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to give exactly lor=2. We immediately see that x must be considerably less than }, or, which is the same thing, 1o is greater than 2. For, squaring both sides, the square of 10! = 101, and that of 2=4, which latter is much less than the former. So } is still too great a value of r, that is to say, 103 is greater than 2; for the cube of lo! is 10, and that of 2 only 8. On the contrary, by making x=}, we give it too small a value; for the fourth power of 10 being 10, and that of 2 being 16, it is evident that 1ot is less than 2: x then, or L.2, is less than }, and yet greater than 1. In the same way we may determine, with respect to every fraction contained between and , whether it be too great or too small.

Trying, for example, with 3, which is a trifle less than }, and greater than }, ; 1o or 103

must = 2, or the seventh power of 10, that is to say, 10° or 100 must be equal to the seventh power of 2; now the latter = 128, and is consequently greater than the former.

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641. It has been shown that the logarithm of 2 is greater than in and less than !, and that therefore the exponent of 10 lies between those two fractions, in order that the power may become = 2. But, although this is known to us, whatever fraction is assumed on this condition, the power resulting from it will always be an irrational number greater or less than 2; the logarithm, therefore, of 2 cannot be accurately expressed by such a fraction: hence we must be content with such an approximation to it as will render the error of no importance. For this purpose decimal fractions are used, which we shall now explain.

642. In the ordinary way of writing numbers by means of the ten figures or chararters

O, 1, 2, 3, 4, 5, 6, 7, 8, 9,

the first figure on the right hand is the only one which has its natural signification; the figures in the second place have ten times the value they would have had in the first; those in the third place have a hundred times the value, and those in the fourth a thousand times, and so on; so that in proportion as they advance towards the left, they acquire a value ten times greater than they had in the preceding rank. Thus, in the number 1849, the figure 9 is in the first place on the right, and is just equal to 9. That in the second place is 4, but this figure, instead of 4, represents 10 x 4 or 40 The figure 8 in the third place is equal to 100 x 8, or 800. Lastly, the 1, which is the fourth to the left, is equal to 1000, hence the number is read as follows, –


One thousand eight hundred and forty-nine.

643. As the value of figures becomes in each rank always ten times greater as we go from the right towards the left, and as it continually becomes less as we proceed from the left to the right, we may by this law advance still further towards the right, and obtain figures whose value may continually decrease and become ten times less; but where this occurs, the place where the figures cease to have their natural value will continue to become ten times less. In this case, however, the place where the figures have their natural value is marked by a point placed after that rank. Thus, if, for instance, we meet with the number 54-76938, it must be thus understood: — the figure 4 in the first place has its natural value, and the second 5 means 50; but the figure 7 which comes after the point expresses only , ; , the figure 6 is equal only to 1%. , the figure 9 is equal to Tú" ; the figure 3 to mism, and the figure 8 to ; thus the more these figures advance towards the right, the more their values diminish, till at last those values become so small, that they may at last be considered as nothing. This species of numbers, then, are what are called decimal fractions, and in this way logarithms are represented in the tables. Thus the logarithm of 2 is expressed by O-3010300, wherein we perceive, as the logarithm does not contain an integer, that its value is # +1% + r.'" + Tw'', 4 miàn 4 Tu'un + Tom'iyun. The last two ciphers might have been omitted; they, however, serve to show that the logarithm quoted contains no parts which have 1000000 and 10000000 for the denominator. It is possible, however, that by continuing the series, smaller parts might have been found, which are neglected, except in extraordinary cases, on account of their extreme minuteness. 644. The logarithm of 3 is known by the tables to be 0.4771213, and, containing no integer, consists of the following fractions:-1: 4-Ti, 4 It's + Tulun+ Tučw 4: Tunkun, 4 Toni'im). This logarithm is, however, not expressed with the utmost exactness; we are only certain that the error is less than Tumkum, one so small, that there are few calculations in which it may not be neglected. 645. By this method of expressing logarithms, that of 1 will be represented by o-ooooooo, since it is really =0. The logarithm of 10 is 1 0000000, or exactly = 1. The logarithm of 100 is 2.0000000, or exactly =2. Hence the logarithms of all numbers between 10 and 100, and, consequently consisting of only two figures, must be comprehended between 1 and 2, and are, therefore, expressed by 1 + a decimal fraction. Thus L-50 = 1 6989700; its value, therefore, is unity added to # 4-1}, + I'm + In on + Inding. So it must be evident that between 100 and 1000 the logarithms of numbers are expressed by two integers with a decimal fraction; the logarithms of numbers between 10000 and 1ooooo by four integers joined to a decimal fraction, and so on. The log. 600, for example, is =2.7781513; that of 2460 is 3:3909351, &c. But the logarithms of numbers less than 10, or those expressed by a single figure, do not contain an integer, and for this reason we find an o before the point. Hence there are two parts of a logarithm which require consideration: the former, that which precedes the point, and denoting the integers, if any; the other, the decimal fractions to be added to the integers. The first part, or integer, of a logarithm, usually called the characteristic, is easily determined from what has been already shown,-that is, it is o for all the numbers having but one figure; I for those which have two; 2 for those which have three, and generally less by one unit than the number of figures. Hence, if the logarithm of 5682 be required, we immediately perceive that the first part, or that of the integers, must be 3. So, reciprocally, when we see the integers of a logarithm, since the number it expresses is greater by one unit than the integer of the logarithm, we know, at once, the number answering to it. Thus, having 4-4771213 for the logarithm of an unknown quantity, it is evident that the number must have five figures, and exceed 10000. Now this number is 30000 for log. 30000= L.3 + L. 10000. Now the logarithm of 3 is known to be equal 0.4771213, and the logarithm of 1OOOO=4, and the sum of those logarithms=4:4771213. 646. From this it will be seen that the first object in considering a logarithm is the decimal fraction following the point, because, when that is known, it will serve for several numbers. For the proof whereof let us take the number 456. Its first part must be 2; and if we represent the decimal fraction which follows it by r, we have L.456=2 + r. If we continue to multiply by 10, we find L.4560=3 + r ; L.45600 = 4 + x; L.456000 =5+ x, and so on. But, if we divide instead of multiply by 10, we shall have 45-6 = 1 + r ; L.4:56 = 0 + r ; L.0:456 = -1 + r ; L.0.0456 = -2 + r ; L.000456 = -3 + x, and so on. 647. Thus, all the numbers arising from the figures 456, whether preceded or followed by ciphers, have the same decimal fraction for the second part of the logarithm, and their differences lie in the integer before the point, which becomes negative when the number is less than 1. As ordinary calculators have difficulty in the use of negative numbers, it is customary to increase the integers of the logarithm by 10, or to write 10

