finally, the depth receive a similar augmentation, changing it from 2 to 3, the solidity will become 81. But 24, the original content, is to 81, the last result, as 4' to 6%, or as 3 to 4.5°, or as 28 : 38. As an additional illustration of the relation between the volumes of similar solids, suppose the middle frustum of a spheroid to have the content of 30516 cubic inches, and the length of a diagonal line drawn from the centre of one of the sides through the middle section of the solid to the bottom of either of the circular ends, to be 36.45 inches. Also suppose the length of a similar diagonal drawn through the middle frustum of another spheroid to be 30.38; to find from these data the content of the latter frustum. Diag. Diag. Content Content 36-45: : 30-388 :: 30516 : 17668. Answer. In practical gauging advantage is taken of this property in the construction of what is called the “diagonal rod," an instrument by means of which, as will be explained in the next chapter, the content of any cask resembling in form the middle frustum of a spheroid, may be determined approximately with great facility. Of all plane figures having perimeters of equal length, that figure contains the largest area in which the perimeter consists of the greatest number of sides. Now, a circle may be conceived to be a polygon of an infinite number of sides (see page 182); it encloses therefore a larger surface than any other plane figure within the same extent of boundary. Thus, if the three sides of a triangle ineasure 9, 12, and 15, its area will be 54 ; if two adjacent sides of a rectangle measure 10 and 8, its area will be 80: if a side of a regular dodecagon measure 3, the area will be 100.76, and if the circumference of a circle be 36, the area will be 103.12. These several figures have the same total length of boundary, viz., 36, but their areas are respectively 54, 80, 100.76, and 103.12. A regular polygon has a greater area than an irregular polygon of the same number of sides and the same length of perimeter; and the area of a square is greater than the area of any other quadrilateral having an equal perimeter. This amounts in effect to saying, that if several equal numbers be added together, and if the same sum be also made up of the same number of unequal parts, then the product of the equal parts will exceed the product of the unequal parts. So when a given number is divided into two parts only, those parts which yield the greatest product are the halves of the number. Example. 24 = 7 + 6 + 5 + 3 + 2 + 1, and the product of these numbers = 1260. But 24 also = 4 + 4 + 4 + 4 + 4 + 4, and the product of these numbers = 4096. Again 24 is made up of the two equal numbers, 12 and 12, the product of which exceeds the product 13 x 11, 14 x 10, 15 x 9, or any other two numbers together = 24. Of all parallelograms of equal perintoter, the rectangle contains the greatest area ; for if the sides of a parallelogram be at all oblique, it is evident that the perpendicular height of the figure cannot be so great as if these sides were at right-angles to the base ; and the area in both cases is found by multiplying the base by the perpendicular height. As the circle includes the greatest surface of all plane figures within the least compass, so it may be easily inferred, the sphere contains the largest volume within the smallest amount of surface, since a sphere is produced by the revolution of a semi-circle around its chord or diameter. Accordingly, the more nearly the form of any solid approaches to that of a sphere, the greater volume will it include within a given surface. Hence, when the spherical ball of a hydrometer becomes indented, the indications of the instrument will be rendered inaccurate by the change in its dimensions causing it to displace less than the proper quantity of fluid. (14.) LOGARITHMS.—Some knowledge of the nature and use of logarithms should be possessed by officers generally; 1st, because logarithms afford in many instances an easy means of abridging the labor of calculation or of verifying the results arrived at by ordinary methods. 2nd, because the construction and extended application of the slide-rule cannot be understood by persons who are totally ignorant of the principles of logarithmic arithmetic, although it is true that the manner of employing that instrument for a few special practical purposes may be acquired without any study of its theory. A very brief and familiar account of logarithms, is, however, all that can be given in the present work. Fractional Exponents. The ordinary notation of the powers and roots of numbers has been fully described in the Chapter on Arithmetic, page 144. It is now necessary to add, that not only the simple extraction of any required root, but also the twofold operation of involution and evolution as performed on the same number, may be very conveniently expressed by the use of a fractional index or exponent, thus dispensing altogether with the radical sign. When an index of this kind is employed, the numerator of the fraction is to be regarded as signifying the order of the power to which the given number is to be raised, and its denominator the order of the root of that power, to be afterwards extracted. Or, since it is immaterial to the result which of these processes is first executed, the denominator of the fractional index may be viewed as a direction to extract a certain root of the number, and the numerator as a direction to involve the root so found to a certain power. For example, the expression 161, means either that the fourth root of the third power of 16, or that the third power of the fourth root of 16 is to be taken. 