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It is because arithmetic is a collection of truths thus arranged, that it is called a science.
69. The first principles or truths upon which any science depends are absolute truths, and having no need of any explanation, are termed self-evident,--they are also called axioms, as for example :“ A part is less than the whole;" and hence in the line
A D С B
Less than is expressed by the sign <; ex. gr., AC <A B.
By the whole of any quantity we understand the sum of all its parts; thus, A B = AD + DC + C B.
70. Things which are equal to the same are equal to one another”; that is, if a = m and b = = m, a is equal to b.
71. In any arithmetical operation, “quantities which are equal the one to the other may be substituted the one for the other.”
72. “ If the same process be applied to two equal quantities, the results will be equal."
In order to show how this self-evident principle may be turned to account, let it be required to find a number which, after we have multiplied it by m, and divided it by d, and added to it c, and subtracted from it a, shall be equal to n. I call x the quantity sought, and I multiply, divide, add to, and subtract from, x, as if it were a known quantity ; ex. gr.:
The product of x and m is ms, and the quotient of this product divided by d is not; if we add to this quotient c, we shall have
d from this sum we have to take a, which is + C a, and as this
т х quantity is equal to n, we have tc - a= n.
d I now add a to both sides or members of the equation, that is,
tc a + a = n + a; but the addition and subtraction of the d same quantity cancels that quantity, or a + a = 0: therefore
+c= n t a. Subtract c from both sides of the equation, and d
+CC=nta - -C; then, because + -c= 0, d
т х we shall have
= n + a
d If we multiply both sides of the equation by d, observing, at the same time, that to multiply by d is to cancel the d, we shall have
d m x = (n + a - c) d; and again, if we divide both sides by m, which
(n + a -c)d, in the case of m x is to cancel the m, we shall have x =
or we see that by means of the above axiom, we have been enabled to have the unknown x by itself on one side of the equation, and all known quantities on the other side; or in other words, a is known, since we know what are the operations to which we must subject certain known quantities in order to have it. The expression
(n + a
which represents the value of x, is
called a formula ; or again, since all these letters, x, m, d, c, a, n, represent certain numbers, let it be required to find a number, which being multiplied by 2, and divided by 3, shall, after the addition of 8, and subtraction of 5, be equal to 9; then a will be equal to (9 + 5 – 8) X 3 6 x 3
2 for 9 + 5 8 = 6, and 6 X 3 18, and 18 ; 2 9, or x equals 9; that is, the number 9 has that property required by the original question. This particular case will serve as an illustration of the principle.
73. Principles which, not being self-evident, require to be explained or proved, are called theorems; ex. gr., in Sect. 44 it was made evident by a figure, that the orders in which two factors are multiplied together, does not affect the result or product: this truth, therefore, is a theorem.
The explanations given are called the demonstration of the theorem. To demonstrate a principle is, therefore, to show why it is true; or “theorem” is a truth which requires demonstration.
74. The steps by which the intellect proceeds in a demonstration, may be exemplified in the theorem already mentioned, and in which it is shown that 4 X 5 5 X 4.
For in Sect. 44, the figure A B C D is formed of 4 rows of 5 squares ; therefore
ABCD 5 X 4. (1.) It is also formed of 5 rows of 4, therefore ABCD 4 x 5. (2.) Or in comparing the two equations, 5 X 4 = 4 x 5. (3.)
For two quantities, 5 X 4 and 4 X 5, being equal to a third, ABCD, are equal to one another.
The last step, or the result of this intellectual process, is called the judgment or conclusion; or the conclusion (3), which I deduce from the equations (1) and (2), proves that the principle enumerated is true, in the case of the two given factors. I could, indeed, have as readily ascertained this by merely multiplying them together, but the general principle, that is, whatever may be the factors, would not have been thereby demonstrated. To arrive at this conclusion, the existence of an analogy must be shown, and from that analogy we are enabled to assert that, whatever may be the factors, a similar figure might be constructed, and that the same conclusion would follow; or in other words, the same cause, under the same circumstances, would always produce the same effect; and hence the general principle.
75. We have drawn two conclusions, one depending on the equations (1) and (2), the other on analogy. The last was the general principle which we wished to establish. It often happens that a much longer
series of arguments or reasonings are requisite, in order to complete a demonstration.
76. To deduce a general principle from a particular case, without regard to analogy, is an error in reasoning; for though it might happen that the conclusion we came to would be true in fact, it would not follow legitimately from the argument, and in most instances we should be wrong altogether; ex. gr., from the fact that 2 + 2 2 X 2, it would be an error to conclude that twice any number is equal to the square
of that number. Again, it is found that 3 + 3 < 3 X 3 ; that 4 + 4 < 4 X 4, &c.; but it would not thence follow that twice any number is always less than its square, for we have (above), one example where these two quantities are equal, and one, also, where twice the number is twice as great as the square of that number, for 1 +1> 1 x l.
77. If, however, we examine the question in the preceding section, we shall perceive the general principle which it involves; for take any number, a, and let us compare a + a and a X a or a?.
Now a + a = 2a; that is to say, in representing the quantity a + a, under the form 2a, there are two factors, 2 and a; but in the case of a X a, a is twice the factor, and it is evident, therefore, that 2a will be equal to a x a only when a is equal to 2, or when 2a and a X a are identical equations. When, however, a= l or is less than 2, 2a must be greater than a X a; for whilst in 2a only the factor a would be less than 2, in a X a both factors would be less than 2.
