siete 1,,..5 4 6 amount at 1 £ per Ton. 4 ‘ ROR Orc sc metee at 4 £2 Bias Oa) 80.) 5 stay at 4/38 £19 17 2 reckoned £19 17 6G — Ay . Vor, I. . Qi 242 OF THE DUTIES OF Example 4. To find the amount of the Duties of Customs, on the following Marble Mortars, of the following dimensions, at 3s 2d per Superficial Foot. No.— 8 .... 7 inches. TO, ...-..12sinches. Ao ea cies. Te: AT Ok Sie 129 3 124 ex nO eleeO 1 OMX WD eae ete La OC S Be, 1k 1086 4x“ 14x 2 = 392 2....4560 1520 144 ) 6080 ( 42 feet. 320 32 | $ d 3 2 6 19 0 | 7 | £6 13 O Duty required. | , N. B. In taking the dimensions of marble mortars, they are measured the shortest distance across, the four nobs not being | measured; but by taking the whole square of this diameter, and | not 7854-10000 ths, no allowance is made for the rounding off. The addition of the 1-3 rd is for the cavity. gf CUSTOMS AND EXCISE. 949 Example 5. To find the amount of the Duties of Customs, on 250 Bundles of Basket Rods, at 3s 2d per standard Bundle, of 3 feet in _ circumference, the medium of 2 girts being 45 inches. 45 A5 36 225 36 180 216 2025 Square of 1 Bundle. 108 250 1296 ) 506250 ( 390 Standard Bundles. 11745 310 £5 “\; 390 Amount at 1 £. ae os Ete lee ae at 2 s + SAIS 8 OR ey at 1 s Bas aera at 2d ee ee £61 15 Duty required. In this calculation, the Standard Bundle, and the given medium Bundle, are considered as cylinders of the same altitude, which are to one another as their bases, or as the squares of either their diameters or circumferences ; and the above may be considered as a contracted mode of expressing the calculation in the form of a question ; thus Sq. in. Bundle. Sq. in. If 1296 produce 1 what will 506250 produce? The answer to which is 390 Standard Bundles, the fractional remainder being rejected. 244 OF THE EXCISE DUTIES ON SPIRITS. PIF ODL OP L MIA PDD AA LPF — These duties on Spirits are charged according to their different degrees of strength, which are now determined in the following | manner. There are two Instruments used for this purpose, one is a common » or Thermometer to denote the temperature of the liquid, and the other is called an Hydrometer to measure its weight or gravity. A ‘Thermometer is used, because all liquids expand, or the same bulks become lighter, as their warmth or temperature is encreased ; and, vice versa, they contract, or the same measures become heavier, the greater the degree of cold that is applied. The principle upon which the Hydrometer acts, is, that by increasing the quantity of pure Spirits, the gravity of the liquid, and consequently, its capability of support, is diminished. The Hydrometer now sanctioned by the authority of Parlia- ment, is called Sikes’s Hydrometer : it consists- of a hollow brass 3all with a Rod fixed below, for steadying the Instrument and_ receiving certain weights, anda graduated stem fixed above, for showing how deep the Hydrometer sinks in the liquid which is the subject of the experiment. This Instrument is so constructed that with two weights, one placed at the bottom and the other at the top, the Instrument sinks to a certain mark on the stem, in distilled water, while without the upper weight, it sinks to the same depth” in what are termed Proof Spirits. Formerly the appellation of proof strength, -was one of a very uncertain nature ; it was generally understood to mean a liquid of such strength as to contain one half of the highest rectified Spirits of Wine, with one half of pure water; it is now defined to a liquid of such a gravity, that 13 equal parts being taken by, measure, the weight is exactly equal to 12 of the same parts of distilled water; and therefore the weight added to the top of the Hydrometer, is exactly 1-12th of the weight both of the Tustrument and the weight in.the liquid ; consequently, admitting we Nort 2 patie uJ DUTIES ON SPIRITS. OAS the specific gravity of distilled water to be 993, or a cubic foot to weigh 993 oz. Avoirdupois, the specific gravity of proof Spirits is 12-13ths of this, or nearly 917, making the difference to be 76 oz. in-each cubic foot; this latter weight being divided into 100 equal parts, each portion is called a degree, or 1 per cent, and thus proof Spirits are said to be 100 per cent above water ; in the same manner, by subtracting corresponding portions, a liquid weighing only 864 oz. per cubic foot, would be said to be 70 per cent above proof. With this Hydrometer, a set of weights numbered from 10 to 90 are used, with a range of 10 divisions on the stem; each divi- sion is subdivided into 5 equal parts, which are called 10ths, making on the whole, 500 subdivisions for a range of 169 degrees, at the standard temperature of 51 degrees; viz, 100 below and 69 above proof. The per-centages shown by these indications, are to be found in a set of Tables constructed from those of Mr. Gilpin, and corresponding to the different variations of tem- perature from 30 to 80 degrees. . Thus supposing the temperature of the Spirits, to be 56°, the Instrument to have on the weight marked 50, and that it then sinks so that the stem is immersed in the fluid to the division 2, and the 4th subdivision or 8; then the indication is called 52.8, which being sought for in the range of the Tables, from 50° to 60°, under 56 is 10.2, shewing the per centage strength above proof. Then by multiplying the number of Gallons by this Per Centage, and dividing the product by 100, the excess is found by which the given quantity is to be increased. Instead of using these Tables and making this calculation, the same result may be obtained by sliding Rules, which are usually - sold with the Instrument. Ié is foreign to the purpose of this Work, to enter into a more particular explanation of the principles upon which this instrument is constructed: besides which it is the Author’s intention to de- velope them in a course of Mathematics, which he intends pub- lishing, and to which the discussion more properly belongs. 