third degree; after which Euclid may be read to the termination. The study of trigonometry may then be introduced; on which subject we have various works of various merits. The treatise prefixed to Brown's Logarithmic Tables may be employed; and when it is understood, and the management of the logarithmic tables acquired, the works of Gregory, Lardner, or Thomson may be consulted; the last is the most simple. After the study of trigonometry, Simpson's conic sections may be read with advantage.

Perhaps it may be a kind of relief at this stage, to see something of the application of mathematics to mechanics, and, for this purpose, the work of Keil on Physics, or the article Mechanics, Hutton's Mathematics, Tegg's edition. The neat little treatise of Mr. Hay of Edinburgh will answer the same purpose exceedingly well. But for the purpose of obtaining a good knowledge of theoretical mechanics, a more extensive knowledge of mathematics than we have hitherto supposed becomes absolutely necessary. A knowledge of the method of fluxions and fluents, or the differential and integral calculus, which bear a strong analogy to each other, and which have been employed for similar purposes. The simplest work on fluxions, and we believe the best, is the treatise of Simpson; and this may be followed by a perusal of Thomson's differential and integral calculus. With this preparation the student may now go on to read the first volume of Gregory's Mechanics, a book in which, we believe, he will find ample satisfaction. The second volume of this excellent work is almost entirely popular, and can cause no difficulty whatever. Another work, well worthy of a perusal, is that of Sir John Leslie: we allude to his Natural Philosophy; a book which, though neither strictly mathematical, nor strictly popular, yet contains much valuable information communicated in both ways. Indeed all the works of this great man, although much has been said against them as to the floridness of their style, will, nevertheless, be found to amply repay the trouble of a perusal.

THE

MODERN MECHANIC.

ARITHMETIC.

VULGAR FRACTIONS.

1. In many cases of division after the quotient is obtained, there is a remainder, which is placed at the end of the quotient, above a small line with the divisor under it: thus-88 divided by 12 gives the quotient 7 and remainder 4, which is written 12) 88 (72. Now, this is called a fraction; and it is written in this way to show that 4 ought to be divided by 12; and in all cases where we meet with numbers written in this form, we conclude that the number above the line is to be divided by that under the line. This should be well borne in mind, as it is of the greatest use in obtaining a clear notion of fractions.

2. A fraction is said to express any number of the equal parts into which one whole is divided. It consists of two numbers-one placed above and the other below a small line. The upper number is called the Numerator, because it numerates how many parts the fraction expresses; and the under number is called the Denominator, because it expresses or denominates of what kind these parts are ;—or, in other words, the denominator shows into how many parts one inch, foot, yard, mile-one whole any thing-is supposed to be divided; and the numerator shows how many of these parts are taken: as of a foot. The denominator shows that the foot is here divided into 12 equal parts (inches ;) and the numerator 4, shows that four of these parts are taken-(4 inches.)

6

12

14

12

3. If the numerator had been equal to the denominator, as 43, then the value of the fraction would have been one whole (foot;) and the numerator, being divided by the denominator, gives 1 as a quotient. In the fraction 1 of a foot, the numerator is greater than the denominator, and the value of the fraction is greater than one: for the foot being divided into twelve equal parts, (inches,) and fourteen such parts (inches) being expressed by this fraction, its value is more than one foot; and the numerator being divided by the denominator, gives 122. Again, of a foot is just six inches, or one-half foot; and had the foot been divided into two equal parts, one of these parts would have been equal to 12, or is equal to . From this we may conclude, that when the numerator is equal to, less, or greater than the denominator, the value of the fraction is equal to, less, or greater than one whole. It is, then, not the numbers which express the numerator and denominator of a fraction, but the relation they bear to each other, that determines the real value of a fraction. 1 2 3 6 24, 6, 12, are alı equal, although expressed by different numbers, the denominators of all the fractions being respectively doubles of their numerators.

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4. From what has been said, it will easily be seen, that, if we multiply or divide both terms of any fraction by the same number, a new fraction will be found, equal to the first; thus, ; multiply both terms by 2, we get, or divide them by 2, 4, and these again by 2, . All who know any thing of a common foot-rule will understand this, at sight.

