Having set the slider as directed, call the 10 on the line D one, and over it you will find its square, viz. 1; or, if you call the 10 on D 10, then the 1 over it must be called 100; or, if you call the 10 on D 100, then the 1 over it will count 1,000, &c.; and over any number on D you will find its square on C, calling the 10 at the middle of the line D, 1, or 100, or 1,000, or 10,000, and counting towards the right, if the number be between 1 and 4, or 10 and 40, or 100 and 400, or 1,000 and 4,000, &c., but if the number, whose square is required, be between 4 and 10, or between 40 and 100, or between 400 and 1,000, or 4,000 and 10,000, find the given number on the line D to the left of the 10 at the middle of the scale. Or you may commence with the 4 at the left end of the D line, and calling it, and counting towards the right, the 10 at the middle of the line will count 1, and the 40 at the end of the line will count 4; or, calling the 4 at the left end of the line D, 4, and counting towards the right, you will find 10 at the middle of the line, and the 40 at the end of the line will count 400; you may then commence again with the 4 at the left end of the D line, and calling it 400, at the middle of the rule you will find 1,000, and at the end 4,000, &c. The square root of any number will be found on D, directly under the number on the C line; but in reading off, to avoid mistake, you must commence with 1 on C, at the middle of the rule, and read towards the right or left as directed above, when reading off the squares of numbers.

When the slider has been set, as directed above, there is a part of the line D on the right, which does not come under the slider, which renders it impossible to read off the square of a number between 3.15 and 4, or between 315 and 400, without resetting the slider, so as to bring this portion of the rule, or scale, under the line of numbers on C. In this, and all similar cases, notice what figure or portion of the slider stands over 4 on the line D at the left end of the line; then set the same figure, or part of the line C, over the 4 or 40 on D, at the right hand, and the difficulty will be overcome.

EXAMPLES.

1. What is the square of 1? what of 9 tenths? 7 tenths? 5 tenths? 35 hundredths? and of 2 tenths?

Having set the slider, as directed in the rule, over 10 on D, calling the 10 one, you will find 1, its square, and running to the left on D, over 9 tenths, you will find its square, 81 hundredths; over 7 tenths, you will find 49 hundredths; over 5 tenths, 25 hundredths; and (sliding the slider to the right, placing 16, or 16 hundredths, on C, over the 40, or 4 tenths, on D, at the right end of the line) over 35 hundredths we have 1225 ten-thousandths; and over 2 tenths you will find 4 hundredths.

2. What is the square of 1.5? of 2? of 3? of 3.5? of 4? of 6? of 10? of 12? of 15? of 25? of 30? of 40? of 60? of 75 ?

Having set the slider, as directed in the rule, over 1.5 you will find 2.25; over 2, 4; over 3, 9; over 3.5, 12.25; over 4, 16; over 6, 36; over 10, 100; over 12, 144; over 15, 225; over 25, 625; over 30, 900; over 40, 1600; over 60, 3,600; and over 75, 5,625.

3. What is the square of 25 hundredths? of 49 hundredths? of 144? and of 33.5 ?

Ans. .0625; .24; 20,736; 1125 nearly.

4. What is the square root of 1? of 81? of 49? of 25? of 2? of 3? of 4? of 10? of 20? of 30? of 40? of 60? of 88? of 144? of 225? of 400? of 625? of 1,000? of 1,400 ? of 2,025? of 2,500 of 3,600? of 6,400? of 8,100? of 40,000? of 90,000?

Having set the slider, as directed in the rule, under 1 we find its root, calling the 10 on D, 1; under 81, we find its root, viz. 9; under 49, 7; under 25, 5; under 2, 1.414; under 3, 1.732; under 4, 2; under 10, 3.16; under 20, 4.47; under 30, 5.47; under 40, 6.32; under 60, 7.75; under 81, 9; under 88, 9.41; under 144, 12; under 225, 15; under 400, 20; under 625, 25; under 1,000, 31.62; under 1,400, (having reset the slider,) 37.4; under 2,025, 45; under 2,500, 50; under 3,600, 60; under 8,100, 90; under 40,000, 200; and under 90,000, we find 300 for the root, &c.