246 OF THE EXCISE DUTIES The principal difference in the construction of this instrument, and of the one before in use, called Clark’s Hydrometer, consists in a rectification of the weights employed ; the theorem for which was furnished to Mr. Sikes’s representatives by the author of this work ; the investigations to which the necessary consideration of the subject gave rise, afforded him the means of making many variations for simplifying the whole plan, but every desideratum — for the purposes in question, seems now to have been fully ac- complished, by an instrument for which a patent has been obtained by Mr. Ashton, of Harp Lane, Tower Street, and we shall close this subject with a brief description of this Hydrometer, which by its great simplicity and accuracy, bids fair to supersede every other at present invented. The instrument generally adapted for Commercial Purposes is the one with two faces; one face is used when the instrument is employed’ for showing the strength of spirits from 20 to 55 per cent over proof, and the other, is used with a proof weight, when the spirits range in strength from 20 per cent over, to 25 per | cent under proof. A four sided instrument with two additional weights, is adapted for going down as low as distilled water, on the one hand, and rising to 65 per cent over proof on the other. In using this Hydrometer, the temperature is first determined by a Thermometer whose ranges are denoted by letters of the alphabet, corresponding with which similarly marked weights are to be used, and then the per centage strength is determined from ~ the numbers on the stem, without any further trouble or any reference to tables. Engravings of these Instruments, with a general account of these and other Hydrometers, are to be found in Burrowes’s New Encyclopedia. It might have been remarked, that, with regard to Sikes’s Hydrometer, the officers of Excise still complain of much trouble and uncertainty in obtaining the results ; frequently, several trials are to be made before the true indication can be obtained ; there being much complication in the management of the instrument and — tables, frequent errors and much loss of time are the necessary consequences ; while many instances are upon record, by which it has occasioned a considerable deterioration in the Revenue. BOIL OOP DEL OCR EDO LOE DDD ON SPIRITS. O47 Example 6. el i ee a To find the duties of Excise on 1218 gallons of British Planta- tion Rum, of the strength denoted by Sikes’s Hydrometer, to be 23.7 over proof. 1218 eb W 8526 3654 2436 Additional 288,666 for extra strength. Gauge quantity 1218 Gallons 1506 .— at 10s 41d per Gall. $ wales wh JUG amount at 10s. See STOR GRRE 3d. Lose ane Woe ae lid s 15624 9 £781 4 9 Excise Duty. * By thus keeping the 1506 one place more to the left, and adding it thus up with the products, the same result is obtained as multiplying it by 10. PIF LEI DPP OIG DPD LIP YAS EXCISE DUTIES ON SPIRITS. With regard to the strength of Spirits imported into this country, — Rum and British Plantation Spirits may be of any strength ; but Brandy, and other foreign Spirits, cannot be imported of a_ strength greater than 9 per cent above proof, unless they have been captured or seized, are regularly entered as such, and are — accompanied by proper certificates. When spirits are seized by officers of the revenue, there is a certain estimation fixed upon the value, and there are certain re- wards allotted to the military whose assistance may be given in ‘making the seizure, or who may be employed in watching and guarding them when seized ; for determining the amounts of these values and rewards, some very accurate and excellently well ar- ranged tables have been calculated by Mr. Lumley, Gauger of Customs. Of the construction of these Tables the following will show the method, while the preceding Example shows the mode in which the duty is calculated upon regularly imported spirits, as well as how they are usually sold when the duty has been paid ; otherwise, the value, particularly of Rum, is determined by a vari- — able rate according to the strength, as may be seen upon consulting any Price-Current. PIF LIF DPE LLEL LLL LE LOL Example 7. PIF SPL POE LEAD ORE ODE LOL The value of seized Brandy of proof strength being £1 3 54 per gallon, to determine the value of such spirits, 10 per cent over and under proof. & a £ 1 3 5f -— = given rate. =i AB, Meow OOD LO Der Cents Rate for 10 over proof £1 #5 94 .600_ inthe Tables. Do. — underprof £1 1 1: .400 do. The above has been chosen at a very simple rate, and affords no specimen of the labour attendant upon forming the Tables. 249 OF AD=VALOREM DUTIES PLDI LIA ALD LAL BIE When duties are payable upon the value of the Goods, and not upon the weight, tale, gauge, or measure, they are said to be levied ad-valorem. The value is to be ascertained by the declaration of the Im- porter, Exporter, Proprietor, or his known Agent, and it is at the option of the Officers in all such cases, to seize them for the use of the Crown, upon paying the entered value, with 10 per cent thereon, and any duty which may have been paid at the time of making the entry, but when the undervaluation for which they are seized is not made with any fraudulent intent, they are allowed to be released upon a moderate compensation being made to the officer. When Goods are seized, the value, if not above £ 20, is to be paid without delay ; if above £20, within 15 days. Oe a A ee a f Example 8. PLL LLL LOL OLLI AOIL LAE AEF To find the Ad-valorem Duty on Bronze Figures, of the entered value of £ 450. 37 10 O Duty per Cent 150 O O For £400 19) ‘15a One. Sa: 50 168 15 O Permanent Duty (one third ) 56 5 O Originally War Duty (one fourth) 9 7.6 Additional Duty 6 S234 7 Total. This is calculated according to the manner of the late distribution of the Duties. Vos t, 2K 250° = OF PREMIUMS ON THE SEIZURE OF VESSELS. DOL LIL FPL APL ACD LOD LOT Vessels or Boats seized for being engaged in Smuggling, not — having a licence, or being constructed contrary to the regulations — of different Acts of Parliament, become forfeited to the Crown, and upon being broken up and destroyed, the oflicers are entitled to the following rewards. “ not above 4 Tons. above 4 Tons. By 28 Geo. III.....£ 2 for Boat. 10s per Ton. Tyrpwied A7 ———~—~ --,+-»40s per Ton. 30s per Ton. TBadlacs ae 28 Geo. III.....