608. Of powers, those of the number 10 are the most remarkable, as being the foundation of our system of arithmetic. We will range in order a few of them, as under : —

To consider which more generally, we may take the powers of any number a, as placed in the following order :

I, II, III, IV, V, VI,

a, aa, aaa, aaaa, aaaaa, aaaaaa, &c.

But in this mode of writing powers there is much inconvenience, because of the trouble of counting the figures and letters; for the purpose of ascertaining the powers intended to be represented, as, for example, the inconvenience of representing the hundredth power would be so great as to incumber almost to impossibility the expression of it. To avoid this inconvenience, an expedient has been devised which is sufficiently convenient, and which we have now to explain. To express, for example, the hundredth power of a, we write just above it to the right the power in question; thus, a100 means, conventionally, a raised to the hundredth power. The number thus written above that whose power or degree it represents is called an exponent, from its expounding the power or degree to which the number is to be raised, which, in the example we have adduced, is 100. Thus, then, a2 represents the square or second power of a, which, as we have seen, may be also represented by aa, either of these expressions being understood with equal facility. To express the cube or third power of a or aaa, a3 is written, by which mode less room is occupied. So a1, a3, aồ, &c. respectively represent the fourth, fifth, and sixth powers of a. We may in this manner represent a by a1, which, in fact, shows nothing more than that this letter is to be written only once. Such a series of powers as we here have noticed is called also a geometrical progression, because each term is once greater than the preceding.

609. As in this series of powers each term increases by multiplying the preceding term by a, thereby increasing the exponent by 1, so where any term is given the preceding one may be found if we divide by a, because it diminishes the exponent by 1: thus showing that the first term a1 must necessarily be or 1; hence, if we proceed according to the exponents, we immediately perceive that the term which precedes the first must be ao, from which follows this remarkable property, that ao is always equal to 1, however great or small the value of the number a may be, even if a be nothing.

610. The series of powers may be continued in a retrograde order, and in two different ways: first, by dividing continually by a; and, secondly, by diminishing the exponent by unity. In either mode the terms will be equal. The decreasing series, exhibited in both forms, is shown in the subjoined table, which is to be read from right to left.

Thus we come to the knowledge of powers whose exponents are negative, and are able to assign the precise value of those powers.

apparent that

a-l

a-2

And hence, from what has been said, it will be

H:

a-3

a-4

aa

a3

or
a2

a4 &c.

This gives us the facility of finding the powers of a product ab; for they must be evidently ab, or a1b1, a2b2, a3b3, a4b4, a5b5, &c.; and the powers of fractions are found in the same manner; for example, those of

a

are

The only matter remaining, then, is the consideration of the powers of negative numbers. Take, for example, the powers of -a, and they will form the following series:

-a, +aa, -a3, +aa, —a3, +a®, &c.;

in which we immediately perceive that those powers are negative whose exponents are odd numbers, and that the powers with even numbers for exponents are positive. Thus the third, fifth, seventh, ninth, &c. powers have the sign ; and the second, fourth, sixth, eighth, &c. powers are affected by the sign +.

CALCULATION OF POWERS.

611. The addition and subtraction of powers is effected by means of the signs and when the powers are different; for example, a3+a2 is the sum of the third and second powers of a; and a-a4 is the remainder when the fourth power of a is subtracted from the fifth; neither of which results can be abridged. If the powers are of the same kind or degree it is not necessary to connect them by signs, thus a3 + a3 makes 2a3, &c. 612. But in the multiplication of powers, we must observe, first, that any power of a multiplied by a, gives the succeeding power, that is to say, the power whose exponent is one unit greater. Thus a multiplied by a produces a3; and a3 multiplied by a produces a1. Similarly, if it be required to multiply by a, the powers of that number having negative exponents, 1 must be added to the exponent. Thus, a multiplied by a produces ao or 1; and this becomes most clearly seen by considering that a is equal to and that the product of being, it is consequently equal to 1. So a 2 multiplied by a produces

or

-1

-1

1 and a-5 multiplied by a produces a1, and so on.

being the

of a2 by a3 is a5; In the case of ne

same as

aa

a

it is

613. If it be required to multiply a power of a by aa or the second power, the exponent then becomes greater by 2. Thus the product of a2 by a2 is at; that that of a by a1 is a6; and, generally, a" multiplied by a2 makes an+2. gative exponents, a' or a is the product of a1 by a2. For a just the same as if we had divided aa by a; hence the product required is or a. In the same way, a multiplied by a produces ao or 1, and a3 multiplied by a2 produces a-1. It is equally clear that to multiply any power of a by as, its exponent must be increased by three units, consequently the product of a" by a3 is a”+3. And as often as it is required to multiply two powers of a, the product must be a power of a whose exponent is equal to the sum of those of the two given powers. For instance, a4 multiplied by a5 will make ao, and a12 multiplied by a7 produces a1o, &c.

