plots; a side of one of which is 10 yards longer than the other; and their areas are as 25 to 9. What are the lengths of the sides? Ans. 25, and 15 yards.

Prob. 7. There are three numbers in arithmetical progression, whose sum is 21; and the sum of the first and second is to the sum of the second and third as 3 to 4. Required the numbers. Ans. 5, 7, 9. Prob. 7. The arithmetical mean of two numbers exceeds the geometrical mean by 13, and the geometrical mean exceeds the harmonical mean by 12. What are the numbers? Ans. 234, and 104.

Prob. 8. Given the sum of three numbers, in harmonical proportion, equal to 26, and their continued product=576; to find the numbers.

Ans. 12, 8, and 6. Prob. 9. It is required to find six numbers in geometrical progression, such, that their sum shall be 315, and the sum of the two extremes 165.

Ans. 5, 10, 20, 40, 80, and 160.

Prob. 10. A number consisting of three digits which are in arithmetical progression, being divided by the sum of its digits, gives a quotient 48; and if 198 be subtracted from it, the digits will be inverted. Required the number. Ans. 432.

Prob. 11. The difference between the first and second of four numbers in geometrical progression is 36, and the difference between the third and fourth is 4; What are the numbers? Ans. 54, 18, 6, and 2.

Prob. 12. There are three numbers in geometrical progression; the sum of the first and second of which is 9, and the sum of the first and third is 15. Required the numbers. Ans. 3, 6, 12. Prob. 13. There are three numbers in geometrical progression, whose continued product is 64, and the sum of their cubes is 584. What are the numbers ?

Ans. 2, 4, 8.

Prob. 14. There are four numbers in geometrical progression, the second of which is less than the fourth

by 24; and the sum of the extremes is to the sum of the means as 7 to 3. Required the numbers. Ans. 1, 3, 9, 27. Prob. 15. There are four numbers in arithmetical progression, whose sum is 28; and their continued product is 585. Required the numbers?

Ans. 1, 5, 9, 13.

Prob. 16. There are four numbers in arithmetical progression; the sum of the squares of the first and second is 34; and the sum of the squares of the third and fourth is 130. Required the numbers.

Ans. 3, 5, 7, 9.

CHAPTER XIV.

ON LOGARITHMS.

495. Previous to the investigation of Logarithms, it may not be improper to premise the two following propositions.

496. Any quantity which from positive becomes negative, and reciprocally, passes through zero, or infinity. In fact, in order that m, which is supposed to be the greater of the two quantities m and n, becomes n, it must pass through n; that is to say, the difference m-n becomes nothing; therefore p, being this difference, must necessarily pass through zero, in order to become negative or-p. But if p becomes-p, the fraction will become; and therefore it passes through, or infinity.

P

497. It may be observed, that in Logarithms, and in some trigonometrical lines, the passage from positive to negative is made through zero; for others of these lines, the transition takes place through infinity: It is only in the first case that we may regard negafive numbers as less than zero; whence there re

sults, that the greater any number or quantity a is, when taken positively, the less is -a; and also, that any negative number is, a fortiori, less than any absolute or positive number whatever.

498. If we add successively different negative quantities to the same positive magnitude, the results shall be so much less according as the negative quantity becomes greater, abstracting from its sign. For instance, 8-1>8-2-8-3, &c.

It is in this sense, that 0>-1>-2>-3, &c.; and 3>0>-1>—2—3—4, &c.

499. Any quantity, which from real becomes imagi nary, or reciprocally, passes through zero, or infinity. This is what may easily be concluded from these expressions,

considered in these three relations,

y2<a2, 'y2=a2, y2a3.

§ I. THEORY OF LOGARITHMS.

500. LOGARITHMS are a set of numbers, which have been computed and formed into tables, for the purpose of facilitating arithmetical calculations; being so contrived, that the addition and subtraction of them answer to the multiplication and division of the natural numbers, with which they are made to correspond.

501. Or, when taken in a similar, but more general sense, logarithms may be considered as the exponents of the powers, to which a given, or invariable number, must be raised, in order to produce all the common, or natural numbers. Thus, if a =y, a*=y', a*"'=y', &c.; then will the indices x, x', 'x", &c. of the several powers of a, be the logarithms of the numbers y, y', y", &c. in the scale, or system, of which a is the base.

502. So that, from either of these formulæ, it appears, that the logarithm of any number, taken separately, is the index of that power of some other number, which, when it is involved in the usual way, is equal to the given number. And since the base a, in the above expressions, can be assumed of any value, greater or less than 1, it is plain that there may be an endless variety of systems of logarithms, answering to the same natural numbers.

503. Let us suppose, in the equation a=y, at first, x=0, we shall have y=1, since (Art. 453), x=1; to x=1, corresponds y=a. Therefore, in every system, the logarithm of unity is zero; and also, the base is the number whose proper logarithm, in the system to which it belongs, is unity. These properties belong essentially to all systems of logarithms.

504. Let + be changed into - in the above equation, and we shall have

1

=y:

Now, the exponent x augmenting continually, the

1

fraction if the base a be greater than unity, will diminish, and may be made to approach continually towards O, as its limit; to this limit corresponds a value of a greater than any assignable number whatever. Hence, it follows, that, when the base a is greater than unity, the logarithm of zero is infinitely negative.

505. Let y and y' be the representatives of two numbers, x and x' the corresponding logarithms for the same base: we shall have these two equations, a=y, and a-y', whose product is a*.a*=y.y', or a*+*=yy', and consequently, by the definition of logarithms, (Art. 501), x+x'=log. yy', or log. yy'= log. y+log. y'.

And, for a like reason, if any number of the equations a*=y, a*=y', a*"'=y", &c. be multiplied together, we shall have a+++etc.yy'y", &c.; and,

consequently, x+x'+x", &c.=log. yy'y", &c.; or log. yy'y", &c.=log. y+log.y'+log.y",

&c.

The logarithm of the product of any number of factors is, therefore, equal to the sum of the logarithms of those factors.

506. Hence, if all the factors y, y', y", &c. are equal to each other, and the number of them be denoted by m, the preceding property will then become log. (ym)=m, log. y.

Therefore the logarithm of the mth power of any number is equal to m times the logarithm of that number.

507. In like manner, if the equation a*=y, be divided by a=y', we shall have, from the nature of

Hence the logarithm of a fraction, or of the quotient arising from dividing one number by another, is equal to the logarithm of the numerator minus the logarithm of the denominator.

508. And if each member of the equation, a*=y, be raised to the fractional power, we shall have

Mx m

a"y"; and consequently, as before,

Therefore the logarithm of a mixed root, or power, of any number, is found by multiplying the logarithm of the given number, by the numerator of the index of that power, and dividing the result by the denominator.

509. And if the numerator m of the fractional index of the number y, be, in this case, taken equal to 1, the preceding formula will then become

log. y=log. Y.