10. ON THE CONSTRUCTION OF THE LINES A, B, C, AND D.

As before stated, in the general description of the Sliding Rule, the divisions on the lines A, B, and C are exactly alike, the length of the spaces being to each other as the logarithms of the numbers which they represent. If the natural numbers be considered as terms in an infinite series of proportionals, beginning at unity, and either increasing or decreasing to infinity, the logarithm of any number is its distance. from unity in that series; and the logarithms of the natural numbers are so related to each other, and to the numbers which they represent, that the sum of any two logarithms is the logarithm of the product of the two numbers for which they stand; and the difference of any two logarithms is the logarithm of the quotient of one of the numbers divided by the other; and twice the logarithm of any number is the logarithm of the square of that number.

Hence the spaces, or the divisions, on the lines A, B, and C, are to each other as the logarithms of the natural numbers which they represent, the distance from 1 to 3 being as much greater than the distance from 1 to 2, as the logarithm of 3 is greater than the logarithm of 2; and the distance from 1 to 4 is as much greater than the distance from 1 to 3, as the logarithm of 4 is greater than the logarithm of 3; and so on through the scale. Four being the square of 2, its logarithm will be double the logarithm of 2; and consequently, the distance from 1 to 4 will be twice the distance from 1 to 2; and because the square of 3 is 9, the distance from 1 to 9 will be twice the distance from 1 to 3; and the distance from 1 to 16 will be, for the same reason, twice the distance from 1 to 4, &c.

Since, therefore, the numbers on the lines A and B are so laid down on the scale, that their distances from unity are as their logarithms compared with unity, it follows that if we bring one of the factors of any number under unity, the other factor will stand over the product; and hence we can multiply on divide any number by bringing the multiplier or divisor under

unity, and seeking for the product under the multiplicand, and for the quotient over the dividend, as has been demonstrated. See 2 and 3.

And since twice the logarithm of any number is the logarithm of the square of that number, and since the numbers on the line C are the squares of those on the line D, it follows that the distances between the numbers on the line D must be twice as great as the distances between the corresponding numbers on the line C. Thus, the distance from 1 to 2 on D, is equal to the distance from 1 to 4 on C; and the distance from 1 to 4 on D, is equal to the distance from 1 to 16 on C, and so on. Hence it follows, that if we place 1 on C over 1 on D, under any number found on C, we shall find its square root on D; or over any number found on D we shall find its square on the line C, as we have shown in the extraction of the square root, and in the squaring of numbers. (See the description of the engineer's rule, ¶ 1.)

11. CUBE ROOT ON THE SLIDING RULE.

To cube any number by the sliding rule :—

Place the given number on the line C over 1 on the line D, and over the given number found on the line D will be found its cube, or third power, on the line C. Or, place the square of any number found on the line C, over 1 on the line D, and over the given number found on the line D, will be found its fourth power on the line C.

To extract the CUBE ROOT by the sliding rule :

Move the slider either way, until the number, whose root is required, stands over the same number on the line D, that 1 on D stands under on C; and then will the number that stands over unity on D, be the root required.

Or, Draw out the slider and reverse its ends, so as to bring the line B, inverted, over the line D; then find the given num

ber on the line B, and place it over 1 on D; then look along the scales, or lines B and D, until you find the numbers or divisions on each coinciding, or the same number on B standing over the same on D; then will that number be the required root.

The slider being reversed, the numbers on the line B will increase from right to left, whilst those on D will increase from left to right; and consequently, the numbers on B must be counted from right to left, and those on D from left to right. It will also occasionally happen, that the numbers on the lines B and D will agree, or coincide, in two different places, only one of which will give the true root; hence some degree of caution is required in reading off.

EXAMPLES.

1. What is the cube of 3? of 12? of 15? of 18? of 5? of 20? of 16? of 25? and of 30?

Answers. 27; 1,728; 3,375; 5,832; 125; 8,000; 4,096; 15,625; and 27,000.

2. What is the cube root of 8 ?

Having reversed the slider, and set 8 on B over 10 on D, look along the scale towards the right, and you will find 2 on B exactly over 2 on D; 2 therefore is the cube root of 8, as you may prove by raising it to the third power.

3. What is the cube root of 27 ?

Having placed 27 on B over 1 on D, look along the scale towards the right, and you will find 3 on D directly under 3 on B; 3 therefore is the cube root of 27.

4. What is the cube root of 125 ?

Having set the slider as directed, look along the scale towards the left, and you will find 5 on B over 5 on D; 5 therefore is the cube root of 125.

5. What is the cube root of 216, or the side of a cubic block which contains 216 solid feet?

Having set the slider, to the left of the middle of the rule you will find the root 6, on B, over 6 on D.