614. On the principles here exhibited, it is easy to determine the highest powers. Thus, to find the twenty-fourth power of 2, multiply the twelfth power by the twelfth power; because 224 is equal to 212 x 212, But we have already seen that 212 is equal to 4096; hence the number 16777216, being the product of 4096 by 4096, is 224, or the required power of 2.

or

-3

615. In division we must observe that to divide a power of a by a the exponent must be diminished by unity. Thus a' divided by a gives a1; ao or 1 divided by a is equal to a-1 ; a divided by a gives a. So, if we have to divide a given power of a by a2, the exponent must be diminished by 2, and if by a3, three units must be subtracted from the exponent of the power proposed; and, generally, if it be required to divide any power of a by any other power of a, the rule is to subtract the exponent of the second from the exponent of the first of those powers. Thus a16 divided by a gives a7; a5 divided by as will give a-1. So a¬3 divided by a1 will give a−7.

616. It is not difficult, then, from what has been said, to find the powers of powers, for it is effected by multiplication. Thus, if we have to seek the square or second power of a3, we find a6, and for the cube or third power of at we have a12 To obtain the square

of a power it is only necessary to double the exponent; for its cube, to triple the exponent, and so on. Thus a2 is the square of a", u3n is the cube of a", and the seventh power of a" is a. The square of a2, or square of the square of a, being aa, is hence called biquadrate. The square of a3 is a; hence the sixth power has received the name of the square-cubed. To conclude, the cube of a3 being ao, the ninth power has received the name of the cubo-cube.

ROOTS RELATIVELY TO POWERS IN GENERAL.

617. The square root of a given number is a number whose square is equal to that number; the cube root, that whose cube is equal to the given number: hence, whatever number be given, such roots of it will exist that their fourth, their fifth, or any other power, will be equal to the given number. For distinction sake, we shall call the square

root the second root, the cube root the third root, the bi-quadrate the fourth root, and so on. As the square or second root is marked by the sign, and the cubic or third root by the sign; so the fourth and fifth roots are respectively marked by the signs and ✅, and so on. It is evident, according to this method of expression, the sign of the square root should be ; but by common consent the figure is always left out; and we are to recollect that when a radical sign has no number prefixed to it, the square root is always meant. To give a still better explanation, we here subjoin some different roots of the number a, with their respective values :

Va is the 4th root of

HHL

5th

6th

a, and so on.

And so, conversely,

618. Whether a be a small or a great number, we know what value to affix to all these roots of different degrees. If unity be substituted for a the roots remain constantly 1; for all powers of 1 have unity for their value. But if the number a be greater than 1, the roots will also all exceed unity; and further, if a represent a less number than 1, all the roots will be less than unity.

619. When the number a is positive, from what has been before said of square and cube roots, we know that all the other roots may be determined, and that they will be real and possible numbers. But if the number a is negative, its second, fourth, sixth, and all even roots become impossible, or imaginary numbers; because all the powers of an even order, whether of positive or of negative numbers, are affected by the sign + ; whereas the third, fifth, seventh, and all odd roots become negative, but rational, because the odd powers of negative numbers are also negative. Hence an inexhaustible source of new kinds of surd or irrational quantities; for, whenever the number a is not a power represented by some one of the foregoing indices, it is impossible to express the root either in whole numbers or fractions, and it must therefore be ranked among the numbers called irrational.

THE REPRESENTATION OF POWERS BY FRACTIONAL EXPONENTS.

620. In the preceding subsections we have seen that the square of any power is found by doubling its exponent, and that in general the square or second power of a" is an. Hence the converse, that the square root of the power a2" is found by dividing the exponent of that power by 2. Thus the square root of a2 is a1; that of a is a3; and as this is general, the square root of a3 is necessarily a1, and that of a7 is a1. Thus we have a for the square root of a1, and hence at is equal to ✅a; a new method of expressing the square root, which requires our particular attention.

α

α

621. To find the cube of a power, as a", we have already shown that its exponent must be multiplied by 3, hence its cube becomes a3"; and, conversely, to find the third or cube root of the power an, we have only to divide the exponent by 3; hence the root is a". Thus, also, al or a is the cube root of a3, a2 that of ao, at that of a12, and so on. The same reasoning is applicable to those cases in which the exponent is not divisible by 3; for it is evident that the cube root of a2 is a3, as the cube root of a1 is as or a13. Hence the third or cube root of a or a1 will be a, which is the same as Va.