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instead of 0 before the point: by which process, instead of -1 we have 9; instead of -2 we have 8; instead of -3 we have 7, &c. But, under these circumstances, it must be recollected that the characteristic has been made ten units too great; nor must we assume that the number consists of ten, mne, or eight figures. We may easily see, in the case in question, that if the characteristic be less than 10, the figures of the number must be written after a point. If the characteristic be 9, we must begin at the first place after a point; if it be 8, we must also place a cipher in the first row, and not begin to write the figures till the second. Thus 9:6589648 would be the logarithm of o-456, and 8.658964.8 the logarithm of 0.0456. This manner of using logarithms is, however, chiefly confined to the use of tables of sines.

648. Ordinary tables do not carry the decimals of logarithms further than seven places or figures, the last whereof must consequently represent the pulmonth part, and we know that they do not err even by so small a part; the error, therefore, is of no importance in ordinary cases. But there are cases, though of no importance in our application of their use, in which still greater exactness is required, and in such cases ordinary tables are not suited to the case.

649. From the circumstance of the characteristic being known at a glance, the tables never give it, but are restricted to the seven figures of the decimal fractions. There are tables wherein the logarithms of all numbers from 1 to 100000 and even those of greater

- numbers are given, by means of small additional tables, showing what is to be added in

proportion to the figures which the proposed numbers have more than those in the tables. But from what has been said, we think the use of them will not be difficult; and, supposing such tables before the reader, we propose the multiplication of the numbers 2401 and 343. The addition of the logarithms of these numbers will, from what has been shown, give the product, as follows:–

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Hence the number sought is 823543. For the sum is the logarithm of the product
required, and its characteristic 5 exhibits a product composed of 6 figures, and they are
found, by the decimal product and the fraction, to be 823543.
650. But it is in the extraction of roots that logarithms are most serviceable; for
example, if we have to extract the square root of 10, we have only to divide the logarithm
of 10, which is 1.0000000, by 2, the quotient 0-5000000 is the logarithm of the root
required, which, in the tables, answers so nearly to 3-16228, whose square answers to 10
so nearly, that it is only one hundred thousandth part too great.
651. The operation, in addition, of expressions consisting of several terms is frequently
represented merely by signs, each expression being placed between two parentheses and
connected with the rest by means of the sign +. Thus, to add the expressions a +b+c
and d-e Ef the sum is thus represented: –

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This, however, is rather representing than performing addition; but, if the parentheses are left out it is then actually performed; for, as the number d4 e4 f is to be added to the other, it is to be done by joining to it +d, then + e, and then +f. If any term is affected by the sign –, it must be joined in the proper way with that sign. To illustrate this, let us consider an example in pure numbers; for example, 16–9 to 13-5. If we begin by adding 16, we shall have 13–5 +16. But this was adding too much, since what was to be added was 16-9, and it is therefore clear that we have added too much by 9; we must, therefore, take away the 9 by writing it with the negative sign, and thus we shall have the true sum –