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For

5. The first use which we shall make of the principle last stated, is to bring two or more fractions to the same denominator, and that without altering their real values. example, take and of a foot. Multiply both terms of the first fraction by the denominator of the second, 4: we get Next multiply both terms of the second fraction by the denominator of the first fraction, that is, by 3: the result is. Now it will be seen (from No. 4) that these two fractions, and, are equal to the two and 4-with this additional advantage, however, that they have the same denominator, 12: the great use of which will be seen here after. A like process is employed in the case of three or more fractions: thus, 3, 4, 4,-multiply the terms of the first fraction by 4 and 5, the denominators of the second

12

and third, we get ; next multiply the second by 3 and 5, the denominators of the first and third, we next get ; lastly, multiply the third by the denominators of the first and second, 3 and 4, we get 8. It will be useful to look over what we have done. In obtaining the numerators of the new fractions, we have multiplied each numerator in the former fractions by all the denominators except its own; and so also for the denominators. But 3 multiplied by 4, and 4 multiplied by 3, are the same thing, viz. 12: so, likewise, 3 multiplied by 4 multiplied by 5 is 60, and will be 60 in whatever order we take them-3 by 4 by 5, or 4 by 3 by 5, or 5 by 3 by 4; when, therefore, we have obtained one denominator, it is sufficient. Hence the usual rule to reduce fractions to a common denominator: Multiply each numerator by all the denominators except its own for new numerators, and all the denominators together for the common denominator.

6. We are now prepared to add two or more fractions together. It is very easy to see how we may add and of an inch, and that their sum is ; but it is not quite so evident how we are to add and of a foot. If we had them, however, of one denomination, the difficulty would vanish. By No. 5, bring them to a common denominatorthey stand thus: and, or 8 and 9 inches; add the numerators, and under their sum place the denominator, 17; divide the numerator by the denominator, (No 1,) the quotient is 1, or one foot five inches. The reason of bringing them to a common denominator is, that we cannot add unlike quantities together: and we do not add the denominators, their only use being to show of what kind the quantities are. The rule, then, is-bring the fractions to a common denominator, add the numerators together, and under their sum place the common denominator.

7. In subtraction we bring the fractions to a common denominator, and taking the lesser from the greater of the two numerators, place under their difference the common denominator. The reason of this may be easily inferred from (No. 6) subtracted from, when brought to a common denominator, from the difference is, equal to

, by No. 4.

8. To take one number as often as there are units in another, is to multiply the one number by the other. To multiply 4 by 2, is to take the number four two times, as

there are two units in 2; and to multiply 4 by, is to take four one-half times, or the half of four, as there is only half a unit in the fraction. This may be thought so simple, that it need not be stated; but, let it be observed, that it explains a fact in the multiplication of fractions, which many excellent practical arithmeticians do not understand; viz. how that, when we multiply by a fraction, the product is less than the number multiplied. If the fraction is to be multiplied by 4, (let the fractions both refer to an inch,) this is taking (inch) 4 times, or taking the one-fourth part of one-half inch, which is one-eighth. The product is obtained by this simple process: multiply the numerators together for a new numerator, and the denominators together for a new denominator; the new fraction will be the product. That this is true in general may be shown by taking other fractions, thus: of,--the product by the rule is, which may be simplified by dividing the numerator and denominator by the same number, on the principle of No. 4; if 4 be the divisor, the result is, which is the same as Now, that is the real product of by, may be shown thus: divide a line AB

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into six equal parts; take two of c these parts, and join them by A CD. Divide CD into four parts,

D

B

and it will be seen that the two parts of this line CD are just equal to one division on the line AB; or 2 of CD is equal to of AB; so that, of is. The rule, then, is general.

9. Division is the reverse of multiplication; hence, to divide in fractions,-invert the divisor, and proceed as in multiplication. Thus, to divide by, invert the divisor. , it becomes, which, multiplied by gives multiplied by , equal to ; and by dividing, to make the fraction less, we obtain, which, by No. 1, is just 2 or twice. This is he quotient; and it is easily seen, if these fractions relate o a foot, that there are 2 quarters or twice of a foot, in one-half foot, or .

10. We have now endeavoured to explain the nature of the fundamental rules of vulgar fractions, as simply as possible; but instances often occur, where it is necessary to prepare for these operations ;-first, where whole numbers are concerned; and secondly, where the fractions are large, and, consequently, not so easily managed.

11. As to the first, where whole numbers are concerned,