These tests are all extremely easy of application, but not one of them can be depended upon for absolute accuracy: thus, in casting out the nines, the digits 0 and 9 may be replaced by each other, and the local values of any or all of the digits may be disarranged whilst the result of the rule remains the same: and in the test last given, the error in one stage of the division may be compensated by another of the same magnitude, but contrary quality in a subsequent one, a defect to which the one just mentioned is likewise subject.

16. In the operation of Division, the number of figures put down may be greatly diminished by what is called the Italian Method, which omits the partial subtrahends and retains only the partial remainders: thus,

257) 392086 531 5 256 2

1350

658

1446

16 15

733

219

and this comprises much fewer figures than the ordinary operation, but it does not furnish the test which has been mentioned in the last article.

Many other contrivances will naturally suggest themselves to the inventive student, but what has already been said will generally be sufficient for ensuring some very considerable degree of practical correctness.

IV. INVOLUTION AND EVOLUTION.

17. The first of these operations, being merely that of Multiplication, is mentioned here, only because the character and circumstances of the direct Arithmetical process constitute a necessary and essential part of the grounds upon which we must endeavour to perform the inverse operation of Evolution.

Since the square of 28 = 28 × 28 = 784, the square root of 784 must be 28: and we have to arrive at the latter of these numbers by means of the former: but as

there appears to be no immediate connection between them, we shall put 28 in the form 20 + 8, and then determine the corresponding form of its square, from the consideration that the product of any two quantities is the sum of the products which arise from multiplying every part of one of them by every part of the other: thus,

the root is 20 + 8

20 + 8

400+ 20 × 8

+ 20 × 8+ 64

the square is 400 + 2 × 20 × 8 +64

which consists of 400 = 20o, together with twice the product of 20 and 8, and 64 = 82: in order therefore to ascertain the square root of 784, expressed in the form 400+ 2 × 20 × 8 + 64, we first find the square root of 400 to be 20: and then from dividing 2 × 20 × 8 by the double of this, or by 2 × 20, the remaining part 8 of the root is obtained; so that 2 × 20 + 8 being now made the divisor, and multiplied by 8, and the product subtracted from 2 × 20 × 8+ 64, it appears that the entire root 20 + 8 or 28 is determined.

Keeping in view the demonstration above given, we may have either of the following operations:

the latter being nearly the same as the former, by omitting the ciphers, as was done in Multiplication and Division: and we observe that the Rule laid down in Article (159) of the text, is here investigated for the particular number under consideration.

Since, the

square of 49

= 492 = (48 + 1)2 = 482 + 2 × 48 + 1,

=

it is obvious that when the root is increased by 1, the corresponding square is increased by twice that root + 1: and the same mode of reasoning being equally applicable

in every other instance, it follows that the remainder at any stage of the process can in no case exceed the double of the root already obtained: agreeably to the observation made at the end of Ex. 4. of Article (159).

The method of Multiplication used in this and some of the preceding Articles of the Appendix will furnish the means of deriving the square of one number from that of another by a very simple proceeding: thus,

the square of 31 = (30 + 1)2 = 900 + 2 × 30 + 1 = 961-: square of 53 = (50 + 3)2 = 2500 + 2 × 50 × 3+9=2809 : and so on.

the

18. The rule for the extraction of the cube root given in Article (191), may be investigated in a similar manner, and the observation at the end of the article may be established upon the same principles; but for the reason stated in the text, it will not be necessary to follow up the inverse processes further in this place, inasmuch as they are rendered much clearer by the use of general Algebraical symbols, and the rules already laid down are quite sufficient for the performance of the operations in every case that can occur.

V. RATIO AND PROPORTION.

19. The relation of two magnitudes may be known by considering how much the one is greater or less than the other, or what is their Difference, as well as by observing how many times the one is contained in the other, or what is their Quotient. The former of these views, called Arithmetical Ratio, constitutes the chief business of the operation of Subtraction; and the latter is termed Geometrical Ratio, because it is generally applied to Geometrical Magnitudes, though it derives its importance from the various uses that are made of it in the calculations of civilized life. In which ever way the comparison may be made, it is evident that no relation can be established between them unless the magnitudes are of the same kind; and consequently Ratio as used in the text must be an abstract quantity, expressing merely the numerical value of one of the magnitudes, with reference to the other considered as an unit of the same kind. See Articles (65) and (96) of the text.

From this it follows that the relation of any two concrete magnitudes of the same kind, as two sums of Money, may be the same as, or equal to, that of two other concrete magnitudes of the same kind, as two bales of Goods: and this Equality of Ratios has been defined to be a Proportion.

20. It is clearly impossible to institute any such comparison between Geometrical Magnitudes without the assistance of their Arithmetical Representatives, which it may not always be in our power accurately to obtain; and accordingly it is stated in the fifth Book of Euclid's Elements, that " Proportion is the Similitude of Ratios; and the first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatever of the first and third being taken, and any equimultiples whatever of the second and fourth; if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth; if equal, equal; and if less, less."

be

This conclusion has been established in the text with respect to numbers forming a proportion, and it may applied immediately to shew whether four numbers taken in order constitute a proportion or not. Thus, if 2 : 3 :: 4: 5;

by taking equimultiples of the first and third, we have

and by taking equimultiples of the second and fourth, we obtain

12 : 12: 24: 20,

in which the condition above enunciated not being fulfilled, we are assured that the numbers 2, 3, 4, 5 do not form a proportion according to the geometrical definition, as the arithmetical definition shews at once, because is not equal to .