£ 2 for Boat. 10s per Ton. — 4 pie hs 20sper Ton. 20s per Ton. — In the making out of the statements of these premiums, the amounts are required to be distributed according to the different Acts by whichthey have been granted, at the separate rates of 10s, 20s, 30s, and 40s per ton; for these purposes, tables were con- structed by the Author of this Work, for the use of the department in — the Customs, in which these calculations are made. For this or any _ similar purpose, the following Table for the 94ths at the rate of 20s per ton, seems fully sufficient for general utility —Of this the following is an Example. PLO LED LP E LELD OB P ELD OOP Example 8. DIL LOOP LLD LOD LLL To find the amount of the premium on 5 tons 73-94 ths, at 30s per ton. LA d 1 10 O 5k For 5 tons 47 > he... O 73-04ths = 5 ‘6.536. -at 20-5. | , do. are D218 a ae ) £8 13 3 54-94ths of a penny. ee re es ee 25] TABLE, Of the amounts of 94-ths of a Ton of Shipping, at 20s per Ton. ggths of a Ton. 5° d. 94ths. g4ths of a Ton. Se d. 94ths. 94ths of a Ton. $. d g4ths, me 2% 52 fae Oe Oe OE 63... 2 tS ded ao me. — 5°10 ee ee Sead A a ae. .— 7 62 4 ant PONG 5 Lr 99 90 ma... — 10. 20 5 —' § $4 6 14— 48 os. J a 72 epee mit eh Si 7 Lots 6 a... 3 | 30 Pipe cc ee MY Bae 8 ROLE Pree a ....— 5° 82 8 ahs LAE 9 Juhi Dig ms ....— 8 40 Oye te, oe 702". 0-210" 6s mo. .,.— 10 ‘92 Ps ic acta | RM i ERT ory epemes oe... 2 1° 50 Too) Ss 64 2 ee a8 me 4 SB yA aeta i alee 5. 6 PP ak ee. m..— 6 .60 Eat Nik: ial Silat 1. eS of mv...— 9, 18 dpe alla OT. Ah, 9 yee oer Tt AG a. .— 11 70 Bn On eae C216 2 4 _...3 2 28 Ge —a'9> 42 Ve, eee CA, io ...— 4 80 Ve ee leet eee ee i mee. .—- 7 38 Ss UL MNS De ON Ge mes 9 90 Gupte W5VOLO 30.5 17a OS moe .4, — AS BOs see, eh OS | a ee oe... .—" 3-6 bi 10 20 5 ene Be Se Se Poet Bui Dare Keen eS 3 ey 86 me. 8! 46 9. RS 30 AO 100 44 oer — 10,68 AGE LR 3S 5G Po) 1S aS meee se” TF ~ 96 FY e S240 65s), Ll’ Nga ees O7E 6. ¢ UL 19 2099 Px SPR ie Ge 6. 36 PN SES BoP O80 8 34 eo) 8) 88 Gellert GAG 9 — 11 .22 mo... —-1l 46 Fs —! 6. 60 90> 19) Tee eT ee 60. s28—— 89 18 Dien 2 ae 30....- 4 56 1 mel biz0 ae TOM Fi 14 2 13 28 9.) eS Noa LIS DOD ODO DID DID OAD LODOD 252 FORM OF AN ENTRY OF SILVER PLATE FOR DRAWBACK, AND THE CALCULATION OF THE AMOUNT. PLL LIL LIL IIL ADF ADF \ WROUGHT SILVER PLATE. PIPPI LI AIL DLE POD LAE Know ye, that William Wright, hath entered | Two hundred and ninety-two ounces of wrought Silver Plate, made since the Ist December, 1784, and before 5th July, 1797— One hundred and sixty-five ounces of wrought Silver Plate, made since 10th October, 1804, and before 1st September, 1815— and Eighty-five ounces of wrought Silver Plate, made since 31st August, 1815, for Drawback; all of the fineness of eleven ounces two pennyweights to each pound troy, on which is the — Goldsmith’s Hall mark, as appears by the oath of William Wright ;” and plated ware and packages value ninety-five pounds; all in the Good Intent, Samuel Jones, for Seville ; ; Paid the duty Dated the 27th March, 1818. In the Fifty- eighth year of King George the Third. Duty 9s 6d PIL LAID DAD OPA BAAD LOD BIG. Drawback ON 229207 AtiOn per 07 seme. oF 22 Tae 165. oz at 18 3. peroz......... 10/40 3 BO.707 WAL Gls (0 4) DOTOZbe ese uuicus 5.) aD 2 23 105 For this Drawback a debenture is obtained at the Custom House | in about six weeks after the shipment, and the amount may be re= | ceived at Goldsmith’s Hall, the following day. PID DAD LOL LED LOD ADD AIG The duty is calculated on £95, at 10s per cent, making 9s 6d. } : PLP DLE PLL DED LIE LLL LAL END OF VOL. i. é Printed by RICHARDS and Co. 3, Grocers’ Hall Court, Poultry, London. - = bOGABRUREMS. PLS LI I LAS IAD AAD LAD LIT LOGARITHMS are numbers serving as indices to other numbers, and they are so constructed, that the sum of the Logarithms of two or more numbers, is the same as the Logarithm of the product of those numbers ; while on the contrary, the sub- traction of the logarithm of one number from the Logarithm of another, produces in the remainder the Logarithm of the quotient of the division of those numbers. LID LFII DOF DOF LAID LAD LIE If a series of numbers be taken, which commences with 0 and increases by the constant addition of the same number, it will form — the Logarithms of any series of numbers commencing with unity, and increasing by the continued multiplication of the same number. mousO ‘2 4 6 8 108 STZ 14 are Logarithms mot 862. 4 Set G 32 64, 128, &c. Et. 3) Oo 27m Sle 243: 729 .:2187;, &e. For in the first and second series, on adding together 6 and 8 which are here the Logarithms of 8 and 16, the amount 14 is the Logarithm of their product 128. Or, in the first and third series the numbers answering to the indices 6 and 8, are 27 and 81, and that answering to the index of their amount, 14, is their product 2187. In this manner any series of numbers and Logarithms may be constructed, but the series from which the tables of Logarithms now in general use have been formed, is that in which the numbers increase by the multiplication by 10; while the indices or Loga- rithms increase by the addition of unity. at i ee | 2 3 4 5 as Logarithms of the numbers 1 10 100 1000 10000 100000, &c. The numbers in these tables being decimally expressed, it follows that the Logarithm of any number from 1 to 10 must be ex- pressed by decimal parts only, from 10 to 100 by 1 with decimal parts, and from 100 to 1000 by 2 with a desimal, &c. Ke. # il LOGARITHMS. In general, these decimal parts alone are called the Logarithms, | and the whole number employed with them is called the Index, but it is evident that this is not quite accurate, and that it is the two, when employed, which form the Logarithm. In the subjoined Tables of Logarithms, The first page contains the whole of the Logarithms of all num- bers from 1 to 100 ;—and In the following pages the decimal part only of the Logarithms is expressed. In taking out the Logarithm of any number, the following directions are to be observed. ist. If the given number contains but two places of figures, its logarithm may be found in the first page ; observing, that if it is not entirely a whole number, the given index is not to be used, but one which is to be found by the 5th Direction. 