be too large, erase the work, and try a less number; (12) add the result to the two numbers above it, and place two dots to the right of the result: this is the new trial divisor for the next figure of the root; (13) add twice the last figure of the root to the number already in column A, or else place in column A three times the portion of the root already found; (14) proceed as before to find the next figure; (15) after five or six figures have been obtained, the others required may be found by the ordinary rules for contracted division of decimals, using as divisor the number in column B, neglecting column A.

Exactly the same method is to be followed in Algebra, care being taken that the quantities be arranged according to the powers of some one letter.

2. Required the cube root of 1 to 10 places of Decimals.

8n3—6n3—{n7—31n©+}}n3+{]n1+13† n3—11 n2—n—1

12n6—6n5—13 n1—5n3+11n2+n+1

8n3 — 6n3 — {n 7 — 3 } n° +7 Z n3 + {{{n1+13† n3—¥ n2—n—! (Root = 2n3n2— }n—1).

8n9

4

-12n6+6n5+13 n1 + 5n3-Un2-n-1 12n665+13 n1 + 5n3—Un2—n—l

5. Extract the cube root of 9 to 15 places of decimals.

Ans. 2.080083823051904.

305

NOTE A.

THE marginal note in page 1 goes upon the principle, that algebraic quantities always represent numbers. Respect for the distinguished mathematicians who think otherwise, requires that our reasons for this view should be stated.

Algebra has confessedly sprung up, by degrees, out of Arithmetic. It was first found convenient to have a symbol, by which an unknown number could be spoken of while the arithmetical process for finding it was going on. Next, since we have often occasion to speak of numbers in general, and since, in Arithmetic, each number is represented by a symbol or combination of symbols appropriated to itself, it was found that circumlocution would be saved by using arbitrary symbols to represent classes of numbers more or less extensive.

This use of general symbols to represent numbers, known or unknown, forms the essential distinction between Arithmetic and Algebra, and out of this all other differences arise. First, in Arithmetic, as well as in Algebra, operations are indicated by conventional marks: we may write 5+3 and 5-3 as well as a+b and a-b. But in Arithmetic there is always at hand a preconcerted symbol, by which, and by no other, the result of an operation must be expressed: therefore we can always perform arithmetical operations; that is, we can always substitute the symbol of the result for the expression which indicates the operation, as 8 for 5+3. In the investigations and even in the results of Algebra, on the contrary, we must rest satisfied with indicating* operations, except in so far as the symbols of particular numbers may be introduced. Hence the signs which are often convenient to the Arithmetician, become indispensable to the Algebraist; no step of his processes can be taken without them, and his very results must be expressed by them: and it is the method of representing numbers by general symbols, which fixes the operator's attention on the meaning of the signs, and raises them to such a degree of prominence and importance. Secondly, in an arithmetical question, since the numbers concerned are particularized, we can modify the form of the problem (its substance being unchanged) so as to avoid a negative result; but when general symbols are used, the relative magnitudes of the numbers which they represent are kept out of view, and in whatever form the problem was at first stated, to that form the work and the answer must be

This is obviously the case when we call a+b the sum, and ab the product of a and b; and though the change of a—(b+c) into a-b-c, and of a (x+y) into ax+ay, or a2 Xa3 into a5 is called actual Subtraction and actual Multiplication, yet it has dispensed but partially with signs of operation, though it disguises more or less the origin of the complex expressions on which it has been made.

strictly adapted; and hence we are often obliged to use the sign of Subtraction when there is nothing from which the quantity

ab

bearing that sign can be taken. Now, let- be the answer to

4

some algebraical question; then according to the first of the above

observations,

ab

4 is merely a requirement, and according to the second, this requirement necessarily arises from our having adhered throughout the operations to the form in which the data were originally presented: or, putting the two observations together, ab that in order to fulfil the conditions of the problem 4 means, as stated, a must be multiplied by b, the product divided by 4, and this result subtracted from 0. What is the meaning of this last demand? How shall we interpret this isolated negative quantity consistently with the original convention, which provided that

should denote Subtraction? The consideration of these queries leads to the views laid down in articles 1...6. Accordingly such views were put forward, though imperfectly, by the older writers on Algebra; but the paradoxical manner in which they spoke of quantities "less than nothing," seeming to use that phrase absolutely and not merely relatively, stirred up Mazeres and Frend to their well-known attacks on the whole doctrine of negative quantities. These authors were not able to banish the isolated +a and -a from Algebra; yet they succeeded in making subsequent writers adopt a different form of expression in speaking of such quantities: the sign began to be spoken of as not always denoting Subtraction, but sometimes indicating that the quantity to which it is prefixed is taken in a "contrary sense"* to that which it would have, if it carried the sign +. The idea of two distinct meanings for each of the signs +, -, having thus gradually crept into Algebra, was at length formally propounded as the basis of certain important deductions, by M. Buée, in his ingenious"Memoire sur les Quantités Imaginaires," read before the Royal Society in June, 1805, and published in the Philosophical Transactions for 1806: his views will be best explained by the following abridged extract from his paper :

"Chacun des signes + et a deux significations tout-à-fait differentes. 10 Mis devant une quantité q ils peuvent designer deux operations arithmetiques opposées, dont cette quantité est le sujet. 2° Devant cette meme quantité ils peuvent designer deux qualités opposées ayant pour sujet les unités dont cette quantité est composée.

"Dans l' Algebre ordinaire, c'est à dire, dans l' Algebre considerée comme l' Arithmetique universelle, une quantité isolée peut porter le signe +, qui dans ce cas n' ajoute rien à l'idée de cette quantité; mais elle ne peut pas porter le signe

En

* This is one of those vague and indefinite phrases, by whose aid difficulties may be slurred over with an air of profundity.