168 = 16 x 16 x 16 = 4096, And 14/4096 = 8 And 28 = 8 Any such indes, it is evident, may be written in the form of a decimal instead of a vulgar fraction without affecting its value. Thus 162 = 1610 = 16°° = 4, that is, the tenth root of the fifth power of 16 is the same number as the squareroot of the first power of 16 or 16 itself. In all other respects fractional exponents follow the same rules of operation as fractions generally. For instance, 48+} = 41$; 48-} = 475; 45x} = 4145; 45:} = 440 or 4* + . By adopting this system, the notation of roots is completely assimilated to that of powers, and the great advantage of uniformity is secured; the term power being here applied to any quantity with an index or exponent, whether that be a whole number or a fraction. Exponents in general.-Exponents, when attached to the same number, possess certain useful properties which the following short table is intended to elucidate. 10% = 10 103 = 3.1623 10$ = 2.1544 103 = 1.7783 10 = 1.4678 1012 = 1.2115 103 = 4.6416 &c., &c. 1st. If it be required to multiply any two or more powers of the same number by one another, we have merely to add together the several exponents, and find the power answering to the sum of these exponents. Example. 102 x 10 x 10* = 100 x 1000 x 10000 = 1000000000. 2nd. If we wish to divide any power of a number by a smaller power of the same number, we have merely to subtract the exponent of the divisor from that of the dividend, and find the power indicated by the difference of these exponents. Example. 108 - 10% = 100000000 = 100000 = 1000. But 1000 = 109 and 3 is the difference of 8 and 5. Again, 10% = 107 = 2.1544 – 1:7783 = 1.2115. But 1.2115 = 1071, and 1=1-1 3rd. To involve any power of a number to any other required power of the same number, it is only necessary to multiply the exponent of the given power by that of the required power, and to find the power answering to the product of such exponents. Example. (10) = 100° = 100000000. But 100000000 = 10°, and 8 is the product of 2 and 4. Similarly, (10%) = 1-4678° = 3-1623. It will be readily inferred that any other number besido 10 would exhibit similar proporties to those which have been exemplified in respect of that particular number. The officer will now have no difficulty in understanding the nature of logarithms, the purposes to which they are made subservient, and the reasons of the rules for their management. Definition of a logarithm.-When an exponent is considered, not with reference to the number of which it signifies a certain power, but with reference to the developed power itself, it is called the logarithm of the latter. Thus in the expression 10'=1000 the number 3 denotes both the exponent of 10 and the logarithm of 1000. So again, when we write 8= 2, the fraction may be taken to mean either the exponent of 8 or the logarithm of 2. To express what that quantity is with respect to which 1 is the logarithm of 2, it is said, that 1 is the logarithm of 2 to the base 8. For a similar reason it is said, that 3 is the logarithm of 1000 to the base 10. The necessity of mentioning a particular number as the base, in all such instances, arises from the fact that the same exponent may serve as the logarithm of an infinite variety of numbers ; e.g., 4' = 64; 78 = 343 ; 10% = 1000, and so on. Here the exponent 3 is equally the logarithm of 64, 343, 1000, &c., but to different bases, and in order to estimate the right value of any given logarithm it is evidently requisite that the number to which it is to be referred as base should be specified. For nearly all purposes the number 10 is the most convenient that can be adopted as the base of a system of logarithms. It is to this base, therefore, that the logarithms in common use are calculated and referred. The word logarithms is derived from the Greek and signifies the ratios of numbers. 10° is said to have to unity, the double or duplicate ratio of 10 to unity; 10°, the triplicate or threefold ratio, and so on. Accordingly, the numbers 2, 3, &c.,—which are the logarithms of 10", 10°, &c., to the base 10-actually express the ratios of 10", 10°, &c., to unity, as compared with the ratio of 10 to unity, and are thus termed with propriety the logarithms of these quantities. Formation of logarithms.-As then logarithms are nothing more than the exponents of some fixed number raised to different powers, it is plain that the same rules which apply to operations with exponents generally, must govern these operations when the exponents are regarded as logarithms. Accordingly, if by any process the various powers of 10, whole or fractional, which are severally equal to the natural numbers from 1 upwards to 10,000 or 100,000, could be computed, and if the exponents of such powers were inserted in a table with the natural numbers opposite each to each, we should be enabled by means of this table, and so far as regarded all numbers comprised within its limits, to substitute addition for multiplication, subtraction for division, and simple multiplication and division for the tedious and complex operations of involution and evolution respectively. Now, since 10% = 10; 10% = 100, &c., the logarithms of the whole powers of 10 may be very readily formed to any