2nd. If the number contains three places of figures, the number is to be found in the first column of the pages following the first, and the decimal part of its Logarithm is to be found in the second column having O at the head. 3rd. If the number contains four places of figures, the three highest are to be found, as before, in the side column, and the other in the heads of the columns; then the decimal of the Logarithm of the four figures will stand on the same line with the first three figures, but under the fourth. N. B. The first two figures are to be taken from the 0 column, but when the left hand figure in either of the interior columns, is a QO, without its being so in the O column, the first two figures are to be taken from the nex: lower line; and the same to be observed of the succeeding numbers on the right of those having this cipher. 4th. If the number contains more than four places of figures, find the decimal of the Logarithm for the four highest according to LOGARITHMS. sis the preceding directions, and multiply the difference between the Logarithms of each given number, as given in the right hand column of the Tables, by the remaining figures; from the product cut off from the right as many figures as those which are in the Multiplier, and add the others to the part of the Logarithm before found. 5th. The decimal part of the Logarithm of any number being thus found.— The index, or whole number to be used with it, is to be thus obtained :-— First—If the given number is entirely a whole number, its Index as before observed will be 0 from 1 to 10 ; 2 from 10 to 100, and, generally, it will be 1 less than the number of places contained in that whole number. Second—If the given number is partly a whole number and partly a decimal, its index is to be regulated only by the whole number. Third—If the given number is a decimal only, its index will be regulated by the distance of the highest figure from the place of units, using —1 if the highest part is 10ths, —2 if 100 ths, and —3 if 1000ths; but marking them thus with what is called the negative “sign, to show that it is to be used in a contrary manner to what it would have been, if the number had been a whole number. 6th. Ifthe given number is a fraction, find its decimal value and use it in its place; or find the Logarithm of the Numerator, snd from it subtract the Logarithm of the Denominator. Example to Direction 1. ower To find the Logarithm of 32. In the 1st page against 32 is 1.505150, which is its Logarithm, PEF DOD AOO LOL IOS , andl iv LOGARITHMS. Example to directions 2 and 5. To find the Logarithm of 327. In page 6 against 327 in the first column, is .514548 and as 327 contains 3 places of figures, its index is 2, and the whole is 2.514548. So for 32.7 the Logarithm is 1.514548 Bt ayolce Ay eae ed eet 0.514548 PBST ALG OY Reet —1.514548 SOT ed semis Gtk Deed 2 —2.514548 OOBET eee Hee ces, —3.514548 Example to directions 3 and 5. ~ PLL LAA ALI LIA D PL EPL ODT To find the Logarithm of 3275. Having found 327 in the left hand column of numbers, proceed to the 6th Logarithmic column, having the 5 at its head, and in the same the Logarithm 5211 is found; to which is to be prefixed 51 from the O column;-and as 3275 contains four places of figures its index is 3, and the whole is 3.515211 So for 327.5 the Logarithm is 2.515211 Sie ELD Us eit aaehore yea estene eters Lo5i5212 N Hie 6 perk Pete eh al 0.515211 RS Sie Rails, ak ee o. 1.515211 N. B. If the Logarithm had been required for 3312, instead of using 51 with the 0090, we should take 52 which is placed against the next lower line of Logarithms. PIP IDL LOE LOE LOD ERPE OID he i ee LOGARITHMS. Example to directions 4 and 6. To find the Logarithm of 327581. Having found the Logarithm answering to 3275, we take the difference 133 in the right hand column, and multiply it by 81; from the product we separate the two lower figures, there being 2 in $1, and adding the others, 107, or rather 108, to the former Loga- rithm 515211 it makes it 515319, which with 5 for its index, there being 6 places of figures, makes the whole of the required Logarithm to be 5.515319. DIL LIS DLO LIO DI ODD Example to direction 6. CLP AIL AAP ALD PDI POP To find the Logarithm of 2, and of 324. zs = .09375 of which’ the Logarithm is .. —2.971971 324 — 32.75 of which the Logarithm is ..... 1.515211, aaa Logatiim of 3s"... ..-. 23.5... a= Orde tie . SRT Merete a acai ohe iol: ars 1.505150 ‘ Get Sale S Remainder, or log. of 3-32nds........... —2.971971 323 — 131-4 ths, and the log. of 131 is ..... 2.197271 MCS OR hy ai he 0.602060 Remainder or log. of 323 .......... ioe tieethoiloae - In the second method of finding the Logarithm of 3-32nds, the index of the remainder is—2, there being a deficiency of 2 in the number from which the subtraction is made. SIL ELE LEG ERE LEA FEE ELIS - > vi ~LOGARITHMS. TO FIND FROM THE TABLES, THE NUMBER CORRESPONDING TO A GIVEN LOGARITHM. j ane LOL LPL ALE LEAR Find in the column with 0 at the head, either the Logarithm corresponding with the decimal part of that which is given, or the next lower. If the decimal part is exactly found, take the figures in the left hand column with O on the right for those which are required ; otherwise, take out the three figures answering to the next lower, and seek for the Logarithm, corresponding either with the one given or the nearest to it, in one of the other columns, the head figure to which will be the fourth figure required ; but if a further number of figures be required, take the lower Logarithm and subtract it from the given one, then divide the remainder decimally by the number belonging to the line in the column of differences, andannex the figures so obtained as a quotient, to the right of the number before obtained ; observing, that more than two extra places can seldom be accurately determined. With respect to the value of the figures so obtained, it is to be observed, that the number of places of whole numbers is one more than that expressed in the index ; and that when the index is negative, the number is a decimal, the highest place of which is so far distant from unity, as the index expresses. Example 1. To find the number answering to the Logarithm 1.361728. In Page 1 the number is found to be 23. Or in Rage 4 it is found to be 230, which as the Index is 1, makes 23.0 or 23. | COLL PPL OLE LIA OIO LAO I OE Example 2. To find the nearest number to four places of figures, corresponding with the Logarithm 1.422919. In the O column of Page 5, the next lower Logarithm is 421604, answerlng to the number 264; and in the 