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13-54-16-9, which shows that the sums result from writing all the terms each with its proper sign. 652. If therefore, it were required to add the expression d-e-f to a-b-c, the sum must be expressed as under: –

a-b-c-d-e-f; wherein it is of no importance in what order we write the terms, if their proper signs be preserved; the sum, for example, might be written

c-e +a–f4-d-b. Hence it will be seen that addition is attended with no difficulty, be the forms of the terms to be added together what they may. Thus, suppose we wished to add the expressions 2a3+6vb-4L.c, and 54/a-7c, they would be written

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655. To represent subtraction each expression is inclosed within two parentheses joining, by the sign –, the expression to be subtracted to that from which it is to be £ Thus, if d-e +f is to be subtracted from a b + c, the remainder is written thus : –

(a-b-c)–(d-c +f);

and this method sufficiently shows which of the two expressions is to be subtracted from the other.

656. But if actual subtraction is to be performed, we must observe, first, that when we subtract a positive quantity + b from another quantity +a, we obtain a -b; and, secondly, when we subtract a negative quantity –b from a, we obtain a + b ; for, to discharge the debt of a person is the same as to give him something. Suppose the expression b–d is to be subtracted from the expression a -c, we must first take away b, which gives a c -b. We have, however, taken away too much by the quantity d. Since we had to subtract only b-d, restoring, then, the value of d, we shall have

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from which it is evident that the terms of the expression to be subtracted must change
their signs, and with such contrary signs be joined to the terms of the other expressions.
657. It is therefore, by means of this rule, easy to perform subtraction; for it is only
necessary to write the expression from which we are to subtract, and join the other to it,
without any change but that of the signs. Thus, in the first example, where it was
required to subtract the expression d-e +f from a-b-c, we obtain a-b-c-d-e-f.
This will be rendered quite clear by an example in numbers. If, for example, we subtract
5–3 + 6 from 7–2+3, we obtain
7 2 + 3-5 + 3-6;
for 7 – 2 + 3 = 8, also 5-3+6 = 8, now 8–8 = 0.
658. Subtraction being then thus easily performed, we have only to observe that if, in
the remainder, two or more terms are found entirely similar with regard to the letters, the
remainder may be reduced to an abridged form by the rules for a similar purpose given in
addition. Suppose we have to subtract from a + b or the sum of two quantities their
difference, or a -b, we shall have a + b-a+ b : now a -a=0 and b +b=2b; the remainder
is therefore 2b, that is to say, double the least of the quantities. The following examples
will further illustrate what we have said: –
aa + ab + bb 3a–4b + 5c a3+ 3aab + 3abb +bs A/a + 2 vb
bb + ab-aa 2b + 4c - 6a a 3-3aab + 3abb-bā A/a-3 vb

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659. Merely to represent compound quantities, each of the expressions to be multiplied together is placed within parentheses, and they are then to be joined together either with

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or without the sign x between them. Thus, to represent the product of the two expressions a-b-c and d-e +f when multiplied together, we write

(a-b-c)x (d–e4-f); and this mode of expressing products is much used, because it shows the factors whereof they are composed. To show, however, how any multiplication is to be actually conducted, let us take for example such a quantity as a-b-c to be multiplied by 2; here each term is separately multiplied by that number, so that for the product we have 2a-2b-2c.;

and the same takes place with all other numbers. Suppose, for example, d had been the number by which we had been required to multiply,

ad-bd 4-cd would have been the product obtained. 660. We have here supposed that d was a positive number, but had the factor been negative, as -e, the rule formerly given must have been applied, namely, that unlike

signs multiplied together produce –, and like signs produce +... We should therefore have had

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661. To show the mode of multiplying a quantity. A by a compound quantity d-e; it will be convenient to take for example one in common numbers: let A, for instance, be multiplied by 7–3. Here it is manifest we have to take 4A; for if we first take A seven times, it will be necessary to subtract 3A from that product. In general, therefore, if we have to multiply by d-e; A must be first multiplied by d and then by e, and the last product must be subtracted from the first, whence we shall have A-eA. Suppose A=a-b, and that this quantity is to be multiplied by d-e, we shall have

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Since, then, we know without doubt the product (a-b)x (d-e), we may now give the same example of multiplication under a different form; thus

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from which we learn that each term of the upper expression must be multiplied by each term of the lower; and that, with regard to the signs, the rule often before given must be strictly observed. From what has been said, we presume no difficulty will arise in calculating the following example, namely, to multiply a +b by a -b.

a + b a -b aa + ab -ab-bb Product = aa-bb For a and b any determinate numbers may be substituted, so that out of the above ex

amples arises the following theorem; viz. the product of the sum of two numbers multi

plied by their difference is equal to the difference of the squares of those numbers; which may be thus expressed

(a+b)x (a-b)=aa-bb From this last follows another theorem; namely, the difference of two square numbers is always a product, and divisible both by the sum and the difference of the roots of those

two squares, consequently the difference of two squares can never be a prime number. We will now present to the reader some other examples:

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