Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

Ex. 2. Find a harmonical mean between 12 and 6.

Ans. 8. Ex. 3. Find a third harmonical proportional to 234 and 144. Ans. 104.

Ex. Find a fourth harmonical proportional to 16, 8, and 3.

Ans. 2.

Ş IV. PROBLEMS IN PROPORTION AND PROGRESSION.

Prob. 1. There are two numbers whose product is 24, and the difference of their cubes : cube of their difference:: 19: 1. What are the numbers.

Let x the greater number, and y= the lesser.

3

Then, xy=24, and x3—y3 : (x−y)3 : : 19 : 1. By expansion, x3-y: x3-3x2y+3xy-y3:: 19:1; .. (Art. 480), 3x2y-3xy2: (x-y): 18:1; and, (Art. 481), dividing by x-y, 3xy: (x-y) :: 18 : 1;

(x—y)2

but xy=24;.. 72: (x-y): 18: 1. Hence, (Art. 190), 18 (x—y)2=72, or (x—y)2=4;

[blocks in formation]

•*. x2+2xy+y2=100, and x+y=10,

but x-y= 2,

..x=6, and y=4.

Re

Prob. 2. Before noon, a clock which is too fast, and points to afternoon time, is put back five hours and forty minutes; and it is observed that the time before shown is to the true time as 29 to 105. quired the true time. Ans. 8 hours, 45 minutes. Prob. 3. Find two numbers, the greater of which shall be to the less as their sum to 42, and as their difference to 6, Ans. 32, and 24. Prob. 4. What two numbers are those, whose difference, sum, and product, are as the numbers 2, 3, and 5, respectively? Ans. 10, and 2. Prob. 5. In a court there are two square grass

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.

4;

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 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 num

bers?

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.

[ocr errors]

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 orp. But if p becomes-p. the fraction will become; and therefore it passes through, or infinity.

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 negative numbers as less than zero; whence there re

[ocr errors]

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 imaginary, or reciprocal, passes through zero, or infinity. This is what may easily be concluded from these expressions,

[blocks in formation]

considered in these three relations,

y2<a2, y2=a2, y2>a3.

§ 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 +x be changed into -x in the above equation, and we shall have

1

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 0, 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,

« ΠροηγούμενηΣυνέχεια »