8 column the nearest Logarithm is 2918, consequently the figures are 2648, and as the Index is 1 the number is 26.48. PLO LOL ODO LOL OOCL LA OE DLA Example 3. To find the number to 6 places of figures, _ corresponding with the Logarithm 4.556976. LOGARITHMS. vii In Page 7 we find .556906 as the next lower Logarithm, the number answering to which is 3605, and as the difference between this Logarithm and the one given is 70, while the difference in the side column is 120, dividing, decimally, the 70 by 120 or7 by 12, we produce 58, which annexed to the former number 3605, makes the required number 36055,8 with 5 places of whole num- bers on account of the given Index being 4. PIL LIS LAA AOL ADA DLT OF CALCULATIONS BY LOGARITHMS. PIS PAP DAD ADA FOF APT For the operation by common numbers of Addition and Sub- traction, no substitution can be made of Logarithms, and conse- quently when they occur in conjunction with the other processes of calculation, in which Logarithms are employed, the value of these Logarithms must be found in numbers, which must be used in the Addition and Subtraction. The Multiplication and Division of numbers, are performed by the Addition and Subtraction of the Logarithms representing those numbers; but when the products contain many places of figures, and when they are required to be exactly expressed, they are better performed by ordinary numbers; and the same is the case in all single Multiplications and Divisions.—W here however several Multiplications succeed each other, and particularly where they are followed by one or more Divisions, the work maybe const- derably abbreviated by employing Logarithms. Particular advan- tages also attend the use of Logarithms in many mathematical operations, where tables of numbers are obliged to be constructed in order to facilitate the calculations; the Logarithms of which being also determined and arranged in Tables, little or no extra trouble is required to take them out, instead of the ordinary numbers ; but in the general operations of Arithmetic, Logarithms are rather troublesome than useful. PII DDD LIL DLP DOD PDP Example 1. To multiply 27 by 31. Bogs OL) 2718.0, 6 i. 1.431364 OC OLY), Vir ete ooo 1.491362 Sum of the Logarithms 2.922726 nearly answering to 837. es Re RS Vill LOGARITHMS. — ee Le xample 2.—To divide 837 by 38. Log. of 837; 208) 2.922725 of ial vids ot): 1.491362 ’ Diff. of the Logarithms 1.431363 nearly answering to 27. Example 3—To find the result of the product of the conti« nued multiplication of 516, 4725, . 4803, and . 05723, divided — by 813 multiplied by 62.81. Log. of 516...... 2.712650 AT 253.00. 2 3.674402 ° -4803....—1.681513 -05723.. . .-——- 2.757624 4.826189 Log. of 813.. 62.81...1.798029 4.708119 Subtractive. remainder 0.118070 answering to 1.31241. In the addition of the first four Logarithms, the amountof the whole numbers or indices is 7, from which ,3, the amount of the two negative indices, is taken. 2.910090 Instead of subtracting the amount of the Logarithms of the divisors, their afithmetical complements, may be added: thus. Log. of 516...... 2.712650 j A725...) .. 3.674402 4803....—9.681513 less 10. -05723....—8.757624 do Comp. of the log. 813....—7.089910 do ty ae oft Be .—8.201971 do 0.118070 answering to 1.31241. - ~ In calculations of this sort it is, as above, better to use the complements of the indices of the Logarithms for the decimal mul- tipliers, and adding all the indices together, to subtract from the amount as many tens as there are complements used. ON THE INVOLUTION AND EVOLUTION OF WUMBERS, AND THE PERFORMANCE OF THESE CALCULATIONS BY LOGARITHMS. PLO LD I PLL POLE LEE LE LDOLR OR When any number is successively multiplied by itself, it is said to be involved; the first product is called its square, the second its cube, the third its biquadrate, &c; and, when to a given number another is found which being thus involved will produce that number, it is said to be evolved, or its root to be extracted... Thus 4 xX 4 = 16 isthe Square of 4, or 4 4.x” 4000 lie 64e srg Come tant. or 4° mx 4 % 4 xX 4:-— 256 ...., Biquadrate,. ons and....8 is the square root of 64 4 is the cube root of 64 PLL LIE LIF LDPE LOL DDE The performance of involution by ordinary numbers, is ex- tremely simple in its plan, but when the number is large it is very tedious; in this case Logarithms are but of little use, since _ the result can seldom be obtained with sufficient precision ; but the - reverse is the case, with the performance of evolution, for except in the extraction of the square root, the difficulty is very consider- able, and in many cases it can hardly be surmounted but by their use. We shall therefore here give directions only for the performing this extraction of the square root, without Logarithms, and them show how the other roots may be extracted with them. POPOL DDL DDILDOL IFE * # xX LOGARITHMS. TO EXTRACT THE SQUARE ROOT. PPP PIP PAL PDP LD DD OD OL OP Separate the given number into places of two figures each, beginning from the right hand with whole numbers, and from the left with decimals. Find the greatest square root of the highest place, and make this root both the divisor and the quotient, | Subtract the product, and to the remainder bring down the next period ; and in this and each succeeding step in the calculation, | double the last figure in the divisor, and to find the quotient figure, find how often the divisor is contained in all the figures | in the partial dividend, excepting the place of units. This being found, place the number of times both in the quotient, and on the © right of the divisor, and thus continue the operation, until either the exact root is obtained, or the calculation is made with as much | precision as is necessary. PPP ALPE POLE LPL POL DLE LOD «i Example. To extract the square root of 1483.7654 3 ) 14,83.76,54 (38,51 &e. 3 9 68 ) 583 Proof. 