622. The application is the same with roots of a higher degree: thus the fourth root of a will be a3, which expression is of the same value as Va. The fifth root of a will be a3, which is equivalent to a, and so on in roots of higher degree. It would be possible, therefore, to dispense altogether with the radical signs, and to substitute fractional exponents for them; but as custom has sanctioned the signs, and as they are met with in all works on algebra, it would be wrong to banish them altogether from calculation. There is, however, sufficient reason to employ, as is frequently done, the other method of calculation; because it clearly corresponds with the nature of the thing. Thus, in fact, it is manifest that a is the square root of a, because we know that its square is equal to a1 or a. 623. What has been said will be sufficient to show how we are to understand fractional exponents; thus, if a should occur, it means that we are first to take the fourth power of a and then extract its cube or third root, and hence a3 is the same as Vat. Again, to find

1 va

10

expresses

the value of a the cube or third power of a or a3 must first be taken, and the fourth root of that power extracted, so that a is the same as Va3. So a is the same as a4, &c. But when the fraction which represents the exponent is greater than unity, the value of the given quantity may be otherwise expressed. Let it, for instance, be a3; now this quantity is equivalent to at which is the product of a2 by at. Now at is equal to va, wherefore a is equal to a√a. So a's, or a3, is equal to a3 a3; and a, that is, a3 a3/a3. From these examples the use of fractional exponents may be properly appreciated. This, however, extends also to fractional numbers, as follows. 624. Suppose is given, we know that it is equal to ; now we have already seen a} that a fraction of the form may be expressed by a ̄ '; and instead of the expression a1. Also, is equal to a. So let the quantity be proposed, it is transformable into which is the product of a2 by a, and this is equivalent to aş' as, or to a13, or, lastly, to ✔a3. These reductions will be facilitated by a little practice. 625. Each root may be variously represented, for a is the same as a, and being equivalent to the fractions,,,, &c., it is clear that a is equal to Va2, to a3, to Vat, and so on. Similarly, a is equal to a3, and to Va2, to Va3, and to a1. It is, moreover, manifest, that the number a, or a1 might be represented by the following radical expressions:

a2

Va2, Ya3, Va1, S'a3, &c.

1

we may use a2 Va3

a property of great use in multiplication and division; for, suppose we have to multiply a by a, we write a3 for Ya, and Va2 instead of Va, thus obtaining the same radical sign for both, and the multiplication being now performed, gives the product √a3. A similar result arises from a+, the product of a multiplied by a3, for + is, and, consequently, the product required is aš, or Va3. If it were required to divide a or a by Va or a}, we should have for the quotient al-, or a—, that is, af, or

METHODS OF CALCULATION AND THEIR MUTUAL CONNECTION.

a.

626. In the foregoing passages have been explained the different methods of calculation in addition, subtraction, multiplication, and division, the involution of powers, and the extraction of roots. We here propose to review the origin of the different methods, and to explain the connection subsisting among them, in order that we may ascertain if it be possible or not for other operations of the same kind to exist; an inquiry which will illustrate the subjects that have been considered. We shall, for this purpose, here introduce a new sign =, which means that equality exists between the quantities it is used to join, and is read equal to. Thus, if I write a=b, it means that a is equal to b; and so 3 x 8

=24.

627. Addition, the process by which we add two numbers together and find their sum, is the first mode of calculation that presents itself to the mind. Thus if a and b be two given numbers whose sum is expressed by c, we shall have a + b = c. So that, knowing the two numbers a and b, we are taught by addition how to find the number c. Recollecting this comparison a+b=c, the question may be reversed by asking in what way b can be found when we know the numbers a and c. Let us, then, ascertain what number must be added to a that the sum may be c. Now, suppose, for instance, a=3, and c=8, it is evident we must have 3+b=8, and b will be found by subtracting 3 from 8. So, generally, to find b, we must subtract a from c, whence arises b=c-a; for, by adding a to both sides again, we have b+a=c-a+a, that is, as was supposed, c. And this is the origin of subtraction, being, indeed, nothing more than an inversion of the question from which addition arises. Now it is possible that it may be required to subtract a greater from a lesser number; as, for example, 9 from 5. In this case we are furnished with the idea of a new kind of numbers, which are called negative numbers, because 5-9=-4.

628. If several equal numbers are to be added together, their sum is found by multiplication, and is called a product. Thus ab expresses the product of the multiplication of a by b, or from a being added to itself b times. If this product be represented by c, we have abe, and we may, by the use of multiplication, determine the number c where the numbers a and b are known. Suppose, for example, a=3, and c=15, so that 3b=15, we have to ascertain what number b represents, merely to find by what number b is to be multiplied, in order that the product may be 15, for to that is the question reduced: and this is division; for the number sought is found by dividing 15 by 3; hence, in general, the number b is found by dividing e by a, whence results the equation b=.