For the same reason, when a > 2, then 2a will always be less than a X a, for in 2a one of the factors would be less than the other factor a.
Hence, when anumber is less than 2, then twice that number is greater than its square; when a number is greater than 2, twice that number is less than its square, and twice 2 is equal to the square of 2.
78. It is by means of axioms, and of theorems already demonstrated to be true, that we are enabled to reason upon and answer questions in arithmetic. These questions are often termed problems.
When we reason from a false principle, the conclusion, so far as it depends on the reasoning, will be false; and whenever our reasoning depends either on the principle itself which we wish to establish, or upon one which follows from it, we are said to “reason in a circle.”
The four fundamental operations in arithmetic, which, from the mode adopted in each, we have severally termed Addition, Subtraction, Multiplication, and Division, are problems; for we have had to find a sum, a difference, a product, and a quotient. These problems enable us to solve others, and we have seen that they give each other mutual assistance.
The answer to a question is called the solution of the problem ; this word also signifies sometimes the course or means we have taken to find the right answer.
Hence, when it is said that we have found the solution of a problem, it may either signify the result itself, or the road by which we have arrived at it.
We cannot be certain that we have given the right answer to a ques.
tion until we can explain the way in which we have arrived at it. Every problem, therefore, requires to be demonstrated.
We must not, however, confound a demonstration with the verification of our work; the first shows that we have reasoned upon principles that are acknowledged to be true; the second, that we have not made any error in our calculations. These verifications are commonly called proofs.
Thus, in the problem (sect. 72), we verify the result (or solution), by performing with the number 9 all the operations which are pointed out; and on finding the number 9, we conclude that we have taken the right course. We shall hereafter see in what consist the proofs of the four fundamental operations in arithmetic.
79. When a problem, of which we have found the most satisfactory solution constantly recurs, and when we have ascertained which is the simplest, shortest, and most ready way of working it, it is frequently called a rule, and is to be committed to memory. It is for this reason that the four fundamental operations of arithmetic of which we have treated, under the names of Addition, Subtraction, Multiplication, and Division, are called rules.
Again, instead of writing down the rule, we find it sometimes translated into algebraic language; that is, into the signs and letters which represent the quantities in the enunciation of the problem.
The algebraic expression for a rule is called a formula. Thus the problem in section 72, gave, as the value of the quantity sought, and
(n+amcd therefore called the unknown quantity, the formula x =
Conversely, we must be able to translate a formula into words at length : that is, we must be able to express, by means of the formula, in what way we should combine the quantities enunciated in the problem, in order to find the quantity sought.
80. It was asserted that, in order to study any subject, we must examine it as a whole, and in all its parts. But in order to solve problems, we must, in addition to this, accustom ourselves to reconstruct that which we have separated into parts.
T'he first of these methods is called an analysis, the second a synthesis.
Ex. gr. If we take a watch to pieces, in order to see the way which it is made, and the relation and use of its several parts, we are performing a kind of analysis ; and if, after having made this examination, we again put the several parts together, so as to reconstruct the watch once more, we are performing a kind of synthesis.
When we formed and named the numbers in the first instance, we took the synthetic method; when we examined the principles of numeration, we took the analytic method.
Generally speaking, the principles of a science are discovered by analysis, and those principles are applied practically by synthesis.
In the problem in sect. 72, we at first took the analytic method, that is, we examined it in all its parts, and observed the relation between these; then, after having named the number sought, x, we effected by synthesis whatever the previous analysis had pointed out to be done.
We found by this process the equality or equation which gave the formula, by means of which we were enabled to find the value of the unknown or number sought.
The first of these operations is termed expressing the problem as an equation, and the second the solution of the equation.
Thus, in order to solve a problem, we commence by expressing it as an equation, after which we proceed to solve this equation. Problems or questions in arithmetic generally lead us to the formula.
F. CURIE. (To be Continued.)
PROPOSED NEW EDITION OF DR. LOWTH'S ENGLISH
Rev. Sir,—There can be no doubt as to the utility of publishing a new edition of Dr. Lowth's Grammar. The chief point connected with it that demands consideration is, whether it would be better to issue it just as the author left it, or that it should undergo those changes and corrections which experience has proved to be so necessary in a work of this description, if used as a school book. Before this could be properly answered, some better understanding as regards the present views existing among teachers on the subject should be obtained, as it is well known that hardly two grammarians
agree in that statement, except upon points where the simplicity is so great as not to give the least chance of a difference of opinion. The grammar of Dr. Lowth, small as it is, is so valuable on account of its critical remarks, and the learning and research employed in the copious notes that accompany it, that it has been a favourite work since its first appearance, although one of the oldest on that subject
The short introduction, independent of its utility in presenting us with a view of grammatical knowledge many years ago, is deserving the attention of every one who may consider the study of the English language a part of good education.
Should it be considered necessary to alter in any way the present work, I do not think a better plan could be adopted, than to adhere. as strictly as possible to the genius of our language, without endeavouring to assimilate it in any degree to another of a more complicated structure, as it is evident nothing can be obtained by so doing in the shape of perspicuity. The fabrication of a passive verb, also, which found a place Dr. Lowth's Grammar, is evidently unnecessary, as the very circumstance of what is called an auxiliary being used, proves there does not exist, per se, such a verb in the language. The object of Dr. Lowth in presenting instances of false composition, was more to prove,
" that our best authors have committed gross mistakes, for want of a due knowledge of English grammar, or at least of proper attention to the rules of it,” than with a view to adopt that method as a principle of instruction. Thus far I do not think any one will disagree with him, especi