8 544 3851 — tt 3851 765 3976 senate ; 5 3825 14830201 | —- —— 7453 ; 7701) 15154 He 7701 14837654 7453 . N. B. When the square root of a fraction is required, and the Square root both of the numerator and of the denominator cannot be exactly found, it is generally better to find its decimal value, and to extract the root of this decimal. ~ LILIA IIIA ALL DOD DOO « Xl OF INVOLUTION AND EVOLUTION BY WHOGARUIMEMS, PLL LOS LILI DIS III IIS The Involution or Evolution of Numbers is performed by mul- tiplying or dividing the Logarithms of those numbers by the number expressing the power or root ; observing, in Evo- lution, that when the Index, is negative, if the Divisor is not exactly contained in the Index, a number must be added to it suf- ficient to make it divisible, which number is also to be used as an Index to the decimal part. “er PEOPLE LILI I AAO II IO Example }. DIPLO LOL IDI IIT To find the Cube number of 27, and of .027. Log. of 27... .1.431364 3 4.294092 — 19683 eS A Thus the cube of 27 is 19683 LOL LIL LIS LOLA AOL OD Log. of .027 — 2.431364 3 t — 5.294092 = .000019683 Thus the cube of .027 is .000019683 In the second calculation, the Product, 6, of the negative index, is diminished by ! which is carried from the decimal part ; the proper index to the product is — 5, which shows that the highest figure in the corresponding number, is five places from the place of units, and therefore that 4 ciphers are required to be prefixed. Xi LOGARITHMS. Example 2. SPP PIP PPL ODO ODO LOO OPO To find the Cube root of 19683, and of .000019683. Log. of 19683 .... 4.294092 Quot. by 3.... 1.431365 = 27 The Cube Root of 19683 is 27. Log. of .000019683 — 5.294092 Quot.by 3 — 2.431364 — .027 The Cube Root of .000019683 is .027 In the Division of this Logarithm, as the index, 5, is not exactly divisible by 3, 1 is added to it and the amount is divided ; then this 1 is used with the 2 in the decimal part, making it 12, which divided by 3 produces the 4. PLL LOL OLE LOL LD OE POD COP Example 3. 4 To find the 4.27th root, or to find the 427 th root of the 100 th power of 836. Log. of 836.... 2.922206 100 427 ) 292.2206 the 100 th power. The 427th root 0.684357 = 4.8345 This decimal form of a root is but rarely met with 3 when it occurs it can be calculated only by Logarithms. GILLIEL IID DOD LOCOPE hOGARLIZHUS OF THE NUMBERS. 1 to 10000. OOWNA HE PB oo © + | 4 Prt femdom O OOOCOOOCO Log. ||N.| Log. || N.| Log. _ ‘000000 * 301030 ‘477121 ‘602060 ‘778151 *845098 *903090 *954243 “000000 *041393 °079181 *113943 °146128 *176091 *204120 "230449 *255273 * 278754 * 301030 * 322219 * 342423 * 361728 * 380211 * 397940 *414973 *431364 *447158 °462398 *477121 *491362 1°505150 33|1°518514 ‘698970 | 1°531479 1° 544068 1° 556303 1+ 568202 1°579784 1*591065 1°602060 1°612734 1°623249 1°633468 1°643453 1°653213 1°662758 1°672098 1°681241 1°690196 1°698970 1°707570 1°716003 1°724276 1°732394 | 1°740363 1°748188 | 1°755875 1‘763428 | 1°770852 1°778151 1°785330 1°792392 67 68 69 1*826075 1°832509 1°838849 1°845098 1°851258 1°857333 1°863323 1° 869232 1°875061 1+880814 1°886491 1-892095 1°897627 1+ 903090 1°908485 1°913814 1°919078 1°924279 1°929419 1°934498 1°939519, 1° 944483 1-949390 1°954243 1°959041 1.963788 1°968483 1°973128 1°977724 1°982271 1‘986772 1°991226 1°995635 LOGARITHMS. 100 101 102 103 104 105 106 {07 108 11] 112 113 114 115 116 117 118 120 121 122 123 124 125 126 127 128 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 0 000000 4321 8600 012837 7033 021189 5306 9384 033424 7426 041393 5323 9218 053078 6905 060698 4458 $186 071882 5547 9181 082785 6360 9905 093422 6910 100371 3804 7210 110590 3943 7271 120574 3852 7105 130334 3539 6721 9879 143015 6128 9219 152288 5336 8362 161368 A353 7317 170262 3186 0434 A751 9026 3259 7451 1603 a7416 9789 3826 7825 5714 9606 3463 7236 1075 4832 8557 2250 5912 9543 3144 6716 0258 3772 7257, 0715 4146 7549 0926 4277 7603 0903 4178 7429 0655 3853 7037 0194 3327 6438 9527 2594 5640 8664 1667 4650 7613 0555 3478 0868 5181 9451 3680 7868 2016 6125 0195 4227 8223 6105 9993 3846 7666 1452 5206 8928 2617 6276 9904 3503 7071 0611 4122 7604 1059 4487 7888 1263 4611 7934 1231 4504 7753 0977 4177 7354 0508 3639 6748 9835 2900 5943 8965 1967 4947 7908 0848 3769 1301 5609 9876 4100 8284 2428 6533 0600 4628 8620 6495 0380 4230 8046 1829 5580 9298 2985 6640 0266 3861 7426 0963 AA7\ 7951 1403 4828 8227 1599 4944 8265 1560 4830 8076 1298 4496 7671 0822 3951 7058 0142 3205 6246 9266 2266 5244 8203 1141 4060 1734 6038 0300 4521 8700 2841 6942 1004 5029 9017 6885 0766 4613 8426 2206 5953 9668 3352 7004 0626 4219 7781 1315 4820 8298 1747 5169 8565 1934 5278 8595 1888 5156 8399 1619 A814 7987 1136 4263 7367 0449 3510 6549 9567 2564 5541 8497 1434 4351 2166 6466 0724 4940 9116 3252 7350 1408 5430 9414 3362 (275 1153 4996 8805 2582 6326 0038 3718 7368 0987 4576 8136 1667 5169 8644 2091 5510 8903 2270 5611 8926 2216 5481 8722 1939 5133 8303 1450 A574 7676 0756 3815 6852 9868 2863 5838 8792 1726 4641 2598 6894 1147 5360 9532 3664 1757 1812 5830 9811 3755 7664 1538 5378 9185 2958 6699 0407 4085 7731 1347 4934 8490 2018 5518 8990 2434 5851 9241 2605 5943 9256 2544 5806 9045 2260 5451 8618 1763 4885 7985 1063 4120 7154 0168 3161 6134 9086 2019 4932 3029 7321 1570 5779 9947 4075 8164 2216 6230 0207 4148 8053 1924 5760 9565 3333 7071 0776 4451 8094 1707 5291 8845 2370 5866 9335 Q777 6191 9579 2940 6276 9586 2871 6131 9368 2580 5769 8934 2076 5196 8294 1370 4424 7457 0469 3460 6430 9380 2311 ees BRET Ree ee rei 3461} 3891 7748} 8174 1993} 2415 6197} 6616 0361 4486 8571 2619 6629 0602 4540 (8442 2309 6142 9942 3709 7443 1145 4816 8457 2067 5647 9198 2721 6215 9681 3119 6531 9916 327 5 6608 9915 3198 6456 9690 2900 6086 9249 2389 5507 8603 1676 4728 7759 0769 3758 6726 9674 2603 5512 0775 4896 8978 3021 7028 0998 4932 8830 2694 6524 0320 4083 7815 1514 5182 8819 2426 6004 9552 3071 6562 0026 3462 6871 0253 3609 6940 0245 3525 6781 0012 3219 6403 9564 2702 5818 8911 1982 5032 3061 1068 4055 7022 9968 2895 5802 bO GO GO OS OO G9 460 OO bo bo is fd feel fet et LOGARITHMS. 3 | 176091 | 6381 | 6670} 6959 | 7248 | 7536 | 7825 |8113| 8401 | 8689 | 289 8977 | 9264 | 9552 | 9839 | 0126 |0413 | 0699 | 0986 | 1272) 1558] 286 181844 | 2129 | 2415 | 2700'| 2985 | 3270 | 3555 | 3839 | 412314407 | 285 4691 | 4975 | 5259 | 5542 | 5825 | 6108 | 6391 | 6674 | 6956 |7239| 283 7521 | 7803 | 8084 | 8366 | 8647 | 8928 | 9209 | 9490 | 9771} 0051! 281 190332 | 0612 | 0892} 1171 | 1451 | 1730 | 2010} 2289 | 2567 | 2846} 280 3125 | 3403 | 3681 | 3959 | 4237 | 4514 | 4792} 5069 | 5346! 5623} 278 5900 | 6176 | 6453 | 6729 | 7005 17281 | 7556 | 7832 | 8107 | 8382} 275 8657 | 8932 | 9206 } 9481 | 9755 | 0029 | 0303] 0577 | 0850| 1124} 274 201397 | 1670 | 1943 | 2216 | 2488 | 2761 | 3033 | 3305 | 3577 | 3848} 272 204120 | 4391 | 4662 | 4933! 