on D. Example.-Find the value of

proportional to /12, √√/27, and 4.

Set 12 on either section of c against 4 on D; and against 27 on the same section of c will be found 6 the result, on D. The setting for this example is the same for all the three forms of rule.

When a problem arises in which the result involves the sums or difference of different quantities, the several portions of the result must be obtained separately from the rule, and then added together. Example. - Find the value of 40 (242 + 322 + 59.52)

Against 46 on D set 40 on c, and against 24, 32, and 59.5 respectively on D will be found on c the numbers 10.9, 19.36, and 66.9; and 97.16, the sum of these, will be the value required.

USE OF THE A, B, C, AND D LINES CONJOINTLY.

In the first and third forms of construction of the rule, explained above, in every position of the slide the same numbers on c and B coincide with numbers upon D and the squares of those numbers upon a. In the third construction the A and в lines will be found on the opposite face of the instrument to the c and D lines. On either of these constructions, then, the A, B, C, and D lines may be used conjointly.

Problem 18. To divide by a Number two Numbers multiplied together, one of which is squared.-Set the unsquared factor on в against the divisor on a, and against the squared factor on D will be found the result on c. Example.-Find the value 52 × 32 Set 32 on B against 6 on a; and against 5 on D will be found 133, the required result, on c.

of

6

With regard to the numeration of the result in these conjoint solutions, it is only necessary to consider the result as always taken on в, and the number on D, if it occur on the second half of this line, as a number with twice as many integral places taken on A, and, if it occur on the first half, as a number on A with twice as many integral places less one, the rule to be applied will, then, be the same as that for solution on the ▲ and в lines (p. 102).

Problem 19. To divide the Product of two Numbers by the Square of a Third Number.-Set the third or squared number on D against one of the factors on c, and against the other factor on a will be found the result on B. Example.-Find

the value of

8 × 16

2.52

Against 2.5 on D set 8 on c; and

against 16 on a will be found 20.5 the result on B.

Solution on First form of Rule :

THE INVERTED SLIDE.

It has been seen that, when the slide is inserted in the groove in the ordinary manner, the numbers on the line a are always proportional to the corresponding numbers on the line B, and the numbers on the line c proportional to the squares of the numbers on the line D. Now, if we invert the slide, the numbers on A and the squares of the numbers on D become reciprocally proportional to the numbers on the slide, and, as a matter of course, division becomes multiplication, and multiplication division: also the products of the corresponding numbers on A and on the slide are everywhere the same, and the products of the numbers on the slide and the squares of the numbers on D are everywhere the same.

The rules for numeration with the inverted slide are deduced from those for the direct slide, by making the differences inverse, so that the term with the greatest number of integral places on the slide coincides with the term with the lesser number of integral places on the rule, and vice versa, also observing that the second section of the slide is now on the left hand, and the first section on the right hand.

EXAMPLES OF THE USE OF THE INVERTED SLIDE.

Problem 20. To find the Reciprocal of a given Number.Set the inverted slide evenly in the groove, so that 10 on c coincides with 1 on a, and 1 on c with 10 on a, and any two coincident numbers will be reciprocals the one of the other, and vice versâ, the numeration being so arranged that the number of zeros after the decimal point in one number shall be one less than the number of integral places in the other, thus―

314 and 3184 being mutually coincident on the inverted slide and the rule, 3184 and 3.14 are mutually reciprocal; i.e.,

Problem 21. To multiply one Number by another.-Set one of the numbers on the slide against the other on a, and against 1 on a will be found the product on the slide. Example.Find the product of 55 by 24. Set the slide so that 24 and 55 on it coincide with 55 and 24 on a, and 1 on a will coincide with the required product, 1320, on the slide.*

Problem 22. To divide one Number by another.-Set the slide so that the dividend and 1 are coincident, and the divisor and quotient will also be coincident. Example.-Divide 58 by 16. Set the slide so that 58 and 1 on A and c coincide, and 16 will coincide with the required quotient, 3.625.

Problem 23. To find the Product of two Numbers, one of which is squared.-Set the unsquared number on the slide against the squared number on D, and against 1 upon D will be found the product on the slide. Example.-Find the value of 17223. Set 23 on the slide against 17 on D, and against 1 on D will be found 6647, the required product, on B.

Problem 24. To cube a given Number.-Set the given number on the slide against the same number on D; and against 1 on D is the cube required. Example.-Find the cube of 8. Set 8 on the slide against 8 on D, and against 1 on D is 512, the required cube, on B.

Problem 25. To find the Cube Root of a given Number.-Set the given number on the slide against 1 on D; and look for a number on the slide standing against the same number on D, and this number will be the root required. Example.-Find the cube root of 7. Set 7 on the slide against 1 on D; and

This double coincidence enables us both to set the rule and to read off the result with greater accuracy.

1.914, the required cube root, will be found coincident on the slide and on D.

Problem 26. To square given Number.-Set the given number on the slide against the same number on a; and against 1 on a will be found on the slide the square required. Example. Find the square of 31.5. Set 31-5 on the slide against 31.5 on A; and against 1 on a will be 992, the required square, on c.

Problem 27. To find the Square Root of a given Number.Set the given number on the slide against 1 on a; and look for a number on the slide standing against the same number on A, and this number will be the root required. Example.The square root of 8. Set 8 an the slide against 1 on a; and 2.828, the required square root, will be found coincident on the slide and on a.

Problem 28. To find a Mean Proportional between two given Numbers. Set one of the numbers on the slide against the other on a; and look for a number on the slide standing against the same number on A, and this number will be the mean proportional required. Example.-Find a mean proportional between 4 and 49. Set 4 on the slide against 49 on A; and 14, the required mean proportional, will be found coincident on the slide and on a.

Problem 29. To find the Fourth Term of an Inverse Proportion. Against the first term on a, set the third term on the slide; and against the second term on a will be found the fourth term on the slide. Example.-Find a number which shall have to 25 the inverse proportion of 42 to 29. Against 42 on a set 25 on c; and against 29 on a will be found 36.2, the result, on c.

Problem 30. To find the Fourth Term of an Inverse Propor