required extent; but when it comes to be inquired what fractional powers of 10 are equal or approximately equal to the numbers, 2, 3, 5, &c., that is, how are we to fill in the exponents in the expression, 10 = 2, 10 = 3, &c., there is some difficulty in answering the question. Unaided arithmetic is here of little avail; it can merely inform us, that as 1 is the logarithm of 10 to the base 10, as 2 is the logarithm of 100 to the same base, and so on, the logarithms of all numbers less than 10 must be proper fractions or decimals, the logarithms of all numbers between 10 and 100 must consist of unity with some fraction attached, &c., &c. Algebraic analysis, however, supplies a general method or formula by which the necessary exponents or logarithms may be computed, and set forth in decimals, without great trouble, to any specified degree of exactness. The details of this investigation, or even a statement of the mode of calculation to be adopted, would be unsuited to the present work. It is sufficient for the practical computist to know, that the labor of determining logarithms answering approximately to all the natural numbers from 1 to 100,000 has been carefully gone through by various persons, independently of one another, and that the results of these operations have been embodied in sets of tables, most of which are trustworthy and inexpensive.* * The tables of five-figure logarithms published by Taylor and Walton, London, price 18. 6d., are accurate, portable, and in every respect well adapted for ordinary use. 0206 100301 10047 10080 10000 10077 10084 10000 10005 co voor 101 Exponents in general.—To facilitate the proper acquisition of all that remains to be understood respecting the nature and application of common logarithms, the following short tables are here introduced: Numbers. Logarithms. 100 1 . .. 0.00000 2 . . . . .30103 7712 .47712 .60206 .69897 7815 .77815 7 . . .84510 •845 O 309 8. . . . .90309 9 . . . 95424 10700000 10 . . . 1.0000 In every system of logarithms, the exponent of that power of the base which is equal to 1, must be 0; for on the principles which have just been shown to apply to the division of a power of a number by another power of the same number, 10' = 101–1 = 10°; but i = 1, consequently, 10o = 1. It is evident that the same property holds good whatever number be taken as the base, and also whatever be the two equal exponents of the dividend and divisor, since the quotient in this case will always = 1, and the difference of the exponents = 0. By any of the expressions in the table to the left, as for instance, 100'30103 = 2, is meant, that if 10 be raised to the 30103rd power, and the 100000th root of this power be extracted, or vice versa, the result will very nearly give the number 2. All the other expressions are to be similarly interpreted. The table to the right is a specimen of the manner in which tables of logarithms are usually arranged. As '30103 denotes that fractional power of 10 which approaches very closely to the number 2, so opposite to the number 2 is placed in the column of logarithms, the decimal 30103. Each of the first nine logarithms in the preceding table consists entirely of decimals, or has 0 for its integral part. Now, if instead of the logarithms of 2, 3, 4, &c., we required the logarithms answering to ten times these numbers, that is, to 20, 30, 40, &c., we should find that it is only necessary to prefix 1 to the logarithms of 2, 3, 4, &c., respectively, to obtain the logarithms in question; for since the exponent of the product formed by multiplying different powers of the same number together is the sum of the exponents of the factors, (page 212) 10 4 7 7 1 2 X 101 = 101'47 712 ; but 1047 712 = 3, and 101 = 10. Therefore, 1014 7 7 12 = 3 X 10 = 30, or, 1.47712 is the logarithm of 30 to the base 10. Similarly, 10*60 206 = 4, and 10-60 206 x 10' = 102'00 206 = 4 x 10 = 40 or 1.60206 is the log. of 40 to the base 10, and so on. Again, if the logs. of 200, 300, 400, &c., or 100 X 2, 100 x 3, 100 X 4, &c., be sought, we have by the same reasoning, 10-30 1 0 3 x 102. = 102'30 103. But 10 30 103 = 2, and 10= 100. Accordingly, 102'30 10 3 = 2 X 100 = 200, that is, the log. of 200 is 2.30103, and so of the other proposed numbers; from which we may infer generally, that having given the logs. of the nine digits, 1, 2, 3, &c., in order to express the log. of the product of any digit by 10, 100, 1000, &c., or any other power of 10, we have merely to write before the given log. the log. of such power of 10; and on inspection of an extended table of logarithms it will be seen that this principle applies to all the values contained in it-that, for instance, the decimal figures placed opposite to the number 4700, are identical with those placed opposito to 47, since 4700 = 100 x 47, and the log. of 4700 must be equal to the log. of 10, added to the log. of 47, that is, to 2 + log. of 47. That part of any log. which stands to the left of the decimal point, and enables us to distinguish the value of the figures in the corresponding natural number, is called the characteristic of the log. because it indicates the lower of the two powers of 10 between which the given number lies, and thus serves to characterise or particularise the log. The characteristic of the log. of 19, for example, is at once seen to be 1, since 19 stands between 1 and 100, the first and second powers of 10. Again, the characteristic of the log. of 658 |