5204 | 5475 | 5745 | 6016 | 6286} 6556} 271 6826 | 7096 | 7365 7634/7903 8172 | 8441 | 8710] 8978 | 9247 | 269 9515| 9783 | 0051} 0318 | 0586 | 0853 | 1120! 1388] 1654} 1921| 267 212188 | 2454 | 2720 | 2986 | 3252 | 3518 | 3783 | 40491 4314| 4579 | 266 4844 | 5109 | 5373] 5638 | 5902 | 6166 | 6430 | 6694 | 6957 | 7221 | 264 7484 | 7747 | 8010} 8273 | 8535 | 8798 | 9060 | 9322 | 9584 | 9826 | 262 220108 | 0370 | 0631 | 0892 | 1153 | 1414 | 1675 | 1936 | 2196 | 2456} 261 2716 | 2976 | 3236 | 3496 | 3755 | 4015 | 4274 | 4533 | 4792 | 5051 | 259 5309 | 5568 | 5826 | 6084 | 6342 | 6600 | 6858 | 7115 | 7372) 7630 | 258 7887 | 8144 | 8400 | 8657 | 8913 | 9170 | 9426 | 9682} 9938 |0193| 256 230449 | 0704 | 0960 | 1215 | 1470} 1724 | 1979 | 2233 | 2488 | 2742| 255 2996 | 3250 | 3504 | 3757 | 4011 | 4264 | 4517 | 4770| 5023 | 5276} 253 ~ 5528) 5781 | 6033 | 6285 | 6537 | 6789 | 7041 | 7292| 7544 | 7795) 252 {| 8046 | 8297 | 8548 | 8799 | 9049 | 9299 | 9550} 9800 | 0050 | 0300} 250 240549 | 0799 | 1048 | 1297 | 1546} 1795 | 2044 | 2293 | 2541 | 2790} 249° 3038 | 3286 | 3534 | 3782 | 4030 | 4277 | 4524 | 4772 | 5019 | 5266} 247 5513 | 5759 | 6006 | 6252 | 6499 | 6745 | 6991 | 7236 | 7482| 7728} 246 7973 | 8219 | 8464] 8709 | 8954] 9198 | 9443 | 9687 | 9932|0176| 245 250420 | 0664 | 0908 | 1151 | 1395| 1638 | 1881 | 2125 | 2368 |2610} 243 2853 | 3096 | 3338 | 3580 | 3822 | 4064 | 4306 | 4548 | 4790 | 5031 | 242 5273 | 5514} 5755 | 5996 | 6237 | 6477 | 6718 | 6958 | 7198 |7438| 241 31} 7679) 7918/8158 | 8398 | 8637 | 8877 | 9116 | 9355 | 9594 | 9833) 239° 26007 1 | 0310 | 0548 | 0787 | 1025 | 1263} 1501 | 1738 | 1976 | 2214} 238 2451 | 2688 | 2925 | 3162 | 3399 | 3636 | 3873 | 4109 | 4346 | 4582} 237 4818} 5054 | 5290} 5525 | 5761 | 5996 | 6232 | 6467 | 6702 | 6937 | 235 5 7172 | 7406 | 7641 | 7875 |8110| 8344 | 8580 | 8812 | 9046 | 9279} 234 36 9513 | 9746 | 9980 | 0213 | 0446 | 0679 | 0912] 1144} 1377 | 1609} 233 37 | 271842 | 2074 | 2306 | 2538 | 2770| 3001 | 3233 | 3465 | 3696 | 3927 | 232 8 | 4158} 4389 | 4620} 4850} 5081 | 5311 | 5542 | 5772 | 6002 | 6232| 230 9 6462 | 6692 | 6921 | 7151] 7380 | 7609 | 7838 | 8067 | 8296 | 8525 | 229 90] 8754) 8982 | 9210] 9439| 9667 | 9895 | 0123 | 0351 | 0578 | 0806 | 228 91 | 281033 | 1261 | 1488] 1715] 1942} 2169 | 2395 | 2622 | 2849 | 3075 | 227 3301 | 3527 | 3753 | 3979 | 4205} 4431 | 4656 | 4882 | 5107 | 5332] 226 93} 5557 | 5782 | 6007 | 6232| 6456 | 6681 | 6905 | 7130} 7354| 7578 | 225 94] 7802] 8026 | 8249 | 8473 | 8696 | 8920| 9143 | 9366 | 9589 | 9812| 223 95 | 290035 | 0257 | 0480 | 0702 | 0925| 1147 | 1369/1591} 1813] 2034} 222 96 | 2256 | 2478 | 2699 | 2920] 3141 | 3363 | 3583 | 3804 | 4025 | 4246 | 221 97} 4466 | 4687 | 4907 | 5127 | 5347 | 5567 | 5787 | 6007 | 6226 | 6446 | 220 38 6665 }6884 | 7104 | 7323 | 7542 | 7761 | 7979 | 8198 | 8416 | 8635} 219 9 - $853 | 9071 | 9289 | 9507 | 9725} 9943 | 0161 | 0378 | 0595 | 0813; 218 - i a nae eee nen nen SL ee ttneennate at catneneesiianeeaGninpmernemtemnemnaemmanmel 92 N. | 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 Q17 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 © 237 238 239 240 241 242 243 244 245 246 247 248 249 0 301030 3196 5351 7496 9630 311754 3867 5970 8063 320146 2219 4282 6336 3380 330414 2438 4454 6460 8456 340444 2423 4392 6353 8305 350248 2183 4108 6026 7935 9835 361728 3612 5488 7356 9216 371068 2912 A748 6577 8398 380211 2017 3815 5606 7390 9166 390935 2697 4452 6199 “LOGARITIMS 4 1898 4059 6211 8351 0481 2600 4710 6809 8898 0977 3046 5105 7155 9194 1225 3246 5257 7260 9253 1237 3212 5178 7135 9083 1023 2954 4876 6790 8696 0593 2482 4363 6236 8101 9958 1806 3647 5481 7306 9124 0934 2737 4533 6321 8101 9875 1641 3400 5152 6896 ba 5 2114 4275 6425 8564 0693 2812 4920 7018 9106 1184 3252 5310 7359 9398 1427 3447 5458 7459 9451 1435 3409 5374 7330 9278 1216 3147 5068 6981 8886 0783 2671 4551 6423 8287 0143 1991 3831 5664 7488 9306 1115 2917 4712 6499 8279 0051 1817 3575 5526 7071 6 by ts {59 2547 | 2764 4706 | 4921 2331 4491 6639 8778 0906 3025 5130 7227 9314 1391 3458 5516 7563 9601 1630 3649 5658 7659 9650 1632 3606 5570 7525 9472 1410 3339 5260 (fee 9076 0972 2859 4739 6610 8473 0328 2175 AO15 5846 7670 9487 1296 3097 4891 6677 8456 0228 1993 3751 5501 7245 6854 8991 1118 3234 5340 7436 9522 1598 3665 5721 7767 9805 1832 3850 5859 7858 9849 1830 3802 5766 7720 9666 1603 3532 5452 7363 9266 1161 3048 4926 6796 8659 0513 2360 4198 6029 7852 9668 1476 3277 5070 6856 8634 0405 2169 3926 5676 7419 7068 9204 1330 3445 5551 7646 9730 1805 3871 5926 7972 0008 2034 405] 6059 8058 0047 2028 3999 5962 7915 9860} 1796 3724 5643 7554 9456 1350 3236 5113 6983 8845 0698 2544 4382 6212 8034 9849 1656 3456 5249 7034 8311 0582 2345 4101 5850 7592 2980 5136 7282 9417 1542 3656 5760 7854 9938 2012 4077 6131 8176 0211 2236 4253 6260 8257 0246 2225 4196 6157 8110 0054 1989 3916 5834 7744 9646 1539 3424 5301 7169 9030 0883 2728 4565 6394 8216 0030 1837 3636 5428 7212 8989 0759 2521 4277 | 6025 7766 89 89 0 9 20 DD © 19 28 c \ > _“ “i ™ tet ta tt a et — on 1g 1g > . 0 50 | 397940 151 9674 52 | 401400 153 3120 154 4834 155 6540 56 8240 S57 9933 858 | 411620 59 3300 60 4973 161 6640 62 8301 063 9956 64 | 421604 v5 3246 166 4882 67 6511 268 8135 269 9752 270 | 431364 271 2969 R72 A569 273 6163 274 7751 275 9333 276 | 440909 177 2480 278 4045 279 5604 280 7158 81 8706 282 | 450249 283 1786 284 3318 285 4845 286 6366 287 7882 88 9392 1289 | 460890 290 2398 291 3893 292 5383 1293 6868 294 8347 295 G822 296 | 471292 1297 2756 '298 A216 5671 OF NUMBERS. iWicat. 