But, frequently, the number c cannot be actually divided by the number a, the letter b having a determinate value; hence a new kind of numbers, called fractions, arises. For, suppose a = 4, c=3, so that 4b-3, in this case b cannot be an integer, but must be a fraction, and we shall find that b can be no more than . Multiplication, then, as we have seen, arises from the addition of equal quantities; so, from the multiplication of several equal quantities together, powers are derived, and they are represented in a general manner by the expres sion a', which signifies that the number a must be multiplied by itself as often as is pointed out by the number b, which is called the exponent, whilst a is called the root, and a the power. If this power be represented by the letter c, we have a=c, an equation in which are found the letters a, b, c. In treating of powers, it has been shown how to find the power itself, that is, the letter c, when the root a and its exponent b are given. Suppose, for instance, a=4, and b=3, we shall have c=43, or the third power of 4, which is 64, whence c=64. If we wish to reverse this question, we shall find that there are two modes of doing it. Let, for instance, two of the three numbers a, b, and c be given. If it be required to find the third, it is clear that the question admits of three different suppositions, and hence, also, of three solutions. The case has been considered in which a and b were the numbers given; we may therefore suppose, further, that c and a or c and bare known, and that it is required to determine the third letter. Now, it must be observed, that between involution and the two operations which lead to it there is a great difference. For when, in addition, we reverse the question, there was only one way of doing it, and it was of no consequence whether we took c and a or c and 6 for the given numbers, for it is quite indifferent to the result whether we write a + b or b + a. And it is quite the same with multiplication; the letters a and b might be placed in each other's places at pleasure, the equation ab=c being exactly the same as ba=c. But in the calculation of powers, we cannot change the places of the letters; for instance, we could on no account write b" for α. This we will illustrate by one example. Thus, let a=4, and b=3, we have a3 = 43 =64. But ba=34=81, two very different results.

629. We may propose two more questions; one to find the root a by means of the given power c, and the exponent b; the other to find the exponent b, the power c and the root a being known. The former of these questions has been answered in the subsection which treats of the extractions of roots: since, if b=2, and a2=c, we know that a is a number whose square is equal to c, and consequently a=c. So, if b = 3 and a3=c, we know that the cube of a is equal to the given number c, and hence that ac. We conclude, generally, from this, how the letter a may be determined by means of the letters c and b; for a must necessarily be /c.

630. We have already seen that if the given number is not a real power (a contingency of frequent occurrence), the required root a can be expressed neither by integers nor fractions; nevertheless, as it must have a determinate value, the same consideration led us to the numbers called surd or irrational numbers, which, on account of the great variety of roots, are divisible into an infinite number of kinds. We were also, by the same enquiry, led to the knowledge of imaginary numbers.

631. Upon the second question, that of determining the exponent by means of the power c and the root a, is founded the very important theory of logarithms; an invention so important that without them scarcely any long calculation could be effected.

LOGARITHMS

632. Resuming, then, the equation abc, we in the doctrine of logarithms assume for the root a number taken at pleasure, but supposed to preserve its assumed value without variation. This being the case, the exponent b is taken, such that the power a becomes equal to a given number c, and this exponent b is said to be the logarithm of the number c. To express this, we shall use the letter L or the initial letters log. Thus, by b➡ L.c or blog.c, we mean that b is equal to the logarithm of the number c, or that the logarithm of c is b.

=

633. If the value of the root a is once established, the logarithm of any number c is but the exponent of that power of a which is equal to c. So that c being a, b is the logarithm of the power of a. If we suppose b=1, we have 1 for the logarithm of al; hence L.a=1. Suppose b=2, we have 2 for the logarithm of a2; that is L.a2=2. Similarly, L. a33, L.a=4, L.a5=5, and so on.

=

-4; &c.

634. Ifb be made 0, 0 must be the logarithm of a0; but a0=1; consequently, L.10, whatever the value of the root a. Ifb1, then -1 will be the logarithm of a1. Now a1=; therefore, L. 1 =-1. So, also, L. = −2; L= 3; L. 635. Thus, then, may be represented the logarithms of all the powers of a, and even those of fractions wherein unity is the numerator, and the denominator a power of a. We see, also, that, in all those cases, the logarithms are integers: but if b were a fraction it would be the logarithm of an irrational number. For suppose b= {}, then is the logarithm of a3, or of a; consequently we have L. va = 1; and in the same way, L. Va=}, L. /a = 4, &c.