3 8114 9847 1573 3292 5005 6710 8410 0102 1788 3467 5140 6807 8467 0121 1768 3410 5045 6674 8297 9914 1525 3129 4728 6322 7909 9491 1066 2636 4201 5760 7313 8861 0403 1940 3471 4997 6518 8033 9543 1048 2548 4042 5532 7016 8495 9969 1438 2903 4362 5816 8287 0020 1745 3464 5175 6881 8579 0271 1956 3635 5307 6973 86335 0286 1933 3573 5208 6836 8459 0075 1685 3290 4888 6481 8067 9648 1224 2793 4357 5915 7468 9015 0557 2093 3624 5149 6670 8184 9694 1198 2697 A19L 5680 7164 8643 0116 1585 3049 4508 5962 8461 0192 1917 3635 5346 7051 8749 0440 2124 3802 5474 7139 8798 0451 2097 3737 5371 6999 8621 0236 1846 3450 5048 6640 8226 9806 1381 2950 4513 6071 7623 9170 0711 2247 3777 5302 6821 8336 9845 1348 2847 4340 5829 7312 8790 0263 1732 3195 4653 6107 4 8634 0365 2089 3807 5517 7221 8918 0608 2292 3970 5641 7306 8964 0616 2261 3901 5534 7161 8782 0398 2007 3610 5207 6800 8384 9964 1538 3106 4669 6226 7778 9394 0865 2400 3930 5454 6973 8487 9995 1498 2997 4489 5977 7460 8938 0410 1878 3341 4799 6252 5 8808 0538 2261 3978 5688 7391 9087 O777 2460 4137 5808 7472 9129 0781 2426 4064 5697 7324 8944 0559 2167 3770 5366 6957 8542 0122 1695 3263 A825 6382 7933 9478 1018 2553 4082 5606 7125 8638 O146 1649 3146 A639 6126 7608 9085 O557 2025 3487 4944 6397 # * % 6 898] O711 2433 4149 5858 7561 9257 0946 2628 4305 5974 7638 9295 0945 2590 4228 5860 7486 9106 0720 2328 3930 5526 7116 8700 0279 1852 3419 4981 6537 8088 9633 1172 2706 4235 5758 7276 8789 0296 1799 3296 4787 6274 7756 9233 0704 AG fl 3633 5090 6542 nT 9154 0883 2605 4320 6029 7731 9426 1114 2796 4472 6141 7804 9460 1110 27 54, 4392 6023 7648 9268 0881 2488 4090 5685 1245 8859 0437 2009 3576 5137 6692 8242 9787 1326 2859 4387 5910 7428 8940 0447 1948 3445 4936 6423 7904 9350 0851 2317 3779 5235 6687 s | 9 | 9327 1056 weed 4492 6199 7901 9595 1283 2964 4639 6308 7970 9625 1275 2918 4555 6186 7811 9429 1042 2649 4249 5844 7433 9017 0594 2165 3732 5293 6848 8397 9941 1479 3012 4540 6062 7579 9091 O597 2098 3594 5085 0571 8052 9527 0998 2465 3925 5381 6832 9501 1228° 2949 4663 6370 8070 9764 1451 3132 4806 6474 8135 9791 1439 3082 4718 6349 7973 9591 1203 2809 4409 6003 7592 9175 0752 2323 3888 5448 7003 8552 0095 1633 3165 4692 6214 7730 9249 0747 2248 3744 5234 6719 8200 9675 1145 2610 4070 5526 6976 173 17 Se 172 170 169 167 167 161 160 158 158 156 155 } 55 153 152 152 151 tol 150 150 149 149 148 147 147 146 146 146 145 N;-+]“—0 300 |477121 301 | 8566 302 |480007 303 | 1443 304 | 2874 305 | 4300 306 | 5721 307 | 7138 308 | 8551 309 | 9958 310 | 491362 311 | 2760 312'| 4155 313 | 5544 314 | ‘6930 315 8311 316 | -9687 317 | 501059 318 | 2427 319 | 3791 320 | 5150 321 | 6505 322 | 7856 323 | 9202 324 | 510545 325 | 1883 326 | 3218 327 | 4548 398 | 5874 329 | 7196 330 | 8514 331 | 9828 332 | 521138 333 | 2444 334 | 3746 335 | 5045 336 | 6339 337 | 7630 338 | 8917 339 | 530200 340 | 1479 341 | 2754 342 | 4026 343 | 5294 344 | 6558 345 | 7819 346 | 9076 347 | 540329 348 | 1579 349 | 2822 7266 2711 O151 1586 3016 4442 5863 7280 8692 0099 1502 2900 4294 5683 7068 8448 9824 1196 2564 3927 5286 6640 7991 9337 0679 2017 3351 4680 6006 7328 9645 9959 1269 2575 3876 5174 6468 |. 7759 9045 0328 1607 2882 4153 5421 6685 7945 , 9202 0455 1704 ate LOGARITUMS 7555 8999 0438 1$72 3302 A727 6147 7563 8973 0380 1782 3179 4572 5960 7344 8724 0099 1470 2837 4199 5957 6911 8260 9606 O947 2284 3617 4946 6271 7592 8909 0221 1530 2835 4136 5435 6727 8016 9302 0584 1862 3136 A407 5674 6937 8197 9452 |O705 1953 3199 7844 9287 0725 2159 3587 5011 6430 7845 9255 0661 2062 3458 4850 6237 7621 8999 0374 1744 3109 4471 5828 7181 8530 9874 1215 2551 3883 5211 6535 7855 9171 0483 1792 3096 4396 5693 6985 8274 9559 0840 2117 3391 4661 5927 7189 8448 9704 0955 2203 3447 7989 9431 Q869 2302 5730 5153 6572 7985 9396 0801 2201 3597 4989 6376 7759 9137 0511 1880 3246 4607 5963 7316 8664 0008 1348 2684 A016 5344 6668 7987 9303 0614 1922 3226 4526 5822 7114 8402 9687 0968 2245 3518 A787 6053 7315 8574 9829 1080 2327 3571 pees |~2 ies ae > COMMER OP 1 GUINEA per Quarter. FRENCH and the CLASSICS each ...........- 1§ GUINEAS per Quarter. GERMAN, DUTCH, and ITALIAN, each...... 2 GUINEAS per Quarter. SPANISH and PORTUGUESE, each ........... 2 GUINEAS per Quarte?- PEA LOL LOD DDD DAD DIL LIF TERMS FOR THE MATHEMATICAL DEPARTMENT, For a Complete Course of Instruction. PILLS IID PPP IPP DED ODO NAVIGATION, with the Lunar Observations, &c ey ooees4 GUINEAS, FORTIFICATION and GUNNERY ...,.. os cabal cMible tila cececet GUINEAS, EVUCLID’S ELEMENTS ......s00c0e Cec eccecesrecececseseeeees) GUINEAS. Pe A Tete oo os oem Mego Boe Sccwsch siieeipe sedecea GUINEAS, PLANE TRIGONOMETRY ........... Valaivie 66's OMY wee evieie athe 2 GUINEAS. SURERICAT. TRIGONOMETRY wow anaes Caviad ites, fhloes ok 4 GUINEAS. CON ICASECELONS 6 00 ccc ox vd tite bance. ete sie crus Te ere 4 GUINEAS, PUTOPR Mirena ee cee ate es ot cas Oued EE ee ee ee ..-5 GUINEAS, MENSURATION, SURVEYING, and GAUGING: ............ 5 GUINEAS, PhaGrlCAL GEOMELRY .... chi eces. Vode shaee sins coebyde os TUINDAS, HUTTON’S COURSE, exclusive of FLUXIONS . ..........12 GUINEAS. PPP PLIPL PLD ILD LOD LODO LLD DRAWING is taught at...............1 GUINEA and a HALF per Quarter. And ONE GUINEA Entrance. PLP DIP BDPP DODLE I DOP DOD TERMS - For a general Course of Instruction. PPP DID PIPL AIS PLL FIRST CLASS .............. 5 Guineas per Quarter. SECOND CLASS ... 0-6 ose ... 4 Guineas per Quarter. THIRD CLASS .............. 3 GuinEAs per Quarter, PPLF DLE PDL BOP DEE LL ID LOE DOP ENTRANCE ...2.. 2 GUINEAS. PPP PIF DIL AIL DLP DLL DDD LIG me When it is preferred to pay the Terms half-yearly in advance, the Entrance is not charged. BALD PILI LIS PIP LLL TERMS For Dinners. ~~ Four Guineas per Quarter. « Mr. TATE respectfully announces, that he receives Four Young Gentlemen as Inmates of his family, at his residence, Shacklewell-House, near Kingsland, who attend his Establishment in Cateaton-Street, in company with himself and his Sons. The arrangements of this department entirely coincide with the plan of a private Family, and are replete with advantages of a very superior nature, both with regard to the domestic comforts of the young persons, and to the opportunities afforded for a rapid progress in their Studies. The Terms for Board only, are Eighty Guineas per Annum ; when a separate Room is required, the Terms are ‘Twenty Guineas extra. The Terms for Tuition are the same as those inthe preceding Prospectus. GID DID LID LOL ODS LOD COO Gentlemen residing at a distance, may be accommodated with Dinners at the Academy. ni aoe fraathae Gib? eiifhalingsti € crib, 9/6 \lleualionult gong | San? ahs as a EMT Seivsliwed si ee poe iat 14 ed ; : Le Sail ais Seta Td eS oak 2, ok, soe ee Oe AN |e FON pee eS & 2. i aoe BOS © VLSTEH); TRUS Shi Bey SE YTS 3 eS bes he Donat isla Ba ae chy hed Wh | Ga k; pea tk. Deby! oY ait hs a Te. 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